The Impact of Dependent Service Times on Large-Scale Service Systems

Abstract

This paper investigates the impact of dependence among successive service times on the transient and steady-state performance of a large-scale service system. This is done by studying an infinite-server queueing model with time-varying arrival rate, exploiting a recently established heavy-traffic limit, allowing dependence among the service times. This limit shows that the number of customers in the system at any time is approximately Gaussian, where the time-varying mean is unaffected by the dependence, but the time-varying variance is affected by the dependence. As a consequence, required staffing to meet customary quality-of-service targets in a large-scale service system with finitely many servers based on a normal approximation is primarily affected by dependence among the service times through this time-varying variance. This paper develops formulas and algorithms to quantify the impact of the dependence among the service times on that variance. The approximation applies directly to infinite-server models but also indirectly to associated finite-server models, exploiting approximations based on the peakedness (the ratio of the variance to the mean in the infinite-server model). Comparisons with simulations confirm that the approximations can be useful to assess the impact of the dependence.