Skip to main content
Log in

Markovian network processes: Congestion-dependent routing and processing

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

A Markovian network process describes the movement of discrete units among a set of nodes that process the units. There is considerable knowledge of such networks, often called queueing networks, in which the nodes operate independently and the routes of the units are independent. The focus of this study, in contrast, is on networks with dependent nodes and routings. Examples of dependencies are parallel processing across several nodes, blocking of transitions because of capacity constraints on nodes, alternate routing of units to avoid congestion, and accelerating or decelerating the processing rate at a node depending on downstream congestion. We introduce a general network process representing the numbers of units at the nodes and derive its equilibrium distribution. This distribution takes the form of a product of functions of vectors in which the arguments of the functions satisfy an interchangeability property. This new type of distribution may apply to other multi-variate processes as well. A basic idea in our approach is a linking of certain micro-level balance properties of the network routing to the processing rates at the nodes. The link is via “ routing-balance partitions” of nodes that are inherent in any network. A byproduct of this approach is a general characterization of blocking of transitions without the restriction that the process is reversible, which had been a standard assumption. We also give necessary and sufficient conditions under which a unit moving in the network sees a time average for the unmoved units (called the MUSTA property). Finally, we discuss when certain flows between nodes in an open network are Poisson processes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. F. Baccelli and P. Brémaud,Palm Probabilities and Stationary Queues, Lecture Notes in Statistics #41 (Springer Verlag, 1987).

  2. E. Basket, K. Chandy, R. Muntz and P. Palacios, Open, closed and mixed networks with different classes of customers, J. ACM 22 (1975) 248–260.

    Google Scholar 

  3. P. Brémaud,Point Processes and Queues (Springer Verlag, 1981).

  4. P. Brémaud, Characteristics of queueing systems observed at events and the connection between stochastic intensity and Palm probability, Queueing systems (1989) to appear.

  5. J.A. Buzacott and D.D. Yao, On queueing network models of flexible manufacturing systems, Queueing Systems 1 (1986) 29–66.

    Google Scholar 

  6. K.E. Chin, Networks of queues with state-dependent flows, Ph.D. Dissertation, Georgia Institute of Technology, 1986.

  7. K.E. Chin and R.F. Serfozo, Networks of queues with blocking and load balancing, in:Proc. Material Handling Research Forum, 1989, to appear.

  8. Y. Dallery, A queueing network model of flexible manufacturing systems consisting of cells,Proc. 1986 IEEE Int. Conf. on Robotics and Automation.

  9. P. Franken, D. König, V. Arndt and V. Schmidt,Queues and Point Processes (Wiley, 1982).

  10. E. Gelenbe and G. Pujolle,Introduction to Queueing Networks (Wiley, 1987).

  11. W.J. Gordon and G.F. Newell (1967), Cyclic queueing systems with restricted queue lengths, Operations Research 15 (1967) 266–278.

    Google Scholar 

  12. A. Hordijk and N. Van Dijk, Adjoint processes, job local balance and insensitivity for stochastic networks, in:Proc. 44th Session of Int. Statistics Institute, 1983.

  13. A. Hordijk and N. Van Dijk, Networks of queues, part I: job-local-balance and the adjoint process; part II: general routing and service characteristics, in:Proc. Int. Seminar on Modelling and Performance Evaluation Methodology, INRIA Vol. I (1983) 79–135.

    Google Scholar 

  14. J.R. Jackson, Networks of waiting lines, Operations Research 5 (1957) 518–521.

    Google Scholar 

  15. O. Kallenberg,Random Measures 3rd ed. (Academic Press, 1983).

  16. F.P. Kelly,Reversibility and Stochastic Networks (Wiley, 1979).

  17. J.F.C. Kingman, Markov population processes, J. Appl. Prob. 6 (1969) 1–18.

    Google Scholar 

  18. A.E. Krzesinski, Multiclass queueing networks with state-dependent routing, Performance Evaluation 7 (1987) 125–143.

    Google Scholar 

  19. B. Melamed, On Poisson traffic processes in discrete-state Markovian systems with applications to queueing theory, Adv. Appl. Prob. 11 (1979) 218–239.

    Google Scholar 

  20. B. Melamed and W. Whitt, On arrivals that see time averages, Operations Research (1989) to appear.

  21. B. Melamed and W. Whitt, On arrivals that see time averages: a Martingale approach, J. Applied Probability (1989) to appear.

  22. P.K. Pollett, Preserving partial balance in continuous-time Markov chains, Adv. Applied Probability 19 (1987) 431–453.

    Google Scholar 

  23. R.F. Serfozo, Poisson functionals of Markov processes and queueing networks, Adv. Appl. Probability (1989) to appear.

  24. D.F. Towsley, Queueing networks with state-dependent routing, J. ACM 27 (1980) 323–337.

    Google Scholar 

  25. N.M. Van Dijk and I. Akyildiz, Networks with mixed processor sharing parallel queues and common pools, Technical Report, Georgia Institute of Technology, 1988.

  26. J. Walrand,An Introduction to Queueing Networks (Prentice Hall, 1988).

  27. J. Walrand and P. Varaiya, Flows in queueing networks: a Martingale approach, Math. Operations Research 6 (1981) 387–404.

    Google Scholar 

  28. P. Whittle, Equilibrium distributions for an open migration process, J. Appl. Prob. 5 (1968) 567–57.

    Google Scholar 

  29. P. Whittle,Systems in Stochastic Equilibrium (Wiley, 1986).

  30. R.W. Wolff, Poisson arrivals see time averages, Operations Research 30 (1982) 223–231.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

This research was sponsored in part by Air Force Office of Scientific Research contract 84-0367.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Serfozo, R.F. Markovian network processes: Congestion-dependent routing and processing. Queueing Syst 5, 5–36 (1989). https://doi.org/10.1007/BF01149184

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01149184

Keywords

Navigation