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In this paper we analyze a single server two-queue model with Bernoulli schedules. This discipline is very flexible and contains the exhaustive and 1-limited disciplines as special cases. We formulate the queueing system as a Riemann boundary value problem with shift. The boundary value problem is solved by exploring a Fredholm integral equation around the unit circle. Some numerical examples are presented at the end of the paper.

This paper deals with the existence of solutions of a stochastic integral equation of the Volterra type and their asymptotic behaviour. Investigations of this paper use the concept of a measure of non-compactness in Banach space and fixed-point theorem of Darbo type. An application to a stochastic model for chemotherapy is also presented.

The paper represents a generalization of one of the main theoretical results of my Ph.D. thesis. The work is an outgrowth of work first begun by E. J. Hannan and a correct 'conjecture' by P. Whittle. The main theorem of this paper proves the existence of, and gives an explicit formula for, the asymptotic best linear predictor of a certain type of non-stationary multivariate time series from noise distorted observations. The non-stationarity arises from the fact that the signal satisfies a difference equation, which when considered as a polynomial, has only elementary divisors. The proof is accomplished by showing, through Hilbert space and harmonic analysis methods, that the generating function is a limit of the generating functions of the stationary analogue; that is, where the difference function has elementary divisors. Finally, it is shown that this asymptotic generating function exactly predicts null solutions to the difference equation. The proof is direct and due to E. J. Hannan.

In this paper we consider a one dimensional stochastic system described by an elliptic equation. A spatially varying random coefficient is introduced to account for uncertainty or imprecise measurements. We model the logarithm of this coefficient by a Gaussian process and provide asymptotic approximations of the tail probabilities of the derivative of the solution.

The risk processes considered in this paper are generated by an underlying Markov process with a regenerative structure and an independent sequence of independent and identically distributed claims. Between the arrivals of claims the process increases at a rate which is a nonnegative function of the present value of the Markov process. The intensity for a claim to occur is another nonnegative function of the value of the Markov process. The claim arrival times are the regeneration times for the Markov process. Two-sided claims are allowed, but the distribution of the positive claims is assumed to have a Laplace transform that is a rational function. The main results describe the joint Laplace transform of the time at ruin and the deficit at ruin. The method used consists in finding partial eigenfunctions for the generator of the joint process consisting of the Markov process and the accumulated claims process, a joint process which is also Markov. These partial eigenfunctions are then used to find a martingale that directly leads to an expression for the desired Laplace transform. In the final section, three examples are given involving different types of the underlying Markov process.

In this paper we consider two classes of reflected Ornstein–Uhlenbeck (OU) processes: the reflected OU process with jumps and the Markov-modulated reflected OU process. We prove that their stationary distributions exist. Furthermore, for the jump reflected OU process, the Laplace transform (LT) of the stationary distribution is given. As for the Markov-modulated reflected OU process, we derive an equation satisfied by the LT of the stationary distribution.

We provide a scaling for compound Poisson distributions that leads (under certain conditions on the Fourier transform) to a weak convergence result as the parameter of the distribution tends to infinity. We show that the limiting probability measure belongs to the class of stable Cauchy laws with Fourier transformt↦ exp(−c|t|− iatlog|t|). We apply this convergence result to the standard discrete Luria–Delbrück distribution and derive an integral representation for the corresponding limiting density, as an alternative to that found in a closely related paper of Kepler and Oprea. Moreover, we verify local convergence and we derive an integral representation for the distribution function of the limiting continuous Luria–Delbrück distribution.

In this paper, we show that the mean comparison theorem, which is valid for Brownian motion, cannot be extended to Poisson processes. A counterexample in the Poisson case for which the mean comparison theorem does not hold is provided.

In this paper, we apply coupling methods to study strong ergodicity for Markov processes, and sufficient conditions are presented in terms of the expectations of coupling times. In particular, explicit criteria are obtained for one-dimensional diffusions and birth-death processes to be strongly ergodic. As a by-product, strong ergodicity implies that the essential spectra of the generators for these processes are empty.

In this paper, the class of periodic Ornstein-Uhlenbeck processes is defined. It is shown that periodic Ornstein-Uhlenbeck processes are stationary Markov random fields and the class of stationary distributions is characterized. In particular, any self-decomposable distribution is the stationary distribution of some periodic Ornstein-Uhlenbeck process. As examples, gamma periodic Ornstein-Uhlenbeck processes and Gaussian periodic Ornstein-Uhlenbeck processes are considered.

In this paper we consider the stochastic wave equation in one spatial dimension driven by a two-parameter Gaussian noise which is white in time and has general spatial covariance. We give conditions on the spatial covariance of the driving noise sufficient for the string to have finite expected energy and calculate this energy as a function of time. We show that these same conditions on the spatial covariance of the driving noise are also sufficient to guarantee that the energy of the string has a version which is continuous almost surely.

Wang and Pötzelberger (1997) derived an explicit formula for the probability that a Brownian motion crosses a one-sided piecewise linear boundary and used this formula to approximate the boundary crossing probability for general nonlinear boundaries. The present paper gives a sharper asymptotic upper bound of the approximation error for the formula, and generalizes the results to two-sided boundaries. Numerical computations are easily carried out using the Monte Carlo simulation method. A rule is proposed for choosing optimal nodes for the approximating piecewise linear boundaries, so that the corresponding approximation errors of boundary crossing probabilities converge to zero at a rate of O(1/n 2).

In this paper, we consider positive linear operators L representable in terms of stochastic processes Z having right-continuous non-decreasing paths. We introduce the equivalent notions of derived operator and derived process of order n of L and Z, respectively. When acting on absolutely continuous functions of order n, we obtain a Taylor's formula of the same order for such operators, thus extending to a positive linear operator setting the classical Taylor's formula for differentiable functions. It is also shown that the operators satisfying Taylor's formula are those which preserve generalized convexity of order n. We illustrate the preceding results by considering discrete time processes, counting and renewal processes, centred subordinators and the Yule birth process.

In this paper, transient characteristics related to excursions of the occupation process ofM/M/∞ queues are studied, when the excursion level is large and close to the mean offered load. We show that the classical diffusion approximation by an Ornstein–Uhlenbeck (OU) process captures well the average values of the transient variables considered, while the asymptotic distributions of these variables depart from those corresponding to the OU process. They exhibit, however, equivalent tail behaviour at infinity and numerical evidence shows that they are amazingly close to each other over the whole half-line.

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n 2). The general case is discussed in terms of operator semigroups.

In this paper we continue the study of the classification problem for random walks in the quarter plane, with zero drifts in the interior of the domain. The necessary and sufficient conditions for these random walks to be ergodic were found earlier in [3]. Here we obtain necessary and sufficient conditions for transience by constructing suitable Lyapounov functions.

In this paper we prove the validity of the Volterra integral equation for the evaluation of first-passage-time probability densities through varying boundaries, given by Buonocore et al. [1], for the case of diffusion processes not necessarily time-homogeneous. We study, specifically those processes that can be obtained from the Wiener process in the sense of [5]. A study of the kernel of the integral equation, in the same way as that by Buonocore et al. [1], is done. We obtain the boundaries for which closed-form solutions of the integral equation, without having to solve the equation, can be obtained. Finally, a few examples are given to indicate the actual use of our method.

This paper presents the form of some characteristics of the Voronoi tessellation which is generated by a stationary Poisson process in . Expressions are given for the spherical and linear contact distribution functions. These formulae lead to numerically tractable double-integral formulae for chord length probability density functions.

Dufresne [1] recently showed that the integral of the exponential of Brownian motion with negative drift is distributed as the reciprocal of a gamma variable. In this paper, it is shown that this result is another formulation of the distribution of last exit times for transient Bessel processes. A bivariate distribution of such integrals of exponentials is also obtained explicitly.

This paper is concerned with asymptotic results for a single-server queue having periodic Poisson input and general service-time distribution, and carries forward the analysis of this model initiated in Harrison and Lemoine. First, it is shown that a theorem of Hooke relating the stationary virtual and actual waiting-time distributions for the GI/G/1 queue extends to the periodic Poisson model; it is then pointed out that Hooke's theorem leads to the extension (developed in [3]) of a related theorem of Takács. Second, it is demonstrated that the asymptotic distribution for the server-load process at a fixed ‘time of day' coincides with the distribution for the supremum, over the time horizon [0,∞), of the sum of a stationary compound Poisson process with negative drift and a continuous periodic function. Some implications of this characterization result for the computation and approximation of the asymptotic distributions are then discussed, including a direct proof, for the periodic Poisson case, of a recent result of Rolski comparing mean asymptotic customer waiting time with that of a corresponding M/G/1 system.

This paper gives an asymptotic evaluation of the probability that the Wiener path first crosses a square root boundary. The result is applied to estimate the moments of the first-passage time distribution of the Ornstein–Uhlenbeck process to a constant boundary.

As has been shown by de Maré, in a stationary Gaussian process the length of the successive zero-crossing intervals cannot be independent, except for the degenerate case of a pure cosine process. However, no closed-form expression of the distribution of these quantities is known at present. In this paper we present an accurate explicit approximative formula, derived by replacing the Slepian model process by its regression curve.

In this paper a random motion on the surface of the 3-sphere whose probability law is a solution of the telegraph equation in spherical coordinates is presented. The connection of equations governing the random motion with Maxwell equations is examined together with some qualitative features of its sample paths. Finally Brownian motion on the 3-sphere is derived as the limiting process of a random walk with latitude-changing probabilities.

In the present paper we deal with two models of the traffic on a divided highway. In both models traffic is flowing in one direction only, in two lanes, of which one (the left hand lane) is used only for overtaking. We suppose that vehicles enter the highway at the same entrance so that the instants at which a vehicle enters the highway form a homogeneous Poisson process, with density λ. Thus λ is the rate at which vehicles enter the highway per unit time. We suppose that there are no junctions, or exits (i.e. the highway extends in one direction to infinity).

The cars are considered as points on an infinite road with no intersections. They can overtake each other without any delay and they travel at constant speeds. These are independent identically distributed random variables also independent of the initial positions of the cars. The main purpose of the paper is the study of the asymptotic distribution for the number of overtakings (and/or meetings) in increasing rectangles in the time-road plane. Under the assumption of (weighted) Poisson distributed cars along the time-axis we deduce the asymptotic distribution of the standardized number of overtakings (and/or meetings) for large rectangles in the time-road plane. Lastly we shall indicate an application of the results.

SummaryA weir of capacity K is considered in which the water inflow is a process with stationary independent increments. Unless the weir is empty, there is a continuous release of water at unit rate; if K is finite the weir may become full in which case the excess water overflows instantaneously. A weir for which K is infinite will be referred to as infinite dam. For the latter the transient behaviour is well known if the input possesses a second moment (cf. e.g., Prabhu [7]) and serves as the starting point for the present paper. This result is first extended to yield the Laplace transform (L.T.) of the trivariate Laplace-Stieltjes transform (L.S.T.) of the content v(t) at time t, the input X(t) in (0, t) and the total time d(t) in the interval (0, t) during which the dam is dry. (Incidentally, the last two quantities, for relevant time intervals, will be carried throughout.) Then we use a relation between the latter and the L.S.T. of the expected number of downward level y crossings of the v(t) process established in Roes [9]. Since the dam processes considered are Markov processes, we have therewith the L.S.T. of the renewal function of the renewal process imbedded at level y. From this, one finds the L.S.T.'s of first entrance and taboo first entrance times (for their definition see introduction). Next we calculate the first skip times for the infinite dam from the first entrance times and the L.T. of the L.S.T. of v(t). It is then a routine matter to determine the taboo first skip times. From the (taboo) first entrance and skip times we derive the first entrance times for the finite dam, which in turn lead to the renewal functions of the renewal processes imbedded in the finite dam content process v*(t) and hence to the transient behaviour of the finite dam.The advantage of the present approach over the one given in Roes [8] is that it is entirely probabilistic and avoids involved analytic arguments. As a result, the question of uniqueness of the solution does not arise, while more insight is obtained in the structure. The L.S.T. of several first entrance times and first skip times have been derived by Cohen [2] for compound Poisson input.

In mathematical models of phenomena occurring in the general areas of the engineering, biological, and physical sciences, random or stochastic equations appear frequently. In this paper we shall formulate a problem in telephone traffic theory which leads to a stochastic integral equation which is a special case of the Volterra type of the form where: (i) ω∊Ω, where Ω is the supporting set of the probability measure space (Ω,B,P); (ii) x(t; ω) is the unknown random variable for t ∊ R +, where R + = [0, ∞); (iii) y(t; ω) is the stochastic free term or free random variable for t ∊ R +; (iv) k(t, τ; ω) is the stochastic kernel, defined for 0 ≦ τ ≦ t < ∞; and (v) f(t, x) is a scalar function defined for t ∊ R + and x ∊ R, where R is the real line.

Let w(t), 0 ≦ t ≦ ∞, be a Brownian motion process, i.e., a zero-mean separable normal process with Pr{w(0) = 0} = 1, E{w(t 1)w(t 2)}= min (t 1, t 2), and let a, b denote the boundaries defined by y = a(t), y = b(t), where b(0) < 0 < a(0) and b(t) < a(t), 0 ≦ t ≦ T ≦ ∞. A basic problem in many fields such as diffusion theory, gambler's ruin, collective risk, Kolmogorov-Smirnov statistics, cumulative-sum methods, sequential analysis and optional stopping is that of calculating the probability that a sample path of w(t) crosses a or b before t = T. This paper shows how this probability may be computed for sufficiently smooth boundaries by numerical solution of integral equations for the first-passage distribution functions. The technique used is to approximate the integral equations by linear recursions whose coefficients are estimated by linearising the boundaries within subintervals. The results are extended to cover the tied-down process subject to the condition w(1) = 0. Some related results for the Poisson process and the sample distribution function are given. The procedures suggested are exemplified numerically, first by computing the probability that the tied-down Brownian motion process crosses a particular curved boundary for which the true probability is known, and secondly by computing the finite-sample and asymptotic powers of the Kolmogorov-Smirnov test against a shift in mean of the exponential distribution.

In the present paper the solutions of two integral equations are derived. One of the integral equations dominates the mathematical description of the stochastic process { v n , n = 1,2, …}, recursively defined by K is a positive constant, τ 1, τ 2, …; Σ 1, Σ 2, …; are independent, non-negative variables, with τ 1, τ 2,…, identically distributed, similarly, the variables Σ 1, Σ 2, …, are identically distributed.