In the present article, we consider a class of reliability structures that can be efficiently described through a finite Markov chain (Markov chain imbeddable systems) and investigate its closeness with respect to the increasing failure rate (IFR) property. More specifically we derive a sufficient condition for the system's lifetime to have increasing failure rate when the identical and independent components comprising it own this property. As an application of the general theory, we establish an alternative proof of the IFR property preservation for the k-out-of-n system and derive some related results for the family of weighted k-out-of-n systems.

An efficient simulation estimator of the expected time until all states of a finite state semi-Markov process have been visited is determined.

For the birth–death process on a finite state space with bilateral boundaries, we give a simpler derivation of the hitting time distributions by h-transform and φ-transform. These transforms can then be used to construct a quick derivation of the hitting time distributions of the minimal birth–death process on a denumerable state space with exit/regular boundaries.

De Bruijn's identity shows a link between two fundamental concepts in information theory: entropy and Fisher information. In the literature, De Bruijn's identity has been stated under the assumption of independence between input signal and an additive noise. However, in the real world, the noise could be highly dependent on the main signal. The main aim of this paper is, firstly, to extend De bruijn's identity for signal-dependent noise channels and, secondly, to study how Stein and heat identities are related to De bruijn's identity. Thus, new versions of De Bruijn's identity are introduced in the case when input signal and additive noise are dependent and are jointly distributed according to Archimedean and Gaussian copulas. It is shown that in this generalized model, the derivatives of the differential entropy can be expressed in terms of a function of Fisher information. Our finding enfolds the conventional De Bruijn's identity as some remarks. Then, the equivalence among the new De Bruijn-type identity, Stein's identity and heat equation identity is established. The paper concludes with an application of the developed results in information theory.

In this article we consider a stochastic game in which each player draws one or two random numbers between 0 and 1. Players can decide to stop after the first draw or to continue for a second draw. The decision is made without knowing the other players’ numbers or whether the other players continue for a second draw. The object of the game is to have the highest total score without going over 1. In the article, we will characterize the optimal stopping rule for each player.

We consider M/G/1-type Markov chains where a transition that decreases the value of the level triggers the phase to a small subset of the phase space. We show how this structure—referred to as restricted downward transitions—can be exploited to speed up the computation of the stationary probability vector of the chain. To this end we define a new M/G/1-type Markov chain with a smaller block size, the G matrix of which is used to find the original chain's G matrix. This approach is then used to analyze the BMAP/PH/1 queue and the BMAP[2]/PH[2]/1 preemptive priority queue, yielding significant reductions in computation time.

We consider the filtering problem for a doubly stochastic Poisson or Cox process, where the intensity follows the Cox–Ingersoll–Ross model. In this article we assume that the Brownian motion, which drives the intensity, is not observed. Using filtering theory for point process observations, we first derive the dynamics for the intensity and its moment-generating function, given the observations of the Cox process. A transformation of the dynamics of the conditional moment-generating function allows us to solve in closed form the filtering problem, between the jumps of the Cox process as well as at the jumps, which constitutes the main contribution of the article. Assuming that the initial distribution of the intensity is of the Gamma type, we obtain an explicit solution to the filtering problem for all t>0. We conclude the article with the observation that the resulting conditional moment-generating function at time t, after Nt jumps, corresponds to a mixture of Nt+1 Gamma distributions. Currently, the model that we analyze has become popular in credit risk modeling, where one uses the intensity-based approach for the modeling of default times of one or more companies. In this approach, the default times are defined as the jump times of a Cox process. In such a model, one only has access to observations of the Cox process, and thus filtering comes in as a natural technique in credit risk modeling.

A large lot is to be sampled by a number of inspectors for the purpose of ascertaining its number of defective units. However, instead of a defective unit always being detected if it is sampled, there are unknown probabilities, depending on the inspectors, that the unit will not be identified as a defective. We offer an approach to estimating the number of defective units in the lot.

Mixture distributions are a frequently used tool in modelling random phenomena. We consider mixtures of densities from a one-parameter exponenvial family of distributions. Using the tools of totally positive functions and the variation-diminishing property of such, we study the effect of sign-crossing properties of two mixing densities μ1 and μ2 on the resulting mixture distributions f1 and f2. The results enable us to make stochastic and variability cornparisons for binomial-beta, mixed Weibull, and mixed gamma distributions.

Let X be a random vector and A a matrix. Let M be the maximal coordinate of the vector AX. For given marginal distributions of the coordinates of X, we present sharp bounds on the expectations of convex increasing functions of M. We derive joint distributions of X that achieve some of these bounds, and under these “worst case” distributions we study the joint distribution of M and the index of the largest coordinate of AX. Some possible applications are PERT network analysis and design of experiments.

This paper analyzes a scheduling system where a fixed number of nonpreemptive jobs is to be processed on multiple parallel processors with different processing speeds. Each processor has an exponential processing time distribution and the processors are ordered in ascending order of their mean processing times. Each job has its own deadline that is exponentially distributed with rate ß1, independent of the deadlines of other jobs and also independent of job processing times. A job departs the system as soon as either its processing completes or its deadline occurs. We show that there exists a simple threshold strategy that slochastically minimizes the total delay of all jobs. The policy depends on distributions of processing times and deadlines, but is independent of the rate of deadlines. When the rate of the deadline distribution is 0 (no deadlines), the total delay reduces to the flowtime (the sum of completion times of all jobs). If each job has its own probability of being correctly processed, then an extension of this policy stochastically maximizes the total number of correctly processed, nontardy jobs. We discuss possible generalizations and limitations of this result.

This paper proposes a pocket-calculator approximation to the normal tail that is more convenient and more accurate than Hamaker's [1] and Lew's [2] over a wide range, when measured in terms of the number of keystrokes and absolute relative error, respectively.

The classical regenerative method of simulation output analysis exploits the regenerative structure of a stochastic process to break up a path into independent and identically distributed cycles based on a single sequence of regeneration times. If a process is regenerative with respect to more than one sequence of regeneration times, the classical regenerative method does not exploit the additional structure, and the variance of the resulting estimator for certain performance measures (e.g., the time-average variance constant) can vary greatly, depending on the particular regeneration sequence chosen. In a previous article, we introduced an efficiency-improvement technique for regenerative simulation of processes having two sequences of regeneration times based on permuting regenerative cycles associated with the second sequence of regeneration points. In this article, we show how to exploit more than two regeneration sequences. In particular, for birth–death Markov chains, the regenerations associated with hitting times to each state can all be exploited. We present empirical results that show significant variance reductions in some cases, and the results seem to indicate that the permuted estimator for the time-average variance constant can have a variance that is independent of the primary regeneration sequence used to run the simulation.

We model the weight (e.g., delay, distance, or cost) from an arbitrary node to the nearest (in weight) peer in a peer-to-peer (P2P) network. The exact probability generating function and an asymptotic analysis is presented for a random graph with independent and identically distributed exponential link weights. The asymptotic distribution function is a Fermi–Dirac distribution that frequently appears in statistical physics. The good agreement with simulation results for relatively small P2P networks makes the asymptotic formula for the probability density function useful for estimating the minimal number of peers to offer an acceptable quality (delay or latency).

In this paper, we discuss the late-arrival discrete-time Geom(n)/G(n)/1/N queue with state-dependent arrival and service processes. Whereas the interarrival times are geometrically distributed, service times are conditioned on the system length at the moment of service initiation. The model has wide applications in computer-communications systems, broadband integrated service digital network, asynchronous transfer mode, and so on. The analysis of the model has been carried out, using the supplementary variable technique, and the final results are presented in the form of recursive equations that can be easily implemented on any personal computer. In addition, relations among state probabilities at prearrival, postdeparture, and random epochs have been developed. To demonstrate the effectiveness of our method, some numerical examples have been presented for service-time distributions such as geometric, deterministic, arbitrary, and mixed. The results obtained in this paper should be found useful by system designers who wish to control the congestion by adjusting the service rate if the arrival traffic changes.

In this paper we consider scheduling n tasks on m parallel machines where the task processing times are i.i.d. random variables with a common distribution function F. Scheduling is done by an a priori assignment of tasks to machines. We show that if the distribution function F is a Pólya frequency function of order 2 (decreasing reverse hazard rate) then the assignment that attempts to place an equal number of tasks on each machine achieves the stochastically smallest makespan among all assignments. The condition embraces many important distributions, such as the gamma and truncated normal distributions. Assuming that the task processing times have a common density that is a Pólya frequency function of order 2 (increasing likelihood ratio), then we find that flatter schedules have stochastically smaller makespans in the sense of the “joint” likelihood ratio.

Stochastic discretization is a technique of representing a continuous random variable as a random sum of i.i.d. exponential random variables. In this article, we apply this technique to study the limiting behavior of a stochastic fluid model. Specifically, we consider an infinite-capacity fluid buffer, where the net input of fluid is regulated by a finite-state irreducible continuous-time Markov chain. Most long-run performance characteristics for such a fluid system can be expressed as the long-run average reward for a suitably chosen reward structure. In this article, we use stochastic discretization of the fluid content process to efficiently determine the long-run average reward. This method transforms the continuous-state Markov process describing the fluid model into a discrete-state quasi-birth–death process. Hence, standard tools, such as the matrix-geometric approach, become available for the analysis of the fluid buffer. To demonstrate this approach, we analyze the output of a buffer processing fluid from K sources on a first-come first-served basis.

Here, we discuss the stochastic comparison of residual lifetimes of parallel systems and inactivity times of series systems by means of the reversed hazard rate order when the components of the systems are independent but not necessarily identically distributed. We also establish some monotonicity properties of such residual lifetimes of parallel systems and inactivity times of series systems. These results extend some of the recent results in this direction due to Zhao, Li, and Balakrishnan [21], Kochar and Xu [12], and Saledi and Asadi [16].

Suppose that there are n types of coupons and that each new coupon collected is type i with probability pi. Suppose, further, that there are m subsets of coupon types and that coupons are collected until all of the types of at least one of these subsets have been collected. When these subsets have no overlap, we derive expressions for the mean and variance of the number of coupons that are needed. In the general case where the subsets can overlap, we derive the mean of the number that are needed. We also note that this number is an increasing failure rate on average random variable and we present a conjecture as to a sufficient condition for it to be an increasing failure rate random variable.

In this work, we establish a connection between the cumulative residual entropy and the Gini mean difference (GMD). Some relationships between the extropy and the GMD, and the truncated GMD and dynamic versions of the cumulative past extropy are also established. We then show that several entropy and extropy measures discussed here can be brought into the framework of probability weighted moments, which would facilitate finding estimators of these measures.