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This paper discusses preventive replacement policies for an independent damage process, in which successive shocks occur at random times, and the independent damages caused by shocks are random variables. It is assumed that a unit fails when the damage has exceeded a prespecified level [Formula: see text]. We consider three models: Model 1: the unit is replaced at the [Formula: see text]th shock for damage [Formula: see text], at damage [Formula: see text], or at time [Formula: see text], whichever occurs first. Model 2: the unit is replaced at the [Formula: see text]th shock for damage [Formula: see text], at damage [Formula: see text] or at shock [Formula: see text], whichever occurs first. Model 3: the unit is replaced at the [Formula: see text]th shock for damage [Formula: see text], at damage [Formula: see text], at shock [Formula: see text] or at time [Formula: see text], whichever occurs first. We obtain the expected cost rates for each model and discuss analytical optimal [Formula: see text] and [Formula: see text] to minimize their expected cost rates. Numerical examples are given when the damage has an exponential distribution and shocks occur at a gamma distribution.

The random age replacement policies discussed in literatures are helpful to complete the nonstopping works with random working cycles, however, maintenance policies are more easily performed at periodic times in real applications. For such a viewpoint, this paper proposes that age replacement policies are planned at periodic times while considering the random working cycles. Using the modeling approaches of replacement first and last policies, we discuss two models such that the unit is replaced at periodic times [Formula: see text] and working cycles [Formula: see text], whichever occurs first and whichever occurs last. The expected cost rate models are obtained and their optimal solutions for [Formula: see text] and [Formula: see text] are discussed. The comparisons between the policies of [Formula: see text] and [Formula: see text], and replacement first and last are made from the point of cost. Numerical examples are illustrated when the failure time has a Weibull distribution.

In order to ensure the data security, plans of incremental and differential backups are usually set up to save the costs which suffer from full backups. In this paper, we suppose that failures occur at data updating times and backups are implemented only at the end of data updates, and full backups are done at time [Formula: see text], over time [Formula: see text] and update [Formula: see text] to balance the costs of data backup and failure recovery. Using the theory of renewal process, we obtain the expected costs of backup and recovery and the expected cost rates for full backups. Optimum solutions of [Formula: see text] and [Formula: see text] to minimize the expected cost rates are obtained, and their comparisons are discussed from the optimizations of the integrated models in analytical ways.

It is well-known in the standard age replacement policy that a finite preventive replacement time does not exist when the failure time is exponential and the optimal preventive replacement time is nonrandom. It is shown that when the failure time is exponential, a finite time exists by introducing the shortage and excess costs. In addition, the random age replacement is proposed and similar discussions are made. Furthermore, the periodic and random inspection policies are taken up, and their optimal policies are shown to correspond theoretically to those of the age replacement ones. It is shown finally that when the random inspection cost is the half of the periodic one, two expected costs are almost the same.

Preventive replacement policy conducted at a routine time [Formula: see text] has been surveyed extensively in literatures. Another two random replacement times [Formula: see text] and [Formula: see text] for time [Formula: see text] are jointly considered in this paper. We formulate the models of replacement first, replacement last and replacement middle, which are based on the respective assumptions of whichever occurs first, whichever occurs last and whichever occurs middle. Next, we modify replacement first and replacement last by using whichever occurs first and last jointly. Furthermore, we give the model of replacement overtime when the assumptions of whichever occurs first, whichever occurs last and replacing the unit over time [Formula: see text] are simultaneously considered. Comparisons among these replacement models are made to find which policy is more economical than the other.

It has been proposed in recent literatures that if replacement time is planned too early prior to failure time, a waste of operation cost, i.e., excess costs, would incur because the system might run for an additional period of time to complete critical operations; and if the replacement time is too late after failure, a great failure cost, i.e., shortage cost, is incurred due to the delay in time of the carelessly scheduled replacement. The above notion of shortage and excess costs are taken into consideration for the replacement first, replacement last and replacement overtime models in this paper. We obtain the expected cost rates and their optimum replacement times. Comparisons among these optimum times are made analytically and numerically.

A new periodic damage model is proposed in this paper, where the damage produced by shocks is only measured at periodic times, without considering the stochastic shocks and the cumulative process of damage in discrete times. We model five replacement policies for the periodic damage models using three approaches of whichever triggering event occurs first, whichever triggering event occurs last, and replacing over a planned measure, which are named as replacement first, replacement last, over-damage replacement, over-damage replacement first, and over-damage replacement last, respectively. In analytical ways, we obtain their optimum replacement policies to minimize the expected cost rates and compare them to find which model is more economical. Finally, numerical examples are given to illustrate the validity of the presented models. This paper was made possible by PDRA Grant # PDRA1-0116-14107 from the Qatar National Research Fund (a member of The Qatar Foundation), and NPRP 4-631-2-233 Scopus

It has been commonly known in a Database Management System that constant plans of incremental/differential backup schemes are set up and their full backups should be done to balance the costs of data backup and failure recovery. In this paper, we consider a database system that is running for 24/7 and has several random busy states of updates, and we suppose that incremental/differential backups are implemented after each update and full backups are triggered at constant time [Formula: see text] and random updates [Formula: see text]. That is, models with decision variables of [Formula: see text] and [Formula: see text] for full backups with random incremental/differential backups are considered. We obtain the expected costs of data backup and failure recovery and the expected cost rates for full backups. Optimum solutions to minimize the expected cost rates are discussed in analytical ways. Furthermore, full backups that are made at time [Formula: see text] and updates [Formula: see text] are considered as constant and random modes respectively and compared analytically. For the discussions, numerical examples in exponential cases are given.

This paper proposes two modified replacement policies which combine random replacement with standard age and periodic policies, where units undergo replacement at the first completion of some random working time over a planned time T, which are called replacement overtime. The expected cost rates of age and periodic replacement overtime polices are obtained, and optimal policies which minimize them are discussed analytically. Comparisons between such two overtime policies and standard age and periodic replacements are made in detail. When two costs for replacement overtime and standard age or periodic policies are the same, it is shown as estimated previously that the standard replacements are better than their overtime policies. Furthermore, two modified replacement overtime costs are discussed and computed numerically when such new policies would be better than the standard age and periodic replacements.