We consider the asymptotic variance of the departure counting process D(t) of the GI/G/1 queue; D(t) denotes the number of departures up to time t. We focus on the case where the system load ϱ equals 1, and prove that the asymptotic variance rate satisfies limt→∞varD(t) / t = λ(1–2/π)(ca2 + cs2), where λ is the arrival rate, and ca2 and cs2 are squared coefficients of variation of the interarrival and service times, respectively. As a consequence, the departures variability has a remarkable singularity in the case in which ϱ equals 1, in line with the BRAVO (balancing reduces asymptotic variance of outputs) effect which was previously encountered in finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multiserver queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue, we present an explicit expression of the variance of D(t) for any t.

We consider spatial stochastic models, which can be applied to, e.g. telecommunication networks with two hierarchy levels. In particular, we consider Cox processes XL and XH concentrated on the edge set T(1) of a random tessellation T, where the points XL,n and XH,n of XL and XH can describe the locations of low-level and high-level network components, respectively, and T(1) the underlying infrastructure of the network, such as road systems, railways, etc. Furthermore, each point XL,n of XL is marked with the shortest path along the edges of T to the nearest (in the Euclidean sense) point of XH. We investigate the typical shortest path length C* of the resulting marked point process, which is an important characteristic in, e.g. performance analysis and planning of telecommunication networks. In particular, we show that the distribution of C* converges to simple parametric limit distributions if a scaling factor κ converges to 0 or ∞. This can be used to approximate the density of C* by analytical formulae for a wide range of κ.

In this paper we demonstrate how some characteristics of queues with the periodic Poisson arrivals can be approximated by the respective characteristics in queues with Markov modulated input. These Markov modulated queues were recently studied by Regterschot and de Smit (1984). The approximation theorems are given in terms of the weak convergence of some characteristics and their uniform integrability. The approximations are applicable for the following characteristics: mean workload, mean workload at the time of day, mean delay, mean queue size.

An almost sure convergence result for the normed population size of a bisexual population model is proved. Properties of the limit random variable are deduced. The derivation of similar results for a general class of such processes is discussed.

The strong law of large numbers of the Marcinkiewicz–Zygmund type is established for the total population in a subcritical branching process with immigration. The moment convergence of the total population is obtained under appropriate moment conditions on the offspring distribution and the immigration distribution.

A region contains n prey labeled 1, 2, …, n. Prey i is captured at the random time Zi; where Z1, Z2, …, Zn are i.i.d. with distribution function F. The statistician must decide when to stop searching, with the goal of maximizing the number of prey captured minus a linear time cost, c. The optimal strategy and its expected payoff are studied asymptotically as n, c →∞, for F a beta or Weibull distribution.