We study percolation and Ising models defined on generalizations of quad-trees used in multiresolution image analysis. These can be viewed as trees for which each mother vertex has2d daughter vertices, and for which daughter vertices are linked together in d-dimensional Euclidean configurations. Retention probabilities and interaction strengths differ according to whether the relevant bond is between mother and daughter or between neighbours. Bounds are established which locate phase transitions and show the existence of a coexistence phase for the percolation model. Results are extended to the corresponding Ising model using the Fortuin-Kasteleyn random-cluster representation.

This paper is devoted to a study of the integral of the workload process of the single server queue, in particular during one busy period. Firstly, we find asymptotics of the area 𝒜 swept under the workload process W(t) during the busy period when the service time distribution has a regularly varying tail. We also investigate the case of a light-tailed service time distribution. Secondly, we consider the problem of obtaining an explicit expression for the distribution of 𝒜. In the general GI/G/1 case, we use a sequential approximation to find the Laplace—Stieltjes transform of 𝒜. In the M/M/1 case, this transform is obtained explicitly in terms of Whittaker functions. Thirdly, we consider moments of 𝒜 in the GI/G/1 queue. Finally, we show asymptotic normality of .

We analyse in this paper the fluid weighted fair queueing system with two classes of customers, who arrive according to Poisson processes and require arbitrarily distributed service times. In a first step, we express the Laplace transform of the joint distribution of the workloads in the two virtual queues of the system by means of unknown Laplace transforms. Such an unknown Laplace transform is related to the distribution of the workload in one queue provided that the other queue is empty. We explicitly compute the unknown Laplace transforms by means of a Wiener—Hopf technique. The determination of the unknown Laplace transforms can be used to compute some performance measures characterizing the system (e.g. the mean waiting time for each class) which we compute in the exponential service case.

We consider a failure-prone system which operates in continuous time and is subject to condition monitoring at discrete time epochs. It is assumed that the state of the system evolves as a continuous-time Markov process with a finite state space. The observation process is stochastically related to the state process which is unobservable, except for the failure state. Combining the failure information and the information obtained from condition monitoring, and using the change of measure approach, we derive a general recursive filter, and, as special cases, we obtain recursive formulae for the state estimation and other quantities of interest. Up-dated parameter estimates are obtained using the EM algorithm. Some practical prediction problems are discussed and an illustrative example is given using a real dataset.

In this paper, the generalized burn-in and replacement model considered by Cha (2001) is further extended to the case in which the probability of Type II failure is time dependent. Two burn-in procedures are considered and they are compared in cases when both the procedures are applicable. Under some mild conditions on the failure rate function r(t) and the Type II failure probability function p(t), the problems of determining optimal burn-in time and optimal replacement policy are considered.

We consider a system with a finite state space subject to continuous-time Markovian deterioration while running that leads to failure. Failures are instantaneously detected. This system is submitted to sequential checking and preventive maintenance: up states are divided into ‘good’ and ‘degraded’ ones and the system is sequentially checked through perfect and instantaneous inspections until it is found in a degraded up state and stopped to allow maintenance (or until it fails). Time between inspections is random and is chosen at each inspection according to the current degradation degree of the system. Markov renewal equations fulfilled by the reliability of the maintained system are given and an exponential equivalent is derived for the reliability. We prove the existence of an asymptotic failure rate for the maintained system, which we are able to compute. Sufficient conditions are given for the preventive maintenance policy to improve the reliability and the asymptotic failure rate of the system. A numerical example illustrates our study.

An analytical expression of the time-dependent probability distribution of M/D/1/N queues initialised in an arbitrary deterministic state is derived. We also obtain a simple analytical expression of the differential equation governing the transient average traffic which only involves probabilities of boundary states. As a by-product, a closed form solution of the departure rate from the system is also determined.

The usual concept of asymptotic independence, as discussed in the context of extreme value theory, requires the distribution of the coordinatewise sample maxima under suitable centering and scaling to converge to a product measure. However, this definition is too broad to conclude anything interesting about the tail behavior of the product of two random variables that are asymptotically independent. Here we introduce a new concept of asymptotic independence which allows us to study the tail behavior of products. We carefully discuss equivalent formulations of asymptotic independence. We then use the concept in the study of a network traffic model. The usual infinite source Poisson network model assumes that sources begin data transmissions at Poisson time points and continue for random lengths of time. It is assumed that the data transmissions proceed at a constant, nonrandom rate over the entire length of the transmission. However, analysis of network data suggests that the transmission rate is also random with a regularly varying tail. So, we modify the usual model to allow transmission sources to transmit at a random rate over the length of the transmission. We assume that the rate and the time have finite mean, infinite variance and possess asymptotic independence, as defined in the paper. We finally prove a limit theorem for the input process showing that the centered cumulative process under a suitable scaling converges to a totally skewed stable Lévy motion in the sense of finite-dimensional distributions.

Several recent papers have shown that for the M/G/1/n queue with equal arrival and service rates, the expected number of lost customers per busy cycle is equal to 1 for every n ≥ 0. We present an elementary proof based on Wald's equation and, for GI/G/1/n, obtain conditions for this quantity to be either less than or greater than 1 for every n ≥ 0. In addition, we extend this result to batch arrivals, where, for average batch size β, the same quantity is either less than or greater than β. We then extend these results to general ways that customers may be lost, to an arbitrary order of service that allows service interruption, and finally to reneging.

This paper studies inspected systems with non-self-announcing failures where the rate of deterioration is governed by a Markov chain. We compute the lifetime distribution and availability when the system is inspected according to a periodic inspection policy. In doing so, we expose the role of certain transient distributions of the environment.

Distributional fixed points of a Poisson shot noise transform (for nonnegative and nonincreasing response functions bounded by 1) are characterized. The tail behavior of fixed points is described. Typically they have either exponential moments or their tails are proportional to a power function, with exponent greater than −1. The uniqueness of fixed points is also discussed.Finally, it is proved that in most cases fixed points are absolutely continuous, apart from the possible atom at zero.

In this paper, we show that the time-average distributions of excess, age, and spread are given by the solution of first-order differential equations. These differential equations can be directly derived in a simple, unified way using a general level-crossing formula based on the balance of up and down crossings on sample paths, which may be helpful for the intuitive interpretation.

For functionals of multitype closed queueing networks, a conditional job-observer property is shown which provides more insight into the classical job-observer property. Applications and examples are given, including the classical job-observer property for the number of customers in a network, a representation of cycle time distributions and a basic formula for sojourn times.

Often in the study of reliability and its applications, the goal is to maximize or minimize certain reliability characteristics or some cost functions. For example, burn-in is a procedure used to improve the quality of products before they are used in the field. A natural question which arises is how long the burn-in procedure should last in order to maximize the mean residual life or the conditional survival probability. In the literature, an upper bound for the optimal burn-in time is obtained by assuming that the underlying distribution of the products has a bathtub-shaped failure rate function; however, no lower bound is available. A similar question arises in studying replacement policy, warranty policy, and inspection models. This article gives a lower bound for the optimal burn-in time, and lower and upper bounds for the optimal replacement and warranty policies, under the same bathtub-shape assumption.

An important class of queueing networks is characterized by the following feature: in contrast with ordinary units, a disaster may remove all work from the network. Applications of such networks include computer networks with virus infection, migration processes with mass exodus and serial production lines with catastrophes. In this paper, we deal with a two-stage tandem queue with blocking operating under the presence of a secondary flow of disasters. The arrival flows of units and disasters are general Markovian arrival processes. Using spectral analysis, we determine the stationary distribution at departure epochs. That distribution enables us to derive the distribution of the number of units which leave the network at a disaster epoch. We calculate the stationary distribution at an arbitrary time and, finally, we give numerical results and graphs for certain probabilistic descriptors of the network.

The benefit obtained by a selfish robot by cheating in a real multirobotic system can be represented by the random variable X n,q : the number of cheating interactions needed before all the members in a cooperative team of robots, playing a TIT FOR TAT strategy, recognize the selfish robot. Stability of cooperation depends on the ratio between the benefit obtained by selfish and cooperative robots. In this paper, we establish the probability model for X n,q . If the values of the parameters n and q are known, then this model allows us to make predictions about the stability of cooperation. Moreover, if these parameters are modifiable, it is possible to tune them to guarantee the viability of cooperation.

Consider a ·/G/k finite-buffer queue with a stationary ergodic arrival process and delayed customer feedback, where customers after service may repeatedly return to the back of the queue after an independent general feedback delay whose distribution has a continuous density function. We use coupling methods to show that, under some mild conditions, the feedback flow of customers returning to the back of the queue converges to a Poisson process as the feedback delay distribution is scaled up. This allows for easy waiting-time approximations in the setting of Poisson arrivals, and also gives a new coupling proof of a classic highway traffic result of Breiman (1963). We also consider the case of nonindependent feedback delays.

We study the closure properties of the family ℒ(α) of classes of life distributions introduced by Lin (1998) under general compounding. We define a discrete analogue of this family and study some properties.

We consider an infinite server resequencing queue, where arrivals are generated by jumps of a semi-Markov process and service times depend on the jumps of this process. The stationary distribution of the sojourn time, conditioned on the state of the semi-Markov process, is obtained both for the case of hyperexponential service times and for the case of a Markovian arrival process. For the general model, an accurate approximation is derived based on a discretisation of interarrival and service times.

We consider one-dimensional processes in which particles annihilate their neighbours, grow until they meet their neighbours or are deposited onto surfaces. All of the models considered have the property that they are connected to exponential series often by an inclusion-exclusion argument.