We analyze the notion of ‘reliability prediction’ by studying in detail a key property that is tacitly assumed to make reliability prediction possible. The analysis leads in turn to a special type of point process for which the connection of future to past can be explicitly displayed. In this type of process, the semi-renewal process, all finite-dimensional distributions are completely determined by the distribution of the time to the first event in the process. The theory provides a heretofore unappreciated unification of the two most commonly used reliability prediction models for maintained systems, namely, the renewal and revival processes. We show that familiar results from renewal theory extend and generalize to semi-renewal processes.

We derive an expansion for the (expected) difference between the continuously monitored supremum and evenly monitored discrete maximum over a finite time horizon of a jump diffusion process with independent and identically distributed normal jump sizes. The monitoring error is of the form a 0 / N 1/2 + a 1 / N 3/2 + · · · + b 1 / N + b 2 / N 2 + b 4 / N 4 + · · ·, where N is the number of monitoring intervals. We obtain explicit expressions for the coefficients {a 0, a 1, …, b 1, b 2, …}. In particular, a 0 is proportional to the value of the Riemann zeta function at ½, a well-known fact that has been observed for Brownian motion in applied probability and mathematical finance.

In dealing with Markov chains, we are sometimes forced by circumstances to make observations on lumped states. We may wish, however, to deal with a random variable which is a function of the numbers in the various (unlumped) states at various times and whose moments, therefore, will involve the probabilities of transition between the unlumped states.

This paper demonstrates that, although these probabilities are immeasurable directly, we can, provided that we restrict our attention to a certain well-defined class of functions as our random variables, reduce all moments and product-moments of such random variables to functions of lumped state probabilities.

A new and simpler proof of Morrison's result that within exponential mixtures only IFR gamma mixing produces linearly increasing mean residual life functions is given. A parallel and new characterization of the DFR gamma laws follows as a consequence. The method of proof used suggests a general result on the infinite divisibility of the mixing distributions in exponential mixtures.

We consider the generalized version in continuous time of the parking problem of Knuth introduced in Bansaye (2006). Files arrive following a Poisson point process and are stored on a hardware identified with the real line, at the right of their arrival point. Here we study the evolution of the endpoints of the data block straddling 0, which is empty at time 0 and is equal to R at a deterministic time.

Processes whose entities would be independent but whose behaviour is influenced by environmental variables, which they in turn affect, are processes with mediated interactions. The independent open migration process with which is associated a random variable, the excitation, is such a process. This self-excited (conditionally) independent open migration process, with finite excitation, is related to a derived process, its piecewise-deterministic analogue. The correspondence gives insight into the process, as well as an equilibrium independence result. A simple example illustrates the presence of paroxysmal phenomena.

Let I denote the fraction of time that an M/D/∞ queue is idle during [0, t]. We give a simple proof that the nth moment of I conditioned on k arrivals equals the kth moment of I conditioned on n arrivals.

A generalized occupancy distribution obtained by Uppuluri and Carpenter (1971) is derived by elementary methods, and extended to a multivariate situation.

The variation of the state vectors p(t) = (pi(t)) of a continuous-time homogeneous Markov system with fixed size is examined. A specific time t0 after which the size order of the elements pi(t) becomes stable provides a criterion of the system's convergence rate. A method is developed to find t0 and a quickly evaluated lower bound for t0. This method is based on the geometric characteristics and the volumes of the attainable structures. Moreover, a condition concerning the selection of starting vectors p(0) is given so that the vector functions p(t) retain the same size order for every time greater than a given time t.

We present a brief summary of some results related to deriving orthogonal representations of second-order random fields and its application in solving linear prediction problems. In the homogeneous and/or isotropic case, the spectral theory provides an orthogonal expansion in terms of spherical harmonics, called spectral decomposition (Yadrenko 1983). A prediction formula based on this orthogonal representation is shown. Finally, an application of this formula in solving a real-data problem related to prospective geophysics techniques is presented.

In this paper, bidimensional stochastic processes defined by ax(t) = y(t)dt and dy(t) = m(y)dt + [2v(y)]1/2dW(t), where W(t) is a standard Brownian motion, are considered. In the first section, results are obtained that allow us to characterize the moment-generating function of first-passage times for processes of this type. In Sections 2 and 5, functions are computed, first by fixing the values of the infinitesimal parameters m(y) and v(y) then by the boundary of the stopping region.

We calculate the mean throughput, number of collisions, successes, and idle slots for random tree algorithms with successive interference cancellation. Except for the case of the throughput for the binary tree, all the results are new. We furthermore disprove the claim that only the binary tree maximizes throughput. Our method works with many observables and can be used as a blueprint for further analysis.

The critical paths of a max-plus linear system with noise are random variables. In this paper we introduce the edge criticalities which measure how often the critical paths traverse each edge in the precedence graph. We also present the parallel path approximation, a novel method for approximating these new statistics as well as the previously studied max-plus exponent. We show that, for low amplitude noise, the critical paths spend most of their time traversing the deterministic maximally weighted cycle and that, as the noise amplitude is increased, the critical paths become more random and their distribution over the edges in the precedence graph approaches a highly uniform measure of maximal entropy.

We prove that for a discrete determinantal process the BK inequality occurs for increasing events generated by simple points. We also give some elementary but nonetheless appealing relationships between a discrete determinantal process and the well-known CS decomposition.

The coupling method of Ball (1986) is extended to carrier-borne epidemics, thus providing a new proof of a result of Daniels (1972) concerning the total size distribution of Downton's stochastic carrier-borne epidemic. The generalization to multipopulation carrier-borne epidemics is immediate.

This paper is devoted to the study of regime-switching jump diffusion processes with countable regimes. It aims to establish Foster–Lyapunov-type criteria for exponential ergodicity of such processes. After recalling results concerning the petiteness of compact sets, this paper presents sufficient conditions for the existence of a Foster–Lyapunov function; this, in turn, helps to establish sufficient conditions for the desired exponential ergodicity for regime-switching jump diffusion processes. Finally, an application to feedback control problems is presented.

Define a random variable

We refer to Yn as a sustained maximum. Under the assumption that the Xi's are i.i.d. the exact distribution of Yn is obtained. Under certain conditions on the underlying distribution F we obtain weak limit results for the Yn as well. Also a combinatorial extreme-value problem is solved.