An exact solution for the problem of M/M/c/k non-preemptive priority queue using state equilibrium equations.

摘要

The calculation of the performance measures of finite capacity M/M/c/k non-preemptive priority queuing systems using state equilibrium equations is challenging especially when it involves more than two priority levels. The number of equilibrium equations increases polynominally with the number of priority levels, the number of servers, or the maximum number of customers allowed in the system. The complexity of equations used to calculate the performance measures of these systems has made the analysis of priority queuing systems very difficult.;The aim of this thesis is to introduce an algorithm to calculate the performance measures for M/M/c/k non-preemptive priority queuing systems. The model is based on deriving the state equilibrium equations by applying the Global Balance Principle and solving them to determine the state probabilities. We use two different solution approaches that rely on two special structures of the solution matrix to reduce the complexity of the calculations. The first approach defines the state transition matrix as sparse while the second approach divides the matrix into block sub-matrices. We then use the state probabilities to calculate the average numbers in queue and in the system for each priority level and for the whole system Finally, we apply Little's Law to calculate the average time in queue and in the system for each priority level and for the system for each priority level and for the whole system.;Tables are provided to show the performance measures of non-preemptive priority queuing systems with two and three priority levels under different utilization rates (rho) for different values of system capacity ( k) and number of servers (c) where the maximum capacity (k) considered is ten and the maximum number of servers ( c) is three.