Approximations for the repairman problem with two repair facilities, II: Spares

1974 ◽  
Vol 6 (1) ◽  
pp. 147-158 ◽  
Author(s):  
Donald L. Iglehart ◽  
Austin J. Lemoine
Keyword(s):  
Markov Chain ◽  
Limit Theorems ◽  
Failure Time ◽  
Failure Time Distribution ◽  
Server Queue ◽  
Finite State ◽  
Repairman Problem ◽  
Repair Facility ◽  
Operating Units

The model considered here consists of n operating units which are subject to stochastic failure according to an exponential failure time distribution. These operating units are backed up by mn spare units. Failures can be of two types. With probability p (q) a failure is of type 1(2) and is sent to repair facility 1(2) for repair. Repair facility 1(2) operates as a -server queue with exponential repair times having parameter μ1 (μ2). The number of units waiting for or undergoing repair at each of the two facilities is a continuous parameter Markov chain with finite state space. This paper derives limit theorems for the stationary distribution of this Markov chain as n becomes large under the assumption that and mn grow linearly with n. These limit theorems give very useful approximations, in terms of the seven parameters characterizing the model, to a distribution that would be difficult to calculate in practice.

Approximations for the repairman problem with two repair facilities, II: Spares

1974 ◽  
Vol 6 (01) ◽  
pp. 147-158 ◽  
Author(s):  
Donald L. Iglehart ◽  
Austin J. Lemoine
Keyword(s):  
Markov Chain ◽  
Limit Theorems ◽  
Failure Time ◽  
Failure Time Distribution ◽  
Server Queue ◽  
Finite State ◽  
Repairman Problem ◽  
Repair Facility ◽  
Operating Units

The model considered here consists of n operating units which are subject to stochastic failure according to an exponential failure time distribution. These operating units are backed up by mn spare units. Failures can be of two types. With probability p (q) a failure is of type 1(2) and is sent to repair facility 1(2) for repair. Repair facility 1(2) operates as a -server queue with exponential repair times having parameter μ 1 (μ 2). The number of units waiting for or undergoing repair at each of the two facilities is a continuous parameter Markov chain with finite state space. This paper derives limit theorems for the stationary distribution of this Markov chain as n becomes large under the assumption that and mn grow linearly with n. These limit theorems give very useful approximations, in terms of the seven parameters characterizing the model, to a distribution that would be difficult to calculate in practice.

Approximations for the repairman problem with two repair facilities, I: No spares

1973 ◽  
Vol 5 (3) ◽  
pp. 595-613 ◽  
Author(s):  
Donald L. Iglehart ◽  
Austin J. Lemoine
Keyword(s):  
Markov Chain ◽  
Limit Theorems ◽  
Failure Time ◽  
Failure Time Distribution ◽  
Continuous Parameter ◽  
Server Queue ◽  
Finite State ◽  
Repairman Problem ◽  
Repair Facility

The model considered here consists of n operating units which are subject to stochastic failure according to an exponential failure time distribution. Failures can be of two types. With probability p(q) a failure is of type 1(2) and is sent to repair facility 1(2) for repair. Repair facility 1(2) operates as a -server queue with exponential repair times having parameter μ1 (μ2). The number of units waiting for or undergoing repair at each of the two facilities is a continuous-parameter Markov chain with finite state space. This paper derives limit theorems for the stationary distribution of this Markov chain as n becomes large under the assumption that both and grow linearly with n. These limit theorems give very useful approximations, in terms of the six parameters characterizing the model, to a distribution that would be difficult to use in practice.

Approximations for the repairman problem with two repair facilities, I: No spares

1973 ◽  
Vol 5 (03) ◽  
pp. 595-613 ◽  
Author(s):  
Donald L. Iglehart ◽  
Austin J. Lemoine
Keyword(s):  
Markov Chain ◽  
Limit Theorems ◽  
Failure Time ◽  
Failure Time Distribution ◽  
Continuous Parameter ◽  
Server Queue ◽  
Finite State ◽  
Repairman Problem ◽  
Repair Facility

The model considered here consists of n operating units which are subject to stochastic failure according to an exponential failure time distribution. Failures can be of two types. With probability p(q) a failure is of type 1(2) and is sent to repair facility 1(2) for repair. Repair facility 1(2) operates as a -server queue with exponential repair times having parameter μ 1 (μ 2). The number of units waiting for or undergoing repair at each of the two facilities is a continuous-parameter Markov chain with finite state space. This paper derives limit theorems for the stationary distribution of this Markov chain as n becomes large under the assumption that both and grow linearly with n. These limit theorems give very useful approximations, in terms of the six parameters characterizing the model, to a distribution that would be difficult to use in practice.

Optimizing replacement time for mining shovel teeth using reliability analysis and Markov chain Monte Carlo simulation

2018 ◽  
Vol 35 (10) ◽  
pp. 2388-2402
Author(s):  
Dilip Sembakutti ◽  
Aldin Ardian ◽  
Mustafa Kumral ◽  
Agus Pulung Sasmito
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Failure Time ◽  
Content Type ◽  
Failure Time Distribution ◽  
Replacement Strategy ◽  
Risk Quantification ◽  
Replacement Time

Purpose The purpose of this paper is twofold: an approach is proposed to determine the optimum replacement time for shovel teeth; and a risk-quantification approached is developed to derive a confidence interval for replacement time. Design/methodology/approach The risk-quantification approach is based on a combination of Monte Carlo simulation and Markov chain. Monte Carlo simulation whereby the wear of shovel teeth is probabilistically monitored over time is used. Findings Results show that a proper replacement strategy has potential to increase operation efficiency and the uncertainties associated with this strategy can be managed. Research limitations/implications The failure time distribution of a tooth is assumed to remain “identically distributed and independent.” Planned tooth replacements are always done when the shovel is not in operation (e.g. between a shift change). Practical implications The proposed approach can be effectively used to determine a replacement strategy, along with the level of confidence level, for preventive maintenance planning. Originality/value The originality of the paper rests on developing a novel approach to monitor wear on mining shovels probabilistically. Uncertainty associated with production targets is quantified.

On the age distribution of a Markov chain

1978 ◽  
Vol 15 (1) ◽  
pp. 65-77 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Limit Theorems ◽  
Age Distribution ◽  
Weak Limit ◽  
Absorbing State ◽  
Absorbing Markov Chain ◽  
Continuous Random Walk ◽  
A Chain ◽  
Watson Process

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain, obtained by restarting the original chain at a fixed state after each absorption. The limiting age, A(j), is the weak limit of the time given Xn = j (n → ∞).A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems for A (J) (J → ∞) are given for these examples.

A QUEUE WITH POSITIVE AND NEGATIVE ARRIVALS GOVERNED BY A MARKOV CHAIN

2003 ◽  
Vol 17 (4) ◽  
pp. 487-501 ◽  
Author(s):  
Yang Woo Shin ◽  
Bong Dae Choi
Keyword(s):  
Markov Chain ◽  
Transient State ◽  
Single Server ◽  
Negative Customers ◽  
Service Distribution ◽  
Finite State ◽  
Sojourn Time Distribution ◽  
Absorbing States ◽  
Negative Arrivals ◽  
Selection Of

We consider a single-server queue with exponential service time and two types of arrivals: positive and negative. Positive customers are regular ones who form a queue and a negative arrival has the effect of removing a positive customer in the system. In many applications, it might be more appropriate to assume the dependence between positive arrival and negative arrival. In order to reflect the dependence, we assume that the positive arrivals and negative arrivals are governed by a finite-state Markov chain with two absorbing states, say 0 and 0′. The epoch of absorption to the states 0 and 0′ corresponds to an arrival of positive and negative customers, respectively. The Markov chain is then instantly restarted in a transient state, where the selection of the new state is allowed to depend on the state from which absorption occurred.The Laplace–Stieltjes transforms (LSTs) of the sojourn time distribution of a customer, jointly with the probability that the customer completes his service without being removed, are derived under the combinations of service disciplines FCFS and LCFS and the removal strategies RCE and RCH. The service distribution of phase type is also considered.

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