An extended replacement policy for a deteriorating system with multi-failure modes

In this paper, the maintenance problem for a deteriorating system with k + 1 failure modes, including an unrepairable failure (catastrophic failure) mode and k repairable failure (non-catastrophic failure) modes, is studied. Assume that the system after repair is not “as good as new” and its deterioration is stochastic. Under these assumptions, an extended replacement policy N is considered: the system will be replaced whenever the number of repairable failures reaches N or the unrepairable failure occurs, whichever occurs first. Our purpose is to determine an optimal extended policy N^∗ such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression of the average cost rate is derived, and the corresponding optimal extended policy N^∗ can be determined analytically or numerically. Finally, a numerical example is given to illustrate some theoretical results of the repair model proposed in this paper.     

A deteriorating cold standby repairable system with priority in use

In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. In this system, it is assumed that the working time distributions and the repair time distributions of the two components are both exponential and component 1 is given priority in use. After repair, component 2 is “as good as new” while component 1 follows a geometric process repair. Under these assumptions, using the geometric process and a supplementary variable technique, some important reliability indices such as the system availability, reliability, mean time to first failure (MTTFF), rate of occurrence of failure (ROCOF) and the idle probability of the repairman are derived. A numerical example for the system reliability R(t) is given. And it is considered that a repair-replacement policy based on the working age T of component 1 under which the system is replaced when the working age of component 1 reaches T. Our problem is to determine an optimal policy T^∗ such that the long-run average cost per unit time of the system is minimized. The explicit expression for the long-run average cost per unit time of the system is evaluated, and the corresponding optimal replacement policy T^∗ can be found analytically or numerically. Another numerical example for replacement model is also given.     

A geometric process maintenance model with preventive repair

In this paper, a geometric process maintenance model with preventive repair is studied. A maintenance policy (T, N) is applied by which the system will be repaired whenever it fails or its operating time reaches T whichever occurs first, and the system will be replaced by a new and identical one following the Nth failure. The long-run average cost per unit time is determined. An optimal policy (T^∗, N^∗) could be determined numerically or analytically for minimizing the average cost. A new class of lifetime distribution which takes into account the effect of preventive repair is studied that is applied to determine the optimal policy (T^∗, N^∗).     

Optimal replacement policy for a deteriorating system with increasing repair times

This paper considers an optimal maintenance policy for a practical and reparable deteriorating system subject to random shocks. Modeling the repair time by a geometric process and the failure mechanism by a generalized δ-shock process, we develop an explicit expression of the long-term average cost per time unit for the system under a threshold-type replacement policy. Based on this average cost function, we propose a finite search algorithm to locate the optimal replacement policy N^∗ to minimize the average cost rate. We further prove that the optimal policy N^∗ is unique and present some numerical examples. Many practical systems fit the model developed in this paper.     

A bivariate optimal replacement policy for a cold standby repairable system with preventive repair

In this paper, the repair-replacement problem for a deteriorating cold standby repairable system is investigated. The system consists of two dissimilar components, in which component 1 is the main component with use priority and component 2 is a supplementary component. In order to extend the working time and economize the running cost of the system, preventive repair for component 1 is performed every time interval T, and the preventive repair is “as good as new”. As a supplementary component, component 2 is only used at the time that component 1 is under preventive repair or failure repair. Assumed that the failure repair of component 1 follows geometric process repair while the repair of component 2 is “as good as new”. A bivariate repair-replacement policy (T, N) is adopted for the system, where T is the interval length between preventive repairs, and N is the number of failures of component 1. The aim is to determine an optimal bivariate policy (T, N)^∗ such that the average cost rate of the system is minimized. The explicit expression of the average cost rate is derived and the corresponding optimal bivariate policy can be determined analytically or numerically. Finally, a Gamma distributed example is given to illustrate the theoretical results for the proposed model.     

Optimal repair–replacement policies for a system with two types of failures

In this paper, the optimal replacement problem is investigated for a system with two types of failures. One type of failure is repairable, which is conducted by a repairman when it occurs, and the other is unrepairable, which leads to a replacement of the system at once. The repair of the system is not “as good as new”. The consecutive operating times of the system after repair form a decreasing geometric process, while the repair times after failure are assumed to be independent and identically distributed. Replacement policy N is adopted, where N is the number of repairable failures. The system will be replaced at the Nth repairable failure or at the unrepairable failure, whichever occurs first. Two replacement models are considered, one is based on the limiting availability and the other based on the long-run average cost rate of the system. We give the explicit expressions for the limiting availability and the long-run average cost rate of the system under policy N, respectively. By maximizing the limiting availability A(N) and minimizing the long-run average cost rate C(N), we theoretically obtain the optimal replacement policies N^∗ in both cases. Finally, some numerical simulations are presented to verify the theoretical results.     

Controlling arrivals for a queueing system with an unreliable server: Newton-Quasi method

This paper deals with the control policy of a removable and unreliable server for an M/M/1/K queueing system, where the removable server operates an F-policy. The so-called F-policy means that when the number of customers in the system reaches its capacity K (i.e. the system becomes full), the system will not accept any incoming customers until the queue length decreases to a certain threshold value F. At that time, the server initiates an exponential startup time with parameter γ and starts allowing customers entering the system. It is assumed that the server breaks down according to a Poisson process and the repair time has an exponential distribution. A matrix analytical method is applied to derive the steady-state probabilities through which various system performance measures can be obtained. A cost model is constructed to determine the optimal values, say (F^∗, μ^∗, γ^∗), that yield the minimum cost. Finally, we use the two methods, namely, the direct search method and the Newton-Quasi method to find the global minimum (F^∗, μ^∗, γ^∗). Numerical results are also provided under optimal operating conditions.     

A bivariate mixed policy for a simple repairable system based on preventive repair and failure repair

In this paper, a simple repairable system (i.e. a one-component repairable system with one repairman) with preventive repair and failure repair is studied. Assume that the preventive repair is adopted before the system fails, when the system reliability drops to an undetermined constant R  , the work will be interrupted and the preventive repair is executed at once. And assume that the preventive repair of the system is “as good as new” while the failure repair of the system is not, and the deterioration of the system is stochastic. Under these assumptions, by using geometric process, we present a bivariate mixed policy (R,N)$(R\text{,}N)$, respectively based on a scale of the system reliability and the failure-number of the system. Our aim is to determine an optimal mixed policy (R,N)^∗$(R\text{,}N{)}^{\ast }$ such that the long-run average cost per unit time (i.e. the average cost rate) is minimized. The explicit expression of the average cost rate is derived, and the corresponding optimal mixed policy can be determined analytically or numerically. Finally, a numerical example is given where the working time of the system yields a Weibull distribution. Some comparisons with a certain existing policy are also discussed by numerical methods.     

An optimal repair–replacement policy for a cold standby system with use priority

In this paper, the maintenance problem for a cold standby system consisting of two dissimilar components and one repairman is studied. Assume that both component 1 and component 2 after repair follow geometric process repair and component 1 is given priority in use when both components are workable. Under these assumptions, using geometric process repair model, we consider a replacement policy N under which the system is replaced when the number of failures of component 1 reaches N. Our purpose is to determine an optimal replacement policy N1 such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression for the average cost rate of the system is derived and the corresponding optimal replacement policy N1 can be determined analytically or numerically. Finally, a numerical example is given to illustrate some theoretical results and the model applicability.     

An extended optimal replacement model of systems subject to shocks

A system is subject to shocks that arrive according to a non-homogeneous Poisson process. As shocks occur a system has two types of failures: type I failure (minor failure) is rectified by a minimal repair, whereas type II failure (catastrophic failure) is removed by replacement. The probability of a type II failure is permitted to depend on the number of shocks since the last replacement. This paper proposes a generalized replacement policy where a system is replaced at the nth type I failure or first type II failure or at age T, whichever occurs first. The cost of the minimal repair of the system at age t depends on the random part C(t) and deterministic paper c(t). The expected cost rate is obtained. The optimal n¹ and optimal T¹ which would minimize the cost rate are derived and discussed. Various special cases are considered and detailed.     

A deteriorating system with its repairman having multiple vacations

This paper considers a repairable system with a repairman, who can take multiple vacations. If the system fails and the repairman is on vacation, it will wait for repair until the repairman is available. Assume that the system cannot be repaired “as good as new” after failures. Under these assumptions, using the geometric process and the supplementary variable technique, some important reliability indexes are derived, such as the system reliability, availability, rate of occurrence of failures, etc. According to the renewal reward theorem, the explicit expression of the expected profit per unit time is obtained. Finally, a numerical example is given to illustrate that there exists an optimal replacement policy N∗, which maximizes the value of the expected profit rate after a long time run.     

A note on replacement policy for a system subject to non-homogeneous pure birth shocks

A system is subject to shocks that arrive according to a non-homogeneous pure birth process. As shocks occur, the system has two types of failures. Type-I failure (minor failure) is removed by a general repair, whereas type-II failure (catastrophic failure) is removed by an unplanned replacement. The occurrence of the failure type is based on some random mechanism which depends on the number of shocks occurred since the last replacement. Under an age replacement policy, a planned (or scheduled) replacement happens whenever an operating system reaches age T. The aim of this note is to derive the expected cost functions and characterize the structure of the optimal replacement policy for such a general setting. We show that many previous models are special cases of our general model. A numerical example is presented to show the application of the algorithm and several useful insights.     

Hyponormal operators with rank-two self-commutators

In this paper it is shown that if T∈L(H) satisfies

(i)

T is a pure hyponormal operator;

(ii)

[T^∗,T] is of rank two; and

(iii)

ker[T^∗,T] is invariant for T,

then T is either a subnormal operator or the Putinar's matricial model of rank two. More precisely, if T_|ker[T^∗,T] has a rank-one self-commutator then T is subnormal and if instead T_|ker[T^∗,T] has a rank-two self-commutator then T is either a subnormal operator or the kth minimal partially normal extension, , of a (k+1)-hyponormal operator Tk which has a rank-two self-commutator for any k∈Z₊. Hence, in particular, every weakly subnormal (or 2-hyponormal) operator with a rank-two self-commutator is either a subnormal operator or a finite rank perturbation of a k-hyponormal operator for any k∈Z₊.     

A Generalized Block Replacement Policy with Minimal Repair and General Random Repair Costs for a Multi-unit System

A generalization of the block replacement policy (BRP) is proposed and analysed for a multi-unit system which has the specific multivariate distribution. Under such a policy an operating system is preventively replaced at times kT (k = 1, 2, 3,...), as in the ordinary BRP, and the replacement of the failed system at failure is not mandatory; instead, a minimal repair to the component of the system can be made. The choice of these two possible actions is based on some random mechanism which is age-dependent. The cost of the ith minimal repair of the component at age y depends on the random part C(y) and the deterministic part C_i(y). The aim of the paper is to find the optimal block interval T which minimizes the long-run expected cost per unit time of the policy.     

A δ-shock maintenance model for a deteriorating system

In this paper, a δ-shock maintenance model for a deteriorating system is studied. Assume that shocks arrive according to a renewal process, the interarrival time of shocks has a Weibull distribution or gamma distribution. Whenever an interarrival time of shocks is less than a threshold, the system fails. Assume further the system is deteriorating so that the successive threshold values are geometrically nondecreasing, and the consecutive repair times after failure form an increasing geometric process. A replacement policy N is adopted by which the system will be replaced by an identical new one at the time following the Nth failure. Then the long-run average cost per unit time is evaluated. Afterwards, an optimal policy N^* for minimizing the long-run average cost per unit time could be determined numerically.     

A Combined Block and Repair Limit Replacement Policy

This paper considers a combined block and repair limit replacement policy. The policy is defined as follows:

• (i) The unit is replaced preventively at times kT(k=1, 2…
• (ii) For failures in [(k - 1)T, kT) the unit undergoes minimal repair if the estimated repair cost is less than x. Otherwise it is replaced by a new one.

The optimal policy is to select T* and x* to minimize the expected cost per unit time for an infinite time span. A numerical example is given to illustrate the method.     

Extended optimal age-replacement policy with minimal repair of a system subject to shocks

An operating system is subject to shocks that arrive according to a non-homogeneous Poisson process. As shocks occur the system has two types of failure: type I failure (minor) or type II failure (catastrophic). A generalization of the age replacement policy for such a system is proposed and analyzed in this study. Under such a policy, if an operating system suffers a shock and fails at age y (⩽t), it is either replaced by a new system (type II failure) or it undergoes minimal repair (type I failure). Otherwise, the system is replaced when the first shock after t arrives, or the total operating time reaches age T (0 ⩽ t ⩽ T), whichever occurs first. The occurrence of those two possible actions occurring during the period [0, t] is based on some random mechanism which depends on the number of shocks suffered since the last replacement. The aim of this paper is to find the optimal pair (t¹, T¹) that minimizes the long-run expected cost per unit time of this policy. Various special cases are included, and a numerical example is given.     

Optimization issues in k-out-of-n systems

The k-out-of-n system is a system consisting of n independent components such that the system works if and only if at least k of these n components are successfully running. Each component of the system is subject to shocks which arrive according to a nonhomogeneous Poisson process. When a shock takes place, the component is either minimally repaired (type 1 failure) or lying idle (type 2 failure). Assume that the probability of type 1 failure or type 2 failure depends on age. First, we investigate a general age replacement policy for a k-out-of-n system that incorporates minimal repair, shortage and excess costs. Under such a policy, the system is replaced at age T or at the occurrence of the (n-k + 1)th idle component, whichever occurs first. Moreover, we consider another model; we assume that the system operates some successive projects without interruptions. The replacement could not be performed at age T. In this case, the system is replaced at the completion of the Nth project or at the occurrence of the (n-k + 1)th idle component, whichever occurs first. For each model, we develop the long term expected cost per unit time and theoretically present the corresponding optimum replacement schedule. Finally, we give a numerical example illustrating the models we proposed. The proposed models include more realistic factors and extend many existing models.     

Optimizing System Availability Under Minimal Repair with Non-Negligible Repair and Replacement Times

Availability measures are given for a repairable system under minimal repair with constant repair times. A new policy and an existing replacement policy for this type of system are discussed. Each involves replacement at the first failure after time T, with T representing total operating time in the existing model and total elapsed time (i.e. operating time + repair time) in the new model. Optimal values of T are found for both policies over a wide range of parameter values. These results indicate that the new and administratively easier policy produces only marginally smaller optimal availability values than the existing policy.