The standard uniformization technique for continuous-time Markov chains is generalized to non-exponential stochastic networks.

The functional differential equation y′(x) = ay(λx) + by(x) arises in many different situations. The purpose of this note is to show how it arises in some multitype branching process cell population models. We also show how its solution can be given an intuitive interpretation as the probability density function of an infinite sum of independent but not identically distributed random variables.

In this paper we study the moment generating function order and the new better than used in the moment generating function order (NBUMG) life distributions. A closure property of this order under an independent random sum is deduced, and stochastic comparisons among the block replacement policy, the age replacement policy, the complete repair policy, and the minimal repair policy of an NBUMG component are investigated.

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

We study a model arising in chemistry where n elements numbered 1, 2, …, n are randomly permuted and if i is immediately to the left of i + 1 then they become stuck together to form a cluster. The resulting clusters are then numbered and considered as elements, and this process keeps repeating until only a single cluster is remaining. In this article we study properties of the distribution of the number of permutations required.

This article describes new estimates for the second largest eigenvalue in absolute value of reversible and ergodic Markov chains on finite state spaces. These estimates apply when the stationary distribution assigns a probability higher than 0.702 to some given state of the chain. Geometric tools are used. The bounds mainly involve the isoperimetric constant of the chain, and hence generalize famous results obtained for the second eigenvalue. Comparison estimates are also established, using the isoperimetric constant of a reference chain. These results apply to the Metropolis-Hastings algorithm in order to solve minimization problems, when the probability of obtaining the solution from the algorithm can be chosen beforehand. For these dynamics, robust bounds are obtained at moderate levels of concentration.

For a spectrally negative self-similar Markov process on $[0,\infty)$ with an a.s. finite overall supremum, we provide, in tractable detail, a kind of conditional Wiener–Hopf factorization at the maximum of the absorption time at zero, the conditioning being on the overall supremum and the jump at the overall supremum. In a companion result the Laplace transform of this absorption time (on the event that the process does not go above a given level) is identified under no other assumptions (such as the process admitting a recurrent extension and/or hitting zero continuously), generalizing some existing results in the literature.

Consider a monotone system with independent alternating renewal processes as component processes, and assume the component uptimes are exponentially distributed. In this paper we study the asymptotic properties of the distribution of the rth downtime of the system, as the failure rates of the components converge to zero. We show that this distribution converges, and the limiting function has a simple form. Thus we have established an easy computable approximation formula for the downtime distribution of the system for highly available systems. We also show that the steady state downtime distribution, i.e. the downtime distribution of a system failure occurring after an infinite run-in period, converges to the same limiting function as the failure rates converge to zero.

Computer analysis of biological sequences often detects deviations from a random model. In the usual model, sequence letters are chosen independently, according to some fixed distribution over the relevant alphabet. Real biological sequences often contain simple repeats, however, which can be broadly characterized as multiple contiguous copies (usually inexact) of a specific word. This paper quantifies inexact simple repeats as local sums in a Markov additive process (MAP). The maximum of the local sums has an asymptotic distribution with two parameters (λ and k), which are given by general MAP formulas. The general MAP formulas are usually computationally intractable, but an essential simplification in the case of repeats permits λ and k to be computed from matrices whose dimension equals the size of the relevant alphabet. The simplification applies to some MAPs where the summand distributions do not depend on consecutive pairs of Markov states as usual, but on pairs with a fixed time-lag larger than one.

Necessary and sufficient conditions are presented under which the point processes equivalent to order statistics of n i.i.d. random variables or superpositions of n i.i.d. renewal processes converge to a non-degenerate limiting process as n approaches infinity. The limiting process must be one of three types of non-homogeneous Poisson process, one of which is the Weibull process. These point processes occur as failure-time models in the reliability theory of repairable systems.

We improve on previous finite time estimates for the simulated annealing algorithm which were obtained from a Cheeger-like approach. Our approach is based on a Poincaré inequality.

For Brownian motion on a convex polyhedral subset of a sphere or torus, the rate of convergence in distribution to uniformity is studied. The main result is a method to take a Markov coupling on the full sphere or torus and create a faster coupling on the convex polyhedral subset. Upper bounds on variation distance are computed, and applications are discussed.

Let r and d be positive integers with r<d. Consider a random d-ary tree constructed as follows. Start with a single vertex, and in each time-step choose a uniformly random leaf and give it d newly created offspring. Let 𝒯d,t be the tree produced after t steps. We show that there exists a fixed δ<1 depending on d and r such that almost surely for all large t, every r-ary subtree of 𝒯d,t has less than t δ vertices. The proof involves analysis that also yields a related result. Consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. In this way, one face is destroyed and three new faces are created. After t steps, we obtain a random triangulated plane graph with t+3 vertices, which is called a random Apollonian network. We prove that there exists a fixed δ<1, such that eventually every path in this graph has length less than t 𝛿, which verifies a conjecture of Cooper and Frieze (2015).

A famous result in renewal theory is the central limit theorem for renewal processes. Since, in applications, usually only observations from a finite time interval are available, a bound on the Kolmogorov distance to the normal distribution is desirable. We provide an explicit non-uniform bound for the renewal central limit theorem based on Stein’s method and track the explicit values of the constants. For this bound the inter-arrival time distribution is required to have only a second moment. As an intermediate result of independent interest we obtain explicit bounds in a non-central Berry–Esseen theorem under second moment conditions.

It is shown that if ϕ is a given function out of a large class satisfying a certain regularity condition, then a supercritical age-dependent branching process {Z(t)} exists with deterministic immigration and given life-length and family-size distributions such that Z(t)/(eat ϕ(t)) converges in probability to a non-zero constant, a being the appropriate Malthusian parameter.

As an easy corollary one discovers the asymptotic behaviour of some processes with random immigration.

The sequential stochastic assignment problem (SSAP) allocates distinct workers with deterministic values to sequentially arriving tasks with stochastic parameters to maximize the expected total reward. In this paper we study an extension of the SSAP, in which the worker values are considered to be random variables, taking on new values upon each task arrival. Several SSAP models with different assumptions on the distribution of the worker values and closed-form expressions for optimal assignment policies are presented.

The accessibility percolation model is investigated on random rooted labeled trees. More precisely, the number of accessible leaves (i.e. increasing paths) Zn and the number of accessible vertices Cn in a random rooted labeled tree of size n are jointly considered in this work. As n → ∞, we prove that (Zn, Cn) converges in distribution to a random vector whose probability generating function is given in an explicit form. In particular, we obtain that the asymptotic distributions of Zn + 1 and Cn are geometric distributions with parameters e/(1 + e) and 1/e, respectively. Much of our analysis is performed in the context of local weak convergence of random rooted labeled trees.

We are concerned with the variation of the supercritical nearest-neighbours contact process such that first infection occurs at a lower rate; it is known that the process survives with positive probability. Regarding the rightmost infected of the process started from one site infected and conditioned to survive, we specify a sequence of space-time points at which its behaviour regenerates and, thus, obtain the corresponding strong law and central limit theorem. We also extend complete convergence in this case.

The purpose of this paper is to analyse the real-time trading of electricity. We address a model for an auction-like trading which captures key features of real-world electricity markets. Our main result establishes that, under certain conditions, the expected total payment for electricity is independent of the particular auction type. This result is analogous to the revenue-equivalence theorem known for classical auctions and could contribute to an improved understanding of different electricity market designs and their comparison.

We consider the Δ(i)/G/1 queue, in which a total of n customers join a single-server queue for service. Customers join the queue independently after exponential times. We consider heavy-tailed service-time distributions with tails decaying as x-α, α ∈ (1, 2). We consider the asymptotic regime in which the population size grows to ∞ and establish that the scaled queue-length process converges to an α-stable process with a negative quadratic drift. We leverage this asymptotic result to characterize the head start that is needed to create a long period of uninterrupted activity (a busy period). The heavy-tailed service times should be contrasted with the case of light-tailed service times, for which a similar scaling limit arises (Bet et al. (2015)), but then with a Brownian motion instead of an α-stable process.