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Bunching in a semi-Markov process

Published online by Cambridge University Press:  14 July 2016

A. G. Hawkes
A. G. Hawkes*
Affiliation:
University of Durham
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Extract

In the type II counter with constant deadtime, particles which arrive within some constant time τ following another particle are unrecorded. We can think of this process as an alternating sequence of gaps and bunches of events. Gaps have duration > τ, while the intervals between any successive pair of events within a bunch are all ≦ τ. Counter theory is usually concerned with the distribution of intervals between recorded events (i.e., the first event of each bunch) and the distribution of the number of recorded events in a given time interval. In the case where the events form a renewal process this has been studied intensively by Pyke [2], Smith [5] and Takács [6].

Type
Research Papers
Information
Journal of Applied Probability , Volume 7 , Issue 1 , April 1970 , pp. 175 - 182
Copyright
Copyright © Applied Probability Trust 

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References

[1] Bartlett, M.S. (1955) An Introduction to Stochastic Processes. Cambridge.Google Scholar
[2] Pyke, R. (1958) On renewal processes related to type I and type II counter models. Ann. Math. Statist. 29, 737–754.CrossRefGoogle Scholar
[3] Pyke, R. (1961) Markov renewal processes: definitions and preliminary properties. Ann. Math. Statist. 32, 1231–1242.CrossRefGoogle Scholar
[4] Pyke, R. (1961) Markov renewal processes with finitely many states. Ann. Math. Statist. 32, 1243–1259.CrossRefGoogle Scholar
[5] Smith, W.L. (1957) On renewal theory, counter problems, and quasi-Poisson processes Proc. Camb. Phil. Soc. 53, 175–193.CrossRefGoogle Scholar
[6] Takács, L. (1956) On a probability problem arising in the theory of counters. Proc. Camb. Phil. Soc. 52, 488–498.CrossRefGoogle Scholar