In this paper we derive conditions on the internal wear process under which the resulting time to failure model will be of the simple collapsible form when the usage accumulation history is available. We suppose that failure occurs when internal wear crosses a certain threshold or a traumatic event causes the item to fail. We model the infinitesimal increment in internal wear as a function of time, accumulated internal wear, and usage history, and we derive conditions on this function to get a collapsible model for the distribution of time to failure given the usage history. We reach the conclusion that collapsible models form the subset of accelerated failure time models with time-varying covariates for which the time transformation function satisfies certain simple properties.

Let F be a distribution function in the domain of attraction of an extreme-value distribution Hγ. If Fu is the distribution function of the excesses over u and Gγ the distribution function of the generalized Pareto distribution, then it is well known that Fu(x) converges to Gγ(x/σ(u)) as u tends to the end point of F, where σ is an appropriate normalizing function. We study the rate of (uniform) convergence to 0 of F̅u(x)-G̅γ((x+u-α(u))/σ(u)), where α and σ are two appropriate normalizing functions.

This work is devoted to asymptotic properties of singularly perturbed Markov chains in discrete time. The motivation stems from applications in discrete-time control and optimization problems, manufacturing and production planning, stochastic networks, and communication systems, in which finite-state Markov chains are used to model large-scale and complex systems. To reduce the complexity of the underlying system, the states in each recurrent class are aggregated into a single state. Although the aggregated process may not be Markovian, its continuous-time interpolation converges to a continuous-time Markov chain whose generator is a function determined by the invariant measures of the recurrent states. Sequences of occupation measures are defined. A mean square estimate on a sequence of unscaled occupation measures is obtained. Furthermore, it is proved that a suitably scaled sequence of occupation measures converges to a switching diffusion.

Denote by αt(μ) the probability law of At(μ) = ∫0texp(2(Bs+μ s))ds for a Brownian motion {Bs, s ≥ 0}. It is well known that αt(μ) is of interest in a number of domains, e.g. mathematical finance, diffusion processes in random environments, stochastic analysis on hyperbolic spaces and so on, but that it has complicated expressions. Recently, Dufresne obtained some remarkably simple expressions for αt(0) and αt(1), as well as an equally remarkable relationship between αt(μ) and αt(ν) for two different drifts μ and ν. In this paper, hinging on previous results about αt(μ), we give different proofs of Dufresne's results and present extensions of them for the processes {At(μ), t ≥ 0}.

Continuous-space-time branching processes (CSBP) are investigated in order to model random energy cascades. CSBPs are based on spectrally positive Lévy processes and, as such, are characterized by their corresponding Laplace exponents. Special emphasis is put on the CSBPs of Feller, Lamperti and Neveu and on their Poisson point process representations. The Neveu model (either supercritical or subcritical) is of particular interest in physics for its connection with the random energy model of Derrida, as revisited by Ruelle. Exploiting some connections between the partition functions of energy and the Poisson-Dirichlet distributions of Pitman and Yor, some information on the zero-temperature limit is extracted. Finally, for the subcritical versions of the three models, we compute the distribution of some of their interesting features: extinction time and probability, area under the profile (total energy) and width (maximal energy).

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.

For arbitrary stationary sequences of random variables satisfying a mild mixing condition, distributional approximations are established for functionals of clusters of exceedances over a high threshold. The approximations are in terms of the distribution of the process conditionally on the event that the first variable exceeds the threshold. This conditional distribution is shown to converge to a nontrivial limit if the finite-dimensional distributions of the process are in the domain of attraction of a multivariate extreme-value distribution. In this case, therefore, limit distributions are obtained for functionals of clusters of extremes, thereby generalizing results for higher-order stationary Markov chains by Yun (2000).

The average delay for the GI/G/1 queue is often approximated as a function of the first two moments of interarrival and service times. For highly irregular arrivals, however, it varies widely among queues with the same first two moments, even in moderately heavy traffic. Empirically, it decreases as the interarrival time third moment increases. For GI/M/1 queues, a heavy-traffic expression for the average delay with this property has been previously obtained. The method, however, sheds little light on why the third moment arises. We analyze the equilibrium idle-period distribution in heavy traffic using real-variable methods. For GI/M/1 queues, we derive the above heavy-traffic result and also obtain conditions under which it is either an upper or lower bound. Our approach provides an intuitive explanation for the result and also strongly suggests that similar results should hold for general service. This is supported by empirical evidence. For any given service distribution, it has been conjectured that the expected delay under pure-batch arrivals, where interarrival times are scaled Bernoulli random variables, is an upper bound on the average delay over all interarrival distributions with the same first two moments. We investigate this conjecture and show, among other things, that pure-batch arrivals have the smallest third moment. We obtain conditions under which this conjecture is true and present a counterexample where it fails. Arrivals that arise as overflows from other queues can be highly irregular. We show that interoverflow distributions in a certain class have decreasing failure rate.

We determine the exact large-buffer asymptotics for a mixture of light-tailed and heavy-tailed input flows. Earlier studies have found a ‘reduced-load equivalence’ in situations where the peak rate of the heavy-tailed flows plus the mean rate of the light-tailed flows is larger than the service rate. In that case, the workload is asymptotically equivalent to that in a reduced system, which consists of a certain ‘dominant’ subset of the heavy-tailed flows, with the service rate reduced by the mean rate of all other flows. In the present paper, we focus on the opposite case where the peak rate of the heavy-tailed flows plus the mean rate of the light-tailed flows is smaller than the service rate. Under mild assumptions, we prove that the workload distribution is asymptotically equivalent to that in a somewhat ‘dual’ reduced system, multiplied by a certain prefactor. The reduced system now consists of only the light-tailed flows, with the service rate reduced by the peak rate of the heavy-tailed flows. The prefactor represents the probability that the heavy-tailed flows have sent at their peak rate for more than a certain amount of time, which may be interpreted as the ‘time to overflow’ for the light-tailed flows in the reduced system. The results provide crucial insight into the typical overflow scenario.

Sums of independent random variables concentrated on the same finite discrete, not necessarily lattice, set of points are approximated by compound Poisson distributions and signed compound Poisson measures. Such approximations can be more accurate than the normal distribution. Short asymptotic expansions are constructed.

We study Taylor series expansions of stationary characteristics of general-state-space Markov chains. The elements of the Taylor series are explicitly calculated and a lower bound for the radius of convergence of the Taylor series is established. The analysis provided in this paper applies to the case where the stationary characteristic is given through an unbounded sample performance function such as the second moment of the stationary waiting time in a queueing system.

This paper studies the first passage times to flat boundaries for a double exponential jump diffusion process, which consists of a continuous part driven by a Brownian motion and a jump part with jump sizes having a double exponential distribution. Explicit solutions of the Laplace transforms, of both the distribution of the first passage times and the joint distribution of the process and its running maxima, are obtained. Because of the overshoot problems associated with general jump diffusion processes, the double exponential jump diffusion process offers a rare case in which analytical solutions for the first passage times are feasible. In addition, it leads to several interesting probabilistic results. Numerical examples are also given. The finance applications include pricing barrier and lookback options.

Generalizing the standard frailty models of survival analysis, we propose to model frailty as a weighted Lévy process. Hence, the frailty of an individual is not a fixed quantity, but develops over time. Formulae for the population hazard and survival functions are derived. The power variance function Lévy process is a prominent example. In many cases, notably for compound Poisson processes, quasi-stationary distributions of survivors may arise. Quasi-stationarity implies limiting population hazard rates that are constant, in spite of the continual increase of the individual hazards. A brief discussion is given of the biological relevance of this finding.

We consider random recursive fractals and prove fine results about their local behaviour. We show that for a class of random recursive fractals the usual multifractal spectrum is trivial in that all points have the same local dimension. However, by examining the local behaviour of the measure at typical points in the set, we establish the size of fine fluctuations in the measure. The results are proved using a large deviation principle for a class of general branching processes which extends the known large deviation estimates for the supercritical Galton-Watson process.

We consider a queue fed by a mixture of light-tailed and heavy-tailed traffic. The two traffic flows are served in accordance with the generalized processor sharing (GPS) discipline. GPS-based scheduling algorithms, such as weighted fair queueing (WFQ), have emerged as an important mechanism for achieving service differentiation in integrated networks. We derive the asymptotic workload behaviour of the light-tailed traffic flow under the assumption that its GPS weight is larger than its traffic intensity. The GPS mechanism ensures that the workload is bounded above by that in an isolated system with the light-tailed flow served in isolation at a constant rate equal to its GPS weight. We show that the workload distribution is, in fact, asymptotically equivalent to that in the isolated system, multiplied by a certain prefactor, which accounts for the interaction with the heavy-tailed flow. Specifically, the prefactor represents the probability that the heavy-tailed flow is backlogged long enough for the light-tailed flow to reach overflow. The results provide crucial qualitative insight in the typical overflow scenario.

We investigate the distribution of the coalescence time (most recent common ancestor) for two individuals picked at random (uniformly) in the current generation of a branching process founded t units of time ago, in both the discrete and continuous (time and state-space) settings. We obtain limiting distributions as t→∞ in the subcritical case. In the continuous setting, these distributions are specified for quadratic branching mechanisms (corresponding to Brownian motion and Brownian motion with positive drift), and we also extend our results for two individuals to the joint distribution of coalescence times for any finite number of individuals sampled in the current generation.

For supercritical multitype Markov branching processes in continuous time, we investigate the evolution of types along those lineages that survive up to some time t. We establish almost-sure convergence theorems for both time and population averages of ancestral types (conditioned on nonextinction), and identify the mutation process describing the type evolution along typical lineages. An important tool is a representation of the family tree in terms of a suitable size-biased tree with trunk. As a by-product, this representation allows a ‘conceptual proof’ (in the sense of Kurtz et al.) of the continuous-time version of the Kesten-Stigum theorem.

Consider N towers each made up of a number of counters. At each step a tower is chosen at random, a counter removed which is then added to another tower also chosen at random. The probability distribution for the time needed to empty one of the towers is obtained in the case N = 3. Arguments are set forward as to why no simple formulae can be expected for N > 3. An asymptotic expression for the mean time before one of the towers becomes empty is derived in the case of four towers when they all initially contain a comparably large number of counters. We then study related problems, in particular the ruin problem for three players. Here we use simple martingale methodology as well as a solution proposed by T. S. Ferguson for a slightly modified problem. Throughout the paper it is our main objective to shed light on the reasons why the case N > 3 is so substantially different from the case N ≤ 3.