Abstract
We analyze a Markovian smart polling model, which is a special case of the smart polling models studied in the work of Boon et al. (Queueing Syst 66:239–274, 2010), as well as a generalization of the gated M / M / 1 queue considered in Resing and Rietman (Stat Neerlandica 58:97–110, 2004). We first derive tractable expressions for the stationary distribution (when it exists) as well as the Laplace transforms of the transition functions of this polling model—while further assuming the system is empty at time zero—and we also present simple necessary and sufficient conditions for ergodicity of the smart polling model. Finally, we conclude the paper by briefly explaining how these techniques can be used to study other interesting variants of this smart polling model.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abate J, Whitt W (1988) Transient behavior of the \(M/M/1\) queue via Laplace transforms. Adv Appl Probab 20:145–178
Abate J, Whitt W (1995) Numerical inversion of Laplace transforms of probability distributions. ORSA J Comput 7:36–43
Adan IJBF (1991) A compensation approach for queueing problems. Ph.D. Thesis, Eindhoven University of Technology
Adan IJBF (2012) Queueing models with multiple waiting lines: direct methods. Slides from talk given at the EURANDOM YEQT-VI workshop, 2 November 2012. https://www.eurandom.tue.nl/events/workshops/2012/YEQT2012/Presentations/Ivo_Adan.pdf. Accessed 20 May 2018
Boon MAA, van Wijk ACC, Adan IJBF, Boxma OJ (2010) A polling model with smart customers. Queueing Syst 66:239–274
Boxma OJ (1994) Polling systems. In: From Universal Morphisms to Megabytes: a Baayen Space Odyssey. Liber Amicorum for P.C. Baayen, CWI, Amsterdam, pp. 215–230 (Liber Amicorum )
Buckingham P, Fralix B (2015) Some new insights into Kolmogorov’s criterion, with applications to hysteretic queues. Markov Process Relat Fields 21:339–368
den Iseger P (2006) Numerical transform inversion using Gaussian quadrature. Probab Eng Inf Sci 20:1–44
Evans GA, Chung KC (2000) Laplace transform inversions using optimal contours in the complex plane. Int J Comput Math 73:531–543
Fralix B (2018) http://bfralix.people.clemson.edu/preprints.htm. Accessed 20 May 2018
Fralix B (2015) When are two Markov chains similar? Stat Probab Lett 107:199–203
He Q-M (2014) Fundamentals of matrix-analytic methods. Springer, New York
Joyner J (2016) A new look at matrix-analytic methods. Ph.D. Dissertation, Clemson University
Joyner J, Fralix B (2016) A new look at Markov processes of \(G/M/1\)-type. Stoch Models 32:253–274
Joyner J, Fralix B (2018) A new look at block-structured Markov processes. Currently under revision: a draft of this preprint can be downloaded from http://bfralix.people.clemson.edu/preprints.htm. Accessed 20 May 2018
Joyner J, Fralix B (2018) On the stationary distribution of the \(M/M/1\) queue in a Markovian environment. In preparation
Latouche G, Ramaswami V (1999) Introduction to matrix-analytic methods in stochastic modeling. ASA-SIAM Publications, Philadelphia
Liu X, Fralix B (2017) New applications of lattice-path counting to Markovian queues. Submitted for publication
Resing JAC (1993) Polling systems and multitype branching processes. Queueing Syst 13:409–426
Resing JAC, Rietman R (2004) The \(M/M/1\) queue with gated random order of service. Stat Neerlandica 58:97–110
Rietman R, Resing JAC (2004) The \(M/G/1\) queue with gated random order of service. Queueing Syst 48:89–102
Selen J, Fralix B (2017) Time-dependent analysis of an \(M/M/c\) preemptive priority system with two priority classes. Queueing Syst 87:379–415
Acknowledgements
The transition rate diagram of a special case of the smart polling system studied in this paper—more specifically, the case where \(N = 4\)—is briefly discussed in the slides of Ivo Adan (2012) from a talk given during the YEQT-VI workshop at EURANDOM on November 2, 2012. The author first learned about the results of Resing and Rietman (2004), as well as the above-mentioned model in Adan (2012) from Johan van Leeuwaarden, during a research visit (funded by a STAR travel grant) to EURANDOM in May 2013. The author would like to thank both EURANDOM for providing a stimulating work environment during this visit, as well as STAR for making this research visit possible. The author also gratefully acknowledges the support of the National Science Foundation, via grant NSF-CMMI-1435261. Finally, the author would like to thank the (anonymous) Associate Editior, as well as two anonymous referees for providing many thoughtful comments and suggestions that helped to improve both the content and the style of this article.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fralix, B. A new look at a smart polling model. Math Meth Oper Res 88, 339–367 (2018). https://doi.org/10.1007/s00186-018-0638-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00186-018-0638-0