Sankhya: The Indian Journal of Statistics

1997, Volume 59, Series B, Pt. 3, 369-383

ON A BATCH ARRIVAL POISSON QUEUE WITH GENERALIZED VACATION

By

ARUN BORTHAKUR, Gauhati University, Guwahati

And

GAUTAM CHOUDHURY, L.C.B. College, Guwahati

SUMMARY. This paper deals with a single server Markovian queue with random batch arrival and generalized vacation. Here the server goes on vacation of random length as soon as the system becomes empty. On return from vacation, if he finds customer(s) waiting in the queue, the server starts serving the customers one by one till the queue size becomes zero again; otherwise he takes another vacation and so on. In this paper the steady state behaviour of this queue under fairly general condition has been studied and we obtain the queue size distributions at stationary point of time, departure point of time and vacation initiation point of time. One of our findings is that the departure point of time queue size distribution decomposes into three independent random variables one of which is the queue size of the standard $M^X/M/1$ queue. An interpretation of the other two random variables is also presented. Also we obtain a simple derivation for Laplace Stieltjes transform of the queue waiting time distribution, analytically explicit expressions for the system state probabilities and provide their appropriate interpretation. Finally we derive some system performance measures.

AMS (1991) subject classification. 60K25, 60J15

Key words and phrases. $M\spx$/$M$/1 queue, queue size, additional queue size, queue waiting time

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