This paper describes a generalization of the classical regenerative method of simulation output analysis. Instead of blocking a generated sample path on returns to a fixed return state, a more general scheme to randomly decompose the path is used. In some cases, this decomposition scheme results in regeneration times that are a supersequence of the classical regeneration times. This “accelerated” regeneration is advantageous in several simulation contexts. It is shown that when this decomposition scheme accelerates regeneration relative to the classical regenerative method, it also yields a smaller asymptotic variance of the regenerative variance estimator than the classical method. Several other contexts in which increased regeneration frequency is beneficial are also discussed.

A simulation method is developed for computing average reward optimal policies, for a finite state and action Markovian decision process. It is shown that the method is consistent; i.e., it produces solutions arbitrarily close to the optimal. Various types of estimation errors and confidence bounds are examined. Finally, it is shown that the probability distribution of the number of simulation cycles required to compute an e-optimal policy satisfies a large deviations property.

The polling network considered here consists of a finite collection of stations visited successively by a single server according to a state-dependent routing scheme. At every visit of a station, a possibly random number of customers are served and depart from the network thereafter. The server takes a possibly random time to walk from one station to the next. The network receives groups of customers at Poisson instants. These customers are distributed randomly over the stations and wait until served. A necessary condition for stability is derived, which proves to be also sufficient in certain special cases. Such cases include a starlike network and a ringlike one with a greedy service policy. Mathematically, positive recurrence of a Markov chain is studied.

Let X be a random variable with characteristic function ϕ. In the case where X is integer-valued and n is a positive integer, a formula (in terms of ϕ) for the probability that n divides X is presented. The derivation of this formula is quite simple and uses only the basic properties of expectation and complex numbers. The formula easily generalizes to one for the distribution of X mod n. Computational simplifications and the relation to the inversion formula are also discussed; the latter topic includes a new inversion formula when the range of X is finite.

When X may take on a more general distribution, limiting considerations of the previous formulas suggest others for the distribution, density, and moments of the fractional part X — [X]. These are easily derived using basic properties of Fourier series. These formulas also yield an alternative inversion formula for ϕ when the range of X is bounded.

Applications to renewal theory and random walks are suggested. A by-product of the approach is a probabilistic method for the evaluation of infinite series.

Let N = [N(t), t ≥ 0] be a point process, and let T1 and T2 be two positive random variables that are independent of N. Define Mk = N(Tk), k = 1,2. In this paper we study conditions on the process N under which some stochastic orders between T1 and T2 are transformed into stochastic orders between M1 and M2. In some cases the converses are also shown to be true. Some applications and examples are given.

Limit theorems are established and relatively simple closed-form approximations are developed for the busy-period distribution in single-server queues. For the M/G/l queue, the complementary busy-period c.d.f. is shown to be asymptotically equivalent as t → ∞ to a scaled version of the heavy-traffic limit (obtained as p → 1), where the scaling parameters are based on the asymptotics as t → ∞. We call this the asymptotic normal approximation, because it involves the standard normal c.d.f. and density. The asymptotic normal approximation is asymptotically correct as t → ∞ for each fixed p and as p → 1 for each fixed t and yields remarkably good approximations for times not too small, whereas the direct heavy-traffic (p → 1) and asymptotic (t → ∞) limits do not yield such good approximations. Indeed, even the approximation based on three terms of the standard asymptotic expansion does not perform well unless t is very large. As a basis for generating corresponding approximations for the busy-period distribution in more general models, we also establish a more general heavy-traffic limit theorem.

Open Jackson networks are studied in which departures can be blocked such as due to a breakdown of an output channel. A simple performance estimate is provided along with an explicit error bound.

For each n ≥ 1, let Πn = (π1, π2, …, πn) be a random permutation of the integers 1,2, …, n. The quantity Bn = Σ measures the degree of oscillation of Πn and is an important measure in the study of sorting algorithms. In this work the asymptotic normality of Bn is derived under the assumption that, for each n ≥ 1, Πn is uniformly distributed over the n! permutations of 1,2, …, n. The proof relies on coupling (π1, π2, …, πn) with a sequence of independent and identically distributed U(0,1) random variables and offers a more probabilistic and computationally simpler alternative to the method of moments proof of Chao, Bai, and Liang (1993, Probability in the Engineering and Informational Sciences 7: 227–235).

In this paper we present a simple method to fit a discrete distribution on the first two moments of a given random variable. With the Fitted distribution we solve approximately Lindley's equation for the D/G/1 queue with discrete service-time distribution using a moment-iteration method. Numerical results show excellent performance of the method.

We consider N unreliable machines that are maintained by M repairmen. The time until failure of machine i, and its repair by repairman j, are exponentially distributed random variable with parameters λi, and μj, respectively. All failure times and repair times are independent. Machine i earns ci, per unit time, while it is working. We wish to maximize the expected total discounted reward. It is shown that when the following conditions are satisfied — c1 ≥ … ≥ cN and c1/λ1≥…≥cN/λN, the policy that assigns the fastest repairman to the machine with the lowest index is optimal. Moreover, it is shown that when the second condition is replaced by λ≤ … ≤λN, then this policy maximizes, stochastically, the number of the most reliable machines at every time t.