Goldstein (1974) derived an upper bound on the variance of certain non-negative functions when the first two moments of the underlying random variable are known. This bound is compared to a simple and fundamental variance bound which requires only that the range of the function be known. It is shown that Goldstein’s bound frequently exceeds the simpler bound. Finally, an interpretation of such bounds in the context of economic risk analysis is given.

A discrete-time model is presented for a system of two queues competing for the service attention of a single server with infinite buffer capacity. The service requirements are geometrically distributed and independent from customer to customer as well as from the arrivals. The allocation of service attention is governed by feedback policies which are based on past decisions and buffer content histories. The cost of operation per unit time is a linear function of the queue sizes. Under the model assumptions, a fixed prioritization scheme, known as the μc-rule, is shown to be optimal for the expected long-run average criterion and for the expected discounted criterion, over both finite and infinite horizons. Two different approaches are proposed for solving these problems. One is based on the dynamic programming methodology for Markov decision processes, and assumes the arrivals to be i.i.d. The other is valid under no additional assumption on the arrival stream and uses direct comparison arguments. In both cases, the sample path properties of the adopted state-space model are exploited.

It is proved that a distribution F is strongly unimodal iff any two quantiles of the convolution of F with any other distribution are further apart than the corresponding quantiles of F itself. Some related characterizations of strong unimodality are also given.

Asymptotic series are derived for two integrals using a Gaussian identity and Laplace’s method, demonstrating an improvement over earlier methods.

An urn contains balls of different colours, which are randomly drawn one at a time. After each draw the number of balls in the urn with the same colour as the ball last drawn is changed. Special cases are sampling with and without replacement, and Pólya sampling. The drawing stops when a given number of colours has been drawn a given number of times. The number of times the different colours have been drawn is studied in this paper by imbedding the urn scheme in birth processes. Both exact and asymptotic results are obtained. In particular, waiting times for sampling with and without replacement and for Pólya sampling are considered.

Suppose segments are distributed at random along a line, their locations being determined by a Poisson process. In the case where segment length is fixed, we compare efficiencies of several different estimates of Poisson intensity. The case of random segment length is also considered, and there we study estimation procedures based on empiric properties. The one-dimensional mosaic may be viewed as an M/G/∞ queue.

We study certain aspects of the sample path behaviour of χ2 processes; in particular, problems related to the behaviour of these processes at their local extrema. Emphasis is placed on behaviour that is qualitatively different to that observed for Gaussian processes, rather than on phenomena common to both classes of processes, such as previously studied (global) extremal type results.

The aim of this note is to prove the following characterization.

We give an upper bound for the average complexity (i.e. the expected number of steps until termination) for a continuous random search algorithm using results from renewal theory. It is thus possible to show that for a predefined accuracy ε, the average complexity of the algorithm is O(–log ε) for ε → 0 which is optimal up to a constant factor.

Bounds obtained by Ramalhoto [1] for the variance of the busy period of an M/G/∞ queue are improved.

A gambler repeatedly plays a game until either he becomes broke or his fortune becomes equal to or exceeds a target amount. The gambler is allowed to make multiple bets, i.e. stake integral amounts on different alternatives of the game, and more than one bet may win simultaneously. The objective is to determine a strategy that maximizes the probability of attaining the target amount. When all bets have the same gain and alternatives of betting exist such that the relevant plays are feasible, the following results are obtained. For unfavorable games, a bold type of policy is shown to be optimal. A timed type of policy is shown to be best within a restricted class of policies for favorable games. In general, optimal policies contain multiple bets. Based on a numerical example, this is established for roulette also.

In this study we assume an inventory system for perishable commodities in which the lifetimes of the items stored are i.i.d. random variable with finite mean.

We utilize the analogy between this inventory system and a queueing system with impatient customers, to study the process of the lost demand, the death of the unused items and the number of items in the system.

The paper starts with a simple direct proof that . A new formula is given for the shape-density for a triangle whose vertices are i.i.d.-uniform in a compact convex set K, and an exact evaluation of that shape-density is obtained when K is a circular disk. An (x, y)-diagram for an auxiliary shape-density is then introduced. When K = circular disk, it is shown that is virtually constant over a substantial region adjacent to the relevant section of the collinearity locus, large enough to contain the work-space for most collinearity studies, and particularly appropriate when the ‘strip’ method is used to assess near-collinearity.

The cμ rule is optimal for arbitrary arrival processes provided that the service times are geometric and the service discipline is preemptive.

A linear mosaic is a process of random line segments distributed according to a Poisson process. This paper presents a wide-ranging treatment of limit theory for the two major macroscopic properties of a linear mosaic: total vacancy, and total number of spacings (or clumps). These quantities do not admit an explicit, exact treatment, and are perhaps most informatively studied by means of limit theory. We permit segment length to have a general distribution, and study the implications of tail properties of this distribution. Necessary and sufficient conditions are given for vacancy on an interval [0, t] to admit a normal approximation as t →∞. An approximate formula is provided for the probability of complete coverage, in the case of general segment length distribution. In all cases, properties of vacancy and number of spacings are studied together, by means of joint limit theorems, rather than individually, as in some earlier work.

Iglehart and Shedler (1983) prove that the ‘labelled jobs’ method for estimation of passage-time characteristics in closed multiclass networks of queues with general service times provides asymptotically shorter confidence intervals than does the ‘marked job’ method. A simple alternative proof of this result, under slightly more restrictive conditions, is given here.

In this paper we define and analyze a class of cumulative shock models associated with a bivariate sequence {Xn, Yn}∞n=0 of correlated random variables. The {Xn} denote the sizes of the shocks and the {Yn} denote the times between successive shocks. The system fails when the cumulative magnitude of the shocks exceeds a prespecified level z. Two models, depending on whether the size of the nth shock is correlated with the length of the interval since the last shock or with the length of the succeeding interval until the next shock, are considered. Various transform results and asymptotic properties of the system failure time are obtained. Further, sufficient conditions are established under which system failure time is new better than used, new better than used in expectation, and harmonic new better than used in expectation.