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*i > 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 -\nu_1 and -\nu_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 - \nu_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 (\nu_1, \nu_2). In particular, traffic at server 1 is then no longer protected from worse behaving (heavier-tailed) traffic at server 2.