Abstract
In this paper we consider a single server finite buffer queue wherein arrivals are governed by Markovian arrival process (MAP) and the service times are arbitrarily distributed. Using the supplementary variable method, the relations among the limiting queue size distributions at post-departure, arbitrary and pre-arrival epochs are obtained. Furthermore, similar results have also been derived for the infinite buffer queue. These relations are consistent with the known results and the method of derivation is simple and elegant.
Similar content being viewed by others
References
Blondia C “Finite capacity vacation models with non-renewal input”. Journal of Applied Porbability 28 (1991) 174–197.
Choi B D Hwang G U and Han D H “Supplementary variable method applied to the MAP/G/1 queueing system”. Journal of Australian Mathematical Society Series B 40, (1998), 86–96.
Gupta U C and Srinivasa Rao T S S “On the analysis server finite queue with state dependent arrival and service processes: M(n)/G/(n)/1/K”. OR Spektrum 20, (1998), 83–89.
Lucantoni D M, Meier-Hellstern K S and Neuts M F “A single-server queue with server vacations and a class of non-renewal process”. Advances in Applied Probability 22, (1990), 676–705.
Lucantoni D M “New results on the single server queue with a batch Markovian arrival process” Commum Statist-Stochastic Models 7, (1991), 1–46.
Neuts M F “Matrix-geometric solutions in stochastic models”. Johns Hopkins University Press, Baltimore (1981).
Neuts M F “Structured stochastic matrices of M/G/1 type and their applications” Marcel Dekker, New York (1989).
Takagi H “Queueing analysis-a foundation of performance evaluation, vol 2. Finite systems” North Holland, Amsterdam (1993).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gupta, U.C., Laxmi, P.V. The Relations among the queue Size Distributions at Departure, Arbitrary and Pre-arrival Epochs in the MAP/G/1 Queue With Finite/infinite Buffer — an Alternative Approach. OPSEARCH 38, 520–530 (2001). https://doi.org/10.1007/BF03398655
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03398655