A discrete single server queue with Markovian arrivals and phase type group services
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Date
1995-1-1
Authors
Alfa, Attahiru Sule
Dolhun, K. Laurie
Chakravarthy, S.
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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.
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Attahiru Sule Alfa, K. Laurie Dolhun, and S. Chakravarthy, “A discrete single server queue with Markovian arrivals and phase type group services,” Journal of Applied Mathematics and Stochastic Analysis, vol. 8, no. 2, pp. 151-176, 1995. doi:10.1155/S1048953395000153