MAINTENANCE OF SYSTEMS BY MEANS OF REPLACEMENTS AND REPAIRS: THE CASE OF PHASE-TYPE DISTRIBUTIONS

2007 ◽  
Vol 24 (03) ◽  
pp. 421-434 ◽  
Author(s):  
DELIA MONTORO-CAZORLA ◽  
RAFAEL PÉREZ-OCÓN
Keyword(s):  
Performance Measures ◽  
Markov Processes ◽  
Stationary Regime ◽  
Numerical Example ◽  
Imperfect Repair ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Comparison Of The Results ◽  
Imperfect Repairs

We present two models for studying a system maintained by means of imperfect repairs before a replacement or a perfect repair is allowed. The operational and repair times follow phase-type distributions. Imperfect repair means that successive operational times decrease and successive repair times increase. Under these assumptions, models that govern the systems are Markov processes, whose structures are determined, and several performance measures are calculated in transient and stationary regime. These models extend other previously studied in the literature. The incorporation of phase-type distributions allows to apply the model to many other distributions. A numerical example illustrates the calculations and allows a comparison of the results.

On some waiting time problems

2000 ◽  
Vol 37 (03) ◽  
pp. 756-764 ◽  
Author(s):  
Valeri T. Stefanov
Keyword(s):  
Survival Analysis ◽  
Finite Number ◽  
Markov Processes ◽  
Waiting Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Waiting Times ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Semi Markov Processes

A unifying technology is introduced for finding explicit closed form expressions for joint moment generating functions of various random quantities associated with some waiting time problems. Sooner and later waiting times are covered for general discrete- and continuous-time models. The models are either Markov chains or semi-Markov processes with a finite number of states. Waiting times associated with generalized phase-type distributions, that are of interest in survival analysis and other areas, are also covered.

TRANSIENT ANALYSIS OF A MULTI-COMPONENT SYSTEM MODELED BY A GENERAL MARKOV PROCESS

2006 ◽  
Vol 23 (03) ◽  
pp. 311-327 ◽  
Author(s):  
RAFAEL PÉREZ-OCÓN ◽  
DELIA MONTORO-CAZORLA ◽  
JUAN ELOY RUIZ-CASTRO
Keyword(s):  
Markov Process ◽  
Performance Measures ◽  
Transient Analysis ◽  
Transition Probabilities ◽  
Dynamic Environment ◽  
General Interest ◽  
System A ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Reliability Systems

An M-unit system in dynamic environment with operational and repair times following phase-type distributions and incorporating geometrical processes is considered. A general Markov process with vectorial states is the appropriate structure for modeling this system. A transient analysis is performed for this complex system and the transition probabilities are calculated. Some performance measures of general interest in the study of systems are obtained using an algorithmic approach, and applied to G-out-of-M systems. A numerical example is presented and the transient performance measures are calculated and compared with the stationary ones. This paper extends previous reliability systems, that can be considered as particular cases of this one. Throughout the paper, the mathematical expressions are given by algorithmic methods, that emphasized the utility of phase-type distributions in the analysis of lifetime data.

On some waiting time problems

2000 ◽  
Vol 37 (3) ◽  
pp. 756-764 ◽  
Author(s):  
Valeri T. Stefanov
Keyword(s):  
Survival Analysis ◽  
Finite Number ◽  
Markov Processes ◽  
Waiting Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Waiting Times ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Semi Markov Processes

A unifying technology is introduced for finding explicit closed form expressions for joint moment generating functions of various random quantities associated with some waiting time problems. Sooner and later waiting times are covered for general discrete- and continuous-time models. The models are either Markov chains or semi-Markov processes with a finite number of states. Waiting times associated with generalized phase-type distributions, that are of interest in survival analysis and other areas, are also covered.

Modelling correlated arrivals with a phase-type process

1984 ◽  
Vol 16 (1) ◽  
pp. 10-10
Author(s):  
Guy Latouche
Keyword(s):  
Markov Processes ◽  
Marginal Distribution ◽  
Type Process ◽  
Correlated Arrivals ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Semi Markov Processes

One considers semi-Markov processes using phase-type distributions, such that the marginal distribution of the interevent times are identically distributed but not independent. By suitably choosing the phase-type distributions, one obtains an exponential marginal distribution.

scholarly journals A Longitudinal Study of the Bladder Cancer Applying a State-Space Model with Non-Exponential Staying Time in States

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 363
Author(s):  
Delia Montoro-Cazorla ◽  
Rafael Pérez-Ocón ◽  
Alicia Pereira das Neves-Yedig
Keyword(s):  
Bladder Cancer ◽  
Longitudinal Study ◽  
State Space Model ◽  
Exponential Distributions ◽  
Significance Level ◽  
Phase Type Distributions ◽  
And Performance ◽  
The Times ◽  
Comparison Of The Results ◽  
Staying Time

A longitudinal study for 847 bladder cancer patients for a period of fifteen years is presented. After the first surgery, the patients undergo successive ones (recurrences). A state-model is selected for analyzing the evolution of the cancer, based on the distribution of the times between recurrences. These times do not follow exponential distributions, and are approximated by phase-type distributions. Under these conditions, a multidimensional Markov process governs the evolution of the disease. The survival probability and mean times in the different states (levels) of the disease are calculated empirically and also by applying the Markov model, the comparison of the results indicate that the model is well-fitted to the data to an acceptable significance level of 0.05. Two sub-cohorts are well-differenced: those reaching progression (the bladder is removed) and those that do not. These two cases are separately studied and performance measures calculated, and the comparison reveals details about the characteristics of the patients in these groups.

Phase Type Distributions with Finite Support

2014 ◽  
Vol 30 (4) ◽  
pp. 576-597 ◽  
Author(s):  
V. Ramaswami ◽  
N. C. Viswanath
Keyword(s):  
Finite Support ◽  
Phase Type ◽  
Phase Type Distributions

Analysis of R-out-of-N repairable systems: the case of phase-type distributions

2004 ◽  
Vol 36 (1) ◽  
pp. 116-138 ◽  
Author(s):  
Yonit Barron ◽  
Esther Frostig ◽  
Benny Levikson
Keyword(s):  
Repairable System ◽  
Repairable Systems ◽  
Regenerative Processes ◽  
Numerical Examples ◽  
Independent Components ◽  
Duality Results ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Two Systems ◽  
Repair Facility

An R-out-of-N repairable system, consisting of N independent components, is operating if at least R components are functioning. The system fails whenever the number of good components decreases from R to R-1. A failed component is sent to a repair facility. After a failed component has been repaired it is as good as new. Formulae for the availability of the system using Markov renewal and semi-regenerative processes are derived. We assume that either the repair times of the components are generally distributed and the components' lifetimes are phase-type distributed or vice versa. Some duality results between the two systems are obtained. Numerical examples are given for several distributions of lifetimes and of repair times.

scholarly journals The Algebraic Degree of Phase-Type Distributions

2010 ◽  
Vol 47 (03) ◽  
pp. 611-629
Author(s):  
Mark Fackrell ◽  
Qi-Ming He ◽  
Peter Taylor ◽  
Hanqin Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Polynomial Algorithm ◽  
Algebraic Degree ◽  
Nonnegative Matrices ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

Efficient fitting of long-tailed data sets into phase-type distributions

2002 ◽  
Vol 30 (3) ◽  
pp. 6-8 ◽  
Author(s):  
Alma Riska ◽  
Vesselin Diev ◽  
Evgenia Smirni
Keyword(s):  
Data Sets ◽  
Phase Type ◽  
Phase Type Distributions

Stochastic inequalities for an overflow model

1987 ◽  
Vol 24 (3) ◽  
pp. 696-708 ◽  
Author(s):  
Arie Hordijk ◽  
Ad Ridder
Keyword(s):  
Failure Rate ◽  
Lower Bounds ◽  
Approximation Method ◽  
Queueing Networks ◽  
Upper And Lower Bounds ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Decreasing Failure Rate ◽  
General Method

A general method to obtain insensitive upper and lower bounds for the stationary distribution of queueing networks is sketched. It is applied to an overflow model. The bounds are shown to be valid for service distributions with decreasing failure rate. A characterization of phase-type distributions with decreasing failure rate is given. An approximation method is proposed. The methods are illustrated with numerical results.

Sign in / Sign up

Export Citation Format

Share Document