Approximate solution for multi-server queueing systems with Erlangian service times

Multi-server queueing systems with Poisson arrivals and Erlangian service times are among the most applicable of what are considered ‘easy’ systems in queueing theory. By selecting the proper order, Erlangian service times can be used to approximate reasonably well many general types of service times which have a unimodal distribution and a coefficient of variation less than or equal to 1. In view of their practical importance, it may be surprising that the existing literature on these systems is quite sparse. The probable reason is that, while it is indeed possible to represent these systems through a Markov process, serious difficulties arise because of (1) the very large number of system states that may be present with increasing Erlang order and/or number of servers, and (2) the complex state transition probabilities that one has to consider. Using a standard numerical approach, solutions of the balance equations describing systems with even a modest Erlang order and number of servers require extensive computational effort and become impractical for larger systems. In this paper we illustrate these difficulties and present the equally likely combinations (ELC) heuristic which provides excellent approximations to typical equilibrium behavior measures of interest for a wide range of stationary multiserver systems with Poisson arrivals and Erlangian service. As system size grows, ELC computational times can be more than 1000 times faster than those for the exact approach. We also illustrate this heuristic's ability to estimate accurately system response under transient and/or dynamic conditions.