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A correspondence between product-form batch-movement queueing networks and single-movement networks

Published online by Cambridge University Press:  14 July 2016

J. L. Coleman*
Affiliation:
University of Adelaide
W. Henderson*
Affiliation:
University of Adelaide
C. E. M. Pearce*
Affiliation:
University of Adelaide
P. G. Taylor*
Affiliation:
University of Adelaide
*
Postal address: Applied Mathematics Department, University of Adelaide, Adelaide 5005, South Australia.
Postal address: Applied Mathematics Department, University of Adelaide, Adelaide 5005, South Australia.
Postal address: Applied Mathematics Department, University of Adelaide, Adelaide 5005, South Australia.
Postal address: Applied Mathematics Department, University of Adelaide, Adelaide 5005, South Australia.

Abstract

A number of recent papers have exhibited classes of queueing networks, with batches of customers served and routed through the network, which have generalised product-form equilibrium distributions. In this paper we look at these from a new viewpoint. In particular we show that, under standard assumptions, for a network to possess an equilibrium distribution that factorises into a product form over the nodes of the network for all possible transition rates, it is necessary and sufficient that it be equivalent to a suitably-defined single-movement network. We consider also the form of the state space for such networks.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1997 

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References

[1] Agnew, J. (1972) Explorations in Number Theory. Brooks/Cole, Monterey.Google Scholar
[2] Anderson, W. J. (1991) Continuous-time Markov Chains: an Applications Oriented Approach. Springer, New York.CrossRefGoogle Scholar
[3] Baskett, F., Chandy, K., Muntz, R. and Palacios, J. (1975) Open closed and mixed networks of queues with different classes of customers. J. ACM 22, 248260.Google Scholar
[4] Boucherie, R. 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, 433455.Google Scholar
[5] Boucherie, R. and Van Dijk, N. M. (1991) Product forms for queueing networks with state-dependent multiple job transitions. Adv. Appl. Prob. 23, 152187.Google Scholar
[6] Buzen, J. P. (1973) Computational algorithms for closed queueing networks with exponential servers. Commun. ACM 16, 527531.Google Scholar
[7] Coleman, J., Henderson, W. and Taylor, P. G. (1996) Product form equilibrium distributions and a convolution algorithm for stochastic Petri nets. Perf. Eval. 26, 159180.Google Scholar
[8] Donatelli, S. and Sereno, M. (1992) On the product form solution for stochastic Petri nets. Proc. 13th Int. Conf. on Appl. and Theory of Petri Nets, Sheffield, UK. pp. 154172.Google Scholar
[9] Frosch, D. (1992) Product form closed synchronized systems of stochastic sequential processes: a convolution algorithm. Preprint. Universität Trier, Germany.Google Scholar
[10] Frosch, D. and Natarajan, K. (1992) Product form solutions for closed synchronized systems of stochastic sequential processes. Proc. 1992 Int. Computer Symp., Taichung, Taiwan. pp. 392402.Google Scholar
[11] Henderson, W., Lucic, D. and Taylor, P. G. (1989) A net level performance analysis of stochastic Petri nets. J. Austral. Math. Soc. B 31, 176187.CrossRefGoogle Scholar
[12] Henderson, W., Pearce, C. E. M., Taylor, P. G. and Van Dijk, N. M. (1990) Closed queueing networks with batch services. Queueing Systems 6, 5970.CrossRefGoogle Scholar
[13] Henderson, W. and Taylor, P. G. (1990) Product form in networks of queues with batch arrivals and batch services. Queueing Systems 6, 7188.Google Scholar
[14] Henderson, W. and Taylor, P. G. (1991) Embedded processes in stochastic Petri nets. IEEE Trans. Soft. Eng. 17, 108116.CrossRefGoogle Scholar
[15] Henderson, W. and Taylor, P. G. (1991) Some new results on queueing networks with batch movement. J. Appl. Prob. 28, 409421.Google Scholar
[16] Jackson, J. (1957) Networks of waiting lines. Operat. Res. 5, 518521.CrossRefGoogle Scholar
[17] Jackson, J. (1963) Jobshop-like queueing systems. Management Sci. 10, 131142.CrossRefGoogle Scholar
[18] Kelly, F. P. (1976) Networks of queues. Adv. Appl. Prob. 8, 416432.Google Scholar
[19] Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, London.Google Scholar
[20] Lazar, A. A. and Robertazzi, T. G. (1991) Markovian Petri net protocols with product form solution. Perf. Eval. 12, 6777.CrossRefGoogle Scholar
[21] Memmi, G. and Roucairol, G. (1980) Linear algebra in net theory. Lecture Notes in Comp. Sci. 84, 213223.Google Scholar
[22] Murata, T. (1989) Petri nets: properties, analysis and applications. Proc. IEEE 77, 541580.Google Scholar
[23] Pollett, P. K. (1987) Preserving partial balance in continuous-time Markov chains. Adv. Appl. Prob. 19, 431–153.Google Scholar
[24] Reiser, M. and Lavenberg, S. S. (1980) Mean-value analysis of closed multichain queueing networks. J. ACM 27, 313322.Google Scholar
[25] Serforzo, R. (1989) Markovian network processes: congestion-dependent routing and processing. Queueing Systems 5, 536.Google Scholar
[26] Walrand, J. (1983) A discrete-time queueing network. J. Appl. Prob. 20, 903909.Google Scholar
[27] Whittle, P. (1986) Systems in Stochastic Equilibrium. Wiley, London.Google Scholar