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Queueing Systems with Structured Markov Chains

  • László Lakatos
  • László Szeidl
  • Miklós Telek
Chapter

Abstract

In the previous chapters we studied queueing systems with different interarrival and service time distributions. Chapter 7 is devoted to the analysis of queueing systems with exponential interarrival and service time distributions. The number of customers in these queueing systems is characterized by CTMCs with a generally nonhomogeneous birth-and-death structure. In contrast, Chap. 8 is devoted to the analysis of queueing systems with nonexponential interarrival and service time distributions. It turns out that far more complex analysis approaches are required for the analysis of queues with nonexponential interarrival and service time distributions. In this chapter we introduce queueing systems whose interarrival and service time distributions are nonexponential, but they can be analyzed with CTMCs. Indeed in this chapter we demonstrate the use of the results of Chap. 5 for the analysis of queueing systems with phase-type (PH) distributed interarrival and service times or with arrival and service processes that are MAPs. The main message of this chapter is that in queueing models the presence of PH or MAP processes instead of exponential distributions results in a generalization of the underlying CTMCs from birth-and-death processes to quasi-birth-and-death (QBDs) processes.

References

  1. 1.
    802.11. IEEE standard for information technology-telecommunications and information exchange between systems-local and metropolitan area networks-specific requirements - part 11: Wireless LAN medium access control (mac) and physical layer (phy) specifications. http://ieeexplore.ieee.org/servlet/opac?punumber=4248376, 2007.
  2. 2.
    N. Abramson. The aloha system: another alternative for computer communications. In: Proceedings Fall Joint Computer Conference. AFIPS Press, 1970.Google Scholar
  3. 3.
    D. Aldous, L. Shepp. The least variable phase type distribution is Erlang. Stoch. Models, 3:467–473, 1987.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    T. Apostol. Calculus I. Wiley, New York, 1967.MATHGoogle Scholar
  5. 5.
    T. Apostol. Calculus II. Wiley, New York, 1969.MATHGoogle Scholar
  6. 6.
    J. R. Artalejo, A. Gómez-Corral. Retrial Queueing Systems: A Computational Approach. Springer, Berlin Heidelberg New York, 2008.MATHCrossRefGoogle Scholar
  7. 7.
    S. Asmussen. Applied Probability and Queues. Springer, Berlin Heidelberg New York, 2003.MATHGoogle Scholar
  8. 8.
    F. Baccelli, P. Brémaud. Elements of Queueing Theory, Applications of Mathematics. Springer, Berlin Heidelberg New York, 2002.Google Scholar
  9. 9.
    F. Baskett, K. Mani Chandy, R. R. Muntz, F. G. Palacios. Open, closed and mixed networks of queues with different classes of customers. J. ACM, 22:248–260, 1975.MATHCrossRefGoogle Scholar
  10. 10.
    S. N. Bernstein. Theory of Probabilities. Moskva, Leningrad, 1946. (in Russian).Google Scholar
  11. 11.
    G. Bianchi. Performance analysis of the IEEE 802.11 distributed coordination function. IEEE J. Select. Areas Commun., 18:535–547, 2000.Google Scholar
  12. 12.
    D. Bini, G. Latouche, B. Meini. Numerical methods for structured Markov chains. Oxford University Press, Oxford, 2005.MATHCrossRefGoogle Scholar
  13. 13.
    A. Bobbio, M. Telek. A benchmark for PH estimation algorithms: results for Acyclic-PH. Stoch. Models, 10:661–677, 1994.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    A. A. Borovkov. Stochastic processes in queueing theory. Applications of Mathematics. Springer, Berlin Heidelberg New York, 1976.MATHCrossRefGoogle Scholar
  15. 15.
    A. A. Borovkov. Asymptotic Methods in Queueing Theory. Wiley, New York, 1984.MATHGoogle Scholar
  16. 16.
    L. Breuer, D. Baum. An Introduction to Queueing Theory and Matrix-Analytic Methods. Springer, Berlin Heidelberg New York, 2005.MATHGoogle Scholar
  17. 17.
    P. J. Burke. The output of a queuing system. Oper. Res., 4:699–704, 1956.MathSciNetCrossRefGoogle Scholar
  18. 18.
    J. Buzen. Computational algorithms for closed queueing networks with exponential servers. Commun. ACM, 16:527–531, 1973.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    V. Ceric, L. Lakatos. Measurement and analysis of input data for queueing system models used in system design. Syst. Anal. Modell. Simul., 11:227–233, 1993.MATHGoogle Scholar
  20. 20.
    Hong Chen, David D. Yao. Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization. Springer, Berlin Heidelberg New York, 2001.MATHGoogle Scholar
  21. 21.
    Y. Chow, H. Teicher. Probability Theory. Springer, Berlin Heidelberg New York, 1978.MATHCrossRefGoogle Scholar
  22. 22.
    K. Chung. Markov chains with stationary transition probabilities. Springer, Berlin Heidelberg New York, 1960.MATHCrossRefGoogle Scholar
  23. 23.
    E. Cinlar. Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ, 1975.MATHGoogle Scholar
  24. 24.
    D. R. Cox. The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables. Proc. Cambridge Philos. Soc., 51:433–440, 1955.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    A. Cumani. On the canonical representation of homogeneous Markov processes modelling failure-time distributions. Microelectron. Reliab., 22:583–602, 1982.CrossRefGoogle Scholar
  26. 26.
    D.J. Daley, D. Vere-Jones. An Introduction to the Theory of Point Process. Springer, Berlin Heidelberg New York, 2008. 2nd edn.Google Scholar
  27. 27.
    Gy. Dallos, Cs. Szabó. Random access methods of communication channels. Akadémiai Kiadó, Budapest, 1984 (in Hungarian).Google Scholar
  28. 28.
    M. De Prycker. Asynchronous Transfer Mode, Solutions for Broadband ISDN. Prentice Hall, Englewood Cliffs, NJ, 1993.Google Scholar
  29. 29.
    P. Erdős, W. Feller, H. Pollard. A theorem on power series. Bull. Am. Math. Soc., 55:201–203, 1949.CrossRefGoogle Scholar
  30. 30.
    G. I. Falin, J. G. C. Templeton. Retrial queues. Chapman and Hall, London, 1997.MATHGoogle Scholar
  31. 31.
    W. Feller. An Introduction to Probability Theory and its Applications, vol. I. Wiley, New York, 1968.MATHGoogle Scholar
  32. 32.
    Chuan Heng Foh, M. Zukerman. Performance analysis of the IEEE 802.11 MAC protocol. In Proceedings of European wireless conference, Florence, February 2002.Google Scholar
  33. 33.
    F. G. Foster. On the stochastic matrices associated with certain queuing processes. Ann. Math. Stat., 24:355–360, 1953.MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    G. Giambene. Queuing Theory and Telecommunications: Networks and Applications. Springer, Berlin Heidelberg New York, 2005.Google Scholar
  35. 35.
    I.I. Gihman, A.V. Skorohod. The Theory of Stochastic Processes, vol. I. Springer, Berlin Heidelberg New York, 1974.MATHCrossRefGoogle Scholar
  36. 36.
    I. I. Gihman, A. V. Skorohod. The Theory of Stochastic Processes, vol. II. Springer, Berlin Heidelberg New York, 1975.MATHCrossRefGoogle Scholar
  37. 37.
    B. Gnedenko, E. Danielyan, B. Dimitrov, G. Klimov, V. Matveev. Priority Queues. Moscow State University, Moscow, 1973 (in Russian).Google Scholar
  38. 38.
    B. V. Gnedenko. Theory of Probability. Gordon and Breach, Amsterdam, 1997. 6th edn.Google Scholar
  39. 39.
    B. V. Gnedenko, I. N. Kovalenko. Introduction to Queueing Theory, 2nd edn. Birkhauser, Boston 1989.CrossRefGoogle Scholar
  40. 40.
    W. J. Gordon, G. F. Newell. Closed queueing systems with exponential servers. Oper. Res., 15:254–265, 1967.MATHCrossRefGoogle Scholar
  41. 41.
    D. Gross, J. F. Shortle, J. M. Thompson, C. M. Harris. Fundamentals of Queueing Theory, 4th edn. Wiley, New York, 2008.Google Scholar
  42. 42.
    W. Henderson. Alternative approaches to the analysis of the M/G/1 and G/M/1 queues. J. Oper. Res. Soc. Jpn., 15:92–101, 1972.MATHGoogle Scholar
  43. 43.
    A. Horváth, M. Telek. PhFit: A general purpose phase type fitting tool. In Tools 2002, pages 82–91, London, April 2002. Lecture Notes in Computer Science, vol. 2324. Springer, Berlin Heidelberg New York.Google Scholar
  44. 44.
    J. R. Jackson. Jobshop-like queueing systems. Manage. Sci., 10:131–142, 1963.CrossRefGoogle Scholar
  45. 45.
    N. K. Jaiswal. Priority Queues. Academic, New York, 1968.MATHGoogle Scholar
  46. 46.
    N. L. Johnson, S. Kotz. Distributions in Statistics: Continuous Multivariate Distributions. Applied Probability and Statistics. Wiley, New York, 1972.MATHGoogle Scholar
  47. 47.
    V.V. Kalashnikov. Mathematical Methods in Queueing Theory. Kluwer, Dordrecht, 1994.Google Scholar
  48. 48.
    S. Karlin, H. M. Taylor. A First Course in Stochastic Processes. Academic, New York, 1975.MATHGoogle Scholar
  49. 49.
    S. Karlin, H. M. Taylor. A Second Course in Stochastic Processes. Academic, New York, 1981.MATHGoogle Scholar
  50. 50.
    J. Kaufman. Blocking in a shared resource environment. IEEE Trans. Commun., 29: 1474–1481, 1981.CrossRefGoogle Scholar
  51. 51.
    F. P. Kelly. Reversibility and Stochastic Networks. Wiley, New York, 1979.MATHGoogle Scholar
  52. 52.
    D. G. Kendall. Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded Markov chain. Ann. Math. Stat., 24:338–354, 1953.MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    A. Khinchin. Mathematisches über die Erwartung vor einem öffentlichen Schalter. Rec. Math., 39:72–84, 1932 (in Russian with German summary).Google Scholar
  54. 54.
    J. F. C. Kingman. Poisson Processes. Clarendon, Oxford, 1993.MATHGoogle Scholar
  55. 55.
    L. Kleinrock. Queuing Systems. Volume 1: Theory. Wiley-Interscience, New York, 1975.Google Scholar
  56. 56.
    G. P. Klimov. Extremal Problems in Queueing Theory. Energia, Moskva, 1964 (in Russian).Google Scholar
  57. 57.
    E. V. Koba. On a retrial queueing system with a FIFO queueing discipline. Theory Stoch. Proc., 8:201–207, 2002.MathSciNetGoogle Scholar
  58. 58.
    V. G. Kulkarni. Modeling and Analysis of Stochastic Systems. Chapman & Hall, London, 1995.MATHGoogle Scholar
  59. 59.
    L. Lakatos. On a simple continuous cyclic waiting problem. Annal. Univ. Sci. Budapest Sect. Comp., 14:105–113, 1994.MathSciNetMATHGoogle Scholar
  60. 60.
    L. Lakatos. A note on the Pollaczek-Khinchin formula. Annal. Univ. Sci. Budapest Sect. Comp., 29:83–91, 2008.MathSciNetMATHGoogle Scholar
  61. 61.
    L. Lakatos. Cyclic waiting systems. Cybern. Syst. Anal., 46:477–484, 2010.MathSciNetCrossRefGoogle Scholar
  62. 62.
    G. Latouche, V. Ramaswami. Introduction to matrix analytic methods in stochastic modeling. SIAM, 1999.Google Scholar
  63. 63.
    A. Lewandowski. Statistical tables. http://www.alewand.de. Nov. 13., 2012.
  64. 64.
    D. V. Lindley. The theory of queues with a single server. Math. Proc. Cambridge Philos. Soc., 48:277–289, 1952.MathSciNetCrossRefGoogle Scholar
  65. 65.
    T. Lindwall. Lectures on the Coupling Method. Wiley, New York, 1992.Google Scholar
  66. 66.
    J. D. C. Little. A proof of the queuing formula: L =AW. Oper. Res., 9:383–387, 1961.MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    A. A. Markov. Rasprostranenie zakona bol’shih chisel na velichiny, zavisyaschie drug ot druga. Izvestiya Fiziko-matematicheskogo obschestva pri Kazanskom universitete, 15:135–156, 1906 (in Russian).Google Scholar
  68. 68.
    L. Massoulie, J. Roberts. Bandwidth sharing: Objectives and Algorithms. In Infocom, 1999.Google Scholar
  69. 69.
    V. F. Matveev, V. G. Ushakov. Queueing systems. Moscow State University, Moskva, 1984 (in Russian).MATHGoogle Scholar
  70. 70.
    P. Medgyessy, L. Takács. Probability Theory. Tankönyvkiadó, Budapest, 1973 (in Hungarian).Google Scholar
  71. 71.
    S. Meyn, R. Tweedie. Markov chains and stochastic stability. Springer, Berlin Heidelberg New York, 1993.MATHCrossRefGoogle Scholar
  72. 72.
    NIST: National Institute of Standards and Technology. Digital library of mathematical functions. http://dlmf.nist.gov. Nov. 13., 2012.
  73. 73.
    M. Neuts. Probability distributions of phase type. In Liber Amicorum Prof. Emeritus H. Florin, pp. 173–206. University of Louvain, Louvain, Belgium, 1975.Google Scholar
  74. 74.
    M.F. Neuts. Matrix Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore, 1981.MATHGoogle Scholar
  75. 75.
    C. Palm. Methods of judging the annoyance caused by congestion. Telegrafstyrelsen, 4:189–208, 1953.Google Scholar
  76. 76.
    A. P. Prudnikov, Y. A. Brychkov, O. I. Marichev. Integrals and series, vol. 2. Gordon and Breach, New York, 1986. Special functions.Google Scholar
  77. 77.
    S. Rácz, M. Telek, G. Fodor. Call level performance analysis of 3rd generation mobile core network. In IEEE International Conference on Communications, ICC 2001, 2:456–461, Helsinki, Finland, June 2001.Google Scholar
  78. 78.
    S. Rácz, M. Telek, G. Fodor. Link capacity sharing between guaranteed- and best effort services on an atm transmission link under GoS constraints. Telecommun. Syst., 17(1–2):93–114, 2001.MATHCrossRefGoogle Scholar
  79. 79.
    M. Reiser, S. S. Lavenberg. Mean value analysis of closed multi-chain queueing networks. J. ACM, 27:313–322, 1980.MathSciNetMATHCrossRefGoogle Scholar
  80. 80.
    J. Roberts. A service system with heterogeneous user requirements - application to multi-service telecommunications systems. In Proceedings of Performance of Data Communications Systems and Their Applications, pp. 423–431, Paris, 1981.Google Scholar
  81. 81.
    K. W. Ross. Multiservice Loss Models for Broadband Telecommunication Networks. Springer, Berlin Heidelberg New York, 1995.MATHCrossRefGoogle Scholar
  82. 82.
    T. Saaty. Elements of Queueing Theory. McGraw-Hill, New York, 1961.MATHGoogle Scholar
  83. 83.
    R. Serfozo. Introduction to Stochastic Networks. Springer, Berlin Heidelberg New York, 1999.MATHCrossRefGoogle Scholar
  84. 84.
    A. N. Shiryaev. Probability. Springer, Berlin Heidelberg New York, 1994.Google Scholar
  85. 85.
    D. L. Snyder. Random Point Processes. Wiley, New York, 1975.MATHGoogle Scholar
  86. 86.
    L. Szeidl. Estimation of the moment of the regeneration period in a closed central-server queueing network. Theory Probab. Appl., 31:309–313, 1986.Google Scholar
  87. 87.
    L. Szeidl. On the estimation of moment of regenerative cycles in a general closed central-server queueing network. Lect. Notes Math., 1233:182–189, 1987.MathSciNetCrossRefGoogle Scholar
  88. 88.
    L. Takács. Investigation of waiting time problems by reduction to Markov processes. Acta Math. Acad. Sci. Hung., 6:101–129, 1955.MATHCrossRefGoogle Scholar
  89. 89.
    L. Takács. The distribution of the virtual waiting time for a single-server queue with Poisson input and general service times. Oper. Res., 11:261–264, 1963.MATHCrossRefGoogle Scholar
  90. 90.
    L. Takács. Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York, 1967.MATHGoogle Scholar
  91. 91.
    H. Takagi. Queueing Analysis. North Holland, Amsterdam, 1991.MATHGoogle Scholar
  92. 92.
    M. Telek. Minimal coefficient of variation of discrete phase type distributions. In G. Latouche, P. Taylor, eds., Advances in algorithmic methods for stochastic models, MAM3, pp. 391–400. Notable Publications, 2000.Google Scholar
  93. 93.
    A. Thümmler, P. Buchholz, M. Telek. A novel approach for fitting probability distributions to trace data with the em algorithm. IEEE Trans. Depend. Secure Comput., 3(3):245–258, 2006. Extended version of DSN 2005 paper.Google Scholar
  94. 94.
    H. Tijms. Stochastic Models: An Algorithmic Approach. Wiley, New York, 1994.MATHGoogle Scholar
  95. 95.
    W. Whitt. A review of l = λw and extensions. Queue. Syst., 9:235–268, 1991.MathSciNetMATHCrossRefGoogle Scholar
  96. 96.
    V. M. Zolotarev. Modern Theory of Summation of Random Variables. VSP, Utrecht, 1997.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • László Lakatos
    • 1
  • László Szeidl
    • 2
    • 3
  • Miklós Telek
    • 4
  1. 1.Eötvös Loránd UniversityBudapestHungary
  2. 2.Óbuda UniversityBudapestHungary
  3. 3.Széchenyi István UniversityGyőrHungary
  4. 4.Budapest University of Technology and EconomicsBudapestHungary

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