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.

We consider sufficient conditions for stochastic equivalence of convex ordered random variables. Our main results apply to all convex ordered distributions on the real line and improve on a recent result of Huang and Lin [8] for equality in distribution of convex ordered survival times. Illustrative applications include testing for equality in distribution with convex ordered alternatives and demonstrating several earlier results on stochastic equivalence as special cases.

Recent developments in stochastic modeling show that enormous analytical advantages can be gained if a general cumulative distribution function (c.d.f.) can be approximated by generalized hyperexponential distributions. In this paper, we introduce a procedure to explicitly construct such approximations of an arbitrary c.d.f. Although our approach can be used in different types of stochastic models, the main motivation comes from queueing theory in obtaining approximations of the idle-period distribution and other performance measures in GI/G/1 queues.

A practical source coding scheme based on approximate string matching is proposed. It is an approximate fixed-length string matching data compression combined with a block-coder based on the empirical distribution. A lemma on approximate string matching, which is an extension of the Kac Lemma, is proved. It is shown, based on the lemma, that the deterministic algorithm converts the stationary and ergodic source, u, into an output process v, and under the assumption that v is a stationary process, after the scheme has run for an infinite time, the optimal compression ratio R(D) is achieved. This reduces the problem of the universal lossy coder to the proof of stationarity of the output process ν in the proposed algorithm. The main advantages of the proposed method are the asymptotic sequential behavior of the encoder and the simplicity of the decoder.

Tasks belonging to N priority classes arrive for processing in a single or multiserver facility. If the processing does not begin by a certain time (deterministic or random), the task is lost and a cost is incurred. We determine properties of dynamic, nonidling, nonpreemptive strategies that minimize an infinite horizon expected cost.

Copula is one of the widely used techniques to describe the dependency structure between components of a system. Among all existing copulas, the family of Archimedean copulas is the popular one due to its wide range of capturing the dependency structures. In this paper, we consider the systems that are formed by dependent and identically distributed components, where the dependency structures are described by Archimedean copulas. We study some stochastic comparisons results for series, parallel, and general $r$-out-of-$n$ systems. Furthermore, we investigate whether a system of used components performs better than a used system with respect to different stochastic orders. Furthermore, some aging properties of these systems have been studied. Finally, some numerical examples are given to illustrate the proposed results.

We consider an infinite series of independent and identical -/GI/∞ queues fed by an arbitrary stationary and ergodic arrival process, A1. Let Ai be the arrival process to the ith node, and let νi be the law of Ai. Denote by (·) the input–output map of the ·/GI/∞ node; that is, νi+1 = (νi). It is known that the Poisson process is a fixed point for . In this paper, we are interested in the asymptotic distribution of the departure process from the nth node, νn+1 = n(ν1), as n ∞. Using couplings for random walks, this limiting distribution is shown to be either a Poisson process or a stationary ν-Poisson process, depending on the joint distribution of A1 and the service process. This generalizes a result of Vere-Jones (1968, Journal of the Royal Statistical Society, Series B 30: 321–333) and is similar in flavor to Mountford and Prabhakar (1995, Annals of Applied Probability 5(1): 121–127), where Poisson convergence is established for departures from a series of exponential server queues using coupling methods.

For discrete versions of red-and-black with a goal, for timid play to be optimal, it suffices, but is not necessary, that no win probability be less than one-half. A condition that is both necessary and sufficient is provided.

We model the virtual contention queue in an ATM nonblocking packet switch with capacity c and input queues by a Geox1, Geox2/D/c Head-of-Line priority queueing system with Random Order Selection within each class and find the joint queue length distribution and the waiting time distribution for each class.

We consider a monotone multistate system with conditionally independent components given the component reliabilities, and random component reliabilities. Upper and lower bounds are derived for the moments of the random reliability function, extending results for binary systems. The second moment is given special attention, as this quantity is used to calculate the standard deviation of the system reliability estimate. The motivation for the paper is to establish a basis for uncertainty analysis and Bayesian estimation in connection with multistate systems.

We give a probabilistic proof of an identity concerning the expectation of an arbitrary function of a compound random variable and then use this identity to obtain recursive formulas for the probability mass function of compound random variables when the compounding distribution is Poisson, binomial, negative binomial random, hypergeometric, logarithmic, or negative hypergeometric. We then show how to use simulation to efficiently estimate both the probability that a positive compound random variable is greater than a specified constant and the expected amount by which it exceeds that constant.

In this paper, we develop a new method for evaluating the reliability polynomial of a hammock network. The method is based on a homogeneous absorbing Markov chain and provides the exact reliability for networks of width less than 5 and arbitrary length. Moreover, it produces a lower bound for the reliability polynomial for networks of width greater than or equal to 5. To investigate how sharp this lower bound is, we compare our method with other approximation methods and it proves to be the most accurate in terms of absolute as well as relative error. Using the fundamental matrix, we also calculate the average time to absorption, which provides the mean length of a network that is expected to work.

The study of the distributions of sums of dependent risks is a key topic in actuarial sciences, risk management, reliability and in many branches of applied and theoretical probability. However, there are few results where the distribution of the sum of dependent random variables is available in a closed form. In this paper, we obtain several analytical expressions for the distribution of the aggregated risks under dependence in terms of copulas. We provide several representations based on the underlying copula and the marginal distribution functions under general hypotheses and in any dimension. Then, we study stochastic comparisons between sums of dependent risks. Finally, we illustrate our theoretical results by studying some specific models obtained from Clayton, Ali-Mikhail-Haq and Farlie-Gumbel-Morgenstern copulas. Extensions to more general copulas are also included. Bounds and the limiting behavior of the hazard rate function for the aggregated distribution of some copulas are studied as well.

In the auto insurance industry, a Bonus-Malus System (BMS) is commonly used as a posteriori risk classification mechanism to set the premium for the next contract period based on a policyholder's claim history. Even though the recent literature reports evidence of a significant dependence between frequency and severity, the current BMS practice is to use a frequency-based transition rule while ignoring severity information. Although Oh et al. [(2020). Bonus-Malus premiums under the dependent frequency-severity modeling. Scandinavian Actuarial Journal 2020(3): 172–195] claimed that the frequency-driven BMS transition rule can accommodate the dependence between frequency and severity, their proposal is only a partial solution, as the transition rule still completely ignores the claim severity and is unable to penalize large claims. In this study, we propose to use the BMS with a transition rule based on both frequency and size of claim, based on the bivariate random effect model, which conveniently allows dependence between frequency and severity. We analytically derive the optimal relativities under the proposed BMS framework and show that the proposed BMS outperforms the existing frequency-driven BMS. Later, numerical experiments are also provided using both hypothetical and actual datasets in order to assess the effect of various dependencies on the BMS risk classification and confirm our theoretical findings.

We study the steady-state queue-length vector in a multi-class queue with relative priorities. Upon service completion, the probability that the next served customer is from class k is controlled by class-dependent weights. Once a customer has started service, it is served without interruption until completion. We establish a state-space collapse for the scaled queue-length vector in the heavy-traffic regime, that is, in the limit the scaled queue-length vector is distributed as the product of an exponentially distributed random variable and a deterministic vector. We observe that the scaled queue length reduces as classes with smaller mean service requirement obtain relatively larger weights. We finally show that the scaled waiting time of a class-k customer is distributed as the product of two exponentially distributed random variables.

Suppose each permutation (πl,…,πn) of ( 1, …, n) has probability 1/n!. The oscillation of (πl; …, πn) is defined as Tn = | πk − πk+1|, where πn+1 = π1. It is proved that (Tn − ETn)/(var Tn)1/2 converges in distribution to N(0,1). The connection between the oscillation and the presortedness measure is also discussed.

The random neural network (RNN) is a probabilitsic queueing theory-based model for artificial neural networks, and it requires the use of optimization algorithms for training. Commonly used gradient descent learning algorithms may reside in local minima, evolutionary algorithms can be also used to avoid local minima. Other techniques such as artificial bee colony (ABC), particle swarm optimization (PSO), and differential evolution algorithms also perform well in finding the global minimum but they converge slowly. The sequential quadratic programming (SQP) optimization algorithm can find the optimum neural network weights, but can also get stuck in local minima. We propose to overcome the shortcomings of these various approaches by using hybridized ABC/PSO and SQP. The resulting algorithm is shown to compare favorably with other known techniques for training the RNN. The results show that hybrid ABC learning with SQP outperforms other training algorithms in terms of mean-squared error and normalized root-mean-squared error.

We establish the quasireversibility of a certain exponential multiserver queue. The queuing discipline imposes restrictions on the numbers of customers of different classes and subclasses that can be served simultaneously. Customer loss is also allowed. Due to its quasireversibility the queue can be used as a node in a product form network. Our work generalizes work of Le Boudec, Krzesinski, and Crosby.

Simulated annealing is a probabilistic algorithm for approximately solving large combinatorial optimization problems. The algorithm can mathematically be described as the generation of a series of Markov chains, in which each Markov chain can be viewed as the outcome of a random experiment with unknown parameters (the probability of sampling a cost function value). Assuming a probability distribution on the values of the unknown parameters (the prior distribution) and given the sequence of configurations resulting from the generation of a Markov chain, we use Bayes's theorem to derive the posterior distribution on the values of the parameters. Numerical experiments are described which show that the posterior distribution can be used to predict accurately the behavior of the algorithm corresponding to the next Markov chain. This information is also used to derive optimal rules for choosing some of the parameters governing the convergence of the algorithm.

It is known that the lifetime of a k−out−of−n system has increasing failure rate (IFR) if all of its components have independent and identically distributed IFR lifetimes. Derman, Lieberman, and Ross raised the same question for consecutive−k−out−of−n systems. But the scarcity of results gives no clue as to whether most of such systems have IFR's. In this paper, we completely solve the k = 2 and k = 3 cases. We prove that these systems, with a handful of exceptions, do not have IFR's (contradicting the result of Derman, Lieberman, and Ross that a consecutive-2-out-of-n cycle has IFR for any n). Our result suggests the conjecture that for every fixed k there exists nk such that for every n ≧ nk a consecutive-k−out−of−n system does not have IFR. We also prove that for every fixed d there exists nd large enough such that for all n ≧ nd a consecuive.(n – d)−out−of−n system has IFR. In particular, if the system is a line, then nd = 3d +1 for d ≧ 1.