Let F be a bathtub-shaped (BT) hazard rate distribution function. It has been shown that the hazard rate function of the order statistics may be BT, increasing, etc. Then, we have carried out a graphical study for some useful lifetime models.

Moreover, we are interested to compare the time that maximizes the mean residual life (MRL) function of F with the one related to a general weighted model in terms of their locations. Also, the times maximizing the conditional reliability proposed by Mi [13] of F have been compared with the corresponding times of a general weighted model. As special cases, we consider order statistics and the proportional hazard rate model.

Let (X1, X2) and (Y1, Y2) be bivariate random vectors with a common marginal distribution (X1, X2) is said to be more positively dependent than (Y1, Y2) if E[h(X1)h(X2)] ≥ E[h(Y1)h(Y2)] for all functions h for which the expectations exist. The purpose of this paper is to study the monotonicity of positive dependence with time for a stationary reversible Markov chain [X1]; that is, (Xs, Xl+s) is less positively dependent as t increases. Both discrete and continuous time and both a denumerable set and a subset of the real line for the state space are considered. Some examples are given to show that the assertions established for reversible Markov chains are not true for nonreversible chains.

We obtain the best possible upper bounds for the moments of a single-order statistic from independent, nonnegative random variables, in terms of the population mean. The main result covers the independent identically distributed case. Furthermore, the case of the sample minimum for merely independent (not necessarily identically distributed) random variables is treated in detail.

Linear birth–death processes with immigration and emigration are major models in the study of population processes of biological and ecological systems, and their transient analysis is important in the understanding of the structural behavior of such systems. The spectral method has been widely used for solving these processes; see, for example, Karlin and McGregor [11]. In this article, we provide an alternative approach: the method of characteristics. This method yields a Volterra-type integral equation for the chance of extinction and an explicit formula for the z-transform of the transient distribution. These results allow us to obtain closed-form solutions for the transient behavior of several cases that have not been previously explicitly presented in the literature.

We study the stochastic optimal control for an assemble-to-order system with multiple products and components that arrive at the system in random batches and according to renewal reward processes. Our purpose is to maximize expected infinite-horizon discounted profit by selecting product prices, component production rates, and a dynamic sequencing rule for assembly. We refine the solution of some static planning problem and a discrete review policy to batch arrival environment and develop an asymptotically optimal policy for the system operating under heavy traffic, which indicates that the system can be approximated by a diffusion process and exhibits a state space collapse property.

Systems are studied that consist of interfering components which are alternatively active (busy) and passive (idle) for random periods. Such systems naturally arise from the performance evaluation of computer models. A concrete invariance condition is imposed on the interferences allowed. Under this condition, the steady-state vector of active components is shown to be of product form as well as to be robust to distributional forms of active and passive periods. The proof is simple and self-contained. A number of concrete and generic examples is provided. These include resource sharing mechanisms, hierarchical circuit switchings, parallel processing, priorities, and breakdowns.

An estimator, based on a simulation, is given for the expected number of events by time t of a renewal process. The estimator is obtained by combining the techniques of control variates and conditional expectation.

Using data obtained by the general capture/recapture sequential sampling process, an exact analytical expression for the maximum likelihood (ML) estimate of the population size, N, is introduced. As a consequence, it is shown that bounded likelihood functions have at most two maxima. For the simple one-by-one case the ML estimate is unique.

A simple random algorithm (SRA) is an algorithm whose behavior is governed by a first-order Markov chain. The expected time complexity of an SRA, given its initial state, is essentially the time to absorption of the underlying chain. The standard approach in computing the expected runtime is numerical. Under certain conditions on the probability transition matrix of an SRA, bounds on its expected runtime can be obtained using simple probabilistic arguments. In particular, one can obtain upper and lower (average time) logarithmic bounds for certain algorithms based on SRAs.

The use of the total hazard variable in simulation is studied. When estimating the mean time until a certain event occurs, this variable can be a powerful variance-reducing control variate. When estimating a small probability, it can be utilized as a direct estimator. Applications to reliability, quality control, and queueing are given.

We consider the problem of admission control to a multiserver finite buffer queue under partial information. The controller cannot see the queue but is informed immediately if an admitted customer is lost due to buffer overflow. Turning away (i.e., blocking) customers is costly and so is losing an admitted customer. The latter cost is greater than that of blocking. The controller's objective is to minimize the average cost of blocking and rejection per incoming customer. Lin and Ross [11] studied this problem for multiserver loss systems. We extend their work by allowing a finite buffer and the arrival process to be of the renewal type. We propose a control policy based on a novel state aggregation approach that exploits the regenerative structure of the system, performs well, and gives a lower bound on the optimal cost. The control policy is inspired by a simulation technique that reduces the variance of the estimators by not simulating the customer service process. Numerical experiments show that our bound varies with the load offered to the system and is typically within 1% and 10% of the optimal cost. Also, our bound is tight in the important case when the cost of blocking is low compared to the cost of rejection and the load offered to the system is high. The quality of the bound degrades with the degree of state aggregation, but the computational effort is comparatively small. Moreover, the control policies that we obtain perform better compared to a heuristic suggested by Lin and Ross. The state aggregation technique developed in this article can be used more generally to solve problems in which the objective is to control the time to the end of a cycle and the quality of the information available to the controller degrades with the length of the cycle.

The paper shows the connections between some importance indices for the components in an engineering coherent system and the performance of the system obtained when a redundancy mechanism is applied to a specific component. A copula approach is used to model the dependency among the components. This approach includes the popular case of independent components. Under some assumptions, it is proved that if component i is more important than component j, then the system obtained by applying a redundancy procedure to the ith component is better, under different stochastic criteria, than that obtained with the jth component. These results can be applied to several redundancy mechanisms. A new importance index is defined to study active redundancies. Some illustrative examples are provided.

We develop a method for computing the optimal double band [b, B] policy for switching between two diffusions with continuous rewards and switching costs. The two switch levels [b, B] are obtained as perturbations of the single optimal switching point a of the control problem with no switching costs. More precisely, we find that in the case of average reward problems the optimal switch levels can be obtained by intersecting two curves: (a) the function, γ(a), which represents the long-run average reward if we were to switch between the two diffusions at a and switches were free, and (b) a horizontal line whose height depends on the size of the transaction costs. Our semianalytical approach reduces, for example, the solution of a problem recently posed by Perry and Bar-Lev (1989, in Stochastic Analysis and Applications 7: 103–115) to the solution of one nonlinear equation.

The paper provides an analysis of G-network with positive customers and signals when signals arriving to the system move customer to another system or destroy in it a group of customers, reducing their number to a random value that is given by a probability distribution. The signal arriving to the system, in which there are no positive customers, does not exert any influence on the queueing network and immediately disappears from it. Streams of positive customers and signals arriving to each of the network systems are independent. Customer in the transition from one system to another brings the latest some revenue, and the revenue of the first system is reduced by this amount. A method of finding the expected revenues of the systems of such a network has been proposed. The case when the revenues from transitions between network states are deterministic functions depending on its states has been considered. A description of the network is given, all possible transitions between network states, transition probabilities, and revenues from state transitions are indicated. A system of difference-differential equations for the expected revenues of network systems has been obtained. To solve it, we propose a method of successive approximations, combined with the method of series. It is proved that successive approximations converge to the stationary solution of such a system of equations, and the sequence of approximations converges to a unique solution of the system. Each approximation can be represented as a convergent power series with an infinite radius of convergence, the coefficients of which are related by recurrence relations. Therefore, it is convenient to use them for calculations on a PC. The obtained results can be applied in forecasting losses in information and telecommunication systems and networks from the penetration of computer viruses into it and conducting computer attacks.

We study G-networks with positive and negative customers and signals. We consider two types of signals: they can make a subnetwork of queues operational or down. As signals are sent by queues after a customer service completion, one can model the availability of a sub-network of queues controlled by another network of queues. We prove that under classical assumptions for G-networks and assumptions on the rerouting probabilities when a subnetwork is not operational, the steady-state distribution, if it exists, has a product form steady state distribution. Some examples are given.

In this article we study the commute and hitting times of simple random walks on spherically symmetric random trees in which every vertex of level n has outdegree 1 with probability 1−qn and outdegree 2 with probability qn. Our argument relies on the link between the commute times and the effective resistances of the associated electric networks when 1 unit of resistance is assigned to each edge of the tree.

We consider Bayes' estimation of the number of independent homogeneous Poisson processes of a superimposed process with unknown rates. The estimation of the total rate of the undetected processes is also considered. Exact posterior distributions are obtained. Monotonicity and asymptotic properties of the Bayes' estimator are also discussed.

We consider the problem of routing and admission control in a loss system featuring two classes of arriving jobs (high-priority and low-priority jobs) and two types of servers, in which decision-making for high-priority jobs is forced, and rewards influence the desirability of each of the four possible routing decisions. We seek a policy that maximizes expected long-run reward, under both the discounted reward and long-run average reward criteria, and formulate the problem as a Markov decision process. When the reward structure favors high-priority jobs, we demonstrate that there exists an optimal monotone switching curve policy with slope of at least −1. When the reward structure favors low-priority jobs, we demonstrate that the value function, in general, lacks structure, which complicates the search for structure in optimal policies. However, we identify conditions under which optimal policies can be characterized in greater detail. We also examine the performance of heuristic policies in a brief numerical study.

Nakamoto doublespend strategy, described in Bitcoin foundational article, leads to total ruin with positive probability. The simplest strategy that avoids this risk incorporates a stopping threshold when success is unlikely. We compute the exact profitability and the minimal double spend that is profitable for this strategy. For a given amount of the transaction, we determine the minimal number of confirmations to be requested by the recipient that makes the double-spend strategy non-profitable. This number of confirmations is only 1 or 2 for average transactions and for a small relative hashrate of the attacker. This is substantially lower than the original Nakamoto number, which is about six confirmations and is widely used. Nakamoto analysis is only based on the success probability of the attack instead of on a profitability analysis that we carry out.

Discrete-time renewal sequences play a fundamental role in the theory of stochastic processes. This article considers periodic versions of such processes; specifically, the length of an interrenewal is allowed to probabilistically depend on the season at which it began. Using only elementary renewal and Markov chain techniques, computational and limiting aspects of periodic renewal sequences are investigated. We use these results to construct a time series model for a periodically stationary sequence of integer counts.