New Better Than Used In Expectation Research Articles

On the Accuracy of the Exponential Approximation to Random Sums of Alternating Random Variables

Using the generalized stationary renewal distribution (also called the equilibrium transform) for arbitrary distributions with a finite non-zero first moment, we prove moment-type error-bounds in the Kantorovich distance for the exponential approximation to random sums of possibly dependent random variables with positive finite expectations, in particular, to geometric random sums, generalizing the previous results to alternating and dependent random summands. We also extend the notions of new better than used in expectation (NBUE) and new worse than used in expectation (NWUE) distributions to alternating random variables in terms of the corresponding distribution functions and provide a criteria in terms of conditional expectations similar to the classical one. As corollary, we provide simplified error-bounds in the case of NBUE/NWUE conditional distributions of random summands.

Some new results on the preservation of the NBUE and NWUE aging classes under the formation of coherent systems

AbstractWe consider the classical problem of whether certain classes of lifetime distributions are preserved under the formation of coherent systems. Under the assumption of independent and identically distributed (i.i.d.) component lifetimes, we consider the NBUE (new better than used in expectation) and NWUE (new worse than used in expectation) classes. First, a necessary condition for a coherent system to preserve the NBUE class is given. Sufficient conditions are then obtained for systems satisfying this necessary condition. The sufficient conditions are satisfied for a collection of systems which includes all parallel systems, but the collection is shown to be strictly larger. We also prove that no coherent system preserves the NWUE class. As byproducts of our study, we obtain the following results for the case of i.i.d. component lifetimes: (a) the DFR (decreasing failure rate) class is preserved by no coherent systems other than series systems, and (b) the IMRL (increasing mean residual life) class is not preserved by any coherent systems. Generalizations to the case of dependent component lifetimes are briefly discussed.

Quantile based tests for exponentiality against DMRQ and NBUE alternatives

Quantile-based reliability analysis has received much attention recently. We propose new quantile-based tests for exponentiality against decreasing mean residual quantile function (DMRQ) and new better than used in expectation (NBUE) classes of alternatives. The exact null distribution of the test statistic is derived when the alternative class is DMRQ. The asymptotic properties of both the test statistics are studied. The performance of the proposed tests with other existing tests in the literature is evaluated through simulation study. Finally, we illustrate our test procedure using real data sets.

Design and Analysis of Medium Access Protocol: Throughput and Short-Term Fairness Perspective

We consider a simple MAC protocol, called the renewal access protocol (RAP), that adopts all of the legacy 802.11 standard but the backoff stage feature. To meet two objectives in the design of the RAP—optimal throughput and high short-term fairness—we develop a mathematical model of the RAP and rigorously analyze the performance of the RAP. First, we show that the throughput performance of the RAP depends only on the expectation of the selection distribution where the backoff counter is selected, provided that the number of terminals is fixed, which is in accordance with a well-known result. Second, with the help of renewal and reliability theories, we analyze the short-term fairness of the RAP. We also show that if the RAP has a selection distribution of the New Better than Used in Expectation (NBUE) type, the RAP can guarantee high short-term fairness. Third, we construct a special binomial distribution that is obviously of the NBUE type that can achieve high short-term fairness as well as optimal throughput when used as the selection distribution of the RAP. Furthermore, by the Poisson approximation for binomial distributions, we propose to use in practice a Poisson distribution corresponding to the special binomial distribution. Numerical and simulation results are provided to validate our analysis.

Generalized Variability Orderings among Nonnegative Fuzzy Random Variables

The variability ordering for more and less variables of fuzzy random variables in terms of its distribution function is defined. A property of new better than used in expectation (NBUE) and new worse than used in expectation (NWUE) is derived as an application to the variability ordering of fuzzy random variables. The concept of generalized variability orderings of nonnegative fuzzy random variables representing lifetime of components is introduced. The<Pdomination is a generalized variability ordering. We proposed an integral inequality to the case of fuzzy random variables using<Pordering. The results included equivalent conditions which justify the generalized variability orderings.

Computing the Throughput of Probabilistic and Replicated Streaming Applications

In this paper, we investigate how to compute the throughput of probabilistic and replicated streaming applications. We are given (i) a streaming application whose dependence graph is a linear chain; (ii) a one-to-many mapping of the application onto a fully heterogeneous target platform, where a processor is assigned at most one application stage, but where a stage can be replicated onto a set of processors; and (iii) a set of random variables modeling the computation and communication times in the mapping. We show how to compute the throughput of the application, i.e., the rate at which data sets can be processed. The problem is easy when application stages are not replicated, i.e., each application stage is assigned to a single processor: in that case the throughput is dictated by the critical hardware resource. However, when stages are replicated, i.e., each application stage may be assigned to several processors, the problem becomes surprisingly complicated: even in the deterministic case, the optimal throughput may be lower than the smallest internal resource throughput.The first contribution of the paper is to provide a general method to compute the throughput when computation and communication times, also called stage parameters, are constant or follow I.I.D. exponential laws. The second contribution is to provide bounds for the throughput when stage parameters form associated random sequences (correlation between communication and processing times of a given data set on the different application stages, i.e., a data set that takes a long time on the first stage is likely to be large, and to take a long time on the next stages), and are N.B.U.E. (New Better than Used in Expectation) variables (if an operation has already been processed for some duration, the remaining time is smaller than the processing time of a fresh operation): the throughput is bounded from below by the exponential case and bounded from above by the deterministic case. An extensive set of simulation allows us to assess the quality of the model, and to observe the actual behavior of several distributions.

Estimation of distributions with the new better than used in expectation property

A lifetime X with survival function S, mean residual life function (MRL) M, and finite mean μ is said to be new better than used in expectation (NBUE) if M(t)≤μ for all t≥0. We propose a new estimator for S, based on a natural estimator of M defined under the NBUE restriction. This is much simpler to implement than the only other restricted estimator in the literature. We also derive some asymptotic properties of the MRL of X and extend our results to the censored case.

Tests for exponentiality against NBUE alternatives: a Monte Carlo comparison

Testing of various classes of life distributions has been addressed in the literature for more than 45 years. In this paper, we consider the problem of testing exponentiality (which essentially implies no ageing) against positive ageing which is captured by the fairly large class of new better than used in expectation (NBUE) distributions. These tests of exponentiality against NBUE alternatives are discussed and compared. The empirical size of the tests is obtained by simulations. Power comparisons for different popular alternatives are done using Monte Carlo simulations. These comparisons are made for both small and large sample sizes. The paper concludes with a discussion in which suggestions are made regarding the choices of the test when a particular alternative is suspected.

Sharp bounds for NBUE distributions

Let F be a new better than used in expectation (NBUE) distribution function with mean μ. In a previous paper (Brown in Probab. Eng. Inf. Sci. 20:195–230, 2006), the author derived the following bound. For any t≥μ, $$\overline{F}(t) = \mathit{Pr}(X \ge t) \le e^{-[{t\over\mu}-1]}. $$ The main result of this paper is to show that this bound is sharp. Other sharp bounds for NBUE distributions are also derived.

SOME NEW RELIABILITY BOUNDS FOR SUMS OF NBUE RANDOM VARIABLES

In this article, we discuss some new upper and lower bounds for the survivor function of the sum ofnindependent random variables each of which has an NBUE (new better than used in expectation) distribution. In some cases, only the means of the random variables are assumed known. These bounds are compared to the sharp bounds given in Cheng and Lam [6], which requires both means and variances known. Although the new bounds are not sharp, they often produce better upper bounds for the survivor function in the extreme right tail of many NBUE lifetime distributions, an important special case in applications. Moreover, a lower bound exists in one case not handled by the lower bounds of Theorem 3 in Cheng and Lam [6]. Numerical studies are presented along with theoretical discussions.

A class of goodness-of-fit tests based on a new characterization of the exponential distribution

A new characterisation of the exponential distribution in the class of new better than used in expectation (NBUE) life distributions is presented. Utilising this characterisation, a new class of goodness-of-fit tests for exponentiality against NBUE alternatives is proposed. The tests are shown to be consistent, and the limiting distributions of the test statistics under the null and alternative hypotheses are derived. The newly proposed tests are compared to existing goodness-of-fit tests by means of Pitman and approximate Bahadur relative efficiencies. A limited Monte Carlo study is conducted to compare the various tests with regard to power for small and moderate sample sizes against a range of alternative distributions. Three members of the class of test statistics are identified as being at least as effective as established tests for exponentiality against NBUE alternatives.

Relating Time and Customer Averages for Queues Using ‘forward’ Coupling from the Past

We give a new method for simulating the time average steady-state distribution of a continuous-time queueing system, by extending a ‘read-once’ or ‘forward’ version of the coupling from the past (CFTP) algorithm developed for discrete-time Markov chains. We then use this to give a new proof of the ‘Poisson arrivals see time averages’ (PASTA) property, and a new proof for why renewal arrivals see either stochastically smaller or larger congestion than the time average if interarrival times are respectively new better than used in expectation (NBUE) or new worse than used in expectation (NWUE).

Relating Time and Customer Averages for Queues Using ‘forward’ Coupling from the Past

We give a new method for simulating the time average steady-state distribution of a continuous-time queueing system, by extending a ‘read-once’ or ‘forward’ version of the coupling from the past (CFTP) algorithm developed for discrete-time Markov chains. We then use this to give a new proof of the ‘Poisson arrivals see time averages’ (PASTA) property, and a new proof for why renewal arrivals see either stochastically smaller or larger congestion than the time average if interarrival times are respectively new better than used in expectation (NBUE) or new worse than used in expectation (NWUE).

Models and Algorithms for Stochastic Online Scheduling

We consider a model for scheduling under uncertainty. In this model, we combine the main characteristics of online and stochastic scheduling in a simple and natural way. Job processing times are assumed to be stochastic, but in contrast to traditional stochastic scheduling models, we assume that jobs arrive online, and there is no knowledge about the jobs that will arrive in the future. The model incorporates both stochastic scheduling and online scheduling as a special case. The particular setting we consider is nonpreemptive parallel machine scheduling, with the objective to minimize the total weighted completion times of jobs. We analyze simple, combinatorial online scheduling policies for that model, and derive performance guarantees that match performance guarantees previously known for stochastic and online parallel machine scheduling, respectively. For processing times that follow new better than used in expectation (NBUE) distributions, we improve upon previously best-known performance bounds from stochastic scheduling, even though we consider a more general setting.

Bounds of moment generating functions of some life distributions

In this article we show that if a life has new better than used in expectation (NBUE) ageing property and if the mean life is finite then the moment generating function exists and is finite. In fact, the moment generating function is shown to be bounded above by that of the exponential distribution with the same mean. Analogous results are also proven for two much bigger families of life distribution, namely, the new better than renewal used in expectation (NBRUE) and the renewal new is better than used in expectation (RNBUE) and the renewal new better than renewal used in expectation (RNBRUE), provided that the life has finite two moments. Further, stronger results are also obtained for the smaller new better than used version of the above classes.

A Kolmogorov-Smirnov Type Test for Shortfall Dominance Against Parametric Alternatives

This paper proposes a Kolmogorov-type test for the shortfall order (also known in the literature as the right-spread or excess-wealth order) against parametric alternatives. In the case of the null hypothesis corresponding to the Negative Exponential distribution, this provides a test for the new better than used in expectation (NBUE) property. Such a test is particularly useful in reliability applications as well as duration and income distribution analysis. The theoretical properties of the testing procedure are established. Simulation studies reveal that the test proposed in this paper performs well, even with moderate sample sizes. Applications to real data, namely chief executive officer (CEO) compensation data and flight delay data, illustrate the empirical relevance of the techniques described in this paper.

Reliability bounds for the NBUE distributions

AbstractIn this paper, we study upper and lower bounds on the reliability in new better than used in expectation (NBUE) life distribution class with fixed first two moments. By a constructive proof, we determine the upper bounds on the reliability analytically in different regions and show that these bounds are sharp. For the lower bounds, similar results are obtained except in one region. For that region, a conjecture is given for further study. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 781–797, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10035

An Optimum Ordering Policy and its Estimation in a Two-Supplier Inventory Model with Uncertainties in Supply

We consider a single-item one-period inventory model where the market demand is assumed to be random and the item under consideration is obtained from two suppliers who also supply random amounts with means equal to the order placed to the supplier concerned. Under new better than used in expectation (NBUE) assumption on the supply distributions, possibly different, a strategy which maximizes a minimum profit has been proposed. An estimate for this maximin order quantity whenever the (customer) demand distribution is unknown has been obtained and strong consistency of the suggested estimator established. AMS (2000) Subject Classification: Primary 90B05; Secondary 62G99, 60F15.

New better than used in expectation processes

In this paper, we study the new better than used in expectation (NBUE) and new worse than used in expectation (NWUE) properties of Markov renewal processes. We show that a Markov renewal process belongs to a more general class of stochastic processes encountered in reliability or maintenance applications. We present sufficient conditions such that the first-passage times of these processes are new better than used in expectation. The results are applied to the study of shock and repair models, random repair time processes, inventory, and queueing models.

New better than used in expectation processes

In this paper, we study the new better than used in expectation (NBUE) and new worse than used in expectation (NWUE) properties of Markov renewal processes. We show that a Markov renewal process belongs to a more general class of stochastic processes encountered in reliability or maintenance applications. We present sufficient conditions such that the first-passage times of these processes are new better than used in expectation. The results are applied to the study of shock and repair models, random repair time processes, inventory, and queueing models.