Page 238 - A First Course In Stochastic Models

REFERENCES 231

et al. (2003), Halfin and Whitt (1981), Jennings et al. (1996) and Whitt (1992).
Influential papers showing Poisson departures for the M/M/c queue are Burke
(1956) and Reich (1957). Insensitivity is a fundamental concept in stochastic ser-
vice systems with no queueing. The illustrative problem from Example 5.4.2 is
adapted from Foschini et al. (1981). A general discussion of the insensitivity phe-
nomenon in stochastic networks can be found in Kelly (1979, 1991) and Van
Dijk (1993). The book of Kelly (1979) makes extensive use of the concept of
time-reversible Markov chains. The method of phases using fictitious stages with
exponentially distributed lifetimes has its origin in the pioneering work of Erlang
on stochastic processes in the early 1900s. The scope of this method was consider-
ably enlarged by Schassberger (1973), who showed that the probability distribution
of any non-negative random variable can be represented as the limit of a sequence
of mixtures of Erlangian distributions with the same scale parameters. This result is
very useful for both analytical and computational purposes. The product-form solu-
tion was first obtained in the paper of R.R.P. Jackson (1954) for a tandem queue
consisting of two single-server stations. This work was considerably extended by
J.R. Jackson (1957, 1963) to produce what have come to be known as Jackson
networks. More material on queueing networks and their applications in computer
and communication networks can be found in the books of Hayes (1984) and
Kleinrock (1976).

REFERENCES

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Burke, P.J. (1956) The output of queueing systems. Operat. Res., 4, 699–704.
Cohen, J.W. (1976) On Regenerative Processes in Queueing Theory. Springer-Verlag, Berlin.
Cohen, J.W. (1979) The multiple phase service network with generalized processor sharing.
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Engset, T. (1918) Die Wahrscheinlichkeitsrechnung zur Bestimmung der W¨ ahleranzahl in
automatischen Fernsprech¨ amtern. Elektrotechn. Zeitschrift, 31, 304–306.
Erlang, A.K. (1917) Solution of some problems in the theory of probabilities of significance
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Foschini, G.J., Gopinath, B. and Hayes, J.F. (1981) Optimum allocation of servers to two
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