Probabilistic analyses of hashing algorithms usually assume that hash values are uniformly distributed over addresses. We study how one of the simplest schemes, hashing with linear probing, behaves in the nonuniform case. A simple measure μ of nonuniformity is the probability two keys hash to the same address, divided by this probability in the uniform case. It turns out that the effect of nonuniformity is to multiply mean search lengths by μ. For high loads, the longest search is multiplied by approximately μ also. Our theoretical results are asymptotics: simulations show good fits with predictions for mean search lengths, but bad fits for longest search lengths.

Given a collection of n points in the plane, the Euclidean matching problem is the task of decomposing the collection into matched pairs connected by line segments in such a way as to minimize the sum of all the segment lengths. The greedy heuristic provides an approximate solution to the Euclidean matching problem by successively matching the two closest unmatched points. We study the behavior of Gn, the sum of the lengths of the segments produced by the greedy heuristic.

A set of ni arms of type i, i = 1,…, L, is available. A pull of arm of type i occupies a duration Vi at the end of which a reward Ci and Ni1,…, NiL new arms are obtained, while all other arms are frozen. A Gittins priority order of types is obtained and shown to yield the maximal discounted reward from this branching process of arms.

Recently, orbits of two-dimensional Markov chains have been used to generate computer images. These chains evolve according to products of i.i.d. affine maps. We deal with mixing models, whereby one mixes together several of these Markov chains, so as to create a mixed image. These mixtures involve starting one Markov chain off at the stationary distribution of another, and then running it for a geometrically distributed number of steps. We use this to analyze various mixing scenarios.

Two different estimators of the failure rate of software are compared. It is shown that one tends to be better when the number of errors is small to moderate and the other when the number of errors is large.

The most probable equilibrium configuration in a circuit-switched network is determined, in the limit of very large scale, by a primal optimization problem whose dual determines the blocking probabilities. This approximation can be inadequate for moderate scale, and so the blocking probabilities are usually determined by the Erlang fixed-point principle. In this paper, it is shown that there is a “softened” form of the dual natural to the problem. This gives determinations of the blocking probabilities agreeing with those derived from the Erlang principle as far as terms of order (scale)-1. It is more attractive than the Erlang principle, however, in that it has a mathematical basis, provides an extremal principle by its nature, yields immediately the geometric distribution of the numbers of free circuits, and clearly agrees with the “hard” dual in the limit of large scale.

We define a notion of order of recurrence for the states and transitions of a general class of time-inhomogeneous Markov chains with transition probabilities proportional to powers of a small vanishing parameter. These orders are shown to satisfy a balance equation across every edge cut in the associated graph. The resulting order balance equations allow computation of the orders of recurrence of the states, and thereby the determination of the asymptotic behavior of the Markov chain.

The method of optimization by simulated annealing is a special case of such Markov processes, and can therefore be treated by means of these balance equations. In particular, in this special situation we show that there holds a detailed balance of order of recurrence across every edge in the graph. Moreover, the sum of the order of recurrence of a state and its cost is shown to be a constant in each connected set of recurrent states. By this approach, we determine the necessary and sufficient condition on the “rate of cooling” to guarantee that a minimum of the optimization problem is hit with probability one. Moreover, the rates of convergence of the probabilities can be deduced from the orders of recurrence.

The mathematical theory of evidence (Shafer et al. [9]) has recently found much interest as an approach to treat uncertainty in expert and knowledge-based systems. Although the theory is very promising, there are not yet many practical applications. Modeling practice has still to be developed. This is a crucial task in view of facilitating the application of evidential modeling. It is the aim of this paper to discuss an important element of evidential modeling–conditional belief–within the scope of the mathematical theory of evidence.

From an urn containing colored balls, one ball is drawn and replaced by a random number of differently colored balls, with the distribution of the added balls depending only on the color of the ball drawn. Under mild regularity conditions, the proportions of different colors will converge to deterministic limits. Two applications of this standard result are described.

System components of communication/computer networks are quite reliable in that their average uptimes are much larger than the average repair/replacement time of a failed unit. By taking this observation into account, a semiMarkov model is developed with a simple regenerative structure, thereby providing strong analytical and computational tractability. Expressions of a variety of dynamic performability measures, such as the cumulative system processing capacity and the task completion time, are explicitly derived. Computational procedures for evaluating such time-dependent performability measures are developed based on these theoretical results combined with the Laguerre transform method. The power and the efficiency of the computational procedures are demonstrated through a numerical example.

Generalized semi-Markov processes and stochastic Petri nets have been proposed as general frameworks for a discrete event simulation on a countable state space. The two formal systems differ, however, with respect to the clock setting (event scheduling) mechanism, the state transition mechanism, and the form of the state space. We obtain conditions under which the marking process of a stochastic Petri net “mimics” a generalized semi-Markov process in the sense that the two processes (and their underlying general state-space Markov chains) have the same finite dimensional distributions. The results imply that stochastic Petri nets have at least the modeling power of generalized semiMarkov processes for discrete event simulation.

We consider the group testing problem for a set of independent items I = [I1,… In] where Ii, has probability pi, of being defective and probability qi = 1 – pi of being good. The problem is to classify all items as good or defective with a minimum expected number of group tests where a group test is a test on a subset S of I with two possible outcomes: either S is pure (contains no defective) or S is contaminated (contains at least one defective, with no information provided about which or how many). No polynomial-time algorithm is known for the group testing problem even for the special case pi = p for all i. Hence, any method that reduces the size of the problem is very helpful. In this paper, we give such a method by providing a simple condition to screen items that should be tested (only) individually. This condition leads to a necessary and sufficient condition for the individual testing algorithm to be optimal, generalizing a result of Unger [1] for the special case of identical pi.

This paper considers a two-person zero-sum stochastic differential game. The dynamics of the game are given by a one-dimensional stochastic differential equation whose diffusion coefficient may be controlled by the players. The drift coefficient is held constant and cannot be controlled. Player l's objective is to maximize the probability that the state at final time, T, is positive, while Player 2's objective is to maximize the probability that the state is negative.

The unfinished work at arrival instants (FCFS waiting time) can be computed via the LCFS preemptive resume discipline. Analyzing the GI/GI/1 queue in this way gives full interpretation in queuing theoretic language to Feller's analysis of the fluctuations of the random walk. Also, this approach leads naturally to stochastic decompositions for the queue with set-up times, and for other variations of the standard queue. For the M/G/1 queue, the derivations are qualitative, and there are additional connections to branching processes.

An improved simulation estimator of the expected total delay of the first n customers in the systems GI/M/k and GI/G/I is obtained by replacing the (raw data) delay of customer i by its expected delay given the state of the system upon its arrival. It is improved in the sense that the sum of the first n such values has the same mean and smaller variance than the usual (raw) estimator. The only constraint on the arrival process is that it be independent of the process of service times.

There are three general methods for system reliability evaluation, namely; (1) inclusion–exclusion, (2) sum of disjoint products, and (3) pivoting. Of these, only pivoting can be applied directly to a logic tree or network graph representation without first finding minimal path (or cut) sets. Domination theory provides the basis for selecting optimal pivoting strategies. Simple proofs of domination-theory results for coherent systems are given, based on the reliability polynomial. These results are related to the problem of finding efficient strategies for computing coherent system reliability. The original results for undirected networks are due to Satyanarayana and Chang [5] (cf. [1]). Many of the original set theoretic results are due to Huseby [3]. However, he does not use the reliability polynomial to prove his results.

This paper derives the steady-state distribution of the virtual wait in GI/M1 queues in which the interarrival times depend on the virtual wait. Applications are queues in which the time it will take for the system to start servicing a new arrival (virtual wait) affects the generation of new arrivals. Such queues arise in airport or large-city freeway traffic control systems. Related state-dependent situations occur in production-inventory systems, cash flow in banking or in insurance systems, dam control systems, and in natural sciences. The method of analysis is by means of system-point level crossing theory.

This paper considers a single machine and n jobs. The machine is subject to breakdown and repair. Job j has a weight wj, a random processing time Xj, and is available for processing from a random time Rj on. The jobs may be subject to precedence constraints. We are interested in optimal policies that minimize the following objective functions: (i) the weighted sum of the completion times; (ii) the weighted sum of an exponential function of the completion times; and (iii) the weighted number of late jobs having due dates.

Recently, Ross [1] proposed an elegant method of approximating transition probabilities and mean occupation times in continuous-time Markov chains based upon recursively inspecting the process at exponential times. The method turned out to be amazingly efficient for the examples investigated. However, no formal rough error bound was provided. Any error bound even though robust is of practical interest in engineering (e.g., for determining truncation criteria or setting up an experiment). This note primarily aims to show that by a simple and standard comparison relation a rough error bound of the method is secured. Also, some alternative approximations are inspected.

The group-testing problem for a binomial set of items is considered. It is desired to classify all items as good or defective with a minimum expected number of group tests. An improvement over the information lower bound, via a weak concavity property, is made for the minimum expected number of group tests.