Transient Solution of an Batch arrival Queue with Two Types of Service, Multiple Vacation, Random breakdown and Restricted Admissibility

service. The server takes vacation each time the system becomes empty and the vacation period is assumed to be general. On returning from vacation if the server finds no customer waiting in the system, then the server again goes for vacation until he finds at least one customer in the system. The system may breakdown at random and repair time follows exponential distribution. In addition we assume restricted admissibility of arriving batches in which not all batches are allowed to join the system at all times. The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been obtained explicitly. Average queue lenth and average system size are also computed.


INTRODUCTION
Queueing systems with vacations have been developed for a wide range applications in production, communication systems, computer networks and etc. Vacation queues have been studied by numerous researchers including Doshi [7], Keilson and Servi [9], Chae et al. [4], Madan et al. [13], Madan and Anabosi [14], Madan and Abu-dayyeh [11] and Badamchi Zadeh and Shankar [2]. Borthakur and Choudhury [3] and Hur and Ahn [8] have studied vacation queues with batch arrivals. Queue with multiple vacations has been studied by Tian and Zhang [20], Srinivasan and Maragatha Sundari [17]. Choudhury et al. [6] have studied M/G/1 queue with two phases of service and Bernoulli vacation schedule under multiple vacation policy. Maraghi et al. [16] have studied batch arrival queueing system with random breakdowns and Bernoulli schedule server vacations having general vacation time.
A queueing system might suddenly break down and hence the server will not be able to continue providing service unless the system is repaired. Takine and Sengupta [18], Aissani and Artalejo [1], Vinck and Bruneel [21] have studied different queueing systems subject to random breakdowns. Kulkarni and Choi [10] and Wang et al. [22] have studied retrial queues with system breakdowns and repairs. Thangaraj and Vanitha [19] discussed the single server model with two stages of heterogeneous service with different service time distributions subject to random breakdowns and compulsory service vacations with arbitrary vacation periods. In some queueing systems with batch arrival there is a restriction such that not all batches are allowed to join the system at all time. This policy is named restricted admissibility. For the first time Madan and Abu-Dayyeh [12], Madan and Choudhury [15] and Choudhury and Madan [5] proposed an queueing system with restricted admissibilty of arriving batches and Bernoulli schedule server vacation. In this paper we consider a batch arrival queue with two types of service where breakdowns may occur at random, and once the system breaks down, it enters a repair process. A single server provides two types of service and each arriving customer has the option of choosing either type of service. If there are no customer waiting in the system then the server goes for vacation with random duration. On returning from vacation, if the server again finds no customer waiting in the system, then the server continues to go for vacation until he finds at least one customer in the system. The service time and the vacation time are generally distributed, while the breakdown and repair times are exponentially distributed. The customers arrive to the system in batches of variable size, but served one by one on a first come -first served basis. This paper is organized as follows. The mathematical description of our model is given in section 2. Equations governing the system are given in section 3. The time dependent solution have been obtained in section 4 and corresponding steady state results have been derived explicitly in section 5. Average queue lenth and average system size are computed in section 6. Conclusion are given in section 7.

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If there are no customers waiting in the system then the server goes for vacation with random duration. On returning from vacation, if the server again finds no customer waiting in the system, then the server continues to go for vacation until he finds at least one customer in the system. Here the server takes multiple vacation. • The server's vacation time follows a general (arbitrary) distribution with distribution function V(t) and density function v(t). Let given that the elapsed vacation time is x , so that The system may break down at random, and breakdowns are assumed to occur according to a Poisson stream with mean breakdown rate 0 >  . Further we assume that once the system breaks down, the customer whose service is interrupted comes back to the head of the queue. Once the system breaks down, it enters a repair process immediately. The repair times are exponentially distributed with mean repair rate There is a policy restricted admissibility of batches in which not all batches are allowed to join the system at all times. Let be the probability that an arriving batch will be allowed to join the system during the period of server's non-vacation period and vacation period respectively. • Various stochastic processes involved in the system are assumed to be independent of each other. We define  ) (t R n : Probability that at time t , the server is inactive due to system breakdown and the system is under repair, while there are n ( 0  n ) customers in the queue.

EQUATIONS GOVERNING THE SYSTEM
The model is then, governed by the following set of differential-difference equations: The above equations are to be solved subject to the following boundary conditions (12) we assume that initially there are no customers in the system and the server is idle. so the initial conditions are

PROBABILITY GENERATING FUNCTIONS OF THE QUEUE LENGTH:THE TIME-DEPENDENT SOLUTION
We define the probability generating functions, which are convergent inside the circle given by z  1 and define the Laplace transform of a function f(t) as We take the Laplace transform of equations (1) to (12) and using equation (13), we get 1 0 Now multiplying equations (19), (21), (23) and (25) by suitable powers of z, adding to equations (18), (20), (22) and (24) and summing over n from 1 to  and using the generating function defined in (14) to (16), we get Integrating equation (30) between 0 to x , we get (1) s z P is given by equation (37).

The steady state results
In this section, we shall derive the steady state probability distribution for our queueing model. To define the steady probabilities we suppress the argument t wherever it appears in the time-dependent analysis. This can be obtained by applying the well-known Tauberian property, In order to determine (1), P P (1), V(1), R(1) denote the steady state probabilities that the server is providing first type of service, second type of service, server on vacation and server under repair without regard to the number of customers in the queue.
denote the probability generating function of the queue size irrespective of the server state. Then adding equations (65) to (68), we obtain and D(z) is given in the equation (69).
hence, the utilization factor  of the system is given by  where primes and double primes in (73) denote first and second derivative at z = 1, respectively. Carrying out the derivative at z = 1 we have

THE AVERAGE QUEUE SIZE AND THE AVERAGE SYSTEM SIZE
where q L has been found by equation (73) and  is obtained from equation (72).

CONCLUSION
In this paper we have studied an