Junji Koyanagi

Optimal Maintenance Problems for Markovian Deteriorating Systems

This chapter deals with optimal maintenance problems for Markovian deteriorating systems. The function of the system deteriorates with time, and the grade of deterioration is classified as one of s+2 discrete states, 0, 1,..., s,s+1, in the order of increasing deterioration. State 0 is a good state, i.e., the system is like new, the states 1,..., s are deterioration states and the state s+ 1 is a failure state. In a normal operation, these states are assumed to constitute a discrete or continuous time Markovian process with an absorbing state s+ 1. In Section 8.1, we first introduce a basic replacement problem for a discrete time Markovian deteriorating system. In Section 8.2, we discuss an optimal inspection and replacement problem for the system in Section 8.1. In Section 8.3, we consider an optimal inspection and replacement problem under incomplete system observation. In Section 8.4, we treat a continuous time Markovian deteriorating system and discuss an optimal inspection and replacement problem. In Section 8.5, we deal with a maintenance problem in queueing system and discuss an optimal maintenance policy based on both the queue length and the server state.

An optimal maintenance policy for a deteriorating server of an M/G/1 queueing system

An optimal maintenance policy for a deteriorating server of an M/G/1 queueing system is considered. The server has multiple states which indicate the deterioration level of the server. The state transitions of the server are governed by a Markov process with one absorbing state - the failure state. When the server fails, corrective maintenance starts immediately. Preventive maintenance can be taken to avoid the failure. After the maintenance, the server becomes as good as new. At the beginning of maintenance, customers in the system are rejected and the arriving customers during the maintenance are also rejected. A cost of one per rejected customer is incurred and our objective is to find the optimal maintenance policy that minimizes the total expected discounted cost over an infinite time horizon. The problem is formulated as a semi-Markov decision process whose state space is the queue length and the server state. The existence of the optimal controls with switch-curve structure is established under some conditions.

An Optimal Maintenance Policy for a Queueing System Server Under Periodic Observation

This paper describes an optimal maintenance policy for an M/M/1 queueing system server. Customers arrive at the system in a Poisson stream and are served by the exponential server. After a random time, the server is interrupted by a failure and this failure is detected through regularly timed observations. We begin corrective maintenance when we detect the failure. Through the failure of the server, we lose the customers in the system at the time of failure, as well as the customers that arrive between the failure and the completion of corrective maintenance. However, it is possible to avoid the failure and subsequent corrective maintenance by performing preventive maintenance at observation time. It is true that customers in the system at the start of preventive maintenance and those that arrive prior to its completion are lost. Since the queueing system should serve as many customers as possible, our objective is to minimize the number of lost customers. We then formulate this problem as a semi-Markov decision process and prove the switch curve structure of the optimal policy.

An assignment problem for a parallel queueing system with two heterogeneous servers

In this paper, we consider an optimization problem for a parallel queueing system with two heterogeneous servers. Each server has its own queue and customers arrive at each queue according to independent Poisson processes. Each service time is independent and exponentially distributed. When a customer arrives at queue 1, the customers in queue 1 can be transferred to queue 2 by paying an assignment cost which is proportional to the number of moved customers. Holding cost is a function of the pair of queue lengths of the two servers. Our objective is to minimize the expected total discounted cost. We use the dynamic programming approach for this problem. Considering the pair of queue lengths as a state space, we show that the optimal policy has a switch over structure under some conditions on the holding cost.

Optimal strategies for some team games

Abstract Consider a game teams A and B, consisting of a sequence of matches, where each match takes place between one player i from A and one player j from B. Given the probability that player i wins over player j, we investigate optimal strategies on how to choose a player for the next match, for the following two types of team games. The first type assumes that after each match, the loser is eliminated from the list of remaining players, while the winner remains in the list. The team from which all players are eliminated loses the game. Assuming the Bradley-Terry model as the probability model, we first show that the winning probability does not depend on the strategy chosen. It is also shown that the Bradley-Terry model is essentially the only model for which this strategy independence holds. The second type of game assumes that both players are eliminated after each match. In this case, it is shown that choosing a player with equal probability is an optimal strategy in the sense of maximizing the expected number of wins of matches, provided that information about the order of players in the other teams is not available. The case in which a team knows the ordering of the other team is also studied.

An optimal policy for joining a queue in processing two kinds of jobs

Abstract We deal with a situation where a worker processes two kinds of jobs, job A and job B. Job A is processed in a certain queueing system, and job B is processed separately from the queueing system if the worker is not in the queueing system. The process of job B consists of several tasks. The number of the tasks is distributed and each task needs a constant time. At each end of task, the worker can know whether all tasks are completed, and a decision is made whether the process of job B is suspended to join the queue. If the process of job B is suspended, the worker joins the queue and the residual tasks are processed after job A has been processed. The objective is to minimize the expected time until two jobs are completed. We prove a monotone property of the optimal policy by a dynamic programming formulation.

An optimal maintenance policy for a server with decreasing arrival rate and non‐cancelable customer

This paper considers a maintenance problem for a queueing system. The arrival rate decreases as the server state becomes worse. The system can be recovered by maintenance, though the system is closed until the end of maintenance. A semi‐Markov decision process is formulated to find the optimal policy that maximizes the total expected discounted income from customers.

An optimal maintenace policy of a discrete time markovian deterioration system

Abstract This paper investigates an optimal inspection and replacement problem for a discrete time Markovian deterioration system. It is assumed that the state of the system cannot be identified without inspection. The problem is to determine an optimal inspection and replacement policy which minimizes the expected long-run discounted cost and is formulated as a semi-Markov decision process. Under some conditions reflecting the practical meaning of deterioration, some structural properties of an optimal policy are obtained.

A DISTRIBUTION OF CHARGERS USED BY ELECTRIC VEHICLES ON AN EXPRESSWAY WITH VARIOUS TYPES OF CHARGING PLACES

In Japan, electric vehicle (EV) is spreading, because EV has several advantages compared with gasoline car. However, EV has two main disadvantages, shorter cruising range and longer charging time. These disadvantages may cause a serious problem on an expressway, because the queue of EV waiting for charge may be long, if enough chargers are not placed at charging places. It is important to estimate the number of chargers in a charging place to avoid the long queue. This paper proposes a model to estimate the number of chargers used by EV. In this model, it is assumed that the driver knows the charge level of EV and decides to charge at charging places according to the type of charging place and charge level.

An optimal maintenance policy for a server with decreasing arrival rate

We study an optimal maintenance policy for the server in a queueing system. Customers arrive at the server in a Poisson stream and are served by an exponential server, which is subject to multiple states indicating levels of popularity. The server state transitions are governed by a Markov process. The arrival rate depends on the server state and it decreases as the server loses popularity. By maintenance the server state recovers completely, though the customers in the system are lost at the beginning of maintenance. The customers who arrive during maintenance are also lost. In this paper, two kinds of such systems are considered. The first system receives a unit reward when a customer arrives at the system and pays a unit cost for each lost customer at the start of maintenance. The second system receives a unit reward at departure, and pays nothing for lost customers at the beginning of maintenance. Our objective is to maximize the total expected discounted profit over an infinite time horizon. We use a semi-Markov decision process to formulate the problem and are able to establish some properties for the optimal maintenance policy under certain conditions.