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Analysis of a Single Server Queue with Interdependence of Arrival and Service Processes – A Semi-Markov Approach

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Information Technologies and Mathematical Modelling. Queueing Theory and Applications (ITMM 2020)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1391))

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Abstract

In many queuing systems, the inter-arrival time and service distributions are dependent. In this paper we analyze such a system where the dependence is through a semi-Markov process. For this we assume that the arrival and service processes evolve in a finite number of phases/stages according to a Markov chain. So, the product space of the two finite sets of states (phases) is considered. The nature of transitions in the states of the combined process are such that transition rates at which the states of the combined process changes depend on the phase in which each ‘marginal’ process is currently in and (the phases of) the state to be visited next. We derive the stability condition and the effect of the interdependence on the stability of the system is brought out. A numerical investigation of the steady state characteristics of the system is also carried out.

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References

  1. Chakravarthy, S. R.: The batch Markovian arrival process: a review and future work. In: Advances in Probability and Stochastic Processes, pp. 21–49 (2001)

    Google Scholar 

  2. Krishnamoorthy A., Joshua A.N.: A BMAP/BMSP/1 queue with Markov dependent arrival and Markov dependent service batches. J. Ind. Manage. Optim. 3(5) (2020). https://doi.org/10.3934/jimo.2020101

  3. Conolly, B.: The waiting time for a certain correlated queue. Oper. Res. 15, 1006–1015 (1968)

    Article  Google Scholar 

  4. Conolly, B., Hadidi, N.: A correlated queue. Appl. Prob. 6, 122–136 (1969)

    Article  MathSciNet  Google Scholar 

  5. Cidon, I., Guerin, R., Khamisy, A., Sidi, M.: On queues with inter-arrival times proportional to service times. Technical repurt. EE PUB, 811, Technion (1991)

    Google Scholar 

  6. Cidon, I., Guerin, R., Khamisy, A., Sidi, M.: Analysis of a correlated queue in communication systems. Technical report. EE PUB, 812, Technion (1991)

    Google Scholar 

  7. Conolly, B., Choo, Q.H.: The waiting process for a generalized correlated queue with exponential demand and service. SIAM J. Appl. Math. 37, 263–275 (1979)

    Article  MathSciNet  Google Scholar 

  8. Hadidi, N.: Queues with partial correlation. SIAM J. Appl. Math. 40, 467–475 (1981)

    Article  MathSciNet  Google Scholar 

  9. Hadidi, N.: Further results on queues with partial correlation. Oper. Res. 33, 203–209 (1985)

    Article  MathSciNet  Google Scholar 

  10. Srikanth, K.I., Manjunath, D.: Queues with dependency between interarrival and service times using mixtures of bivariates. Stoch. Models 22(1), 3–20 (2006)

    Article  MathSciNet  Google Scholar 

  11. Fendick, K.W., Saksena, V.R., Whitt, W.: Dependence in packet queues. IEEE Trans. Commun. 37(11), 1173–1183 (1989)

    Article  MathSciNet  Google Scholar 

  12. Combe, M.B., Boxma, O.J.: BMAP modeling of a correlated queue. In: Walrand, J., Bagchi, K., Zobrist, G.W. (eds.) Network Performance Modeling and Simulation, pp. 177–196. Gordon and Breach Science Publishers, Philadelphia (1999)

    Google Scholar 

  13. Boxma, O.J., Perry, D.: A queueing model with dependence between service and interarrival time. Eur. J. Oper. Res. 128(13), 611–624 (2001)

    Article  MathSciNet  Google Scholar 

  14. Adan, I.J.B.F., Kulkarni, V.G.: Single-server queue with Markov-dependent inter-arrival and service times. Queueing Syst. 45(2), 113–134 (2003)

    Article  MathSciNet  Google Scholar 

  15. Valsiou, M., Adan, I.J.B.F., Boxma, O.J.: A two-station queue with dependent preparation and service times. Eur. J. Oper. Res. 195(1), 104–116 (2009)

    Article  MathSciNet  Google Scholar 

  16. Badila, E.S., Boxma, O.J., Resing, J.A.C.: Queues and risk processes with dependencies. Stoch. Models. 30(3), 390–419 (2014)

    Article  MathSciNet  Google Scholar 

  17. Neuts, M.F.: Markov chains with applications to queueing theory, which have a matrix-geometric invariant probability vector. Adv. Appl. Prob. 10, 125–212 (1978)

    Article  MathSciNet  Google Scholar 

  18. Neuts, M.F.: Matrix Geometric Solutions for Stochastic Models. John Hopkins University Press, Baltimore (1981)

    MATH  Google Scholar 

  19. Neuts, M.F., Pagano, M.E.: Generating random variates from a distribution of phase type. In: Oren, T.I., Delfosse, C.M., Shub, C.M. (eds.) Winter Simulation Conference, pp. 381–387. IEEE (1981)

    Google Scholar 

  20. Sengupta, B.: The semi-Markovian queue: theory and applications. Stoch. Models 6(3), 383–413 (1990)

    Article  MathSciNet  Google Scholar 

  21. Sengupta, B.: Markov processes whose steady state distribution is matrix exponential with an application to the GI/PH/1 queue. Adv. Appl. Prob. 21, 159–180 (1989)

    Article  MathSciNet  Google Scholar 

  22. Lambert, J., van Houdt, B., Blondia, C.: Queue with correlated service and inter-arrival times and its application to optical buffers. Stoch. Models 22(2), 233–251 (2006)

    Article  MathSciNet  Google Scholar 

  23. van Houdt, B.: A matrix geometric representation for the queue length distribution of multitype semi-Markovian queues. Perform. Eval. 69(7–8), 299–314 (2012)

    Article  Google Scholar 

  24. Buchholz, P., Kriege, J.: Fitting correlated arrival and service times and related queueing performance. Queueing Syst. 85, 337–359 (2017)

    Article  MathSciNet  Google Scholar 

  25. Asmussen, S., Nerman, O., Olsson, M.: Fitting phase-type distributions via the EM algorithm. Scand. J. Stat. 23(4), 419–441 (1996)

    MATH  Google Scholar 

  26. Breuer, L.: An EM algorithm for batch Markovian arrival processes and its comparison to a simpler estimation procedure. Ann. Oper. Res. 112, 123–138 (2002). https://doi.org/10.1023/A:1020981005544

    Article  MathSciNet  MATH  Google Scholar 

  27. Horváth, G., Okamura, H.: A fast EM algorithm for fitting marked Markovian arrival processes with a new special structure. In: Balsamo, M.S., Knottenbelt, W.J., Marin, A. (eds.) EPEW 2013. LNCS, vol. 8168, pp. 119–133. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40725-3_10

    Chapter  Google Scholar 

  28. Ozawa, T.: Analysis of queues with Markovian service process. Stoch. Models. 20(4), 391–413 (2004)

    Article  MathSciNet  Google Scholar 

  29. Whitt, W., You, W.: Using Robust queueing to expose the impact of dependence in single-server queues. Oper. Res. 66(1), 184–199 (2017)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Government Engineering College, Thrissur, India

    K. R. Ranjith, B. Gopakumar & Sajeev S. Nair

  2. Centre for Research in Mathematics, CMS College, Kottayam, India

    Achyutha Krishnamoorthy

Authors
  1. K. R. Ranjith
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  2. Achyutha Krishnamoorthy
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  3. B. Gopakumar
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  4. Sajeev S. Nair
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Editor information

Editors and Affiliations

  1. Belarusian State University, Minsk, Belarus

    Alexander Dudin

  2. Tomsk State University, Tomsk, Russia

    Anatoly Nazarov

  3. Tomsk State University, Tomsk, Russia

    Alexander Moiseev

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Ranjith, K.R., Krishnamoorthy, A., Gopakumar, B., Nair, S.S. (2021). Analysis of a Single Server Queue with Interdependence of Arrival and Service Processes – A Semi-Markov Approach. In: Dudin, A., Nazarov, A., Moiseev, A. (eds) Information Technologies and Mathematical Modelling. Queueing Theory and Applications. ITMM 2020. Communications in Computer and Information Science, vol 1391. Springer, Cham. https://doi.org/10.1007/978-3-030-72247-0_31

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  • DOI: https://doi.org/10.1007/978-3-030-72247-0_31

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-72246-3

  • Online ISBN: 978-3-030-72247-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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