We study propagation of avalanches in a certain excitable network. The model is a particular case of the one introduced by Larremore et al. (Phys. Rev. E, 2012) and is mathematically equivalent to an endemic variation of the Reed–Frost epidemic model introduced by Longini (Math. Biosci., 1980). Two types of heuristic approximation are frequently used for models of this type in applications: a branching process for avalanches of a small size at the beginning of the process and a deterministic dynamical system once the avalanche spreads to a significant fraction of a large network. In this paper we prove several results concerning the exact relation between the avalanche model and these limits, including rates of convergence and rigorous bounds for common characteristics of the model.

We consider a special version of random sequential adsorption (RSA) with nearest-neighbor interaction on infinite tree graphs. In classical RSA, starting with a graph with initially inactive nodes, each of the nodes of the graph is inspected in a random order and is irreversibly activated if none of its nearest neighbors are active yet. We generalize this nearest-neighbor blocking effect to a degree-dependent threshold-based blocking effect. That is, each node of the graph is assumed to have its own degree-dependent threshold and if, upon inspection of a node, the number of active direct neighbors is less than that node's threshold, the node will become irreversibly active. We analyze the activation probability of nodes on an infinite tree graph, given the degree distribution of the tree and the degree-dependent thresholds. We also show how to calculate the correlation between the activity of nodes as a function of their distance. Finally, we propose an algorithm which can be used to solve the inverse problem of determining how to set the degree-dependent thresholds in infinite tree graphs in order to reach some desired activation probabilities.

We analyze the long-term stability of a stochastic model designed to illustrate the adaptation of a population to variation in its environment. A piecewise deterministic process modeling adaptation is coupled to a Feller logistic diffusion modeling population size. As the individual features in the population become further away from the optimal ones, the growth rate declines, making population extinction more likely. Assuming that the environment changes deterministically and steadily in a constant direction, we obtain the existence and uniqueness of the quasi-stationary distribution, the associated survival capacity, and the Q-process. Our approach also provides several exponential convergence results (in total variation for the measures). From this synthetic information, we can characterize the efficiency of internal adaptation (i.e. population turnover from mutant invasions). When the latter is lacking, there is still stability, but because of the high level of population extinction. Therefore, any characterization of internal adaptation should be based on specific features of this quasi-ergodic regime rather than the mere existence of the regime itself.

The G/G/1 queueing system is studied by means of the regeneration point method, exploiting the concept of busy cycles. Recurrence relations are set-up for the distribution of the queue length at the arrival epochs. The same method is used to obtain the correlation structure of arrivals and departures for the G/M/1 queue.

We showed in [2] that if an object of initial size x (x large) is subjected to a succession of random partitions, then the object is decomposed into a large number of terminal cells, each of relatively small size, where if Z(x, B) denotes the number of such cells whose sizes are points in the set B, then there exists c, (0 < ≦ 1), such that Z(x, B)x−c converges in probability, as x → ∞, to a random variable W. We show here that if a parent object of size x produces k offspring of sizes y1, y2, ···, yk and if for each k x - y1 - y2 - ··· - yk (the ‘waste’ or the ‘cover’, depending on the point of view) is relatively small, then for each n the nth cumulant, Ψn (x, B), of Z(x, B) satisfies Ψn (x, B)x-c → κn (B), as x → ∞, for some κn(B). Thus, writing N = xc, Z(x, B) has approximately the same distribution as the sum of N independent and identically distributed random variables (The determination of the distribution of the individual appears to be a difficult problem.) The theory also applies when an object of moderate size is broken down into very fine particles or granules.

Let X1, X2, …, Xn, … be mutually independent with common CDF F and, for each m, n, let Xm:n be the mth largest among the first n. We consider max1≤n<∞ (X1:n – cn) and the ‘optimal stopping rule' N which maximizes where all l and In particular, we consider and All of these are considered for c ϵ (0,∞) as well as asymptotically as c → 0+.

We consider a birth and growth model where points (‘seeds’) arrive on a line randomly in time and space and proceed to ‘cover’ the line by growing at a uniform rate in both directions until an opposing branch is met; points which arrive on covered parts of the line do not contribute to the process. Existing results concerning the number of seeds assume that points arrive according to a Poisson process, homogeneous on the line, but possibly inhomogeneous in time. We derive results under less stringent assumptions, namely that the arrival process be a stationary simple point process.

In the series system (competing risks) set-up the observed data are generally accepted as the lifetime (T) and the identifier (δ) of the component causing the failure of the system. Peterson (1976) has provided bounds for the joint survival function of the component lifetimes in terms of the joint distribution of (T, δ). In the case of more complex coherent systems, there are various schemes of observation in the literature. In this paper we provide bounds for the joint and marginal survival functions of the component lifetimes in terms of the joint distribution of the data as obtained under existing and new schemes of observation. We also tackle the reverse problem of obtaining bounds for the joint distributions of the data for given marginal distributions of the component lifetimes and the distribution of the system lifetimes.