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The Cesaro Limit of Departures from Certain ./GI/1 Queueing Tandems

Mountford, Tom; Prabhakar, Balaji


HPL-BRIMS-96-17

Keyword(s): cesaro limits; increasing hazardous services; couplings

Abstract: We consider an infinite tandem of independent identical ./GI/1 queues with mean service rate equal to 1 subjected to stationary and ergodic inputs of rate r less than 1. Of some interest in the study of such queueing tandems are the following three inter-related questions: (1) For each r less than 1, does there exist a rate r stationary and ergodic process which is an invariant distribution for the queue? (2) For a fixed r, is this invariant distribution unique? (3) When a stationary and ergodic arrival process of rate r less than 1 is input to the first queue, do the successive departure processes converge in distribution to the invariant distribution (assuming it exists)? For general non-exponential server queues, it is not yet known if invariant distributions exist. However for each r less than 1, should one exist, it is known to be unique. This note contributes to the third question when the service time distribution of each queue in the tandem has an {\em increasing hazard rate}. It is shown that when a stationary and ergodic arrival process of rate r less than 1 is passed through a tandem of such queues, the Cesaro averages of the successive departure processes converge weakly to a limit which is an invariant distribution for the queue.

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