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...

In a client-server type system, the server software is required to run continuously for very long periods. Due to repeated and potentially faulty usage by many clients, such software "ages" with time and eventually fails. Huang et. al. proposed a technique called "software rejuvenation" [9] in which the software is periodically stopped and then restarted in a "robust" state after proper maintenance. This "renewal" of software prevents (or at least postpones) the crash failure. As the time lost (or the cost incurred) due to the software failure is typically more than the time lost (or the cost incurred) due to rejuvenation, the technique reduces the expected unavailability of the software. In this paper, we present a quantitative analysis of software rejuvenation. The behavior of the system is represented through a Markov Regenerative Stochastic Petri Net (MRSPN) model which is solved both for steady state as well as transient conditions. We provide a closedform analytical solution for ...

Random quantities of interest in operations research models can often be determined conveniently in the form of transforms. Hence, numerical transform inversion can be an effective way to obtain desired numerical values of cumulative distribution functions, probability density functions and probability mass functions. However, numerical transform inversion has not been widely used. This lack of use seems to be due, at least in part, to good simple numerical inversion algorithms not being well known. To help remedy this situation, in this paper we present a version of the Fourier-series method for numerically inverting probability generating functions. We obtain a simple algorithm with a convenient error bound from the discrete Poisson summation formula. The same general approach applies to other transforms. Key Words: numerical inversion of transforms, computational probability, generating functions, Fourier-series method, Poisson summation formula, discrete Fourier transform.

With the increasing complexity of multiprocessor and distributed processing systems, the need to develop efficient and accurate modeling methods is evident. Fault-tolerance and degradable performance of such systems has given rise to considerable interest in models for the combined evaluation of performance and reliability [1, 2]. Most of these models are based upon Markov or semi-Markov reward processes. Beaudry [1] proposed a simple method for computing the distribution of performability in a Markov reward process. We present two extensions of Beaudry's approach. First, we generalize the method to a semi-Markov reward process. Second, we remove the restriction requiring the association of zero reward to absorbing states only. Such reward models can be used to evaluate the effectiveness of degradable fault-tolerant systems. We illustrate the use of the approach with three interesting applications.

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The recent literature on Markov Regenerative Stochastic Petri Nets (MRSPN) assumes that the random firing time associated to each transition is resampled each time the transition fires or is disabled by the firing of a competitive transition. This modeling assumption does not cover the case of preemption mechanisms of repeat identical nature (pri). In this policy, an interrupted job must be repeated with an identical requirement so that its associated random variable must not be resampled. The paper investigates the implication of a pri policy into a MRSPN and describes an analytical procedure for the derivation of expressions for the transient probabilities. Key words: Stochastic Petri Nets, Semi-Markov Reward Models, Markov regenerative processes, preemptive repeat identical policy. 1 Introduction The analysis of stochastic systems with nonexponential timing is of increasing interest in the literature and requires the development of suitable modeling tools. Choi et al. have shown ...

. Markov Regenerative Stochastic Petri Nets (MRSPN) have been recently introduced in the literature with the aim of combining exponential and non-exponential ring times into a single model. However, the realizations of the general MRSPN model, so far discussed, require that at most a single non-exponential transition is enabled in each marking and that its associated memory policy is of enabling type. The present paper extends the previous models by allowing the memory policy to be of age type and by allowing multiple general transitions to be simultaneously enabled, provided that their enabling intervals do not overlap. A nal completely developed example, that couldn't have been considered in previous formulations, derives the closed form expressions for the transient state probabilities for a queueing system with preemptive resume (prs) service policy. Key words: Markov regenerative processes, Stochastic Petri Nets, Queueing systems with preemptive resume service, Transient analys...

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The analysis of stochastic systems with non-exponential timing requires the development of suitable modeling tools. Recently, some eort has been devoted to generalize the concept of Stochastic Petri nets, by allowing the ring times to be generally distributed. The evolution of the PN in time becomes a stochastic process, for which in general, no analytical solution is available. The paper surveys suitable restrictions of the PN model with generally distributed transition times, that have appeared in the literature, and compares these models from the point of view of the modeling power and the numerical complexity. Key words: Stochastic Petri Nets, Non-exponential Distributions, Phase-type Distributions, Markov and Semimarkov Reward Models, Markov Regenerative Processes, Queueing Systems with Preemption. 1 Introduction The usual denition of Stochastic Petri Net (SPN) implies that all the timed activities associated to the transitions are represented by exponential random ...

Many processes must complete in the presence of failures. Different systems respond to task failure in different ways. The system may resume a failed task from the failure point (or a saved checkpoint shortly before the failure point), it may give up on the task and select a replacement task from the ready queue, or it may restart the task. The behavior of systems under the first two scenarios is well documented, but the third (RESTART) has resisted detailed analysis. In this paper we derive tight asymptotic relations between the distribution of task times without failures to the total time when including failures, for any failure distribution. In particular, we show that if the task time distribution has an unbounded support then the total time distribution H is always heavy-tailed. Asymptotic expressions are given for the tail of H in various scenarios. The key ingredients of the analysis are the Cramér–Lundberg asymptotics for geometric sums and integral asymptotics, that in some cases are obtained via Tauberian theorems and in some cases by bare-hand calculations. Key words Cramér-Lundberg approximation, failure recovery, geometric sums, heavy tails, logarithmic asymptotics, mixture distribution, power tail, RESTART, Tauberian theorem