Markov Processes with Constant Transition Rates of Size One & Two
The two classical single server queueing systems, M/M/1 and M/M/1/N are generalized to allow constant transition rates of size two in addition to the standard constant transitions rates of size one. In terms of the queueing models, these new systems each allow customers to arrive or be served instantly in pairs as well as individually. The steady state distributions are explicitly determined and a condition for the existence of a steady state distribution is established in the infinite-state space case. Assuming that a steady state condition prevails, the canonical performance measures are determined. Expressions for the average number of customers in either system or queue are derived. Formulae for the average waiting time that a customer spends in each system or queue are also developed.
AMS Subject Classification60K25 60J27
KeywordsSingle server queue Batch queueing systems Markov processes with transition steps of size one and two Performance measures Bulk queue M/M/1 M/M/1/N Steady state distributions
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- Dela, M., 2011. A Sample Path Approach to Little’s Formula. Masters Thesis, Department of Mathematics and Statistics, California State Polytechnic University Pomona.Google Scholar
- El-Taha, M., Stidham Jr., S., 1999. Sample-Path Analysis of Queueing Systems (International Series in Operations Research & Management Science). Kluwer, Boston.Google Scholar
- Harbottle, L.M., Hunter, B., Krinik., A., 2011. The irreducible four-state Markov process. Integration, Mathematical Theory and Applications, Volume 2, Issue 3.Google Scholar
- Shun, K., 2009. Markov processes with transitions of size one and two. Masters Thesis, Department of Mathematics and Statistics, California State Polytechnic University Pomona.Google Scholar