ON AN M/G/1 QUEUE WITH A RANDOM SET UP TIME, RANDOM BREAKDOWNS AND DELAYED DETERMINISTIC REPAIRS

We study a single server queue with Poisson arrivals in batches of variable size. The server provides one by one general service to customers with a set-up time of random length before starting the first service at the start of the system as well as after every idle period of the system. The set-up time has been assumed to be general. Further, the server is subject to random breakdowns. The repair time has been assumed to be deterministic with a further delay time before starting repairs. The delay time in starting repairs has been assumed to be general. We find steady state queue length of various states of the system in terms of probability generating functions. Steady state results of a few interesting special cases have been derived.


INTRODUCTION
In real life situations, most of the queuing systems are subject to interruptions and delays in service 6569 ON AN / /1 QUEUE WITH A RANDOM SET UP TIME due to many factors such as random breakdowns due to power failure or failure of the vital parts of the system, server vacations, delays in starting the service, etc. Such interruptions and delays have a definite effect on the system efficiency, the quality and cost of service and waiting time of customers. When all customers waiting for service in a system are served, the system becomes idle and waits for a new batch of customers to arrive. At the end of such idle periods, the system needs extra time termed as 'set-up time' or 'warming up time' before serving the first customer of the next busy period of the system. We refer the reader to Choudhury and Madan [2] who dealt with such a set-up time in their work. In this paper, we study a / /1 queueing system with a random set up time preceding the first service after each idle period.
This random set up time has been assumed to be general. In addition, we assume that the system is subject to random breakdowns from time to time and when a breakdown occurs the repairs on the system do not start immediately. We further assume that there is a delay time in starting the repairs. This delay time too has been assumed to be general. Such a delay time was assumed by Madan [7]. On completion of delay time, the system undergoes repairs immediately. waiting for service in a system are served, the system becomes idle and waits for a new batch of customers to arrive. At the end of such idle periods, the system needs extra time termed as 'setup time' or 'warming up time' before serving the first customer of the next busy period of the system. We refer the reader to Choudhury and Madan [2] who dealt with such a set-up time in their work. In this paper, we study a M(X)/G/1 queueing system with a random set up time, random breakdowns and delayed deterministic repairs.

THE MATHEMATICAL MODEL
• Customers (units) arrive ; and therefore, • It is assumed that the server is subject to random breakdowns. Let ∝ be the first order probability that the server may breakdown while providing service to a customer during the short interval of time( , + ].
• As soon as the server fails, the customer whose service is interrupted comes back to the head of the queue and would be taken up first for service immediately alter the server becomes operative.
• We assume that as soon as the server breaks down, its repairs do not start immediately. There is a delay in the start of its repairs.
• We assume that the repair time of the server is deterministic with constant repair time 'd'.
• All stochastic processes involved in the system are independent of each other.

DEFINITIONS AND NOTATIONS
• Let ( , ) be the probability that at time t there are n (0) customers in the queue excluding one customer in service with elapsed service time x. Accordingly, denotes the probability that at time t there are ≥0 customers in the queue excluding one customer in service irrespective of the value of x. • Let Q (t) be the steady state probability that there is no customer in the system and the server is idle.
• We assume that all stochastic processes involved in the system are independent of each other.

STEADY STATE EQUATIONS
Based on the underlying assumptions of the model, we obtain the following steady state equations Above equations must be solved subject to the following boundary conditions: +1 (0) =∝ , ≥ 0 (4.10)

STEADY STATE SOLUTION
Multiplying both sides of equation ( Which is the steady state probability that the server is proving service to customers Which is the steady sate probability that the system is under set-up time

CONCLUSION
The paper studies a new queueing model in which a single server providers general service to customers arriving at the system in a compound Poisson process. The system needs warming up time at the start of service of the first customer whenever the system starts first time or before serving the first customer after every idle period. Further, the system is subject to random breakdowns and there is a delay time before starting the repairs. We assume that the service time, the set-up time and the delay time all follow a general distribution and the repair time is deterministic. We obtain new, clear and explicit tractable steady state results of the system. The results of many interesting particular cases have been derived from the main results.