Bivariate Tempered Space-Fractional Poisson Process and Shock Models

TL;DR

This paper introduces a new bivariate tempered space-fractional Poisson process, explores its properties, and applies it to shock models for reliability analysis, deriving key distributional results and special cases.

Contribution

It presents the first study of BTSFPP, linking it to differential equations and reliability models, with detailed distributional and shock process analysis.

Findings

BTSFPP distributional properties derived

Failure time is exponential under certain conditions

Reliability functions and hazard rates analyzed

Abstract

In this paper, 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 L\'{e}vy measure for the BTSFPP is also derived. A bivariate competing risks and shock model based on the BTSFPP for predicting the failure times of the items that undergo two random shocks is also explored. The system is supposed to break when the sum of two types of shocks reaches a certain random threshold. Various results related to reliability such as reliability function, hazard rates, failure density, and the probability that the failure occurs due to a certain type of shock are studied. We show that for a general L\'{e}vy subordinator, the failure time of the system is exponentially distributed with mean depending on the Laplace exponent of the L\'{e}vy subordinator when the threshold has geometric distribution. Some special cases and several typical examples are also demonstrated.