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Analysis of the ${M}^{X}/G/1$ queueing system with vacation times. (English) Zbl 1192.60101
Summary: We consider an ${M}^{X}/G/1$ quening system, where batches of customers are assumed to arrive the system according to a compound Poisson process. As soon as the system becomes empty, the server takes a vacation for a random length of time called vacation time to do other jobs, which is uninterruptible. After returning from that vacation, there are two possibilities viz. (i) he keeps on taking vacations till he finds at least one unit in the queue (multiple vacations) or (ii) he may take only one vacation between two successive busy periods (single vacation). The steady state behaviour of this ${M}^{X}/G/1$ queueing system is derived by an analytic approach to study the queue size distribution at a stationary (random) as well as a departure point of time under multiple vacation policy. Also, attempts have been made to obtain the queue size distribution of a more generalized model at a departure point to cover both the cases multiple and single vacations.
##### MSC:
 60K25 Queueing theory 90B22 Queues and service (optimization)