We reduce the upper bound for the bond percolation threshold of the cubic lattice from 0.447 792 to 0.347 297. The bound is obtained by a growth process approach which views the open cluster of a bond percolation model as a dynamic process. A three-dimensional dynamic process on the cubic lattice is constructed and then projected onto a carefully chosen plane to obtain a two-dimensional dynamic process on a triangular lattice. We compare the bond percolation models on the cubic lattice and their projections, and demonstrate that the bond percolation threshold of the cubic lattice is no greater than that of the triangular lattice. Applying the approach to the body-centered cubic lattice yields an upper bound of 0.292 893 for its bond percolation threshold.

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.

Signatures are useful in analyzing and evaluating coherent systems. However, their computation is a challenging problem, especially for complex coherent structures. In most cases the reliability of a binary coherent system can be linked to a tail probability associated with a properly defined waiting time random variable in a sequence of binary trials. In this paper we present a method for computing the minimal signature of a binary coherent system. Our method is based on matrix-geometric distributions. First, a proper matrix-geometric random variable corresponding to the system structure is found. Second, its probability generating function is obtained. Finally, the companion representation for the distribution of matrix-geometric distribution is used to obtain a matrix-based expression for the minimal signature of the coherent system. The results are also extended to a system with two types of components.

In this paper we consider a new generalized finite mixture model formed by dependent and identically distributed (d.i.d.) components. We then establish results for the comparisons of lifetimes of two such generalized finite mixture models in two different cases: (i) when the two mixture models are formed from two random vectors $\textbf{X}$ and $\textbf{Y}$ but with the same weights, and (ii) when the two mixture models are formed with the same random vectors but with different weights. Because the lifetimes of k-out-of-n systems and coherent systems are special cases of the mixture model considered, we used the established results to compare the lifetimes of k-out-of-n systems and coherent systems with respect to the reversed hazard rate and hazard rate orderings.

A class of controlled branching processes with continuous time is introduced and some limiting distributions are obtained in the critical case. An extension of this class as regenerative controlled branching processes with continuous time is proposed and some asymptotic properties are considered.

Let $\{Y_{1},\ldots ,Y_{n}\}$ be a collection of interdependent nonnegative random variables, with $Y_{i}$ having an exponentiated location-scale model with location parameter $\mu _i$, scale parameter $\delta _i$ and shape (skewness) parameter $\beta _i$, for $i\in \mathbb {I}_{n}=\{1,\ldots ,n\}$. Furthermore, let $\{L_1^{*},\ldots ,L_n^{*}\}$ be a set of independent Bernoulli random variables, independently of $Y_{i}$'s, with $E(L_{i}^{*})=p_{i}^{*}$, for $i\in \mathbb {I}_{n}.$ Under this setup, the portfolio of risks is the collection $\{T_{1}^{*}=L_{1}^{*}Y_{1},\ldots ,T_{n}^{*}=L_{n}^{*}Y_{n}\}$, wherein $T_{i}^{*}=L_{i}^{*}Y_{i}$ represents the $i$th claim amount. This article then presents several sufficient conditions, under which the smallest claim amounts are compared in terms of the usual stochastic and hazard rate orders. The comparison results are obtained when the dependence structure among the claim severities are modeled by (i) an Archimedean survival copula and (ii) a general survival copula. Several examples are also presented to illustrate the established results.

Distinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton–Watson trees. For example, let $\mathcal{T}_1$ be the event that a Galton–Watson tree is infinite and let $\mathcal{T}_2$ be the event that it contains an infinite binary tree starting from its root. These events satisfy similar recursive properties: $\mathcal{T}_1$ holds if and only if $\mathcal{T}_1$ holds for at least one of the trees initiated by children of the root, and $\mathcal{T}_2$ holds if and only if $\mathcal{T}_2$ holds for at least two of these trees. The probability of $\mathcal{T}_1$ has a continuous phase transition, increasing from 0 when the mean of the child distribution increases above 1. On the other hand, the probability of $\mathcal{T}_2$ has a first-order phase transition, jumping discontinuously to a non-zero value at criticality. Given the recursive property satisfied by the event, we describe the critical child distributions where a continuous phase transition takes place. In many cases, we also characterise the event undergoing the phase transition.

In this paper we study a large system of N servers, each with capacity to process at most C simultaneous jobs; an incoming job is routed to a server if it has the lowest occupancy amongst d (out of N) randomly selected servers. A job that is routed to a server with no vacancy is assumed to be blocked and lost. Such randomized policies are referred to JSQ(d) (Join the Shortest Queue out of d) policies. Under the assumption that jobs arrive according to a Poisson process with rate $N\lambda^{(N)}$ where $\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}\,}$ , $\sigma\in\mathbb{R}_+$ and $\beta\in\mathbb{R}$ , we establish functional central limit theorems for the fluctuation process in both the transient and stationary regimes when service time distributions are exponential. In particular, we show that the limit is an Ornstein–Uhlenbeck process whose mean and variance depend on the mean field of the considered model. Using this, we obtain approximations to the blocking probabilities for large N, where we can precisely estimate the accuracy of first-order approximations.

Consider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir (2005) Ann. Prob.33, 2036 for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.

The performance and effectiveness of an age replacement policy can be assessed by its mean time to failure (MTTF) function. We develop shock model theory in different scenarios for classes of life distributions based on the MTTF function where the probabilities $\bar{P}_k$ of surviving the first k shocks are assumed to have discrete DMTTF, IMTTF and IDMTTF properties. The cumulative damage model of A-Hameed and Proschan [1] is studied in this context and analogous results are established. Weak convergence and moment convergence issues within the IDMTTF class of life distributions are explored. The preservation of the IDMTTF property under some basic reliability operations is also investigated. Finally we show that the intersection of IDMRL and IDMTTF classes contains the BFR family and establish results outlining the positions of various non-monotonic ageing classes in the hierarchy.

In open Kelly and Jackson networks, servers are assigned to individual stations, serving customers only where they are assigned. We investigate the performance of modified networks where servers cooperate. A server who would be idle at the assigned station will serve customers at another station, speeding up service there. We assume interchangeable servers: the service rate of a server at a station depends only on the station, not the server. This gives work conservation, which is used in various ways. We investigate three levels of server cooperation, from full cooperation, where all servers are busy when there is work to do anywhere in the network, to one-way cooperation, where a server assigned to one station may assist a server at another, but not the converse. We obtain the same stability conditions for each level and, in a series of examples, obtain substantial performance improvement with server cooperation, even when stations before modification are moderately loaded.

In this work, we study a new model for continuum line-of-sight percolation in a random environment driven by the Poisson–Voronoi tessellation in the d-dimensional Euclidean space. The edges (one-dimensional facets, or simply 1-facets) of this tessellation are the support of a Cox point process, while the vertices (zero-dimensional facets or simply 0-facets) are the support of a Bernoulli point process. Taking the superposition Z of these two processes, two points of Z are linked by an edge if and only if they are sufficiently close and located on the same edge (1-facet) of the supporting tessellation. We study the percolation of the random graph arising from this construction and prove that a 0–1 law, a subcritical phase, and a supercritical phase exist under general assumptions. Our proofs are based on a coarse-graining argument with some notion of stabilization and asymptotic essential connectedness to investigate continuum percolation for Cox point processes. We also give numerical estimates of the critical parameters of the model in the planar case, where our model is intended to represent telecommunications networks in a random environment with obstructive conditions for signal propagation.

We consider a stochastic matching model with a general compatibility graph, as introduced by Mairesse and Moyal (2016). We show that the natural necessary condition of stability of the system is also sufficient for the natural ‘first-come, first-matched’ matching policy. To do so, we derive the stationary distribution under a remarkable product form, by using an original dynamic reversibility property related to that of Adan, Bušić, Mairesse, and Weiss (2018) for the bipartite matching model.

We analyze average-based distributed algorithms relying on simple and pairwise random interactions among a large and unknown number of anonymous agents. This allows the characterization of global properties emerging from these local interactions. Agents start with an initial integer value, and at each interaction keep the average integer part of both values as their new value. The convergence occurs when, with high probability, all the agents possess the same value, which means that they all know a property of the global system. Using a well-chosen stochastic coupling, we improve upon existing results by providing explicit and tight bounds on the convergence time. We apply these general results to both the proportion problem and the system size problem.

We study the transient and limiting behavior of a queue with a Pólya arrival process. The Pólya process is interesting because it exhibits path-dependent behavior, e.g. it satisfies a non-ergodic law of large numbers: the average number of arrivals over time [0, t] converges almost surely to a nondegenerate limit as $t \rightarrow \infty$. We establish a heavy-traffic diffusion limit for the $\sum_{i=1}^{n} P_i/GI/1$ queue, with arrivals occurring exogenously according to the superposition of n independent and identically distributed Pólya point processes. That limit yields a tractable approximation for the transient queue-length distribution, because the limiting net input process is a Gaussian Markov process with stationary increments. We also provide insight into the long-run performance of queues with path-dependent arrival processes. We show how Little’s law can be stated in this context, and we provide conditions under which there is stability for a queue with a Pólya arrival process.

We study ergodic properties of a class of Markov-modulated general birth–death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken to be large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the ‘averaged’ fluid limit exists. Here, we establish convergence rates for the moments of the approximating diffusion process to those of the Markov-modulated birth–death process. This is accomplished by comparing the generator of the approximating diffusion and that of the joint Markov process. We also provide several examples which demonstrate how the theory can be applied.

In this paper, we introduce a family of processes with values on the nonnegative integers that describes the dynamics of populations where individuals are allowed to have different types of interactions. The types of interactions that we consider include pairwise interactions, such as competition, annihilation, and cooperation; and interactions among several individuals that can be viewed as catastrophes. We call such families of processes branching processes with interactions. Our aim is to study their long-term behaviour under a specific regime of the pairwise interaction parameters that we introduce as the subcritical cooperative regime. Under such a regime, we prove that a process in this class comes down from infinity and has a moment dual which turns out to be a jump-diffusion that can be thought as the evolution of the frequency of a trait or phenotype, and whose parameters have a classical interpretation in terms of population genetics. The moment dual is an important tool for characterizing the stationary distribution of branching processes with interactions whenever such a distribution exists; it is also an interesting object in its own right.

The peeling process, which describes a step-by-step exploration of a planar map, has been instrumental in addressing percolation problems on random infinite planar maps. Bond and face percolations on maps with faces of arbitrary degree are conveniently studied via so-called lazy-peeling explorations. During such explorations, distinct vertices on the exploration contour may, at latter stage, be identified, making the process less suited to the study of site percolation. To tackle this situation and to explicitly identify site-percolation thresholds, we come back to the alternative “simple” peeling exploration of Angel and uncover deep relations with the lazy-peeling process. Along the way, we define and study the random Boltzmann map of the half-plane with a simple boundary for an arbitrary critical weight sequence. Its construction is nontrivial especially in the “dense regime,” where the half-planar random Boltzmann map does not possess an infinite simple core.

Extended gamma processes have been seen as a flexible extension of standard gamma processes in the recent reliability literature, for the purpose of cumulative deterioration modeling. The probabilistic properties of the standard gamma process have been well explored since the 1970s, whereas those of its extension remain largely unexplored. In particular, stochastic comparisons between degradation levels modeled by standard gamma processes and ageing properties for the corresponding level-crossing times are now well understood. The aim of this paper is to explore similar properties for extended gamma processes and see which ones can be broadened to this new context. As a by-product, new stochastic comparisons for convolutions of gamma random variables are also obtained.

Let $\boldsymbol{f}$ be a square-integrable, zero-mean, random vector with observable realizations in a Hilbert space H, and let $\boldsymbol{g}$ be an associated square-integrable, zero-mean, random vector with realizations which are not observable in a Hilbert space K. We seek an optimal filter in the form of a closed linear operator X acting on the observable realizations of a proximate vector $\boldsymbol{f}_{\epsilon } \approx \boldsymbol{f}$ that provides the best estimate $\widehat{\boldsymbol{g}}_{\epsilon} = X \boldsymbol{f}_{\epsilon}$ of the vector $\boldsymbol{g}$ . We assume the required covariance operators are known. The results are illustrated with a typical example.