Hostname: page-component-74656f9884-2dmjx Total loading time: 0 Render date: 2026-11-09T00:12:58.896Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterization of the Generalized Pólya Process and its Applications

Part of: Applications Special processes

Published online by Cambridge University Press:  22 February 2016

Ji Hwan Cha
Ji Hwan Cha*
Affiliation:
Ewha Womans University
*
∗ Postal address: Department of Statistics, Ewha Womans University, Seoul, 120-750, Korea. Email address: jhcha@ewha.ac.kr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In this paper some important properties of the generalized Pólya process are derived and their applications are discussed. The generalized Pólya process is defined based on the stochastic intensity. By interpreting the defined stochastic intensity of the generalized Pólya process, the restarting property of the process is discussed. Based on the restarting property of the process, the joint distribution of the number of events is derived and the conditional joint distribution of the arrival times is also obtained. In addition, some properties of the compound process defined for the generalized Pólya process are derived. Furthermore, a new type of repair is defined based on the process and its application to the area of reliability is discussed. Several examples illustrating the applications of the obtained properties to various areas are suggested.

Keywords

Generalized Pólya processrestarting propertydistribution of the number of eventsconditional arrival time distributionrepair typereliability application

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 62P30: Applications in engineering and industry

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 4 , December 2014 , pp. 1148 - 1171
Copyright
© Applied Probability Trust 

References

Aven, T. (1996). Condition based replacement policies—A counting process approach. Reliab. Eng. System Safety 51, 275–281.Google Scholar
Aven, T. and Jensen, U. (1999). Stochastic Models in Reliability. Springer, New York.CrossRefGoogle Scholar
Aven, T. and Jensen, U. (2000). A general minimal repair model. J. Appl. Prob. 37, 187–197.CrossRefGoogle Scholar
Badı´a, F. G., Berrade, M. D. and Campos, C. A. (2001). Optimization of inspection intervals based on cost. J. Appl. Prob. 38, 872–881.Google Scholar
Barbu, V. S. and Limnios, N. (2008). Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications. Their Use in Reliability and DNA Analysis. Springer, New York.Google Scholar
Cha, J. H. (2001). Burn-in procedures for a generalized model. J. Appl. Prob. 38, 542–553.CrossRefGoogle Scholar
Cha, J. H. (2003). A further extension of the generalized burn-in model. J. Appl. Prob. 40, 264–270.CrossRefGoogle Scholar
Cha, J. H. and Finkelstein, M. (2009). On a terminating shock process with independent wear increments. J. Appl. Prob. 46, 353–362.CrossRefGoogle Scholar
Cha, J. H. and Finkelstein, M. (2011a). On new classes of extreme shock models and some generalizations. J. Appl. Prob. 48, 258–270.CrossRefGoogle Scholar
Cha, J. H. and Finkelstein, M. (2011b). Stochastic intensity for minimal repairs in heterogeneous populations. J. Appl. Prob. 48, 868–876.CrossRefGoogle Scholar
Cox, D. R. (1962). Renewal Theory. Methuen, London.Google Scholar
Cox, D. R. and Isham, V. (2000). Point Processes. Chapman & Hall, London.Google Scholar
Ebrahimi, N. (1997). Multivariate age replacement. J. Appl. Prob. 34, 1032–1040.Google Scholar
Finkelstein, M. and Cha, J. H. (2013). Stochastic Modelling for Reliability. Shocks, Burn-in and Heterogeneous Populations. Springer, London.Google Scholar
Kao, E. P. C. (1997). An Introduction to Stochastic Processes. Duxbury, Belmont, CA.Google Scholar
Konno, H. (2010). On the exact solution of a generalized Pólya process. Adv. Math. Phys. 2010, 504267.Google Scholar
Limnios, N. and Oprişan, G. (2001). Semi-Markov Processes and Reliability. Birkhäuser, Boston, MA.CrossRefGoogle Scholar
Mi, J. (1994). Burn-in and maintenance policies. Adv. Appl. Prob. 26, 207–221.CrossRefGoogle Scholar
Nakagawa, T. (2005). Maintenance Theory of Reliability. Springer, London.Google Scholar
Ross, S. M. (1996). Stochastic Processes, 2nd edn. John Wiley, New York.Google Scholar
Ross, S. M. (2003). Introduction to Probability Models, 8th edn. Academic Press, San Diego, CA.Google Scholar
Sherif, Y. S. and Smith, M. L. (1981). Optimal maintenance models for systems subject to failure—a review. Naval Res. Logistics Quart. 28, 47–74.Google Scholar
Tadj, L., Ouali, M.-S., Yacout, S. and Ait-Kadi, D. (eds) (2011). Replacement Models with Minimal Repair. Springer, London.Google Scholar
Valdez-Flores, C. and Feldman, R. M. (1989). A survey of preventive maintenance models for stochastically deteriorating single-unit systems. Naval Res. Logistics [Quart. to 36,] 419–446.3.0.CO;2-5>CrossRefGoogle Scholar
Wang, H. (2002). A survey of maintenance policies of deteriorating systems. Europ. J. Operat. Res. 139, 469–489.Google Scholar