In many applications, the Gaussian convolution is approximately computed by means of recursive filters, with a significant improvement of computational efficiency. We are interested in theoretical and numerical issues related to such an use of recursive filters in a three-dimensional variational data assimilation (3Dvar) scheme as it appears in the software OceanVar. In that context, the main numerical problem consists in solving large linear systems with high efficiency, so that an iterative solver, namely the conjugate gradient method, is equipped with a recursive filter in order to compute matrix-vector multiplications that in fact are Gaussian convolutions. Here we present an error analysis that gives effective bounds for the perturbation on the solution of such linear systems, when is computed by means of recursive filters. We first prove that such a solution can be seen as the exact solution of a perturbed linear system. Then we study the related perturbation on the solution and we demonstrate that it can be bounded in terms of the difference between the two linear operators associated to the Gaussian convolution and the recursive filter, respectively. Moreover, we show through numerical experiments that the error on the solution, which exhibits a kind of edge effect, i.e. most of the error is localized in the first and last few entries of the computed solution, is due to the structure of the difference of the two linear operators.

Many boundary value problems occur in a natural way while studying fluid flow problems in a channel. The solutions of two such boundary value problems are obtained and analysed in the context of flow problems involving three layers of fluids of different constant densities in a channel, associated with an impermeable bottom that has a small undulation. The top surface of the channel is either bounded by a rigid lid or free to the atmosphere. The fluid in each layer is assumed to be inviscid and incompressible, and the flow is irrotational and two-dimensional. Only waves that are stationary with respect to the bottom profile are considered in this paper. The effect of surface tension is neglected. In the process of obtaining solutions for both the problems, regular perturbation analysis along with a Fourier transform technique is employed to derive the first-order corrections of some important physical quantities. Two types of bottom topography, such as concave and convex, are considered to derive the profiles of the interfaces. We observe that the profiles are oscillatory in nature, representing waves of variable amplitude with distinct wave numbers propagating downstream and with no wave upstream. The observations are presented in tabular and graphical forms.

The problem of surface water wave scattering by two thin nearly vertical barriers submerged in deep water from the same depth below the mean free surface and extending infinitely downwards is investigated here assuming linear theory, where configurations of the two barriers are described by the same shape function. By employing a simplified perturbational analysis together with appropriate applications of Green's integral theorem, first-order corrections to the reflection and transmission coefficients are obtained. As in the case of a single nearly vertical barrier, the first-order correction to the transmission coefficient is found to vanish identically, while the correction for the reflection coefficient is obtained in terms of a number of definite integrals involving the shape function describing the two barriers. The result for a single barrier is recovered when two barriers are merged into a single barrier.

In this paper, motivated by the problem of the coexistence on transmission links of telecommunications networks of elastic and unresponsive traffic, we study the impact on the busy period of an M/M/1 queue of a small perturbation in the service rate. The perturbation depends upon an independent stationary process (X(t)) and is quantified by means of a parameter ε ≪ 1. We specifically compute the two first terms of the power series expansion in ε of the mean value of the busy period duration. This allows us to study the validity of the reduced service rate approximation, which consists in comparing the perturbed M/M/1 queue with the M/M/1 queue whose service rate is constant and equal to the mean value of the perturbation. For the first term of the expansion, the two systems are equivalent. For the second term, the situation is more complex and it is shown that the correlations of the environment process (X(t)) play a key role.

We study the parametric perturbation of Markov chains with denumerable state spaces. We consider both regular and singular perturbations. By the latter we mean that transition probabilities of a Markov chain, with several ergodic classes, are perturbed such that (rare) transitions among the different ergodic classes of the unperturbed chain are allowed. Singularly perturbed Markov chains have been studied in the literature under more restrictive assumptions such as strong recurrence ergodicity or Doeblin conditions. We relax these conditions so that our results can be applied to queueing models (where the conditions mentioned above typically fail to hold). Assuming ν-geometric ergodicity, we are able to explicitly express the steady-state distribution of the perturbed Markov chain as a Taylor series in the perturbation parameter. We apply our results to quasi-birth-and-death processes and queueing models.

We define and analyse a random coverage process of the d-dimensional Euclidean space which allows us to describe a continuous spectrum that ranges from the Boolean model to the Poisson–Voronoi tessellation to the Johnson–Mehl model. As for the Boolean model, the minimal stochastic setting consists of a Poisson point process on this Euclidean space and a sequence of real valued random variables considered as marks of this point process. In this coverage process, the cell attached to a point is defined as the region of the space where the effect of the mark of this point exceeds an affine function of the cumulative effect of all marks. This cumulative effect is defined as the shot-noise process associated with the marked point process. In addition to analysing and visualizing this spectrum, we study various basic properties of the coverage process such as the probability that a point or a pair of points be covered by a typical cell. We also determine the distribution of the number of cells which cover a given point, and show how to provide deterministic bounds on this number. Finally, we also analyse convergence properties of the coverage process using the framework of closed sets, and its differentiability properties using perturbation analysis. Our results require a pathwise continuity property for the shot-noise process for which we provide sufficient conditions. The model in question stems from wireless communications where several antennas share the same (or different but interfering) channel(s). In this case, the area where the signal of a given antenna can be received is the area where the signal to interference ratio is large enough. We describe this class of problems in detail in the paper. The results obtained allow us to compute quantities of practical interest within this setting: for instance the outage probability is obtained as the complement of the volume fraction; the law of the number of cells covering a point allows us to characterize handover strategies, and so on.

We derive formulas for the first- and higher-order derivatives of the steady state performance measures for changes in transition matrices of irreducible and aperiodic Markov chains. Using these formulas, we obtain a Maclaurin series for the performance measures of such Markov chains. The convergence range of the Maclaurin series can be determined. We show that the derivatives and the coefficients of the Maclaurin series can be easily estimated by analysing a single sample path of the Markov chain. Algorithms for estimating these quantities are provided. Markov chains consisting of transient states and multiple chains are also studied. The results can be easily extended to Markov processes. The derivation of the results is closely related to some fundamental concepts, such as group inverse, potentials, and realization factors in perturbation analysis. Simulation results are provided to illustrate the accuracy of the single sample path based estimation. Possible applications to engineering problems are discussed.

We study a fundamental feature of the generalized semi-Markov processes (GSMPs), called event coupling. The event coupling reflects the logical behavior of a GSMP that specifies which events can be affected by any given event. Based on the event-coupling property, GSMPs can be classified into three classes: the strongly coupled, the hierarchically coupled, and the decomposable GSMPs. The event-coupling property on a sample path of a GSMP can be represented by the event-coupling trees. With the event-coupling tree, we can quantify the effect of a single perturbation on a performance measure by using realization factors. A set of equations that specifies the realization factors is derived. We show that the sensitivity of steady-state performance with respect to a parameter of an event lifetime distribution can be obtained by a simple formula based on realization factors and that the sample-path performance sensitivity converges to the sensitivity of the steady-state performance with probability one as the length of the sample path goes to infinity. This generalizes the existing results of perturbation analysis of queueing networks to GSMPs.

We use the fact that the Palm measure of a stationary random measure is invariant to phase space change to generalize the light traffic formula initially obtained for stationary processes on a line to general spaces. This formula gives a first-order expansion for the expectation of a functional of the random measure when its intensity vanishes. This generalization leads to new algorithms for estimating gradients of functionals of geometrical random processes.

Given a parametric family of regenerative processes on a common probability space, we investigate when the derivatives (with respect to the parameter) are regenerative. We primarily consider sequences satisfying explicit, Lipschitz recursions, such as the waiting times in many queueing systems, and show that derivatives regenerate together with the original sequence under reasonable monotonicity or continuity assumptions. The inputs to our recursions are i.i.d. or, more generally, governed by a Harris-ergodic Markov chain. For i.i.d. input we identify explicit regeneration points; otherwise, we use coupling arguments. We give conditions for the expected steady-state derivative to be the derivative of the steady-state mean of the original sequence. Under these conditions, the derivative of the steady-state mean has a cycle-formula representation.

Starting with some mild assumptions on the parametrization of the service process, perturbation analysis (PA) estimates are obtained for stationary and ergodic single-server queues. Besides relaxing the stochastic assumptions, our approach solves some problems associated with the traditional regenerative approach taken in most of the previous work in this area. First, it avoids problems caused by perturbations interfering with the regenerative structure of the system. Second, given that the major interest is in steady-state performance measures, it examines directly the stationary version of the system, instead of considering performance measures expressed as Cesaro limits. Finally, it provides new estimators for general (possibly discontinuous) functions of the workload and other steady-state quantities.

We consider a finite Markov chain with nearly-completely decomposable stochastic matrix. We determine bounds for the error, when the stationary probability vector is approximated via a perturbation analysis.

The paper studies the sensitivity of the throughput with respect to a mean service rate in a closed queueing network with exponentially distributed service requirements and state-dependent service rates. The study is based on perturbation analysis of queueing networks. A new concept, the realization factor of a perturbation, is introduced. The properties of realization factors are discussed, and a set of equations specifying the realization factors are derived. The elasticity of the steady state throughput with respect to a mean service rate equals the product of the steady state probability and the corresponding realization factor. This elasticity can be estimated by applying a perturbation analysis algorithm to a sample path of the system. The sample path elasticity of the throughput with respect to a mean service rate converges with probability 1 to the elasticity of the steady state throughput. The theory provides an analytical method of calculating the throughput sensitivity and justifies the application of perturbation analysis.

Realization probability is a new concept pertaining to perturbation analysis of closed queuing networks. The sensitivities of throughputs in a closed single-class Jackson network can be expressed in terms of realization probabilities. In this paper, based on a discussion of perturbation analysis for networks with state-dependent service rates, we derive some new formulas for sensitivities of throughputs using realization probability.

Perturbation analysis is an efficient approach to estimating the sensitivities of the performance measures of a queueing network. A new notion, called the realization probability, provides an alternative way of calculating the sensitivity of the system throughput with respect to mean service times in closed Jackson networks with single class customers and single server nodes (Cao (1987a)). This paper extends the above results to systems with finite buffer sizes. It is proved that in an indecomposable network with finite buffer sizes a perturbation will, with probability 1, be realized or lost. For systems in which no server can directly block more than one server simultaneously, the elasticity of the expected throughput can be expressed in terms of the steady state probability and the realization probability in a simple manner. The elasticity of the throughput when each customer’s service time changes by the same amount can also be calculated. These results provide some theoretical background for perturbation analysis and clarify some important issues in this area.

We consider a queue which is obtained by slightly modifying the M/PH/1 queue, so that weak correlations are introduced among the interarrival times. We show how the stationary probability distribution may be studied by a perturbation analysis.

Perturbation analysis is a new technique which yields the sensitivities of system performance measures with respect to parameters based on one sample path of a system. This paper provides some theoretical analysis for this method. A new notion, the realization probability of a perturbation in a closed queueing network, is studied. The elasticity of the expected throughput in a closed Jackson network with respect to the mean service times can be expressed in terms of the steady-state probabilities and realization probabilities in a very simple way. The elasticity of the throughput with respect to the mean service times when the service distributions are perturbed to non-exponential distributions can also be obtained using these realization probabilities. It is proved that the sample elasticity of the throughput obtained by perturbation analysis converges to the elasticity of the expected throughput in steady-state both in mean and with probability 1 as the number of customers served goes to This justifies the existing algorithms based on perturbation analysis which efficiently provide the estimates of elasticities in practice.