New Worse 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 $(d=1)$.

Monotonicity properties and the deficit at ruin in the Sparre Andersen model

Let H u (y) be the (proper) distribution function of the deficit at ruin, given that ruin occurs with initial surplus u, in the Sparre Andersen model of risk theory. Dickson & dos Reis (1996) discussed the monotonicity of H u (y) as a function of u. In this paper, we obtain various monotonicity results for H u (y) and other related quantities for the decreasing/increasing failure rate (DFR/IFR) and the increasing/decreasing mean residual lifetime (IMRL/DMRL) classes of distributions. These results in particular extend and make more concrete the results of Dickson & dos Reis (1996) and Willmot & Lin (1998). A new class of distributions (increasing convolution ratio; ICR) is introduced. This class extends the well-known class of distributions with IFR. Specifically, we show that if the ladder height distribution F in the model is ICR, the ratio is a non-decreasing function of u, where ψ(u) denotes the ruin probability and . Further, we obtain generalizations (expressed in terms of the distribution of the deficit) of the well-known new worse than used (NWU) property of the probability of non-ruin.

Resource allocation in grid computing

Grid computing, in which a network of computers is integrated to create a very fast virtual computer, is becoming ever more prevalent. Examples include the TeraGrid and Planet-lab.org, as well as applications on the existing Internet that take advantage of unused computing and storage capacity of idle desktop machines, such as Kazaa, SETI@home, Climateprediction.net, and Einstein@home. Grid computing permits a network of computers to act as a very fast virtual computer. With many alternative computers available, each with varying extra capacity, and each of which may connect or disconnect from the grid at any time, it may make sense to send the same task to more than one computer. The application can then use the output of whichever computer finishes the task first. Thus, the important issue of the dynamic assignment of tasks to individual computers is complicated in grid computing by the option of assigning multiple copies of the same task to different computers. We show that under fairly mild and often reasonable conditions, maximizing task replication stochastically maximizes the number of task completions by any time. That is, it is better to do the same task on as many computers as possible, rather than assigning different tasks to individual computers. We show maximal task replication is optimal when tasks have identical size and processing times have a NWU (New Worse than Used; defined later) distribution. Computers may be heterogeneous and their speeds may vary randomly, as is the case in grid computing environments. We also show that maximal task replication, along with a c μ rule, stochastically maximizes the successful task completion process when task processing times are exponential and depend on both the task and computer, and tasks have different probabilities of completing successfully.

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.

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.

NWU property of a class of random sums

In this note, we derive an inequality for the renewal process. Then, using this inequality, together with an identity in terms of the renewal process for the tails of random sums, we prove that a class of random sums is always new worse than used (NWU). Thus, the well-known NWU property of geometric sums is extended to the class of random sums. This class is illustrated by some examples, including geometric sums, mixed geometric sums, certain mixed Poisson distributions and certain negative binomial sums.

NWU property of a class of random sums

In this note, we derive an inequality for the renewal process. Then, using this inequality, together with an identity in terms of the renewal process for the tails of random sums, we prove that a class of random sums is always new worse than used (NWU). Thus, the well-known NWU property of geometric sums is extended to the class of random sums. This class is illustrated by some examples, including geometric sums, mixed geometric sums, certain mixed Poisson distributions and certain negative binomial sums.

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.

An Optimal Design Problem for Limited Processor Sharing Systems

An optimal design problem is considered for processor-sharing service systems, where the number of customers served simultaneously is limited to a fixed finite number (called multiplicity). The decision variable is the multiplicity and the optimal criterion is to minimize the mean number of customers in the system, or, alternatively, to minimize the mean sojourn time for a customer. It is proved that the processor-sharing discipline with any multiplicity ameliorates the system performance if a service-requirement distribution is an NWU (new worse than used) type and deteriorates the performance if the distribution type is Erlangian. Moreover, it is conjectured through numerical results that the limited processor-sharing system will ameliorate (deteriorate) the performance if the coefficient of variation of the service-requirement distribution is larger (smaller) than 1.

Characterization of Some First Passage Times Using Log-Concavity and Log-Convexity as Aging Notions

An interpretation of log-concavity and log-convexity as aging notions is given in this paper. It imitates a stochastic ordering characterization of the NBU (new better than used) and the NWU (new worse than used) notions but stochastic ordering is now replaced by the likelihood ratio ordering. The new characterization of log-concavity and log-convexity sheds new light on these properties and enables one to obtain intuitively simple proofs of the log-convexity and log-concavity of some first passage times of interest in branching processes and in reliability theory.

Stochastic ordering and a class of multivariate new better than used distributions

A new characterization for the univariate class of new better than used ‘NBU’ distributions in terms of stochastic ordering is introduced. A multivariate version of this characterization is then used to define a multivariate class of NBU distributions. Basic properties of this class are derived. Comparisons and relationships of this new class with earlier classes are developed. Two multivariate new worse than used (NWU) classes of life distributions are defined and compared and their basic properties are studied.