QBD approximations of a call center queueing model with general patience distribution

Abstract

This paper investigates a computationally practical way for analyzing a call center queueing model, i.e., a finite-capacity, multi-server queueing model, where each server goes on a single vacation. Poisson arrival process and exponential service and vacation times are assumed. We also assume that each customer may leave the queue due to impatience. Customers’ patience times are i.i.d. random variables with a general distribution. Level-dependent finite QBD (quasi-birth–death) processes are employed to approximate such a queueing model. Two approaches are considered. The first one uses the phase-type (PH) distribution to approximate the general patience distribution, whereas the second one is based on the idea of replacing the eventual reneging of customers with balking. We find that the first approach is almost impossible to compute numerically due to the exponential growth of the size of the block matrices in a level-dependent finite QBD. We examine the validity and applicability of the approximation based on the second approach and show that it gives us a practical way to obtain performance measures of call center systems in practical scale with sufficiently reasonable accuracy.