Mingchih Chen

A summary of periodic and random inspection policies

Some systems work for a job with random working times. It would be useful for such systems that they are checked at the completion of working times to detect their failures, which is called a random inspection policy. Two inspection policies, where a system is checked at periodic or successive times and also at every completion of working times, are considered. The total expected costs until the detection of failure are obtained, and when the random working time is exponential, optimal inspection policies which minimize them are derived analytically. Furthermore, the inspection policy where the system is checked only at every completion of N th working time is proposed. Finally, as one of modified random inspection models, the backup model where the system goes back to the latest checking time when it has failed is taken up and analyzed, by using the inspection policy.

Random and age replacement policies

This paper proposes a random and age replacement policy for an operating unit which works at random times. First, the unit is replaced before failure at a planned time T or at the completion of a working time, whichever occurs first. The expected cost rate is obtained. Next, as one extended model, the unit is replaced before failure at a number N of working times or at a planned time T. An optimal policy which minimizes the expected cost rate is discussed analytically, and its numerical example is given. Two modified models, where the unit is replaced at the first completion of the working time over time T or at number N, and it is replaced at time T or number N, whichever occurs last, are considered. Furthermore, we show that this corresponds to a cumulative damage model by replacing shock with work. Finally, one optimization problem of how much time to preset for scheduling the completion of N works is proposed.

Optimal redundant systems for works with random processing time

This paper studies the optimal redundant policies for a manufacturing system processing jobs with random working times. The redundant units of the parallel systems and standby systems are subject to stochastic failures during the continuous production process. First, a job consisting of only one work is considered for both redundant systems and the expected cost functions are obtained. Next, each redundant system with a random number of units is assumed for a single work. The expected cost functions and the optimal expected numbers of units are derived for redundant systems. Subsequently, the production processes of N tandem works are introduced for parallel and standby systems, and the expected cost functions are also summarized. Finally, the number of works is estimated by a Poisson distribution for the parallel and standby systems. Numerical examples are given to demonstrate the optimization problems of redundant systems.

Optimal Scheduling Of Random Works With Reliability Application

This paper addresses a scheduling problem with random processing time when the operating system is subject to stochastic failure. First, scheduling of a single job is reviewed and the expected cost rate is obtained. Single scheduling time and multiple scheduling times for one job are discussed and the expected cost functions are obtained to setup the optimal due date. Numerical example is given to demonstrate the scheme of multiple scheduling times. Next, the scheduling problems of N tandem jobs and N parallel jobs are formulated and the expected cost functions are also derived. The optimal scheduling times which minimize the expected cost functions are discussed analytically.

Periodic and Random Inspection Policies for Computer Systems

Faults in computer systems sometimes occur intermittently. This paper applies a standard inspection policy with imperfect inspection to a computer system: The system is checked at periodic times and its failure is detected at the next checking time with a certain probability. The expected cost until failure detection is obtained, and when the failure time is exponential, an optimal inspection time to minimize it is derived. Next, when the system executes computer processes, it is checked at random processing times and its failure is detected at the next checking time with a certain probability. The expected cost until failure detection is obtained, and when random processing times are exponential, an optimal inspection time to minimize it is derived. Finally, this paper compares optimal times for two inspection policies and shows that if the random inspection cost is the half of the periodic one, then two expected costs are almost the same.

Replacement policies for a parallel system with shortage and excess costs

It has been assumed in most maintenance models that (i) the number of units for a parallel system can be predefined precisely, (ii) maintenance cost after failure should be avoided, and (iii) age replacement can always be performed at its optimized times. However, these assumptions are challenged from practical perspectives in this paper. For this purpose, shortage and excess costs, which claim that replacement done too early before failure involves a waste of operation, are introduced into replacement models, such as replacement plans with time T or distribution G(t) for constant n or random N of units. Furthermore, the number N of working times and the number n of available units for a scheduled replacement time T are discussed. For each model, the expected replacement cost for one cycle and expected replacement cost rate for a long run are optimized, and numerical examples are given.

Extended maintenance and inspection models for a parallel system

This paper discusses an optimal maintenance policy for a parallel system with periodic inspection. The system is composed of several units which have an independent and identical distribution. When the number of failed unit exceeds a threshold level, the system incurs a loss cost per unit of time from a failure to the next inspection. When the next inspection finds the failure, the system is overhauled, and the system becomes like new. Further, the system is overhauled at planned time in spite of the state of the system. Introducing the loss cost for an inspection and overhaul, the expected cost is obtained and an optimal number of inspections which minimizes the expected cost rate is derived analytically. Numerical examples are given when the failure distribution of each unit is exponential.

Comparisons of standby and parallel systems in reliability, replacement, scheduling and application

This paper compares the standby and parallel systems through the following analyses: reliability measures such as reliabilities, mean times to failure, and failure rates are first compared. These comparisons are reconsidered when the number of units of the two systems cannot be predefined constantly but is a random variable that can be estimated. Replacement policies in which how many number of units should be provided for replacements are compared second. When the two systems work for the same job with a random processing time, their optimal scheduling problems are compared third by employing excess and shortage costs. Finally, the standby and parallel modes are applied into data transmission system, and their mean transmission times are compared. All comparative discussions are given analytically and numerically in exponential distributions, and their potential values could be explored in future studies.

Comparisons of standby and parallel systems for processing times

Two types of redundant systems subject to stochastic failures during the continuous production process are investigated for the best performance with minimum costs. This paper studies the optimal redundant units for parallel systems and standby systems when processing jobs with random working times. First, the reliability and mean time to failure for both redundant systems are derived and compared. Next, the expected cost functions and the optimal numbers of units are obtained for the two redundant systems. Subsequently, availabilities of parallel and standby systems are defined. Finally, the expected cost functions of the parallel and standby systems are discussed under the case of N tandem works with random processing times.

Replacement Policies with a Random Threshold Number of Faults

Most systems fail when a certain amount of reliability quantities have exceeded their threshold levels. The typical example is cumulative damage model in which a system is subjected to shocks and suffers some damage due to shocks, and fails when the total damage has exceeded a failure level K. This paper proposes the following reliability model: Faults occur at a nonhomogeneous Poisson process and the system fails when N faults have occurred, which could be applied to optimization problems in computer systems with fault tolerance, and we suppose that the system is replaced before failure at a planned time T. Two cases where the threshold fault number N is constantly given and is a random variable are considered, we obtain the expected cost rates and discuss their optimal policies.