Let denote a rectangular lattice in the Euclidean plane E2, generated by (a × b) rectangles. In this paper we consider the probability that a random ellipse having main axes of length 2α and 2ß, with intersects . We regard the lattice as the union of two orthogonal sets and of equidistant lines and evaluate the probability that the random ellipse intersects or . Moreover, we consider the dependence structure of the events that the ellipse intersects or . We study further the case when the main axes of the ellipse are parallel to the lines of the lattice and satisfy 2ß = min (a, b) < 2α = max (a, b). In this case, the probability of intersection is 1, and there exist almost surely two perpendicular segments in within the ellipse. We evaluate the distribution function, density, mean and variance of the length of these segments. We conclude with a generalization of this problem in three dimensions.

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.

A large number n of sensors (finite connected intervals) are placed randomly on the real line so that the distances between the consecutive midpoints are independent random variables with expectation inversely proportional to n. In this work we address two fundamental sensor allocation problems. The interference problem tries to reallocate the sensors from their initial positions to eliminate overlaps. The coverage problem, on the other hand, allows overlaps, but tries to eliminate uncovered spaces between the originally placed sensors. Both problems seek to minimize the total sensor movement while reaching their respective goals. Using tools from queueing theory, Skorokhod reflections, and weak convergence, we investigate asymptotic behaviour of optimal costs as n increases to ∞. The introduced methodology is then used to address a more complicated, modified coverage problem, in which the overlaps between any two sensors can not exceed a certain parameter.

During DNA replication, small fragments of DNA are formed. These have been observed experimentally and the mechanism of their formation modelled mathematically. Using the stochastic model of Cowan and Chiu (1992), (1994), we find the probability distribution of the number of fragments. A new discrete distribution arises. The work has interest as an application of the recent theory on quasirenewal equations in Piau (2000).

Analogues of the central limit theorem and iterated logarithm law have recently been obtained for the Galton-Watson process; similar results are established in this paper for the temporally homogeneous Markov branching process and for the associated increasing process consisting of the number of splits in the original process up to time t.

The size order problem of the probability state vector elements of a homogeneous Markov system is examined. The time t0 is evaluated, after which the order of the state vector probabilities remains unchanged, and a formula to quickly find a lower bound for t0 is given. A formula for approximating the mode of the state sizes ni(t) as a function of the means Eni(t), and a relation to evaluate P(ni(t) = x+1) by means of certain terms which constitute P(ni(t) = x) are derived.

A spatial epidemic process where the individuals are located at positions in the Euclidean space R2 is considered. The infective individuals, with an infection period that is exponentially distributed with parameter µ, move in R2 according to a Brownian motion with a diffusion coefficient σ2. The susceptible individuals may also move. But we shall use the approximation that they remain unchanged in numbers and therefore assume that the averaged ‘density' of susceptibles per unit area is the same throughout space and time. The transition probability rate of infection of a susceptible in the infinitesimal element of area dy by an infective in dx is assumed to be a function h(x – y |) of the distance | x – y | between x and y. Then our process can be considered as a two-dimensional birth and death Brownian motion. Let be the number of infective individuals in the set D at time t and . The almost everywhere convergence of the random variables to a limit random variable W(D) is established.

The distribution of the interpoint distance process of a sequence of pairwise interaction point processes is considered. It is shown that, if the interaction function is piecewise-continuous, then the sequence of interpoint distance processes converges weakly to an inhomogeneous Poisson process under certain sparseness conditions. Convergence of the expectation of the interpoint distance process to the mean of the limiting Poisson process is also established. This suggests a new nonparametric estimator for the interaction function if independent identically distributed samples of the point process are available.

A continuum structure function (CSF) y is a non-decreasing mapping from the unit hypercube to the unit interval. Define whereas γ (γ) < α for all y < x}, the set of minimal vectors to level α. This paper examines CSFs for which each Pα is finite. It is shown that if γ is such a CSF and X is a vector of independent random variables, the distribution of γ (X) is readily calculated. Further, if γ is an arbitrary right-continuous CSF, the distribution of γ (X) may be approximated arbitrarily closely by that of γ′(X) where γ′ is a right-continuous CSF for which each minimal vector set is finite.

Cottrell et al. (1983) have indicated how ideas from the large deviations theory lead to fast simulation schemes that estimate the mean time taken by the slotted ALOHA protocol to saturate starting empty. Such fast simulation schemes are particularly useful when the attempt probability is small. The remaining time to saturation when the protocol has been operating for a time is more accurately described by the quasi-stationary exit time from the stable regime. The purpose of this article is to prove that the ratio of the quasi-stationary exit time to the exit time starting empty approaches 1 as the attempt probability becomes small.

We derive explicit formulas for the second moments of the absorption time matrices in the Markov renewal branching process. These formulas may easily be computationally implemented and are useful in the iterative computation of the semi-Markov matrices, which give the distributions of the duration and of the number of customers served in a busy period of a great variety of complex queueing models.

Consider a homogeneous Poisson process in with density ρ, and add the origin as an extra point. Now connect any two points x and y of the process with probability g(x − y), independently of the point process and all other pairs, where g is a function which depends only on the Euclidean distance between x and y, and which is nonincreasing in the distance. We distinguish two critical densities in this model. The first is the infimum of all densities for which the cluster of the origin is infinite with positive probability, and the second is the infimum of all densities for which the expected size of the cluster of the origin is infinite. It is known that if , then the two critical densities are non-trivial, i.e. bounded away from 0 and ∞. It is also known that if g is of the form , for some r > 0, then the two critical densities coincide. In this paper we generalize this result and show that under the integrability condition mentioned above the two critical densities are always equal.

We derive some limit theorems associated with the Ewens sampling formula when its parameter is increasing together with a sample size. Moreover, the limit results are applied in order to investigate asymptotic properties of the maximum likelihood estimator.

Let N be a stationary Markov-modulated marked point process on ℝ with intensity β∗ and consider a real-valued functional ψ(N). In this paper we study expansions of the form Eψ(N) = a0 + β∗a1 + ·· ·+ (β∗)nan + o((β∗)n) for β∗→ 0. Formulas for the coefficients ai are derived in terms of factorial moment measures of N. We compute a1 and a2 for the probability of ruin φ u with initial capital u for the risk process in the Markov-modulated environment; a0 = 0. Moreover, we give a sufficient condition for ϕu to be an analytic function of β∗. We allow the premium rate function p(x) to depend on the actual risk reserve.

We study the asymptotic behavior as α → 0 of the number of independent random arcs of length a needed to cover at least 1 − Kα (K ≧ 1) of a circle of unit circumference.

Certain limit theorems due to Berman involve the total time spent by Brownian motion with positive drift below an independent exponentially distributed level. Imhof has calculated the density function and shown that this random variable has two interesting probabilistic properties. We give sample path arguments which explain these two facts.

Although the usual optimal stopping problem is described as a Markov decision process with two decisions, stop and continue, we shall consider a model which distinguishes the observer's strategy from the system's two decisions. The observer can select a strategy defined on an action space, and the decision of the system to stop or continue is determined by a prescribed conditional probability. For this model, it may happen that the strategy (a) to stop is refused, or (b) to continue is forcibly stopped. This is a slight modification of the one-dimensional stopping problem by involving refusal and forced stopping. The model is motivated by the uncertain secretary choice problem of Smith (1975) and the multivariate stopping problem of Kurano, Yasuda and Nakagami (1980), (1982).

In [1], [4] and [6] simple and intuitively obvious bounds were suggested for various overflow systems and claimed to be insensitive, that is to be valid also in the non-exponential case. This insensitivity was formalized in [4] and [6] when the service distributions have a monotone failure rate. This technical note provides another proof which for the case of a pure overflow system with one customer class extends the insensitivity to the general case.

In this paper, we present a perturbation method of solving a second-order difference equation with variable coefficients with some additional assumptions. This method can be used to devise an algorithmic solution for the moments of sojourn times in some processor sharing queues. In particular, we examine three queues. The first has exponential service and a fairly general interrupted arrival mechanism. The second is a cyclic queue. The third is a model for a computer system in which finite and infinite sources interact.

Expressions for the mean and variance of the recurrence time of non-overlapping draft-patterns of draft from a Moran Reservoir Model (discrete-state and discrete-time Markov chain) are derived using Feller's Renewal argument. In addition an expression for the mean recurrence time for self-overlapping patterns of draft is derived using run-theory.