A Light-Traffic Theorem for Multi-Server Queues

Several approximations for the expected delay in an M/G/c queue depend on both its light- and heavy-traffic behavior. Although the required heavy-traffic result has been proved, the light-traffic result has only been conjectured by Boxma, Cohen, and Huffels. We prove this conjecture when the service is of phase-type; intuitively, any M/G/c queue can be arbitrarily closely approximated by such a system. In particular, as the traffic goes to zero, we show that the probability of delay depends only on the mean of the service-time distributions and that the delay (when positive) converges in distribution to the minimum of c independent equilibrium-excess service-times. This result justifies an efficient computational approach to obtain numerical results for M/G/c queues and provides useful methodology for the approximation of other complicated stochastic systems.