Hostname: page-component-7479d7b7d-q6k6v Total loading time: 0 Render date: 2024-07-12T18:47:42.009Z Has data issue: false hasContentIssue false
  • English
  • Français

On some waiting time problems

Part of: Stochastic processes Distribution theory Distribution theory - Probability Special processes

Published online by Cambridge University Press:  14 July 2016

Valeri T. Stefanov
Valeri T. Stefanov*
Affiliation:
The University of Western Australia
*
∗Postal address: Department of Mathematics and Statistics, The University of Western Australia, Nedlands, WA 6907, Australia. Email address: stefanov@maths.uwa.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

A unifying technology is introduced for finding explicit closed form expressions for joint moment generating functions of various random quantities associated with some waiting time problems. Sooner and later waiting times are covered for general discrete- and continuous-time models. The models are either Markov chains or semi-Markov processes with a finite number of states. Waiting times associated with generalized phase-type distributions, that are of interest in survival analysis and other areas, are also covered.

Keywords

Markov chainMarkov renewal processstopping timerunexponential familygeneralized phase-type distribution

MSC classification

Primary: 60E10: Characteristic functions; other transforms 60G40: Stopping times; optimal stopping problems; gambling theory 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Secondary: 62E15: Exact distribution theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 3 , September 2000 , pp. 756 - 764
Copyright
Copyright © Applied Probability Trust 2000 

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

Aalen, O. O. (1995). Phase type distributions in survival analysis. Scand. J. Statist. 22, 447–463.Google Scholar
Antzoulakos, D. L. (1999). On waiting time problems associated with runs in Markov dependent trials. Ann. Inst. Statist. Math. 51, 323–330.Google Scholar
Antzoulakos, D. L., and Philippou, A. (1997). Probability distribution functions of succession quotas in the case of Markov dependent trials. Ann. Inst. Statist. Math. 49, 531–539.Google Scholar
Aki, S. (1992). Waiting time problems for a sequence of discrete random variables. Ann. Inst. Statist. Math. 44, 363–378.Google Scholar
Aki, S., Balakrishnan, N., and Mohanty, S. G. (1996). Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials. Ann. Inst. Statist. Math. 48, 773–787.Google Scholar
Aki, S., and Hirano, K. (1993). Discrete distributions related to succession events in two-state Markov chain. Statistical Science and Data Analysis. Ed. Matisita, K. et al. VSP, Amsterdam, pp. 467–474.Google Scholar
Aki, S., and Hirano, K. (1999). Sooner and later waiting time problems for runs in Markov dependent bivariate trials. Ann. Inst. Statist. Math. 51, 17–29.CrossRefGoogle Scholar
Balasubramanian, K., Viveros, R., and Balakrishnan, N. (1993). Sooner and later waiting time problems for Markovian Bernoulli trials. Statist. Prob. Lett. 18, 153–161.Google Scholar
Barndorff-Nielsen, O. (1978). Information and Exponential Families. John Wiley, Chichester.Google Scholar
Barndorff-Nielsen, O. (1980). Conditionality resolutions. Biometrika 67, 293–310.CrossRefGoogle Scholar
Biggins, J. D. (1987). A note on repeated sequences in Markov chains. Adv. Appl. Prob. 19, 739–742.CrossRefGoogle Scholar
Biggins, J. D., and Cannings, C. (1987). Markov renewal processes, counters and repeated sequences in Markov chains. Adv. Appl. Prob. 19, 521–545.CrossRefGoogle Scholar
Blom, G., and Thorburn, D. (1982). How many random digits are required until given sequences are obtained? J. Appl. Prob. 19, 518–531.Google Scholar
Brown, L. (1986). Fundamentals of Statistical Exponential Families. IMS, Hayward.Google Scholar
Butler, R. W., and Huzurbazar, A. V. (1997). Stochastic network models for survival analysis. J. Amer. Statist. Assoc. 92, 246–257.Google Scholar
Chryssaphinou, O., and Papastavridis, S. (1990). The occurrence of a sequence of patterns in repeated dependent experiments. Theory Prob. Appl. 35, 167–173.Google Scholar
Ҫinlar, E. (1975). Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs.Google Scholar
Ebneshahrashoob, M., and Sobel, M. (1990). Sooner and later waiting time problems for Bernoulli trials. Statist. Prob. Lett. 9, 5–11.CrossRefGoogle Scholar
Fu, J. C. (1996). Distribution theory of runs and patterns associated with a sequence of multistate trials. Statist. Sinica 6, 957–974.Google Scholar
Grimmett, G. R., and Stirzaker, D. R. (1994). Probability and random processes. 2nd edn. Clarendon Press, Oxford.Google Scholar
Huzurbazar, A. V. (1999). Flowgraph models for generalized phase type distribution having non-exponential waiting times. Scand. J. Statist. 26, 145–157.Google Scholar
Koutras, M. V., and Alexandrou, V. A. (1997). Sooner waiting time problems in a sequence of trinary trials. J. Appl. Prob. 34, 593–609.Google Scholar
Ling, K. D. (1990). On geometric distributions of order (k_1,…,k_m). Statist. Prob. Lett. 9, 163–171.Google Scholar
Ling, K., and Low, T. (1993). On the soonest and latest waiting time distributions: succession quotas. Commun. Statist. –- Theory Meth. 22, 2207–2221.CrossRefGoogle Scholar
Mood, A. M. (1940). The distribution theory of runs. Ann. Math. Statist. 11, 367–392.Google Scholar
Stefanov, V. T. (1991). Noncurved exponential families associated with observations over finite state Markov chains. Scand. J. Statist. 18, 353–356.Google Scholar
Stefanov, V. T. (1999). On the occurrence of composite events and clusters of points. J. Appl. Prob. 36, 1012–1018.CrossRefGoogle Scholar
Stefanov, V. T., and Pakes, A. G. (1997). Explicit distributional results in pattern formation. Ann. Appl. Prob. 7, 666–678.Google Scholar
Uchida, M. (1998). On generating functions of waiting time problems for sequence patterns of discrete random variables. Ann. Inst. Statist. Math. 50, 655–671.Google Scholar
Uchida, M., and Aki, S. (1995). Sooner and later waiting time problems in a two-state Markov chain. Ann. Inst. Statist. Math. 47, 415–433.Google Scholar