The so called “Israeli Queue” is a single server polling system with batch service of an unlimited size, where the next queue to be visited is the one in which the first customer in line has been waiting for the longest time. The case with finite number of queues (groups) was introduced by Boxma, Van der Wal and Yechiali [3]. In this paper we extend the model to the case with a (possibly) infinite number of queues. We analyze the M/M/1, M/M/c, and M/M/1/N—type queues, as well as a priority model with (at most) M high-priority classes and a single lower priority class. In all models we present an extensive probabilistic analysis and calculate key performance measures.

In this paper we show how the permutation Monte Carlo method, originally developed for reliability networks, can be successfully adapted for stochastic flow networks, and in particular for estimation of the probability that the maximal flow in such a network is above some fixed level, called the threshold. A stochastic flow network is defined as one, where the edges are subject to random failures. A failed edge is assumed to be erased (broken) and, thus, not able to deliver any flow. We consider two models; one where the edges fail with the same failure probability and another where they fail with different failure probabilities. For each model we construct a different algorithm for estimation of the desired probability; in the former case it is based on the well known notion of the D-spectrum and in the later one—on the permutational Monte Carlo. We discuss the convergence properties of our estimators and present supportive numerical results.

This paper carries out stochastic comparisons of series and parallel systems with independent and heterogeneous components in the sense of the hazard rate order, the reversed hazard rate order, and the likelihood ratio order. The main results extend and strengthen the corresponding ones by Misra and Misra [18] and by Ding, Zhang, and Zhao [8]. Meanwhile, the results on the hazard rate order of parallel systems and the reversed hazard order of series systems serve as nice supplements to Theorem 16.B.1 of Boland and Proschan [4] and Theorem 3.2 of Nanda and Shaked [20], respectively.

In Dynamic Programing, mixed strategies consist of randomizing the choice of actions. In some problems, such as portfolio management, it makes sense to diversify actions rather than choosing among them purely or randomly. Optimal betting in casinos and roulette by a gambler with fixed goal was studied by Dubins and Savage [9] and their school without the element of diversification (betting simultaneously on different holes of the roulette), once it was proved (Smith's theorem - Smith [16], Dubins [8] and Gilat and Weiss [10]) that diversification does not increase the probability of reaching the goal. We question the scope of this finding, that was based on the assumption that the holes on which gamblers can bet are disjoint, such as 1 and BLACK in regular roulette. A counter example is provided in which holes are nested, such as 1 and RED. Thus, it may be rational for gamblers with a fixed goal to place chips on more than one hole at the table.

This note is related to a joint work with Michèle Cohen on the preference for safety in the Choquet Expected Utility model.

Many modern networks grow from blocks. We study the probabilistic behavior of parameters of a blocks tree, which models several kinds of networks. It grows from building blocks that are themselves rooted trees. We investigate the number of leaves, depth of nodes, total path length, and height of such trees. We use methods from the theory of Pólya urns and martingales.

This paper considers generalized birthday problems, in which there are d classes of possible outcomes. A fraction fi of the N possible outcomes has probability αi/N, where $\sum_{i=1}^{d} f_{i} =\sum_{i=1}^{d} f_{i}\alpha_{i}=1$. Sampling k times (with replacements), the objective is to determine (or approximate) the probability that all outcomes are different, the so-called uniqueness probability (or: no-coincidence probability). Although it is trivial to explicitly characterize this probability for the case d=1, the situation with multiple classes is substantially harder to analyze.

Parameterizing k≡ aN, it turns out that the uniqueness probability decays essentially exponentially in N, where the associated decay rate ζ follows from a variational problem. Only for small d this can be solved in closed form. Assuming αi is of the form 1+φiɛ, the decay rate ζ can be written as a power series in ɛ; we demonstrate how to compute the corresponding coefficients explicitly. Also, a logarithmically efficient simulation procedure is proposed. The paper concludes with a series of numerical experiments, showing that (i) the proposed simulation approach is fast and accurate, (ii) assuming all outcomes equally likely would lead to estimates for the uniqueness probability that can be orders of magnitude off, and (iii) the power-series based approximations work remarkably well.

We consider the social welfare model of Naor [20] and revenue-maximization model of Chen and Frank [7], where a single class of delay-sensitive customers seek service from a server with an observable queue, under state dependent pricing. It is known that in this setting both revenue and social welfare can be maximized by a threshold policy, whereby customers are barred from entry once the queue length reaches a certain threshold. However, no explicit expression for this threshold has been found. This paper presents the first derivation of the optimal threshold in closed form, and a surprisingly simple formula for the (maximum) revenue under this optimal threshold. Utilizing properties of the Lambert W function, we also provide explicit scaling results of the optimal threshold as the customer valuation grows. Finally, we present a generalization of our results, allowing for settings with multiple servers.

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.