Consider the following service system.

Network dynamics with point-process-based interactions are of paramount modeling interest. Unfortunately, most relevant dynamics involve complex graphs of interactions for which an exact computational treatment is impossible. To circumvent this difficulty, the replica-mean-field approach focuses on randomly interacting replicas of the networks of interest. In the limit of an infinite number of replicas, these networks become analytically tractable under the so-called ‘Poisson hypothesis’. However, in most applications this hypothesis is only conjectured. In this paper we establish the Poisson hypothesis for a general class of discrete-time, point-process-based dynamics that we propose to call fragmentation-interaction-aggregation processes, and which are introduced here. These processes feature a network of nodes, each endowed with a state governing their random activation. Each activation triggers the fragmentation of the activated node state and the transmission of interaction signals to downstream nodes. In turn, the signals received by nodes are aggregated to their state. Our main contribution is a proof of the Poisson hypothesis for the replica-mean-field version of any network in this class. The proof is obtained by establishing the propagation of asymptotic independence for state variables in the limit of an infinite number of replicas. Discrete-time Galves–Löcherbach neural networks are used as a basic instance and illustration of our analysis.

We study a simple first-order nonnegative bilinear time-series model and give conditions under which the model is stationary. The probability density function of the stationary distribution (when it exists) is found. We also discuss the tail behaviour of the stationary distribution and calculate the probability density function by a numerical method. Simulation is used to check the calculation.

For most repairable systems, the number N(t) of failed components at time t appears to be a good quality parameter, so it is critical to study this random function. Here the components are assumed to be independent and both their lifetime and their repair time are exponentially distributed. Moreover, the system is considered new at time 0. Our aim is to compare the random variable N(t) with N(∞), especially in terms of total variation distance. This analysis is used to prove a cut-off phenomenon in the same way as Ycart (1999) but without the assumption of identical components.

In this paper we investigate the properties of stationary point processes motivated by the following traffic model. Suppose there is a dichotomy of slow and fast points (cars) on a road with limited overtaking. It is assumed that fast points are delayed behind (or are clustered at) a slow point in accordance with the principles of a GI/G/s queue, the order of service being irrelevant. Thus each slow point represents a service station, with the input into each station consisting of a fixed (but random) displacement of the output of the previous queueing station. It is found that tractable results for stationary point processes occur for the cases M/M/s (s = 1, 2, ···, ∞) and M/G/∞. In particular, it is found that for these cases the steady state point processes are compound Poisson and that for the M/M/1 case the successive headways form a two state Markov renewal process. In addition it is shown that the input, output, and queue size processes in a steady state M/G/∞ queue are independent at any fixed time; this is a result I have been unable to find in the literature.

We study the Bayesian disorder problem for a negative binomial process. The aim is to determine a stopping time which is as close as possible to the random and unknown moment at which a sequentially observed negative binomial process changes the value of its characterizing parameter p ∈ (0, 1). The solution to this problem is explicitly derived through the reduction of the original optimal stopping problem to an integro-differential free-boundary problem. A careful analysis of the free-boundary equation and of the probabilistic nature of the boundary point allows us to specify when the smooth fit principle holds and when it breaks down in favour of the continuous fit principle.

An Ornstein-Uhlenbeck process subject to a quadratic killing rate is analyzed. The distribution for the process killing time is derived, generalizing the analogous result for Brownian motion. The derivation involves the use of Hermite polynomials in a spectral expansion.

In this article, we generalize results by Dimitrov and Khalil (1990), Khalil et al. (1991), and van Harn and Steutel (1991) and obtain some characterizations of the exponential and geometric laws.

A gambler, with an initial fortune less than 1, wants to buy a house which sells today for 1. Due to inflation, the price of the house tomorrow will be 1 + α, where α is a nonnegative constant, and will continue to go up at this rate, becoming (1 + α) n on the nth day. Once each day, he can stake any amount of fortune in his possession, but no more than he possesses, on a primitive casino. It is well known that, in a subfair primitive casino without the presence of inflation, the gambler should play boldly. The presence of inflation would motivate the gambler to recognize the time value of his fortune and to try to reach his goal as quickly as possible; intuitively, we would conjecture that the gambler should again play boldly. However, in this note we will show that, unexpectedly, bold play is not necessarily optimal.

We prove polynomial ergodicity for the one-dimensional Zig-Zag process on heavy-tailed targets and identify the exact order of polynomial convergence of the process when targeting Student distributions.

Consider a finite Markov process {Xn} described by the one-step transition probabilities In describing the transition probabilities in the above manner we are adopting the convention that (0)0 = 1 so that the states 0 and M are absorbing, and the states 1,2,···,M-1 are transient.

As a consequence of the convergence of the age distribution, several other limit results follow for supercritical Bellman-Harris processes. These concern the distribution of split times, the growth ‘rate' of the process, and the distribution of ancestral lifelengths.

In this paper mathematical methods are given for finding the distributions of the virtual waiting time and the occupation time for single-server queues in which the interarrival times and the service times are discrete random variables taking on positive or non-negative integral values. First we shall consider deterministic queues in which the interarrival times and the service times are numerical (non-random) quantities and deduce a fundamental relation for such queues. Then we shall consider random queues with recurrent input and general service times and study the time dependent behavior of such queues.

In this paper, we obtain a closed form for the covariance function of a general stationary regenerative process. It is used to derive exact asymptotics of the covariance function of stationary ON/OFF and workload processes, when ON and OFF periods are heavy-tailed and mutually dependent. The case of a G/G/1/0 queueing system with heavy-tailed arrival and/or service times is studied in detail.

This paper derives an expression for the correlation function of a continuous parameter branching process, and obtains conditions for the convergence of this function to a zero or positive value.

We consider a marking procedure of the vertices of a tree where each vertex is marked independently from the others with a probability that depends only on its out-degree. We prove that a critical Galton–Watson tree conditioned on having a large number of marked vertices converges in distribution to the associated size-biased tree. We then apply this result to give the limit in distribution of a critical Galton–Watson tree conditioned on having a large number of protected nodes.

Given a finite collection of strings of letters from a fixed alphabet, it is of interest, in the contexts of data compression and DNA sequencing, to find the length of the shortest string which contains each of the given strings as a consecutive substring. In order to analyze the average behavior of the optimal superstring length, substrings of specified lengths are considered with the letters selected independently at random. An asymptotic expression is obtained for the savings from compression, i.e. the difference between the uncompressed (concatenated) length and the optimal superstring length.

We study the rare-event behavior of the workload process in a transitory queue, where the arrival epochs (or 'points') of a finite number of jobs are assumed to be the ordered statistics of independent and identically distributed (i.i.d.) random variables. The service times (or 'marks') of the jobs are assumed to be i.i.d. random variables with a general distribution, that are jointly independent of the arrival epochs. Under the assumption that the service times are strictly positive, we derive the large deviations principle (LDP) satisfied by the workload process. The analysis leverages the connection between ordered statistics and self-normalized sums of exponential random variables to establish the LDP. In this paper we present the first analysis of rare events in transitory queueing models, supplementing prior work that has focused on fluid and diffusion approximations.

A model is proposed for a manpower system in which promotion probabilities are functions of the seniority structure within the grades. The aim is to investigate the adequacy of a strategy which appeared to be being used by the management of the nurses in an Athens hospital. It is shown that the strategy is capable of restoring the system to normal state but that the problem will usually recur.

We compare R1(t), the reliability function of a redundant m-of-n system operating within the laboratory, with RD(t), the reliability function of the same system operating subject to environmental effects. Within the laboratory, all component lifetimes are independent and identically distributed according to G(α + 1, λ), a gamma distribution with index α + 1 and scale λ. Outside the laboratory, we adopt the model of Lindley and Singpurwalla (J. Appl. Prob. 23 (1986), 418-431) and assume that, conditional on a positive random variable η which models the effect of the common environment, all component lifetimes are independent and identically distributed according to G(α + 1, λη). When α is a non-negative integer we prove that for RD(t) to underestimate (resp. overestimate) R1(t) for all t sufficiently close to zero, it is necessary and sufficient that E(η(n-m + 1)(α+1)) > 1 (resp. E(η(n-m + 1)(α+1)) < 1). In the case in which n = 2, m= 1 and α = 0 we obtain a special case of a result of Currit and Singpurwalla (J. Appl. Prob. 26 (1988), 763-771). As an application, we obtain a necessary and sufficient condition under which RD(t) initially understimates (or overestimates) R1(t) when η follows a gamma or an inverse Gaussian distribution.