Ageing properties of certain dependent geometric sums

1992 ◽  
Vol 29 (3) ◽  
pp. 655-666 ◽  
Author(s):  
Antal Kováts ◽  
Tamás F. Móri
Keyword(s):  
Failure Time ◽  
Random Variables ◽  
Working Time ◽  
System Failure ◽  
Random Sum ◽  
Unit System ◽  
Repair Facility ◽  
Geometric Sums ◽  
First Time ◽  
Operating Unit

We study some distribution properties of a random sum of i.i.d. non-negative random variables, where the number of terms is geometrically distributed and not independent of the summands. The results are applied to the system failure time of a one-unit system with a single spare and repair facility. In such a system when the operating unit fails it is immediately replaced by the spare and sent to the repair facility. The system continues operating until the first time when the failed unit has not yet been repaired by the failure of the operating unit. Certain ageing properties such as NBU, NWU, NBUE, NWUE, HNBUE, HNWUE, L+ and L– are shown to be inheritable from the working time of the operating unit to the system lifetime.

Ageing properties of certain dependent geometric sums

1992 ◽  
Vol 29 (03) ◽  
pp. 655-666 ◽  
Author(s):  
Antal Kováts ◽  
Tamás F. Móri
Keyword(s):  
Failure Time ◽  
Random Variables ◽  
Working Time ◽  
System Failure ◽  
Random Sum ◽  
Unit System ◽  
Repair Facility ◽  
Geometric Sums ◽  
First Time ◽  
Operating Unit

We study some distribution properties of a random sum of i.i.d. non-negative random variables, where the number of terms is geometrically distributed and not independent of the summands. The results are applied to the system failure time of a one-unit system with a single spare and repair facility. In such a system when the operating unit fails it is immediately replaced by the spare and sent to the repair facility. The system continues operating until the first time when the failed unit has not yet been repaired by the failure of the operating unit. Certain ageing properties such as NBU, NWU, NBUE, NWUE, HNBUE, HNWUE, L+ and L– are shown to be inheritable from the working time of the operating unit to the system lifetime.

A class of correlated cumulative shock models

1985 ◽  
Vol 17 (2) ◽  
pp. 347-366 ◽  
Author(s):  
Ushio Sumita ◽  
J. George Shanthikumar
Keyword(s):  
Failure Time ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Random Variables ◽  
System Failure ◽  
Shock Models ◽  
The Times ◽  
New Better Than Used ◽  
Better Than

In this paper we define and analyze a class of cumulative shock models associated with a bivariate sequence {Xn, Yn}∞n=0 of correlated random variables. The {Xn} denote the sizes of the shocks and the {Yn} denote the times between successive shocks. The system fails when the cumulative magnitude of the shocks exceeds a prespecified level z. Two models, depending on whether the size of the nth shock is correlated with the length of the interval since the last shock or with the length of the succeeding interval until the next shock, are considered. Various transform results and asymptotic properties of the system failure time are obtained. Further, sufficient conditions are established under which system failure time is new better than used, new better than used in expectation, and harmonic new better than used in expectation.

A class of correlated cumulative shock models

1985 ◽  
Vol 17 (02) ◽  
pp. 347-366 ◽  
Author(s):  
Ushio Sumita ◽  
J. George Shanthikumar
Keyword(s):  
Failure Time ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Random Variables ◽  
System Failure ◽  
Shock Models ◽  
The Times ◽  
New Better Than Used ◽  
Better Than

In this paper we define and analyze a class of cumulative shock models associated with a bivariate sequence {Xn , Yn }∞ n =0 of correlated random variables. The {Xn } denote the sizes of the shocks and the {Yn } denote the times between successive shocks. The system fails when the cumulative magnitude of the shocks exceeds a prespecified level z. Two models, depending on whether the size of the nth shock is correlated with the length of the interval since the last shock or with the length of the succeeding interval until the next shock, are considered. Various transform results and asymptotic properties of the system failure time are obtained. Further, sufficient conditions are established under which system failure time is new better than used, new better than used in expectation, and harmonic new better than used in expectation.

EVALUATION OF THE RELIABILITY OF COMPLEX TECHNICAL RECONFIGURATION SYSTEMS BY STATISTICAL MODELING METHOD

2019 ◽  
pp. 84-88
Author(s):  
A.Yu. Kulakov
Keyword(s):  
Normal Distribution ◽  
Statistical Modeling ◽  
Quantitative Estimate ◽  
Failure Time ◽  
Modeling Method ◽  
System Failure ◽  
Similar System ◽  
Orbit Control ◽  
Reliability Characteristics ◽  
Complex Technical System

Goal. Assess the reliability of a complex technical system with periodic reconfiguration and compare the results obtained a similar system, but without reconfiguration. Materials and methods. In this article uses the method of statistical modeling (Monte Carlo) to assess the reliability of complex system. We using the normal and exponential distribution of failure time for modeling failures of system elements. Reconfiguration algorithm is the algorithm proposed for the attitude and orbit control system of spacecraft. Results. A computer program has been developed for assessing reliability on the basis of a statistical modeling method, which makes it possible to evaluate systems of varying complexity with exponential and normal distribution, as well as with and without periodic reconfiguration. A quantitative estimate of the reliability as a function of the probability of system failure is obtained. Conclusion. It has been demonstrated that a system with reconfiguration has the best reliability characteristics, both in the case of exponential and normal distribution of failures.

A note on the Probability that first component fails before the second component

2016 ◽  
Vol 5 (2) ◽  
Author(s):  
Ehtesham Husain ◽  
Masood ul Haq
Keyword(s):  
Probability Distribution ◽  
Failure Time ◽  
Random Variables ◽  
Life Time ◽  
Life Testing ◽  
Human Organs

<p><span>The reliability (unreliability) and life testing are important topics in the field of engineering, electronic, <span>medicine, economic and many more, where we are interested in, life of components, human organs, <span>subsystem and system. Statistically, a probability distribution failure time (life time) of a certain form is <span>usually assumed to give reliability of a component for a system for each time t. Some well known <span>parametric life time models (T ≥ 0) are Exponential, Weibull, Inverse Weibull, Gamma, Lognormal, <span>normal ( T&gt;0 ; left truncated ) etc. </span></span></span></span></span></span></p><p><span><span><span><span><span><span><span>In this paper we consider a system that, has two components with independent but non-identical life time <span>probabilities explained by two distinct random variables say T<span>1 <span>and T<span>2 <span>, where T<span>1 <span>has a constant hazard <span>rate and T<span>2 <span>has an increasing hazard respectively </span></span></span></span></span></span></span></span><br /><br class="Apple-interchange-newline" /></span></span></span></span></span></span></span></span></span></p>

scholarly journals EWTD: maintaining the pressure

2009 ◽  
Vol 91 (8) ◽  
pp. 258-259 ◽  
Author(s):  
John Black
Keyword(s):  
Medical Association ◽  
Working Time ◽  
European Working Time Directive ◽  
News Item ◽  
Junior Doctors ◽  
Public Concern ◽  
European Working ◽  
Working Time Directive ◽  
First Time ◽  
Radio Programme

The ill-judged introduction of the 48-hour week demanded by the European Working Time Directive (EWTD) was the first news item on every television and radio programme on Saturday 1 August, indicating just how successful the College has been in raising public concern. It was encouraging that the British Medical Association (BMA) speakers were for the first time expressing serious anxiety, largely about the effects on training and about pressure being put on junior doctors to falsify their hours returns. All conversions to the cause are welcome, however late in the day. It is disappointing that the BMA is not yet stressing the dangers to patients, which they are surely hearing about from their members working in the acute specialties.

The tail behaviour of a random sum of subexponential random variables and vectors

Extremes ◽  
2007 ◽  
Vol 10 (1-2) ◽  
pp. 21-39 ◽  
Author(s):  
D. J. Daley ◽  
Edward Omey ◽  
Rein Vesilo
Keyword(s):  
Random Variables ◽  
Random Sum ◽  
Tail Behaviour

scholarly journals Uniform renewal theory with applications to expansions of random geometric sums

2007 ◽  
Vol 39 (4) ◽  
pp. 1070-1097 ◽  
Author(s):  
J. Blanchet ◽  
P. Glynn
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Renewal Theory ◽  
Random Variable ◽  
Higher Order ◽  
Order Correction ◽  
Diffusion Approximations ◽  
Geometric Sums ◽  
Correction Terms

Consider a sequence X = (Xn: n ≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variable M with parameter p. The random variable SM = X1 + ∙ ∙ ∙ + XM is called a geometric sum. In this paper we obtain asymptotic expansions for the distribution of SM as p ↘ 0. If EX1 > 0, the asymptotic expansion is developed in powers of p and it provides higher-order correction terms to Renyi's theorem, which states that P(pSM > x) ≈ exp(-x/EX1). Conversely, if EX1 = 0 then the expansion is given in powers of √p. We apply the results to obtain corrected diffusion approximations for the M/G/1 queue. These expansions follow in a unified way as a consequence of new uniform renewal theory results that are also developed in this paper.

scholarly journals Failure-Correlated Opportunity-based Age Replacement Models

2019 ◽  
Vol 27 (02) ◽  
pp. 2040008
Author(s):  
Tadashi Dohi ◽  
Hiroyuki Okamura
Keyword(s):  
Arrival Time ◽  
Failure Time ◽  
Unit System ◽  
Hazard Gradient ◽  
Age Replacement ◽  
The Cost ◽  
Bivariate Copula ◽  
Bivariate Probability ◽  
Vector Valued ◽  
Replacement Time

In this paper, we extend the existing opportunity-based age replacement policies by taking account of dependency between the failure time and the arrival time of a replacement opportunity for one-unit system. Based on the bivariate probability distribution function of the failure time and the arrival time of the opportunity, we focus on two opportunity-based age replacement problems and characterize the cost-optimal age replacement policies which minimize the relevant expected costs, with the hazard gradient, which is a vector-valued bivariate hazard rate. Through numerical examples with the Farlie–Gumbel–Morgenstern bivariate copula and the Gaussian bivariate copula having the general marginal distributions, we investigate the dependence of correlation between the failure time and the opportunistic replacement time on the age replacement policies.

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