We obtain a limit theorem for the joint distribution of the maximum value and sample mean of a random length sequence of independent and identically distributed random variables. This extends a previous bivariate convergence result for fixed length sequences and incidentally yields a new proof of Berman's classical limit theorem for the maximum value of a random number of random variables. Our approach uses a property of record time sequences and leads to probabilistically intuitive proofs. We also consider the partition of a finite interval into a random number of subintervals by the points of a non-delayed renewal process. Using the bivariate convergence result for random length sequences, we establish a limit theorem for the joint distribution of the number and maximum length of the subintervals as the interval length becomes large. This leads to limiting results for the ratio of the maximum to the mean subinterval length. Such results are of interest in connection with a simple model of parallel processing.

This paper presents the form of some characteristics of the Voronoi tessellation which is generated by a stationary Poisson process in . Expressions are given for the spherical and linear contact distribution functions. These formulae lead to numerically tractable double-integral formulae for chord length probability density functions.

The Connors–Kumar notion of recurrence orders for state and state transitions of time-inhomogeneous Markov chains is redefined ‘pathwise' and applied to the sample path analysis of the simulated annealing algorithm.

We consider the Mk/M/∞ queue with k heterogeneous customers in a batch where the customer of type i in a batch requires an exponential service time with parameter µi. In steady state, the joint generating function of the number of customers of type i being served in the system is derived explicitly by solving a partial differential equation.