An LR Method Approach to Transient Analysis of M/M/1 Jackson Type Queueing Network Systems

Abstract

The effective analysis of queueing networks under stochastic situations is very important for understanding congestion, waiting time and service performance of real-life systems. In this study, an approach for transient analysis of M/M/1 Jackson type network queueing system with fuzzy parameters is proposed based on LR method. The suggested model is composed of two service nodes and one Central Server with an arrival stream following the Poisson process and service time follows the exponential distribution. To represent uncertainty in arrival and service rates, triangular and trapezoidal fuzzy numbers are considered. The lower and upper bounds of the performance measures based on α-cut are transformed in Left–Right (LR) form by central values, left spreads and right spreads. The transient state probabilities are derived by solving Kolmogorov forward differential equations, numerically by Runge–Kutta method. The expected system length and mean waiting time key performance measures are calculated for Node-1, Node-2 and Central Server. From the results, it is noted that the LR method is one of the simple, compact and meaningful representation of the fuzzy queueing outcomes. The left and right spreads decrease progressively with the increase of the α-cut value, which means that the uncertainty is reduced and the values are moved towards crisp values. The Central Server exhibits more variability because it has both direct and routed arrivals. Thus, the LR method is considered to be an effective method to simplify fuzzy calculations and to understand the transient queueing system performance under fuzzy environment.