Departures from a Queue with Many Busy Servers

Abstract

To analyze networks of queues, it is important to be able to analyze departure processes from single queues. For the M/M/s and M/G/∞ models, the stationary departure process is simple (Poisson), but in general the stationary departure process is quite complicated. As a basis for approximations, this paper shows that the stationary departure process is approximately Poisson when there are many busy slow servers in a large class of stationary G/GI/s congestion models having s servers, infinite waiting room, the first-come first-served discipline, and mutually independent and identically distributed service times that are independent of a stationary arrival process. Limit theorems are proved for the departure process in a G/GI/s system in which the number of servers and the offered load (arrival rate divided by the service rate) both increase. The asymptotic behavior of the departure process depends on the way the arrival rate changes. If the arrival rate is held fixed, so that the offered load increases by slowing down the service rate, then the departure process converges to a Poisson process. For this result, the service-time distribution is assumed to be phase-type. Other limiting behavior occurs if the arrival rate approaches zero or infinity. Convergence is established in each case by applying previous heavy-traffic limit theorems.