A model of a stochastic froth is introduced in which the rate of random coalescence of a pair of bubbles depends on an inverse power law of their sizes. The main question of interest is whether froths with a large number of bubbles can grow in a stable fashion; that is, whether under some time-varying change of scale the distributions of rescaled bubble sizes become approximately stationary. It is shown by way of a law of large numbers for the froths that the question can be re-interpreted in terms of a measure flow solving a nonlinear Boltzmann equation that represents an idealized deterministic froth. Froths turn out to be stable in the sense that there are scalings in which the rescaled measure flow is tight and, for a particular case, stable in the stronger sense that the rescaled flow converges to an equilibrium measure. Precise estimates are also given for the degree of tightness of the rescaled measure flows.

A neural model with N interacting neurons is considered. A firing of neuron i delays the firing times of all other neurons by the same random variable θ(i), and in isolation the firings of the neuron occur according to a renewal process with generic interarrival time Y (i). The stationary distribution of the N-vector of inhibitions at a firing time is computed, and involves waiting distributions of GI/G/1 queues and ladder height renewal processes. Further, the distribution of the period of activity of a neuron is studied for the symmetric case where θ(i) and Y (i) do not depend upon i. The tools are probabilistic and involve path decompositions, Palm theory and random walks.

The paper studies a model of repairable systems which is flexible enough to incorporate the standard imperfect repair and many other models from the literature. Palm stationarity of virtual ages, inter-failure times and degrees of repair is studied. A Loynes-type scheme and Harris recurrent Markov chains combined with coupling methods are used. Results on the weak total variation and moment convergences are obtained and illustrated by examples with IFR, DFR, heavy-tailed and light-tailed lifetime distributions. Some convergences obtained are monotone and/or at a geometric rate.

Queueing networks have been rather restricted in order to have product form distributions for network states. Recently, several new models have appeared and enlarged this class of product form networks. In this paper, we consider another new type of queueing network with concurrent batch movements in terms of such product form results. A joint distribution of the requested batch sizes for departures and the batch sizes of the corresponding arrivals may be arbitrary. Under a certain modification of the network and mild regularity conditions, we give necessary and sufficient conditions for the network state to have the product form distribution, which is shown to provide an upper bound for the one in the original network. It is shown that two special settings satisfy these conditions. Algorithms to calculate their stationary distributions are considered, with numerical examples.

This paper finds the first and second moments of the number of arrivals in a stable M/M/k queue during an idle period, i.e. in a period when at least one server is idle. These and similar results are used along with renewal theory asymptotics to find the first and second moments of the proportion of arrivals occurring while there are at least ξ customers waiting for service. The asymptotics of the second moment for large ξ are established.

A set of jobs is to be processed on a machine which is subject to breakdown and repair. When the processing of a job is interrupted by a machine breakdown, the processing later resumes at the point at which the breakdown occurred. We assume that the machine uptime is Erlang distributed and that processing and repair times follow general distributions. Simple permutation policies on both machine parameters and the processing distributions are given which minimize the weighted number of tardy jobs, weighted flow times and the weighted sum of the job delays.

In this paper we develop policies for scheduling dynamically arriving jobs to a broad class of parallel-processing queueing systems. We show that in heavy traffic the policies asymptotically minimize a measure of the expected system backlog, which we call system work. Our results yield succinct, closed-form expressions for optimal system work in heavy traffic.

In this paper we study finite clusters in a high density Boolean model with balls of two distinct sizes. Alexander (1993) studied the geometric structures of finite clusters in a high density Boolean model with balls of fixed size and showed that the only possible structure admitted by such events is that all Poisson points comprising the cluster are packed tightly inside a small sphere. When the balls are of varying sizes, the event that the cluster consists of k 1 big balls and k 2 small balls (both k 1, k 2 ≥ 1) occurs only when the centres of all big balls are compressed in a small sphere and the centres of the small balls are distributed uniformly inside the region formed by the big balls in such a way that the small balls are totally contained inside the big balls. We also show that it is most likely that a finite cluster in a high density Boolean model with varying ball sizes is made up only of small balls.

An optimal repair/replacement problem for a single-unit repairable system with minimal repair and random repair cost is considered. The existence of the optimal policy is established using results of the optimal stopping theory, and it is shown that the optimal policy is a ‘repair-cost-limit’ policy, that is, there is a series of repair-cost-limit functions g n (t), n = 1, 2,…, such that a unit of age t is replaced at the nth failure if and only if the repair cost C(n, t) ≥ g n (t); otherwise it is minimally repaired. If the repair cost does not depend on n, then there is a single repair cost limit function g(t), which is uniquely determined by a first-order differential equation with a boundary condition.

Olivier and Walrand (1994) claimed that the departure process of an MMPP/M/1 queue is not an MAP unless the queue is a stationary M/M/1 queue. They also conjectured that the departure process of an MAP/PH/1 queue is not an MAP unless the queue is a stationary M/M/1 queue. We show that their proof of the first result has an algebraic error, which leaves open the above question of whether the departure process of an MMPP/M/1 can be an MAP.

Coffman, Courtois, Gilbert and Piret (1991) have introduced a flow process in graphs, where each vertex gets an initial random resource, and where at each time vertices with large resources tend to attract resources from neighbours. The initial resources are assumed to be i.i.d., with a continuous distribution.

We are mainly interested in the following question: does, with probability 1, the resource of each vertex change only finitely many times?

Coffman et al. concentrate mainly on the case where the graph corresponds with the integer points on the line, in which case it is easily seen that the answer to the above question is positive. For higher-dimensional lattices they make general remarks which suggest that the answer to the above question is still positive. However, no proof seems to be known.

We restrict to the case of the square lattice, and, by a percolation approach, we reduce the question above to the question whether a certain quantity, which can be obtained from a finite computation, is sufficiently small. This computation is, however, still too long to be executed in an acceptable time. We therefore apply Monte Carlo simulation for this finite problem, which gives overwhelming evidence that, for the square lattice, the answer to the main question is positive.

This paper studies the queueing process in a class of D-policy models with Poisson bulk input, general service time, and four different vacation scenarios, among them a multiple vacation, single vacation and idle server. The D-policy specifies a busy period discipline, which requires an idle or vacationing server to resume his service when the workload process crosses some fixed level D. The analysis of the queueing process is based on the theory of fluctuations for three-dimensional marked counting processes presented in the paper. For all models, we derive the stationary distributions for the embedded and continuous time parameter queueing processes in closed analytic forms and illustrate the results by a number of examples and applications.

We consider the problem of self-organizing a linear list when the list is subjected to a random number of requests. Three separate move-to-front heuristics are considered. The difference between them lies in whether the items in the requested set are moved to the front in random order, or the same order as they were in originally, or in the order opposite to the one they were in originally. The eigenvalues of the transition probability matrices corresponding to the three heuristics are explicitly derived.

The stability of polling models is examined using associated fluid limit models. Examples are presented which generalize existing results in the literature or provide new stability conditions while in both cases providing simple and intuitive proofs of stability. The analysis is performed for both general single server models and specific multiple server models.

The asymptotic behaviour of the cumulative mean of a reward process 𝒵ρ, where the reward function ρ belongs to a rather large class of functions, is obtained. It is proved that E𝒵ρ (t) = C 0 + C 1 t + o(1), t → ∞, where C 0 and C 1 are fully specified. A section is devoted to the dual process of a semi-Markov process, and a formula is given for the mean of the first passage time from a state i to a state j of the dual process, in terms of the means of passage times of the original process.

Much of the literature in reliability and survival analysis considers failure models indexed by a single scale. There are situations which require that failure be described by several scales. An example from reliability is items under warranty whose failure is recorded by time and amount of use. An example from survival analysis is the death of a mine worker which is noted by age and the duration of exposure to dust.

This paper proposes an approach for developing probabilistic models indexed by two scales: time, and usage, a quantity that is related to time. The relationship between the scales is described by an additive hazards model. The evolution of usage is described by stochastic processes like the Poisson, the gamma and the Markov additive. The paper concludes with an application involving the setting of warranties. Two features differentiate this work from related efforts: a use of specific processes for describing usage, and a use of Monte Carlo techniques for generating the models.

We show that the GEM process has strong ordering properties: the probability that one of the k largest elements in the GEM sequence is beyond the first ck elements (c > 1) decays superexponentially in k.

An interacting particle system modelling competing growth on the ℤ2 lattice is defined as follows. Each x ∈ ℤ2 is in one of the states {0,1,2}. 1's and 2's remain in their states for ever, while a 0 flips to a 1 (a 2) at a rate equal to the number of its neighbours which are in state 1 (2). This is a generalization of the well-known Richardson model. 1's and 2's may be thought of as two types of infection, and 0's as uninfected sites. We prove that if we start with a single site in state 1 and a single site in state 2, then there is positive probability for the event that both types of infection reach infinitely many sites. This result implies that the spanning tree of time-minimizing paths from the origin in first passage percolation with exponential passage times has at least two topological ends with positive probability.

We consider a single server queue where the arrival process is a Markov-modulated Poisson process and service times are independent and identically distributed and independent from arrivals. The underlying intensity process is assumed ergodic with generator cQ, c > 0. We prove under some monotonicity assumptions on Q that the stationary workload W(c) is decreasing in c with respect to the increasing convex ordering.

We present new closed form solutions for partition functions used to normalize the steady-state flow balance equations of certain Markovian type queueing networks. The results focus on single class closed product form networks with state space constraints at the queueing stations. They are achieved by combining the partition function of the open network, having finite local buffers with a delta function in order to fix the number of customers in the system.