On Properties of the Phase-type Mixed Poisson Process and its Applications to Reliability Shock Modeling

Goyal, Dheeraj; Hazra, Nil Kamal; Finkelstein, Maxim

doi:10.1007/s11009-022-09961-2

Cited by 8 publications

(7 citation statements)

References 41 publications

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“…The PPHP, introduced by Goyal et al [12], is the mixed NHPP with the continuous phase type (PH) mixing distribution. It is well known that "The set of PH distributions is dense in the set of probability distributions on the non-negative real half-line".…”

Section: Pphpmentioning

confidence: 99%

“…In the following lemma we give the pmf of the random variable N (t) following the PPHP (see Goyal et al [12]).…”

Section: Pphpmentioning

confidence: 99%

“…This process is mathematically tractable and contains many well-known processes (namely, the HPP), the NHPP, the Pólya process, and the generalized Pólya process (GPP)) as the particular cases. The PPHP was defined and studied by Goyal et al [12]. Due to the denseness property of the mixing distribution (namely, the PH distribution), the PPHP can be used to approximate any mixed Poisson process.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On survival of coherent systems subject to random shocks

Goyal

Hazra

Finkelstein

2022

Preprint

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We consider coherent systems subject to random shocks that can damage a random number of components of a system. Based on the distribution of the number of failed components, we discuss three models, namely, (i) a shock can damage any number of components (including zero) with the same probability, (ii) each shock damages, at least, one component, and (iii) a shock can damage, at most, one component. Shocks arrival times are modeled using three important counting processes, namely, the Poisson generalized gamma process, the Poisson phase-type process and the renewal process with matrix Mittag-Leffler distributed inter-arrival times. For the defined shock models, we discuss relevant reliability properties of coherent systems. An optimal replacement policy for repairable systems is considered as an application of the proposed modeling.

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Section: Pphpmentioning

confidence: 99%

“…In the following lemma we give the pmf of the random variable N (t) following the PPHP (see Goyal et al [12]).…”

Section: Pphpmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On survival of coherent systems subject to random shocks

Goyal

Hazra

Finkelstein

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…This process is mathematically tractable and contains many well-known processes (namely, the HPP), the NHPP, the Pólya process, and the generalized Pólya process (GPP)) as the particular cases. The PPHP was defined and studied by Goyal et al (2022b). Due to the denseness property of the mixing distribution (namely, the PH distribution), the PPHP can be used to approximate any mixed Poisson process.…”

Section: Introductionmentioning

confidence: 99%

On Survival of Coherent Systems Subject to Random Shocks

Goyal,

Kamal Hazra,

Finkelstein

2024

Methodol Comput Appl Probab

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“…Their optimal maintenance policy is obtained by determining the system deterioration thresholds for deciding to perform maintenance. Goyal et al (2022) focused on a phase-type mixed Poisson process shock model and derived the failure rate of the system. They regarded the shock as a discrete-time event that affects the system failure.…”

Section: Introductionmentioning

confidence: 99%

Optimal Maintenance Policy for Deteriorating Systems Subject to External Shock Damage Under a Semi-Markov Decision Process

Zirui,

Lu,

Yamamoto

2023

TQS

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Condition-based maintenance of systems that deteriorates in accordance with a semi-Markov process was investigated. In addition to considering the system deterioration state as a criterion for maintenance decisionmaking, the cumulative damage caused by random external shocks was considered as well. The occurrence of random shocks was assumed to follow a Poisson process. Furthermore, the damage from random external shocks was assumed to be affected by the system deterioration state, and the system deterioration process was assumed to be affected by the cumulative damage from random external shocks. The system operator chooses the maintenance actions (continue to operate, repair, or replace) that minimize the total expected discounted cost over an infinite time horizon at the time when the system deterioration state changes. The maintenance policy optimization problem is formulated as a semi-Markov decision process considering cumulative damage from random external shocks. The properties of the total expected discounted cost with respect to the system deterioration state and cumulative damage are proven to be monotone. A threshold-type maintenance policy is derived on the basis of the monotone properties. Numerical examples illustrate the threshold-type maintenance policy.

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On Properties of the Phase-type Mixed Poisson Process and its Applications to Reliability Shock Modeling

Cited by 8 publications

References 41 publications

On survival of coherent systems subject to random shocks

On survival of coherent systems subject to random shocks

On Survival of Coherent Systems Subject to Random Shocks

Optimal Maintenance Policy for Deteriorating Systems Subject to External Shock Damage Under a Semi-Markov Decision Process

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