Abstract
An appropriate model for a number of systems is a network of queues in which the output of one queue is fed into another. Under a wide range of assumptions, these networks may be modeled and analyzed by means of multidimensional birth-death processes. The salient result of this work is the product form solution in which the joint distribution of queue occupancies is the product of functions of the number in the individual queues. Networks satisfying the proper set of assumptions are called Jackson networks after J. R. Jackson, who discovered the product form solution.(1) In this chapter the model is applied to store-and-forward message-switched networks. Using the theory of Jackson networks we shall find queue occupancy and delay in message-switched networks. These results enable us to allocate transmission capacity in an optimum fashion. In the next chapter these same ideas are extended in order to model flow control in a store-and-forward network.
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References
J. R. Jackson, “Networks of Waiting Lines,” Operations Research, 5, 518–521 (1957).
J. R. Jackson, “Jobshop-like Queueing Systems,” Management Science, 10,131–142 (1963).
H. Kobayashi, Modeling and Analysis—An Introduction to System Performance Methodology. Addison-Wesley, Reading, Massachusetts (1978).
E. Reich, “Waiting Time when Queues are in Tandem,” Annals of Mathematical Statistics, 28, 768–773 (1957).
P. J. Burke, “The Output Process of a Stationary M/M/S Queueing System,” Annals of Mathematical Statistics, 37(4), 1144–1152 (1968).
P. J. Burke, “The Dependence of Delays in Tandem Queues,” Annals of Mathematical Statistics, 35, 874–875 (1964).
B. Melamed, “On Poisson Traffic Processes in Discrete State Markovian Systems with Applications to Queueing Theory,” Technical Report 77–7, Department of Industrial and Operations Engineering, University of Michigan (1977).
F. J. Beutler and B. Melamed, “Decomposition and Customer Streams of Feedback Networks of Queues in Equilibrium,” Operations Research, 26(6), 1059–1072, November-December (1978).
A. J. Lemoine, “Networks of Queues—A Survey of Equilibrium Analysis,” Management Science, 24(4), 464–481, December (1977).
F. Baskette et al., “Open, Closed and Mixed Networks of Queues with Different Classes of Customers,” Journal of the Association for Compting Machinery, 22, 248–260 (1975).
S. M. Ross, Stochastic Processes, John Wiley, New York (1983).
L. Kleinrock, Communication Nets: Stochastic Message Flow and Delay. McGraw-Hill, New York (1964), out of print; reprinted Dover, New York (1972).
B. Avi-Itzhak, “A Sequence of Service Stations with Arbitrary Input and Regular Service Times,” Management Science, 11(5), 565–571, March (1965).
H. D. Friedman, “Reduction Methods for Tandem Queueing Systems,” Operations Research, 13, 121–133 (1965).
I. Rubin, “Message Path Delays in Packet Switching Communication Networks,” IEEE Transactions on Communications, COM-23(2), 186–192, February (1975).
M. Kaplan, “A Two-fold Tandem Net with Deterministic Link and Source Interfaces,” Operations Research, 28, 512–526, May-June (1980).
B. Meister, H. R. Mueller, and H. Rudin, “New Optimization Criteria for Message-Switched Networks,” IEEE Transactions on Communications Technology, COM-19(3), 256–260, June (1971).
R. E. Bellman, Dynamic Programming, Princeton University Press, Princeton, New Jersey (1957).
S. E. Dreyfus, Dynamic Programming and the Calculus of Variations, Academic Press, New York (1965).
J. F. Hayes, “The Viterbi Algorithm Applied to Digital Data Transmission,” IEEE Communication Society, 13(2), 15–20, March (1975).
D. G. Cantor and M. Gerla, “Capacity Allocation in Distributed Computer Networks,” Proceedings of 7th Hawaii International Conference on System Science (1974), pp. 115–117.
L. Kleinrock, Communication Nets: Stochastic Message Flow and Delay, McGraw-Hill, New York (1964), out of print; reprinted by Dover, New York (1972).
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© 1984 Plenum Press, New York
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Hayes, J.F. (1984). Networks of Queues. In: Modeling and Analysis of Computer Communications Networks. Applications of Communications Theory. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-4841-2_10
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DOI: https://doi.org/10.1007/978-1-4684-4841-2_10
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