Geni Gupur Functional analysis method for the M / G / 1 queueing model with single working vacation

In this paper, we study the asymptotic property of underlying operator corresponding to the M/G/1 queueing model with single working vacation, where both service times in a regular busy period and in a working vacation period are function. We obtain that all points on the imaginary axis except zero belong to the resolvent set of the operator and zero is an eigenvalue of both the operator and its adjoint operator with geometric multiplicity one. Therefore, we deduce that the time-dependent solution of the queueing model strongly converges to its steady-state solution. We also study the asymptotic behavior of the time-dependent queueing system’s indices for the model.


Introduction
Queueing system with server working vacations have arisen many researchers' attention because the working vacation policy is more appropriate to model the real system in which the server has additional task during a vacation [1][2][3]. Unlike a classical vacation policy, the working vacation policy requires the server working at a lower rate rather than completely stopping service during a vacation. Therefore, compared with the classical vacation model, there are also customers who leave the system due to the completion of the services during working vacation. In this way, the number of customers in the system may be reduced. For example, an agent in a call center is required to do additional work after speaking with a customer. The agent may provide service to the next customer at a lower rate while performing additional tasks. In 2002, Servi and Finn [1] rst introduced the M/M/1 queueing system with multiple working vacation. Since then, many researchers have extended their work to various type of queueing system (see Chandrasekaran et al. [4] ). Kim et al. [5] and Wu and Takagi [6] extended Servi and Finn's [1] M/M/1 queueing system to an M/G/1 queueing system. Xu et al. [7] and Baba [8] studied a batch arrival M X M queueing with working vacation. Gao and Yao [9] generalized it to an M X G queueing system. Baba [10] introduced the general input GI/M/1 queueing model with working vacation. Du [11] and Arivudainambi et al. [12] developed retrial queueing model with the concept of working vacation, etc. In 2012, Zhang and Hou [13] established the mathematical model of the M/G/1 queueing system with single working vacation by using the supplementary variable technique and studied the queueing length distribution and service status at the arbitrary epoch in the steady-state case under the following hypothesis: lim t→∞ p ,v (t) = p ,v , lim t→∞ p n,v (x, t) = p n,v (x), n ≥ , lim t→∞ p ,b (t) = p ,b , lim t→∞ p n,b (x, t) = p n,b (x), n ≥ . (H) According to Zhang and Hou [13], the M/G/1 queueing model with single working vacation can be described by the following partial di erential equations: with the integral boundary conditions (1.2) If we assume the system states when there are no customers in the system and the server is in vacation, i.e., represents the probability that there is no customer in the system and the server is in a working vacation period at time t; p n,v (x, t)dx (n ≥ ) is the probability that at time t the server is in a working vacation period and there are n customers in the system with elapsed service time of the customer undergoing service lying in (x, x + dx]; p ,b (t) represents the probability that there is no customer in the system and the server is in a regular busy period at time t; p n,b (x, t)dx (n ≥ ) is the probability that at time t the server is in a regular busy period and there are n customers in the system with elapsed service time of the customer undergoing service lying in (x, x + dx]; λ is the mean arrival rate of customers; θ is the vacation duration rate of the server; µ v (x) is the service rate of the server while the server is in a working vacation period and satis es is the service rate of the server while the server is in a regular busy period and satisfying In fact, the above hypothesis (H) implies the following two hypotheses in view of partial di erential equations: Hypothesis 1. The model has a unique time-dependent solution.
Hypothesis 2. The time-dependent solution converges to its steady-state solution.
In 2016, Kasim and Gupur [14] did the dynamic analysis for the above model and gave the detailed proof of the hypothesis 1. Moreover, when the service rates in a working vacation period and in a regular busy period are constant, by using the C − semigroup theory they obtained that the hypothesis 2 also hold. In the general case, the service rates are function, the hypothesis 2 does not always hold, see Gupur [15] and Kasim and Gupur [16], and it is necessary to study the asymptotic behavior of the time-dependent solution of the model. This paper is an e ort on this subject. The rest of this paper is organized as follows. In Section 2 we convert the model into an abstract Cauchy problem. In Section 3, by investigating the spectral properties of the underlying operator we give the main results of this paper. Firstly, we prove that is an eigenvalue of the underlying operator with geometric multiplicity one by using the probability generating function. Next, to obtain the resolvent set of the underlying operator we apply the boundary perturbation method. We obtain that all points on the imaginary axis except zero belong to the resolvent set of the operator. Last, we determine the adjoint operator and verify that is an eigenvalue of the adjoint operator with geometric multiplicity one. Finally, based on these results we present the desired result in this paper: the time-dependent solution of the model strongly converges to its steady-state solution. In addition, the asymptotic behavior of the queueing system's indices are discussed. A conclusion is given in Section 4. Section 5 provides a detail proof of some lemmas.

Abstract Setting for the system
In this section, we reformulate the equation (1.1)-(1.3) as an abstract Cauchy problem. We start by introducing the state space as follows.
It is obvious that X × Y is a Banach space. De ne an operator and its domain. where p n,v (x) and p n,b (x)(n ≥ ) are absolutely continuous and We choose the boundary space of X × Y ∂(X × Y) = l × l and de ne two boundary operators as

Now we de ne operator A and its domain as
Then the above equations (1.1)-(1.3) can be written as an abstract Cauchy problem: Kasim and Gupur [14] have obtained the following results.

Main results
In this section, rstly we prove that is an eigenvalue of A with geometric multiplicity one, next we study the resolvent set of operator A by using the Greiner's idea [17] and obtain that all points on the imaginary axis except zero belong to the resolvent set of A. Thirdly, we determine the expression of A * , the adjoint operator of A, and verify that is an eigenvalue of the adjoint operator A * with geometric multiplicity one. Thus, we conclude that the time-dependent solution of the system (2.1) strongly converges to its steady-state solution.
then is an eigenvalue of A with geometric multiplicity one.
Proof. We consider the equation A(p v , p b ) = , which is equivalent to with the boundary conditions By solving (3.1) we obtain The probability generating functions of these sequences are given by, for z < Then (3.5) can be written as Now, we introduce the row-vector generating functions Hence, from (3.7)-(3.9) we deduce An easy computation shows that This together with (3.10) yields where e = ( , ) T . In the following, by using the Rouche's theorem we conclude that z − C(z) has a unique zero point inside unit circle z = . Let this root be denoted by γ, this must be root of the numerator of the equation (3.12) too. So, substituting z = γ into (3.12) we get (3.6) and (3.13) give (3.14) By using the L'Hospital rule and (3.12)-(3.14), we determine Thus, is an eigenvalue of A. Moreover, from (3.5) it easy to see that the eigenvectors corresponding to zero span one dimensional linear space, i.e., the geometric multiplicity of is one.
In order to obtain the asymptotic behavior of the time-dependent solution of the system (2.1) we need to know the spectrum of A on the imaginary axis (see Theorem 14 in Gupur et al. [18]). For that purpose we use boundary perturbation method, which is developed by the Greiner [17], through which the spectrum of the operator can be deduced by discussing the boundary operator. It is related to the resolvent set of operator A and spectrum of ΦD γ , where D γ is inverse of L in ker(γI − A m ). Hence, we rst consider the operator and discuss its inverse. For any given (y, z) ∈ X × Y , we consider the equation (γI − A )(p v , p b ) = (y, z), i.e., If we introduce the following two operators as , Similarly, we have Therefore, we obtain the following two lemmas and their proof given in the appendix. , (a ,b , a ,b , a ,b , ⋯) ∈ l .

Lemma 3.3. Let
(3.31) Using the results in Greiner [17], observe that the operator L is surjective. So, is invertible if γ ∈ ρ(A ). Its inverse will play an important role in the characterization of the spectrum of A on the imaginary axis and we denote its inverse by and call it the Dirichlet operator. Furthermore, Lemma 3.3 gives the explicit formula of D γ for all γ ∈ ρ(A ), From the expression of D γ and the de nition of Φ, it is easy to determine the explicit form of ΦD γ as follows.
Haji and Radl [19] gave the following result, which indicates the relations between the spectrum of A and spectrum of ΦD γ .

Lemma 3.4.
If γ ∈ ρ(A ) and there exists γ ∈ C such that ∈ σ(ΦD γ ), then From Lemma 3.4 and Nagel [20], we obtain the resolvent set of A on the imaginary axis.

Lemma 3.5. If
then all points on the imaginary axis except zero belong to the resolvent set of A.
implies that there exists a positive constant K > such that ∀ β > K, In this formula, by replacing and using the fact Theorem 2.1 and Lemma 3.1 ensures that T(t) is a positive contraction C −semigroup and its spectral bound is zero. By Nagel [20] we know that σ(A) is imaginary additively cyclic (see also Thorem 1.88 in [21]) which states that iβ ∈ σ(A) ⇒ iβh ∈ σ(A), all positive integer h.
A trivial veri cation shows that X * × Y * , the dual space of X × Y , is as follows.
It is evident that X * × Y * is a Banach space.
Lemma 3.6. A * , the adjoint operator of A, is as follows.
here α in D(A * ) is a constant which is independent of n.
then the time-dependent solution of the system (2.1) strongly converges to its steady-state solution, i.e., is the eigenvector in Lemma 3.1 and ω is decided by the eigenvector in Lemma 3.7 and the initial value (p v , p b )( ).
In the following, by applying the Theorem 3.8 we brie y discuss the queueing system's indices. It is easily seen that the time-dependent queueing size at the departure point converges to a positive number, i.e., and the time-dependent queueing length L(t) converges to the steady-state queueing length L, that is, From this we can obtain that other queuing indices L q (t), W(t) and W q (t) also converge to a positive number L q , W and W q respectively.

Conclusion
In this paper, we study an M/G/1 queueing model with single working vacation, in which the service time is generally distributed. The system is described by in nite number of partial di erential equations with integral boundary conditions which we have converted into an abstract Cauchy problem in the Banach space. Then, by investigating the spectrum of the operator on the imaginary axis, which corresponds to the M/G/1 queueing model with single working vacation, we proved that the time-dependent solution of the model strongly converges to its steady-state solution. In other words, we veri ed that the hypothesis 2 holds in the sense of strong convergence. In this paper and our previous paper, we only studied spectra of the operator on the right half complex plane and imaginary axis, which corresponds to the M/G/1 queueing model with single working vacation, so it is worth studying spectra of the operator on the left half complex plane.
From (A.1) and (A.2) together with condition of this lemma and using φ v ≤ µ v , φ b ≤ µ b we deduce, for any (y, z) ∈ X × Y (γI − A ) − (y, z) This shows that the result of this lemma is right.
Proof of Lemma 3.3. If (p v , p b ) ∈ ker(γI − A m ), then (γI − A m )(p v , p b ) = , which is equivalent to