scholarly journals On the Canonical Representation of Order 3 Discrete Phase Type Distributions

2015 ◽  
Vol 318 ◽  
pp. 143-158 ◽  
Author(s):  
Illés Horváth ◽  
János Papp ◽  
Miklós Telek
Keyword(s):  
Canonical Representation ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Discrete Phase

A closure characterisation of phase-type distributions

1992 ◽  
Vol 29 (01) ◽  
pp. 92-103 ◽  
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.

scholarly journals Preservation of phase-type distributions under Poisson shock models

1992 ◽  
Vol 24 (01) ◽  
pp. 223-225 ◽  
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.

scholarly journals Fitting traffic traces with discrete canonical phase type distributions and Markov arrival processes

2014 ◽  
Vol 24 (3) ◽  
pp. 453-470 ◽  
Author(s):  
András Meszáros ◽  
János Papp ◽  
Miklós Telek
Keyword(s):  
Stochastic Models ◽  
Model Fitting ◽  
Canonical Representation ◽  
Distance Measures ◽  
Moment Matching ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Canonical Representations ◽  
Arrival Processes ◽  
Markov Arrival

Abstract Recent developments of matrix analytic methods make phase type distributions (PHs) and Markov Arrival Processes (MAPs) promising stochastic model candidates for capturing traffic trace behaviour and for efficient usage in queueing analysis. After introducing basics of these sets of stochastic models, the paper discusses the following subjects in detail: (i) PHs and MAPs have different representations. For efficient use of these models, sparse (defined by a minimal number of parameters) and unique representations of discrete time PHs and MAPs are needed, which are commonly referred to as canonical representations. The paper presents new results on the canonical representation of discrete PHs and MAPs. (ii) The canonical representation allows a direct mapping between experimental moments and the stochastic models, referred to as moment matching. Explicit procedures are provided for this mapping. (iii) Moment matching is not always the best way to model the behavior of traffic traces. Model fitting based on appropriately chosen distance measures might result in better performing stochastic models. We also demonstrate the efficiency of fitting procedures with experimental results

scholarly journals On the canonical representation of phase type distributions

2009 ◽  
Vol 66 (8) ◽  
pp. 396-409 ◽  
Author(s):  
Gábor Horváth ◽  
Miklós Telek
Keyword(s):  
Canonical Representation ◽  
Phase Type ◽  
Phase Type Distributions

A closure characterisation of phase-type distributions

1992 ◽  
Vol 29 (1) ◽  
pp. 92-103 ◽  
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.

Modelling a reliability system governed by discrete phase-type distributions

2008 ◽  
Vol 93 (11) ◽  
pp. 1650-1657 ◽  
Author(s):  
Juan Eloy Ruiz-Castro ◽  
Rafael Pérez-Ocón ◽  
Gemma Fernández-Villodre
Keyword(s):  
Phase Type ◽  
Phase Type Distributions ◽  
Discrete Phase

scholarly journals Preservation of phase-type distributions under Poisson shock models

1992 ◽  
Vol 24 (1) ◽  
pp. 223-225 ◽  
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.

Sign in / Sign up

Export Citation Format

Share Document