Matrix analytic solution to a quorum queueing system

A quorum queueing system is considered here. It is assumed that the server activation is triggered only if the queue size is larger than or equal to r. Furthermore, after a service completion, if the queue size is less than r the server is assumed to wait for the queue to be filled. Once the queue reaches r, a random sample of customers, drawn from the sample space {1,2,h.,r}, is taken for service. A Markov formulation of the system observed at service completions is presented. Then, the matrix analytic method developed by Neuts is used to solve for the steady state probabilities of the number of customers in the system. The performance of the system is then studied under various operational conditions, some illustrative examples are provided, and conclusions are presented.