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Product Forms for Stochastic Interference Systems

Published online by Cambridge University Press:  27 July 2009

N.M. Van Dijk
Affiliation:
Department of Econometrics The Free University Amsterdam, The Netherlands
J.P. Veltkamp
Affiliation:
Twente University, The Netherlands

Abstract

Systems are studied that consist of interfering components which are alternatively active (busy) and passive (idle) for random periods. Such systems naturally arise from the performance evaluation of computer models. A concrete invariance condition is imposed on the interferences allowed. Under this condition, the steady-state vector of active components is shown to be of product form as well as to be robust to distributional forms of active and passive periods. The proof is simple and self-contained. A number of concrete and generic examples is provided. These include resource sharing mechanisms, hierarchical circuit switchings, parallel processing, priorities, and breakdowns.

Type
Articles
Copyright
Copyright © Cambridge University Press 1988

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References

Barbour, A. (1976). Networks of queues and the method of stages. Advances in Applied Probability 8: 584591.CrossRefGoogle Scholar
Baskett, F., Chandy, M., Muntz, R., & Palacios, J. (1975). Open, closed, and mixed networks of queues with different classes of customers. J.A.C.M. 22.CrossRefGoogle Scholar
Burman, D.Y., Lehoczky, J.P., & Lim, Y.Insensitivity of blocking probabilities in a circuit switching network. Journal of Applied Probability 21: 850859.CrossRefGoogle Scholar
Chandy, K.M., Howard, J.H., & Towsley, D.F. (1977). Product form and local balance in queuing networks. J.A.C.M. 24: 250263.Google Scholar
Chandy, K.M. & Martin, A.J. (1983). A characterization of product-form queuing networks. J.A.C.M. 30: 286299.Google Scholar
Foshini, G.J. & Gopinath, B. (1983). Sharing memory optimally. IEEE Transactions on Cornnlunication 31: 352359.Google Scholar
Hordijk, A. & Schassberger, R. (1982). Weak convergence of generalized semi-Markov processes. Stochastic Processes and their Applications 12: 271291.Google Scholar
Hordijk, A. & van Dijk, N.M. (1981). Networks of queues with blocking. In Kylstra, K.J. (ed.), Performance 81. North Holland, pp. 5165.Google Scholar
Hordijk, A. & van Dijk, N.M. (1982). Stationary probabilities for networks of queues. Applied Probability–Computer Sciences; The Interface (eds. Disney, L. and Ott, T.J.), Birkhäuser 2: 423451.Google Scholar
Hordijk, A. & 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, 60: 158205.Google Scholar
Hordijk, A. & van Dijk, N.M. (1983). Adjoint process, job-local-balance and insensitivity of stochastic networks. Bulletin of the 44th Session of the International Statistical Institute 50: 776788.Google Scholar
Jackson, J.R. (1963). Jobshop-like queuing systems. Management Science 10: 131142.CrossRefGoogle Scholar
Kamoun, P. & Kleinrock, L. (1980). Analysis of finite storage in a computer network node environment under general traffic conditions. IEEE Transactions on Communication 28: 9921003.Google Scholar
Kaufman, J. (1981). Blocking in a shared resource environment. IEEE Transactions on Communication 29: 14741481.CrossRefGoogle Scholar
Kelly, F.P. (1979). Reversibility and stochastic networks. Wiley.Google Scholar
Lam, S.S. (1976). “Store- and forward-buffer requirements in a packet switching network. IEEE Transactions Communication 24: 394403.Google Scholar
Lavenberg, S.S. & Reiser, M. (1980). Stationary-state probabilities at arrival instants for closed queuing networks with multiple types of customers. Journal of Applied Probability 17: 10481061.CrossRefGoogle Scholar
Mitra, D. & McKenna, J. (1986). Asymptotic expansions for closed Markovian networks with state-dependent service rates. J.A.C.M. 33: 568592.Google Scholar
Onvural, R.O. & Perros, H.G. (1986). On equivalencies of blocking mechanisms in queuing networks with blocking. Operations Research Letters 5: 293297.Google Scholar
Pinsky, E. & Yemini, Y. (1984). A statistical mechanics of some interconnection networks. Performance '84. North Holland.Google Scholar
Sevcik, K.C. & Mitrani, I. (1981). The distribution of queuing networks states at input and output instants. J.A.C.M. 28: 358371.Google Scholar
Swiderski, J. (1984). Unified analysis of local flow-control mechanisms in message-switched networks. IEEE Transactions on Communication 32: 12861293.Google Scholar
Tijms, H.C. (1986). Stochastic modelling analysis. Wiley.Google Scholar
van Dijk, N.M. & Tijms, H.C. (1986). Insensitivity in two-node blocking models with applications. Teletraffic Analysis and Computer Performance Evaluation, North Holland, 329340.Google Scholar
Walrand, J. (1987). Introduction to queuing networks. Prentice Hall (to appear).Google Scholar
Whitt, W. (1985). Blocking when service is required from several facilities simultaneously. AJ&J Technical Journal 64(8): 18071856.Google Scholar
Whittle, P. (1985). Partial balance and insensitivity. Journal of Applied Probability 22: 168176.Google Scholar
Yemini, Y. (1982). A–statistical mechanics of distributed resource-sharing mechanisms. Research report, Columbia University, New York.Google Scholar
Yemini, Y. & Kleinrock, L. (1980). Interfering queuing processes in packet broadcoast communications. IFIP, Tokyo.Google Scholar
Zahorjan, J., Sevcik, K.C., Eager, D.L., & Galler, B. (1982). Balanced job bound analysis of queuing networks. Communications of the Association for Computing Machine 25: 134141.CrossRefGoogle Scholar