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.

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.

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.

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.

We consider a K-out-of-N system maintained by R repairmen, where the lifetime of the ith component is an exponentially distributed random variable with parameter μi. Repairmen are distinguishable, and the time it takes the rth repairman to repair a failed component is an exponentially distributed random variable with parameter λr. Repaired components are as good as new and preemptions are allowed. We show that the policy that assigns the faster repairmen to the most reliable components is optimal with respect to several optimality criteria.

The approach we take in establishing stochastic optimality with respect to the number of functioning components is of wide applicability to different classes of stochastic optimization problems.

A consecutive-k−out−of−n system is a graph with n vertices where the system fails if and only if some path of k consecutive vertices all fail. The assumption is made that qi, is the failing probability of the ith vertex and the states of the vertices are statistically independent. The problem is to find the best system, i.e., a set {q1, …,qn}, and its assignment to the vertices of the system graph that maximizes the reliability of the system, under the constraint that the product of qi's is a constant. We solve the problem for all lines and some cycles. We then extend our results to trees and show that for given n and k the best tree is a star and the worst is a line.

We study a model of queue storage in which items (requests for single units of storage) arrive in a Poisson stream and are accommodated by the first available location in a linear scan of storage. The processing times of items are independent, exponentially distributed random variables. The set of occupied locations (identified by their indices) at time t forms a random subset Si, of [1,2,.…]. The extent of the fragmentation in Si, i.e., the alternating holes and occupied regions of storage, is measured by Wt, = max St, – |St|.

Simultaneous estimation problems that deal with the estimation of several means for normal distributions are considered under the squared-error loss. Heuristic procedures are presented which can be applied to parameter estimation problems for a wide class of distributions specified only by their means and variances. Explicit results are obtained for the heuristic shrinkage estimators in the normal distribution case. Limiting behavior of the relative risk savings of these estimators is studied. The performances of the proposed estimators for normal distribution means are compared with other existing estimators by a computer simulation.

When confronted by a sequence of independent, identical, favorable-though- risky investments, a risk-averse agent will in fact elect to invest as part of a large syndicate even if he or she would not undertake even one of the investments by himself or herself. This willingness to invest emanates from the risk reduction associated with the risk pooling effected by the syndicate. Our first two theorems describe the syndicate size needed to induce the agent's participation.

The consequence of adding risks is considered next. We demonstrate that the agent will undertake a large number of these independent investments by himself or herself even if each investment taken by itself constitutes a poor risk. The conditions needed to ensure this somewhat surprising result are (a) moment bounds on the investments, and (b) either a polynomial or an exponential bound on the lower tail of the agent's utility function. Finally, we draw the connection between this result, proper risk aversion, and monotonicity of the agent's absolute aversion to risk.

Simple and insensitive lower and upper bounds are proposed for the call congestion of M/GI/c/n queues. To prove them we establish the general monotonicity property that increasing the waiting room and/or the number of servers in a /GI/c/n queue increases the throughput. An asymptotic result on the number of busy servers is obtained as a consequence of the bounds. Numerical evidence as well as an application to optimal design illustrates the potential usefulness for engineering purposes. The proof is based on a sample path argument.

Two simple approximation formulas for approximating the areas under the t distribution are presented here which are easily programmable on pocket calculators. Their accuracy should be sufficient for most of the practical applications.