Hostname: page-component-6898cfcb55-th4rj Total loading time: 0 Render date: 2026-11-11T06:33:07.772Z Has data issue: false hasContentIssue false
  • English
  • Français

Spatial birth-death processes with multiple changes and applications to batch service networks and clustering processes

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, Postbus 7161, 1007 MC Amsterdam, The Netherlands.
Postal address for both authors: Faculteit der Economische Wetenschappen en Econometrie, Vrije Universiteit, Postbus 7161, 1007 MC Amsterdam, The Netherlands.
Get accessRights & Permissions [Opens in a new window]

Abstract

Reversible spatial birth-death processes are studied with simultaneous jumps of multi-components. A relationship is established between (i) a product-form solution, (ii) a partial symmetry condition on the jump rates and (iii) a solution of a deterministic concentration equation. Applications studied are reversible networks of queues with batch services and blocking and clustering processes such as those found in polymerization chemistry. As illustrated by examples, known results are hereby unified and extended. An expectation interpretation of the transition rates is included.

Keywords

QUEUEING NETWORKPRODUCT FORMCONCENTRATION EQUATIONSYMMETRY PROPERTY

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 2 , June 1990, pp. 433 - 455
Copyright
Copyright © Applied Probability Trust 1990 

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.)

Article purchase

Temporarily unavailable

References

[1] Bertsekas, D. P. and Gallager, R. G. (1987) Data Networks. Prentice-Hall, Englewood Cliffs, NJ. Google Scholar
[2] Chandy, K. M. and Martin, A. J. (1983) A characterization of product-form queueing networks. JACM 30, 286–299.Google Scholar
[3] Daduna, H. and Schassberger, R. (1983) Networks of queues in discrete time. Z. Operat. Res. 27, 159–175.Google Scholar
[4] Van Dongen, P. G. J. and Ernst, M. H. (1987) Fluctuations in coagulating systems. J. Statist. Phys. 49, 879–926.Google Scholar
[5] Van Dongen, P. G. J. and Ernst, M. H. (1984) Kinetics of reversible polymerization. J. Statist. Phys. 37, 301–324.CrossRefGoogle Scholar
[6] Ernst, M. H. (1984) Kinetic theory of clustering. In International Summer School on Fundamental Problems in Statistical Mechanics, VI Trondheim, Norway, June 1984, ed. Cohen, E. G. D.. North-Holland, Amsterdam.Google Scholar
[7] Foschini, G. J. and Gopinath, B. (1983) Sharing memory optimally. IEEE Trans. Comm. 31, 352–360.Google Scholar
[8] Gihman, I. I. and Skorohod, A. V. (1975) The Theory of Stochastic Processes II. Springer-Verlag, Berlin.Google Scholar
[9] 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. In Lecture Notes in Control and Informational Sciences 60, Springer-Verlag, Berlin, 158–205.Google Scholar
[10] Van Kampen, N. G. (1981) Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam.Google Scholar
[11] Kaufman, J. S. (1981) Blocking in a shared resource environment. IEEE Trans. Commun. 29, 1474–1481.Google Scholar
[12] Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
[13] Krzesinski, A. E. (1987) Multiclass queueing networks with state-dependent routing. Performance Evaluation 7, 125–143.Google Scholar
[14] Lam, S. S. (1977) Queueing networks with population size constraints. IBM J. Res. Develop. 21, 370–378.Google Scholar
[15] Lushnikov, A. A. (1978) Coagulation in finite systems. J. Coll. Interf. Sci. 65, 276–285.CrossRefGoogle Scholar
[16] 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.Google Scholar
[17] Pujolle, G. (1988) Discrete-time queueing systems with a product form solution. MASI Research Report. Google Scholar
[18] 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
[19] Serfozo, R. F. (1988) Markovian network processes with system-dependent transition rates. Research Report, Georgia Institute of Technology. Google Scholar
[20] Stockmayer, W. H. (1943) Theory of molecular size distributions and gel formation in branched chain polymers. J. Chem. Phys. 11, 45–55.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] Towsley, D. F. (1980) Queueing networks with state-dependent routing. JACM 27, 323–337.CrossRefGoogle Scholar
[23] Van Dijk, N. M. (1988) Product forms for random access schemes. Computer Networks and ISDN Systems. Google Scholar
[24] Walrand, J. (1983) A discrete-time queueing network. J. Appl. Prob. 20, 903–909.Google Scholar
[25] Whittle, P. (1986) Systems in Stochastic Equilibrium. Wiley, New York.Google Scholar
[26] Yao, D. D. and Buzacott, J. A. (1987) Modeling a class of flexible manufacturing systems with reversible routing. Operat. Res. 35, 87–93.Google Scholar