Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-30T21:49:30.051Z Has data issue: false hasContentIssue false
  • English
  • Français

Steady-state Markov chain models for the Heine and Euler distributions

Part of: Basic hypergeometric functions Markov processes

Published online by Cambridge University Press:  14 July 2016

A. W. Kemp
A. W. Kemp*
Affiliation:
University of St Andrews
*
∗ Postal address: Department of Mathematical and Computational Sciences, University of St Andrews, St Andrews KY16 9SS, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper puts forward steady-state Markov chain models for the Heine and Euler distributions. The models for oil exploration strategies that were discussed by Benkherouf and Bather (1988) are reinterpreted as current-age models for discrete renewal processes. Steady-state success-runs processes with non-zero probabilities that a trial is abandoned, Foster processes, and equilibrium random walks corresponding to elective M/M/1 queues are also examined.

Keywords

q-ANALOGUESSUCCESS-RUNSFOSTER PROCESSELECTIVE M/M/1 QUEUE

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 33D15: Basic hypergeometric functions in one variable, $_rphi_s$
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 4 , December 1992 , pp. 869 - 876
Copyright
Copyright © Applied Probability Trust 1992 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andrews, G. E. (1986) q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. American Mathematical Society, Providence, RI.Google Scholar
Benkherouf, L. and Bather, J. A. (1988) Oil exploration: sequential decisions in the face of uncertainty. J. Appl. Prob. 25, 529–543.CrossRefGoogle Scholar
Bharucha-Reid, A. T. (1960) Elements of the Theory of Markov Processes and Their Applications. McGraw-Hill, New York.Google Scholar
Foster, F. G. (1952) A Markov chain derivation of discrete distributions. Ann. Math. Statist. 23, 624–627.CrossRefGoogle Scholar
Haight, F. A. (1957) Queueing with balking. Biometrika 44, 360–369.CrossRefGoogle Scholar
Kemp, A. W. (1979) Convolutions involving binomial pseudo-variables. Sankhya A41, 232–243.Google Scholar
Kemp, A. W. (1992a) Heine-Euler extensions of the Poisson distribution. Commun. Statist.-Theory Meth. 21, 791–798.CrossRefGoogle Scholar
Kemp, A. W. (1992b) On counts of organisms able to signal the presence of an observer. Biom. J. 34, 595–604.CrossRefGoogle Scholar
Morse, P. M. (1958) Queues, Inventories and Maintenance. Wiley, New York.Google Scholar
Slater, L. J. (1966) Generalized Hypergeometric Functions. Cambridge University Press, London.Google Scholar