Simulation of queueing network with time varying arrival rates

Queueing models are important in the analysis of many aspects of system or human behaviour. In such widely divergent fields as administrative processes, health services, traffic control, production control, computer communication networks, time sharing, and architecture, queueing analysis is indisposable for better understanding and ultimately synthesis of the system.However straight mathematical analysis is difficult, as the difference equations describing a queueing network are non-linear and of infinite dimensional. Due to this difficulty, time varying cases are seldomly included in the study of analytic solutions of queueing systems. In existing literature, the mean arrival rates are assumed independent of time. This assumption is of dubious validity as the arrival rate in actual operations varies with a twenty four hours period. While one can assume further that at every instant in time, the equilibrium condition for constant arrival rate is reached for the current arrival rate, it has never been verified that such is the case.An alternative approach is to reduce the queueing problem to finite dimensional and to simulate it on computer as is proposed in the present paper. The basic assumptions are (1) time-varying random arrival rate with Poisson statistics and (2) exponentially distributed service time. An algorithm and flow diagram for doing this is given. With the simulation techniques developed in this paper, we can study the dynamics of varying arrival rate. Simulation of a simple network shows that as the arrival rate approaches capacity, the divergence between simulated results and previous analytical results for the equilibrium condition become increasingly large. A conclusion is therefore that except for systems with very light loading, analytical results based on constant arrival rate are not accurate. Simulation is necessary if one is to obtain a true picture of the system.