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Bivariate tempered space-fractional Poisson process and shock models

Part of: Special processes Survival analysis and censored data Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  23 May 2024

Ritik Soni ,
Ashok Kumar Pathak ,
Antonio Di Crescenzo Open the ORCID record for Antonio Di Crescenzo [Opens in a new window]  and
Alessandra Meoli
Ritik Soni*
Affiliation:
Central University of Punjab, Bathinda
Ashok Kumar Pathak*
Affiliation:
Central University of Punjab, Bathinda
Antonio Di Crescenzo*
Affiliation:
Università degli Studi di Salerno, Fisciano, Italy
Alessandra Meoli*
Affiliation:
Università degli Studi di Salerno, Fisciano, Italy
*
*Postal address: Department of Mathematics and Statistics, Central University of Punjab, Bathinda, Punjab 151401, India.
*Postal address: Department of Mathematics and Statistics, Central University of Punjab, Bathinda, Punjab 151401, India.
****Postal address: Dipartimento di Matematica, Università degli Studi di Salerno, I-84084 Fisciano (SA), Italy.
****Postal address: Dipartimento di Matematica, Università degli Studi di Salerno, I-84084 Fisciano (SA), Italy.
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Abstract

We introduce a bivariate tempered space-fractional Poisson process (BTSFPP) by time-changing the bivariate Poisson process with an independent tempered $\alpha$-stable subordinator. We study its distributional properties and its connection to differential equations. The Lévy measure for the BTSFPP is also derived. A bivariate competing risks and shock model based on the BTSFPP for predicting the failure times of items that undergo two random shocks is also explored. The system is supposed to break when the sum of two types of shock reaches a certain random threshold. Various results related to reliability, such as reliability function, hazard rates, failure density, and the probability that failure occurs due to a certain type of shock, are studied. We show that for a general Lévy subordinator, the failure time of the system is exponentially distributed with mean depending on the Laplace exponent of the Lévy subordinator when the threshold has a geometric distribution. Some special cases and several typical examples are also demonstrated.

Keywords

Tempered space-fractional Poisson processLévy subordinatorshock modelfailure distributionreliability

MSC classification

Primary: 60G22: Fractional processes, including fractional Brownian motion 60G51: Processes with independent increments; L'evy processes 60G55: Point processes 60E05: Distributions
Secondary: 60K10: Applications (reliability, demand theory, etc.) 62N05: Reliability and life testing
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 17
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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