The elementary problem of exhaustively sampling a finite population without replacement is used as a nonreversible test case for comparing two recently proposed MCMC algorithms for perfect sampling, one based on backward coupling and the other on strong stationary duality. The backward coupling algorithm runs faster in this case, but the duality-based algorithm is unbiased for user impatience. An interesting by-product of the analysis is a new and simple stochastic interpretation of a mixing-time result for the move-to-front rule.

We develop an algorithm for simulating approximate random samples from the invariant measure of a Markov chain using backward coupling of embedded regeneration times. Related methods have been used effectively for finite chains and for stochastically monotone chains: here we propose a method of implementation which avoids these restrictions by using a “cycle-length” truncation. We show that the coupling times have good theoretical properties and describe benefits and difficulties of implementing the methods in practice.

We explicitly give the probability mass function and the probability generating function of the first k-record index for a sequence of independent and identically distributed random variables that take on a finite set of possible values. We also compute its factorial moments.

This note presents an efficient algorithm for the simulation of the number of invariant positions in a random permutation of the integers 1,..., n. An application to the famous matching problem of Montmort is discussed.

A significant amount of recent research has been focused on the stability of multiclass open networks of queues (MONQs). It has been shown that these networks may be unstable under various queueing disciplines even when at each one of the nodes the arrival rate is less than the service rate. Clearly, in such cases the expected delay and the expected number of customers in the system are infinite. In this paper we propose a new class of scheduling rules that can be used in multiclass queueing networks. We refer to this class as the stable shortfall-based priority (SSBP) rules. This SSBP class itself belongs to a larger class of rules, which we refer to as the shortfall-based priority (SBP) rules. SBP is a generalization of the standard non-preemptive priority rule in which customers of the same priority class are served first-come, first-served (FCFS). Rules from SBP can mimic FCFS as well as the so-called strict or head-of-the-line priority disciplines. We show that the use of any rule from the SSBP class ensures stability in a broad class of MONQs found in practice. We proceed with the construction of a sample path inequality for the work done by an SSBP server and show how this inequality can be used to derive upper bounds for the delay when service times are bounded. Bounds for the expected delay of each class of customers in an MONQ are then obtained under the assumptions that the external arrival processes have i.i.d. interarrival times, the routings are deterministic and the service times at each step of the route are bounded. In order to derive these bounds for the average delay in an MONQ we make use of some of the classical ideas of Kingman.

A machining process produces a batch of n units every time period. At the end of the period, the units are inspected (provided inspection is cost free). Based on the inspection and the quality data as well as the system history, a decision is made as to whether the process is in control. If not, the process is “revised,” in terms of machine recalibration, maintenance, or repair. In the presence of inspection cost, there is also the need to decide whether to inspect the produced batch. We formulate the problem as Markov decision programs, considering both discounted and average costs. We prove the optimality of certain threshold policies, and characterize the monotone behavior of the optimal thresholds.

Let S0,…,Sn be a symmetric random walk that starts at the origin (S0 = 0) and takes steps uniformly distributed on [— 1,+1]. We study the large-n behavior of the expected maximum excursion and prove the estimate

,

where c = 0.297952.... This estimate applies to the problem of packing n rectangles into a unit-width strip; in particular, it makes much more precise the known upper bound on the expected minimum height, O(n½), when the rectangle sides are 2n independent uniform random draws from [0,1].

Given a Markov chain on a countable state space, we present a Lyapunov (sufficient) condition for existence of an invariant probability with a geometric tail.

We study an optimal stopping problem for a nonhomogeneous Markov process, with a reward function that is lower semicontinuous everywhere and smooth in certain regions. We prove that the payoff (value function) is lower semicontinuous as well and solves a so-called generalized Stefan problem in each of these regions. We provide some results for the geometry of the “stopping observations” set. Our results generalize those in Bassan, Brezzi, and Scarsini (1996). The problem we consider stems from an economic model in which several self-interested agents desire information, whereas a social planner, although benevolent toward the agents, might decide to withhold information in order to induce diversification in their behavior.

A statistic for testing the null hypothesis that F is the exponential distribution against the alternative of harmonic new better than used in expectation (HNBUE) is proposed. The asymptotic distribution theory for this statistic is derived under the null hypothesis and asymptotic relative efficiency (ARE) with respect to other competing tests for HNBUE is evaluated. This test is applied to the leukemia data described in Bryson and Siddiqui (1969).