Let (X1, …, Xn) be a multivariate normal random vector with any mean vector, variances equal to 1 and covariances equal and positive. Turner and Whitehead [9] established that the largest order statistic max{X1, …, Xn} is less than the standard normal random variable in the dispersive order. In this paper, we give a new and straightforward proof for this result. Several possible extensions of this result are also discussed.

We consider a system consisting of a single server serving a fixed number of stations. At each station, there is an infinite queue of customers that have to undergo a preparation phase before being served. This model is connected to layered queueing networks, to an extension of polling systems and surprisingly to random graphs. We are interested in the waiting time of the server. For the case where the server polls the stations cyclically, we give a sufficient condition for the existence of a limiting waiting-time distribution and we study the tail behavior of the stationary waiting time. Furthermore, assuming that preparation times are exponentially distributed, we describe in depth the resulting Markov chain. We also investigate a model variation where the server does not necessarily poll the stations in a cyclic order, but always serves the customer with the earliest completed preparation phase. We show that the mean waiting time under this dynamic allocation never exceeds that of the cyclic case, but that the waiting-time distributions corresponding to both cases are not necessarily stochastically ordered. Finally, we provide extensive numerical results investigating and comparing the effect of the system's parameters to the performance of the server for both models.

The modified-offered-load approximation can be used to choose a staffing function (the time-varying number of servers) to stabilize delay probabilities at target levels in multi-server delay models with time-varying arrival rates, with or without customer abandonment. In contrast, as we confirm with simulations, it is not possible to stabilize blocking probabilities to the same extent in corresponding loss models, without extra waiting space, because these probabilities necessarily change dramatically after each staffing change. Nevertheless, blocking probabilities can be stabilized provided that we either randomize the times of staffing changes or average the blocking probabilities over a suitably small time interval. We develop systematic procedures and study how to choose the averaging parameters.

The objective of this note is to study the distribution of the running maximum of the level in a level-dependent quasi-birth-death process. By considering this running maximum at an exponentially distributed “killing epoch” T, we devise a technique to accomplish this, relying on elementary arguments only; importantly, it yields the distribution of the running maximum jointly with the level and phase at the killing epoch. We also point out how our procedure can be adapted to facilitate the computation of the distribution of the running maximum at a deterministic (rather than an exponential) epoch.

A combined optimal dividend/reinsurance problem with two types of insurance claims, namely the expected premium principle and the variance premium principle, is discussed. Dividend payments are considered with both fixed and proportional transaction costs. The objective of an insurer is to determine an optimal dividend–reinsurance policy so as to maximize the expected total value of discounted dividend payments to shareholders up to ruin time. The problem is formulated as an optimal regular-impulse control problem. Closed-form solutions for the value function and optimal dividend–reinsurance strategy are obtained in some particular cases. Finally, some numerical analysis is given to illustrate the effects of safety loading on optimal reinsurance strategy.

Binary series-parallel (BSP) graphs have applications in transportation modeling, and exhibit interesting combinatorial properties. This work is limited to the second aspect. The BSP graphs of a given size – measured in edges – are enumerated. Under a uniform probability model, we investigate the asymptotic distribution of the order (number of vertices) and the asymptotic average length of a random walk (under different strategies) of large graphs of the same size. The systematic method throughout is to define the graphs, and the features we evaluate by a structural equation (equivalent to a regular expression). The structural equation is translated into an equation for a generating function, which is then analyzed.

For non-negative random variables with finite means we introduce an analogous of the equilibrium residual-lifetime distribution based on the quantile function. This allows us to construct new distributions with support (0, 1), and to obtain a new quantile-based version of the probabilistic generalization of Taylor's theorem. Similarly, for pairs of stochastically ordered random variables we come to a new quantile-based form of the probabilistic mean value theorem. The latter involves a distribution that generalizes the Lorenz curve. We investigate the special case of proportional quantile functions and apply the given results to various models based on classes of distributions and measures of risk theory. Motivated by some stochastic comparisons, we also introduce the “expected reversed proportional shortfall order”, and a new characterization of random lifetimes involving the reversed hazard rate function.

This paper deals with the mean residual life function (MRLF) and its monotonicity in the case of additive and multiplicative hazard rate models. It is shown that additive (multiplicative) hazard rate does not imply reduced (proportional) MRLF and vice versa. Necessary and sufficient conditions are obtained for the two models to hold simultaneously. In the case of non-monotonic failure rates, the location of the turning points of the MRLF is investigated in both the cases. The case of random additive and multiplicative hazard rate is also studied. The monotonicity of the mean residual life is studied along with the location of the turning points. Examples are provided to illustrate the results.

It is pointed out that the weighted generalized gamma distribution of Priyadarshani and Oluyede [2] is a generalized gamma distribution.