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 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.     

Optimal Control of an <Emphasis Type="Italic">M/E</Emphasis><Subscript><Emphasis Type="Italic">k</Emphasis></Subscript>/1 Queueing System with a Removable Service Station

This paper studies the optimal operation of an M/E _k /1 queueing system with a removable service station under steady-state conditions. Analytic closed-form solutions of the controllable M/E _k /1 queueing system are derived. This is a generalization of the controllable M/M/1, the ordinary M/E _k /1, and the ordinary M/M/1 queueing systems in the literature. We prove that the probability that the service station is busy in the steady-state is equal to the traffic intensity. Following the construction of the expected cost function per unit time, we determine the optimal operating policy at minimum cost.     

Condition monitoring: a new perspective

We present a novel approach to support preventive replacement decisions based on condition monitoring. It is assumed that the condition of a technical unit is measured continuously, with the measurement indicating in which of k≥2 states the unit is. A new unit is in state S₁, and failure occurs the moment a unit leaves state S _k . A unit will only go from state S _j to state S_j+1, and these immediately observed transitions occur at random times. At such transition moments a decision is required on preventive replacement of the unit, which would require preparation time. Such a decision needs to be based on inferences about the total residual time till failure. We present a method for such inferences that relies on minimal model assumptions added to data on state transitions. We illustrate the approach via simulated examples, and discuss possible use of the method and related aspects.     

Mean First-passage Times for a <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> Sytem with Repair

Consider a k-out-of-n system where the lifetimes of the components are independent and identically distributed exponential (λ) random variables. Each component has its own repair facility, the repair times being independent and identically distributed exponential (μ) random variables, independent of the failure times. The mean operating time and mean repair time during the cycle between two successive breakdowns are found using renewal theory and the expression for the system availability. Using these, the mean first-passage times from any of the operating states of the system to the down state, and the mean first-passage times from any of the down states to the operating state are found recursively.     

Reliability Analysis of a One-Unit System with Unrepairable Spare Units and its Optimization Applications

It is assumed that a unit is either in operation or is in repair. When the main unit is under repair, spare units which cannot be repaired are used. In this system the following quantities are of interest: (i) The time distribution and the mean time to first-system failure, given that the n spare units are provided at time 0. (ii) The probability that the number of the failed spare units are equal to exactly n during the interval (0, t], and its expected number during the interval (0, t]. These quantities are derived by solving the renewal-type equations.Two optimization problems are discussed using the results obtained, viz.: (i) The expected cost of two systems, one with both a main unit and spare units and the other with only spare units is considered. (ii) A preventive maintenance policy of the main unit is considered in order to minimize the expected cost rate. Some policies of the two problems are discussed under suitable conditions. Numerical examples are also presented.     

Costing a Finite Minimal Repair Replacement Policy

In this paper an integral equation technique is used to evaluate the expected cost for the period (0, t] of a policy involving minimal repair at failure with replacement after N failures. This cost function provides an appropriate criterion to determine the optimal replacement number N* for a system required for use over a finite time horizon. In an example, it is shown that significant cost savings can be achieved using N* from the new finite time horizon model rather than the value predicted by the usual asymptotic model.     

<Emphasis Type="Italic">km</Emphasis>th price sealed-bid auctions with general independent values and equilibrium linear mark-ups

A generalised bidding model is developed to calculate a bidder’s expected profit and auctioners expected revenue/payment for both a General Independent Value and Independent Private Value (IPV) kmth price sealed-bid auction (where the mth bidder wins at the kth bid payment) using a linear (affine) mark-up function. The Common Value (CV) assumption, and highbid and lowbid symmetric and asymmetric First Price Auctions and Second Price Auctions are included as special cases. The optimal n bidder symmetric analytical results are then provided for the uniform IPV and CV models in equilibrium. Final comments concern implications, the assumptions involved and prospects for further research.     

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.     

Repair Replacement Modelling over Finite Time Horizons

In this paper an integral equation approach is given for evaluating the expected cost of repair replacement policies over finite time horizons. An asymptotic estimate of this expected cost is also obtained. The policy involving imperfect repair on failure with replacement after N failures is taken as an illustrative example and optimal policies N* are found for both infinite and finite time horizons of use.     

A <Emphasis Type="Italic">p</Emphasis>-adic hard-core model with three states on a Cayley tree

We examine the p-adic hard-core model with three states on a Cayley tree. Translationinvariant and periodic p-adic Gibbs measures are studied for the hard-core model for k = 2. We prove that every p-adic Gibbs measure is bounded for p ≠ 2. We show in particular that there is no strong phased transition for a hard-core model on a Cayley tree of order k.     

Markov decision processes with noise-corrupted and delayed state observations

We consider the partially observed Markov decision process with observations delayed by k time periods. We show that at stage t, a sufficient statistic is the probability distribution of the underlying system state at stage t - k and all actions taken from stage t - k through stage t - 1. We show that improved observation quality and/or reduced data delay will not decrease the optimal expected total discounted reward, and we explore the optimality conditions for three important special cases. We present a measure of the marginal value of receiving state observations delayed by (k - 1) stages rather than delayed by k stages. We show that in the limit as k →∞ the problem is equivalent to the completely unobserved case. We present numerical examples which illustrate the value of receiving state information delayed by k stages.     

A Maintenance Model with Opportunities and Interrupt Replacement Options

A component has time to failure X with density f(·). Opportunities arise as a Poisson process of rate λ. At an opportunity the component may at choice, be replaced at cost c₂. At failure the components can either be minimally repaired at cost c₁ or replaced at cost c₄. Finally the component may be replaced at any time at an ‘interrupt’ cost of c₃. We derive expressions for the long-run expected cost rate, and give examples of numerical optimisation with respect to control limits on age at an opportunity, and at an interrupt replacement.     

Optimal Reliability Design of <Emphasis Type="Italic">K</Emphasis>-out-of-<Emphasis Type="Italic">N</Emphasis> Systems Subject to Two Kinds of Failure

K-out-of-N systems formed from N identical and independent components are considered in which the components can take two states: 0 (open) or 1 (closed) on command (e.g. Electromagnetic Relays and Solid State Switches). The components are subject to two kinds of failure on command: failure to open and failure to close. A K-out-of-N system is closed if and only if at least K of its components are closed. The system is considered open or closed depending on the states of its components. The optimum system is taken to be that system which maximizes the reliability. This paper finds the optimum K-out-of-N system given a fixed number of components.     

Continuity of the Asymptotics of Expected Zeros of Fewnomials

In “Random complex fewnomials, I,” the limiting formula of the (normalized) expected distribution of complex zeros of a system of k random m-variate fewnomials where the coefficients are taken from the \({\mathrm {SU}}(m+1)\) ensemble and the spectra are chosen uniformly at random is determined. We recall this result and show the limiting formula is a (k, k)-form with continuous coefficients.     

Analysis of the stochastic cash balance problem using a level crossing technique

The simple cash management problem includes the following considerations: the opportunity cost of holding too much cash versus the penalty cost of not having enough cash to meet current needs; the cost incurred (or profit generated) when making changes to cash levels by increasing or decreasing them when necessary; the uncertainty in timing and magnitude of cash receipts and cash disbursements; and the type of control policy that should be used to minimize the required level of cash balances and related costs. In this paper, we study a version of this problem in which cash receipts and cash disbursements occur according to two independent compound Poisson processes. The cash balance is monitored continuously and an order-point, order-up-to-level, and keep-level \( \left( {s, S, M} \right) \) policy is used to monitor the content, where \( s \le S \le M \). That is, (a) if, at any time, the cash level is below s, an order is immediately placed to raise the level to S; (b) if the cash level is between s and M, no action is taken; (c) if the cash level is greater than M, the amount in excess of M is placed into an earning asset. We seek to minimize the expected total costs per unit time of running the cash balance. We use a level-crossing approach to develop a solution procedure for finding the optimal policy parameters and costs. Several numerical examples are given to illustrate the tradeoffs.     

Reliability Bounds for Multicriteria Systems

This paper deals with general structure functions, where arbitrary degrees of performance between perfect functioning and complete failure are allowed for each component and the n-component system itself. We make the assumption that the n-component system can be modelled as a structure function given by a mapping φ:L ⁿ →L ^k ₀, L and L₀ being two linearly ordered sets, so that the performance of the system is evaluated according to k single criteria. Global concepts of minimal path and minimal cut are discussed for these multicriteria systems; general reliability bounds based on them are deduced and compared with those given in previous papers.     

When does the zero-one <Emphasis Type="Italic">k</Emphasis>-law fail?

The limit probabilities of the first-order properties of a random graph in the Erd?s–Rényi model G(n, n^?α), α ∈ (0, 1), are studied. A random graph G(n, n^?α) is said to obey the zero-one k-law if, given any property expressed by a formula of quantifier depth at most k, the probability of this property tends to either 0 or 1. As is known, for α = 1? 1/(2^k?1 + a/b), where a > 2^k?1, the zero-one k-law holds. Moreover, this law does not hold for b = 1 and a ≤ 2^k?1 ? 2. It is proved that the k-law also fails for b > 1 and a ≤ 2^k?1 ? (b + 1)².     

Maximal <Emphasis Type="Italic">k</Emphasis>-Intersecting Families of Subsets and Boolean Functions

A family of subsets of an n-element set is k-intersecting if the intersection of every k subsets in the family is nonempty. A family is maximalk-intersecting if no subset can be added to the family without violating the k-intersection property. There is a one-to-one correspondence between the families of subsets and Boolean functions defined as follows: To each family of subsets, assign the Boolean function whose unit tuples are the characteristic vectors of the subsets.We show that a family of subsets is maximal 2-intersecting if and only if the corresponding Boolean function is monotone and selfdual. Asymptotics for the number of such families is obtained. Some properties of Boolean functions corresponding to k-intersecting families are established fork > 2.     

An axiomatization of the Choquet integral in the context of multiple criteria decision making without any commensurability assumption

An axiomatization of the Choquet integral is proposed in the context of multiple criteria decision making without any commensurability assumption. The most essential axiom—named Commensurability Through Interaction—states that the importance of an attribute i takes only one or two values when a second attribute k varies. When the importance takes two values, the point of discontinuity is exactly the value on the attribute k that is commensurate to the fixed value on attribute i. If the weight of criterion i does not depend on criterion k, for any value of the other criteria than i and k, then criteria i and k are independent. Applying this construction to any pair i, k of criteria, one obtains a partition of the set of criteria. In each block, the criteria interact one with another, and it is thus possible to construct vectors of values on the attributes that are commensurate. There is complete independence between the criteria of any two blocks in this partition. Hence one cannot ensure commensurability between two blocks in the partition. But this is not a problem since the Choquet integral is additive between subsets of criteria that are independent.