Healthcare inventory management in the presence of supply disruptions and a reliable secondary supplier

Abstract

We study the inventory review policy for a healthcare facility to minimize the impact of inevitable drug shortages. Usually, healthcare facilities do not rely on a single source of supply, and alternative mechanisms are present. When the primary supplier is not available, items are produced in-house or supplied through another supplier, albeit with additional cost. Our aim in this study is to determine how optimal inventory parameters are adjusted depending on the availability of the primary supplier. We show that an approximation provides trivial results, yet fails to capture the nuances therein. Our proposed Markov chain model overcomes these issues, and numerical results illustrate the significant economic impact of inventory parameter optimization. Furthermore, we simulate uncertainty scenarios and provide sensitivity analyses concerning fixed ordering cost for the secondary supplier, shortage frequency, shortage duration, and demand rates.

Appendices

Limiting probabilities and expected inventory levels for rare shortages with fast recovery, $\mu <\alpha$

1.1 Case 1: $Q_2 < R_1$

In order to obtain the limiting probabilities of the states, we divide the Markov chain states into four parts. The first part includes all the available states (all the states on the left side of the transition diagram). The second part consists of all the states from $(Q_1+R_1,U)$ to $(R_1+1,U)$; the third includes states $(R_1,U)$ to $(Q_2+1,U)$, and the remaining states are in the fourth part. Next, we obtain the limiting probabilities of these parts separately.

All the states of the first part can be written as a function of the limiting probability of the $(Q_1+R_1,A)$ state:

$$\begin{aligned} P_{Q_1+R_1-j,A}= \left( \frac{\lambda }{\lambda +\mu }\right) ^{j}P_{Q_1+R_1,A},\qquad {j=0,\ldots ,Q_1-1}. \end{aligned}$$

(7)

The sum of the probabilities of these states is

$$\begin{aligned} \sum _{j=0}^{Q_1-1} P_{Q_1+R_1-j,A}=\left( \frac{\lambda +\mu }{\mu }\right) \left[ 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right] P_{Q_1+R_1,A}. \end{aligned}$$

(8)

Now, let us try to obtain the limiting probabilities of the second part, that is, the states between $(Q_1+R_1,U)$ and $(R_1+1,U)$. Similarly, we can write the limiting probabilities of all the states in this part as a function of the limiting probabilities of $(Q_1+R_1,U)$ and $(Q_1+R_1,A)$ states. We have the following equation for the $(Q_1+R_1,U)$ state:

$$\begin{aligned} P_{Q_1+R_1,U}= \left( \frac{\mu }{ \lambda +\alpha }\right) P_{Q_1+R_1,A}. \end{aligned}$$

(9)

For the second state of this part, we have the following:

$$\begin{aligned} P_{Q_1+R_1-1,U}= \left( \frac{\mu }{ \lambda +\alpha }\right) P_{Q_1+R_1-1,A}+\left( \frac{\lambda }{ \lambda +\alpha }\right) P_{Q_1+R_1,U}. \end{aligned}$$

Using Eqs. (7) and (9), we substitute $P_{Q_1+R_1-1,A}$ and $P_{Q_1+R_1,U}$ respectively:

$$\begin{aligned} P_{Q_1+R_1-1,U}= \left( \left( \frac{\mu }{ \lambda +\alpha }\right) \left( \frac{\lambda }{\lambda +\mu }\right) +\left( \frac{\lambda }{ \lambda +\alpha }\right) \left( \frac{\mu }{ \lambda +\alpha }\right) \right) P_{Q_1+R_1,A}. \end{aligned}$$

The generalized form is as follows:

$$\begin{aligned} P_{Q_1+R_1-j,U}&=\frac{\mu \lambda ^j}{\lambda +\alpha }\left[ \sum _{k=0}^{j}\frac{1}{(\lambda +\alpha )^k(\lambda +\mu )^{j-k}} \right] P_{Q_1+R_1,A}. \end{aligned}$$

(10)

Let us try to find the sum of series in the last equation. We have

$$\begin{aligned} P_{Q_1+R_1-j,U}&=\frac{\mu }{\lambda +\alpha }\left( \frac{\lambda }{\lambda +\mu } \right) ^{j} \sum _{k=0}^{j}\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{k} P_{Q_1+R_1,A} \nonumber \\&=\frac{\mu }{\lambda +\alpha }\left( \frac{\lambda }{\lambda +\mu } \right) ^{j} \frac{1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{j+1} }{\frac{\alpha -\mu }{\lambda +\alpha } }P_{Q_1+R_1,A} \nonumber \\&=\frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{j} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{j+1} \right) P_{Q_1+R_1,A}, \qquad {j=0,\ldots ,Q_1-1}. \end{aligned}$$

(11)

We can obtain the following equation for $(R_1+1,U)$ state:

$$\begin{aligned} P_{R_1+1,U}&= \frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1} \right) P_{Q_1+R_1,A} \nonumber \\&=\frac{\mu (\lambda +\mu )}{\lambda (\alpha -\mu )}\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1} \right) P_{Q_1+R_1,A}. \end{aligned}$$

(12)

The sum of the probabilities of the states in this part is

$$\begin{aligned} \sum _{j=0}^{Q_1-1} P_{Q_1+R_1-j,U}&=\frac{\mu }{\alpha -\mu }\sum _{j=0}^{Q_1-1} \left( \frac{\lambda }{\lambda +\mu } \right) ^{j}\left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{j+1} \right) P_{Q_1+R_1,A} \nonumber \\&=\frac{\mu }{\alpha -\mu }\left[ \frac{1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}}{1 -\frac{\lambda }{\lambda +\mu }} -\frac{\lambda +\mu }{\lambda +\alpha }\frac{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} }{1-\frac{\lambda }{\lambda +\alpha } } \right] P_{Q_1+R_1,A} \nonumber \\&=\left[ \frac{\lambda +\mu }{\alpha -\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) -\frac{\mu (\lambda +\mu )}{\alpha (\alpha -\mu )}\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) \right] P_{Q_1+R_1,A} \end{aligned}$$

(13)

For the third part, we obtain the limiting probabilities in a similar way with what we have done in the first part. All the limiting probabilities of these states can be written as a function of the limiting probability of $(R_1+1,U)$ state:

$$\begin{aligned} P_{R_1-j,U} =\left( \frac{\lambda }{\lambda +\alpha }\right) ^{j+1}P_{R_1+1,U}, \qquad {j=0,\ldots ,R_1-Q_2-1}. \end{aligned}$$

(14)

The sum of the probabilities of third part is

$$\begin{aligned} \sum _{j=0}^{R_1-Q_2-1} P_{R_1-j,U}&=\frac{\lambda }{\alpha }\left[ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1-Q_2} \right] P_{R_1+1,U}. \end{aligned}$$

(15)

Now, let us try to obtain the limiting probabilities of the last part. The structure is similar to previous part, and the only thing we have to consider is to write the probabilities of these states in terms of the probability of $(Q_2,U)$ state. We, then, would have the following:

$$\begin{aligned} P_{Q_2-j,U} = \left( \frac{\lambda }{\lambda +\alpha }\right) ^{j}P_{Q_2,U}, \qquad {j=0,\ldots ,Q_2-1}. \end{aligned}$$

(16)

We can write the following equation for the limiting probability of $(Q_2,U)$ state:

$$\begin{aligned} \left( \lambda +\alpha \right) P_{Q_2,U} =\lambda \left[ P_{1,U} +P_{Q_2+1,U} \right] . \end{aligned}$$

(17)

Using Eqs. (14) and (16), we substitute for $P_{Q_2+1,U}$ and $P_{1,U}$, respectively, in the last equation:

$$\begin{aligned} \left( \lambda +\alpha \right) P_{Q_2,U}&=\lambda \left[ \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-1} P_{Q_2,U} +\left( \frac{\lambda }{\lambda +\alpha }\right) ^{ R_1-Q_2}P_{R_1+1,U} \right] \nonumber \\ P_{Q_2,U}&= \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2} P_{Q_2,U} +\left( \frac{\lambda }{\lambda +\alpha }\right) ^{ R_1-Q_2+1} P_{R_1+1,U} \nonumber \\ P_{Q_2,U}&= \frac{\left( \frac{\lambda }{\lambda +\alpha }\right) ^{ R_1-Q_2+1}}{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}}P_{R_1+1,U}. \end{aligned}$$

(18)

Then, we have the following equation to obtain the limiting probabilities of the last part:

$$\begin{aligned} P_{Q_2-j,U} = \frac{\left( \frac{\lambda }{\lambda +\alpha }\right) ^{ R_1-Q_2+1+j}}{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}}P_{R_1+1,U}, \qquad {j=0,\ldots ,Q_2-1}. \end{aligned}$$

(19)

The sum of the probabilities of these states is

$$\begin{aligned} \sum _{j=0}^{Q_2-1}P_{Q_2-j,U}&= \frac{\left( \frac{\lambda }{\lambda +\alpha }\right) ^{ R_1-Q_2+1}}{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}}. \frac{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2} }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) } P_{R_1+1,U} \nonumber \\&=\frac{\left( \frac{\lambda }{\lambda +\alpha }\right) ^{ R_1-Q_2+1} }{\frac{\alpha }{\lambda +\alpha } }P_{R_1+1,U} \nonumber \\&=\frac{\lambda }{\alpha }\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1-Q_2} P_{R_1+1,U} \end{aligned}$$

(20)

Considering the fact that the limiting probabilities of all of the states must add up to 1, we obtain the limiting probability of $(Q_1+R_1,A)$ state for case 1 as follows:

$$\begin{aligned}&\left[ \frac{\lambda +\mu }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) + \frac{\lambda +\mu }{\alpha -\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) \right. \nonumber \\&\quad \left. -\frac{\mu (\lambda +\mu )}{\alpha (\alpha -\mu )}\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) \right] P_{Q_1+R_1,A} \nonumber \\&\quad +\frac{\lambda }{\alpha }\left[ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1-Q_2} \right] P_{R_1+1,U} +\frac{\lambda }{\alpha }\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1-Q_2} P_{R_1+1,U} =1 \nonumber \\&\quad 1=\left[ \frac{\alpha (\lambda +\mu )}{\mu (\alpha -\mu )}\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) -\frac{\mu (\lambda +\mu )}{\alpha (\alpha -\mu )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) \right] \nonumber \\&\quad P_{Q_1+R_1,A} +\frac{\lambda }{\alpha }P_{R_1+1,U} \nonumber \\&\quad 1=\Bigg [\frac{\alpha (\lambda +\mu )}{\mu (\alpha -\mu )}\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) -\frac{\mu (\lambda +\mu )}{\alpha (\alpha -\mu )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) \nonumber \\&\quad +\frac{\mu (\lambda +\mu )}{\alpha (\alpha -\mu )}\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1} -\frac{\mu (\lambda +\mu )}{\alpha (\alpha -\mu )} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \Bigg ] P_{Q_1+R_1,A} \nonumber \\&\quad 1=\frac{(\lambda +\mu )(\alpha +\mu )}{\alpha \mu }\Bigg ( 1-\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1} \Bigg )P_{Q_1+R_1,A} \nonumber \\&\quad P_{Q_1+R_1,A}=\frac{\alpha \mu }{(\lambda +\mu )(\alpha +\mu )} \Bigg ( 1-\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1} \Bigg )^{-1}. \end{aligned}$$

(21)

Plugging (21) in (12) we obtain $P_{1,U}$ to be used in the expected total cost function in (6) as follows:

$$\begin{aligned} P_{1,U}= & {} \frac{\left( \frac{\lambda }{\lambda +\alpha }\right) ^{ R_1}}{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}}\frac{\mu (\lambda +\mu )}{\lambda (\alpha -\mu )}\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1}\left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1} \right) \nonumber \\{} & {} \times \frac{\alpha \mu }{(\lambda +\mu )(\alpha +\mu )} \Bigg ( 1-\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1} \Bigg )^{-1}. \end{aligned}$$

(22)

Expected inventory level of this case can be computed as follows:

$$\begin{aligned} I(Q_1;Q_2;R_1)&= \sum _{j=0}^{Q_1-1} \left( Q_1+R_1-j\right) P_{Q_1+R_1-j,A} + \sum _{j=0}^{Q_1-1} \left( Q_1+R_1-j\right) P_{Q_1+R_1-j,U} \nonumber \\&\quad + \sum _{j=0}^{R_1-Q_2-1} \left( R_1-j\right) P_{R_1-j,U} + \sum _{j=0}^{Q_2-1}(Q_2-j) P_{Q_2-j,U}, \end{aligned}$$

(23)

which can be rewritten as

$$\begin{aligned} I(Q_1;Q_2;R_1)&=(Q_1+R_1) \left( \sum _{j=0}^{Q_1-1} P_{Q_1+R_1-j,A} +\sum _{j=0}^{Q_1-1} P_{Q_1+R_1-j,U} \right) \nonumber \\&\quad - \sum _{j=0}^{Q_1-1} j P_{Q_1+R_1-j,A} -\sum _{j=0}^{Q_1-1} j P_{Q_1+R_1-j,U} + R_1 \sum _{j=0}^{R_1-Q_2-1} P_{R_1-j,U} \nonumber \\&\quad -\sum _{j=0}^{R_1-Q_2-1}j P_{R_1-j,U} + Q_2 \sum _{j=0}^{Q_2-1} P_{Q_2-j,U} -\sum _{j=0}^{Q_2-1}j P_{Q_2-j,U}. \end{aligned}$$

(24)

Knowing that the sum of the limiting probabilities of the all the states is 1, we rewrite the last equation as

$$\begin{aligned} I(Q_1;Q_2;R_1)&=(Q_1+R_1)-Q_1\sum _{j=0}^{R_1-Q_2-1} P_{R_1-j,U} -(Q_1+R_1-Q_2)\sum _{j=0}^{Q_2-1} P_{Q_2-j,U} \nonumber \\&\quad - \sum _{j=0}^{Q_1-1} j P_{Q_1+R_1-j,A} -\sum _{j=0}^{Q_1-1} j P_{Q_1+R_1-j,U}\nonumber \\&\quad -\sum _{j=0}^{R_1-Q_2-1}j P_{R_1-j,U} -\sum _{j=0}^{Q_2-1}j P_{Q_2-j,U}. \end{aligned}$$

(25)

We replace the sum of probabilities from Eqs. (15) and (20), and $P_{Q_1+R_1-j,A}$, $P_{Q_1+R_1-j,U}$, $P_{R_1-j,U}$, and $P_{Q_2-j,U}$ from Eqs. (8), (13), (15), and (20):

$$\begin{aligned} I(Q_1;Q_2;R_1)&=(Q_1+R_1)\nonumber \\&\quad -\left[ Q_1\frac{\lambda }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1-Q_2} \right) +(Q_1+R_1-Q_2)\frac{\lambda }{\alpha }\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1-Q_2} \right] P_{R_1+1,U} \nonumber \\&\quad -\left[ \sum _{j=0}^{Q_1-1} j \left( \frac{\lambda }{\lambda +\mu }\right) ^{j}+\sum _{j=0}^{Q_1-1} j \frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{j}\left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{j+1} \right) \right] P_{Q_1+R_1,A} \nonumber \\&\quad -\left[ \sum _{j=0}^{R_1-Q_2-1}j \left( \frac{\lambda }{\lambda +\alpha }\right) ^{j+1}+\sum _{j=0}^{Q_2-1}j \frac{\left( \frac{\lambda }{\lambda +\alpha }\right) ^{ R_1-Q_2+1+j}}{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}}\right] P_{R_1+1,U}. \end{aligned}$$

(26)

Knowing that

$$\begin{aligned} \sum _{i=0}^{n}i p^i&=p\sum _{i=0}^{n}i p^{i-1}=p\sum _{i=0}^{n}\frac{\partial }{\partial p}p^i=p \frac{\partial }{\partial p}\sum _{i=0}^{n}p^i \nonumber \\&=\frac{np^{n+2}-(n+1)p^{n+1}+p}{(1-p)^2}, \end{aligned}$$

(27)

we can rewrite the last equation as

$$\begin{aligned} I(Q_1;Q_2;R_1)&=(Q_1+R_1)-\frac{\lambda }{\alpha } \left[ Q_1+(R_1-Q_2)\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1-Q_2} \right] P_{R_1+1,U} \nonumber \\&\quad -\Bigg [\left( \frac{\alpha }{\alpha -\mu }\right) \left( \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+1}-Q_1\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2}\right) \nonumber \\&\quad -\left( \frac{\mu (\lambda +\mu )}{(\alpha -\mu )(\lambda +\alpha )}\right) \left( \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+1}-Q_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2} \right) \Bigg ]P_{Q_1+R_1,A} \nonumber \\&\quad -\Bigg [\left( \frac{\lambda }{\lambda +\alpha }\right) \left( \frac{(R_1-Q_2-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1-Q_2+1}-(R_1-Q_2)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1-Q_2}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2} \right) \nonumber \\&\quad +\left( \frac{\left( \frac{\lambda }{\lambda +\alpha }\right) ^{ R_1-Q_2+1}}{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}}\right) \left( \frac{(Q_2-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2+1}-Q_2\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2} \right) \Bigg ]P_{R_1+1,U}. \end{aligned}$$

(28)

We can plug in limiting probabilities and obtain the final equation to be used in (6):

$$\begin{aligned} \begin{aligned} I(Q_1;Q_2;R_1)&=(Q_1+R_1)-\frac{\lambda }{\alpha } \left[ Q_1+(R_1-Q_2)\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1-Q_2} \right] \\&\quad \frac{\mu (\lambda +\mu )}{\lambda (\alpha -\mu )}\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1} \right) \frac{\alpha \mu }{(\lambda +\mu )(\alpha +\mu )}\Bigg ( 1-\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1} \Bigg )^{-1} \\&\quad -\left[ \left( \frac{\alpha }{\alpha -\mu }\right) \left( \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+1}-Q_1\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2}\right) \right. \\&\quad \left. -\left( \frac{\mu (\lambda +\mu )}{(\alpha -\mu )(\lambda +\alpha )}\right) \left( \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+1}-Q_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2} \right) \right] \\&\quad \times \frac{\alpha \mu }{(\lambda +\mu )(\alpha +\mu )} \Bigg ( 1-\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1} \Bigg )^{-1}\\&\quad -\left[ \left( \frac{\lambda }{\lambda +\alpha }\right) \left( \frac{(R_1-Q_2-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1-Q_2+1}-(R_1-Q_2)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1-Q_2}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2} \right) \right. \\&\quad \left. +\left( \frac{\left( \frac{\lambda }{\lambda +\alpha }\right) ^{ R_1-Q_2+1}}{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}}\right) \left( \frac{(Q_2-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2+1}-Q_2\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2} \right) \right] \\&\quad \times \frac{\mu (\lambda +\mu )}{\lambda (\alpha -\mu )}\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1}\left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1} \right) \frac{\alpha \mu }{(\lambda +\mu )(\alpha +\mu )} \Bigg ( 1-\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1} \Bigg )^{-1}. \end{aligned} \end{aligned}$$

(29)

1.2 Case 2: $R_1\le Q_2 \le Q_1+R_1$

Similar to Case 1, we divide the Markov chain states into four parts. The first part includes all the available states, similar to Case 1. However, the second part consists of all the states from $(Q_1+R_1,U)$ to $(Q_2+1,U)$; the third includes states $(Q_2,U)$ to $(R_1+1,U)$, and the remaining states are in the fourth part, states between $(R_1,U)$ and (1, U). Next, we would obtain the limiting probabilities of these parts separately.

We can obtain the limiting probabilities of the first part from those of case 1:

$$\begin{aligned} P_{Q_1+R_1-j,A} = \left( \frac{\lambda }{\lambda +\mu }\right) ^{j}P_{Q_1+R_1,A}, \qquad {j=0,\ldots ,Q_1-1}. \end{aligned}$$

(30)

The sum of the probabilities of these states is

$$\begin{aligned} \sum _{j=0}^{Q_1-1} P_{Q_1+R_1-j,A}&=\frac{\lambda +\mu }{\mu }\left[ 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right] P_{Q_1+R_1,A}. \end{aligned}$$

(31)

The equations for the second part are similar to part two of case 1, but the range of the states, j, has to be updated. We have the following equation for the $(Q_1+R_1,U)$ state:

$$\begin{aligned} P_{Q_1+R_1-j,U}&=\left( \frac{\mu }{\alpha -\mu }\right) \left( \frac{\lambda }{\lambda +\mu } \right) ^{j}\left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{j+1} \right) P_{Q_1+R_1,A}, \nonumber \\&\qquad {j=0,\ldots ,Q_1+R_1-Q_2-1}. \end{aligned}$$

(32)

The sum of the states in this part, for case 2, is

$$\begin{aligned}&\sum _{j=0}^{Q_1+R_1-Q_2-1}P_{Q_1+R_1-j,U} \nonumber \\&\quad =\frac{\lambda +\mu }{\alpha -\mu } \left[ 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} -\frac{\mu }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \right) \right] P_{Q_1+R_1,A}. \end{aligned}$$

(33)

For the third part, we have the following equation for $(Q_2,U)$ state:

$$\begin{aligned} P_{Q_2,U}=\frac{\lambda }{\lambda +\alpha }\left( P_{Q_2+1,U}+P_{1,U}\right) +\frac{\mu }{\lambda +\alpha }P_{Q_2,A}. \end{aligned}$$

(34)

For the remaining states of this part, we obtain the limiting probabilities as a function of $P_{Q_2,U}$ and $P_{Q_1+R_1,A}$:

$$\begin{aligned} P_{Q_2-j,U}&= \left( \frac{\lambda }{\lambda +\alpha }\right) ^{j}P_{Q_2,U} + \frac{\mu }{\lambda +\alpha }\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2}\nonumber \\&\quad \sum _{k=0}^{j-1}\left( \frac{\lambda }{\lambda +\mu }\right) ^{j-k}\left( \frac{\lambda }{\lambda +\alpha }\right) ^{k} P_{Q_1+R_1,A} \nonumber \\ P_{Q_2-j,U}&= \left( \frac{\lambda }{\lambda +\alpha }\right) ^{j}P_{Q_2,U} + \frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2+j}\nonumber \\&\quad \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{j} \right) P_{Q_1+R_1,A}, \nonumber \\&\quad {j=1,\ldots ,Q_2-R_1-1}. \end{aligned}$$

(35)

Note that the range is starting from one, not zero. $P_{R_1+1,U}$ is

$$\begin{aligned} P_{R_1+1,U}&= \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1}P_{Q_2,U}\nonumber \\&\quad + \frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1-1}\left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \right) P_{Q_1+R_1,A}. \end{aligned}$$

(36)

The sum of the probabilities of third part is

$$\begin{aligned} \sum _{j=0}^{Q_2-R_1-1} P_{Q_2-j,U}&=\sum _{j=0}^{Q_2-R_1-1}\left( \frac{\lambda }{\lambda +\alpha }\right) ^{j}P_{Q_2,U} \nonumber \\&\quad +\sum _{j=1}^{Q_2-R_1-1} \frac{\mu }{\alpha -\mu } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2+j}\left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{j} \right) P_{Q_1+R_1,A} \nonumber \\&=\frac{\lambda +\alpha }{\alpha }\left[ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1} \right] P_{Q_2,U}\nonumber \\&\quad +\frac{\lambda }{\alpha -\mu } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \Bigg [1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-R_1-1} \Bigg ]P_{Q_1+R_1,A} \nonumber \\&\quad -\frac{\lambda \mu }{\alpha (\alpha -\mu )} \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \Bigg [1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \Bigg ]P_{Q_1+R_1,A}. \end{aligned}$$

(37)

Now, let us try to obtain the limiting probabilities of the last part. As mentioned above, this part would be considered only when the reorder point is greater than zero. The structure is similar to part one, and the only thing we have to consider is to write the probabilities of these states in terms of the probability of $(R_1+1,U)$ state. We, then, have the following:

$$\begin{aligned} P_{R_1-j,U} = \left( \frac{\lambda }{\lambda +\alpha }\right) ^{j+1}P_{R_1+1,U}, \qquad {j=0,\ldots ,R_1-1}. \end{aligned}$$

(38)

The sum of the probabilities of these states is

$$\begin{aligned} \sum _{j=0}^{R_1-1}P_{R_1-j,U}=\frac{\lambda }{\alpha }\left( 1- \left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}\right) P_{R_1+1,U}. \end{aligned}$$

(39)

In order to obtain the closed form equation for the limiting probability of $(Q_2,U)$ state, we have the following equations:

$$\begin{aligned} P_{1,U}&= \left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}P_{R_1+1,U} \nonumber \\&=\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-1}P_{Q_2,U} + \frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1-1}\nonumber \\&\quad \left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \right) P_{Q_1+R_1,A}, \end{aligned}$$

(40)

$$\begin{aligned} P_{Q_2+1,U}&=\frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2-1}\left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \right) P_{Q_1+R_1,A}, \end{aligned}$$

(41)

$$\begin{aligned} P_{Q_2,A}&= \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2}P_{Q_1+R_1,A}. \end{aligned}$$

(42)

Then from Eq. (34),

$$\begin{aligned} P_{Q_2,U}&= \frac{\mu }{\alpha -\mu } \left[ \frac{\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1-1}\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1+1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \right) }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}} \right. \nonumber \\&\quad \left. +\frac{\frac{\lambda }{\lambda +\alpha } \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2-1}\left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \right) + \frac{\alpha -\mu }{\lambda +\alpha }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2} }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}} \right] P_{Q_1+R_1,A}. \end{aligned}$$

(43)

Considering the fact that the sum of the limiting probabilities of all of the states must add up to 1, we obtain the limiting probability of $(Q_1+R_1,A)$ state for case 2 as follows:

$$\begin{aligned} 1&=\Bigg [\frac{\lambda +\mu }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) +\frac{\lambda +\mu }{\alpha -\mu } \left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2}\right. \nonumber \\&\quad \left. -\frac{\mu }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \right) \right) \Bigg ] P_{Q_1+R_1,A} \nonumber \\&\quad +\frac{\lambda +\alpha }{\alpha }\left[ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1} \right] P_{Q_2,U}\nonumber \\&\quad +\frac{\lambda }{\alpha -\mu } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \Bigg [1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-R_1-1} \Bigg ]P_{Q_1+R_1,A} \nonumber \\&\quad -\frac{\lambda \mu }{\alpha (\alpha -\mu )} \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \Bigg [1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \Bigg ]P_{Q_1+R_1,A}\nonumber \\&\quad + \frac{\lambda }{\alpha }\left( 1- \left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}\right) P_{R_1+1,U}. \end{aligned}$$

(44)

Then,

$$\begin{aligned} P_{Q_1+R_1,A}&=\Bigg [\frac{\alpha +\mu }{\alpha \mu } -\frac{(\alpha +\mu )(\lambda +\mu )}{\mu \alpha } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}\Bigg ]^{-1} \nonumber \\&=\frac{\alpha \mu }{(\alpha +\mu )(\lambda +\mu )}\left( \frac{1}{\lambda +\mu } -\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) ^{-1}. \end{aligned}$$

(45)

Expected inventory level can be computed as follows:

$$\begin{aligned} I(Q_1;Q_2;R_1)&= \sum _{j=0}^{Q_1-1} \left( Q_1+R_1-j\right) P_{Q_1+R_1-j,A} \nonumber \\&\quad + \sum _{j=0}^{Q_1+R_1-Q_2-1} \left( Q_1+R_1-j\right) P_{Q_1+R_1-j,U} \nonumber \\&\quad + \sum _{j=0}^{Q_2-R_1-1} \left( Q_2-j\right) P_{Q_2-j,U} + \sum _{j=0}^{R_1-1}(R_1-j) P_{R_1-j,U}, \end{aligned}$$

(46)

which can be rewritten as

$$\begin{aligned} I(Q_1;Q_2;R_1)&=(Q_1+R_1)\left( \sum _{j=0}^{Q_1-1} P_{Q_1+R_1-j,A} +\sum _{j=0}^{Q_1+R_1-Q_2-1} P_{Q_1+R_1-j,U} \right) \nonumber \\&\quad - \sum _{j=0}^{Q_1-1} j P_{Q_1+R_1-j,A} \nonumber \\&\quad -\sum _{j=0}^{Q_1+R_1-Q_2-1} j P_{Q_1+R_1-j,U} + Q_2 \sum _{j=0}^{Q_2-R_1-1} P_{Q_2-j,U} \nonumber \\&\quad -\sum _{j=0}^{Q_2-R_1-1}j P_{Q_2-j,U} + R_1 \sum _{j=0}^{R_1-1} P_{R_1-j,U} -\sum _{j=0}^{R_1-1}j P_{R_1-j,U} \end{aligned}$$

(47)

$$\begin{aligned}&=(Q_1+R_1) -Q_1 \frac{\lambda }{\alpha }\left( 1- \left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}\right) P_{R_1+1,U}\nonumber \\&\quad - (Q_1+R_1-Q_2)\Bigg [\frac{\lambda +\alpha }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1} \right) P_{Q_2,U} \nonumber \\&\quad +\frac{\lambda }{\alpha -\mu } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \Bigg (1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-R_1-1} \Bigg )P_{Q_1+R_1,A} \nonumber \\&\quad -\frac{\lambda \mu }{\alpha (\alpha -\mu )} \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \Bigg (1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \Bigg )P_{Q_1+R_1,A} \Bigg ] \nonumber \\&\quad - \sum _{j=0}^{Q_1-1} j P_{Q_1+R_1-j,A}-\sum _{j=0}^{Q_1+R_1-Q_2-1} j P_{Q_1+R_1-j,U} \nonumber \\&\quad -\sum _{j=0}^{Q_2-R_1-1}j P_{Q_2-j,U} -\sum _{j=0}^{R_1-1}j P_{R_1-j,U}. \end{aligned}$$

(48)

We replace the probabilities from Eqs. (15) and (20), and $P_{Q_1+R_1-j,A}$, $P_{Q_1+R_1-j,U}$, $P_{R_1-j,U}$, and $P_{Q_2-j,U}$ from Eqs. (31), (33), (37), and (39):

$$\begin{aligned} I(Q_1;Q_2;R_1)&=-Q_1 \frac{\lambda }{\alpha }\left( 1- \left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}\right) P_{R_1+1,U}\nonumber \\&\quad - (Q_1+R_1-Q_2)\Bigg [\frac{\lambda +\alpha }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1} \right) P_{Q_2,U} \nonumber \\&\quad +\frac{\lambda }{\alpha -\mu } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \Bigg (1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-R_1-1} \Bigg )P_{Q_1+R_1,A} \nonumber \\&\quad -\frac{\lambda \mu }{\alpha (\alpha -\mu )} \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \Bigg (1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \Bigg )P_{Q_1+R_1,A} \Bigg ] \nonumber \\&\quad +(Q_1+R_1)-\sum _{j=0}^{Q_1-1}j \left( \frac{\lambda }{\lambda +\mu }\right) ^{j}P_{Q_1+R_1,A}\nonumber \\&\quad -\sum _{j=0}^{Q_1+R_1-Q_2-1} j \frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{j} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{j+1} \right) P_{Q_1+R_1,A} \nonumber \\&\quad -\sum _{j=0}^{Q_2-R_1-1}j \left( \frac{\lambda }{\lambda +\alpha }\right) ^{j}P_{Q_2,U} -\sum _{j=0}^{Q_2-R_1-1}j \frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2+j}\nonumber \\&\quad \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{j} \right) P_{Q_1+R_1,A} -\sum _{j=0}^{R_1-1}j \left( \frac{\lambda }{\lambda +\alpha }\right) ^{j+1}P_{R_1+1,U} \end{aligned}$$

(49)

$$\begin{aligned}&I(Q_1;Q_2;R_1)\nonumber \\&\quad =-Q_1 \frac{\lambda }{\alpha }\left( 1- \left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}\right) P_{R_1+1,U}\nonumber \\&\qquad - (Q_1+R_1-Q_2)\Bigg [\frac{\lambda +\alpha }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1} \right) P_{Q_2,U} \nonumber \\&\qquad + \frac{\lambda }{\alpha -\mu } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \Bigg (1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-R_1-1} \Bigg )P_{Q_1+R_1,A} \nonumber \\&\qquad -\frac{\lambda \mu }{\alpha (\alpha -\mu )} \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \Bigg (1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \Bigg )P_{Q_1+R_1,A} \Bigg ] \nonumber \\&\qquad +(Q_1+R_1) -\Bigg [ \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+1}-Q_1 \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2} \nonumber \\&\qquad -\left( \frac{\mu (\lambda +\mu )}{(\alpha -\mu )(\lambda +\alpha )}\right) \frac{(Q_1+R_1-Q_2-1) \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2+1}-(Q_1+R_1-Q_2)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2} \nonumber \\&\qquad +\left( \frac{\mu }{\alpha -\mu }\right) \frac{(Q_1+R_1-Q_2-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2+1}-(Q_1+R_1-Q_2)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2} +\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2} \Bigg ]P_{Q_1+R_1,A} \nonumber \\&\qquad -\frac{(Q_2-R_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-R_1+1}-(Q_2-R_1) \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-R_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}P_{Q_2,U} \nonumber \\&\qquad - \frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \frac{(Q_2-R_1-1) \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-R_1+1}-(Q_2-R_1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-R_1} +\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2}P_{Q_1+R_1,A} \nonumber \\&\qquad +\frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2}\frac{(Q_2-R_1-1) \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-R_1+1}-(Q_2-R_1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-R_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}P_{Q_1+R_1,A} \nonumber \\&\qquad -\left( \frac{\lambda }{\lambda +\alpha }\right) \frac{(R_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1+1} -R_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}P_{R_1+1,U}. \end{aligned}$$

(50)

For this case, $I(Q_1;Q_2;R_1)$ and $P_{1,U}$ that is used when calculating $SF(Q_1;Q_2;R_1)$ can be written in terms of $\alpha$, $\lambda$, $\mu$, $Q_1$, $Q_2$, and $R_1$ as follows:

$$\begin{aligned} I(Q_1;Q_2;R_1)= & {} (Q_1+R_1) + \frac{\alpha \mu }{(\alpha +\mu )(\lambda +\mu )}\left( \frac{1}{\lambda +\mu }-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) ^{-1}\\{} & {} \left[ -Q_1 \frac{\lambda }{\alpha }\left( 1- \left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}\right) \Bigg [\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1}\frac{\mu }{\alpha -\mu }\right. \\{} & {} \times \left[ \frac{\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1-1}\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1+1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \right) }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}}\right. \\{} & {} \left. +\frac{\frac{\lambda }{\lambda +\alpha } \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2-1}\left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \right) +\frac{\alpha -\mu }{\lambda +\alpha }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2} }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}} \right] + \frac{\mu }{\alpha -\mu } \\{} & {} \left. \times \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1-1}\left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \right) \right] \\{} & {} -(Q_1+R_1-Q_2)\left[ \frac{\lambda +\alpha }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1} \right) \frac{\mu }{\alpha -\mu } \right. \\{} & {} \left[ \frac{\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1-1}\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1+1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \right) }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}} \right. \\{} & {} \left. +\frac{\frac{\lambda }{\lambda +\alpha } \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2-1}\left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \right) +\frac{\alpha -\mu }{\lambda +\alpha }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2} }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}} \right] \\{} & {} +\frac{\lambda }{\alpha -\mu } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \Bigg (1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-R_1-1} \Bigg )\\ \end{aligned}$$

$$\begin{aligned}{} & {} \quad \left. -\frac{\lambda \mu }{\alpha (\alpha -\mu )} \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \Bigg (1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \Bigg ) \right] \\{} & {} \quad -\left[ \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+1}-Q_1 \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2} \right. \\{} & {} \quad -\left( \frac{\mu (\lambda +\mu )}{(\alpha -\mu )(\lambda +\alpha )}\right) \frac{(Q_1+R_1-Q_2-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2+1}-(Q_1+R_1-Q_2)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2} \\{} & {} \quad \left. +\left( \frac{\mu }{\alpha -\mu }\right) \frac{(Q_1+R_1-Q_2-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2+1}-(Q_1+R_1-Q_2)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2} \right] \\{} & {} -\frac{(Q_2-R_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-R_1+1}-(Q_2-R_1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-R_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}\\{} & {} \frac{\mu }{\alpha -\mu } \left[ \frac{\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1-1}\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1+1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \right) }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}} \right. \\{} & {} \left. +\frac{\frac{\lambda }{\lambda +\alpha } \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2-1}\left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \right) +\frac{\alpha -\mu }{\lambda +\alpha }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2} }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}} \right] \\{} & {} - \frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \frac{(Q_2-R_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-R_1+1}-(Q_2-R_1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-R_1}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2}\\{} & {} +\frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2}\frac{(Q_2-R_1-1) \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-R_1+1}-(Q_2-R_1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-R_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}\\{} & {} -\left( \frac{\lambda }{\lambda +\alpha }\right) \frac{(R_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1+1}-R_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2} \left[ \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1}\right. \\{} & {} \frac{\mu }{\alpha -\mu } \left[ \frac{\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1-1} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1+1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \right) }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}} \right. \\{} & {} \left. +\frac{\frac{\lambda }{\lambda +\alpha } \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2-1}\left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \right) +\frac{\alpha -\mu }{\lambda +\alpha }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2} }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}} \right] \\{} & {} \left. \left. +\frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \right) \right] \right] ,\\{} & {} P_{1,U}=\frac{\alpha \mu }{(\alpha +\mu )(\lambda +\mu )}\left( \frac{1}{\lambda +\mu }-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) ^{-1}\Bigg [\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-1}\\{} & {} \frac{\mu }{\alpha -\mu } \left[ \frac{\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1-1}\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1+1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \right) }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}} \right. \\{} & {} \left. +\frac{\frac{\lambda }{\lambda +\alpha } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2-1}\left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \right) + \frac{\alpha -\mu }{\lambda +\alpha }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2} }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}} \right] \\{} & {} \left. \frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1-1}\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \right) \right] . \end{aligned}$$

1.3 Case 3: $Q_2>Q_1+R_1$

In order to obtain the limiting probabilities of the states we divide the Markov chain states into five parts. The first part includes all the available states from state $(Q_2,A)$ to $(Q_1+R_1+1,A)$. The second part includes all the states from $(Q_2,U)$ to $(Q_1+R_1+1,U)$; the third consists of all the available states from $(Q_1+R_1,A)$ to $(R_1+1,A)$, fourth part includes states $(Q_1+R_1,U)$ to $(R_1+1,U)$, and fifth part includes all the remaining states. Next, we obtain the limiting probabilities of these parts separately.

All the states of the first part can be written as an equation of the limiting probability of $(Q_2,A)$ state. By doing so, we write the following equation for the limiting probabilities of the first part:

$$\begin{aligned} P_{Q_2-j,A}= & {} \left( \frac{\lambda }{\lambda +\mu }\right) ^{j} P_{Q_2,A} +\frac{\alpha }{\lambda +\mu }P_{Q_2-j,U}, \qquad {j=1,\ldots ,Q_2-Q_1-R_1-1}, \end{aligned}$$

(51)

$$\begin{aligned} P_{Q_2-j,U}= & {} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{j} P_{Q_2,U} +\frac{\mu }{\lambda +\alpha }P_{Q_2-j,A}, \qquad {j=1,\ldots ,Q_2-Q_1-R_1-1}, \end{aligned}$$

(52)

$$\begin{aligned} P_{Q_2,A}= & {} \frac{\alpha }{\lambda +\mu }P_{Q_2,U}. \end{aligned}$$

(53)

From Eqs. (51), (52), and (53) we obtain the following for limiting probabilities of the first part:

$$\begin{aligned} P_{Q_2-j,A}&=\frac{\alpha (\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )}\left[ \left( \frac{\lambda }{\lambda +\mu }\right) ^{j}+ \left( \frac{\lambda }{\lambda +\alpha }\right) ^{j} \right] P_{Q_2,U},\nonumber \\&\quad {j=1,\ldots ,Q_2-Q_1-R_1-1}. \end{aligned}$$

(54)

The sum of the probabilities of these states is

$$\begin{aligned}&\sum _{j=0}^{Q_2-Q_1-R_1-1} P_{Q_2-j,A}\nonumber \\&\quad =\frac{\alpha }{\lambda +\mu }P_{Q_2,U} +\sum _{j=1}^{Q_2-Q_1-R_1-1}\frac{\alpha (\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )}\left[ \left( \frac{\lambda }{\lambda +\mu }\right) ^{j}+ \left( \frac{\lambda }{\lambda +\alpha }\right) ^{j} \right] P_{Q_2,U}\nonumber \\&\quad =\frac{\lambda +\alpha }{\lambda +\alpha +\mu }\left[ \frac{\alpha }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) +\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1}\right) \right. \nonumber \\&\qquad \left. +\frac{\alpha (\lambda +\alpha +\mu )}{(\lambda +\mu )(\lambda +\alpha )}\right] P_{Q_2,U}. \end{aligned}$$

(55)

For the second part, we follow steps similar to part one:

$$\begin{aligned} P_{Q_2-j,U}&=\left[ \frac{(\mu +\lambda )(\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{j} + \frac{\mu \alpha }{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\mu } \right) ^{j} \right] P_{Q_2,U}, \nonumber \\&\quad {j=1,\ldots ,Q_2-Q_1-R_1-1}. \end{aligned}$$

(56)

For the sum of these states, we have

$$\begin{aligned}&\sum _{j=0}^{Q_2-Q_1-R_1-1}P_{Q_2-j,U}\nonumber \\&\quad = P_{Q_2,U}+\sum _{j=1}^{Q_2-Q_1-R_1-1}\left[ \frac{(\mu +\lambda ) (\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{j} + \frac{\mu \alpha }{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\mu } \right) ^{j} \right] P_{Q_2,U} \nonumber \\&\quad =\frac{\lambda +\mu }{\lambda +\alpha +\mu }\left[ \frac{\lambda +\alpha }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1} \right) +\frac{\alpha }{\lambda +\mu } \left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) \right. \nonumber \\&\quad \left. +\frac{\lambda +\alpha +\mu }{\lambda +\mu } \right] P_{Q_2,U}. \end{aligned}$$

(57)

Now, let us try to obtain the limiting probabilities of the third part:

$$\begin{aligned} P_{Q_1+R_1-j,A}= \left( \frac{\lambda }{\lambda +\mu } \right) ^{j} P_{Q_1+R_1,A}, \qquad {j=0,\ldots ,Q_1-1}. \end{aligned}$$

(58)

The sum of these states is

$$\begin{aligned} \sum _{j=0}^{Q_1-1}P_{Q_1+R_1-j,A}&=\frac{1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}}{1-\frac{\lambda }{\lambda +\mu }}P_{Q_1+R_1,A} \nonumber \\&=\frac{\lambda +\mu }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) P_{Q_1+R_1,A}. \end{aligned}$$

(59)

For the fifth part, we have

$$\begin{aligned} P_{R_1-j,U} = \left( \frac{\lambda }{\lambda +\alpha }\right) ^{j+1}P_{R_1+1,U}, \qquad {j=0,\ldots ,R_1-1}. \end{aligned}$$

(60)

The sum of the probabilities of the states in the fifth part is

$$\begin{aligned} \sum _{j=0}^{R_1-1} P_{R_1-j,U}&=\frac{\lambda }{\lambda +\alpha } \frac{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}}{1-\frac{\lambda }{\lambda +\alpha }}P_{R_1+1,U} \nonumber \\&=\frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \right) P_{R_1+1,U}. \end{aligned}$$

(61)

For the fourth part, we have the following:

$$\begin{aligned} P_{Q_1+R_1,U}&= \frac{\mu }{ \lambda +\alpha } P_{Q_1+R_1,A} +\frac{\lambda }{ \lambda +\alpha } P_{Q_1+R_1+1,U}. \end{aligned}$$

(62)

Then,

$$\begin{aligned} P_{Q_1+R_1-1,U}&=\frac{\mu }{\lambda +\alpha }P_{Q_1+R_1-1,A}+ \frac{\lambda }{\lambda +\alpha }P_{Q_1+R_1,U} \nonumber \\&=\left( \frac{\mu }{\lambda +\alpha }\right) \left( \frac{\lambda }{\lambda +\mu }\right) P_{Q_1+R_1,A}\nonumber \\&\quad +\left( \frac{\mu }{\lambda +\alpha }\right) \left( \frac{\lambda }{\lambda +\alpha }\right) P_{Q_1+R_1,A} +\left( \frac{\lambda }{\lambda +\alpha }\right) ^{2}P_{Q_1+R_1+1,U}, \end{aligned}$$

(63)

$$\begin{aligned} P_{Q_1+R_1-2,U}&=\frac{\mu }{\lambda +\alpha }P_{Q_1+R_1-2,A}+ \frac{\lambda }{\lambda +\alpha }P_{Q_1+R_1-1,U} \nonumber \\&= \frac{\mu }{\lambda +\alpha }\left( \frac{\lambda }{\lambda +\mu }\right) ^{2} P_{Q_1+R_1,A}\nonumber \\&\quad + \frac{\mu }{\lambda +\alpha }\frac{\lambda }{\lambda +\mu }\frac{\lambda }{\lambda +\alpha }P_{Q_1+R_1,A} +\frac{\mu }{\lambda +\alpha }\left( \frac{\lambda }{\lambda +\alpha }\right) ^{2}P_{Q_1+R_1,A} \nonumber \\&\quad +\left( \frac{\lambda }{\lambda +\alpha }\right) ^{3}P_{Q_1+R_1+1,U}. \end{aligned}$$

(64)

The generalized form is as follows:

$$\begin{aligned} P_{Q_1+R_1-j,U}&=\frac{\mu \lambda ^j}{\lambda +\alpha }\left[ \sum _{k=0}^{j}\frac{1}{(\lambda +\alpha )^k(\lambda +\mu )^{j-k}} \right] P_{Q_1+R_1,A} \nonumber \\&\quad +\left( \frac{\lambda }{\lambda +\alpha } \right) ^{j+1}P_{Q_1+R_1+1,U} \nonumber \\&=\frac{\mu }{\lambda +\alpha }\left( \frac{\lambda }{\lambda +\mu } \right) ^{j} \left[ \sum _{k=0}^{j} \left( \frac{\lambda +\mu }{\lambda +\alpha } \right) ^k \right] P_{Q_1+R_1,A} \nonumber \\&\quad +\left( \frac{\lambda }{\lambda +\alpha } \right) ^{j+1}P_{Q_1+R_1+1,U} \nonumber \\ P_{Q_1+R_1-j,U}&=\frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{j} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{j+1}\right) P_{Q_1+R_1,A}\nonumber \\&\quad + \left( \frac{\lambda }{\lambda +\alpha } \right) ^{j+1}P_{Q_1+R_1+1,U}, \nonumber \\&\quad j=0,\ldots ,Q_1-1. \end{aligned}$$

(65)

For $(R_1+1,U)$ state, we have

$$\begin{aligned} P_{R_1+1,U}&=\frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1}\right) P_{Q_1+R_1,A} \nonumber \\&\quad +\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}P_{Q_1+R_1+1,U}. \end{aligned}$$

(66)

Note that we can obtain $P_{Q_1+R_1+1,U}$ from Eq. (56) as follows:

$$\begin{aligned} P_{Q_1+R_1+1,U}&=\left[ \frac{(\lambda +\mu )(\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1-1} \right. \nonumber \\&\quad \left. + \frac{\mu \alpha }{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1} \right] P_{Q_2,U}. \end{aligned}$$

(67)

Let us try to find the sum of series in last equation. We have

$$\begin{aligned} \sum _{j=0}^{Q_1-1}P_{Q_1+R_1-j,U}&=\left[ \frac{\lambda +\mu }{\alpha -\mu } \left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}\right) \right. \nonumber \\&\quad \left. -\frac{\mu (\lambda +\mu )}{\alpha (\alpha -\mu )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) \right] P_{Q_1+R_1,A} \nonumber \\&\quad +\frac{\lambda }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) P_{Q_1+R_1+1,U}. \end{aligned}$$

(68)

For $(Q_2,U)$ state, we have the following equation:

$$\begin{aligned} P_{Q_2,U}=\frac{\lambda }{\lambda +\alpha }P_{1,U}+\frac{\mu }{\lambda +\alpha }P_{Q_2,A}. \end{aligned}$$

(69)

We replace $P_{Q_2,A}$ from Eq. (53):

$$\begin{aligned} P_{Q_2,U}=\frac{\lambda +\mu }{\lambda +\mu +\alpha }P_{1,U}. \end{aligned}$$

(70)

From the equation for the fifth part, we obtain $P_{1,U}$:

$$\begin{aligned} P_{1,U}=\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}P_{R_1+1,U}. \end{aligned}$$

(71)

Then,

$$\begin{aligned} P_{Q_2,U}=\frac{\lambda +\mu }{\lambda +\mu +\alpha } \left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}P_{R_1+1,U}. \end{aligned}$$

(72)

For $P_{R_1+1,U}$, we have,

$$\begin{aligned} P_{R_1+1,U}&=\frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1}\right) P_{Q_1+R_1,A} +\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1} \times \nonumber \\&\quad \left[ \frac{(\lambda +\mu )(\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )} \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1-1} \right. \nonumber \\&\quad \left. + \frac{\mu \alpha }{\lambda (\lambda +\alpha +\mu )} \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1} \right] \nonumber \\&\quad \times \frac{\lambda +\mu }{\lambda +\mu +\alpha }\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}P_{R_1+1,U} \nonumber \\ P_{R_1+1,U}&=\frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1}\right) P_{Q_1+R_1,A}+ \nonumber \\&\quad \frac{(\lambda +\mu )^2}{(\lambda +\mu +\alpha )^2} \left[ \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-2} + \frac{\mu \alpha }{\lambda ^2} \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1} \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] P_{R_1+1,U} \nonumber \\ P_{R_1+1,U}&=\frac{\frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{(\lambda +\mu +\alpha )^2} \left[ \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-2} + \frac{\mu \alpha }{\lambda ^2} \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } P_{Q_1+R_1,A} \end{aligned}$$

(73)

Then for $P_{Q_2,U}$,

$$\begin{aligned} P_{Q_2,U}=\frac{\frac{\lambda +\mu }{\lambda +\mu +\alpha }\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{(\lambda +\mu +\alpha )^2} \left[ \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-2} + \frac{\mu \alpha }{\lambda ^2} \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1} \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } P_{Q_1+R_1,A} \end{aligned}$$

(74)

$$\begin{aligned} 1&=\frac{\lambda +\alpha }{\lambda +\alpha +\mu }\left[ \frac{\alpha }{\mu } \left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) \right. \nonumber \\&\quad +\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1}\right) +\frac{\alpha (\lambda +\alpha +\mu )}{(\lambda +\mu )(\lambda +\alpha )}\Bigg ] P_{Q_2,U} \nonumber \\&\quad +\frac{\lambda +\mu }{\lambda +\alpha +\mu }\Bigg [\frac{\lambda +\alpha }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1} \right) \nonumber \\&\quad \left. +\frac{\alpha }{\lambda +\mu } \left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) +\frac{\lambda +\alpha +\mu }{\lambda +\mu } \right] P_{Q_2,U} \nonumber \\&\quad +\frac{\lambda +\mu }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) P_{Q_1+R_1,A} +\frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \right) P_{R_1+1,U} \nonumber \\&\quad +\left[ \frac{\lambda +\mu }{\alpha -\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}\right) - \frac{\mu (\lambda +\mu )}{\alpha (\alpha -\mu )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) \right] \nonumber \\&\quad P_{Q_1+R_1,A} +\frac{\lambda }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) P_{Q_1+R_1+1,U} \end{aligned}$$

(75)

$$\begin{aligned} 1&=\left[ \frac{\alpha }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) +\frac{\lambda +\alpha }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1}\right) \right. \nonumber \\&\quad \left. +\frac{\alpha +\lambda +\mu }{\lambda +\mu }\right] P_{Q_2,U} + \frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \right) P_{R_1+1,U} +\frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) P_{Q_1+R_1+1,U} \nonumber \\&\quad +\left[ \frac{\alpha (\lambda +\mu )}{\mu (\alpha -\mu )}\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}\right) - \frac{\mu (\lambda +\mu )}{\alpha (\alpha -\mu )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) \right] P_{Q_1+R_1,A}. \end{aligned}$$

(76)

Then,

$$\begin{aligned}&\left[ \left[ \frac{\alpha }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) +\frac{\lambda +\alpha }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1}\right) +\frac{\alpha +\lambda +\mu }{\lambda +\mu }\right] \right. \nonumber \\&\quad \times \frac{\frac{\lambda +\mu }{\lambda +\mu +\alpha }\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}\frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{(\lambda +\mu +\alpha )^2} \left[ \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-2} + \frac{\mu \alpha }{\lambda ^2} \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1} \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \nonumber \\&\quad +\frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \right) \frac{\frac{\mu }{\alpha -\mu } \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{(\lambda +\mu +\alpha )^2} \left[ \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-2} + \frac{\mu \alpha }{\lambda ^2} \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \nonumber \\&\quad +\frac{\lambda }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) \left[ \frac{(\lambda +\mu )(\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1-1}\right. \nonumber \\&\quad \left. + \frac{\mu \alpha }{\lambda (\lambda +\alpha +\mu )} \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1} \right] \nonumber \\&\quad \times \frac{\frac{\lambda +\mu }{\lambda +\mu +\alpha }\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{(\lambda +\mu +\alpha )^2} \left[ \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-2} + \frac{\mu \alpha }{\lambda ^2} \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \nonumber \\&\quad \left. +\frac{\alpha (\lambda +\mu )}{\mu (\alpha -\mu )}\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}\right) - \frac{\mu (\lambda +\mu )}{\alpha (\alpha -\mu )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) \right] ^{-1}=P_{Q_1+R_1,A}. \end{aligned}$$

(77)

Expected inventory level can be computed as follows:

$$\begin{aligned} I(Q_1;Q_2;R_1)&=\sum _{j=0}^{Q_2-Q_1-R_1-1} (Q_2-j)(P_{Q_2-j,A}+ P_{Q_2-j,U}) + \sum _{j=0}^{R_1-1}(R_1-j) P_{R_1-j,U} \nonumber \\&\quad +\sum _{j=0}^{Q_1-1}(Q_1+R_1-j)(P_{Q_1+R_1-j,A}+P_{Q_1+R_1-j,U}). \end{aligned}$$

(78)

Then,

$$\begin{aligned} I(Q_1;Q_2;R_1)&= Q_2 \sum _{j=0}^{Q_2-Q_1-R_1-1}(P_{Q_2-j,A}+P_{Q_2-j,U})\nonumber \\&\quad +(Q_1+R_1)\sum _{j=0}^{Q_1-1}(P_{Q_1+R_1-j,A}+ P_{Q_1+R_1-j,U}) \nonumber \\&\quad +R_1\sum _{j=0}^{R_1-1}P_{R_1-j,U} - \sum _{j=0}^{Q_2-Q_1-R_1-1} j (P_{Q_2-j,A}+P_{Q_2-j,U})\nonumber \\&\quad - \sum _{j=0}^{R_1-1} j P_{R_1-j,U} \nonumber \\&\quad - \sum _{j=0}^{Q_1-1} j (P_{Q_1+R_1-j,A}+ P_{Q_1+R_1-j,U}). \end{aligned}$$

(79)

We replace the probabilities $P_{Q_2-j,A}$, $P_{Q_1+R_1-j,A}$, $P_{Q_2-j,U}$, $P_{Q_1+R_1-j,U}$, and $P_{R_1-j,U}$ from Eqs. (55), (59), (57), (61), and (68):

$$\begin{aligned} I(Q_1;Q_2;R_1)&=Q_2 \left[ \frac{\alpha }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) +\frac{\lambda +\alpha }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1}\right) \right. \nonumber \\&\quad \left. + \frac{\alpha +\lambda +\mu }{\lambda +\mu }\right] P_{Q_2,U} +(Q_1+R_1) \frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) P_{Q_1+R_1+1,U} \nonumber \\&\quad +(Q_1+R_1)\left[ \frac{\alpha (\lambda +\mu )}{\mu (\alpha -\mu )} \left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}\right) - \frac{\mu (\lambda +\mu )}{\alpha (\alpha -\mu )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) \right] P_{Q_1+R_1,A} \nonumber \\&\quad +R_1 \frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \right) P_{R_1+1,U} -\frac{\alpha }{\lambda } \times P_{Q_2,U} \times \nonumber \\&\quad \left[ \frac{(Q_2-Q_1-R_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1+1}-(Q_2-Q_1-R_1) \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1} +\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2}\right] \nonumber \\&\quad -\frac{(Q_2-Q_1-R_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1}-(Q_2-Q_1-R_1) \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1-1}+1}{(1-\frac{\lambda }{\lambda +\alpha })^2} P_{Q_2,U} \nonumber \\&\quad -\left( \frac{\lambda }{\lambda +\alpha }\right) \left( \frac{(R_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1+1}-R_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}\right) P_{R_1+1,U} \nonumber \\&\quad -\left( \frac{\alpha }{\alpha -\mu }\right) \left( \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+1} -Q_1\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2}\right) P_{Q_1+R_1,A} \nonumber \\&\quad +\left( \frac{\mu (\mu +\lambda )}{(\lambda +\alpha )(\alpha -\mu )}\right) \left( \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+1}-Q_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1} +\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}\right) P_{Q_1+R_1,A} \nonumber \\&\quad -\left( \frac{\lambda }{\lambda +\alpha } \right) \left( \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+1}-Q_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}\right) P_{Q_1+R_1+1,U}. \end{aligned}$$

(80)

Note that the equations for this case would not bear any changes for $R_1=0$.

For this case, $I(Q_1;Q_2;R_1)$ and $P_{1,U}$ that is used when calculating $SF(Q_1;Q_2;R_1)$ can be written in terms of $\alpha$, $\lambda$, $\mu$, $Q_1$, $Q_2$, and $R_1$ as follows:

$$\begin{aligned} P_{1,U}= & {} \left[ \frac{\frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{(\lambda +\mu +\alpha )^2} \left[ \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-2} + \frac{\mu \alpha }{\lambda ^2} \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \left[ \left[ \frac{\alpha }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) \right. \right. \right. \nonumber \\{} & {} \left. +\frac{\lambda +\alpha }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1}\right) +\frac{\alpha +\lambda +\mu }{\lambda +\mu }\right] \frac{\lambda +\mu }{\lambda +\mu +\alpha }\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} +\frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \right) \nonumber \\{} & {} +\frac{\lambda }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) \left[ \frac{(\lambda +\mu )(\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1-1} + \frac{\mu \alpha }{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1} \right] \nonumber \\{} & {} \left. \left. \times \frac{\lambda +\mu }{\lambda +\mu +\alpha }\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \right] +\frac{\alpha (\lambda +\mu )}{\mu (\alpha -\mu )}\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}\right) - \frac{\mu (\lambda +\mu )}{\alpha (\alpha -\mu )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) \right] ^{-1} \nonumber \\{} & {} \times \left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}\frac{\frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{(\lambda +\mu +\alpha )^2} \left[ \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-2} + \frac{\mu \alpha }{\lambda ^2} \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \end{aligned}$$

$$\begin{aligned} I(Q_1;Q_2;R_1)= & {} \left[ \frac{\frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{(\lambda +\mu +\alpha )^2} \left[ \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-2} + \frac{\mu \alpha }{\lambda ^2} \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \left[ \left[ \frac{\alpha }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) \right. \right. \right. \nonumber \\{} & {} \left. +\frac{\lambda +\alpha }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1}\right) +\frac{\alpha +\lambda +\mu }{\lambda +\mu }\right] \frac{\lambda +\mu }{\lambda +\mu +\alpha } \left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} +\frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \right) \nonumber \\{} & {} +\frac{\lambda }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) \left[ \frac{(\lambda +\mu )(\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )} \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1-1} + \frac{\mu \alpha }{\lambda (\lambda +\alpha +\mu )} \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1} \right] \nonumber \\{} & {} \left. \left. \times \frac{\lambda +\mu }{\lambda +\mu +\alpha }\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \right] +\frac{\alpha (\lambda +\mu )}{\mu (\alpha -\mu )}\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}\right) - \frac{\mu (\lambda +\mu )}{\alpha (\alpha -\mu )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) \right] ^{-1} \nonumber \\{} & {} \times \left[ \frac{\frac{\lambda +\mu }{\lambda +\mu +\alpha }\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{(\lambda +\mu +\alpha )^2} \left[ \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-2} + \frac{\mu \alpha }{\lambda ^2} \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \right. \nonumber \\{} & {} \times \left[ Q_2 \Bigg [\frac{\alpha }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) +\frac{\lambda +\alpha }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1}\right) + \frac{\alpha +\lambda +\mu }{\lambda +\mu }\right] \nonumber \\{} & {} +(Q_1+R_1) \frac{\lambda }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) \left[ \frac{(\lambda +\mu )(\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )} \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1-1}\right. \nonumber \\{} & {} \left. + \frac{\mu \alpha }{\lambda (\lambda +\alpha +\mu )} \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1} \right] \nonumber \\ \end{aligned}$$

$$\begin{aligned}{} & {} \quad -\frac{\alpha }{\lambda } \times \left[ \frac{(Q_2-Q_1-R_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1+1} -(Q_2-Q_1-R_1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2}\right] \nonumber \\{} & {} \quad -\frac{(Q_2-Q_1-R_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1}-(Q_2-Q_1-R_1) \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1-1}+1}{(1-\frac{\lambda }{\lambda +\alpha })^2} \nonumber \\{} & {} \quad -\left( \frac{\lambda }{\lambda +\alpha } \right) \left( \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+1} -Q_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}\right) \nonumber \\{} & {} \quad \left. \left[ \frac{(\lambda +\mu )(\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )} \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1-1} + \frac{\mu \alpha }{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1} \right] \right] \nonumber \\{} & {} \quad \left. +(Q_1+R_1)\Bigg [\frac{\alpha (\lambda +\mu )}{\mu (\alpha -\mu )}\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}\right) - \frac{\mu (\lambda +\mu )}{\alpha (\alpha -\mu )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) \right] \nonumber \\{} & {} \quad +R_1 \frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \right) \frac{\frac{\mu }{\alpha -\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{(\lambda +\mu +\alpha )^2} \left[ \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-2} + \frac{\mu \alpha }{\lambda ^2} \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1} \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \nonumber \\{} & {} \quad -\left( \frac{\lambda }{\lambda +\alpha }\right) \left( \frac{(R_1-1) \left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1+1}-R_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1} +\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}\right) \frac{\frac{\mu }{\alpha -\mu } \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\mu }{\lambda +\alpha }\right) ^{Q_1}\right) }{\nonumber }\\{} & {} \quad {1-\frac{(\lambda +\mu )^2}{(\lambda +\mu +\alpha )^2} \left[ \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-2} + \frac{\mu \alpha }{\lambda ^2} \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \nonumber \\{} & {} \quad -\left( \frac{\alpha }{\alpha -\mu }\right) \left( \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+1} -Q_1\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2}\right) \nonumber \\{} & {} \quad \left. +\left( \frac{\mu (\mu +\lambda )}{(\lambda +\alpha )(\alpha -\mu )}\right) \left( \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+1}-Q_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}\right) \right] \end{aligned}$$

Limiting probabilities and expected inventory levels for frequent shortages with slow recovery, $\mu >\alpha$

1.1 Case 1: $Q_2<R_1$

All the equations of Case 1 for frequent shortage case are the same as the Case 1 of rare shortage except the second part. To obtain the limiting probabilities of the second part, we write the limiting probabilities of all the states in this part as a function of the limiting probabilities of $(Q_1+R_1,U)$ and $(Q_1+R_1,A)$ states. We have the following equation for the $(Q_1+R_1,U)$ state:

$$\begin{aligned} P_{Q_1+R_1,U}= \left( \frac{\mu }{ \lambda +\alpha }\right) P_{Q_1+R_1,A}. \end{aligned}$$

(81)

For the second state of this part, we have the following:

$$\begin{aligned} P_{Q_1+R_1-1,U}= \left( \frac{\mu }{ \lambda +\alpha }\right) P_{Q_1+R_1-1,A}+\left( \frac{\lambda }{ \lambda +\alpha }\right) P_{Q_1+R_1,U}. \end{aligned}$$

Using Eqs. (7) and (81), we substitute $P_{Q_1+R_1-1,A}$ and $P_{Q_1+R_1,U}$ respectively:

$$\begin{aligned} P_{Q_1+R_1-1,U}= \left( \frac{\mu }{ \lambda +\alpha }\left( \frac{\lambda }{\lambda +\mu }\right) +\frac{\lambda }{ \lambda +\alpha } \left( \frac{\mu }{ \lambda +\alpha }\right) \right) P_{Q_1+R_1,A}. \end{aligned}$$

The generalized form of the probabilities of this case is as follows:

$$\begin{aligned} P_{Q_1+R_1-j,U}&=\frac{\mu \lambda ^j}{\lambda +\alpha }\left[ \sum _{k=0}^{j}\frac{1}{(\lambda +\alpha )^{j-k}(\lambda +\mu )^{k}} \right] P_{Q_1+R_1,A}. \end{aligned}$$

(82)

Let us try to find the sum of series in last equation. We have,

$$\begin{aligned} P_{Q_1+R_1-j,U}&=\frac{\mu }{\lambda +\alpha }\left( \frac{\lambda }{\lambda +\alpha } \right) ^{j}\sum _{k=0}^{j}\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{k} P_{Q_1+R_1,A} \nonumber \\&=\frac{\mu }{\lambda +\alpha }\left( \frac{\lambda }{\lambda +\alpha } \right) ^{j}\frac{1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{j+1} }{\frac{\mu -\alpha }{\lambda +\mu } }P_{Q_1+R_1,A} \nonumber \\&=\frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{j}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{j+1} \right) P_{Q_1+R_1,A}, \nonumber \\&\qquad {j=0,\ldots ,Q_1-1}. \end{aligned}$$

(83)

We can obtain the following equation for $(R_1+1,U)$ state when $\alpha <\mu$ and $Q_2<R_1$:

$$\begin{aligned} P_{R_1+1,U}&= \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1-1}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1} \right) P_{Q_1+R_1,A} \nonumber \\&=\frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )}\left[ \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right] P_{Q_1+R_1,A}. \end{aligned}$$

(84)

The sum of the states in this part is

$$\begin{aligned} \sum _{j=0}^{Q_1-1} P_{Q_1+R_1-j,U}&=\sum _{j=0}^{Q_1-1} \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{j}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{j+1} \right) P_{Q_1+R_1,A} \nonumber \\&=\frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )} \left[ \frac{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}}{1-\frac{\lambda }{\lambda +\alpha }} -\frac{\lambda +\alpha }{\lambda +\mu }\frac{1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} }{1-\frac{\lambda }{\lambda +\mu } } \right] P_{Q_1+R_1,A} \nonumber \\&=\left[ \frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) -\frac{\lambda +\mu }{\mu -\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) \right] P_{Q_1+R_1,A}. \end{aligned}$$

(85)

Considering the fact the sum of the limiting probabilities of all of the states must add up to 1, we obtain the limiting probability of $Q_1+R_1,A$ state as follows:

$$\begin{aligned}&\left[ \frac{\lambda +\mu }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) + \frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) \right. \nonumber \\&\quad \left. -\frac{\lambda +\mu }{\mu -\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) \right] P_{Q_1+R_1,A} \nonumber \\&\quad +\frac{\lambda }{\alpha }\left[ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1-Q_2} \right] P_{R_1+1,U} +\frac{\lambda }{\alpha }\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1-Q_2} P_{R_1+1,U} =1 \end{aligned}$$

(86)

$$\begin{aligned}&1=\left[ \frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) - \frac{\alpha (\lambda +\mu )}{\mu (\mu -\alpha )}\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) \right] \nonumber \\&\quad P_{Q_1+R_1,A} +\frac{\lambda }{\alpha }P_{R_1+1,U} \end{aligned}$$

(87)

$$\begin{aligned}&1=\Bigg [\frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) - \frac{\alpha (\lambda +\mu )}{\mu (\mu -\alpha )}\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) \nonumber \\&\quad +\frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )}\left[ \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1} -\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right] \Bigg ] P_{Q_1+R_1,A} \end{aligned}$$

(88)

$$\begin{aligned}&\quad 1=\frac{(\lambda +\mu )(\alpha +\mu )}{\alpha \mu }\Bigg ( 1-\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1} \Bigg )P_{Q_1+R_1,A} \end{aligned}$$

(89)

$$\begin{aligned}&\quad P_{Q_1+R_1,A}=\frac{\alpha \mu }{(\lambda +\mu )(\alpha +\mu )} \Bigg ( 1-\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1} \Bigg )^{-1} \end{aligned}$$

(90)

As it can be seen, the equations are the same in case 1 for both conditions. As a result, the inventory level of this case is be equal to the inventory level of case 1 for the rare shortage condition.

For this case, $I(Q_1;Q_2;R_1)$ and $P_{1,U}$ that is used when calculating $SF(Q_1;Q_2;R_1)$ can be written in terms of $\alpha$, $\lambda$, $\mu$, $Q_1$, $Q_2$, and $R_1$ as follows:

$$\begin{aligned} \begin{aligned} P_{1,U}&=\frac{\left( \frac{\lambda }{\lambda +\alpha }\right) ^{ R_1-Q_2+1+j}}{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}}\frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )}\left[ \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right] \\&\quad \frac{\alpha \mu }{(\lambda +\mu )(\alpha +\mu )} \Bigg ( 1-\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1} \Bigg )^{-1}\\ I(Q_1;Q_2;R_1)&=(Q_1+R_1)-\frac{\lambda }{\alpha } \left[ Q_1+(R_1-Q_2)\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1-Q_2} \right] \\&\quad \frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )}\left[ \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right] \frac{\alpha \mu }{(\lambda +\mu )(\alpha +\mu )} \Bigg ( 1-\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1} \Bigg )^{-1} \\&\quad -\Bigg [\left( \frac{\alpha }{\alpha -\mu }\right) \left( \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+1} -Q_1\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2}\right) \\&\quad -\left( \frac{\mu (\lambda +\mu )}{(\alpha -\mu )(\lambda +\alpha )}\right) \left( \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+1}-Q_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2} \right) \Bigg ]\\&\quad \frac{\alpha \mu }{(\lambda +\mu )(\alpha +\mu )} \Bigg ( 1-\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1} \Bigg )^{-1} \\&\quad -\Bigg [\left( \frac{\lambda }{\lambda +\alpha }\right) \left( \frac{(R_1-Q_2-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1-Q_2+1}-(R_1-Q_2)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1-Q_2} +\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2} \right) \\&\quad +\left( \frac{\left( \frac{\lambda }{\lambda +\alpha }\right) ^{ R_1-Q_2+1}}{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}}\right) \left( \frac{(Q_2-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2+1} -Q_2\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2} \right) \Bigg ]\\&\quad \frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )}\left[ \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right] \frac{\alpha \mu }{(\lambda +\mu )(\alpha +\mu )} \Bigg ( 1-\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1} \Bigg )^{-1} \end{aligned} \end{aligned}$$

1.2 Case 2: $R_1 \le Q_2\le Q_1+R_1$

All the equations of Case 2 for the frequent shortage case are the same as Case 2 for the rare shortage condition except the second and third parts. The equations for the second part are similar to part two of case 1, but the range of the states, j, has to be updated. We have the following equation for the $(Q_1+R_1,U)$ state:

$$\begin{aligned} P_{Q_1+R_1-j,U}&=\frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{j}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{j+1} \right) P_{Q_1+R_1,A}, \nonumber \\&\quad {j=0,\ldots ,Q_1+R_1-Q_2-1}. \end{aligned}$$

(91)

The sum of the states in this part, for case 2, is

$$\begin{aligned}&\sum _{j=0}^{Q_1+R_1-Q_2-1} P_{Q_1+R_1-j,U} \nonumber \\&\quad =\left[ \frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \right) \right. \nonumber \\&\qquad \left. -\frac{\lambda +\mu }{\mu -\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \right) \right] P_{Q_1+R_1,A}. \end{aligned}$$

(92)

For the third part, we obtain the limiting probabilities as a function of $P_{Q_2,U}$ and $P_{Q_1+R_1,A}$:

$$\begin{aligned} \begin{aligned} P_{Q_2-j,U}&= \left( \frac{\lambda }{\lambda +\alpha }\right) ^{j}P_{Q_2,U} + \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2+j}\\&\quad \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{j} \right) P_{Q_1+R_1,A}, \\&\quad {j=1,\ldots ,Q_2-R_1-1}. \end{aligned} \end{aligned}$$

(93)

Note that the range is starting from one, not zero. The sum of the probabilities of the third part is

$$\begin{aligned} \sum _{j=0}^{Q_2-R_1-1} P_{Q_2-j,U}&=\sum _{j=1}^{Q_2-R_1-1} \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2+j}\nonumber \\&\quad \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{j} \right) P_{Q_1+R_1,A} \nonumber \\&\quad + \sum _{j=0}^{Q_2-R_1-1}\left( \frac{\lambda }{\lambda +\alpha }\right) ^{j}P_{Q_2,U} \nonumber \\&=\frac{\lambda +\alpha }{\alpha }\left[ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1} \right] P_{Q_2,U}\nonumber \\&\quad +\frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \left[ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \right] P_{Q_1+R_1,A} \nonumber \\&\quad - \frac{(\lambda +\mu )^2}{(\mu -\alpha )(\lambda +\alpha )} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \left[ 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right] P_{Q_1+R_1,A}. \end{aligned}$$

(94)

For $P_{R_1+1,U}$, we have,

$$\begin{aligned} P_{R_1+1,U}&= \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1}P_{Q_2,U} + \frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}\nonumber \\&\quad \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right) P_{Q_1+R_1,A}. \end{aligned}$$

(95)

In order to obtain the closed form equation for the limiting probability of $(Q_2,U)$ state, we use the equations from (38) and (95):

$$\begin{aligned} P_{1,U}&=\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}P_{R_1+1,U} \nonumber \\&= \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-1}P_{Q_2,U} + \frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1}\nonumber \\&\quad \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right) P_{Q_1+R_1,A}. \end{aligned}$$

(96)

From (34), we also know that

$$\begin{aligned} P_{Q_2,U}=\frac{\lambda }{\lambda +\alpha }\left( P_{Q_2+1,U}+P_{1,U}\right) +\frac{\mu }{\lambda +\alpha }P_{Q_2,A}. \end{aligned}$$

(97)

From (91), we have the following for $P_{Q_2+1,U}$:

$$\begin{aligned} P_{Q_2+1,U}&=\frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2-1}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \right) P_{Q_1+R_1,A}. \end{aligned}$$

(98)

From the second case of rare shortage condition, we know that

$$\begin{aligned} P_{Q_2,A} = \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2}P_{Q_1+R_1,A}. \end{aligned}$$

(99)

Then,

$$\begin{aligned} P_{Q_2,U}&=\left[ \frac{ \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right) }{ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2} } + \frac{\frac{\mu }{\lambda +\alpha } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}}\right. \nonumber \\&\quad \left. +\frac{ \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \right) }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2} } \right] P_{Q_1+R_1,A}. \end{aligned}$$

(100)

Now, for $P_{Q_1+R_1,A}$, we have,

$$\begin{aligned} 1&=\frac{\lambda +\mu }{\mu }\left[ 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right] P_{Q_1+R_1,A}\nonumber \\&\quad + \frac{\lambda }{\alpha }\left( 1- \left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}\right) P_{R_1+1,U} \nonumber \\&\quad + \left[ \frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \right) \right. \nonumber \\&\quad \left. -\frac{\lambda +\mu }{\mu -\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \right) \right] P_{Q_1+R_1,A} \nonumber \\&\quad + \frac{\lambda +\alpha }{\alpha }\left[ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1} \right] P_{Q_2,U}\nonumber \\&\quad +\frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \left[ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \right] P_{Q_1+R_1,A} \nonumber \\&\quad - \frac{(\lambda +\mu )^2}{(\mu -\alpha )(\lambda +\alpha )} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \left[ 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right] P_{Q_1+R_1,A}\nonumber \\ P_{Q_1+R_1,A}&=\left[ \frac{\lambda +\mu }{\mu } \left( 1 - \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) + \frac{\lambda +\mu }{\alpha }- \frac{\lambda }{\alpha } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2}\right. \nonumber \\&\quad +\frac{\lambda (\lambda +\mu )}{\alpha (\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2} \nonumber \\&\quad -\frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} +\frac{(\lambda +\mu )^2}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda +\mu }{\lambda }-\frac{\mu }{\alpha } \right) \nonumber \\&\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}\left( \frac{\lambda +\alpha }{\lambda +\mu } \right) ^{Q_2-R_1}\bigg ]^{-1}\nonumber \\ P_{Q_1+R_1,A}&=\left[ \frac{\lambda +\mu }{\mu } \left( 1 - \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) + \frac{\lambda +\mu }{\alpha }- \frac{\lambda }{\alpha } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2}\right. \nonumber \\&\quad +\frac{\lambda +\mu }{\lambda +\alpha }\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2} \nonumber \\&\quad \times \left( \frac{\lambda }{\alpha }+\frac{\lambda +\mu }{\mu -\alpha } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right) - \frac{\mu (\lambda +\mu )}{\mu -\alpha }\nonumber \\&\quad \left. \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}\left( \frac{1}{\lambda } +\frac{1}{\alpha }\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right) \right] ^{-1} \end{aligned}$$

(101)

Expected inventory level can be computed as follows:

$$\begin{aligned} I(Q_1;Q_2;R_1)&= \sum _{j=0}^{Q_1-1} \left( Q_1+R_1-j\right) P_{Q_1+R_1-j,A} \nonumber \\&\quad + \sum _{j=0}^{Q_1+R_1-Q_2-1} \left( Q_1+R_1-j\right) P_{Q_1+R_1-j,U} \nonumber \\&\quad + \sum _{j=0}^{Q_2-R_1-1} \left( Q_2-j\right) P_{Q_2-j,U} + \sum _{j=0}^{R_1-1}(R_1-j) P_{R_1-j,U}. \end{aligned}$$

(102)

Then,

$$\begin{aligned} I(Q_1;Q_2;R_1)&=(Q_1+R_1)\left( \sum _{j=0}^{Q_1-1} P_{Q_1+R_1-j,A} +\sum _{j=0}^{Q_1+R_1-Q_2-1} P_{Q_1+R_1-j,U} \right) \nonumber \\&\quad -\sum _{j=0}^{Q_1-1} j P_{Q_1+R_1-j,A} \nonumber \\&\quad -\sum _{j=0}^{Q_1+R_1-Q_2-1} j P_{Q_1+R_1-j,U} + Q_2 \sum _{j=0}^{Q_2-R_1-1} P_{Q_2-j,U} -\sum _{j=0}^{Q_2-R_1-1}j P_{Q_2-j,U} \nonumber \\&\quad + R_1 \sum _{j=0}^{R_1-1} P_{R_1-j,U} -\sum _{j=0}^{R_1-1}j P_{R_1-j,U} \nonumber \\&\quad =(Q_1+R_1) -Q_1\left( \frac{\lambda }{\alpha }\right) \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}\right) P_{R_1+1,U} \nonumber \\&\quad -(Q_1+R_1-Q_2) \left( \frac{\lambda +\alpha }{\alpha }\right) \left[ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1} \right] P_{Q_2,U} \nonumber \\&\quad -(Q_1+R_1-Q_2) \left[ \frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \right) \right] P_{Q_1+R_1,A} \nonumber \\&\quad +(Q_1+R_1-Q_2)\left[ \frac{(\lambda +\mu )^2}{(\mu -\alpha )(\lambda +\alpha )} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right) \right] P_{Q_1+R_1,A} \nonumber \\&\quad - \sum _{j=0}^{Q_1-1} j P_{Q_1+R_1-j,A} -\sum _{j=0}^{Q_1+R_1-Q_2-1} j P_{Q_1+R_1-j,U} \nonumber \\&\quad -\sum _{j=0}^{Q_2-R_1-1}j P_{Q_2-j,U} -\sum _{j=0}^{R_1-1}j P_{R_1-j,U}. \end{aligned}$$

(103)

We replace the sum of probabilities from Eqs. (94) and (39), and $P_{Q_1+R_1-j,A}$, $P_{Q_1+R_1-j,U}$, $P_{R_1-j,U}$, and $P_{Q_2-j,U}$ from Eqs. (30), (91), (93), and (38):

$$\begin{aligned} I(Q_1;Q_2;R_1)&= (Q_1+R_1) -Q_1\frac{\lambda }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}\right) P_{R_1+1,U}\nonumber \\&\quad -(Q_1+R_1-Q_2) \frac{\lambda +\alpha }{\alpha }\left[ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1} \right] P_{Q_2,U} \nonumber \\&\quad -(Q_1+R_1-Q_2) \left[ \frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \right) \right] P_{Q_1+R_1,A} \nonumber \\&\quad +(Q_1+R_1-Q_2)\left[ \frac{(\lambda +\mu )^2}{(\mu -\alpha )(\lambda +\alpha )} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2}\right. \nonumber \\&\quad \left. \left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right) \right] P_{Q_1+R_1,A} \nonumber \\&\quad -\left[ \sum _{j=0}^{Q_1-1} j \left( \frac{\lambda }{\lambda +\mu }\right) ^{j} +\sum _{j=0}^{Q_1+R_1-Q_2-1} j\left( \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\right) \right. \nonumber \\&\quad \left. \left( \frac{\lambda }{\lambda +\alpha } \right) ^{j}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{j+1} \right) \right] P_{Q_1+R_1,A} \nonumber \\&\quad -\sum _{j=0}^{Q_2-R_1-1}j \left( \frac{\lambda }{\lambda +\alpha }\right) ^{j}P_{Q_2,U} +j \left( \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\right) \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2+j}\nonumber \\&\quad \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{j} \right) P_{Q_1+R_1,A} +\sum _{j=0}^{R_1-1}j \left( \frac{\lambda }{\lambda +\alpha }\right) ^{j+1}P_{R_1+1,U}. \end{aligned}$$

(104)

We can rewrite the last equation as

$$\begin{aligned} I(Q_1;Q_2;R_1)&= (Q_1+R_1) -Q_1\frac{\lambda }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}\right) P_{R_1+1,U} \nonumber \\&\quad -(Q_1+R_1-Q_2) \frac{\lambda +\alpha }{\alpha }\left[ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1} \right] P_{Q_2,U} \nonumber \\&\quad -(Q_1+R_1-Q_2) \left[ \frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \right) \right] P_{Q_1+R_1,A} \nonumber \\&\quad +(Q_1+R_1-Q_2)\left[ \frac{(\lambda +\mu )^2}{(\mu -\alpha )(\lambda +\alpha )} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right) \right] P_{Q_1+R_1,A} \nonumber \\&\quad -\left[ \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+1}-Q_1\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2}\right. \nonumber \\&\quad +\left( \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\right) \frac{(Q_1+R_1-Q_2-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2+1}-(Q_1+R_1-Q_2)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2} \nonumber \\&\quad \left. -\left( \frac{\mu }{\mu -\alpha }\right) \frac{(Q_1+R_1-Q_2-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2+1}-(Q_1+R_1-Q_2)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2} \right] P_{Q_1+R_1,A} \nonumber \\&\quad -\frac{(Q_2-R_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-R_1+1}-(Q_2-R_1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-R_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2} P_{Q_2,U} \nonumber \\&\quad - \left( \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\right) \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2}\left[ \frac{(Q_2-R_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-R_1+1}-(Q_2-R_1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-R_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}\right. \nonumber \\&\quad \left. - \frac{(Q_2-R_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-R_1+1}-(Q_2-R_1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-R_1}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2} \right] P_{Q_1+R_1,A} \nonumber \\&\quad - \left( \frac{\lambda }{\lambda +\alpha }\right) \frac{(R_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1+1}-R_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}P_{R_1+1,U}. \end{aligned}$$

(105)

For this case, $I(Q_1;Q_2;R_1)$ and $P_{1,U}$ that is used when calculating $SF(Q_1;Q_2;R_1)$ can be written in terms of $\alpha$, $\lambda$, $\mu$, $Q_1$, $Q_2$, and $R_1$ as follows:

$$\begin{aligned} \begin{aligned} P_{1,U}&= \bigg [\frac{\lambda +\mu }{\mu } \bigg (1 - \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \bigg ) + \frac{\lambda +\mu }{\alpha }- \frac{\lambda }{\alpha } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \\&\quad +\frac{\lambda (\lambda +\mu )}{\alpha (\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2} -\frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} +\frac{(\lambda +\mu )^2}{(\mu -\alpha )(\lambda +\alpha )} \\&\quad \times \left( \frac{\lambda +\mu }{\lambda }-\frac{\mu }{\alpha } \right) \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1} \left( \frac{\lambda +\alpha }{\lambda +\mu } \right) ^{Q_2-R_1}\bigg ]^{-1} \\&\quad \Bigg [\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-1} \Bigg [ \frac{ \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )} \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right) }{ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2} } \\&\quad + \frac{\frac{\mu }{\lambda +\alpha } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}} \\&\quad +\frac{ \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \right) }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2} } \Bigg ] \\&\quad + \frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right) \Bigg ]. \end{aligned} \end{aligned}$$

$$\begin{aligned}{} & {} I(Q_1;Q_2;R_1)=(Q_1+R_1) + \bigg [\frac{\lambda +\mu }{\mu } \bigg (1 - \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \bigg ) \\{} & {} \quad + \frac{\lambda +\mu }{\alpha }- \frac{\lambda }{\alpha } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} +\frac{\lambda (\lambda +\mu )}{\alpha (\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2} \\{} & {} \quad -\frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} +\frac{(\lambda +\mu )^2}{(\mu -\alpha )(\lambda +\alpha )} \\{} & {} \quad \times \left( \frac{\lambda +\mu }{\lambda }-\frac{\mu }{\alpha } \right) \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1} \left( \frac{\lambda +\alpha }{\lambda +\mu } \right) ^{Q_2-R_1}\bigg ]^{-1} \\{} & {} \quad \Bigg [ -Q_1\frac{\lambda }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}\right) \Bigg [\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1}\\{} & {} \quad \Bigg [ \frac{ \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right) }{ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2} }\\{} & {} \quad + \frac{\frac{\mu }{\lambda +\alpha } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}} +\frac{ \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \right) }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2} } \Bigg ] \\{} & {} \quad + \frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right) \Bigg ] \\{} & {} \quad -(Q_1+R_1-Q_2) \frac{\lambda +\alpha }{\alpha } \left[ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1} \right] \\{} & {} \quad \Bigg [ \frac{ \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right) }{ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2} } + \frac{\frac{\mu }{\lambda +\alpha } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}} \\{} & {} \quad +\frac{ \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \right) }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2} } \Bigg ] \\{} & {} \quad -(Q_1+R_1-Q_2) \Bigg [ \frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \Bigg (1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1} \Bigg ) \Bigg ] \\{} & {} \quad +(Q_1+R_1-Q_2)\Bigg [ \frac{(\lambda +\mu )^2}{(\mu -\alpha )(\lambda +\alpha )} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2} \Bigg (1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-R_1-1} \Bigg )\Bigg ]\\{} & {} \quad -\Bigg [ \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+1}-Q_1\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2}\\{} & {} \quad +\left( \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\right) \frac{(Q_1+R_1-Q_2-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2+1}-(Q_1+R_1-Q_2)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}\\ \end{aligned}$$

$$\begin{aligned}{} & {} \quad -\left( \frac{\mu }{\mu -\alpha }\right) \frac{(Q_1+R_1-Q_2-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2+1} -(Q_1+R_1-Q_2)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+R_1-Q_2}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2} \Bigg ] \\{} & {} \quad -\frac{(Q_2-R_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-R_1+1}-(Q_2-R_1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-R_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}\\{} & {} \quad \Bigg [ \frac{ \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right) }{ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2} } + \frac{\frac{\mu }{\lambda +\alpha } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}}\\{} & {} \quad +\frac{ \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \right) }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2} } \Bigg ] \\{} & {} \quad - \left( \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\right) \left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1+R_1-Q_2}\Bigg [ \frac{(Q_2-R_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-R_1+1}-(Q_2-R_1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-R_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}\\{} & {} \quad - \frac{(Q_2-R_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-R_1+1}-(Q_2-R_1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-R_1}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2} \Bigg ] \\{} & {} \quad - \left( \frac{\lambda }{\lambda +\alpha }\right) \frac{(R_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1+1}-R_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}\\{} & {} \quad \times \Bigg [\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-R_1-1}\Bigg [ \frac{ \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right) }{ 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2} } \\{} & {} \quad +\frac{\frac{\mu }{\lambda +\alpha } \left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2}} \\{} & {} \quad +\frac{ \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1-Q_2}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1+R_1-Q_2} \right) }{1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2} } \Bigg ] \\{} & {} \quad + \frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_2-R_1-1} \right) \Bigg ] \Bigg ], \end{aligned}$$

1.3 Case 3: $Q_2>Q_1+R_1$

All parts of this model, except part 4, are the same as the Case 3 of rare shortage condition. So, we just try to find the equations for fourth part and replace the other parts from case 3 of rare-shortage model. For part 4, we have

$$\begin{aligned} P_{Q_1+R_1-j,U}&=\frac{\mu \lambda ^j}{\lambda +\alpha }\left[ \sum _{k=0}^{j}\frac{1}{(\lambda +\alpha )^{j-k}(\lambda +\mu )^k} \right] P_{Q_1+R_1,A} \nonumber \\&\quad +\left( \frac{\lambda }{\lambda +\alpha } \right) ^{j+1}P_{Q_1+R_1+1,U} \end{aligned}$$

(106)

$$\begin{aligned} P_{Q_1+R_1-j,U}&=\frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{j} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{j+1}\right) P_{Q_1+R_1,A} \nonumber \\&\quad +\left( \frac{\lambda }{\lambda +\alpha } \right) ^{j+1}P_{Q_1+R_1+1,U},\nonumber \\&\quad j=0,\ldots ,Q_1-1. \end{aligned}$$

(107)

Let us try to find the sum of series in last equation. We have

$$\begin{aligned} \sum _{j=0}^{Q_1-1}P_{Q_1+R_1-j,U}&=\Bigg [\frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) \nonumber \\&\quad - \frac{\lambda +\mu }{\mu -\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}\right) \Bigg ] P_{Q_1+R_1,A} \end{aligned}$$

(108)

$$\begin{aligned}&\quad +\frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) P_{Q_1+R_1+1,U}. \end{aligned}$$

(109)

From Eq. (107), we have the following equation for $P_{R_1+1,U}$:

$$\begin{aligned} P_{R_1+1,U}&=\frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) \nonumber \\&\quad P_{Q_1+R_1,A} +\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}P_{Q_1+R_1+1,U}. \end{aligned}$$

(110)

We know the probability of $(Q_1+R_1+1,U)$ and $(Q_2,U)$ from the probability equations for case 3 of rare shortage model, (56) and (72). Then for $P_{R_1+1,U}$:

$$\begin{aligned} P_{R_1+1,U}=&\frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1-1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) P_{Q_1+R_1,A} +\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1} \nonumber \\&\quad \times \Bigg [\frac{(\lambda +\mu )(\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1-1} \nonumber \\&\quad + \frac{\mu \alpha }{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1} \Bigg ] \nonumber \\&\quad \times \frac{\lambda +\mu }{\lambda +\mu +\alpha } \left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}P_{R_1+1,U} \nonumber \\ P_{R_1+1,U}&=\frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1-1}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) P_{Q_1+R_1,A} \nonumber \\&\quad + \frac{(\lambda +\mu )^2}{\lambda (\lambda +\mu +\alpha )^2} \Bigg [(\lambda +\alpha ) \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-1}\nonumber \\&\quad + \frac{\mu \alpha }{\lambda +\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1} \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \Bigg ] P_{R_1+1,U} \nonumber \\ P_{R_1+1,U}&=\frac{\frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}\left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{\lambda (\lambda +\mu +\alpha )^2} \left[ (\lambda +\alpha )\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-1} + \frac{\mu \alpha }{\lambda +\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] }P_{Q_1+R_1,A}. \end{aligned}$$

(111)

We also know that

$$\begin{aligned} P_{Q_2,U}=\frac{\frac{\mu (\lambda +\mu )^2}{\lambda (\mu -\alpha )(\lambda +\mu +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{\lambda (\lambda +\mu +\alpha )^2} \left[ (\lambda +\alpha )\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-1} + \frac{\mu \alpha }{\lambda +\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] }P_{Q_1+R_1,A}. \end{aligned}$$

(112)

For $P_{Q_1+R_1,A}$, we have

$$\begin{aligned} 1&=\frac{\lambda +\alpha }{\lambda +\alpha +\mu }\Bigg [ \frac{\alpha }{\mu } \left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) \nonumber \\&\quad +\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1}\right) +\frac{\alpha (\lambda +\alpha +\mu )}{(\lambda +\mu )(\lambda +\alpha )}\Bigg ] P_{Q_2,U} \nonumber \\&\quad +\frac{\lambda +\mu }{\lambda +\alpha +\mu }\Bigg [\frac{\lambda +\alpha }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1} \right) \nonumber \\&\quad +\frac{\alpha }{\lambda +\mu } \left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) +\frac{\lambda +\alpha +\mu }{\lambda +\mu } \Bigg ] P_{Q_2,U} \nonumber \\&\quad +\frac{\lambda +\mu }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1} \right) P_{Q_1+R_1,A} +\frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \right) P_{R_1+1,U} \nonumber \\&\quad +\Bigg [\frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) -\frac{\lambda +\mu }{\mu -\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}\right) \Bigg ] \nonumber \\&\quad P_{Q_1+R_1,A} +\frac{\lambda }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) P_{Q_1+R_1+1,U}. \end{aligned}$$

(113)

Then,

$$\begin{aligned} 1&=\Bigg [ \frac{\alpha }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) +\frac{\lambda +\alpha }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1}\right) \nonumber \\&\quad +\frac{\alpha +\lambda +\mu }{\lambda +\mu }\Bigg ] P_{Q_2,U} + \frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \right) P_{R_1+1,U} \nonumber \\&\quad +\frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) P_{Q_1+R_1+1,U} \nonumber \\&\quad +\Bigg [\frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) -\frac{\alpha (\lambda +\mu )}{\mu (\mu -\alpha )}\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}\right) \Bigg ] \nonumber \\&\quad P_{Q_1+R_1,A}, \end{aligned}$$

(114)

$$\begin{aligned} P_{Q_1+R_1,A}&=\Bigg [ \Bigg [ \frac{\alpha }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) +\frac{\lambda +\alpha }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1}\right) \nonumber \\&\quad +\frac{\alpha +\lambda +\mu }{\lambda +\mu }\Bigg ] \times \frac{\frac{\mu (\lambda +\mu )^2}{\lambda (\mu -\alpha )(\lambda +\mu +\alpha )} \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{\lambda (\lambda +\mu +\alpha )^2} \left[ (\lambda +\alpha )\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-1} + \frac{\mu \alpha }{\lambda +\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1} \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \nonumber \\&\quad + \frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \right) \frac{\frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{\lambda (\lambda +\mu +\alpha )^2} \left[ (\lambda +\alpha )\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-1} + \frac{\mu \alpha }{\lambda +\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \nonumber \\&\quad +\frac{\lambda }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) \Bigg [\frac{(\lambda +\mu )(\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )} \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1-1} + \frac{\mu \alpha }{\lambda (\lambda +\alpha +\mu )} \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1} \Bigg ] \nonumber \\&\quad \times \frac{\frac{\mu (\lambda +\mu )^2}{\lambda (\mu -\alpha )(\lambda +\mu +\alpha )} \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{\lambda (\lambda +\mu +\alpha )^2} \left[ (\lambda +\alpha )\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-1} + \frac{\mu \alpha }{\lambda +\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1} \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \nonumber \\&\quad +\Bigg [\frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) -\frac{\alpha (\lambda +\mu )}{\mu (\mu -\alpha )}\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}\right) \Bigg ] \Bigg ] ^{-1}. \end{aligned}$$

(115)

Expected inventory level can be computed as follows:

$$\begin{aligned} I(Q_1;Q_2;R_1)&=\sum _{j=0}^{Q_2-Q_1-R_1-1} (Q_2-j,A)P_{Q_2-j,A}+ \sum _{j=0}^{Q_2-Q_1-R_1-1} \left( Q_2-j\right) P_{Q_2-j,U} \nonumber \\&\quad + \sum _{j=0}^{R_1-1}(R_1-j) P_{R_1-j,U} \nonumber \\&\quad +\sum _{j=0}^{Q_1-1} \left( Q_1+R_1-j\right) P_{Q_1+R_1-j,A} \nonumber \\&\quad +\sum _{j=0}^{Q_1-1} \left( Q_1+R_1-j\right) P_{Q_1+R_1-j,U}. \end{aligned}$$

(116)

Then,

$$\begin{aligned} I(Q_1;Q_2;R_1)&=Q_2 \sum _{j=0}^{Q_2-Q_1-R_1-1}(P_{Q_2-j,A}+P_{Q_2-j,U})\nonumber \\&\quad +(Q_1+R_1)\sum _{j=0}^{Q_1-1}(P_{Q_1+R_1-j,A}+ P_{Q_1+R_1-j,U}) \nonumber \\&\quad +R_1\sum _{j=0}^{R_1-1}P_{R_1-j,U} - \sum _{j=0}^{Q_2-Q_1-R_1-1} j (P_{Q_2-j,A}+P_{Q_2-j,U})\nonumber \\&\quad - \sum _{j=0}^{R_1-1} j P_{R_1-j,U} \nonumber \\&\quad - \sum _{j=0}^{Q_1-1} j (P_{Q_1+R_1-j,A}+ P_{Q_1+R_1-j,U}). \end{aligned}$$

(117)

We replace the probabilities $P_{Q_2-j,A}$, $P_{Q_1+R_1-j,A}$, $P_{Q_2-j,U}$, $P_{Q_1+R_1-j,U}$, and $P_{R_1-j,U}$ from Eqs. (55), (59), (57), (108), and (61):

$$\begin{aligned} I(Q_1;Q_2;R_1)&=Q_2\Bigg [\frac{\alpha }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) \nonumber \\&\quad +\frac{\lambda +\alpha }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1}\right) \nonumber \\&\quad + \frac{\alpha +\lambda +\mu }{\lambda +\mu }\Bigg ] P_{Q_2,U} +(Q_1+R_1)\frac{\lambda }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) P_{Q_1+R_1+1,U} \nonumber \\&\quad +(Q_1+R_1)\Bigg [\frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) -\frac{\alpha (\lambda +\mu )}{\mu (\mu -\alpha )}\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}\right) \Bigg ] P_{Q_1+R_1,A} \nonumber \\&\quad +R_1 \frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \right) P_{R_1+1,U} \nonumber \\&\quad -\sum _{j=1}^{Q_2-Q_1-R_1-1}j\frac{\alpha (\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )} \left[ \left( \frac{\lambda }{\lambda +\mu }\right) ^{j} + \left( \frac{\lambda }{\lambda +\alpha }\right) ^{j} \right] P_{Q_2,U} \nonumber \\&\quad -\sum _{j=1}^{Q_2-Q_1-R_1-1}j\Bigg [\frac{(\mu +\lambda )(\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )} \left( \frac{\lambda }{\lambda +\alpha } \right) ^{j} + \frac{\mu \alpha }{\lambda (\lambda +\alpha +\mu )} \left( \frac{\lambda }{\lambda +\mu } \right) ^{j} \Bigg ] P_{Q_2,U} \nonumber \\&\quad -\sum _{j=0}^{Q_1-1}j \left( \frac{\lambda }{\lambda +\mu } \right) ^{j} P_{Q_1+R_1,A}-\sum _{j=0}^{R_1-1} j \left( \frac{\lambda }{\lambda +\alpha }\right) ^{j+1}P_{R_1+1,U} \nonumber \\&\quad -\sum _{j=0}^{Q_1-1} j\frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )} \left( \frac{\lambda }{\lambda +\alpha } \right) ^{j} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu } \right) ^{j+1}\right) \nonumber \\&\quad P_{Q_1+R_1,A} +j \left( \frac{\lambda }{\lambda +\alpha } \right) ^{j+1}P_{Q_1+R_1+1,U} \end{aligned}$$

(118)

$$\begin{aligned} I(Q_1;Q_2;R_1)&=Q_2 \Bigg [\frac{\alpha }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) \nonumber \\&\quad +\frac{\lambda +\alpha }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1}\right) \nonumber \\&\quad + \frac{\alpha +\lambda +\mu }{\lambda +\mu }\Bigg ] P_{Q_2,U} +(Q_1+R_1) \frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) P_{Q_1+R_1+1,U} \nonumber \\&\quad +(Q_1+R_1)\Bigg [\frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) -\frac{\alpha (\lambda +\mu )}{\mu (\mu -\alpha )} \left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}\right) \Bigg ] P_{Q_1+R_1,A} \nonumber \\&\quad +R_1\left( \frac{\lambda }{\alpha }\right) \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \right) P_{R_1+1,U} -\left( \frac{\alpha }{\lambda }\right) P_{Q_2,U} \times \nonumber \\&\quad \Bigg [ \frac{(Q_2-Q_1-R_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1+1}-(Q_2-Q_1-R_1) \left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2}\Bigg ] \nonumber \\&\quad -\frac{(Q_2-Q_1-R_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1}-(Q_2-Q_1-R_1) \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1-1}+1}{(1-\frac{\lambda }{\lambda +\alpha })^2} P_{Q_2,U} \nonumber \\&\quad -\left( \frac{\lambda }{\lambda +\alpha }\right) \frac{(R_1-1) \left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1+1}-R_1 \left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}P_{R_1+1,U} \nonumber \\&\quad +\left( \frac{\alpha }{\mu -\alpha } \right) \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+1}-Q_1\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2} P_{Q_1+R_1,A} \nonumber \\&\quad -\left( \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\right) \frac{(Q_1-1) \left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+1}-Q_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2} P_{Q_1+R_1,A} \nonumber \\&\quad -\left( \frac{\lambda }{\lambda +\alpha } \right) \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+1}-Q_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2} P_{Q_1+R_1+1,U}. \end{aligned}$$

(119)

Note that this equation is the same with case 3 of rare shortage condition.

For this case, $I(Q_1;Q_2;R_1)$ and $P_{1,U}$ that is used when calculating $SF(Q_1;Q_2;R_1)$ can be written in terms of $\alpha$, $\lambda$, $\mu$, $Q_1$, $Q_2$, and $R_1$ as follows:

$$\begin{aligned} P_{1,U}= & {} \left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1}\\{} & {} \frac{\frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{\lambda (\lambda +\mu +\alpha )^2} \left[ (\lambda +\alpha )\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-1} + \frac{\mu \alpha }{\lambda +\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] }P_{Q_1+R_1,A} \\ I(Q_1;Q_2;R_1)= & {} \left[ \left[ \frac{\alpha }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) +\frac{\lambda +\alpha }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1}\right) \right. \right. \\{} & {} \left. +\frac{\alpha +\lambda +\mu }{\lambda +\mu }\right] \\{} & {} \times \frac{\frac{\mu (\lambda +\mu )^2}{\lambda (\mu -\alpha )(\lambda +\mu +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{\lambda (\lambda +\mu +\alpha )^2} \left[ (\lambda +\alpha )\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-1} + \frac{\mu \alpha }{\lambda +\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \\{} & {} + \frac{\lambda }{\alpha } \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \right) \\{} & {} \times \frac{\frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{\lambda (\lambda +\mu +\alpha )^2} \left[ (\lambda +\alpha )\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-1} + \frac{\mu \alpha }{\lambda +\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \\{} & {} +\frac{\lambda }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) \\{} & {} \times \left[ \frac{(\lambda +\mu )(\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1-1} + \frac{\mu \alpha }{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1} \right] \\{} & {} \times \frac{\frac{\mu (\lambda +\mu )^2}{\lambda (\mu -\alpha )(\lambda +\mu +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{\lambda (\lambda +\mu +\alpha )^2} \left[ (\lambda +\alpha )\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-1} + \frac{\mu \alpha }{\lambda +\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \\{} & {} \left. +\left[ \frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) -\frac{\alpha (\lambda +\mu )}{\mu (\mu -\alpha )}\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}\right) \right] \right] ^{-1} \\{} & {} \left[ Q_2 \left[ \frac{\alpha }{\mu }\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_2-Q_1-R_1-1}\right) +\frac{\lambda +\alpha }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_2-Q_1-R_1-1}\right) + \frac{\alpha +\lambda +\mu }{\lambda +\mu }\right] \right. \\ \end{aligned}$$

$$\begin{aligned}{} & {} \quad \times \frac{\frac{\mu (\lambda +\mu )^2}{\lambda (\mu -\alpha )(\lambda +\mu +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{\lambda (\lambda +\mu +\alpha )^2} \left[ (\lambda +\alpha )\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-1} + \frac{\mu \alpha }{\lambda +\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \\{} & {} \quad +(Q_1+R_1) \frac{\lambda }{\alpha }\left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1} \right) \left[ \frac{(\lambda +\mu )(\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1-1} + \frac{\mu \alpha }{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1} \right] \\{} & {} \quad \times \frac{\frac{\mu (\lambda +\mu )^2}{\lambda (\mu -\alpha )(\lambda +\mu +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{\lambda (\lambda +\mu +\alpha )^2} \left[ (\lambda +\alpha )\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-1} + \frac{\mu \alpha }{\lambda +\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \\{} & {} \quad +(Q_1+R_1)\left[ \frac{\mu (\lambda +\mu )}{\alpha (\mu -\alpha )} \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{Q_1}\right) -\frac{\alpha (\lambda +\mu )}{\mu (\mu -\alpha )}\left( 1-\left( \frac{\lambda }{\lambda +\mu }\right) ^{Q_1}\right) \right] \\{} & {} \quad +R_1\left( \frac{\lambda }{\alpha }\right) \left( 1-\left( \frac{\lambda }{\lambda +\alpha }\right) ^{R_1} \right) \frac{\frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{\lambda (\lambda +\mu +\alpha )^2} \left[ (\lambda +\alpha )\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-1} + \frac{\mu \alpha }{\lambda +\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \\{} & {} \quad -\left( \frac{\alpha }{\lambda }\right) \frac{\frac{\mu (\lambda +\mu )^2}{\lambda (\mu -\alpha )(\lambda +\mu +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{\lambda (\lambda +\mu +\alpha )^2} \left[ (\lambda +\alpha )\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-1} + \frac{\mu \alpha }{\lambda +\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \\{} & {} \quad \times \left[ \frac{(Q_2-Q_1-R_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1+1}-(Q_2-Q_1-R_1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2}\right] \\{} & {} \quad -\frac{(Q_2-Q_1-R_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1}-(Q_2-Q_1-R_1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1-1}+1}{(1-\frac{\lambda }{\lambda +\alpha })^2} \\{} & {} \quad \times \frac{\frac{\mu (\lambda +\mu )^2}{\lambda (\mu -\alpha )(\lambda +\mu +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{\lambda (\lambda +\mu +\alpha )^2} \left[ (\lambda +\alpha )\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-1} + \frac{\mu \alpha }{\lambda +\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \\{} & {} \quad -\left( \frac{\lambda }{\lambda +\alpha }\right) \frac{(R_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1+1}-R_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{R_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}\\{} & {} \quad \times \frac{\frac{\mu (\lambda +\mu )}{\lambda (\mu -\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{\lambda (\lambda +\mu +\alpha )^2} \left[ (\lambda +\alpha )\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-1} + \frac{\mu \alpha }{\lambda +\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] } \\{} & {} \quad +\left( \frac{\alpha }{\mu -\alpha } \right) \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1+1}-Q_1\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_1}+\frac{\lambda }{\lambda +\mu }}{(1-\frac{\lambda }{\lambda +\mu })^2}\\{} & {} \quad -\left( \frac{\mu (\lambda +\mu )}{(\mu -\alpha )(\lambda +\alpha )}\right) \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+1}-Q_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2} \\{} & {} \quad -\left( \frac{\lambda }{\lambda +\alpha } \right) \frac{(Q_1-1)\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+1}-Q_1\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1}+\frac{\lambda }{\lambda +\alpha }}{(1-\frac{\lambda }{\lambda +\alpha })^2}\\{} & {} \quad \times \left[ \frac{(\lambda +\mu )(\lambda +\alpha )}{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-Q_1-R_1-1} + \frac{\mu \alpha }{\lambda (\lambda +\alpha +\mu )}\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1} \right] \\{} & {} \quad \left. \times \frac{\frac{\mu (\lambda +\mu )^2}{\lambda (\mu -\alpha )(\lambda +\mu +\alpha )}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \left( 1-\left( \frac{\lambda +\alpha }{\lambda +\mu }\right) ^{Q_1}\right) }{1-\frac{(\lambda +\mu )^2}{\lambda (\lambda +\mu +\alpha )^2} \left[ (\lambda +\alpha )\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_2-1} + \frac{\mu \alpha }{\lambda +\mu }\left( \frac{\lambda }{\lambda +\mu } \right) ^{Q_2-Q_1-R_1-1}\left( \frac{\lambda }{\lambda +\alpha } \right) ^{Q_1+R_1} \right] }\right] \end{aligned}$$