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A coincidence problem in telephone traffic with non-recurrent arrival process

Published online by Cambridge University Press:  09 April 2009

P. D. Finch
Affiliation:
Department of StatisticsUniversity of Melbourne.
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We consider the following problem. Calls arrive at a telephone exchange at the instants t0, t1, … tm …. The telephone exchange contains a denumerable infinity of channels. The holding times of calls are non-negative random variables distributed independently of the times at which calls arrive, independently of which channel a call engages and independently of each other with a common distribution function B(x). Takacs [3Τm = tm+1tm, m ≧ are identically and independently distributed non-negative random variables with common distribution function, A (x). Finch [1] has studied the transient behaviour in the case of a recurrent arrival process and exponential holding time, that is when the common distribution of holding time is given by In this paper we make no assumption about the arrival process {tm}. The underlying principle of this paper is the same as that of Finch [2]. We consider the instants of arrival t0, t1,…, tm,… as given and determine various probabilities of interest conditionally as functions of the inter-arrival intervals Τ1,…, Τm,… When the arrival process is a stochastic process we can then determine the relevant unconditional probabilities by integration.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1963

References

[1]Finch, P. D., On the transiet behaviour of a variate in telephone traffic, Annals of Math. Stat., 32 (1961), 230234.CrossRefGoogle Scholar
[2]Finch, P. D., The single-server queueing system with nonrecurrent input process and Erlang service time, This Journal 3 (1983), 220238.Google Scholar
[3]Takacs, L., On a coincidence problem concerning telephone traffic, Acta Math. Sci. Hung. 9 (1958), 4581.CrossRefGoogle Scholar