The goal of this work is twofold. We show the following:

1. 1. In spite of the common consensus on the classic Markov chain Monte Carlo (MCMC) as a universal tool for generating samples on complex sets, it fails to generate points uniformly distributed on discrete ones, such as that defined by the constraints of integer programming. In fact, we will demonstrate empirically that not only does it fail to generate uniform points on the desired set, but typically it misses some of the points of the set.

2. 2. The splitting, also called the cloning method – originally designed for combinatorial optimization and for counting on discrete sets and presenting a combination of MCMC, like the Gibbs sampler, with a specially designed splitting mechanism—can also be efficiently used for generating uniform samples on these sets. Without introducing the appropriate splitting mechanism, MCMC fails. Although we do not have a formal proof, we guess (conjecture) that the main reason that the classic MCMC is not working is that its resulting chain is not irreducible. We provide valid statistical tests supporting the uniformity of generated samples by the splitting method and present supportive numerical results.

In this paper we consider stochastic flow-shop scheduling with reference to certain lateness-related performance measures. We show that for various assumptions on the distribution of job-processing times of a flow shop, certain scheduling policies following the stochastic analogy of the Earliest Due Date (EDD) rule yield optimal results.

In this paper, we exactly analyze the performance of the slotted ALOHA access scheme with capture in a multichannel packet radio communication environment for the IFT (immediate-first-transmission) protocol and the DFT (delayed-first-transmission) protocol. We derive four moment generating functions for the following performance measures: (1) the number of packet deparures in each group of capture level in any slot, (2) the interval time between wo consecutive slot ends with the same number of departures, (3) the interval time between two consecutive slot ends with at least one departure and the number of departures in each group in that slot, and (4) the packet delay for each group. We calculate the averages and higher moments of these performance measures by differentiating the moment generating functions and numerically compare the systems with and without ower capture. The system consists of a finite population of Nstations, both fixed and mobile, that are divided into L different capture groups and access a set of parallel M channels to transmit their packets. Capture effect means that a packet transmitted by a station with a highest capture level can be received accurately, even when other packets in lower capture levels are simultaneously transmitted on the same channel and in the same slot. Numerical comparison to a multichannel system without capture is made. Capture effects on channel utilization, mean packet delay, and coefficients of variation of packet delay and interdeparture time are examined.

The standard regenerative method for estimating steady-state parameters is extended to permit cycles that begin and end in different states. This result is established using the Dynkin martingale and a related solution to Poisson's equation. We compare the variance constant that appears in the associated central limit theorem with that arising from cycles that begin and end in the same state. The standard regenerative method has a smaller variance constant than does the alternative.

Bivariate Lomax distribution due to Lindley and Singpurwalla has been characterized through the bivariate extension of a univariate characterizing property of the Lomax distribution.

Consider a group of players playing a sequence of games. There are k players, having arbitrary initial fortunes. Each game consists of each remaining player putting 1 in a pot, which is then won (with equal probability) by one of them. Players whose fortunes drop to 0 are eliminated. Let T(i) be the number of games played by i, and let T=max iT(i). For the case k=3, martingale stopping theory can be used to derive E[T] and E[T(i)]. When k>3, we obtain upper bounds on E[T] and, in the case in which all players have the same initial fortune, on E[T(i)]. Efficient simulation methods for estimating E[T] and E[T(i)] are discussed.

This paper is concerned with a certain property of the stationary distribution of a generalized semi-Markov process (GSMP) known as insensitivity. It is well-known that the so-called Matthes' conditions form a necessary and sufficient algebraic criterion for insensitivity. Most proofs of these conditions are basically algebraic. By interpreting a GSMP as a simple queueing network, we are able to show that Matthes' conditions are equivalent to the quasi-reversibility of the network, thus obtaining another simple proof of the sufficiency of these conditions. Furthermore, we apply our method to find a simple criterion for the insensitivity of GSMP's with generalized routing (in a sense that is introduced in the paper).

In this paper we consider the use of coupling ideas in efficiently computing a certain class of transient performance measures. Specifically, we consider the setting in which the stationary distribution is unknown, and for which no exact means of generating stationary versions of the process is known. In this context, we can approximate the stationary distribution from empirical data obtained from a first-stage steady-state simulation. This empirical approximation is then used in place of the stationary distribution in implementing our coupling-based estimator. In addition to the empirically based coupling estimator itself, we also develop an associated confidence interval procedure.

In a previous paper, we defined and studied a branching process input stream that can be used to model the organization of packets (in packetized communication) in messages. In this paper we derive a central result on the effect that such streams have on a packet switch (or queue) and further study one particular model (the case of so-called short-range dependence), where the distribution of the number of packets in a buffer can explicitly be found (albeit in the form of a transform). The solution to this particular model suggests an approximation for the average buffer content in the general case.

This paper is devoted to the performance analysis and optimization of blood testing procedures. We present a queueing model of two queues in series, representing the two stages of a blood-testing procedure. Service (testing) in stage 1 is performed in batches, whereas it is done individually in stage 2. Since particular elements of blood can only be stored and used within a finite time window, the sojourn time of blood units in the system of two queues in series is an important performance measure, which we study in detail. We also introduce a profit objective function, taking into account blood acquisition and screening costs as well as profits for blood units, which were found uncontaminated and were tested fast enough. We optimize that profit objective function w.r.t. the batch size and the length of the time window.

For a bivariate random vector $(X, Y)$, suppose $X$ is some interesting loss variable and $Y$ is a benchmark variable. This paper proposes a new variability measure called the joint tail-Gini functional, which considers not only the tail event of benchmark variable $Y$, but also the tail information of $X$ itself. It can be viewed as a class of tail Gini-type variability measures, which also include the recently proposed tail-Gini functional. It is a challenging and interesting task to measure the tail variability of $X$ under some extreme scenarios of the variables by extending the Gini's methodology, and the two tail variability measures can serve such a purpose. We study the asymptotic behaviors of these tail Gini-type variability measures, including tail-Gini and joint tail-Gini functionals. The paper conducts this study under both tail dependent and tail independent cases, which are modeled by copulas with so-called tail order property. Some examples are also shown to illuminate our results. In particular, a generalization of the joint tail-Gini functional is considered to provide a more flexible version.

Almost stochastic dominance has been receiving a great amount of attention in the financial and economic literatures. In this paper, we characterize the properties of almost first-order stochastic dominance (AFSD) via distorted expectations and investigate the conditions under which AFSD is preserved under a distortion transform. The main results are also applied to establish stochastic comparisons of order statistics and receiver operating characteristic curves via AFSD.

We study 2-stage game-theoretic problem oriented 3-stage service policy computing, convolutional neural network (CNN) based algorithm design, and simulation for a blockchained buffering system with federated learning. More precisely, based on the game-theoretic problem consisting of both “win-lose” and “win-win” 2-stage competitions, we derive a 3-stage dynamical service policy via a saddle point to a zero-sum game problem and a Nash equilibrium point to a non-zero-sum game problem. This policy is concerning users-selection, dynamic pricing, and online rate resource allocation via stable digital currency for the system. The main focus is on the design and analysis of the joint 3-stage service policy for given queue/environment state dependent pricing and utility functions. The asymptotic optimality and fairness of this dynamic service policy is justified by diffusion modeling with approximation theory. A general CNN based policy computing algorithm flow chart along the line of the so-called big model framework is presented. Simulation case studies are conducted for the system with three users, where only two of the three users can be selected into the service by a zero-sum dual cost game competition policy at a time point. Then, the selected two users get into service and share the system rate service resource through a non-zero-sum dual cost game competition policy. Applications of our policy in the future blockchain based Internet (e.g., metaverse and web3.0) and supply chain finance are also briefly illustrated.

The purpose of this paper is to establish some limit theorems of delayed averages for countable nonhomogeneous Markov chains. The definition of the generalized C-strong ergodicity and the generalized uniformly C-strong ergodicity for countable nonhomogeneous Markov chains is introduced first. Then a theorem about the generalized C-strong ergodicity and the generalized uniformly C-strong ergodicity for the nonhomogeneous Markov chains is established, and its applications to the information theory are given. Finally, the strong law of large numbers of delayed averages of bivariate functions for countable nonhomogeneous Markov chains is proved.

It has long been observed that independent laboratories differ in reporting the results of repeated experiments. The problem is to detect those laboratories that might be considered aberrant. Previous analyses have been based on an analysis of variance framework or on subset selection for the detection of aberrant laboratories. The present procedure is also based on an ANOVA model but uses a Bayesian estimate of the number of aberrant laboratories. Subsequent to the determination of the aberrant laboratories, a linear model is used to separate sampling and laboratory effects.

In a family, parameterized by θ, of non-negative random variables with finite, positive second moment, Taylor's law (TL) asserts that the population variance is proportional to a power of the population mean as θ varies: σ2 (θ) = a[μ(θ)]b, a > 0. TL, sometimes called fluctuation scaling, holds widely in science, probability theory, and stochastic processes. Here we report diverse examples of TL with b = 2 (equivalent to a constant coefficient of variation) arising from a difference of random variables in normed vector spaces of dimension 1 and larger. In these examples, we compute a exactly using, in some cases, a simple, new technique. These examples may prove useful in future models that involve differences of random variables, including models of the spatial distribution and migration of human populations.

In the fall of 1948 in my course on Least Squares in the Department of Mathematical Statistics at Columbia University (and in the spring in Correlation and Chi-Square), I was particularly impressed by one of the students— Gerald J. Lieberman. I was disappointed that this promising student left Columbia after one year, but it was not long until our paths met again. It is a pleasure to dedicate this paper to my colleague and close friend!

We consider in this article an urn and ball problem with replacement, where balls are with different colors and are drawn uniformly from a unique urn. The numbers of balls with a given color are independent and identically distributed random variables with a heavy tailed probability distribution—for instance a Pareto or a Weibull distribution. We draw a small fraction p≪1 of the total number of balls. The basic problem addressed in this article is to know to which extent we can infer the total number of colors and the distribution of the number of balls with a given color. By means of Le Cam's inequality and the Chen–Stein method, bounds for the total variation norm between the distribution of the number of balls drawn with a given color and the Poisson distribution with the same mean are obtained. We then show that the distribution of the number of balls drawn with a given color has the same tail as that of the original number of balls. Finally, we establish explicit bounds between the two distributions when each ball is drawn with fixed probability p.