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ON THE TIME-DEPENDENT BEHAVIOR OF A MARKOVIAN REENTRANT-LINE MODEL

Published online by Cambridge University Press:  03 August 2018

Brian Fralix
Brian Fralix*
Affiliation:
Department of Mathematical Sciences Clemson University, Clemson, SC, USA E-mail: bfralix@clemson.edu
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Abstract

We use the random-product technique from [5] to study both the steady-state and time-dependent behavior of a Markovian reentrant-line model, which is a generalization of the preemptive reentrant-line model studied in the work of Adan and Weiss [2]. Our results/observations yield additional insight into why the stationary distribution of the reentrant-line model from [2] exhibits an almost-geometric product-form structure: indeed, our generalized reentrant-line model, when stable, admits a stationary distribution with a similar product-form representation as well. Not only that, the Laplace transforms of the transition functions of our reentrant-line model also have a product-form structure if it is further assumed that both Buffers 2 and 3 are empty at time zero.

Keywords

random-product representationreentrant-line modeltime-dependent behavior
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 33 , Issue 3 , July 2019 , pp. 367 - 386
Copyright
Copyright © Cambridge University Press 2018 

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References

1.Abate, J. & Whitt, W. (1995). Numerical inversion of Laplace transforms of probability distributions. ORSA Journal of Computing 7: 36–43.Google Scholar
2.Adan, I. & Weiss, G. (2006). Analysis of a simple Markovian re-entrant line with infinite supply of work under the LBFS policy. Queueing Systems 54: 169–183.Google Scholar
3.Adan, I., van Leeuwaarden, J. & Selen, J. (2017). Analysis of Structured Markov Chains. Draft available at https://arxiv.org/abs/1709.09060.Google Scholar
4.Brown, J.W. & Churchill, R.V. (2009). Complex variables and applications, 8th ed. New York: McGraw-Hill Companies.Google Scholar
5.Buckingham, P. & Fralix, B. (2015). Some new insights into Kolmogorov's criterion, with applications to hysteretic queues. Markov Processes and Related Fields 21: 339–368.Google Scholar
6.Doroudi, S., Fralix, B. & Harchol-Balter, M. (2016). Clearing Analysis on Phases: exact limiting probabilities for skip-free, unidirectional, quasi-birth-death processees. Stochastic Systems 6: 420–458.Google Scholar
7.Fralix, B. (2015). When are two Markov chains similar? Statistics and Probability Letters 107: 199–203.Google Scholar
8.Fralix, B. (2018). A new look at a smart polling model. Mathematical Methods of Operations Research, to appear.Google Scholar
9.Gelenbe, E. (1991). Product-form queueing networks with negative and positive customers. Journal of Applied Probability 28: 656–663.Google Scholar
10.Joyner, J. & Fralix, B. (2015). A new look at Markov processes of G/M/1-type. Stochastic Models 32: 253–274.Google Scholar
11.Joyner, J. & Fralix, B. (2017). A new look at block-structured Markov processes. Under revision: a draft of this paper can be accessed at http://bfralix.people.clemson.edu/preprints.htm.Google Scholar
12.Latouche, G. & Ramaswami, V. (1999). Introduction to matrix-analytic methods in stochastic modeling. Philadelphia: ASA-SIAM Publications.Google Scholar
13.Liu, X. & Fralix, B. (2017). New applications of lattice-path counting to Markovian queues. Submitted for publication: a draft of this work can be downloaded from http://bfralix.people.clemson.edu/preprints.htm.Google Scholar
14.Selen, J. & Fralix, B. (2017). Time-dependent analysis of a multi-server priority system. Queueing Systems 87: 379–415.Google Scholar