We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a finite linear combination of geometric terms and present conditions on the structure of these linear combinations such that the resulting measure may yield an invariant measure of a random walk. We demonstrate that each geometric term must individually satisfy the balance equations in the interior of the state space and further show that the geometric terms in an invariant measure must have a pairwise-coupled structure. Finally, we show that at least one of the coefficients in the linear combination must be negative.

For each n ≥ 1, let Πn = (π1, π2, …, πn) be a random permutation of the integers 1,2, …, n. The quantity Bn = Σ measures the degree of oscillation of Πn and is an important measure in the study of sorting algorithms. In this work the asymptotic normality of Bn is derived under the assumption that, for each n ≥ 1, Πn is uniformly distributed over the n! permutations of 1,2, …, n. The proof relies on coupling (π1, π2, …, πn) with a sequence of independent and identically distributed U(0,1) random variables and offers a more probabilistic and computationally simpler alternative to the method of moments proof of Chao, Bai, and Liang (1993, Probability in the Engineering and Informational Sciences 7: 227–235).

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.

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.

Let $\left\{ {{\bi X}_k = {(X_{1,k},X_{2,k})}^{\top}, k \ge 1} \right\}$ be a sequence of independent and identically distributed random vectors whose components are allowed to be generally dependent with marginal distributions being from the class of extended regular variation, and let $\left\{ {{\brTheta} _k = {(\Theta _{1,k},\Theta _{2,k})}^{\top}, k \ge 1} \right\}$ be a sequence of nonnegative random vectors that is independent of $\left\{ {{\bi X}_k, k \ge 1} \right\}$. Under several mild assumptions, some simple asymptotic formulae of the tail probabilities for the bidimensional randomly weighted sums $\left( {\sum\nolimits_{k = 1}^n {\Theta _{1,k}} X_{1,k},\sum\nolimits_{k = 1}^n {\Theta _{2,k}} X_{2,k}} \right)^{\rm \top }$ and their maxima $({{\max} _{1 \le i \le n}}\sum\nolimits_{k = 1}^i {\Theta _{1,k}} X_{1,k},{{\max} _{1 \le i \le n}}\sum\nolimits_{k = 1}^i {\Theta _{2,k}} X_{2,k})^{\rm \top }$ are established. Moreover, uniformity of the estimate can be achieved under some technical moment conditions on $\left\{ {{\brTheta} _k, k \ge 1} \right\}$. Direct applications of the results to risk analysis are proposed, with two types of ruin probability for a discrete-time bidimensional risk model being evaluated.

We prove Hwang and Yao's conjecture about failure of consecutive-k-out-of-n systems whose components have independent and identically distributed increasing failure rate (IFR) lifetimes, namely, that for each k ≥ 2 there exists nk such that for every n ≥ nk the system does not preserve IFR. For the cases k = 4 and 5, we present complete solutions. We present further conjectures.

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.

Chebyshev inequality estimates the probability for exceeding the deviation of a random variable from its mathematical expectation in terms of the variance of the random variable. In modern probability theory, the Chebyshev inequality is the most frequently used tool for proving different convergence processes; for example, it plays a fundamental role in proofs of various forms of laws of large numbers. The mathematical expression of the bound on the probability in the Chebyshev inequality is very simple and can be modified easily for different kinds of sequence of random variables (e.g., for the case of sums of independent random variables). This fact lies behind these frequent applications. In this setting, the Chebyshev inequality has pure theoretical “applications” in probability theory and its role is to provide “a guarantee” of convergence but not to give a bound on concrete probability content.

In the present article we consider the Chebyshev inequality as a probability bound that is essential for the translation from its conventional theoretical applications to the practical setting if easy-to-compute multivariate generalizations are derived.

Such an inequality for the random vectors having multivariate Normal distribution is proved. The new inequality gives a lower bound in terms of variances on the probability that the random vector in question falls into an Euclidean ball with center at mean vector. The need and importance of consideration of this kind of multivariate Chebyshev inequality stemmed from several problems in engineering and informational sciences (Hassibi and Boyd [9], Jeng [10], Jeng and Woods [11], Molina, Katseggelos, Mateos, Hermoso, and Segall [13]). Jeng [10] derived an inequality that gives an upper bound for the probability in question. The simultaneous application of the established multivariate Chebyshev inequality and Jeng's inequality is useful in practical problems by providing lower and upper bounds on the probability content.

The inequality is attractive by its being easy to compute and its similarity to the original Chebyshev inequality, in contrast to well-known complicated multivariate Chebyshev inequalities. The present article also gives some insights into the very origin of the Chebyshev inequality, which makes the article self-contained.

An improved simulation estimator of the expected total delay of the first n customers in the systems GI/M/k and GI/G/I is obtained by replacing the (raw data) delay of customer i by its expected delay given the state of the system upon its arrival. It is improved in the sense that the sum of the first n such values has the same mean and smaller variance than the usual (raw) estimator. The only constraint on the arrival process is that it be independent of the process of service times.

Recently, Sherman et al. [14] analyzed an M/G/1 retrial queuing model in which customers are forced to retry their service if interrupted by a server failure. Using classical techniques, they provided a stability analysis, queue length distributions, key performance parameters, and stochastic decomposition results. We analyze the system under a static Bernoulli routing policy that routes a proportion of arriving customers directly to the orbit when the server is busy or failed. In addition to providing the key performance parameters, we show that this system exhibits a dual stability structure, and we characterize the optimal Bernoulli routing policy that minimizes the total expected holding costs per unit time.

Tasks belonging to N classes arrive for processing in a multi-server facility. Each arriving task has a due date (deterministic or random) associated with the completion of its service. If the service of a task is completed at a time other than the task's due date, an earliness or tardiness penalty is incurred. We determine properties of dynamic nonidling nonpreemptive and dynamic nonidling preemptive scheduling strategies that minimize an infinite horizon expected discounted cost due to the earliness and tardiness penalties. We provide examples that illustrate the properties of the optimal strategies.

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.

In this article, we observe that processes with the uniform order statistics property (UOSP) can be characterized by the condition that their first n epoch times have a joint [ell ]∞≤-spherical density, n ≥ 1. Some related results, and some further properties of [ell ]∞≤-spherical densities, are also given. We also extend some of the results regarding the UOSP to the more general (not necessarily uniform) order statistics property. Finally, we develop a theory of discrete-time discrete-state processes with the UOSP, where the need to consider multiple jumps, at a single time point, arises.

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.

We use the Energy Packet Network (EPN) to investigate an optimal energy distribution problem for the computer-communication system which is powered by intermittent renewable energy sources. The objective is to find an optimal energy distribution to minimize the proposed cost function which computes penalty costs caused by the overall average response time of jobs and the energy loss. In this EPN system, we consider the energy can be lost through storage leakages, or due to empty workstations which will consume energy even no job needs to be processed. Related numerical examples with different sets of parameter values are presented in the paper to evaluate the system performance and to examine the obtained analytical solution. Then a special case is considered to study the optimal system performance when the total energy harvesting rate is sufficiently large.

The relative aging is an important notion which is useful to measure how a system ages relative to another one. Among the existing stochastic orders, there are two important orders describing the relative aging of two systems, namely, aging faster orders in the cumulative hazard and the cumulative reversed hazard rate functions. In this paper, we give some sufficient conditions under which one coherent system ages faster than another one with respect to the aforementioned stochastic orders. Further, we show that the proposed sufficient conditions are satisfied for k-out-of-n systems. Moreover, some numerical examples are given to illustrate the applications of proposed results.

In many scheduling applications, minimizing delays is of high importance. One adverse effect of such delays is that the reward for completion of a job may decay over time. Indeed in healthcare settings, delays in access to care can result in worse outcomes, such as an increase in mortality risk. Motivated by managing hospital operations in disaster scenarios, as well as other applications in perishable inventory control and information services, we consider non-preemptive scheduling of jobs whose internal value decays over time. Because solving for the optimal scheduling policy is computationally intractable, we focus our attention on the performance of three intuitive heuristics: (1) a policy which maximizes the expected immediate reward, (2) a policy which maximizes the expected immediate reward rate, and (3) a policy which prioritizes jobs with imminent deadlines. We provide performance guarantees for all three policies and show that many of these performance bounds are tight. In addition, we provide numerical experiments and simulations to compare how the policies perform in a variety of scenarios. Our theoretical and numerical results allow us to establish rules-of-thumb for applying these heuristics in a variety of situations, including patient scheduling scenarios.

We consider a single-item, two-echelon spare parts inventory model for repairable parts for capital goods with high downtime costs. The inventory system consists of multiple local warehouses, a central warehouse, and a central repair facility. When a part at a customer fails, if possible his request for a ready-for-use part is fulfilled by his local warehouse. Also, the failed part is sent to the central repair facility for repair. If the local warehouse is out of stock, then, via an emergency shipment, a ready-for-use part is sent from the central warehouse if it has a part in stock. Otherwise, it is sent via a lateral transshipment from another local warehouse, or via an emergency shipment from the external supplier. We assume Poisson demand processes, generally distributed leadtimes for replenishments, repairs, and emergency shipments, and a basestock policy for the inventory control.

Our inventory system is too complex to solve for a steady-state distribution in closed form. We approximate it by a network of Erlang loss queues with hierarchical jump-over blocking. We show that this network has a product-form steady-state distribution. This enables an efficient heuristic for the optimization of basestock levels, resulting in good approximations of the optimal costs.

In Balaji and Mahmoud [1], the authors introduced a distance-based Gini index for rooted trees. In this paper, we introduce a degree-based Gini index (or just simply degree Gini index) for graphs. The latter index is a topological measure on a graph capturing the proximity to regular graphs. When applied across the random members of a class of graphs, we can identify an average measure of regularity for the class. Whence, we can compare the classes of graphs from the vantage point of closeness to regularity.

We develop a simplified computational formula for the degree Gini index and study its extreme values. We show that the degree Gini index falls in the interval [0, 1). The main focus in our study is the degree Gini index for the class of binary trees. Via a left-packing transformation, we show that, for an arbitrary sequence of binary trees, the Gini index has inferior and superior limits in the interval [0, 1/4]. We also show, via the degree Gini index, that uniform rooted binary trees are more regular than binary search trees grown from random permutations.

The Rate Conservation Law (RCL) of Miyazawa [18] is generalized to what we call a General Rate Conservation Law (GRCL) to cover processes of unbounded variation such as Brownian motion and more general Levy processes. The general setup is that of a time-stationary semimartingale Y = [Yt: t ≥ 0], which is allowed to have jumps. From an elementary application of Ito's formula together with the Palm inversion formula, we obtain a law that includes Miya-zawa's RCL as a special case. A variety of applications and connections with the RCL are given. For example, we show that using the GRCL, one can immediately obtain the noted steady-state decomposition results for vacation queueing models, including those obtained by Kella and Whitt [13] for Jump-Levy processes. Other examples include state-dependent diffusion processes such as the Ornstein-Uhlenbeck process.