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 r1 *⩾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 -v1 and -v2, namely, 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-v1. 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(v1,v2 ). In particular, traffic at server 1 is then no longer protected from worse-behaved (heavier-tailed) traffic at server 2