Optimal replacement under a general failure model

1978 ◽  
Vol 10 (2) ◽  
pp. 431-451 ◽  
Author(s):  
Bo Bergman
Keyword(s):  
Threshold Value ◽  
Rate Function ◽  
The State ◽  
Failure Rate Function ◽  
State Variable ◽  
Optimal Replacement ◽  
State Dependent ◽  
Long Run ◽  
Special Cases ◽  
Dependent Failure

Replacement policies based on measurements of some increasing state variable, e.g. wear, accumulated damage or accumulated stress, are studied in this paper. It is assumed that the state measurements may be regarded as realizations of some stochastic process and that the proneness to failure of an active unit may be described by an increasing state-dependent failure rate function. Average long-run cost per unit time is considered. The optimal replacement rule is shown to be a control limit rule, i.e. it is optimal to replace either at failure or when the state variable has reached some threshold value, whichever occurs first. The optimal rule is determined. Some generalizations and special cases are given.

Optimal replacement under a general failure model

1978 ◽  
Vol 10 (02) ◽  
pp. 431-451 ◽  
Author(s):  
Bo Bergman
Keyword(s):  
Threshold Value ◽  
Rate Function ◽  
The State ◽  
Failure Rate Function ◽  
State Variable ◽  
Optimal Replacement ◽  
State Dependent ◽  
Long Run ◽  
Special Cases ◽  
Dependent Failure

Replacement policies based on measurements of some increasing state variable, e.g. wear, accumulated damage or accumulated stress, are studied in this paper. It is assumed that the state measurements may be regarded as realizations of some stochastic process and that the proneness to failure of an active unit may be described by an increasing state-dependent failure rate function. Average long-run cost per unit time is considered. The optimal replacement rule is shown to be a control limit rule, i.e. it is optimal to replace either at failure or when the state variable has reached some threshold value, whichever occurs first. The optimal rule is determined. Some generalizations and special cases are given.

scholarly journals A Markovian growth-collapse model

2006 ◽  
Vol 38 (1) ◽  
pp. 221-243 ◽  
Author(s):  
Onno Boxma ◽  
David Perry ◽  
Wolfgang Stadje ◽  
Shelemyahu Zacks
Keyword(s):  
General Theory ◽  
Rate Function ◽  
Current Level ◽  
Hitting Times ◽  
Dependent Rate ◽  
State Dependent ◽  
Long Run ◽  
Special Cases ◽  
Collapse Model ◽  
Markov Modulated

We consider growth-collapse processes (GCPs) that grow linearly between random partial collapse times, at which they jump down according to some distribution depending on their current level. The jump occurrences are governed by a state-dependent rate function r(x). We deal with the stationary distribution of such a GCP, (Xt)t≥0, and the distributions of the hitting times Ta = inf{t ≥ 0 : Xt = a}, a > 0. After presenting the general theory of these GCPs, several important special cases are studied. We also take a brief look at the Markov-modulated case. In particular, we present a method of computing the distribution of min[Ta, σ] in this case (where σ is the time of the first jump), and apply it to determine the long-run average cost of running a certain Markov-modulated disaster-ridden system.

scholarly journals A Markovian growth-collapse model

2006 ◽  
Vol 38 (01) ◽  
pp. 221-243 ◽  
Author(s):  
Onno Boxma ◽  
David Perry ◽  
Wolfgang Stadje ◽  
Shelemyahu Zacks
Keyword(s):  
General Theory ◽  
Rate Function ◽  
Current Level ◽  
Hitting Times ◽  
Dependent Rate ◽  
State Dependent ◽  
Long Run ◽  
Special Cases ◽  
Collapse Model ◽  
Markov Modulated

We consider growth-collapse processes (GCPs) that grow linearly between random partial collapse times, at which they jump down according to some distribution depending on their current level. The jump occurrences are governed by a state-dependent rate functionr(x). We deal with the stationary distribution of such a GCP, (Xt)t≥0, and the distributions of the hitting timesTa= inf{t≥ 0 :Xt=a},a> 0. After presenting the general theory of these GCPs, several important special cases are studied. We also take a brief look at the Markov-modulated case. In particular, we present a method of computing the distribution of min[Ta, σ] in this case (where σ is the time of the first jump), and apply it to determine the long-run average cost of running a certain Markov-modulated disaster-ridden system.

scholarly journals Generalising Exponential Distributions Using an Extended Marshall–Olkin Procedure

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 464
Author(s):  
Victoriano García ◽  
María Martel-Escobar ◽  
F.J. Vázquez-Polo
Keyword(s):  
Failure Rate ◽  
Real Data ◽  
Rate Function ◽  
Exponential Distributions ◽  
Symmetric Distributions ◽  
Failure Rate Function ◽  
Numerical Examples ◽  
Special Cases ◽  
The Common ◽  
Family Of Distributions

This paper presents a three-parameter family of distributions which includes the common exponential and the Marshall–Olkin exponential as special cases. This distribution exhibits a monotone failure rate function, which makes it appealing for practitioners interested in reliability, and means it can be included in the catalogue of appropriate non-symmetric distributions to model these issues, such as the gamma and Weibull three-parameter families. Given the lack of symmetry of this kind of distribution, various statistical and reliability properties of this model are examined. Numerical examples based on real data reflect the suitable behaviour of this distribution for modelling purposes.

A further extension of the generalized burn-in model

2003 ◽  
Vol 40 (1) ◽  
pp. 264-270 ◽  
Author(s):  
Ji Hwan Cha
Keyword(s):  
Probability Function ◽  
Rate Function ◽  
Replacement Policy ◽  
Time Dependent ◽  
Type Ii ◽  
Failure Rate Function ◽  
Mild Conditions ◽  
Optimal Replacement ◽  
Replacement Model ◽  
Optimal Replacement Policy

In this paper, the generalized burn-in and replacement model considered by Cha (2001) is further extended to the case in which the probability of Type II failure is time dependent. Two burn-in procedures are considered and they are compared in cases when both the procedures are applicable. Under some mild conditions on the failure rate function r(t) and the Type II failure probability function p(t), the problems of determining optimal burn-in time and optimal replacement policy are considered.

A further extension of the generalized burn-in model

2003 ◽  
Vol 40 (01) ◽  
pp. 264-270 ◽  
Author(s):  
Ji Hwan Cha
Keyword(s):  
Probability Function ◽  
Rate Function ◽  
Replacement Policy ◽  
Time Dependent ◽  
Type Ii ◽  
Failure Rate Function ◽  
Mild Conditions ◽  
Optimal Replacement ◽  
Replacement Model ◽  
Optimal Replacement Policy

In this paper, the generalized burn-in and replacement model considered by Cha (2001) is further extended to the case in which the probability of Type II failure is time dependent. Two burn-in procedures are considered and they are compared in cases when both the procedures are applicable. Under some mild conditions on the failure rate function r(t) and the Type II failure probability function p(t), the problems of determining optimal burn-in time and optimal replacement policy are considered.

Government spending multipliers over business cycle

2019 ◽  
Vol 11 (1) ◽  
pp. 18-29 ◽  
Author(s):  
Naser Yenus Nuru
Keyword(s):  
Fiscal Policy ◽  
Government Spending ◽  
Developing Economies ◽  
Threshold Value ◽  
Economic Cycle ◽  
Content Type ◽  
Vector Autoregressive ◽  
State Dependent ◽  
Long Run ◽  
Spending Shocks

Purpose The purpose of this paper is to show the asymmetric effects of government spending shocks for South Africa over the period 1960Q1–2014Q2. Design/methodology/approach A threshold vector autoregressive model that allows parameters to switch according to whether a threshold variable crosses an estimated threshold is employed to address the objective of this paper. The threshold value is determined endogenously using Hansen (1996) test. Generalized impulse responses introduced by Koop et al. (1996) are used to study the effects of government spending shocks on growth depending on their size, sign and timing with respect to the economic cycle. The author also uses a Cholesky decomposition identification scheme in order to identify discretionary government spending shocks in the non-linear model. Findings The empirical findings support the state-dependent effects of fiscal policy. In particular, the effects of 1 or 2 standard deviations expansionary or contractionary government spending shock on output are very small both on impact and in the long run; and a bit larger in downturns but has only a very limited effect or no effect in times of expansion. This result gives support to the evidence in the recent literature that fiscal policy in developing countries is overwhelmingly procyclical. Originality/value It adds to the scarce empirical fiscal literature of the South African economy in particular and developing economies in general by allowing non-linearities to estimate the effect of government spending shocks over economic cycle.

scholarly journals A New Flexible Generalized Lindley Model: Properties, Estimation and Applications

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1678
Author(s):  
Abdulrahman Abouammoh ◽  
Mohamed Kayid
Keyword(s):  
Maximum Likelihood ◽  
Likelihood Estimation ◽  
Real Data ◽  
Moment Generating Function ◽  
Rate Function ◽  
Failure Rate Function ◽  
Right Censored Data ◽  
Special Cases ◽  
The Moment ◽  
The Right

A new method for generalizing the Lindley distribution, by increasing the number of mixed models is presented formally. This generalized model, which is called the generalized Lindley of integer order, encompasses the exponential and the usual Lindley distributions as special cases when the order of the model is fixed to be one and two, respectively. The moments, the variance, the moment generating function, and the failure rate function of the initiated model are extracted. Estimation of the underlying parameters by the moment and the maximum likelihood methods are acquired. The maximum likelihood estimation for the right censored data has also been discussed. In a simulation running for various orders and censoring rates, efficiency of the maximum likelihood estimator has been explored. The introduced model has ultimately been fitted to two real data sets to emphasize its application.

State-dependent signalling in queueing networks

1994 ◽  
Vol 26 (02) ◽  
pp. 436-455 ◽  
Author(s):  
W. Henderson ◽  
B. S. Northcote ◽  
P. G. Taylor
Keyword(s):  
Queueing Networks ◽  
State Dependence ◽  
Product Form ◽  
Batch Size ◽  
Single Server ◽  
Negative Customers ◽  
Affect Control ◽  
State Dependent ◽  
Special Cases ◽  
Equilibrium Distributions

It has recently been shown that networks of queues with state-dependent movement of negative customers, and with state-independent triggering of customer movement have product-form equilibrium distributions. Triggers and negative customers are entities which, when arriving to a queue, force a single customer to be routed through the network or leave the network respectively. They are ‘signals' which affect/control network behaviour. The provision of state-dependent intensities introduces queues other than single-server queues into the network. This paper considers networks with state-dependent intensities in which signals can be either a trigger or a batch of negative customers (the batch size being determined by an arbitrary probability distribution). It is shown that such networks still have a product-form equilibrium distribution. Natural methods for state space truncation and for the inclusion of multiple customer types in the network can be viewed as special cases of this state dependence. A further generalisation allows for the possibility of signals building up at nodes.

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