Optimal time to repair a broken server

1989 ◽  
Vol 21 (2) ◽  
pp. 376-397 ◽  
Author(s):  
Awi Federgruen ◽  
Kut C. So
Keyword(s):  
Rate Function ◽  
Operating Costs ◽  
Optimal Time ◽  
Single Server ◽  
Maintenance Policy ◽  
Stationary Policy ◽  
Cost Rate ◽  
Long Run ◽  
Poisson Arrivals ◽  
Number Of Customers

We consider a single-server queueing system with Poisson arrivals and general service times. While the server is up, it is subject to breakdowns according to a Poisson process. When the server breaks down, we may either repair the server immediately or postpone the repair until some future point in time. The operating costs of the system include customer holding costs, repair costs and running costs. The objective is to find a corrective maintenance policy which minimizes the long-run average operating costs of the system. The problem is formulated as a semi-Markov decision process. Under some mild conditions on the repair time and service time distributions and the customer holding cost rate function, we prove that there exists an optimal stationary policy which is characterized by a single threshold parameter: a repair is initiated if and only if the number of customers in the system exceeds this threshold. We also show how the average cost under such policies may be computed and how an optimal policy may efficiently be determined.

Optimal time to repair a broken server

1989 ◽  
Vol 21 (02) ◽  
pp. 376-397 ◽  
Author(s):  
Awi Federgruen ◽  
Kut C. So
Keyword(s):  
Rate Function ◽  
Operating Costs ◽  
Optimal Time ◽  
Single Server ◽  
Maintenance Policy ◽  
Stationary Policy ◽  
Cost Rate ◽  
Long Run ◽  
Poisson Arrivals ◽  
Number Of Customers

We consider a single-server queueing system with Poisson arrivals and general service times. While the server is up, it is subject to breakdowns according to a Poisson process. When the server breaks down, we may either repair the server immediately or postpone the repair until some future point in time. The operating costs of the system include customer holding costs, repair costs and running costs. The objective is to find a corrective maintenance policy which minimizes the long-run average operating costs of the system. The problem is formulated as a semi-Markov decision process. Under some mild conditions on the repair time and service time distributions and the customer holding cost rate function, we prove that there exists an optimal stationary policy which is characterized by a single threshold parameter: a repair is initiated if and only if the number of customers in the system exceeds this threshold. We also show how the average cost under such policies may be computed and how an optimal policy may efficiently be determined.

Optimality of threshold policies in single-server queueing systems with server vacations

1991 ◽  
Vol 23 (02) ◽  
pp. 388-405 ◽  
Author(s):  
Awi Federgruen ◽  
Kut C. So
Keyword(s):  
Queueing Systems ◽  
Operating Cost ◽  
Rate Function ◽  
Batch Size ◽  
Critical Threshold ◽  
Single Server ◽  
Queue Size ◽  
Cost Rate ◽  
Holding Cost ◽  
Poisson Arrivals

In this paper we consider a class of single-server queueing systems with compound Poisson arrivals, in which, at service completion epochs, the server has the option of taking off for one or several vacations of random length. The cost structure consists of a holding cost rate specified by a general non-decreasing function of the queue size, fixed costs for initiating and terminating service, and a variable operating cost incurred for each unit of time that the system is in operation. We show under some weak conditions with respect to the holding cost rate function and the service time, vacation time and arrival batch size distributions that it is either optimal among all feasible (stationary and non-stationary) policies never to take a vacation, or it is optimal to take a vacation when the system empties out and to resume work when, upon completion of a vacation, the queue size is equal to or in excess of a critical threshold. These optimality results are generalized for several variants of this model.

Optimality of threshold policies in single-server queueing systems with server vacations

1991 ◽  
Vol 23 (2) ◽  
pp. 388-405 ◽  
Author(s):  
Awi Federgruen ◽  
Kut C. So
Keyword(s):  
Queueing Systems ◽  
Operating Cost ◽  
Rate Function ◽  
Batch Size ◽  
Critical Threshold ◽  
Single Server ◽  
Queue Size ◽  
Cost Rate ◽  
Holding Cost ◽  
Poisson Arrivals

In this paper we consider a class of single-server queueing systems with compound Poisson arrivals, in which, at service completion epochs, the server has the option of taking off for one or several vacations of random length. The cost structure consists of a holding cost rate specified by a general non-decreasing function of the queue size, fixed costs for initiating and terminating service, and a variable operating cost incurred for each unit of time that the system is in operation. We show under some weak conditions with respect to the holding cost rate function and the service time, vacation time and arrival batch size distributions that it is either optimal among all feasible (stationary and non-stationary) policies never to take a vacation, or it is optimal to take a vacation when the system empties out and to resume work when, upon completion of a vacation, the queue size is equal to or in excess of a critical threshold. These optimality results are generalized for several variants of this model.

Availability analysis and maintenance optimization for multiple failure mode systems considering imperfect repair

2021 ◽  
pp. 1748006X2110127
Author(s):  
Qingan Qiu ◽  
Baoliang Liu ◽  
Cong Lin ◽  
Jingjing Wang
Keyword(s):  
Failure Modes ◽  
Maintenance Cost ◽  
Imperfect Maintenance ◽  
Maintenance Policy ◽  
Cost Rate ◽  
Availability Analysis ◽  
Long Run ◽  
Optimal Maintenance ◽  
Maintenance Policies ◽  
Periodic Inspections

This paper studies the availability and optimal maintenance policies for systems subject to competing failure modes under continuous and periodic inspections. The repair time distribution and maintenance cost are both dependent on the failure modes. We investigate the instantaneous availability and the steady state availability of the system maintained through several imperfect repairs before a replacement is allowed. Analytical expressions for system availability under continuous and periodic inspections are derived respectively. The availability models are then utilized to obtain the optimal inspection and imperfect maintenance policy that minimizes the average long-run cost rate. A numerical example for Remote Power Feeding System is presented to demonstrate the application of the developed approach.

scholarly journals Large Deviations for the Single-Server Queue and the Reneging Paradox

2021 ◽  
Author(s):  
Rami Atar ◽  
Amarjit Budhiraja ◽  
Paul Dupuis ◽  
Ruoyu Wu
Keyword(s):  
Large Deviations ◽  
Decay Rate ◽  
Sample Path ◽  
Arrival Rate ◽  
Rate Function ◽  
Service Rate ◽  
Single Server ◽  
Large Deviations Principle ◽  
Long Run ◽  
The Individual

For the M/M/1+M model at the law-of-large-numbers scale, the long-run reneging count per unit time does not depend on the individual (i.e., per customer) reneging rate. This paradoxical statement has a simple proof. Less obvious is a large deviations analogue of this fact, stated as follows: the decay rate of the probability that the long-run reneging count per unit time is atypically large or atypically small does not depend on the individual reneging rate. In this paper, the sample path large deviations principle for the model is proved and the rate function is computed. Next, large time asymptotics for the reneging rate are studied for the case when the arrival rate exceeds the service rate. The key ingredient is a calculus of variations analysis of the variational problem associated with atypical reneging. A characterization of the aforementioned decay rate, given explicitly in terms of the arrival and service rate parameters of the model, is provided yielding a precise mathematical description of this paradoxical behavior.

Average Cost Semi-Markov Decision Processes and the Control of Queueing Systems

1989 ◽  
Vol 3 (2) ◽  
pp. 247-272 ◽  
Author(s):  
Linn I. Sennott
Keyword(s):  
Markov Decision Processes ◽  
Average Cost ◽  
Queueing Systems ◽  
Decision Processes ◽  
Single Server ◽  
Stationary Policy ◽  
Markov Decision ◽  
Optimal Stationary Policy ◽  
Poisson Arrivals ◽  
Action Spaces

Semi-Markov decision processes underlie the control of many queueing systems. In this paper, we deal with infinite state semi-Markov decision processes with nonnegative, unbounded costs and finite action sets. Axioms for the existence of an expected average cost optimal stationary policy are presented. These conditions generalize the work in Sennott [22] for Markov decision processes. Verifiable conditions for the axioms to hold are obtained. The theory is applied to control of the M/G/l queue with variable service parameter, with on-off server, and with batch processing, and to control of the G/M/m queue with variable arrival parameter and customer rejection. It is applied to a timesharing network of queues with a single server and finally to optimal routing of Poisson arrivals to parallel exponential servers. The final section extends the existence result to compact action spaces.

scholarly journals A Perishable Inventory System with Postponed Demands and Multiple Server Vacations

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
R. Jayaraman ◽  
B. Sivakumar ◽  
G. Arivarignan
Keyword(s):  
Steady State ◽  
Joint Probability ◽  
Life Time ◽  
Inventory System ◽  
Inventory Level ◽  
Joint Probability Distribution ◽  
Single Server ◽  
Cost Rate ◽  
Stochastic Inventory ◽  
Number Of Customers

A mathematical modelling of a continuous review stochastic inventory system with a single server is carried out in this work. We assume that demand time points form a Poisson process. The life time of each item is assumed to have exponential distribution. We assume(s,S)ordering policy to replenish stock with random lead time. The server goes for a vacation of an exponentially distributed duration at the time of stock depletion and may take subsequent vacation depending on the stock position. The customer who arrives during the stock-out period or during the server vacation is offered a choice of joining a pool which is of finite capacity or leaving the system. The demands in the pool are selected one by one by the server only when the inventory level is aboves, with interval time between any two successive selections distributed as exponential with parameter depending on the number of customers in the pool. The joint probability distribution of the inventory level and the number of customers in the pool is obtained in the steady-state case. Various system performance measures in the steady state are derived, and the long-run total expected cost rate is calculated.

scholarly journals Optimal Condition-Based Maintenance Strategy for Multi-Component Systems under Degradation Failures

Energies ◽  
2020 ◽  
Vol 13 (17) ◽  
pp. 4346
Author(s):  
Kui Wang ◽  
Chao Deng ◽  
Lili Ding
Keyword(s):  
Critical Level ◽  
Condition Based Maintenance ◽  
Maintenance Strategy ◽  
Maintenance Policy ◽  
Cost Rate ◽  
Optimal Monitoring ◽  
Long Run ◽  
Component Systems ◽  
Maintenance Decision ◽  
The Impact

This paper proposes a condition-based maintenance strategy for multi-component systems under degradation failures. The maintenance decision is based on the minimum long-run average cost rate (LACR) and the maximum residual useful lifetime (RUL), respectively. The aim of this paper is to determine the optimal monitoring interval and critical level for multi-component systems under different optimization objectives. A preventive maintenance (PM) is triggered when the degradation of component exceeds the corresponding critical level. Afterwards, the paper discusses the relationship between the critical level and the monitoring interval with regards to the LACR and RUL. Methods are also proposed to determine the optimal monitoring interval and the critical level under two decision models. Finally, the impact of maintenance decision variables on the LACR and RUL is discussed through a case study. A comparison with conventional maintenance policy shows an outstanding performance of the new model.

scholarly journals A Stochastic Failure Model with Dependent Competing Risks and its Applications to Condition-Based Maintenance

2015 ◽  
Vol 52 (2) ◽  
pp. 558-573 ◽  
Author(s):  
Ji Hwan Cha ◽  
Inma T. Castro
Keyword(s):  
Competing Risks ◽  
Common Source ◽  
Maintenance Strategy ◽  
Failure Model ◽  
Maintenance Policy ◽  
Cost Rate ◽  
Long Run ◽  
Degradation Failure ◽  
Average Cost Rate ◽  
Optimal Maintenance Policy

In this paper a stochastic failure model for a system with stochastically dependent competing failures is analyzed. The system is subject to two types of failure: degradation failure and catastrophic failure. Both types of failure share an initial common source: an external shock process. This implies that they are stochastically dependent. In our developments of the model, the type of dependency between the two kinds of failure will be characterized. Conditional properties of the two competing risks are also investigated. These properties are the fundamental basis for the development of the maintenance strategy studied in this paper. Considering this maintenance strategy, the long-run average cost rate is derived and the optimal maintenance policy is discussed.

A Stochastic Failure Model with Dependent Competing Risks and its Applications to Condition-Based Maintenance

2015 ◽  
Vol 52 (02) ◽  
pp. 558-573 ◽  
Author(s):  
Ji Hwan Cha ◽  
Inma T. Castro
Keyword(s):  
Competing Risks ◽  
Common Source ◽  
Maintenance Strategy ◽  
Failure Model ◽  
Maintenance Policy ◽  
Cost Rate ◽  
Long Run ◽  
Degradation Failure ◽  
Average Cost Rate ◽  
Optimal Maintenance Policy

In this paper a stochastic failure model for a system with stochastically dependent competing failures is analyzed. The system is subject to two types of failure: degradation failure and catastrophic failure. Both types of failure share an initial common source: an external shock process. This implies that they are stochastically dependent. In our developments of the model, the type of dependency between the two kinds of failure will be characterized. Conditional properties of the two competing risks are also investigated. These properties are the fundamental basis for the development of the maintenance strategy studied in this paper. Considering this maintenance strategy, the long-run average cost rate is derived and the optimal maintenance policy is discussed.

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