Several notions of stochastic convexity and concavity and their properties are studied in this paper. Efficient sample path approaches are developed in order to verify the occurrence of these notions in various applications. Numerous examples are given. The use of these notions in several areas of probability and statistics is demonstrated. In queueing theory, the convexity (as a function of c) of the steady-state waiting time in a GI/D/c queue, and (as a function of the arrival or service rates) in a GI/G/1 queue, is established. Also the convexity of the queue length in the M/M/c case (as a function of the arrival rate) is shown, thus strengthening previous results while simplifying their derivation. In reliability theory, the convexity of the payoff in the success rate of an imperfect repair is obtained and used to find an optimal repair probability. Also the convexity of the damage as a function of time in a cumulative damage shock model is shown. In branching processes, the convexity of the population size as a function of a parameter of the offspring distribution is proved. In non-parametric statistics, the stochastic concavity (or convexity) of the empirical distribution function is established. And, for applications in the theory of probability inequalities, we identify several families of distributions which are convexly parametrized.

The main characteristics of the time-nonhomogeneous Johnson–Mehl tessellation with specified nucleation intensity α (t) = αtß, β > −1 and α > 0 being constants and t time, are investigated as functions of β.

This paper deals with the single-server queue with batch arrivals. We show that under suitable conditions the waiting-time distribution of an individual customer has an asymptotically exponential expansion. Computationally useful characterizations of the amplitude factor and the decay parameter are given for the practically important case in which the interarrival time and the service time have phase-type distributions.

In this paper we consider the observed lifetime of a multistate monotone system and ‘the critical lower set' which causes the system deterioration. Then, under suitable conditions, we identify the component lifetime distribution using a Newton–Kantorovic iterative procedure as in Meilijson (1981).

We consider a queue which is obtained by slightly modifying the M/PH/1 queue, so that weak correlations are introduced among the interarrival times. We show how the stationary probability distribution may be studied by a perturbation analysis.

Asymptotic approximations are developed herein for the optimal policies in discounted multi-armed bandit problems in which new projects are continually appearing, commonly known as ‘open bandit problems’ or ‘arm-acquiring bandits’. It is shown that under certain stability assumptions the open bandit problem is asymptotically equivalent to a closed bandit problem in which there is no arrival of new projects, as the discount factor approaches 1. Applications of these results to optimal scheduling of queueing networks are given. In particular, Klimov&s priority indices for scheduling queueing networks are shown to be limits of the Gittins indices for the associated closed bandit problem, and extensions of Klimov&s results to preemptive policies and to unstable queueing systems are given.

A consecutive k-within-m-out-of-n system consists of n identical and stochastically independent components arranged on a line. The system will fail if and only if within m consecutive components, there are at least k failures. Let Tn be the system's lifetime. Then, under quite general conditions we prove that there is a positive constant a such that the random variable n1/kaTn converges to a Weibull distribution as n →∞.

We consider a stochastic process in continuous time and two point processes on the real line, all jointly stationary. We show that under a certain mixing condition the values of the process at the points of the second point process converge weakly under the Palm distribution with respect to the first point process, and we identify the limit. This result is a supplement to two other known results which are mentioned below.

It is shown by a simple sample path argument that the ruin probabilities for a risk reserve process with premium rate p(r) depending on the reserve r and finite or infinite horizon are related in a simple way to the state probabilities of a compound Poisson dam with the same release rate p(r) at content r. In the infinite horizon case, this result has been established by Harrison and Resnick (1978), and in the finite horizon case with constant p it extends well-known relations to the M/G/1 virtual waiting time.

Let [An, Bn] be random subintervals of [0, 1] defined recursively as follows. Let A1 = 0, B1 = 1 and take Cn, Dn to be the minimum and maximum of k, i.i.d. random points uniformly distributed on [An, Bn]. Choose [An+1, Bn+1] to be [Cn, Bn], [Any Dn] or [Cn, Dn] with probabilities p, q, r respectively, p + q + r = 1. It is shown that the limiting distribution of [Any Bn] has the beta distribution on [0,1] with parameters k(p + r) and k(q + r). The result is used to consider a randomized version of Golden Section search.

In this note, we study a new repair replacement model for a deteriorating system, in which the successive survival times of the system form a geometric process and are stochastically non-increasing, whereas the consecutive repair times after failure also constitute a geometric process but are stochastically non-decreasing. Two kinds of replacement policy are considered, one based on the working age of the system and the other one determined by the number of failures. The explicit expressions of the long-run average costs per unit time under these two kinds of policy are calculated.