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Product forms for queueing networks with state-dependent multiple job transitions

Published online by Cambridge University Press:  01 July 2016

Richard J. Boucherie  and
Nico M. Van DIJK
Richard J. Boucherie*
Affiliation:
Free University, Amsterdam
Nico M. Van DIJK*
Affiliation:
Free University, Amsterdam
*
∗Postal address for both authors: Faculteit der Economische Wetenschappen en Econometrie, Vrije Universiteit, de Boelelaan 1105, 1081 HV Amsterdam, The Netherlands.
∗Postal address for both authors: Faculteit der Economische Wetenschappen en Econometrie, Vrije Universiteit, de Boelelaan 1105, 1081 HV Amsterdam, The Netherlands.
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Abstract

A general framework of continuous-time queueing networks is studied with simultaneous state dependent service completions such as due to concurrent servicing or discrete-time slotting and with state dependent batch routings such as most typically modelling blocking. By using a key notion of group-local-balance, necessary and sufficient conditions are given for the stationary distribution to be of product form. These conditions and a constructive computation of the product form are based upon merely local solutions of the group-local-balance equations which can usually be solved explicitly for concrete networks. Moreover, a decomposition theorem is presented to separate service and routing conditions. General batch service and batch routing examples yielding a product form are hereby concluded. As illustrated by various examples, known results on both discrete- and continuous-time queueing networks are unified and extended.

Keywords

DISCRETE-TIME QUEUEING NETWORKBATCH SERVICEBATCH ROUTINGBLOCKINGGROUP-LOCAL-BALANCE
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 1 , March 1991 , pp. 152 - 187
Copyright
Copyright © Applied Probability Trust 1991 

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References

1. Baskett, F., Chandy, K. M., Muntz, R. R. and Palacios, F. G. (1975) Open, closed and mixed networks of queues with different classes of customers. J. Assoc. Comput. Mach. 22, 248–260.CrossRefGoogle Scholar
2. Boucherie, R. J. and Van Dijk, N. M. (1990) Spatial birth-death processes with multiple changes and applications to batch service networks and clustering processes. Adv. Appl. Prob. 22, 433–455.CrossRefGoogle Scholar
3. Burman, D. Y., Lehoczky, J. P. and Lim, Y. (1982) Insensitivity of blocking probabilities in a circuit switching network. J. Appl. Prob. 21, 850–859.CrossRefGoogle Scholar
4. Chandy, K. M., Howard, J. H. and Towsley, D. F. (1977) Product form and local balance in queueing networks. J. Assoc. Comput. Mach. 24, 250–263.CrossRefGoogle Scholar
5. Chandy, K. M. and Martin, A. J. (1983) A characterization of product-form queueing networks. J. Assoc. Comput. Mach. 30, 286–299.CrossRefGoogle Scholar
6. Daduna, H. and Schassberger, R. (1983) Networks of queues in discrete time. Z. Operat. Res. 27, 159–175.Google Scholar
7. Henderson, W. and Taylor, P. G. (1990) Product form in networks of queues with batch arrivals and batch services. QUESTA 6, 71–88.Google Scholar
8. Hordijk, A. and Van Dijk, N. M. (1983) Networks of queues Part I: Job-local-balance and the adjoint process. Part II: General routing and service characteristics. Lecture Notes in Control and Informational Sciences , Springer-Verlag, Berlin, 60, 158–205.Google Scholar
9. Jackson, J. R. (1957) Networks of waiting lines. Operat. Res. 5, 518–521.CrossRefGoogle Scholar
10. Jackson, J. R. (1963) Jobshop-like queueing systems. Management Sci. 10, 131–142.CrossRefGoogle Scholar
11. Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, Chichester.Google Scholar
12. Krzesinsky, A. E. (1987) Multiclass queueing networks with state-dependent routing. Performance Eval. 7, 125–143.CrossRefGoogle Scholar
13. Noetzel, A. S. (1979) A generalized queueing discipline for product form network solutions. J. Assoc. Comput. Mach. 26, 779–793.CrossRefGoogle Scholar
14. Pittel, B. (1979) Closed exponential networks of queues with saturation: the Jackson-type stationary distribution and its asymptotic analysis. Math. Operat. Res. 4, 357–378.CrossRefGoogle Scholar
15. Pujolle, G., Claude, J. P. and Seret, D. (1985) A discrete tandem queueing system with a product form solution. Proc. Internat . Seminar on Computer Networking and Performance Evaluation, North-Holland, Kyoto, 139–147.Google Scholar
16. Pujolle, G. (1988) Discrete-time queueing systems with a product form solution. MASI Research Report.Google Scholar
17. Schassberger, R. (1978) The insensitivity of stationary probabilities in networks of queues. Adv. Appl. Prob. 10, 906–912.CrossRefGoogle Scholar
18. Schassberger, R. (1979) A definition of discrete product form distributions. Z. Operat. Res. 23, 189–195.Google Scholar
19. Schassberger, R. and Daduna, H. (1983) A discrete-time technique for solving closed queueing models of computer systems. Technical Report, Technische Universität Berlin.CrossRefGoogle Scholar
20. Serfozo, R. F. (1989) Markovian network processes: congestion dependent routing and processing. QUESTA 5, 5–36.Google Scholar
21. Taylor, P. G., Henderson, W., Pearce, C. E. M. and Van Dijk, N. M. (1988) Closed queueing networks with batch services. Technical Report, University of Adelaide.Google Scholar
22. Van Dijk, N. M. (1988) Product forms for queueing networks with limited clusters. Research Report, Free University, Amsterdam, The Netherlands. Performance Eval. To appear.Google Scholar
23. Van Dijk, N. M. (1989) ‘Stop = recirculate' for exponential product form queueing networks with departure blocking.’ Research Report, Free University, Amsterdam, The Netherlands. Operat. Res. Lett. To appear.Google Scholar
24. Walrand, J. (1983) A discrete-time queueing network. J. Appl. Prob. 20, 903–909.CrossRefGoogle Scholar
25. Walrand, J. (1988) An Introduction to Queueing Networks. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
26. Whittle, P. (1986) Systems in Stochastic Equilibrium. Wiley, Chichester.Google Scholar