Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T00:25:46.420Z Has data issue: false hasContentIssue false
  • English
  • Français

Preserving partial balance in continuous-time Markov chains

Published online by Cambridge University Press:  01 July 2016

P. K. Pollett
P. K. Pollett*
Affiliation:
The University of Adelaide
*
∗Present address: School of Mathematical and Physical Sciences, Murdoch University, Murdoch, WA 6150, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

Recently a number of authors have considered general procedures for coupling stochastic systems. If the individual components of a system, when considered in isolation, are found to possess the simplifying feature of either reversibility, quasireversibility or partial balance they can be coupled in such a way that the equilibrium analysis of the system is considerably simpler than one might expect in advance. In particular the system usually exhibits a product-form equilibrium distribution and this is often insensitive to the precise specification of the individual components. It is true, however, that certain kinds of components lose their simplifying feature if the specification of the coupling procedure changes. From a practical point of view it is important, therefore, to determine if, and then under what conditions, the revelant feature is preserved.

In this paper we obtain conditions under which partial balance in a component is preserved and these often amount to the requirement that there exists a quantity which is unaffected by the internal workings of the component in question. We give particular attention to the components of a stratified clustering process as these most often suffer from loss of partial balance.

Keywords

REVERSIBILITYQUASIREVERSIBILITYWEAK COUPLINGINSENSITIVITYVARIED ARRIVAL RATESCLUSTERING PROCESSPRODUCT FORM
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 2 , June 1987 , pp. 431 - 453
Copyright
Copyright © Applied Probability Trust 1987 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barbour, A. D. (1976) Networks of queues and the method of stages. Adv. Appl. Prob. 8, 585–591.CrossRefGoogle Scholar
Burman, D. Y., Lehoczky, J. P. and Lim, Y. (1984) Insensitivity of blocking probabilities in a circuit-switching network. J. Appl. Prob. 21, 850–859.CrossRefGoogle Scholar
Jansen, U. and König, D. (1980) Insensitivity and steady-state probabilities in product form for queueing networks. Elektron. Informationsverarb. u. Kybernetik 16, 385–397.Google Scholar
Kelly, F. P. (1976) Networks of queues. Adv. Appl. Prob. 8, 416–432.CrossRefGoogle Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Kelly, F. P. (1981) Networks of quasi-reversible nodes. In Applied Probability-Computer Science, the Interface: Proceedings of the ORSA-TIMS Boca Raton Symposium , ed. Disney, R., Birkhauser Boston, Cambridge, Ma.Google Scholar
Kelly, F. P. (1983) Invariant measures and the q-matrix. In Probability, Statistics and Analysis , ed. Kingman, J. F. C. and Reuter, G. E. H., London Mathematical Society Lecture Notes Series No. 79, Cambridge University Press, 143–160.CrossRefGoogle Scholar
Kelly, F. P. (1986) Blocking probabilities in large circuit-switched networks. Adv. Appl. Prob. 18, 473–505.CrossRefGoogle Scholar
Kelly, F. P. and Pollett, P. K. (1983) Sojourn times in closed queueing networks. Adv. Appl. Prob. 15, 638–656.CrossRefGoogle Scholar
König, D., Matthes, K. and Nawrotzki, K. (1967) Verallgemeinerungen der Erlangschen und Engsetschen Formeln (Eine Methode in der Bedienungstheorie). Akademie-Verlag, Berlin.Google Scholar
Mcquarrie, D. A. (1967) Stochastic approach to chemical kinetics. J. Appl. Prob. 4, 413–478.CrossRefGoogle Scholar
Muntz, R. R. (1972) Poisson departure processes and queueing networks, IBM Research Report RC4145, IBM Thomas J. Watson Research Centre, Yorktown Heights, New York. (A shortened version of this paper appeared in Proc. 7th Ann. Conf. Information Science and Systems, Princeton (1973), 436-440.) Google Scholar
Pollett, P. K. (1985) Altering the q-matrix: the problem of varied arrival rates. Proc. 7th Nat. Conf. Austral. Soc. Operat. Res. , 206–234.Google Scholar
Pollett, P. K. (1986) Connecting reversible Markov processes. Adv. Appl. Prob. 18, 880–900.CrossRefGoogle Scholar
Schassberger, R. (1977) Insensitivity of steady-state distributions of generalized semi-Markov processes. Part I. Ann. Prob. 5, 87–99.CrossRefGoogle Scholar
Schassberger, R. (1978) The insensitivity of stationary probabilities in networks of queues. Adv. Appl. Prob. 10, 906–912.CrossRefGoogle Scholar
Walrand, J. and Varaiya, P. (1980) Interconnections of Markov chains and quasi-reversible queueing networks, Stoch. Proc. Appl. 10, 209–219.CrossRefGoogle Scholar
Whittle, P. (1965a). Statistical processes of aggregation and polymerisation. Proc. Camb. Phil. Soc. 61, 475–495.CrossRefGoogle Scholar
Whittle, P. (1965b) The equilibrium statistics of a clustering process in the uncondensed phase. Proc. R. Soc. London A285, 501–519.Google Scholar
Whittle, P. (1967) Nonlinear migration processes, Bull. Internat. Inst. Statist. 42, 642–647.Google Scholar
Whittle, P. (1972) Statistical and critical points of polymerisation processes. Proc. Symp. Statistical and Probabilistic Problems in Metallurgy, Suppl. Adv. Appl. Prob. , 199–215.CrossRefGoogle Scholar
Whittle, P. (1980a) Polymerisation processes with intrapolymer bonding I. One type of unit. Adv. Appl. Prob. 12, 94–115.CrossRefGoogle Scholar
Whittle, P. (1980b) Polymerisation processes with intrapolymer bonding II. Stratified processes. Adv. Appl. Prob. 12, 116–134.CrossRefGoogle Scholar
Whittle, P. (1980c) Polymerisation processes with intrapolymer bonding III. Several types of unit. Adv. Appl. Prob. 12, 135–153.CrossRefGoogle Scholar
Whittle, P. (1981) A direct derivation of the equilibrium distribution for a polymerisation process. Theory Prob. Appl. 26, 344–355.CrossRefGoogle Scholar
Whittle, P. (1983) Relaxed Markov processes. Adv. Appl. Prob. 15, 769–782.CrossRefGoogle Scholar
Whittle, P. (1984) Weak coupling in stochastic systems. Proc. R. Soc. London A395, 141–151.Google Scholar
Whittle, P. (1985a) Partial balance and insensitivity. J. Appl. Prob. 22, 168–176.CrossRefGoogle Scholar
Whittle, P. (1985b) Scheduling and characterization problems for stochastic networks. J.R. Statist. Soc. B 47, 407–415.Google Scholar
Whittle, P. (1986) Partial balance, insensitivity and weak coupling. Adv. Appl. Prob. 18, 706–723.CrossRefGoogle Scholar