On the equivalence of the (E_h/M/c) and (M/M/c) finite source queues

Abstract

This paper demonstrates the similarity, in most respects, of the (EJM/c) and (M/M/c) finite source queues and, in particular, the equivalence of the steady state balance equations and the steady state probabilities of these queueing models. The approach, an alternative to existing approaches and less mathematically oriented than others, relies on a state definition that is one-dimensional for the out-of-system times that follow an Erlang distribution. For the (Eh/M/c) model an automatic procedure is outlined for establishing the transfer rates, corresponding to customer out-of-system movements, between states. The balance equations can be handled analytically for small h and source size values but a numerical solution for larger h and source size values may be preferred. There is little change in the calculations between single server and multiserver cases. With equivalent steady state probabilities for the (Eh/M/c) and M/M/c models, equivalent system characteristics, namely average waiting times, customer numbers, output and production can be demonstrated for the two finite source models.

Keywords:

• finite source queueing theory
• Erlang distributions