Approximation of Discrete Phase-Type Distributions

We characterise the classes of continuous and discrete phase-type distributions in the following way. They are known to be closed under convolutions, mixtures, and the unary ‘geometric mixture' operation. We show that the continuous class is the smallest family of distributions that is closed under these operations and contains all exponential distributions and the point mass at zero. An analogous result holds for the discrete class. We also show that discrete phase-type distributions can be regarded as ℝ+-rational sequences, in the sense of automata theory. This allows us to view our characterisation of them as a corollary of the Kleene–Schützenberger theorem on the behavior of finite automata. We prove moreover that any summable ℝ+-rational sequence is proportional to a discrete phase-type distribution.

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Preservation of phase-type distributions under Poisson shock models

Advances in Applied Probability ◽

10.1017/s0001867800024265 ◽

1992 ◽

Vol 24 (01) ◽

pp. 223-225 ◽

Cited By ~ 1

Author(s):

M. Manoharan ◽

Harshinder Singh ◽

Neeraj Misra

Keyword(s):

Continuous Phase ◽

Finite Mixture ◽

Life Distribution ◽

Mean Values ◽

Shock Models ◽

Phase Type ◽

Phase Type Distributions ◽

Discrete Phase ◽

The Mean ◽

The Relationship

In this paper, we consider the life distribution H(t) of a device subject to shocks governed by a finite mixture of homogeneous Poisson processes. It will be shown that if (pk ), the probabilities that the device fails on the kth shock, has a discrete phase-type (DPH) distribution, then H(t) is continuous phase-type (CPH). The relationship between the mean values of (pk ) and H(t) is established.

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On the Canonical Representation of Order 3 Discrete Phase Type Distributions

Electronic Notes in Theoretical Computer Science ◽

10.1016/j.entcs.2015.10.024 ◽

2015 ◽

Vol 318 ◽

pp. 143-158 ◽

Cited By ~ 2

Author(s):

Illés Horváth ◽

János Papp ◽

Miklós Telek

Keyword(s):

Canonical Representation ◽

Phase Type ◽

Phase Type Distributions ◽

Discrete Phase

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A closure characterisation of phase-type distributions

Journal of Applied Probability ◽

10.2307/3214794 ◽

1992 ◽

Vol 29 (1) ◽

pp. 92-103 ◽

Cited By ~ 22

Author(s):

Robert S. Maier ◽

Colm Art O'Cinneide

Keyword(s):

Analogous Result ◽

Finite Automata ◽

Automata Theory ◽

Phase Type Distribution ◽

Exponential Distributions ◽

Phase Type ◽

Phase Type Distributions ◽

Discrete Phase ◽

Rational Sequence ◽

Family Of Distributions

We characterise the classes of continuous and discrete phase-type distributions in the following way. They are known to be closed under convolutions, mixtures, and the unary ‘geometric mixture' operation. We show that the continuous class is the smallest family of distributions that is closed under these operations and contains all exponential distributions and the point mass at zero. An analogous result holds for the discrete class.We also show that discrete phase-type distributions can be regarded as ℝ+-rational sequences, in the sense of automata theory. This allows us to view our characterisation of them as a corollary of the Kleene–Schützenberger theorem on the behavior of finite automata. We prove moreover that any summable ℝ+-rational sequence is proportional to a discrete phase-type distribution.

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Modelling a reliability system governed by discrete phase-type distributions

Reliability Engineering & System Safety ◽

10.1016/j.ress.2008.01.005 ◽

2008 ◽

Vol 93 (11) ◽

pp. 1650-1657 ◽

Cited By ~ 21

Author(s):

Juan Eloy Ruiz-Castro ◽

Rafael Pérez-Ocón ◽

Gemma Fernández-Villodre

Keyword(s):

Phase Type ◽

Phase Type Distributions ◽

Discrete Phase

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Preservation of phase-type distributions under Poisson shock models

Advances in Applied Probability ◽

10.2307/1427739 ◽

1992 ◽

Vol 24 (1) ◽

pp. 223-225 ◽

Cited By ~ 5

Author(s):

M. Manoharan ◽

Harshinder Singh ◽

Neeraj Misra

Keyword(s):

Continuous Phase ◽

Finite Mixture ◽

Life Distribution ◽

Mean Values ◽

Shock Models ◽

Phase Type ◽

Phase Type Distributions ◽

Discrete Phase ◽

The Mean ◽

The Relationship

In this paper, we consider the life distribution H(t) of a device subject to shocks governed by a finite mixture of homogeneous Poisson processes. It will be shown that if (pk), the probabilities that the device fails on the kth shock, has a discrete phase-type (DPH) distribution, then H(t) is continuous phase-type (CPH). The relationship between the mean values of (pk) and H(t) is established.

Download Full-text