A model for the general repair of a system subject to stochastic failure is introduced and studied. Properties of the stochastic point process generated by a sequence of general repairs are investigated. An application of the model to logistics is described

Suppose is a sequence of independent, identically distributed random vectors in . Say is a record if there is a record simultaneously in both coordinates at index n. For , let be the number of records falling in region A. If A≷Bare two regions of such that every point of Ais south-west of every point of B, then and are independent. We call this property ordered independent scattering and discuss which other notions of record give rise to it

The product-form stationary distributions of open Markovian networks of queues are characterised in terms of eigenvalues and eigenvectors of appropriately defined rate matrices in an operator-analytic formulation. For a Jackson network, the conditions enforced by such a formulation are shown to be equivalent to the conditions imposed by the standard traffic equations. Generalisations to a wider class of product-form networks are also given.

We consider a generating matrix function H(z)of jumps of a random walk controlled by a finite-state Markov chain. Under the so-called “natural” conditions we prove that the total and partial indices of the matrix function F(z)(which is—H(z)with removed singularity) are zeroes, if the ergodicity condition holds and prove that this condition is necessary and sufficient for ergodicity. Based on the indices being zeroes, we prove that ifF(z)is meromorphic in then the number of roots of det F(z)there is equal to the number of its poles. We show on an example how the partial indices may shape up the complex analytic factorization components of F(z). The roots of on the unit circle are related to a special form of H(z).

The basic problem in diffraction tomography is the problem of reconstructing the index of refraction of a scattering object. A new algorithm for its solution in the special case where the index of refraction is assumed to be real and not less than unity is proposed. An estimate of this algorithm's rate of convergence is given. The limited-view problem has assumed a central place in recent investigations in diffraction tomography. The estimate of our algorithm's rate of convergence allows one to estimate the degree of uncertainty in the reconstruction of the index of refraction from incomplete data for special directions of wave propagation

A sample-path duality is shown between the stationary service time in progress for single-server queues with general stationary input operating under the preemptive last-in-first-out (LIFO) discipline, and the first strictly descending ladder height in risk processes with a general stationary input. In addition to giving new insight, the duality allows one to easily obtain some known probabalistic formulas for risk processes by calculating corresponding elementary queueing quantities. The duality thus “explains” why these risk formulas hold under such generality. For example, under time stationarity, the first descending ladder height distribution for the risk process is shown to be the same as the time-stationary remaining service time in a preemptive LIFO queue. The duality is very simple and based on basic queueing ideas and time reversal of input. An arrival-stationary analog is also given as is an extension covering all descending ladder heights jointly

We develop methods for the computation of some distribution functions or their Laplace transforms, associated with a time homogeneous diffusion with killing under the assumption that its generator and the killing-rate function are piece-wise constant. The main result is the representation of the Laplace-transform of the joint distribution of the killing time and of the state at killing. In particular, it is shown that the marginal density of the state at killing has an explicit representation. We also consider the computation of the analogous distributions for a countable time-homogeneous Markov chain with killing. This work was motivated by applications of Markov processes to model the evolution of markers; see Jewell and Kalbfleisch[3]

We consider a generalized birth death process that represents a multiserver (closed or open) overflow queueing system with n primary servers and c-n secondary servers. An arrival to the system joins a server of the primary group, if available, otherwise it overflows to the secondary group. If all servers are busy, arrivals are queued, provided the queue buffer is not full, and served as servers become free. Arrivals that find the queue buffer full are lost. Our overflow model differs from models in the open literature in that it combines state dependent arrival rates with group dependent service rates. We present a general formulation that allows a simple derivation of the joint stationary probabilities of i busy primary servers and j busy secondary severs. This main result easily lends itself to an efficient iterative algorithm to evaluate the joint probabilities. We apply our basic theorem to produce new results for several overflow models. A distinctive feature of our approach is that it uses transform-free analysis.

For the partial sum process of i.i.d. random variables we derive sequences of new upper and lower bounds for the probability that it will exit from a given compact interval to the left and for the expected first-exit time. The sequences are generated by the same iterative procedure starting from different initial values. They converge monotonically to the desired first-exit quantities at an exponential rate