We introduce an incremental growth model for directed binary series-parallel (SP) graphs. The vertices of a directed binary SP graphs can only have outdgrees 1 or 2. We show that the number of vertices of outdegree 1 have a normal distribution (so, necessarily, the vertices of outdegree 2 have a normal distribution, too). Furthermore, we study the average length of a random walk between the poles of the graph. The asymptotic equivalent of the latter property includes the golden ratio. Pólya urns will systematically provide a coding method to initiate the studies.

Various large-scale suppliers frequently use web-based spot markets, along with discount stores and foreign distributors, for inventory liquidation. Recognizing the potential benefits of such practices, we consider a multi-period, integrated replenishment, and liquidation problem for a capacitated supplier facing stochastic demand from a spot market along with its primary market (with higher priority contractual customers). In each period, the supplier must decide: (i) how much to produce, and (ii) if there are excess units left after sales to the primary market, how many of these to liquidate. We show that the optimal policy is characterized by two quantities: the critical produce-up-to level and the critical retain-up-to level. We establish bounds for these two quantities. We identify two practical benchmark policies and establish thresholds on the unit revenue earned from the spot market such that one of the two benchmark policies is optimal. We provide closed form expressions to determine these thresholds for the infinite horizon problem under specific conditions on the available production capacity. In general, it is difficult, if not impossible, to theoretically determine these thresholds in closed form for the finite horizon problem. Hence, we report results of a computational study to gain insights regarding the behavior of the optimal policy with respect to the spot market revenue. Our computational results also quantify the benefits of the optimal policy relative to the benchmark policies and examine the effects of demand correlation.

Unimodality and strong unimodality of the distribution of ascendingly ordered random variables have been extensively studied in the literature, whereas these properties have not received much attention in the case of descendingly ordered random variates. In this paper, we show that log concavity of the reversed hazard rate implies that of the density function. Using this fundamental result, we establish some convexity properties of such random variables. To do this, we first provide a counterexample showing that a claim of Basak & Basak [7] about the lower record values is not valid. Then, we provide conditions under which unimodality properties of the distribution of lower k-record values would hold. Finally, some extensions to dual generalized order statistics in both univariate and multivariate cases are discussed.

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.

This paper focuses on a discrete-time risk model in which both insurance risk and financial risk are taken into account. We study the asymptotic behavior of the ruin probability and the tail probability of the aggregate risk amount. Precise asymptotic formulas are derived under weak moment conditions of involved risks. The main novelty of our results lies in the quantification of the impact of the financial risk.

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.

A recent paper by Pozdnyakov and Steele [6] is devoted to distributed sensor networks in which the sensors send summarized information to some remotely located fusion agents. Main focus is on sensors that send their information according to a so-called binary-plus-passive design. Some exact relations in a sequential approach, aimed at making decisions before a given budget is exhausted are presented. In this paper, we provide asymptotic results in the case of more general sensors for increasing budgets. The essential ingredient is the observation that the setting can be modeled within the framework of certain stopped two-dimensional random walks. We also provide asymptotics for the cost for large decision boundaries, and, finally, some comments on the probability of a positive/negative decision as well as on the duration of the process until a decision is made in the binary-plus-passive design.

Skellam's name is traditionally attached to the distribution of the difference of two independent Poisson random variables. Many bivariate extensions of this distribution are possible, e.g., through copulas. In this paper, the authors focus on a probabilistic construction in which two Skellam random variables are affected by a common shock. Two different bivariate extensions of the Skellam distribution stem from this construction, depending on whether the shock follows a Poisson or a Skellam distribution. The models are nested, easy to interpret, and yield positive quadrant-dependent distributions, which share the convolution closure property of the univariate Skellam distribution. The models can also be adapted readily to account for negative dependence. Closed form expressions for Pearson's correlation between the components make it simple to estimate the para-meters via the method of moments. More complex formulas for Kendall's tau and Spearman's rho are also provided.

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.