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Abstract

The central approach used in this chapter, just as in Section 2.6, is to examine a stationary Markov chain together with its time-reversal: another Markov chain defined through reversing the time index of the original chain. Applying this technique to the Jackson network in Section 2.6, we not only recovered the product-form distributions derived earlier in Chapter 2 but also established the Poisson property of the exit processes from the network. In particular, we showed that in equilibrium the Jackson network preserves the Poisson property of the input processes to the network, such that the exit processes are also Poisson. This “Poisson-in-Poisson-out” property is the main motivation here for defining and studying a class of so-called quasi-reversible queues. This class is an extension of the basic M/M(n)/1 queues, which constitute the nodes in a Jackson network, to allow multiclass jobs and more general arrival and service disciplines, while still preserving the Poisson-in-Poisson-out property. It turns out that a general multiclass network, known as a Kelly network, which connects a set of quasi-reversible queues through some very general routing schemes, still enjoys the basic properties of the Jackson network, namely, the product-form equilibrium distribution and the Poisson-in-Poisson-out property.

Keywords

Invariant Distribution Stationary Markov Chain Jackson Network Independent Poisson Process Ergodic Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Barbour, A.D. 1976. Networks of Queues and the Methods of Stages. Adv. Appl. Prob., 8, 584–591.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Baskett, F., Chandy, M., Muntz, R., And Palacios, J. 1975. Open, Closed, and Mixed Networks of Queues with Different Classes of Customers. J. Assoc. Comput. Mach., 22, 248–260.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    BRÉMaud, P. 1989. Characteristics of Queueing Systems Observed at Events and the Connection between Stochastic Intensity and Palm Probability. Queueing Systems: Theory and Applications, 5, 99–112.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    BRÉMaud, P., Kannurpatti, R., And Mazumdar, R. 1992. Event and Time Averages: A Review and Some Generalizations. Adv. Appl. Prob., 24, 377–411.zbMATHCrossRefGoogle Scholar
  5. [5]
    Burke, P.J. 1956. The Output of a Queueing System. Operations Res., 4, 699–704.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Burke, P.J. 1957. The Output of a Stationary M/M/s Queueing System. Ann. Math. Stat., 39, 1144–1152.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Chao, X., Miyazawa, M., And Pinedo, M., Queueing Networks: Customers, Signals and Product Form Solutions, Wiley, New York, 1999.zbMATHGoogle Scholar
  8. [8]
    Kelly, F.P. 1975. Networks of Queues with Customers of Different Types. J. Appl. Prob., 12, 542–554.zbMATHCrossRefGoogle Scholar
  9. [9]
    Kelly, F.P. 1976. Networks of Queues. Adv. Appl. Prob., 8, 416–432.zbMATHCrossRefGoogle Scholar
  10. [10]
    Kelly, F.P. 1979. Reversibility and Stochastic Networks. Wiley, New York.zbMATHGoogle Scholar
  11. [11]
    Kelly, F.P. 1982. Networks of Quasi-Reversible Nodes. In Applied Probability—Computer Science, the Interface, Proc. of the Orsa/Tims Boca Raton Symposium, R. Disney and T. Ott (eds.), Birkhäuser, Boston, MA.Google Scholar
  12. [12]
    Lavenberg, S.S. And Reiser, M. 1980. Stationary State Probabilities at Arrival Instants for Closed Queueing Networks with Multiple Types of Customers. J. Appl. Prob., 17, 1048–1061.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Melamed, B. And Whitt, W. 1990. On Arrivals That See Time Averages: A Martingale Approach. J. Appl. Prob., 27, 376–384.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Melamed, B. And Whitt, W. 1990. On Arrivals That See Time Averages. Operations Res., 38, 156–172.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Melamed, B. And Yao, D.D. 1995. The Asta Property. In: Advances in Queueing, J.H. Dshalalow (ed.), Crc Press, Boca Raton, FL, 195–224.Google Scholar
  16. [16]
    Neuts, M.F. 1981. Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore, MD.Google Scholar
  17. [17]
    Serfozo, R.F. 1989. Poisson Functionals of Markov Processes and Queueing Networks. Adv. Appl. Prob., 21, 595–611.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Serfozo, R., Introduction to Stochastic Networks, Springer-Verlag, New York, 1999.zbMATHCrossRefGoogle Scholar
  19. [19]
    Sevcik, K.C. And Mitrani, I. 1981. The Distribution of Queueing Network States at Input and Output Instants. J. Assoc. Comput. Mach., 28, 358–371.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Walrand, J. 1988. An Introduction to Queueing Networks. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
  21. [21]
    Whittle, P. 1968. Equilibrium Distributions for an Open Migration Process. J. Appl. Prob., 5, 567–571.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Whittle, P. 1985. Partial Balance and Insensitivity. J. Appl. Prob., 22, 168–176.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Whittle, P. 1986. Systems in Stochastic Equilibrium. Wiley, New York.zbMATHGoogle Scholar
  24. [24]
    Wolff, R.W. 1982. Poisson Arrivals See Time Averages. Operations Res., 30, 223–231.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Hong Chen
    • 1
  • David D. Yao
    • 2
  1. 1.Faculty of Commerce and Business AdministrationUniversity of British ColumbiaVancouverCanada
  2. 2.Department of Operations Research and Industrial EngineeringColumbia UniversityNew YorkUSA

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