The Error in Steady-State Approximations for the Time-Dependent Waiting Time Distribution

Abstract

Arrival processes to queues often exhibit time dependence, but time-dependent queues are notoriously difficult to analyze. Usually one adopts some kind of steady-state approximation for time-dependent performance. We consider a range of queueing models that see wide use in practice, particularly in the modeling of service industry applications, and develop approximations for the error involved in using a steady-state approximation for time-dependent quantities. We focus on the distribution of the waiting time before reaching service, although the approach we use could be extended to many other performance measures. The results reinforce and extend what is known about such errors. For example, customer abandonment leads to dramatic reductions in the error in steady-state approximations as compared to a system without customer abandonment.

Keywords:

• Call center
• Nonhomogeneous Poisson process
• Poisson's equation

Mathematics Subject Classification:

• 90B22
• 60K25
• 60K30

ACKNOWLEDGMENT

We would like to thank Ger Koole for a suggestion that led to the generalization to τ ≥ 0 from τ = 0, and the referees and editors for constructive criticism that improved the paper. This research was partially supported by NSF Grant Number DMI-0400287.