Time-dependent analysis of virtual waiting time behaviour in discrete time queues

Abstract

Discrete time queueing models have been shown previously to be of practical use for modelling the approximate time-dependent behaviour of queue length in systems of the form M( t)/ G/ c. In this paper we extend these models to include the time-dependent behaviour of virtual waiting time. Key components of this research are the derivation of exact (algorithmic) expressions for virtual waiting time behaviour. Statistical approximations for the distributional form of virtual waiting times are then developed and tested. These approximations reduce the computational effort involved in the evaluation of the exact expressions by a factor of over 1000 whilst still maintaining very high accuracy levels. The models have been implemented for use on a PC and are shown to be capable of modelling the time-dependent virtual waiting time behaviour of realistic queueing systems in just a few seconds. Results from the models are compared with results obtained by the Simple Stationary Approximation, the Pointwise Stationary Approximation and Modified Offered Load approximations. They clarify further some of the strengths and weaknesses of these approaches and demonstrate the potential of the Discrete Time Modelling approach.