The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ ℝd and a gauge body B ⊂ ℝd, such a generalized contact distribution is the conditional distribution of the random vector (dB(L,Z),uB(L,Z),pB(L,Z),lB(L,Z)) given that Z∩L = ∅, where Z is a Boolean model, dB(L,Z) is the distance of L from Z with respect to B, pB(L,Z) is the boundary point in L realizing this distance (if it exists uniquely), uB(L,Z) is the corresponding boundary point of B (if it exists uniquely) and lB(L,·) may be taken from a large class of locally defined functionals. In particular, we pursue the question of the extent to which the spatial density and the grain distribution underlying an inhomogeneous Boolean model Z are determined by the generalized contact distributions of Z.

A (generalized) stochastic fluid system Q is defined as the one-dimensional Skorokhod reflection of a finite variation process X (with possibly discontinuous paths). We write X as the (not necessarily minimal) difference of two positive measures, A, B, and prove an alternative ‘integral representation’ for Q. This representation forms the basis for deriving a ‘Little's law’ for an appropriately constructed stationary version of Q. For the special case where B is the Lebesgue measure, a distributional version of Little's law is derived. This is done both at the arrival and departure points of the system. The latter result necessitates the consideration of a ‘dual process’ to Q. Examples of models for X, including finite variation Lévy processes with countably many jumps on finite intervals, are given in order to illustrate the ideas and point out potential applications in performance evaluation.

Families of Poisson processes defined on general state spaces and with the intensity measure scaled by a positive parameter are investigated. In particular, mean value relations with respect to the scale parameter are established and used to derive various Gamma-type results for certain geometric characteristics determined by finite subprocesses. In particular, we deduce Miles' complementary theorem. Applications of the results within stochastic geometry and particularly for random tessellations are discussed.

A generalized semi-Markov process with reallocation (RGSMP) was introduced to accommodate a large class of stochastic processes which cannot be analyzed by the well-known model of an ordinary generalized semi-Markov process (GSMP). For stationary RGSMP whose initial distribution has a product form, we show that, for a randomly chosen clock of a fixed insensitive type, if the lifetime of this clock is changed to infinity, then the background process is stationary under a certain time change. This implies that the expected time required for the tagged clock to consume a given amount x of resource, called the attained sojourn time, is a linear function of x. Such stationarity and linearity results are known for two special RGSMPs: ordinary GSMP and Kelly's symmetric queue. Our results not only extend them to a general RGSMP but also give more detailed formulas, which allow us to calculate for instance the expected attained sojourn time while the background process is in a given state. Furthermore, we remark that analogous results hold for GSMP with point-process input, in which the lifetimes of clocks of a fixed type form an arbitrary stationary sequence (of not necessarily independent random variables).

This article reviews results related to event and time averages (EATA) for point process models, including PASTA, ASTA and ANTIPASTA under general hypotheses. In particular, the results for the stationary case relating the Palm and martingale approach are reviewed. The non-stationary case is discussed in the martingale framework where minimal conditions for ASTA generalizing earlier work are presented in a unified framework for the discrete- and continuous-time cases. In addition, necessary and sufficient conditions for ASTA to hold in the stationary case are discussed in the case even when stochastic intensities may not exist and a short proof of the ANTIPASTA results known to date are given.

We introduce multivariate partial orderings related with the Palm and time-stationary probabilities of a point process. Using these orderings, we give conditions for the monotonicity of a random sequence, with respect to some integral stochastic ordering, to be inherited with a continuous time process in which this sequence is imbedded. This type of inheritance is also discussed for the property of association.

A sequence Xn, 1 of independent and identically distributed random vectors is considered. Under a condition of regular variation, the number of vertices of the convex hull of {X1, …, Xn} converges in distribution to the number of vertices of the convex hull of a certain Poisson point process. In this paper, it is proved without sharpening the conditions that the expectation of this number also converges; expressions are found for its limit, generalizing results of Davis et al. (1987). We also present some results concerning other quantities of interest, such as area and perimeter of the convex hull and the probability that a given point belongs to the convex hull.

For a continuous time Markov chain the time points of transitions, belonging to a subset of the set of all transitions, are observed. Special cases include the point process generated by all transitions and doubly stochastic Poisson processes with a Markovian intensity. Equations are derived for the conditional distribution of the state of the Markov chain, given observations of the point process. This distribution may be used for prediction. For the forward recurrence time of the point process, distributions corresponding to synchronous and asynchronous sampling are also derived. The Palm distribution for the point process is specified in terms of the corresponding initial distribution for the Markov chain. In examples the point processes of arrivals and departures in a queueing system are studied. Two biological applications deal with estimation of population size and detection of epidemics.