Coupled Processors with Regularly Varying Service Times

Boxma, Onno (Eindhoven, The Netherlands)
boxma@win.tue.nl

(joint work with Sem Borst and Predrag Jelenkovic)

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 $r_1^*\gt 1$ 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 .