A subfair discounted red-and-black game with a house limit is introduced and the optimality of the modified bold strategy is shown.

A needle of length l dropped at random on a grid of parallel lines of distance d apart can have multiple intersections if l > d. The distribution of the number of intersections and approximate moments for large l are derived. The distribution is shown to converge weakly to an arc sine law as l/d→∞.

The queueing system GI/G/1 with group arrivals and individual service of the customers is considered. For the stable situation the limiting distribution of the waiting time distribution of the kth served customer for k → ∞ is derived by using the theory of regenerative processes. It is assumed that the group sizes are i.i.d. variables of which the distribution is aperiodic. The relation between this limiting distribution and the stationary distribution of the virtual waiting time is derived.

In 1968 Lampard determined some limit distributions for a counter process in which the input and output are independent Poisson processes. In 1974 Phatarfod dealt with the generalizations of Lampard's formulas for the case when the input and output form a bivariate Poisson process; however, his reasoning is erroneous. The object of this paper is to determine the correct limit distributions for the generalized process.

A two-server loss system is considered with N classes of Poisson arrivals, where the service distribution function and server preferences are arrival-class dependent. The stationary state probabilities are derived and found to be independent of the form of the service distributions.

This note draws attention to a practical problem of interest to physicists, namely the distribution of the inter-arrival times of those customers accepted by the M/D/1/K + 1 queue. The form of the distribution is found for all K and the generalisation to M/G/1/K + 1 is discussed. The approach follows the contour integral method described by Cohen.