Abstract
A new class of models, which combines closed queueing networks with branching processes, is introduced. The motivation comes from MIMD computers and other service systems in which the arrival of new work is always triggered by the completion of former work, and the amount of arriving work is variable. In the variant of branching/queueing networks studied here, a customer branches into a random and state-independent number of offspring upon completing its service. The process regenerates whenever the population becomes extinct. Implications for less rudimentary variants are discussed. The ergodicity of the network and several other aspects are related to the expected total number of progeny of an associated multitype Galton-Watson process. We give a formula for that expected number of progeny. The objects of main interest are the stationary state distribution and the throughputs. Closed-form solutions are available for the multi-server single-node model, and for homogeneous networks of infinite-servers. Generally, branching/queueing networks do not seem to have a product-form state distribution. We propose a conditional product-form approximation, and show that it is approached as a limit by branching/queueing networks with a slowly varying population size. The proof demonstrates an application of the nearly complete decomposability paradigm to an infinite state space.
Similar content being viewed by others
References
S. Asmussen, Applied Probability and Queues (Wiley, New York, 1987).
O.I. Aven, E.G. Coffman Jr. and Y.A. Kogan,Stochastic Analysis of Computer Storage (Reidel, Dordrecht, 1987).
N. Bayer, Closed queueing networks with branching populations, D.Sc.thesis, Technion - Israel Institute of Technology, Department of Industrial Engineering and Management (1994).
N. Bayer and Y. Kogan, Branching modelsof MIMD architectures, Operations Research Statistics and Economics Mimeiograph Series 404, Technion - Israel Institute of Technology, Department of Industrial Engineering and Management (1992).
A. Berman and R.J. Plemmons,Nonnegative Matrices in the Mathematical Sciences (Academic Press, New York, 1979).
K.L. Chung, A Course in Probability Theory(Academic Press, New York, 1974).
P.J. Courtois,Decomposability: Queueing and Computer System Applications (Academic Press, New York, 1977).
M.J. Flynn, Very high speed computingsystems, Proceedings of the IEEE 54(12) (1966) 1901–1909.
W.J. Gordon and G.F. Newell, Closed queueing systems with exponentialservers, Oper. Res. 15 (1967) 254–265.
T.E. Harris, TheTheory of Branching Processes (Springer/ Prentice-Hall, 1963).
J.R. Jackson, Jobshop-like queueing systems, Management Science 10(1) (1963) 131–142.
H. Kobayashi, Modeling and Analysis: AnIntroduction to System Performance Evaluation Methodology (Addison-Wesley, Reading, MA, 1978).
S. Kotz, N.L. Johnson and C.B. Read, eds.,Encyclopedia of Statistical Sciences (Wiley, New York, 1985).
M. Kuijper, First-order Representations of Linear Systems (Birkhauser, Basel, 1994).
S. Orey, Lecture Notes on Limit Theorems for MarkovChain Transition Probabilities (Van-Nostard Reinhold, New York, 1971).
B.A. Sevastyanov, Branching Processes (Nauka, Moscow,1971) (in Russian).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bayer, N., Kogan, Y. Branching/queueing networks: Their introduction and near-decomposability asymptotics. Queueing Systems 27, 251–269 (1997). https://doi.org/10.1023/A:1019170216562
Issue Date:
DOI: https://doi.org/10.1023/A:1019170216562