Role of Poisson Distribution in Queuing Systems with Application to Simulation Data

Abstract

Queuing systems are very popular in modeling service processes in sectors like health care, banking and telecommunications. Poisson distribution is the dominant distribution in the study of random arrival processes, but it needs both theoretical and empirical testing to be more practical. The purpose of the study is to compare the performance of Poisson distribution to describe the arrival processes in an M/M/1 queuing system both analytically and through simulation. A theoretical model involving M/M/1 model was coupled with a discrete-event Monte Carlo simulation, written in R. The exponential distributions based on Poisson assumptions generated arrival and service processes. The modeling included the time to add a schedule of events and the elimination of warm-up and repeated runs (n = 10,000). Measures of performance such as average waiting time (W), length of queue (Lq), and system utilization (rho) were calculated. The measures of model accuracy were 95% confidence interval, mean absolute error (MAE) and root mean square error (RMSE). Theoretical values were found to be in high agreement with simulated results in all traffic conditions. The confidence intervals were low and always contained theoretical estimates, which means high precision. There were low error measures, which testified to the model accuracy. There were nonlinear increases in system performance as traffic intensity neared unity with significant increases in waiting time and queue length. The M/M/1 model using Poisson gives a good reliable and statistically valid model of arrival processes, although the performance depends greatly on the utilization of the system.