Em/Ek/S queuing systems with state-dependent arrivals

This paper presents a solution to a class of queuing system problems that are a generalization of the type Em/Ek/S. The queuing system has s servers. The service time for each customer has a gamma distribution with parameters k and μ, where the expected value of the service time is k/μ. Customer interarrival times have gamma distributions with parameters m and m/λn, where n is the number of customers in the system and the expected value of the interarrival time is m/λn. The methodology advances an effective approach to solving these problems which has received little attention in the literature—possibly because the direction of earlier research was largely established before recent advances in computer software and hardware enabled the approach used herein to be effectively pursued. A computer implementation of the methodology described in this paper allows one to quickly—in seconds or minutes of PC time—obtain exact results from a microcomputer to a wider range of queuing problems than has hitherto been possible.