Batch arrival discrete time queue with gated vacation system

Abstract

A class of single server vacation queues, which have batch arrivals and single server, is considered in discrete time. Here the server goes on vacation of random length as soon as the system becomes empty. On return from vacation, if he finds any customers waiting in the queue, the server starts serving the customers one by one until the queue size is zero (the queue discipline is FIFO); otherwise he takes another vacation and so on. The vacation model understudy here is the Gated systems: In a gated system, as soon as the server returns from vacation it places a gate behind the last waiting customer. It then begins to serve only the customers who are within the gate, based on some rules of how many or how long it could serve. It is shown here that the interarrival, service, vacation and server operation time can be cast with markov based representation then this class of vacation models can then be studied as matrix-product problem which belongs to a class of matrix analytic family - thereby allowing us to use result from [2] to solve the resulting matrix product problem. Most importantly it is shown that using discrete time modelling approach to study some vacation model is more appropriate and makes the model much more algorithmically tractable.

Journal of the Nigerian Association of Mathematical Physics, Volume 15 (November, 2009), pp 415 - 424