New Better Than Used Research Articles

A New Generalization of the Class of NBU Distributions

It is well-known that the increasing-failure-rate-average (IFRA), new-better-than-used (NBU), and new-better-than-used-in-expectation (NBUE) classes of life distributions are successive generalizations of the class of increasing-failure-rate (IFR) distributions. Further, each of the IFR, IFRA and NBU classes can be characterized in terms of the exponential distribution and certain partial orderings on the family of all distributions. It is not known, however, if the same holds for the NBUE class. This paper introduces a new generalization (called NBAFR) of the NBU class that allows a characterization in terms of the exponential distribution and a (new) partial ordering. The NBAFR class neither contains nor is contained in the NBUE class. Like the latter, the NBAFR class is preserved under the reliability operation of addition of life lengths. But unlike the NBUE, the NBAFR class is also preserved under formation of s-coherent systems - just like the IFR, IFRA, and NBU classes.

Processes with new better than used first-passage times

Let with Z(0) = 0 be a random process under investigation and N be a point process associated with Z. Both Z and N are defined on the same probability space. Let with R 0 = 0 denote the consecutive positions of points of N on the half-line . In this paper we present sufficient conditions under which (Z, R) is a new better than used (NBU) process and give several examples of NBU processes satisfying these conditions. In particular we consider the processes in which N is a renewal and a general point process. The NBU property of some semi-Markov processes is also presented.

Processes with new better than used first-passage times

Let with Z(0) = 0 be a random process under investigation and N be a point process associated with Z. Both Z and N are defined on the same probability space. Let with R0 = 0 denote the consecutive positions of points of N on the half-line . In this paper we present sufficient conditions under which (Z, R) is a new better than used (NBU) process and give several examples of NBU processes satisfying these conditions. In particular we consider the processes in which N is a renewal and a general point process. The NBU property of some semi-Markov processes is also presented.

Estimating a Survival Curve when New Is Better Than Used

Let F be a distribution function on (0, ∞), and let S = 1 − F be its corresponding survival function. F is New Better than Used (NBU) if S(x)S(y) ≥ S(x + y) for all x and y. Let Sn(x) be the empirical survival function based on a random sample of size n from an NBU distribution function F. This paper studies the estimator Ŝn(x) defined as sup{Sn(x + y)/Sn(y)}, where the supremum is taken over all y for which Sn(y) > 0. We show that Ŝn is an NBU survival curve, and that it is strongly uniformly consistent for S when the underlying distribution has compact support (for example, when sampling is subject to type I censoring). Moreover, in such problems, we show that the rate of convergence of Ŝn is optimal.

New better than used processes

A stochastic process , such that P{Z(0) = 0} = 1, is said to be new better than used (NBU) if, for every x, the first-passage time Tx = inf {t: Z(t) > x} satisfies P{TX > s + t} for everys . In this paper it is shown that many useful processes are NBU. Examples of such processes include processes with shocks and recovery, processes with random repair-times, various Gaver–Miller processes and some strong Markov processes. Applications in reliability theory, queueing, dams, inventory and electrical activity of neurons are indicated. It is shown that various waiting times for clusters of events and for short and wide gaps in some renewal processes are NBU random variables. The NBU property of processes and random variables can be used to obtain bounds on various probabilistic quantities of interest; this is illustrated numerically.

New better than used processes

A stochastic process , such that P{Z(0) = 0} = 1, is said to be new better than used (NBU) if, for every x, the first-passage time Tx = inf {t: Z(t) > x} satisfies P{TX > s + t} for everys . In this paper it is shown that many useful processes are NBU. Examples of such processes include processes with shocks and recovery, processes with random repair-times, various Gaver–Miller processes and some strong Markov processes. Applications in reliability theory, queueing, dams, inventory and electrical activity of neurons are indicated. It is shown that various waiting times for clusters of events and for short and wide gaps in some renewal processes are NBU random variables. The NBU property of processes and random variables can be used to obtain bounds on various probabilistic quantities of interest; this is illustrated numerically.

Testing Whether New is Better than Used with Randomly Censored Data

A life distribution $F$, with survival function $\bar{F} \equiv 1 - F$, is new better than used (NBU) if $\bar{F}(x + y) \leq \bar{F}(x)\bar{F}(y)$ for all $x, y \geq 0$. We propose a test of $H_0 : F$ is exponential, versus $H_1 : F$ is NBU, but not exponential, based on a randomly censored sample of size $n$ from $F$. Our test statistic is $J^c_n = \int \int \bar{F}_n(x + y) dF_n(x) dF_n(y)$, where $F_n$ is the Kaplan-Meier estimator. Under mild regularity on the amount of censoring, the asymptotic normality of $J^c_n$, suitably normalized, is established. Then using a consistent estimator of the null standard deviation of $n^{1/2}J^c_n$, an asymptotically exact test is obtained. We also study, using tests for the censored and uncensored models, the efficiency loss due to the presence of censoring.

A Multivariate New Better than Used Class Derived from a Shock Model

We introduce a new class of multivariate new better than used (MNBU) life distributions based on a shock model similar to that yielding the Marshall-Olkin multivariate exponential distribution. Let T1, …, TM be independent new better than used (NBU) life lengths. Let F̄(t1, …, tn) be the joint survival function of minI∈A1 T1, I = 1, …, n, where A1, …, An are nonempty subsets of {1, …, M} and Ur=1n A1 = {1, …, M} F̄(t1, …, tn) is said to be a MNBU survival function. Basic properties of MNBU survival functions are derived. Comparisons and relationships of this new class of MNBU survival functions are developed with earlier classes.

Determining sample size when searching for rare items : Richard W. Madsen and James E. Holstein. IEEE Trans. Reliab.R-31 (5), 451 (1982)

In this paper multicomponent reliability systems are considered where component failure rate and repair completion rate depend on the environment state and, ages and current repair durations of the other components. this is a generalization of the model of Shaked M. and Shanthikumar J.G.[3]. Sufficient conditions on the set of the rates which imply stochastic ordering between first failure times of two such systems are found. Sufficient conditions on rates which imply that the first failure time of such a system is new better than used(NBU) are given. Some results of [3] may be regarded as the special cases of our assertions.

Minimizing expected makespan in stochastic open shops

Suppose that two machines are available to process n tasks. Each task has to be processed on both machines; the order in which this happens is immaterial. Task j has to be processed on machine 1 (2) for random time Xj (Yj) with distribution Fj (Gj). This kind of model is usually called an open shop. The time that it takes to process all tasks is normally called the makespan. Every time a machine finishes processing a task the decision-maker has to decide which task to process next. Assuming that Xj and Yj have the same exponential distribution we show that the optimal policy instructs the decision-maker, whenever a machine is freed, to start processing the task with longest expected processing time among the tasks still to be processed on both machines. If all tasks have been processed at least once, it does not matter what the decision-maker does, as long as he keeps the machines busy. We then consider the case of n identical tasks and two machines with different speeds. The time it takes machine 1 (2) to process a task has distribution F (G). Both distributions F and G are assumed to be new better than used (NBU) and we show that the decision-maker stochastically minimizes the makespan when he always gives priority to those tasks which have not yet received processing on either machine.

Minimizing expected makespan in stochastic open shops

Suppose that two machines are available to process n tasks. Each task has to be processed on both machines; the order in which this happens is immaterial. Task j has to be processed on machine 1 (2) for random time Xj (Yj ) with distribution Fj (Gj ). This kind of model is usually called an open shop. The time that it takes to process all tasks is normally called the makespan. Every time a machine finishes processing a task the decision-maker has to decide which task to process next. Assuming that Xj and Yj have the same exponential distribution we show that the optimal policy instructs the decision-maker, whenever a machine is freed, to start processing the task with longest expected processing time among the tasks still to be processed on both machines. If all tasks have been processed at least once, it does not matter what the decision-maker does, as long as he keeps the machines busy. We then consider the case of n identical tasks and two machines with different speeds. The time it takes machine 1 (2) to process a task has distribution F (G). Both distributions F and G are assumed to be new better than used (NBU) and we show that the decision-maker stochastically minimizes the makespan when he always gives priority to those tasks which have not yet received processing on either machine.

A Class of Multivariate New Better Than Used Distributions

A class of multivariate distributions with new better than used (NBU) marginals is introduced. A number of necessary and sufficient conditions for a distribution to be a member in the class are given. Closure results, which are useful for the identification or construction of members in the class, are derived. In particular, simple proofs of some well known preservation properties of the class of the univariate NBU distributions are obtained. Two examples of replacement models, that give rise to multivariate NBU distributions in the class, are discussed.

Multivariate Classes of Life Distributions in Reliability Theory

Four classes of life distributions which have been useful in describing aging where systems are assumed to have independent univariate component lifetimes are: (1) the increasing failure rate (IFR) class; (2) the increasing failure rate average (IFRA) class; (3) the new better than used (NBU) class; and (4) the new better than used in expectation (NBUE) class. These classes together with multivariate analogs of the IFR and IFRA classes are reviewed and discussed. An up to date survey of the literature on this topic is included. New multivariate definitions of the NBU and NBUE classes are introduced.

On the first time a separately maintained parallel system has been down for a fixed time

AbstractConsider a system consisting of n separately maintained independent components where the components alternate between intervals in which they are “up” and in which they are “down”. When the ith component goes up [down] then, independent of the past, it remains up [down] for a random length of time, having distribution Fi[Gi], and then goes down [up]. We say that component i is failed at time t if it has been “down” at all time points s ϵ[t‐A.t]: otherwise it is said to be working. Thus, a component is failed if it is down and has been down for the previous A time units. Assuming that all components initially start “up,” let T denote the first time they are all failed, at which point we say the system is failed. We obtain the moment‐generating function of T when n = l, for general F and G, thus generalizing previous results which assumed that at least one of these distributions be exponential. In addition, we present a condition under which T is an NBU (new better than used) random variable. Finally we assume that all the up and down distributions Fi and Gi i = l,….n, are exponential, and we obtain an exact expression for E(T) for general n; in addition we obtain bounds for all higher moments of T by showing that T is NBU.

On the time to first failure in multicomponent exponential reliability systems

Consider an n-component reliability system having the property that at any time each of its components is either up (i.e., working) or down (i.e., being repaired). Each component acts independently and we suppose that each time the i th component goes up it remains up for an exponentially distributed time having mean μ i , and each time it goes down it remains down for an exponentially distributed time having mean υ i . We further suppose that whether or not the system itself is up at any time depends only on which components are up at that time. We are interested in the distribution of the time of first system failure when all components are initially up at time zero. In section 2 we show that this distribution has the NBU (i.e., new better than used) property, and in Section 3 we make use of this and other results to obtain a lower bound to the mean time until first system failure.