A discrete single server queue with Markovian arrivals and phase type group services. (1995)

• Record Type: Journal Article
• Authors: Alfa, Attahiru Sule; Dolhun, K. Laurie; Chakravarthy, S.
• Is Part Of: Journal of applied mathematics and stochastic analysis. Volume 8:Number 2(1995)
• Page Start: 151
• Page End: 176
• Publication Date: 1995
• Subjects: Markovian arrival process -- phase type distributions -- matrix geometric method -- group service; Mathematical models -- Periodicals; Computer simulation -- Periodicals; Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Computer simulation; Mathematical models; Applied Mathematics; Periodicals; Electronic journals; 519.22
• DOI: 10.1155/S1048953395000153
• Languages: English
• ISSNs: 1048-9533

Abstract

Abstract : We consider a single-server discrete queueing system in which arrivals occur according to a Markovian arrival process. Service is provided in groups of size no more than M customers. The service times are assumed to follow a discrete phase type distribution, whose representation may depend on the group size. Under a probabilistic service rule, which depends on the number of customers waiting in the queue, this system is studied as a Markov process. This type of queueing system is encountered in the operations of an automatic storage retrieval system. The steady-state probability vector is shown to be of (modified) matrix-geometric type. Efficient algorithmic procedures for the computation of the rate matrix, steady-state probability vector, and some important system performance measures are developed. The steady-state waiting time distribution is derived explicitly. Some numerical examples are presented.