On the heavy-traffic limit theorem for GI/G/∞ queues

Abstract

A revealing alternate proof is provided for the Iglehart (1965), (1973)–Borovkov (1967) heavy-traffic limit theorem for GI/G/s queues. This kind of heavy traffic is obtained by considering a sequence of GI/G/s systems with the numbers of servers and the arrival rates going to ∞ while the service-time distributions are held fixed. The theorem establishes convergence to a Gaussian process, which in general is not Markov, for an appropriate normalization of the sequence of stochastic processes representing the number of customers in service at arbitrary times. The key idea in the new proof is to consider service-time distributions that are randomly stopped sums of exponential phases, and then work with the discrete-time vector-valued Markov chain representing the number of customers in each phase of service at arrival epochs. It is then easy to show that this sequence of Markov chains converges to a multivariate O–U (Ornstein–Uhlenbeck) diffusion process by applying simple criteria in Stroock and Varadhan (1979). The Iglehart–Borovkov limit for these special service-time distributions is the sum of the components of this multivariate O–U process. Heavy-traffic convergence is also established for the steady-state distributions of GI/M/s queues under the same conditions by exploiting stochastic-order properties.