N Parallel Queueing Systems with Server Breakdown and Repair

摘要

A model that can be used to study the influence of machine breakdown and limitedrepair capacity on the performance of a system that has to provide service continuously is presented. A system consisting of N single server queueing stations, each serving its own stream of customers, is considered. The servers of the stations are subject to breakdown. Broken servers are repaired by a joint repair facility consisting of K parallel repairmen. Whenever K is less than N, the repair facility is causing interference between the N stations. The behavior of the queue length at a particular station (say station 1) is studied in the long run. By modeling station 1 as an M/M/1 queue in a Markovian environment, an exact matrix geometric solution for the marginal queue length distribution and for the distributions of the queue length at the beginning of an up (down) period of the server is obtained. Since the matrix geometric solution becomes rather time and memory consuming when N is large, approximating closed form expressions are also developed. By assuming the lengths of subsequent down periods to be independent, station 1 is approximated by an M/M/1 queue with independent interruptions. Stochastic decompositions are employed to obtain approximations for the queue length distributions. With this model several design issues can be investigated such as the number of repairmen that is needed to maintain a certain pool of machines, or the number of machines that can be assigned to a certain crew of repairmen. Several numerical examples illustrate how the queue at station 1 is afffected by breakdowns of server 1 and by the other stations due to congestion at the repair facility.