We present a detailed analysis of random motions moving in higher spaces with a natural number of velocities. In the case of the so-called minimal random dynamics, under some broad assumptions, we give the joint distribution of the position of the motion (for both the inner part and the boundary of the support) and the number of displacements performed with each velocity. Explicit results for cyclic and complete motions are derived. We establish useful relationships between motions moving in different spaces, and we derive the form of the distribution of the movements in arbitrary dimension. Finally, we investigate further properties for stochastic motions governed by non-homogeneous Poisson processes.

We create a framework to analyze the timing and frequency of instantaneous interactions between pairs of entities. This type of interaction data is especially common nowadays and easily available. Examples of instantaneous interactions include email networks, phone call networks, and some common types of technological and transportation networks. Our framework relies on a novel extension of the latent position network model: we assume that the entities are embedded in a latent Euclidean space and that they move along individual trajectories which are continuous over time. These trajectories are used to characterize the timing and frequency of the pairwise interactions. We discuss an inferential framework where we estimate the individual trajectories from the observed interaction data and propose applications on artificial and real data.

In a multitype branching process, it is assumed that immigrants arrive according to a non-homogeneous Poisson or a contagious Poisson process (both processes are formulated as a non-homogeneous birth process with an appropriate choice of transition intensities). We show that the normalized numbers of objects of the various types alive at time t for supercritical, critical, and subcritical cases jointly converge in distribution under those two different arrival processes. Furthermore, we provide some transient expectation results when there are only two types of particles.

Consider two systems, labeled system 1 and system 2, each with m components. Suppose component i in system k, k = 1, 2, is subjected to a sequence of shocks occurring randomly in time according to a non-explosive counting process {Γ i(t), t > 0}, i = 1, ···, m. Assume that Γ1, · ··, Γm are independent of Mk = (Mk,1, · ··, Mk,m), the number of shocks each component in system k can sustain without failure. Let Zk,i be the lifetime of component i in system k. We find conditions on processes Γ1, · ··, Tm such that some stochastic orders between M1 and M2 are transformed into some stochastic orders between Z1 and Z2. Most results are obtained under the assumption that Γ1, · ··, Γm are independent Poisson processes, but some generalizations are possible and can be seen from the proofs of theorems.

First, some basic concepts from the theory of point processes are recalled and expanded. Then some notions of stochastic comparisons, which compare whole processes, are introduced. The use of these notions is illustrated by stochastically comparing renewal and related processes. Finally, applications of the different notions of stochastic ordering of point processes to many replacement policies are given.

The reliability of many stochastic repairable systems depends on several characteristics that are time dependent. In this paper, we develop general repair models for a repairable system by using auxiliary stochastic processes which describe physical characteristics of the system and derive various properties of resulting models. We also obtain an inference procedure to assess the number of failures and expected number of failures for our proposed models by observing auxiliary processes.

Clearly, our inference procedure is different from the traditional approach, where successive times of occurrence of failures are observed.

In this paper we consider parallel and series systems, the components of which can be ‘improved'. The ‘improvement' consists of supplying the components with cold or hot standby spares or by allotting to them fixed budgets for minimal repairs. A fixed total resource of spares or minimal repairs is available. We find the optimal allocation of the resource items in several commonly encountered settings.

We consider models of systems of components with dependent life-lengths having certain multivariate distributions. Upon failure, components are repaired. Two types of repair are possible. After perfect repair, a unit has the same life distribution as a new item. After imperfect repair, a unit has the life distribution of an item which is of the same age but has never failed. Different sources of failure are distinguished and affect the probabilities of perfect and imperfect repair.

Singh and Jain (1989) have proved some preservation results for partial orderings of life distributions assuming that shocks occur according to a homogeneous Poisson process. It is shown that their results hold under less restrictive conditions.

The problem of how to compare repairable systems has long been of interest to those who are engaged in reliability analysis. In this article we have introduced a type of comparison of two counting processes. Various properties and applications of our stochastic comparison are also studied.

We consider a class of secretary problems in which the order of arrival of candidates is no longer uniformly distributed. By a suitable embedding in a time-transformed extremal process it is shown that the asymptotic winning probability is again 1/e as in the classical situation. Extensions of the problem to more than one choice are also considered.

A direct approach to derive dependence properties among the hitting times of bivariate processes has been initiated by Ebrahimi (1987) and explored further by Ebrahimi and Ramallingam (1988). In this paper, new results are obtained for multivariate processes, which help us to identify positive and negative dependence structures among the hitting times of the processes. Applications of our theorems to reliability of systems are given.

The general cost structure of a unit on line is assumed to be governed by a stochastic process , where R(t) denotes the operating cost on [0, t)and ζ denotes the time of an unscheduled (or unplanned) replacement by a new unit at a cost c1. For an age replacement maintenance policy, scheduled (or planned) replacements occur whenever an operating unit reaches age T, whereas in the block replacement case, scheduled replacements occur every T units of time. Such scheduled replacements cost c2. The expected long-run cost per unit time can then be expressed in the form A(T)/Ε [min(ζ, T)] and B(T)/T respectively. Our main result shows that where U is the associated renewal function generated by ζ.

A system is subject to shocks; each shock at time t increases the cumulative damage λ (t) by a constant amount, while the system is subject to repair in between the shocks which brings down λ (t) at a constant rate. The shock arrival process is an inhomogeneous Poisson process with intensity function λ (t) and each shock weakens the system making it more expensive to run. The long-run expected cost per unit time of running the system is obtained as well as the variance of the cost which are used to get optimal times of replacement of the system.

In this article we examine an R & D project in which the project status changes according to a diffusion process. The decision variables include a resource expenditure strategy and a stopping policy which determines when the project should be terminated. The drift and the diffusion parameters of the project status process are assumed to be functions of the resource expenditure rate.

The terminal reward from the project is a non-decreasing function of the project status. Our purpose is to select optimal investment strategies under the discounted return criterion.

The value of the project is shown to be a solution of a second order, non-linear differential equation. Finally, we derive the optimal investment strategies for an R & D project in which the project status changes according to a non-homogeneous compound Poisson process by using diffusion approximation.

Necessary and sufficient conditions are presented under which the point processes equivalent to order statistics of n i.i.d. random variables or superpositions of n i.i.d. renewal processes converge to a non-degenerate limiting process as n approaches infinity. The limiting process must be one of three types of non-homogeneous Poisson process, one of which is the Weibull process. These point processes occur as failure-time models in the reliability theory of repairable systems.