This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourier-series method are remarkably easy to use, requiring programs of less than fifty lines. The Fourier-series method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourier-series method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this...

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Abstract. The proliferation of client–server systems in business continues unabated, as applications are split into local tasks run on ‘client ’ workstations and resource-intensive computations run on a ‘server ’ mainframe. The complexity of such systems requires quantitative modelling for their efficient design and reconfiguration throughout their lifetime. The tools and techniques that are needed for the effective performance management of distributed client–server systems are discussed and illustrated by a case study taken from the financial sector. 1.

Abstract. We evaluate the performance of a class of two-hop relay protocols for mobile ad hoc networks. The interest is on the multicopy two-hop relay (MTR) protocol, where the source may generate multiple copies of a packet and use relay nodes to deliver the packet (or a copy) to its destination, and on the twohop relay protocol with erasure coding. Performance metrics of interest are the time to deliver a single packet to its destination, the number of copies of the packet at delivery instant, and the total number of copies that the source generates. The packet copies at relay nodes have limited lifetime (time-to-live TTL). Via a Markovian analysis, the three performance metrics of the MTR protocol are obtained in closed-from in the case where the number of the copies in the network is limited. Also, we develop an approximation analysis in the case where the inter-meeting times between nodes are arbitrarily distributed and the TTLs of the copies are constant and all equal. In particular, we show that exponential intermeeting times yield stochastically smaller delivery delays than hyper-exponential inter-meeting times, and that exponential TTLs yield stochastically larger delivery delays than constant TTLs. Finally, we characterize the delivery delay and the number of transmissions in the two-hop relay protocol with erasure coding and compare this scheme with the multicopy scheme. Keywords: Mobile ad hoc network, Two-hop relay protocol, Erasure coding, Mobility

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We evaluate the performance of a class of two-hop relay protocols for mobile ad hoc networks via simple models. The focus is on the multicopy two-hop relay protocol, where the source may generate multiple copies of a packet and use relay nodes to transmit the packet (or a copy) to its destination, and on the two-hop relay protocol with erasure coding, where a piece of information is fragmented into n blocks in such a way that the destination may decode the data if it receives at least k blocks. Performance metrics of interest are the time to deliver a single packet to its destination, the number of copies of the packet at delivery instant, and the total number of copies that the source generates; the latter number will be larger when TTLs are associated with the copies of a packet, a situation that we address. We also investigate the case where the number of copies of a packet currently in the network is limited so as to limit the energy consumption. Performance metrics are obtained in closed-from for the multicopy two-hop relay protocol in the case of exponential inter-meeting times, exponential TTLs and when the number of copies of the packet in the network is limited. We evaluate the impact of constant TTLs as opposed to exponential TTLs, and we develop an approximation analysis in the case where the inter-meeting times are arbitrarily distributed. In particular, we show that exponential inter-meeting times yield stochastically smaller delivery delays than hyper-exponential inter-meeting times; we also show that exponential TTLs yield larger expected delivery delays than constant TTLs. Finally, we characterize the delivery delay in the two-hop relay protocol with erasure coding and compare this scheme with the multicopy routing scheme.

Transient heat conduction in homogeneous and non-homogeneous

Abstract. Response time quantiles reflect user-perceived quality of service more accurately than mean or average response time measures. Consequently, on-line transaction processing benchmarks, telecommunications Service Level Agreements and emergency services legislation all feature stringent 90th percentile response time targets. This chapter describes a range of techniques for extracting response time densities and quantiles from large-scale Markov and semi-Markov models of real-life systems. We describe a method for the computation of response time densities or cumulative distribution functions which centres on the calculation and subsequent numerical inversion of their Laplace transforms. This can be applied to both Markov and semi-Markov models. We also review the use of uniformization to calculate such measures more efficiently in purely Markovian models. We demonstrate these techniques by using them to generate response time quantiles in a semi-Markov model of a high-availability web-server. We show how these techniques can be used to analyse models with state spaces of O 10 7 states and above. 1. Introduction. A

. We consider the integrodifferential equation u(t; x) = R t 0 a(t \Gamma s)uxx (s; x)ds with initial and boundary conditions corresponding to the Rayleigh problem. The kernel has the form a(t) = a 0 + a1 t + R t 0 a 1 (s)ds, where a 0 0, a1 0, and a 1 2 L 1 loc (R+ ) is of positive type and satisfies the condition R 1 0 e \Gammafflt ja 1 (t)jdt ! 1 for every ffl ? 0. Solving the equation numerically and performing a careful error analysis we show that the solution u(t; x) need not be nondecreasing in t 0 for fixed x ? 0, if a 1 is nonnegative, nonincreasing, and convex. The same result is shown to hold under the assumption that a 1 is completely positive. This answers a question that remained unsolved in [J. Pruß, Math. Ann., 279 (1987), p. 330]. In the case where a1 is convex, piecewise linear, the solution is shown to be almost everywhere equal to a function which is discontinuous across infinitely many parallel lines. Key words. viscoelasticity, integrodifferentia...