Coupled Processors with Regularly Varying Service Times

Consider two M=G=1 queues that are coupled in the following way. Whenever both queues are non-empty, each server serves its own queue at unit speed. However, if server 2 has no work in its own queue, then it assists server 1, resulting in an increased service speed r 1 1 in the first queue. This kind of coupling is related to generalized processor sharing. We assume that the service request distributions at both queues are regularly varying at infinity of index 1 and 2, viz., they are heavy-tailed. Under this assumption, we present a detailed analysis of the tail behaviour of the workload distribution at each queue. If the guaranteed unit speed of server 1 is already sufficient to handle its offered traffic, then the workload distribution at the first queue is shown to be regularly varying at infinity of index 1 1. But if it is not sufficient, then the workload distribution at the first queue is shown to be regularly varying at infinity of index 1 min( 1; 2). In particular, traffic at server 1 is then no longer protected from worse behaving (heavier-tailed) traffic at server 2. Keywords— Coupled processors, Generalized Processor Sharing, workload, tail behaviour, regular variation.