General shock models associated with correlated renewal sequences

1983 ◽  
Vol 20 (3) ◽  
pp. 600-614 ◽  
Author(s):  
J. G. Shanthikumar ◽  
U. Sumita
Keyword(s):  
Limit Theorem ◽  
Renewal Process ◽  
Threshold Level ◽  
Numerical Results ◽  
Shock Model ◽  
Failure Times ◽  
Clearing System ◽  
Shock Models ◽  
Subsequent Interval

In this paper we define and analyze a general shock model associated with a correlated pair (Xn, Yn) of renewal sequences, where the system fails when the magnitude of a shock exceeds (or falls below) a prespecified threshold level. Two models, depending on whether the nth shock Xn is correlated to the length Yn of the interval since the last shock, or to the length Yn of the subsequent interval until the next shock, are considered. The transform results, an exponential limit theorem, and properties of the associated renewal process of the failure times are obtained. An application in a stochastic clearing system with numerical results is also given.

General shock models associated with correlated renewal sequences

1983 ◽  
Vol 20 (03) ◽  
pp. 600-614 ◽  
Author(s):  
J. G. Shanthikumar ◽  
U. Sumita
Keyword(s):  
Limit Theorem ◽  
Renewal Process ◽  
Threshold Level ◽  
Numerical Results ◽  
Shock Model ◽  
Failure Times ◽  
Clearing System ◽  
Shock Models ◽  
Subsequent Interval

In this paper we define and analyze a general shock model associated with a correlated pair (Xn, Yn ) of renewal sequences, where the system fails when the magnitude of a shock exceeds (or falls below) a prespecified threshold level. Two models, depending on whether the nth shock Xn is correlated to the length Yn of the interval since the last shock, or to the length Yn of the subsequent interval until the next shock, are considered. The transform results, an exponential limit theorem, and properties of the associated renewal process of the failure times are obtained. An application in a stochastic clearing system with numerical results is also given.

Distribution properties of the system failure time in a general shock model

1984 ◽  
Vol 16 (2) ◽  
pp. 363-377 ◽  
Author(s):  
J. G. Shanthikumar ◽  
Ushio Sumita
Keyword(s):  
Failure Time ◽  
Sufficient Conditions ◽  
System Failure ◽  
Shock Model ◽  
Completely Monotone ◽  
Shock Models ◽  
Subsequent Interval ◽  
New Better Than Used ◽  
Better Than

In this paper we study some distribution properties of the system failure time in general shock models associated with correlated renewal sequences (Xn, Yn) . Two models, depending on whether the magnitude of the nth shock Xn is correlated to the length Yn of the interval since the last shock, or to the length of the subsequent interval to the next shock, are considered. Sufficient conditions under which the system failure time is completely monotone, new better than used, new better than used in expectation, and harmonic new better than used in expectation are given for these two models.

Distribution properties of the system failure time in a general shock model

1984 ◽  
Vol 16 (02) ◽  
pp. 363-377 ◽  
Author(s):  
J. G. Shanthikumar ◽  
Ushio Sumita
Keyword(s):  
Failure Time ◽  
Sufficient Conditions ◽  
System Failure ◽  
Shock Model ◽  
Completely Monotone ◽  
Shock Models ◽  
Subsequent Interval ◽  
New Better Than Used ◽  
Better Than

In this paper we study some distribution properties of the system failure time in general shock models associated with correlated renewal sequences (X n, Y n) . Two models, depending on whether the magnitude of the nth shock X n is correlated to the length Y n of the interval since the last shock, or to the length of the subsequent interval to the next shock, are considered. Sufficient conditions under which the system failure time is completely monotone, new better than used, new better than used in expectation, and harmonic new better than used in expectation are given for these two models.

scholarly journals Asymptotic results for the first and second moments and numerical computations in discrete-time bulk-renewal process

2019 ◽  
Vol 29 (1) ◽  
pp. 135-144
Author(s):  
James Kim ◽  
Mohan Chaudhry ◽  
Abdalla Mansur
Keyword(s):  
Discrete Time ◽  
Renewal Process ◽  
Generating Functions ◽  
Continuous Time ◽  
Constant Term ◽  
Numerical Results ◽  
Asymptotic Results ◽  
Numerical Computations ◽  
Second Moments ◽  
Simplified Solution

This paper introduces a simplified solution to determine the asymptotic results for the renewal density. It also offers the asymptotic results for the first and second moments of the number of renewals for the discrete-time bulk-renewal process. The methodology adopted makes this study distinguishable compared to those previously published where the constant term in the second moment is generated. In similar studies published in the literature, the constant term is either missing or not clear how it was obtained. The problem was partially solved in the study by Chaudhry and Fisher where they provided a asymptotic results for the non-bulk renewal density and for both the first and second moments using the generating functions. The objective of this work is to extend their results to the bulk-renewal process in discrete-time, including some numerical results, give an elegant derivation of the asymptotic results and derive continuous-time results as a limit of the discrete-time results.

A general shock model for a reliability system

2000 ◽  
Vol 37 (04) ◽  
pp. 925-935 ◽  
Author(s):  
Georgios Skoulakis
Keyword(s):  
Distribution Function ◽  
Point Process ◽  
Renewal Process ◽  
General System ◽  
System Structure ◽  
Random Numbers ◽  
System Failure ◽  
Shock Model ◽  
Stieltjes Transform ◽  
Special Cases

We study a reliability system subject to shocks generated by a renewal point process. When a shock occurs, components fail independently of each other with equal probabilities that are random numbers drawn from a distribution that may differ from shock to shock. We first consider the case of a parallel system and derive closed expressions for the Laplace-Stieltjes transform and the expectation of the time to system failure and for its density in the case that the distribution function of the renewal process possesses a density. We then treat a more general system structure, which has some very important special cases, such as k-out-of-n:F systems, and derive analogous formulae.

Failure distributions of shock models

1980 ◽  
Vol 17 (03) ◽  
pp. 745-752 ◽  
Author(s):  
Gary Gottlieb
Keyword(s):  
Failure Rate ◽  
Sufficient Conditions ◽  
Cumulative Damage ◽  
Shock Model ◽  
Life Distribution ◽  
Increasing Failure Rate ◽  
Shock Models ◽  
Damage Process

A single device shock model is studied. The device is subject to some damage process. Under the assumption that as the cumulative damage increases, the probability that any additional damage will cause failure increases, we find sufficient conditions on the shocking process so that the life distribution will be increasing failure rate.

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.

A conservation property for general GI/G/1 queues with an application to tandem queues

1979 ◽  
Vol 11 (03) ◽  
pp. 660-672 ◽  
Author(s):  
E. Nummelin
Keyword(s):  
Limit Theorem ◽  
Total Variation ◽  
Renewal Process ◽  
Markov Renewal Process ◽  
Output Process ◽  
Tandem Queues ◽  
Tandem Queue ◽  
Input Process ◽  
Conservation Property ◽  
Positive Recurrent

We show that, if the input process of a generalGI/G/1 queue is a positive recurrent Markov renewal process then the output process, too, is a positive recurrent Markov renewal process (the conservation property). As an application we consider a general tandem queue and prove a total variation limit theorem for the associated waiting and service times.

A general shock model for a reliability system

2000 ◽  
Vol 37 (4) ◽  
pp. 925-935 ◽  
Author(s):  
Georgios Skoulakis
Keyword(s):  
Distribution Function ◽  
Point Process ◽  
Renewal Process ◽  
General System ◽  
System Structure ◽  
Random Numbers ◽  
System Failure ◽  
Shock Model ◽  
Stieltjes Transform ◽  
Special Cases

We study a reliability system subject to shocks generated by a renewal point process. When a shock occurs, components fail independently of each other with equal probabilities that are random numbers drawn from a distribution that may differ from shock to shock. We first consider the case of a parallel system and derive closed expressions for the Laplace-Stieltjes transform and the expectation of the time to system failure and for its density in the case that the distribution function of the renewal process possesses a density. We then treat a more general system structure, which has some very important special cases, such as k-out-of-n:F systems, and derive analogous formulae.

Semi-Markov shock models with additive damage

1986 ◽  
Vol 18 (3) ◽  
pp. 772-790 ◽  
Author(s):  
M. J. M. Posner ◽  
D. Zuckerman
Keyword(s):  
Sufficient Conditions ◽  
Control Limit ◽  
Shock Model ◽  
Markov Modelling ◽  
Shock Models ◽  
Replacement Model ◽  
Computational Aspects ◽  
Replacement Problem

We examine a replacement model for a semi-Markov shock model with additive damage. Sufficient conditions are given for the optimality of control limit policies. The paper generalizes and unifies previous research in the area.In addition, we investigate in detail the practical modelling and computational aspects of the replacement problem using a semi-Markov modelling structure.

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