AN M/M/1/N QUEUE WITH WORKING BREAKDOWNS AND VACATIONS-AN APPROACH TO COST ANALYSIS

In this paper, we consider an M/M/1/N queue with working breakdown and two types of server vacations. Sensitivity analysis of the model is performed to aissue with the point of streamlining service rates by limiting the normal expense per unit time using the direct search method.


Introduction
In our every day life we often face the queueing situations where the server is unavailable for a random period of time due to server failures. For instance, in manufacturing systems and communication systems the server may breakdown due to machine or job related problems. Queueing models with unreliable servers have been studied by various authors in the past since the perfectly reliable server is virtually nonexistent. Studies on the above topic are called queueing systems with server breakdowns. There is an extensive literature on queueing systems with server breakdowns and for an exhaustive review on this topic we refer the readers to Krishnamoorhty et al. [8]. In most of the previous research, it is generally assumed that the server completely stops service during the lengthy and unpredictable breakdown period. Kalidass and Kasturi [5] studied a new class of breakdowns policy in which a customer is served at a slower rate when the system is in partial defective. Such kind of breakdowns are called working breakdowns. Li [9] discussed equilibrium customer strategies in M/M/1 queue with working breakdowns. Kim and Lee [7] studied M/G/1 queueing system with disaster and working breakdowns. Recently, Yang and Wu [15], numerically derived transient state probabilities for an M/M/1/N queue with working breakdowns and server vacations. Recently Kalidass and Pavithra [6] studied M/M/1/N queue with working breakdowns and Bernoulli's feedbacks.
In some situations, an idle server will start some other uninterruptible task which is referred to as a 'vacation period'. For a comprehensive and complete review on vacation queueing systems, we refer the readers to Doshi (1986) [3], Ke et al. [4] and Shweta Upadhyaya [12]. In this paper, we consider an M/M/1/N queue with working breakdowns and vacations. From the practical point of view this type of model can be used to study many real life situations. Deepa and Kalidass [1] studied the steady state probabilities of the model by matrix analytic method. Obtained by the above analysis, some performance measures and numerical examples of the system was presented. In this paper the cost optimization for the above model is discussed.

The model
In this model, we consider a single server queue system with arrival rate λ. The service times during the normal period are exponentially distributed independent and identically distributed random variables with mean service time 1 µ . The server proceeds on a vacation whenever the system is empty. The duration of the server vacation is assumed to be exponentially distributed with a parameter γ. The server may get partial breakdowns while providing the service. The partial breeakdown time follows an exponential distribution with parameter α. After the partial breakdowns server provides service in a slow manner. The partial breakdown service times are exponentially distributed independent and identically distributed random variables with mean service rate 1 µv > 1 µ . After every (breakdown) busy period, the server goes for a vacation. The vacation time follows an exponential distribution with γ v . The repair times while the server is in partial breakdown and vacation follows an exponential distribution with parameter β v (> β). All inter arrival times, inter (normal or breakdown) service times, and inter vacation times are independent of each other. Let C(t) be the server state at time t. Then 1, the server is in normal state, 2, the server is in working breakdown, 3, the server is in normal and vacation, 4, the server is in working breakdown and vacation. Let N (t) be the number of customers in the system at time t. Then {(C(t), N (t)), t ≥ 0} is a continuous time Markov chain.

Steady state analysis
Here we consider the model studied by  [1] where the steady state probability equations governing the model are as follows: Using the normalizing condition, where e = (1 1 · · · 1) T

The cost model and its optimization
In this segment, we study the comletel expected cost work with choice variable µ and µ v for the above examined model. Our primary point is to acquire the ideal qualities for the rates µ and µ v so as to limit the expense.

Cost function
Let us characterize the accomanying cost comonents.
C 1 : holding cost per unit time for every client present in the framework. C 2 : cost per unit time when the server is in working brakdown and vacation period. C 3 : cost per unit time when the server is busy during the ordinary assistance time frame. C 4 : cost per unit time when the server is busy during the working breakdown assistance time frame. C 5 : cost per unit time when one client sits tight for ordinary or working breakdown assistance in the framework. C 6 : fixed cost for slow service rate and C 7 : fixed cost for quick service rate.  Using the definition of these cost elements listed above, the expected cost function per unit time is given by The cost minimization problem can be formulated as The unpredictability of articultions P 1 , P 2 , P 3 and P 4 confound the cost work in the condition.
Tragically, it is difficult to infer the scientific answers for the ideal help rate at the base anticipated expense. Thus, we develop the approximation to obtain the optimal service rates µ * , µ * v by Direct search method.

Direct search method
We assume the following cost parameters as C 1 = 10, C 2 = 80, C 3 = 75, Numerical examples are presented to determine the optimal value µ by means of the direct search method. From the following cost function figures and tables we come to know that, Table  1 shows that for various values of µ v = 0.8, 0.85, 0.9 the minimum expected cost Rs.84.9859 is achieved at µ = 1.5, for µ v = 0.8 Rs. 83.5948 is achieved at µ = 1.5, for µ v = 0.85 and Rs.82.2808 is achieved at µ = 1.5, for µ v = 0.9.
From figure 2 it is seen that initially the total cost diminishes and begins expanding with the ward of µ for fixed estimations of µ v . The raised nature of the cost work concerning µ show the pattern for the ideal expense by expanding the typical assistance sace of the clients.
From figure 3, it is evident that total cost function decreases first and then increases. Consequently, the convex nature arises in the total cost function. This confirms the possibility of obtaining the optimum service rates.
As we hoped, the figure 6 shows the convexity in the total cost function. From Table 6, we concluded that the minimum expected cost Rs.84.9859 is achieved at µ = 1.5 for β v = 3, Rs.85.1305 is achieved at µ = 1.5, for β v = 6 and Rs.85.1919 is achieved at µ = 1.5, for β v = 9.
As we hoped, the figure 7 shows the convexity in the total cost function.       Table 3. The effect of µ on the cost function for various values of γ

Conclusion
In this paper, Economic analysis of a finite capacity queueing system with working breakdowns have been discussed by plotting graphical diagrams with the corresponding values as tables for     Table 5. The effect of µ on the cost function for various values of α λ = 0.7, γ = 2, β = 1, γ v = 5, α = 0.2, ν = 1, µ v = 0.8, and c 1 = 10, c 2 = 80, c 3 = 75, c 4 = 60, c 5 = 40, c 6 = 7, c 7 = 2, c 8 = 15  Table 6. The effect of µ on the cost function for various values of β v the various parameters involved in the model.Our model can be seen as an expansion of the model [1].Additionally utilizing direct search method the ideal qualities for service rates with the objective of limiting the normal expense per unit time is inferred. It would be enthusiasm to  Table 7. The effect of µ on the cost function for various values of µ v build up this examination by including the idea of working breakdowns and retrial customers, in the model.