Exactly Computing the Tail of the Poisson-Binomial Distribution

We present ShiftConvolvePoibin, a fast exact method to compute the tail of a Poisson-binomial distribution (PBD). Our method employs an exponential shift to retain its accuracy when computing a tail probability, and in practice we find that it is immune to the significant relative errors that other methods, exact or approximate, can suffer from when computing very small tail probabilities of the PBD. The accompanying R package is also competitive with the fastest implementations for computing the entire PBD.

Analysis of a Discrete-Time Queueing Model with Disasters

Queueing models with disasters can be used to evaluate the impact of a breakdown or a system reset in a service facility. In this paper, we consider a discrete-time single-server queueing system with general independent arrivals and general independent service times and we study the effect of the occurrence of disasters on the queueing behavior. Disasters occur independently from time slot to time slot according to a Bernoulli process and result in the simultaneous removal of all customers from the queueing system. General probability distributions are allowed for both the number of customer arrivals during a slot and the length of the service time of a customer (expressed in slots). Using a two-dimensional Markovian state description of the system, we obtain expressions for the probability, generating functions, the mean values, variances and tail probabilities of both the system content and the sojourn time of an arbitrary customer under a first-come-first-served policy. The customer loss probability due to a disaster occurrence is derived as well. Some numerical illustrations are given.

The critical window in random digraphs

We consider the component structure of the random digraph D(n,p) inside the critical window $p = n^{-1} + \lambda n^{-4/3}$ . We show that the largest component $\mathcal{C}_1$ has size of order $n^{1/3}$ in this range. In particular we give explicit bounds on the tail probabilities of $|\mathcal{C}_1|n^{-1/3}$ .

Delay in a 2-State Discrete-Time Queue with Stochastic State-Period Lengths and State-Dependent Server Availability and Arrivals

In this paper, we consider a discrete-time multiserver queueing system with correlation in the arrival process and in the server availability. Specifically, we are interested in the delay characteristics. The system is assumed to be in one of two different system states, and each state is characterized by its own distributions for the number of arrivals and the number of available servers in a slot. Within a state, these numbers are independent and identically distributed random variables. State changes can only occur at slot boundaries and mark the beginnings and ends of state periods. Each state has its own distribution for its period lengths, expressed in the number of slots. The stochastic process that describes the state changes introduces correlation to the system, e.g., long periods with low arrival intensity can be alternated by short periods with high arrival intensity. Using probability generating functions and the theory of the dominant singularity, we find the tail probabilities of the delay.

TAIL BEHAVIOR OF STOPPED LÉVY PROCESSES WITH MARKOV MODULATION

This article concerns the tail probabilities of a light-tailed Markov-modulated Lévy process stopped at a state-dependent Poisson rate. The tails are shown to decay exponentially at rates given by the unique positive and negative roots of the spectral abscissa of a certain matrix-valued function. We illustrate the use of our results with an application to the stationary distribution of wealth in a simple economic model in which agents with constant absolute risk aversion are subject to random mortality and income fluctuation.

Multivariate Tail Probabilities: Predicting Regional Pertussis Cases in Washington State

In disease modeling, a key statistical problem is the estimation of lower and upper tail probabilities of health events from given data sets of small size and limited range. Assuming such constraints, we describe a computational framework for the systematic fusion of observations from multiple sources to compute tail probabilities that could not be obtained otherwise due to a lack of lower or upper tail data. The estimation of multivariate lower and upper tail probabilities from a given small reference data set that lacks complete information about such tail data is addressed in terms of pertussis case count data. Fusion of data from multiple sources in conjunction with the density ratio model is used to give probability estimates that are non-obtainable from the empirical distribution. Based on a density ratio model with variable tilts, we first present a univariate fit and, subsequently, improve it with a multivariate extension. In the multivariate analysis, we selected the best model in terms of the Akaike Information Criterion (AIC). Regional prediction, in Washington state, of the number of pertussis cases is approached by providing joint probabilities using fused data from several relatively small samples following the selected density ratio model. The model is validated by a graphical goodness-of-fit plot comparing the estimated reference distribution obtained from the fused data with that of the empirical distribution obtained from the reference sample only.

Tail asymptotics for the area under the excursion of a random walk with heavy-tailed increments

We study the tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and heavy-tailed increments. We determine the asymptotics for tail probabilities for the area.