Exact Solutions for Open, Closed and Mixed Queueing Networks with Rejection Blocking

Abstract Open, mixed and closed queueing networks with multiple job classes, reversible routing and rejection blocking are investigated in this paper. Jobs may change class membership and general service requirement distributions that depend on the job class are allowed. We prove that the equilibrium state probabilities have product form if at all stations either the scheduling discipline is symmetric or all service requirements at the station have the same exponential distribution. The solution implies insensitivity in this kind of blocking networks, ie. the distribution of the jobs in equilibrium, irrespective of their remaining service requirements, depends only on their mean service requirement.

[1]  Frank Kelly,et al.  Reversibility and Stochastic Networks , 1979 .

[2]  Donald F. Towsley,et al.  Product Form and Local Balance in Queueing Networks , 1977, JACM.

[3]  K. Mani Chandy,et al.  Open, Closed, and Mixed Networks of Queues with Different Classes of Customers , 1975, JACM.

[4]  Stephen S. Lavenberg,et al.  Mean-Value Analysis of Closed Multichain Queuing Networks , 1980, JACM.

[5]  A. Barbour.  Networks of queues and the method of stages , 1976, Advances in Applied Probability.

[6]  Jeffrey R. Spirn.  Queuing Networks with Random Selection for Service , 1979, IEEE Transactions on Software Engineering.

[7]  Boris G. Pittel,et al.  Closed Exponential Networks of Queues with Saturation: The Jackson-Type Stationary Distribution and Its Asymptotic Analysis , 1979, Math. Oper. Res..

[8]  B. Melamed.  Note-A Note on the Reversibility and Duality of Some Tandem Blocking Queueing Systems , 1986 .

[9]  Frank Kelly,et al.  Networks of queues with customers of different types , 1975, Journal of Applied Probability.

[10]  F. Kelly,et al.  Networks of queues , 1976, Advances in Applied Probability.

[11]  Simonetta Balsamo,et al.  Some Equivalence Properties for Queueing Networks with and without Blocking , 1983, Performance.

[12]  N M van Dijk,et al.  Insensitivity in two-node blocking models with applications , 1986 .

[13]  Harry G. Perros,et al.  On equivalencies of blocking mechanisms in queueing networks with blocking , 1986 .

[14]  K. Mani Chandy,et al.  Computational algorithms for product form queueing networks , 1980 .

[15]  Isi Mitrani,et al.  The Distribution of Queuing Network States at Input and Output Instants , 1979, JACM.

[16]  Jeffrey P. Buzen,et al.  Computational algorithms for closed queueing networks with exponential servers , 1973, Commun. ACM.

[17]  Benjamin Melamed,et al.  On the reversibility of queueing networks , 1982 .

[18]  D. Cox.  A use of complex probabilities in the theory of stochastic processes , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.

[19]  Henk Tijms,et al.  Stochastic modelling and analysis: a computational approach , 1986 .

[20]  Stephen S. Lavenberg,et al.  Stationary state probabilities at arrival instants for closed queueing networks with multiple types of customers , 1980, Journal of Applied Probability.

[21]  Ian F. Akyildiz,et al.  Queueing networks with blocking , 1987 .

[22]  Oj Onno Boxma,et al.  Teletraffic Analysis and Computer Performance Evaluation , 1988 .

[23]  Hisashi Kobayashi,et al.  Queuing Networks with Multiple Closed Chains: Theory and Computational Algorithms , 1975, IBM J. Res. Dev..

[24]  G. Pujolle,et al.  Throughput Capacity of a Sequence of Queues with Blocking Due to Finite Waiting Room , 1979, IEEE Transactions on Software Engineering.

[25]  W. J. Gordon,et al.  Closed Queuing Systems with Exponential Servers , 1967, Oper. Res..

[26]  Andrew S. Noetzel.  A Generalized Queueing Discipline for Product Form Network Solutions , 1979, JACM.

[27]  K. Mani Chandy,et al.  A Characterization of Product-Form Queuing Networks , 1983, JACM.