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Biometrika 1960 47(3-4):245-251; doi:10.1093/biomet/47.3-4.245
© 1960 by Biometrika Trust
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Hierarchical birth and death processes II. Applications

GUNNAR BLOM

University of Lund Sweden

In Part I of this paper (pp. 235–44), the theory of hierarchical birth and death processes was expounded. In the present part, two kinds of applications will be given, namely, to machine interference and telephone trunking problems.

In § 1, a review is given of the well-known queueing problem arising when a group of identical machines is serviced by one operator. Stoppages are assumed to follow the simple Poisson scheme, and the service-time distribution is supposed to be exponential. In § 2, an analysis is made of the corresponding problem for machines with different needs of service but the same service-time distribution. This problem does not seem to have been discussed before in the literature. In § 3, the result is generalized to the case of several operators.

Palm (1937) determined the equilibrium probabilities of the states of a busy-signal telephone system used by a finite number of subscribers with different call frequencies. It is shown in § 4 that this problem can be solved by means of the theory of hierarchical birth and death processes.


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