Suppose you repeatedly play a game of chance in which you have the advantage. Your return on investment is your net gain divided by the total amount that you have bet. It is shown that the ratio of your return on investment under optimal proportional betting to your return on investment under constant betting converges to an exponential distribution with mean as your advantage tends to 0. The case of non-optimal proportional betting is also treated.

The paper presents sufficient conditions for certain functions to be convex. Functions of this type often appear in Markov decision processes, where their maximum is the solution of the problem. Since a convex function takes its maximum at an extreme point, the conditions may greatly simplify a problem. In some cases a full solution may be obtained after the reduction is made. Some illustrative examples are discussed.

Consider a random triangle formed by vertices which are independent and distributed uniformly inside a given bounded convex region K. Then a well-known result of W. Blaschke states that, among all regions of fixed area, the area AK of this random triangle is stochastically bounded by Acircle (Ac) and A triangle (At). The distribution of the latter was obtained by Alagar (1977). We use the classical Crofton technique to derive an integral representation for the distribution function Fc of Ac from which Fc is calculated using numerical quadrature. The maximum deviation between Fc and the distribution function of At is less than 0.04. Additionally, we obtain an explicit form for the distribution of Aparallelogram which may be represented as a mixture of functions involving Dirichlet-distributed random vectors.

Arnold and Laguna introduced a model for income distributions in which the income of the present generation of individuals has the same distribution as the minimum of a random number Nn of independent copies of some random variable and {Nn} is independent. The present paper gives a fairly complete analysis of this model and a number of extensions of it.

The Weibull distribution for the strength of a long fiber is reviewed, with emphasis on its derivation via extreme-value theory. It is shown that a corresponding Weibull distribution, but with different scale and shape parameters, is appropriate for the lower tail distribution of a bundle of parallel fibers under general load-sharing assumptions. This result is applied to series-parallel systems. The paper contains discussion of the application of these results to the strength of fibrous materials. It is concluded that the Weibull approximation, developed for series-parallel systems, is appropriate for long thin materials consisting of a small number of parallel fibers.

When two-dimensional figures, called laminae, are randomly placed on a plane domains result that can either be aggregates or individual laminae. The intersection of the union, U, of these domains with a specified field of view, F, in the plane is considered. The separate elements of the intersection are called clumps; they may be laminae, aggregates or partial laminae and aggregates. A formula is derived for the expected number of clumps minus enclosed voids. For bounded laminae homeomorphic to a closed disc with isotropic random direction the formula contains only their mean area and mean perimeter, the area and perimeter of F, and the intensity of the Poisson process.

In this paper a point-process approach is given for determining the Palm-type (number-weighted) distribution of the size factor of the grains of a stationary grain model in the plane with non-overlapping, identically shaped and identically orientated convex grains starting from a suitably chosen characteristic of the grain model observed in fixed points of the plane.

Some new characterizations of the geometric distribution are studied. A generalization of the characterization by the well-known ‘lack-of-memory' property is given together with some closely related characterizations. Furthermore the modified geometric distribution is characterized by a distributional property of the difference of two successive order statistics. The latter result extends work of Puri and Rubin (1970). Finally the geometric distribution is characterized by a conditional distribution property of the difference of two arbitrary order statistics, which generalizes a result by Arnold (1980). Some of the results given answer open questions put in earlier papers.

By considering continuous-time Markov decision processes where decisions can be made at any time, we show in the case of M/M/1 queues with discounted costs that there exists a monotone optimal policy among all the regular policies.

This is the sequel to a previous paper (Baudin (1981)). The joint probability generating functional of two point processes is introduced as a tool to compute the conditional intensity of the process of cluster centers of a multidimensional Poisson cluster process when a realization is given in a bounded observation window. An explicit formula is derived but it is too complicated for actual use; a linear method for practical estimation is discussed.

We consider the Darvey, Ninham and Staff model for reversible chemical reactions, in the case where the ratio of the rate constants is either very large or very small. It is shown that the distribution of the number of molecules at equilibrium may sometimes be closely approximated by the Poisson distribution, and on other occasions, by a distribution with much smaller tails than the Poisson. The second type of approximating distribution is termed a Bessel distribution, and its properties are studied.

A device is repaired at failure. With probability p, it is returned to the ‘good-as-new' state (perfect repair), with probability 1 – p, it is returned to the functioning state, but it is only as good as a device of age equal to its age at failure (imperfect repair). Repair takes negligible time. We obtain the distribution Fp of the interval between successive good-as-new states in terms of the underlying life distribution F. We show that if F is in any of the life distribution classes IFR, DFR, IFRA, DFRA, NBU, NWU, DMRL, or IMRL, then Fp is in the same class. Finally, we obtain a number of monotonicity properties for various parameters and random variables of the stochastic process. The results obtained are of interest in the context of stochastic processes in general, as well as being useful in the particular imperfect repair model studied.

A general result for queueing systems with retrials is presented. This result relates the expected total number of retrials conducted by an arbitrary customer to the expected total number of retrials that take place during an arbitrary service time. This result is used in the analysis of a special system where two types of customer arrive in an independent Poisson fashion at a single-server service station with no waiting room. The service times of the two types of customer have independent general distributions with finite second moments. When the incoming customer finds the server busy he immediately leaves and tries his luck again after an exponential amount of time. The retrial rates are different for different types of customers. Expressions are derived for the expected number of retrial customers of each type.

For several queueing systems, sufficient conditions are given ensuring that from the coincidence of some time-stationary and customer-stationary characteristics of the number of customers in the system such as idle or loss probabilities it follows that the arrival process is Poisson.

In this paper we define and analyze a general shock model associated with a correlated pair (Xn, Yn) of renewal sequences, where the system fails when the magnitude of a shock exceeds (or falls below) a prespecified threshold level. Two models, depending on whether the nth shock Xn is correlated to the length Yn of the interval since the last shock, or to the length Yn of the subsequent interval until the next shock, are considered. The transform results, an exponential limit theorem, and properties of the associated renewal process of the failure times are obtained. An application in a stochastic clearing system with numerical results is also given.

Consider a many-server queueing system in which the servers are numbered. If a customer arrives when two or more servers are idle he selects the server with lowest index (this is called the ordered entry selection rule). An explicit expression for the traffic handled by the various servers in a GI/M/s queueing system with ordered entry is derived. For the M/M/s queueing system the probability distribution of the number of busy servers among the first k(k = 1, 2,…, s) servers will be given. Finally, a formula for the traffic handled by the first server in an M/D/s system will be derived. All results are derived under steady-state conditions. As an application some numerical data for the server utilizations will be given and compared to data obtained from simulation studies of a closed-loop continuous belt-conveyor.

The class of life distributions for which , where , and , is studied. We prove that this class is larger than the HNBUE (HNWUE) class (consisting of those life distributions for which for x ≧ 0) and present results concerning closure properties under some usual reliability operations. We also study some shock models and a certain cumulative damage model. The class of discrete life distributions for which for 0 ≦ p ≦ 1, where , is also studied.

Consider m distinguishable urns with k distinguishable cells each and suppose that n indistinguishable balls are randomly allocated into these urns. When the capacity of each cell is limited to one ball (restricted occupancy) or unlimited (pseudo-contagious occupancy) and empty urns are not permitted, the probability function and factorial moments of the number Mt = Mt (n, m, k) of urns containing exactly t balls are expressed in terms of numbers related to the Stirling numbers. The limiting distributions of Mt = Mt (n, m, k) when k → 0 or k →∞ are derived; these distributions are expressed in terms of the Stirling numbers of the first and second kind respectively. Moreover a modification of the occupancy models of Barton and David is shown to lead to the preceding distributions.

For a stochastic epidemic of the type considered by Bailey [1] and Kendall [3], Daniels [2] showed that ‘when the threshold is large but the population size is much larger, the distribution of the number remaining uninfected in a large epidemic has approximately the Poisson form.' A simple, intuitive proof is given for this result without use of Daniels's assumption that the original number of infectives is ‘small'. The proof is based on a construction of the epidemic process which is more explicit than the usual description.

We prove a threshold theorem for the Reed–Frost chain-binomial model which is analogous to the threshold theorem of Williams (1971) for the general stochastic epidemic. We show that when the population size is large a ‘true epidemic’ occurs with a non-zero probability if and only if an initial infective individual infects on average more than one susceptible individual.