In this article, we study the order statistics from heterogenous negative binomial random variables. Sufficient conditions are provided for comparing the extreme order statistics according to the usual stochastic order. For the special case of geometric distribution, a sufficient condition is established for comparing order statistics in the sense of multivariate stochastic order. Applications in the Poisson–Gamma shock model and redundant systems have been described as well.

Sequential order statistics describe the ordered failure times in a k-out-of-n system, where the failures of components might affect the performance of remaining working components. In this article aging properties of sequential order statistics are examined and conditions are given such that the distribution of a sequential order statistic is ILR, IFR, IFRA, or NBU. Moreover, conditions for aging properties of spacings of sequential order statistics are obtained.

We specify the smallest mode of the ordinary multinomials leading to the expression of the maximal probability of convolution powers of the discrete uniform distribution. The generating function for an extension of the maximal probability is given.

We extend the classic sequential stochastic assignment problem to include arrivals of workers. When workers are all of the same type, we show that the socially optimal policy is the same as the individually optimal policy for which workers are given priority according to last come–first served. This result also holds under several variants in the model assumptions. When workers have different types, we show that the socially optimal policy is determined by thresholds such that more valuable jobs are given to more valuable workers, but now the individually optimal policy is no longer socially optimal. We also show that the overall value increases when worker or job values become more variable.

We consider M/G/1-type Markov chains where a transition that decreases the value of the level triggers the phase to a small subset of the phase space. We show how this structure—referred to as restricted downward transitions—can be exploited to speed up the computation of the stationary probability vector of the chain. To this end we define a new M/G/1-type Markov chain with a smaller block size, the G matrix of which is used to find the original chain's G matrix. This approach is then used to analyze the BMAP/PH/1 queue and the BMAP[2]/PH[2]/1 preemptive priority queue, yielding significant reductions in computation time.

We consider a discrete-time DAR(1)/D/1 queue and provide an analysis on the variances of both its system size and sojourn time. Our approach is simple, but the results are nice, as these variances are found in closed form. We first establish the relation between these variances, based on which we then use the conditioning technique to analyze the expected cross terms that come from its system recurrence relation. The closed-form results allow us to explicitly examine the effect from the batch size distribution and the autocorrelation parameter p. It is observed that as p grows toward 1, the standard deviations of the two performance measures will blow up in same asymptotic order of O(1/(1−p)) as their means. These are demonstrated through numerical examples.

In this article we study optimal admission to an M/M/k/N queue with several customer types. The reward structure consists of revenues collected from admitted customers and holding costs, both of which depend on customer types. This article studies average rewards per unit time and describes the structures of stationary optimal, canonical, bias optimal, and Blackwell optimal policies. Similar to the case without holding costs, bias optimal and Blackwell optimal policies are unique, coincide, and have a trunk reservation form with the largest optimal control level for each customer type. Problems with one holding cost rate have been studied previously in the literature.