Tail asymptotics of the solution R to a fixed-point problem of the type R=DQ+∑1NRm are derived under heavy-tailed conditions allowing both dependence between Q and N and the tails to be of the same order of magnitude. Similar results are derived for a K-class version with applications to multi-type branching processes and busy periods in multi-class queues.

We study a variant of Gilbert's disc model, in which discs are positioned at the points of a Poisson process in ℝ2 with radii determined by an underlying stationary and ergodic random field φ:ℝ2→[0,∞), independent of the Poisson process. This setting, in which the random field is independent of the point process, is often referred to as geostatistical marking. We examine how typical properties of interest in stochastic geometry and percolation theory, such as coverage probabilities and the existence of long-range connections, differ between Gilbert's model with radii given by some random field and Gilbert's model with radii assigned independently, but with the same marginal distribution. Among our main observations we find that complete coverage of ℝ2 does not necessarily happen simultaneously, and that the spatial dependence induced by the random field may both increase as well as decrease the critical threshold for percolation.

We consider a Markov-modulated fluid flow production model under the D-policy, that is, as soon as the storage reaches level 0, the machine becomes idle until the total storage exceeds a predetermined threshold D. Thus, the production process alternates between a busy and an idle machine. During the busy period, the storage decreases linearly due to continuous production and increases due to supply; during the idle period, no production is rendered by the machine and the storage level increases by only supply arrivals. We consider two types of model with different supply process patterns: continuous inflows with linear rates (fluid type), and batch inflows, where the supplies arrive according to a Markov additive process (MAP) and their sizes are independent and have phase-type distributions depending on the type of arrival (MAP type). Four types of cost are considered: a setup cost, a production cost, a penalty cost for an idle machine, and a storage cost. Using tools from multidimensional martingale and hitting time theory, we derive explicit formulae for these cost functionals in the discounted case. Numerical examples, a sensitivity analysis, and insights are provided.

In this paper we develop renewal theorems for point processes with interarrival times ξ(Xn+1Xn…), where (Xn)n∈ℤ is a stochastic process with finite state space Σ and ξ:ΣA→ℝ is a Hölder continuous function on a subset ΣA⊂Σℕ. The theorems developed here unify and generalise the key renewal theorem for discrete measures and Lalley's renewal theorem for counting measures in symbolic dynamics. Moreover, they capture aspects of Markov renewal theory. The new renewal theorems allow for direct applications to problems in fractal and hyperbolic geometry, for instance to the problem of Minkowski measurability of self-conformal sets.

Vibration energy harvesting aims to harness the energy of ambient random vibrations for power generation, particularly in small-scale devices. Typically, stochastic excitation driving the harvester is modelled as a Brownian process and the dynamics are studied in the equilibrium state. However, non-Brownian excitations are of interest, particularly in the nonequilibrium regime of the dynamics. In this work we study the nonequilibrium dynamics of a generic piezoelectric harvester driven by Brownian as well as (non-Brownian) Lévy flight excitation, both in the linear and the Duffing regimes. Both the monostable and the bistable cases of the Duffing regime are studied. The first set of results demonstrate that Lévy flight excitation results in higher expectation values of harvested power. In particular, it is shown that increasing the noise intensity leads to a significant increase in power output. It is also shown that a linearly coupled array of nonlinear harvesters yields improved power output for tailored values of coupling coefficients. The second set of results show that Lévy flight excitation fundamentally alters the bifurcation characteristics of the dynamics. Together, the results underscore the importance of non-Brownian excitation characterised by Lévy flight in vibration energy harvesting, both from a theoretical viewpoint and from the perspective of practical applications.

We study a class of growing systems of random walks on regular trees, known as frog models with geometric lifetime in the literature. With the help of results from renewal theory, we derive new bounds for their critical parameters. Our approach also improves the existing bounds for the critical parameter of a percolation model on trees known as cone percolation.

The Holme‒Kim random graph process is a variant of the Barabási‒Álbert scale-free graph that was designed to exhibit clustering. In this paper we show that whether the model does indeed exhibit clustering depends on how we define the clustering coefficient. In fact, we find that the local clustering coefficient typically remains positive whereas global clustering tends to 0 at a slow rate. These and other results are proven via martingale techniques, such as Freedman's concentration inequality combined with a bootstrapping argument.

We consider a simple model for the evolution of a limit order book in which limit orders of unit size arrive according to independent Poisson processes. The frequencies of buy limit orders below a given price level, respectively sell limit orders above a given level, are described by fixed demand and supply functions. Buy (respectively, sell) limit orders that arrive above (respectively, below) the current ask (respectively, bid) price are converted into market orders. There is no cancellation of limit orders. This model has been independently reinvented by several authors, including Stigler (1964), and Luckock (2003), who calculated the equilibrium distribution of the bid and ask prices. We extend the model by introducing market makers that simultaneously place both a buy and sell limit order at the current bid and ask price. We show that introducing market makers reduces the spread, which in the original model was unrealistically large. In particular, we calculate the exact rate at which market makers need to place orders in order to close the spread completely. If this rate is exceeded, we show that the price settles at a random level that, in general, does not correspond to the Walrasian equilibrium price.

We investigate the asymptotic properties of the transient queue length process Q(t)=max(Q(0)+X(t)-ct, sup0≤s≤t(X(t)-X(s)-c(t-s))), t≥0, in the Gaussian fluid queueing model, where the input process X is modeled by a centered Gaussian process with stationary increments and c>0 is the output rate. More specifically, under some mild conditions on X and Q(0)=x≥0, we derive the exact asymptotics of πx,Tu(u)=ℙ(Q(Tu)>u) as u→∞. The interplay between u and Tu leads to two qualitatively different regimes: short-time horizon when Tu is relatively small with respect to u, and moderate- or long-time horizon when Tu is asymptotically much larger than u. As a by-product, we discuss the implications for the speed of convergence to stationarity of the model studied.

One model of real-life spreading processes is that of first-passage percolation (also called the SI model) on random graphs. Social interactions often follow bursty patterns, which are usually modelled with independent and identically distributed heavy-tailed passage times on edges. On the other hand, random graphs are often locally tree-like, and spreading on trees with leaves might be very slow due to bottleneck edges with huge passage times. Here we consider the SI model with passage times following a power-law distribution ℙ(ξ>t)∼t-α with infinite mean. For any finite connected graph G with a root s, we find the largest number of vertices κ(G,s) that are infected in finite expected time, and prove that for every k≤κ(G,s), the expected time to infect k vertices is at most O(k1/α). Then we show that adding a single edge from s to a random vertex in a random tree 𝒯 typically increases κ(𝒯,s) from a bounded variable to a fraction of the size of 𝒯, thus severely accelerating the process. We examine this acceleration effect on some natural models of random graphs: critical Galton--Watson trees conditioned to be large, uniform spanning trees of the complete graph, and on the largest cluster of near-critical Erdős‒Rényi graphs. In particular, at the upper end of the critical window, the process is already much faster than exactly at criticality.

We consider a polling system with two queues, exhaustive service, no switchover times, and exponential service times with rate µ in each queue. The waiting cost depends on the position of the queue relative to the server: it costs a customer c per time unit to wait in the busy queue (where the server is) and d per time unit in the idle queue (where there is no server). Customers arrive according to a Poisson process with rate λ. We study the control problem of how arrivals should be routed to the two queues in order to minimize the expected waiting costs and characterize individually and socially optimal routeing policies under three scenarios of available information at decision epochs: no, partial, and complete information. In the complete information case, we develop a new iterative algorithm to determine individually optimal policies (which are symmetric Nash equilibria), and show that such policies can be described by a switching curve. We use Markov decision processes to compute the socially optimal policies. We observe numerically that the socially optimal policy is well approximated by a linear switching curve. We prove that the control policy described by this linear switching curve is indeed optimal for the fluid version of the two-queue polling system.

In this paper we are concerned with the reliability properties of two coherent systems having shared components. We assume that the components of the systems are two overlapping subsets of a set of n components with lifetimes X1,...,Xn. Further, we assume that the components of the systems fail according to the model of sequential order statistics (which is equivalent, under some mild conditions, to the failure model corresponding to a nonhomogeneous pure-birth process). The joint reliability function of the system lifetimes is expressed as a mixture of the joint reliability functions of the sequential order statistics, where the mixing probabilities are the bivariate signature matrix associated to the structures of systems. We investigate some stochastic orderings and dependency properties of the system lifetimes. We also study conditions under which the joint reliability function of systems with shared components of order m can be equivalently written as the joint reliability function of systems of order n (n>m). In order to illustrate the results, we provide several examples.

In this paper we study the number of customers in infinite-server queues with a self-exciting (Hawkes) arrival process. Initially we assume that service requirements are exponentially distributed and that the Hawkes arrival process is of a Markovian nature. We obtain a system of differential equations that characterizes the joint distribution of the arrival intensity and the number of customers. Moreover, we provide a recursive procedure that explicitly identifies (transient and stationary) moments. Subsequently, we allow for non-Markovian Hawkes arrival processes and nonexponential service times. By viewing the Hawkes process as a branching process, we find that the probability generating function of the number of customers in the system can be expressed in terms of the solution of a fixed-point equation. We also include various asymptotic results: we derive the tail of the distribution of the number of customers for the case that the intensity jumps of the Hawkes process are heavy tailed, and we consider a heavy-traffic regime. We conclude by discussing how our results can be used computationally and by verifying the numerical results via simulations.

In this paper we are concerned with a model in econophysics, the subfield of statistical physics that applies concepts from traditional physics to economics. Our model is an example of an interacting particle system with disorder, meaning that some of the transition rates are not identical but rather drawn from a fixed distribution. Economical agents are represented by the vertices of a connected graph and are characterized by the number of coins they possess. Agents independently spend one coin at rate one for their basic need, earn one coin at a rate chosen independently from a distribution ϕ, and exchange money at rate µ with one of their nearest neighbors, with the richest neighbor giving one coin to the other neighbor. If an agent needs to spend one coin when his/her fortune is at 0, he/she dies, i.e. the corresponding vertex is removed from the graph. Our first results focus on the two extreme cases of lack of cooperation µ=0 and perfect cooperation µ = ∞ for finite connected graphs. These results suggest that, when overall the agents earn more than they spend, cooperation is beneficial for the survival of the population, whereas when overall the agents earn less than they spend, cooperation becomes detrimental. We also study the infinite one-dimensional system. In this case, when the agents earn less than they spend on average, the density of agents that die eventually is bounded from below by a positive constant that does not depend on the initial number of coins per agent or the level of cooperation.

The purpose of this paper is to establish mixing rates for infinite measure preserving almost Anosov diffeomorphisms on the two-dimensional torus. The main task is to establish regular variation of the tails of the first return time to the complement of a neighbourhood of the neutral fixed point.

We develop shock model theory in different scenarios for the ℳ-class of life distributions introduced by Klar and Müller (2003). We also study the cumulative damage model of A-Hameed and Proschan (1975) in the context of ℳ-class and establish analogous results. We obtain moment bounds and explore weak convergence issues within the ℳ-class of life distributions.

For many practical situations in reliability engineering, components in the system are usually dependent since they generally work in a collaborative environment. In this paper we build sufficient conditions for comparing two coherent systems under different random environments in the sense of the usual stochastic, hazard rate, reversed hazard rate, and likelihood ratio orders. Applications and numerical examples are provided to illustrate all the theoretical results established here.

In this paper we derive nonasymptotic upper bounds for the size of reachable sets in random graphs. These bounds are subject to a phase transition phenomenon triggered by the spectral radius of the hazard matrix, a reweighted version of the adjacency matrix. Such bounds are valid for a large class of random graphs, called local positive correlation (LPC) random graphs, displaying local positive correlation. In particular, in our main result we state that the size of reachable sets in the subcritical regime for LPC random graphs is at most of order O(√n), where n is the size of the network, and of order O(n2/3) in the critical regime, where the epidemic thresholds are driven by the size of the spectral radius of the hazard matrix with respect to 1. As a corollary, we also show that such bounds hold for the size of the giant component in inhomogeneous percolation, the SIR model in epidemiology, as well as for the long-term influence of a node in the independent cascade model.

In this paper we prove a quenched functional central limit theorem for a biased random walk on a supercritical Galton–Watson tree with leaves. This extends a result of Peres and Zeitouni (2008) where the case without leaves was considered. A conjecture of Ben Arous and Fribergh (2016) suggests an upper bound on the bias which we observe to be sharp.

Consider the process which starts with N ≥ 3 distinct points on ℝd, and fix a positive integer K < N. Of the total N points keep those N - K which minimize the energy amongst all the possible subsets of size N - K, and then replace the removed points by K independent and identically distributed points sampled according to some fixed distribution ζ. Repeat this process ad infinitum. We obtain various quite nonrestrictive conditions under which the set of points converges to a certain limit. This is a very substantial generalization of the `Keynesian beauty contest process' introduced in Grinfeld et al. (2015), where K = 1 and the distribution ζ was uniform on the unit cube.