Abstract
In this paper we study Markovian queueing networks in which the service and the routing characteristics have a particular form which leads to a product form stationary distribution for the number of customers in the various queues of the network. We show that if certain transitions are prohibited due to blocking conditions, then the form of the stationary distribution is preserved under a certain rerouting protocol. Several examples are presented which illustrate the wide applicability of the model.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I.F. Akyildiz, Exact analysis of queueing networks with rejection blocking, in: Proc. of the 1st Internat. Workshop on Queueing Networks with Blocking, eds. H.G. Perros and T. Atliok (North-Holland, Amsterdam, 1989) pp. 19-29.
F. Baskett, M. Chandy, R. Muntz and J. Palacios, Open, closed and mixed networks of queues with different classes of customers, J. Assoc. Comput. Mach. 22 (1975) 248-260.
W.J. Gordon and G.F. Newell, Closed queueing systems with exponential servers, Oper. Res. 15 (1967) 254-265.
W.J. Gordon and G.F. Newell, Cyclic queueing systems with restricted length queues, Oper. Res. 15 (1967) 266-277.
A. Hordijk and N.M. van Dijk, Networks of queues with blocking, in: Performance '81, ed. K.J. Kylstra (North-Holland, Amsterdam, 1981) pp. 51-65.
A. Hordijk and N.M. van Dijk, Stationary probabilities for networks of queues, in: Applied Probability-Computer Science, Vol. II, The Interface, eds. R.L. Disney and T.J. Ott (Birkhäuser, Boston, 1982) pp. 423-451.
A. Hordijk and N.M. van Dijk, 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 Information Sciences, eds. F. Bacceli and G. Fayoll (Springer, Berlin, 1983) pp. 158-205.
J.R. Jackson, Networks of waiting lines, Oper. Res. 5 (1957) 518-521.
J.R. Jackson, Jobshop-like queueing systems, Mgmt. Sci. 10 (1963) 131-142.
F.P. Kelly, Networks of queues with customers of different types, J. Appl. Probab. 12 (1975) 542-554.
F.P. Kelly, Networks of queues, Adv. in Appl. Probab. 8 (1976) 416-432.
F.P. Kelly, Reversibility and Stochastic Networks (Wiley, New York, 1979).
J.G. Kemmeny and J.L. Snell, Finite Markov Chains (Springer, New York, 1976).
R. Serfozo, Markovian network processes: congestion dependent routing and processing, Queueing Systems 5 (1989) 5-36.
N.M. van Dijk, Stop=recirculate for exponential product form queueing networks with departure blocking, Oper. Res. Lett. 10 (1991) 343-351.
N.M. van Dijk, Product form for queueing networks with limited clusters, Part I: Theory, Part II: Applications, Research Report, Free University of Amsterdam (1991).
N.M. van Dijk, Queueing Networks and Product Forms, A System Approach (Wiley, Chichester, 1993).
D.D. Yao and J.A. Buzacott, Modelling a class of flexible manufacturing systems with reversible routing, Oper. Res. 35 (1987) 87-93.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Economou, A., Fakinos, D. Product form stationary distributions for queueing networks with blocking and rerouting. Queueing Systems 30, 251–260 (1998). https://doi.org/10.1023/A:1019117121530
Issue Date:
DOI: https://doi.org/10.1023/A:1019117121530