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How Multiserver Queues Scale with Growing Congestion-Dependent Demand

Department of Industrial Engineering and Operations Research, Columbia University, 304 S.W. Mudd Building, 500 West 120th Street, New York, New York 10027-6699
ward.whitt{at}columbia.edu

We investigate how performance scales in the standardM/M/nqueue in the presence of growing congestion-dependent customer demand. We scale the queue by letting the potential (congestion-free) arrival rate be proportional to the number of servers,n, and lettingnincrease. We let the actual arrival rate withnservers be of the form _n=f(_n)n, wherefis a strictly-decreasing continuous function and _nis a steady-state congestion measure. We consider several alternative congestion measures, such as the mean waiting time and the probability of delay. We show, under minor regularity conditions, that for each nthere is a unique equilibrium pair (*_n*_n) such that *_nis the steady-state congestion associated with arrival rate *n, *n. Moreover, we show that, asnincreases, the queue with the equilibrium arrival rate *nis brought into heavy traffic, but the three different heavy-traffic regimes for multiserver queues identified by Halfin and Whitt (1981) each can arise depending on the congestion measure used. In considerable generality, there is asymptotic service efficiency: the server utilization approaches one asnincreases. Under the assumption of growing congestion-dependent demand, the service efficiency can be achieved even if there is significant uncertainty about the potential demand, because the actual arrival rate adjusts to the congestion.

Subject classifications: Queues, multichannel: congestion-dependent demand; Queues, limit theorems: heavy traffic; Queues, Markovian: multiserver.
History: Received September 2001; revision received May 2002; accepted July 2002.

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