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September, 1959 Imbedded Markov Chain Analysis of a Waiting-Line Process in Continuous Time
Donald P. Gaver Jr.
Ann. Math. Statist. 30(3): 698-720 (September, 1959). DOI: 10.1214/aoms/1177706200


Bunches of individual customers approach a single servicing facility according to a stationary compound Poisson process. The resulting waiting line process is studied in continuous time by the method of the imbedded Markov chain, cf. Kendall [7], [8], and of renewal theory, cf. Blackwell [3], Feller [5], and Smith [9]. Busy period phenomena are discussed, cf. Theorem 1, in which the transform of the joint d.f. of busy period duration and the number of departures in that duration is expressed as the root $x_1(s, z)$ of a functional equation, a generalization of a result of Takacs [12]. In terms of $x_1(s, z)$ "zero-avoiding" transition probabilities are characterized. A simple model for "instantaneous defection" is analyzed. Using renewal theory, ergodic properties of waiting line lengths and waiting times are discussed for the "general" process, in which idle and busy periods recur.


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Donald P. Gaver Jr.. "Imbedded Markov Chain Analysis of a Waiting-Line Process in Continuous Time." Ann. Math. Statist. 30 (3) 698 - 720, September, 1959.


Published: September, 1959
First available in Project Euclid: 27 April 2007

zbMATH: 0087.33601
MathSciNet: MR107913
Digital Object Identifier: 10.1214/aoms/1177706200

Rights: Copyright © 1959 Institute of Mathematical Statistics

Vol.30 • No. 3 • September, 1959
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