New Better Than Used Research Articles

Achievable Stability in Redundancy Systems

We investigate the achievable stability region for redundancy systems and a quite general workload model with different job types and heterogeneous servers, reflecting job-server affinity relations which may arise from data locality issues and soft compatibility constraints. Under the assumption that job types are known beforehand we establish for New-Better-than-Used (NBU) distributed speed variations that no replication gives a strictly larger stability region than replication. Strikingly, this does not depend on the underlying distribution of the intrinsic job sizes, but observing the job types is essential for this statement to hold. In case of non-observable job types we show that for New-Worse-than-Used (NWU) distributed speed variations full replication gives a larger stability region than no replication.

Achievable Stability in Redundancy Systems

We consider a system with N~parallel servers where incoming jobs are immediately replicated to, say, d~servers. Each of the N servers has its own queue and follows a FCFS discipline. As soon as the first job replica is completed, the remaining replicas are abandoned. We investigate the achievable stability region for a quite general workload model with different job types and heterogeneous servers, reflecting job-server affinity relations which may arise from data locality issues and soft compatibility constraints. Under the assumption that job types are known beforehand we show for New-Better-than-Used (NBU) distributed speed variations that no replication $(d=1)$ gives a strictly larger stability region than replication $(d>1)$. Strikingly, this does not depend on the underlying distribution of the intrinsic job sizes, but observing the job types is essential for this statement to hold. In case of non-observable job types we show that for New-Worse-than-Used (NWU) distributed speed variations full replication ($d=N$) gives a larger stability region than no replication $(d=1)$.

Minimizing the Age of Information Through Queues

In this paper, we investigate scheduling policies that minimize the age of information in single-hop queueing systems. We propose a Last-Generated, First-Serve (LGFS) scheduling policy, in which the packet with the earliest generation time is processed with the highest priority. If the service times are i.i.d. exponentially distributed, the preemptive LGFS policy is proven to be age-optimal in a stochastic ordering sense. If the service times are i.i.d. and satisfy a New-Better-than-Used (NBU) distributional property, the non-preemptive LGFS policy is shown to be within a constant gap from the optimum age performance. These age-optimality results are quite general: (i) they hold for arbitrary packet generation times and arrival times (including out-of-order packet arrivals); (ii) they hold for multi-server packet scheduling with the possibility of replicating a packet over multiple servers; (iii) and they hold for minimizing not only the time-average age and mean peak age, but also for minimizing the age stochastic process and any non-decreasing functional of the age stochastic process. If the packet generation time is equal to the packet arrival time, the LGFS policies reduce to the Last-Come, First-Serve (LCFS) policies. Hence, the age optimality results of LCFS-type policies are also established.

The Age of Information in Multihop Networks

Information updates in multihop networks such as Internet of Things (IoT) and intelligent transportation systems have received significant recent attention. In this paper, we minimize the age of a single information flow in interference-free multihop networks. When preemption is allowed and the packet transmission times are exponentially distributed, we prove that a preemptive last-generated, first-served (LGFS) policy results in smaller age processes across all nodes in the network than any other causal policy (in a stochastic ordering sense). In addition, for the class of new-better-than-used (NBU) distributions, we show that the non-preemptive LGFS policy is within a constant age gap from the optimum average age. In contrast, our numerical result shows that the preemptive LGFS policy can be very far from the optimum for some NBU transmission time distributions. Finally, when preemption is prohibited and the packet transmission times are arbitrarily distributed, the non-preemptive LGFS policy is shown to minimize the age processes across all nodes in the network among all work-conserving policies (again in a stochastic ordering sense). Interestingly, these results hold under quite general conditions, including 1) arbitrary packet generation and arrival times, and 2) for minimizing both the age processes in stochastic ordering and any non-decreasing functional of the age processes.

Stability of Constant Retrial Rate Systems with NBU Input*

We study the stability of a single-server retrial queueing system with constant retrial rate, general input and service processes. First, we present a review of some relevant recent results related to the stability criteria of similar systems. Sufficient stability conditions were obtained by Avrachenkov and Morozov (2014), which hold for a rather general retrial system. However, only in the case of Poisson input is an explicit expression provided; otherwise one has to rely on simulation. On the other hand, the stability criteria derived by Lillo (1996) can be easily computed but only hold for the case of exponential service times. We present new sufficient stability conditions, which are less tight than the ones obtained by Avrachenkov and Morozov (2010), but have an analytical expression under rather general assumptions. A key assumption is that interarrival times belongs to the class of new better than used (NBU) distributions. We illustrate the accuracy of the condition based on this assumption (in comparison with known conditions when possible) for a number of non-exponential distributions.

On testing exponentiality against NBARFR life distribution

This paper considers testing exponentiality against new better than average renewal failure rate (NBARFR) alternatives. The percentiles of this test statistic are tabulated for simple sizes 5(1)50. Pitman's asymptotic efficiencies relative to the tests of the new better than used (NBU), new better than used failure rate (NBUFR) and new better than used renewal failure rate (NBURFR) (Amhmed, 1994; Hendi et al., 2000 and Mahmoud and Abdul Alim, 2002). The powers of this test are also calculated for some used life distributions. The problem when right-censored data is available is handled. Pratical applications of our tests in the medical sciences are present.

A general family of NBU classes of life distributions

In this paper we introduce and study a family of new better than used ageing notions parameterized by a function h . The new better than used ( NBU ) , the new better than used of a specified age t 0 ( NBU - t 0 ) and the new better than used in total time on test transform order ( NBUT ) , are special cases of this new family. We study some properties of the new family, and we give some applications of it in actuarial science, reliability theory and statistics.

TESTS FOR DETECTING MORE NBU-NESS AT A SPECIFIC AGE

Summary This paper proposes a class of non-parametric test procedures for testing the null hypothesis that two distributions, F and G, are equal versus the alternative hypothesis that F is ‘more NBU (new better than used) at specified age t0’ than G. Using Hoeffding's two-sample U-statistic theorem, it establishes the asymptotic normality of the test statistics and produces a class of asymptotically distribution-free tests. Pitman asymptotic efficacies of the proposed tests are calculated with respect to the location and shape parameters. A numerical example is provided for illustrative purposes.

A Simple and More Efficient New Approach to Life Testing

Testing of various classes of life distributions has long been a subject of investigation. Most of these procedures, however, are not easy to grasp by non-specialized users of statistics. Additionally, not all of these procedures have good efficiencies. In the present investigation, a new and much simpler approach is presented for testing exponentiality (the common null hypothesis in life testing) against two perennial life classes, namely the class of increasing failure rate (IFR) distributions, the class of new better than used (NBU) distributions and the class of new better than used of specified age (NBU − t s ). This new approach is very simple and is also more efficient for some commonly used alternatives than older and more involved methodologies in the sense of Pitman's asymptotic relative efficiency.

On applications of residual lifetimes of compound geometric convolutions

We demonstrate that the residual lifetime distribution of a compound geometric distribution convoluted with another distribution, termed a compound geometric convolution, is again a compound geometric convolution. Conditions under which the compound geometric convolution is new worse than used (NWU) or new better than used (NBU) are then derived. The results are applied to ruin probabilities in the stationary renewal risk model where the convolution components are of particular interest, as well as to the equilibrium virtual waiting time distribution in the G/G/1 queue, an approximation to the equilibrium M/G/c waiting time distribution, ruin in the classical risk model perturbed by diffusion, and second-order reliability classifications.

On applications of residual lifetimes of compound geometric convolutions

We demonstrate that the residual lifetime distribution of a compound geometric distribution convoluted with another distribution, termed a compound geometric convolution, is again a compound geometric convolution. Conditions under which the compound geometric convolution is new worse than used (NWU) or new better than used (NBU) are then derived. The results are applied to ruin probabilities in the stationary renewal risk model where the convolution components are of particular interest, as well as to the equilibrium virtual waiting time distribution in the G/G/1 queue, an approximation to the equilibrium M/G/c waiting time distribution, ruin in the classical risk model perturbed by diffusion, and second-order reliability classifications.

On the Coordinated Random Group Replacement Policy in Multivariate Repairable Systems

We study a system with multiple components and preventive maintenance. At predetermined times, some of the components are replaced with new ones. We permit the set of components that are chosen for replacement to be random and study the effect of dependency in this selection. For example, we show that it is often better to have simultaneous replacements of components, rather than replacing each component independently of the others. We also show that preventive maintenance only makes sense for components whose lifetimes are new-better-than-used (NBU).

Comparison of maintenance policies with monotone failure rate distributions

AbstractWe compare different maintenance policies assuming that the system lifetime has either increasing failure rate (IFR), or decreasing failure rate (DFR) distribution. We show that these assumptions yield strongly stochastic comparisons between the process' intensities. This yields weaker stochastic comparison between the processes than the stochastic comparison that hold when the system lifetime is new better than used (NBU) or new worse than used (NWU). Copyright © 2002 John Wiley & Sons, Ltd.

CLASSIFICATION OF MULTIVARIATE LIFE DISTRIBUTIONS BASED ON PARTIAL ORDERING

Barlow and Proschan presented some interesting connections between univariate classifications of life distributions and partial orderings where equivalent definitions for increasing failure rate (IFR), increasing failure rate average (IFRA), and new better than used (NBU) classes were given in terms of convex, star-shaped, and superadditive orderings. Some related results are given by Ross and Shaked and Shanthikumar. The introduction of a multivariate generalization of partial orderings is the object of the present article. Based on that concept of multivariate partial orderings, we also propose multivariate classifications of life distributions and present a study on more IFR-ness.

Testing whether a life distribution is new better than used in the tail

A new class of life distributions, called new better than used of age t 0 and beyond (NBU-[t 0, ∞)), is introduced. It includes the class of new better than used (NBU) life distribution and is included in the class of new better than used of a specified age t 0(NBU − t 0). Testing whether a life distribution is exponential in [t 0, ∞) versus that it is NBU there is presented both when t 0 is known or when it is unknown but can be estimated. The special case when t 0 is the p-th percentile of the distribution leads to an asymptotically distribution-free procedure whether it is known or not.

Aging properties and bounds for ruin probabilities and stop-loss premiums

In this paper, we derive relations between stop-loss premiums and their associated ruin probabilities by use of the new worse than used (NWU) aging property of the ruin probability. General upper and lower bounds for the stop-loss premium are derived. Also, we get a general upper bound for the ruin probability, the general upper bound is sharper than that of Willmot [Refinements and distributional generalizations of Lundberg's inequalities, Insurance: Mathematics and Economics 15 (1994) 49–63] and asymptotically sharper than that of Broeckx et al. [Ordering of risks and ruin probabilities, Insurance: Mathematics and Economics 5 (1986) 35–40]. The asymptotical behavior of these bounds is discussed. The relationships among sub-exponential, new better than used (NBU) distributions and upper bounds of the ruin probability and stop-loss premiums are considered.

On the relationship between bounds on the tails of compound distributions

A comparison of two upper bounds for the tails of compound distributions, both defined in terms of a new worse than used (NWU) distribution, reveals that one is sharper in the decreasing failure rate (DFR) case. An inductive argument is employed to construct a lower bound in terms of a new better than used (NBU) distribution which is a dual to the upper bound. It is also sharper than the corresponding lower bound in the increasing failure rate (IFR) case. Applications to ruin theory are then given.

Tail of compound distributions and excess time

Bounds on the tail of compound distributions are considered. Using a generalization of Wald's fundamental identity, we derive upper and lower bounds for various compound distributions in terms of new worse than used (NWU) and new better than used (NBU) distributions respectively. Simple bounds are obtained when the claim size distribution is NWUC, NBUC, NWU, NBU, IMRL, DMRL, DFR and IFR. Examples on how to use these bounds are given.

Tail of compound distributions and excess time

Bounds on the tail of compound distributions are considered. Using a generalization of Wald's fundamental identity, we derive upper and lower bounds for various compound distributions in terms of new worse than used (NWU) and new better than used (NBU) distributions respectively. Simple bounds are obtained when the claim size distribution is NWUC, NBUC, NWU, NBU, IMRL, DMRL, DFR and IFR. Examples on how to use these bounds are given.

Bounding a Probability Measure Over a Polymatroid with an Application to Transportation Problems

For M = {1, …, m} and a supermodular set function f: 2M → ℛ, define the polymatroid P(M, f) = {x: ∑j∈A xj ≥ f(A), A ⊆ M; x ∈ ℛm}. We develop a bound for the probability P{X ∈ P(M, f)}, where X is a random m-vector. In particular we show that if X, X1, …, Xm, are independent and identically distributed nonnegative random variables with new better than used (NBU) distribution function, then P{X ∈ P(M, f)} ≥ P{X ≥ f(M̂)}, where M̂ = arg max {f(A): A ⊆ M}. We apply this result to transportation problems with random supply and/or demand. Suppose we have a set K = {1, …, k} of k supply nodes with a random supply Si, i ∈ K and a set N = {1, …, n} of n demand nodes with a random demand Dj, j ∈ N. Let β be the probability that the random demand can be met by the random supply. If the demand and supply are mutually independent, D1, …, Dn, are independent, and S1, …, Sk, are independent and have NBU distribution function, then β ≤ ∏j=1n P{Zj ≥ Dj}, where Zj = ∑i∈K Si rij, j∈N and rij takes the value one if node i can supply the demand node j and zero otherwise (i ∈ K and j ∈ N).