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On a batch arrival Poisson queue with generalized vacation. (English) Zbl 0916.60079
Summary: A single server Markovian queue with random batch arrival and generalized vacation is considered where the server goes on vacation of random length as soon as the system becomes empty. On return, if he finds customers waiting in the queue, the server starts serving them one by one till the queue size becomes zero again; otherwise he takes another vacation. 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. The departure point of time queue size distribution decomposes into three independent random variables one of which is the queue size of the standard $\text{M}^X/\text{M}/1$ queue. 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.

60K25Queueing theory
60G50Sums of independent random variables; random walks