In this paper, we review some recent results on the stochastic comparison of order statistics and related statistics from independent and heterogeneous proportional hazard rates models, gamma variables, geometric variables, and negative binomial variables. We highlight the close connections that exist between some classical stochastic orders and majorization-type orders.

Suppose we have n objects of different weights. We randomly sample pairs of objects, and for each sampled pair use a balance scale to determine which of the two objects is heavier. It is assumed that the sequence of sampled pairs is iid, each selection uniformly distributed on the set of n(n−1)/2 pairs. We continue sampling until the first time that we can definitively identify the heaviest of the n objects. The problem of interest is to compute the expected number of selected pairs.

We consider the problem of finding the optimal routing of a single vehicle that delivers K different products to N customers that are served according to a particular order. It is assumed that the demands of the customers for each product are discrete random variables, and the total demand of each customer for all products cannot exceed the vehicle capacity. The joint probability mass function of the demands of each customer is known. It is assumed that all products are stored together in the vehicle's single compartment. The policy that serves all customers with the minimum total expected cost is found by implementing a suitable dynamic programming algorithm. We prove that this policy has a specific threshold-type structure. Furthermore, we study a corresponding infinite-time horizon problem in which the service of the customers is not completed when the last customer has been serviced but it continues periodically with the same customer order. The demands of each customer for the products have the same distributions at different periods. The discounted cost optimal policy and the average-cost optimal policy have the same structure as the optimal policy in the finite-horizon problem. Numerical results are given that illustrate the structural results.

The stochastic sequential assignment problem (SSAP) allocates distinct workers to sequentially arriving tasks with stochastic parameters to maximize the expected total reward. In this paper, the assignment of tasks is performed under the threshold criterion, which seeks a policy that minimizes the probability of the total reward failing to achieve a target value. A Markov-decision-process approach is employed to model the problem, and sufficient conditions for the existence of a deterministic Markov optimal policy are derived, along with fundamental properties of the optimal value function. An algorithm to approximate the optimal value function is presented, and convergence results are established.

We introduce a natural growth model for directed series-parallel (SP) graphs and look at some of the graph properties under this stochastic model. Specifically, we look at the degrees of certain types of nodes in the random SP graph. We examine the degree of a pole and will find its exact distribution, given by a probability formula with alternating signs. We also prove that, for a fixed value s, the number of nodes of outdegree 1, …, s asymptotically has a joint multivariate normal distribution. Pólya urns will systematically provide a working tool.

N. Balakrishnan and Peng Zhao have prepared an outstanding survey of recent results that stochastically compare various order statistics and some ranges based on two collections of independent heterogeneous random variables. Their survey focuses on results for heterogeneous exponential random variables and their extensions to random variables with proportional hazard rates. In addition, some results that stochastically compare order statistics based on heterogeneous gamma, Weibull, geometric, and negative binomial random variables are also given. In particular, the authors of have listed some stochastic comparisons that are based on one heterogeneous collection of random variables, and one homogeneous collection of random variables. Personally, I find these types of comparisons to be quite fascinating. Balakrishnan and Zhao have done a thorough job of listing all the known results of this kind.

We investigate the evolution of an urn of balls of two colors, where one chooses a pair of balls and observes rules of ball addition according to the outcome. A nonsquare ball addition matrix of the form $\left( \matrix{a & b \cr c & d \cr e & f}\right)$ corresponds to such a scheme, in contrast to pólya urn models that possess a square ball addition matrix. We look into the case of constant row sum (the so-called balanced urns) and identify a linear case therein. Two cases arise in linear urns: the nondegenerate and the degenerate. Via martingales, in the nondegenerate case one gets an asymptotic normal distribution for the number of balls of any color. In the degenerate case, a simpler probability structure underlies the process. We mention in passing a heuristic for the average-case analysis for the general case of constant row sum.

The sequential and stochastic assignment problem (SSAP) has wide applications in logistics, finance, and health care management, and has been well studied in the literature. It assumes that jobs with unknown values arrive according to a stochastic process. Upon arrival, a job's value is made known and the decision-maker must immediately decide whether to accept or reject the job and, if accepted, to assign it to a resource for a reward. The objective is to maximize the expected reward from the available resources. The optimal assignment policy has a threshold structure and can be computed in polynomial time. In reality, there exist situations in which the decision-maker may postpone the accept/reject decision. In this research, we study the value of postponing decisions by allowing a decision-maker to hold a number of jobs which may be accepted or rejected later. While maintaining this queue of arrivals significantly complicates the analysis, optimal threshold policies exist under mild assumptions when the resources are homogeneous. We illustrate the benefits of delaying decisions through higher profits and lower risk in both cases of homogeneous and heterogeneous resources.

Professors Balakrishnan and Zhao have written an excellent survey on the recent developments of stochastic comparisons of order statistics, which cover almost every aspect of ordering properties of order statistics from both continuous and discrete heterogeneous populations. My discussion will be limited to the skewness of order statistics and order statistics from heterogeneous populations with different shape parameters.

Consider two independent Markov chains having states 0, 1, and identical transition probabilities. At each stage one of the chains is observed, and a reward equal to the observed state is earned. Assuming prior probabilities on the initial states of the chains it is shown that the myopic policy that always chooses to observe the chain most likely to be in state 1 stochastically maximizes the sequence of rewards earned in each period.

We examine the time-dependent behavior of a birth–death process, whose birth rates and death rates are decreasing and increasing, respectively, with respect to the current state. Such models can be used to describe Markovian queueing systems with exponential reneging, where potential arrivals balk with a certain probability that depends on the number of customers observed upon arrival. Our results are derived by interpreting the birth–death process as the queue-length process of what we refer to as the “knockout queue.”

One of the interesting problems on the stochastic behavior of random recurrent events in a random time interval is to obtain the conditions under which the reliability properties of a random time T are inherited by N(T), where {N(t):t≥0} is a stochastic process. Most of the studies on the topic has been done under the assumption that the random time T and the stochastic process {N(t):t≥0} are stochastically independent. However, in practice, there can be different cases when appropriate dependence structure is more appropriate. In this paper, we study the preservation of a renewal process stopped at a random time when they are “stochastically dependent.” We discuss the stochastic ordering properties and the preservation of reliability classes for the random counting variables N(T) when the corresponding counting process is a renewal process. Furthermore, we study the preservation of NBUE (NWUE) reliability class when the counting process is a homogeneous Poisson process.

We present analytic results for warehouse systems involving pairs of carousels. Specifically, for various picking strategies, we show that the sojourn time of the picker satisfies an integral equation that is a contraction mapping. As a result, numerical approximations for performance measures such as the throughput of the system are extremely accurate and converge fast (e.g., within five iterations) to their real values. We present simulation results validating our results and examining more complicated strategies for pairs of carousels.

Expansions for moments of $\overline{X}$, the mean of a random sample of size n, are given for both the univariate and multivariate cases. The coefficients of these expansions are simply Bell polynomials. An application is given for the compound Poisson variable SN, where $S_{n} = n \overline{X}$ and N is a Poisson random variable independent of X1, X2, …, yielding expansions that are computationally more efficient than the Panjer recursion formula and Grubbström and Tang's formula.

Balakrishnan and Zhao does an excellent job in this issue at reviewing the recent advances on stochastic comparison between order statistics from independent and heterogeneous observations with proportional hazard rates, gamma distribution, geometric distribution, and negative binomial distributions, the relation between various stochastic order and majorization order of concerned heterogeneous parameters is highlighted. Some examples are presented to illustrate main results while pointing out the potential direction for further discussion.

In this paper, we suggest a new class of counting processes, called the Class of Geometric Counting Processes (CGCP), where each member of the counting process in the class has increments described by the geometric distribution. Distinct from the Poisson process, they do not possess the property of independent increments, which usually complicates probabilistic analysis. The suggested CGCP is defined and the dependence structure shared by the members of the class is discussed. As examples of useful applications, we consider stochastic survival models under external shocks. We show that the corresponding survival probabilities under reasonable assumptions can be effectively described by the CGCP without specifying the dependence structure.

Linear combinations of independent random variables have been extensively studied in the literature. Xu & Hu [21] and Pan, Xu, & Hu [16] unified the study of linear combinations of independent random variables under the general setup. This paper is a companion one of these two papers. In this paper, we will further study this topic. The results are further generalized to the cases of permutation invariant random variables and of independent but not necessarily identically distributed random variables which are ordered in the likelihood ratio or the hazard ratio order.

In this paper, we study three delay systems where different classes of impatient customers arrive according to independent Poisson processes. In the first system, a single server receives two classes of customers with general service time requirements, and follows a non-preemptive priority policy in serving them. Both classes of customers abandon the system when their exponentially distributed patience limits expire. The second system comprises parallel and identical servers providing the same type of service for both classes of impatient customers under the non-preemptive priority policy. We assume exponential service times and consider two cases depending on the time-to-abandon distribution being exponentially distributed or deterministic. In either case, we permit different reneging rates or patience limits for each class. Finally, we consider the first-come-first-served policy in single- and multi-server settings. In all models, we obtain the Laplace transform of the virtual waiting time for each class by exploiting the level-crossing method. This enables us to compute the steady-state system performance measures.

I thank the authors for providing another excellent review on stochastic comparisons of order statistics after the review paper written by Kochar and Xu (2007).

The unified multivariate counting process (UMCP), previously studied by the same authors, enables one to describe most of the existing counting processes in terms of its components, thereby providing a comprehensive view for such processes often defined separately and differently. The purpose of this paper is to study a multivariate reward process defined on the UMCP. By examining the probabilistic flow in its state space, various transform results are obtained. The asymptotic behavior, as t→∞, of the expected univariate reward process in a form of a product of components of the multivariate reward process is studied. As an application, a manufacturing system is considered, where the cumulative profit given a preventive maintenance policy is described as a univariate reward process defined on the UMCP. The optimal preventive maintenance policy is derived numerically by maximizing the cumulative profit over the time interval [0, T].