scholarly journals Reliability assessment of system under a generalized run shock model

2018 ◽  
Vol 55 (4) ◽  
pp. 1249-1260 ◽  
Author(s):  
Min Gong ◽  
Min Xie ◽  
Yaning Yang
Keyword(s):  
Arrival Time ◽  
Reliability Assessment ◽  
State System ◽  
Shock Model ◽  
External Shocks ◽  
Phase Type Distribution ◽  
Type Distribution ◽  
Markovian Property ◽  
Phase Type ◽  
Real Practice

Abstract In this paper we are concerned with modelling the reliability of a system subject to external shocks. In a run shock model, the system fails when a sequence of shocks above a threshold arrive in succession. Nevertheless, using a single threshold to measure the severity of a shock is too critical in real practice. To this end, we develop a generalized run shock model with two thresholds. We employ a phase-type distribution to model the damage size and the inter-arrival time of shocks, which is highly versatile and may be used to model many quantitative features of random phenomenon. Furthermore, we use the Markovian property to construct a multi-state system which degrades with the arrival of shocks. We also provide a numerical example to illustrate our results.

Reliability assessment of system under a generalized cumulative shock model

2019 ◽  
Vol 234 (1) ◽  
pp. 129-137 ◽  
Author(s):  
Min Gong ◽  
Serkan Eryilmaz ◽  
Min Xie
Keyword(s):  
Reliability Assessment ◽  
Shock Model ◽  
External Shocks ◽  
Phase Type Distribution ◽  
Internal Factors ◽  
Reliability Characteristics ◽  
Shock Models ◽  
The One ◽  
Random Shocks ◽  
Real Practice

Reliability assessment of system suffering from random shocks is attracting a great deal of attention in recent years. Excluding internal factors such as aging and wear-out, external shocks which lead to sudden changes in the system operation environment are also important causes of system failure. Therefore, efficiently modeling the reliability of such system is an important applied problem. A variety of shock models are developed to model the inter-arrival time between shocks and magnitude of shocks. In a cumulative shock model, the system fails when the cumulative magnitude of damage caused by shocks exceed a threshold. Nevertheless, in the existing literatures, only the magnitude is taken into consideration, while the source of shocks is usually neglected. Using the same distribution to model the magnitude of shocks from different sources is too critical in real practice. To this end, considering a system subject to random shocks from various sources with different probabilities, we develop a generalized cumulative shock model in this article. We use phase-type distribution to model the variables, which is highly versatile to be used for modeling quantitative features of random phenomenon. We will discuss the reliability characteristics of such system in some detail and give some clear expressions under the one-dimensional case. Numerical example for illustration is also provided along with a summary.

scholarly journals A Parameter Estimation Method of Shock Model Constructed with Phase-Type Distribution on the Condition of Interval Data

2020 ◽  
Vol 2020 ◽  
pp. 1-14 ◽  
Author(s):  
Long Wang ◽  
Yue Li ◽  
Yanling Qian ◽  
Xu Luo
Keyword(s):  
Parameter Estimation ◽  
Likelihood Function ◽  
Estimation Method ◽  
Interval Data ◽  
Shock Model ◽  
Phase Type Distribution ◽  
Type Distribution ◽  
Ph Distribution ◽  
Parameter Estimation Method ◽  
Phase Type

The phase-type distribution (also known as PH distribution) has mathematical properties of denseness and closure in calculation and is, therefore, widely used in shock model constructions describing occurrence time of a shock or its damage. However, in the case of samples with only interval data, modeling with PH distribution will cause decoupling issues in parameter estimation. Aiming at this problem, an approximate parameter estimation method based on building PH distribution with dynamic order is proposed. Firstly, the shock model established by PH distribution and the likelihood function under samples with only interval data are briefly introduced. Then, the principle and steps of the method are introduced in detail, and the derivation processes of some related formulas are also given. Finally, the performance of the algorithm is illustrated by a case with three different types of distributions.

A Note on Characterizing Service Interruptions with Phase-Type Distribution

2013 ◽  
Vol 31 (4) ◽  
pp. 671-683 ◽  
Author(s):  
A. Krishnamoorthy ◽  
P. K. Pramod ◽  
S. R. Chakravarthy
Keyword(s):  
Phase Type Distribution ◽  
Type Distribution ◽  
Service Interruptions ◽  
Phase Type

scholarly journals A generalized class of correlated run shock models

2018 ◽  
Vol 6 (1) ◽  
pp. 131-138 ◽  
Author(s):  
Femin Yalcin ◽  
Serkan Eryilmaz ◽  
Ali Riza Bozbulut
Keyword(s):  
Stopping Time ◽  
Random Variables ◽  
Random Variable ◽  
Phase Type Distribution ◽  
Type Distribution ◽  
Shock Models ◽  
Phase Type ◽  
Over Time

AbstractIn this paper, a generalized class of run shock models associated with a bivariate sequence {(Xi, Yi)}i≥1 of correlated random variables is defined and studied. For a system that is subject to shocks of random magnitudes X1, X2, ... over time, let the random variables Y1, Y2, ... denote times between arrivals of successive shocks. The lifetime of the system under this class is defined through a compound random variable T = ∑Nt=1 Yt , where N is a stopping time for the sequence {Xi}i≤1 and represents the number of shocks that causes failure of the system. Another random variable of interest is the maximum shock size up to N, i.e. M = max {Xi, 1≤i≤ N}. Distributions of T and M are investigated when N has a phase-type distribution.

Presentation of phase-type distributions as proper mixtures

1985 ◽  
Vol 22 (01) ◽  
pp. 247-250 ◽  
Author(s):  
David Assaf ◽  
Naftali A. Langberg
Keyword(s):  
Phase Type Distribution ◽  
Type Distribution ◽  
Phase Type ◽  
Phase Type Distributions

It is shown that any phase-type distribution can be represented as a proper mixture of two distinct phase-type distributions. Using different terms, it is shown that the class of phase-type distributions does not include any extreme ones. A similar result holds for the subclass of upper-triangular phase-type distributions.

scholarly journals Ruin Probability for Stochastic Flows of Financial Contract under Phase-Type Distribution

Risks ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 53
Author(s):  
Franck Adékambi ◽  
Kokou Essiomle
Keyword(s):  
Ruin Probability ◽  
Upper And Lower Bounds ◽  
Ruin Probabilities ◽  
Adjustment Coefficient ◽  
Phase Type Distribution ◽  
Type Distribution ◽  
Andersen Model ◽  
Phase Type ◽  
Financial Contract ◽  
The Impact

This paper examines the impact of the parameters of the distribution of the time at which a bank’s client defaults on their obligated payments, on the Lundberg adjustment coefficient, the upper and lower bounds of the ruin probability. We study the corresponding ruin probability on the assumption of (i) a phase-type distribution for the time at which default occurs and (ii) an embedding of the stochastic cash flow or the reserves of the bank to the Sparre Andersen model. The exact analytical expression for the ruin probability is not tractable under these assumptions, so Cramér Lundberg bounds types are obtained for the ruin probabilities with concomitant explicit equations for the calculation of the adjustment coefficient. To add some numerical flavour to our results, we provide some numerical illustrations.

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