We consider the problem of a firm facing failures with weak forewarning signals. In the base model that we study, the firm watches for signals of a random arrival of a disruptive innovation and continuously updates the posterior probability that a disruptive innovation has already happened. A disruptive innovation is marked by a rapid increase in the growth rate of the market for a new technology, and it is followed by a random arrival of catastrophic failure of the firm. The firm can invest capital to adopt the innovation to prevent failure. The optimal policy is to adopt it when the posterior probability exceeds an optimally chosen threshold. We investigate the probability of failure under the optimal policy when the cost of failure is large and the arrival rate of disruptive innovation is low. The probability of failure is close to one if the arrival rate is extremely low while it is close to zero if the arrival rate is moderate. We also consider an extension of the base model to incorporate recurrence of disruptive innovation; when the arrival rate is moderate, the optimal threshold and the failure probability can be significantly larger than those of the base model.

In this article, we study comparison theorems for stochastic functionals like V(∞;C) = sup0≤t {M(t) − C(t)} or V(T;C) = sup0≤t≤T {M(t) − C(t)}, where {M(t)} and {C(t)} are two independent nondecreasing processes with stationary increments. We will tacitly assume that the considered stochastic functionals are proper random variables. We prove that V(T;C′) ≤icxV(T;C) ≤icxV(T;C′′), where and C′′(dt) = c(0) dt, provided dC(t) is absolute continuous with density c(t). Similarly, we show that V(∞;C′) ≤icxV(∞;C) ≤icxV(∞;C′′). For proofs, we develop the theory of the ≤idcv ordering defined by increasing directionally concave functions. We apply the theory to M/G/1 priority queues and M/G/1 queues with positive and negative customers.

Mobile networks are universally used for personal communications, but also increasingly used in the Internet of Things and machine-to-machine applications in order to access and control critical services. However, they are particularly vulnerable to signaling storms, triggered by malfunctioning applications, malware or malicious behavior, which can cause disruption in the access to the infrastructure. Such storms differ from conventional denial of service attacks, since they overload the control plane rather than the data plane, rendering traditional detection techniques ineffective. Thus, in this paper we describe the manner in which storms happen and their causes, and propose a detection framework that utilizes traffic measurements and key performance indicators to identify in real-time misbehaving mobile devices. The detection algorithm is based on the random neural network which is a probabilistic computational model with efficient learning algorithms. Simulation results are provided to illustrate the effectiveness of the proposed scheme.

This article provides analytical results on which are the implications of the statistical dependencies among certain random parameters on the variability of the number of susceptibles of the carrier-borne epidemic model with heterogeneous populations and of the number of infectives under the Reed–Frost model with random infection rates. We consider dependencies among the random infection rates, among the random infectious times, and among random initial susceptibles of several carrier-borne epidemic models. We obtain conditions for the variability ordering between the number of susceptibles for carrier-borne epidemics under two different random environments, at any time-scale value. These results are extended to multivariate comparisons of the random vectors of populations in the strata. We also obtain conditions for the increasing concave order between the number of infectives in the Reed–Frost model under two different random environments, for any generation. Variability bounds are obtained for different epidemic models from modeling dependencies for a range of special cases that are useful for risk assessment of disease propagation.

Motivated by hydrological applications, the exact distributions of R = X + Y, P = XY, and W = X/(X + Y) and the corresponding moment properties are derived when X and Y follow Block and Basu's bivariate exponential distribution. An application of the results is provided to drought data from Nebraska.

We consider the optimal scheduling of a finite capacity shuttle in a two-node network with imperfect information. When shuttle trips do not depend on the number of passengers carried, we prove optimality and monotonicity of threshold policies. We derive conditions for dispatching that reduce the computational effort required to compute an optimal threshold policy. We prove a counter-example to the optimality of threshold policies for finite horizon problems where trip lengths increase stochastically in the number of passengers carried.

In this paper, we introduce a new class of set-valued risk measures which satisfies cash sub-additivity. Dual representation for them is provided. Moreover, we also investigate dynamic set-valued cash sub-additive risk measures and discuss the corresponding multi-portfolio time consistency. The equivalent characterization of the multi-portfolio time consistency is given. Finally, an example is also given to illustrate the introduction of set-valued cash sub-additive risk measures. The present paper can be considered as a set-valued extension of scalar cash sub-additive risk measures introduced by El Karouii and Ravanelli [8].

A pool design is random if it varies according to a probability distribution. There are four types of random design proposed in the literature: random incidence design, random k-set design, random distinct k-set design, and random k-size design. Recently Hwang gave an approximation to estimate the number of unresolved positives for random distinct k-set design. In this article, we give exact formulas for all four types of random designs for estimating the number of unresolved positives. We also do some numerical comparisons of the four designs.

This paper shows that the equilibrium distribution of a queueing network with batch routing continues to be of product form if a batch that cannot enter its destination — for example, as a consequence of capacity constraints — immediately triggers a new transition that takes up the whole batch.

In this article notions of exponential dominance and uncertainty for weighted and unweighted distributions are explored and used to compare values of the informational energy function and the differential entropy. Stochastic inequalities and bounds for cross-discrimination and uncertainty measures in weighted and unweighted residual life distribution functions and related reliability measures are presented. Moment-type inequalities for the comparisons of weighted and unweighted residual life distributions are also presented.

We consider asset allocation strategies for the case where an investor can allocate his wealth dynamically between a risky stock, whose price evolves according to a geometric Brownian motion, and a risky bond, whose price is subject to negative jumps due to its credit risk and therefore has discontinuous sample paths. We derive optimal policies for a number of objectives related to growth. In particular, we obtain the policy that minimizes the expected time to reach a given target value of wealth in an exact explicit form. We also show that this policy is exactly equivalent to the policy that is optimal for maximizing logarithmic utility of wealth and, hence, the expected average rate at which wealth grows, as well as to the policy that maximizes the actual asymptotic rate at which wealth grows. Our results generalize and unify results obtained previously for cases where the bond was risk-free in both continuous- and discrete-time.

This paper considers the customers’ equilibrium and socially optimal joining–balking behavior in a single-server Markovian queue with a single working vacation and Bernoulli interruptions. The model is motivated by practical service systems where the service rate can be adjusted according to whether or not the system is empty. Specifically, we focus on a single-server queue in which the server's service rate is reduced from a regular to a lower one when the system becomes empty. This lower rate period is called a working vacation for the server which may represent that part of the service facility is under a maintenance process or works on other non-queueing job, or simply for saving the energy (for a machine server case). In this paper, we assume that the working vacation period is terminated after a random period or with probability p after serving a customer in a non-empty system. Such a system is called a queue with single working vacation and Bernoulli interruptions. Customers are strategic and can make choice of joining or balking based on different levels of system information. We consider four scenarios: fully observable, almost observable, almost unobservable, and fully unobservable queue cases. Under a reward-cost structure, we analyze the customer's equilibrium and social-optimal strategies. In addition, the effects of system parameters on optimal strategies are illustrated by numerical examples.

We consider the problem of routing incoming airplanes to two runways of an airport. Due to air turbulence, the necessary separation time between two successive landing operations depends on the type of airplane. When viewed as a queuing problem, this means that we have dependent service times. The aim is to minimize the waiting times of aircrafts. We consider here a model in which arrivals form a stochastic process and the decision-maker does not know anything about future arrivals. We formulate this as a problem of stochastic dynamic programming and investigate the monotonicity of optimal routing strategies with respect to the workload of the runways, for example. We show that an optimal strategy is monotone (i.e., of switching type) only in a restricted case where decisions depend on the state of the runways only and not on the type of the arriving aircraft. Surprisingly, in the more realistic case where this type is also known to the decision-maker, monotonicity need not hold.

We present a decision procedure to determine the optimal maintenance intervals of a computer software throughout its operational phase. Our model accounts for the average cost per each maintenance activity and the damage cost per failure with the future cost discounted. Our decision policy is optimal in the sense that it minimizes the expected total cost. Our model assumes that the total number of errors in the software has a Poisson distribution with known mean λ and each error causes failures independently of other errors at a known constant failure rate. We study the structures of the optimal policy in terms of λ and present efficient numerical algorithms to compute the optimal maintenance time intervals, the optimal total number of maintenances, and the minimal total expected cost throughout the maintenance phase.

Recently, Ross [1] proposed an elegant method of approximating transition probabilities and mean occupation times in continuous-time Markov chains based upon recursively inspecting the process at exponential times. The method turned out to be amazingly efficient for the examples investigated. However, no formal rough error bound was provided. Any error bound even though robust is of practical interest in engineering (e.g., for determining truncation criteria or setting up an experiment). This note primarily aims to show that by a simple and standard comparison relation a rough error bound of the method is secured. Also, some alternative approximations are inspected.

Discrete-time queues are frequently used in telecommunication networks. This paper proposes a new method which is based only on probabilistic arguments. It derives the distributions of numbers of customers in the discrete-time systems Geom/G/1 and Geom/G/1/M. Though the derivation does not involve use of the transforms, the transforms may be obtained, if desired. Another advantage of this is that the numerical results obtained are stable as there are no negative signs involved in summations. Further, the method can be easily used to solve more complex problems in discrete- and continuous-time queues.

This article considers the optimization and optimality of single-item/location, infinite-horizon, (s,S) inventory models. Departing from the conventional approach, we do not assume the loss function describing holding and shortage costs per period to be quasiconvex. As the existing optimization algorithms have been established on the condition of quasiconvexity, our goal in this article is to develop a computational procedure for obtaining optimal (s,S) policies for models with general loss functions. Our algorithm is based on the parametric method commonly used in fractional programming and is intuitive, exact, and efficient. Moreover, this method allows us to extend the optimality of (s,S) policies to a broader class of loss functions that can be non-quasiconvex.

Since the introduction of the concept of coherent systems and the description of the reliability of such systems in terms of the reliabilities of the components, the concept of importance of a component has created a new and fruitful area of research. Two distinct concepts of importance can be found in the literature. We take the view that the importance of a component or a module that is part of a system can be derived directly from the role of the component or the module in the failure of the system. Here again, it is possible that there will be several definitions of role. In this paper we define the role of a module (or component) to be the probability that the module is among all the modules (or components) that failed at the time of system failure. The role of a module depends on the structure of the system in terms of the modules, the structure of the module in terms of its components and the distribution of lifetimes of the components. In this paper we study the role of a module under several structures and distributions for lifetimes. We establish various monotonicity properties and indicate applications of these properties to optimal allocation. Another quantity that describes the nature of the components in sustaining the system is the number of components that fail at the time of the failure of the system. We establish monotonicity properties for the expected number of failed components and also indicate applications to optimal allocation.

For a Markov chain on a finite partially ordered state space, we show that its Siegmund dual exists if and only if the chain is Möbius monotone. This is an extension of Siegmund's result for totally ordered state spaces, in which case the existence of the dual is equivalent to the usual stochastic monotonicity. Exploiting the relation between the stationary distribution of an ergodic chain and the absorption probabilities of its Siegmund dual, we present three applications: calculating the absorption probabilities of a chain with two absorbing states knowing the stationary distribution of the other chain; calculating the stationary distribution of an ergodic chain knowing the absorption probabilities of the other chain; and providing a stable simulation scheme for the stationary distribution of a chain provided we can simulate its Siegmund dual. These are accompanied by concrete examples: the gambler's ruin problem with arbitrary winning/losing probabilities; a non-symmetric game; an extension of a birth and death chain; a chain corresponding to the Fisher–Wright model; a non-standard tandem network of two servers, and the Ising model on a circle. We also show that one can construct a strong stationary dual chain by taking the appropriate Doob transform of the Siegmund dual of the time-reversed chain.