Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T04:04:59.215Z Has data issue: false hasContentIssue false
  • English
  • Français

A stochastic process whose successive intervals between events form a first order Markov chain — I

Published online by Cambridge University Press:  14 July 2016

D. G. Lampard
D. G. Lampard*
Affiliation:
Department of Electrical Engineering, Monash University, Clayton, Victoria
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we discuss a counter system whose output is a stochastic point process such that the time intervals between pairs of successive events form a first order Markov chain. Such processes may be regarded as next, in order of complexity, in a hierarchy of stochastic point processes, to “renewal” processes, which latter have been studied extensively. The main virtue of the particular system which is studied here is that virtually all its important statistical properties can be obtained in closed form and that it is physically realizable as an electronic device. As such it forms the basis for a laboratory generator whose output may be used for experimental work involving processes of this kind.

Such statistical properties as the one and two-dimensional probability densities for the time intervals are considered in both the stationary and nonstationary state and also discussed are corresponding properties of the successive numbers arising in the stores of the counter system. In particular it is shown that the degree of coupling between successive time intervals may be adjusted in practice without altering the one dimensional probability density for the interval lengths. It is pointed out that operation of the counter system may also be regarded as a problem in queueing theory involving one server alternately serving two queues. A generalization of the counter system, whose inputs are normally a pair of statistically independent Poisson processes, to the case where one of the inputs is a renewal process is considered and leads to some interesting functional equations.

Type
Research Papers
Information
Journal of Applied Probability , Volume 5 , Issue 3 , December 1968 , pp. 648 - 668
Copyright
Copyright © Applied Probability Trust 1968 

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

[1] Barrett, J. F. and Lampard, D. G. (1955) An expansion for some second order probability distributions and its applications to noise problems. I.R.E. Trans. PGIT IT–1, No. 1, 10–15.Google Scholar
[2] Copson, E. T. (1948) Theory of Functions of a Complex Variable. Oxford University Press.Google Scholar
[3] Cox, D. R. (1962) Renewal Theory. Methuen Monograph in Applied Probability and Statistics, London.Google Scholar
[4] Cox, D. R. and Smith, W. L. (1954) On the superposition of renewal processes. Biometrika 41, 91–99.Google Scholar
[5] Cox, D. R. and Smith, W. L. (1961) Queues. Methuen Monograph in Applied Probability and Statistics, London.Google Scholar
[6] Cramer, H. (1951) Mathematical Methods of Statistics. Princeton University Press. p. 249.Google Scholar
[7] Dunne, M. C. Computer control of an isolated intersection. Draft report, Mathematics Department, University of Adelaide.Google Scholar
[8] Erdélyi, A. et al. (1955) Higher Transcendental Functions. McGraw-Hill, New York. Vol. 1.Google Scholar
[9] Erdélyi, A. et al. (1955) Higher Transcendental Functions. McGraw-Hill, New York. Vol. 2.Google Scholar
[10] Feller, W. (1958) An Introduction to Probability Theory and its Applications. John Wiley & Sons Inc. Vol. 1, Chapter XV.Google Scholar
[11] Gaver, D. P. (1964) On priority type disciplines in queueing. Proceedings of the Symposium on Congestion Theory. University of North Carolina Monograph Series in Probability and Statistics.Google Scholar
[12] Hawkes, A. G. (1966) Delay at traffic intersections. J. R. Statist. Soc. B 28, No. 1, 202–212.Google Scholar
[13] Lampard, D. G. (1965) A stochastic process whose successive intervals between events form a first-order Markov chain. I. Report No. MEE 65–1, Electrical Engineering Dept., Monash University.Google Scholar
[14] Lampard, D. G. and Redman, S. J. (1963) Statistical properties of the integral of a binary random process. I.E.E.E. Trans. PGCT CT–10, No. 3, 413–427.Google Scholar
[15] Lancaster, H. O. (1958) The structure of bivariate distributions. Ann. Math. Statist. 29, No. 3, 719–736.Google Scholar
[16] Lee, P. A. (1965) Properties of the Meixner polynomials. Report No. MEE 65–6, Electrical Engineering Department, Monash University.Google Scholar
[17] Lee, P. A. (1966) A stochastic process whose successive intervals between events form a first-order Markov chain. II. Report No. MEE 66–2, Electrical Engineering Department, Monash University.Google Scholar
[18] McFadden, J. A. (1959) The probability density of the output of an RC filter when the input is a binary random process. I.R.E. Trans. PGIT IT–5, No. 4, 174–178.Google Scholar
[19] Meixner, J. (1934) Orthogonale Polynomsysteme mit einer besonderen Gestalt der Erzeugenden Funktion. J. London Math. Soc., 9, 6–13.CrossRefGoogle Scholar
[20] Moran, P. A. P. (1967) Testing for correlation between non-negative variates. Biometrika, 54, Parts 3 & 4, 385–394.Google Scholar
[21] Redman, S. J. & Lampard, D. G. (1963) Stochastic sampling of a binary random process. I.E.E.E. Trans. PGCT CT–10, No. 1, 3–24.Google Scholar
[22] Watson, G. N. (1948) A Treatise on the Theory of Bessel Functions. Cambridge University Press.Google Scholar
[23] Watson, G. N. (1933) Notes on generating functions of polynomials — Laguerre polynomials. J. London Math. Soc. 8, 189–192.Google Scholar