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Stability, queue length and delay of deterministic and stochastic queueing networks
 IEEE Transactions on Automatic Control
, 1994
"... Motivated by recent development in high speed networks, in this paper we study two types of stability problems: (i) conditions for queueing networks that render bounded queue lengths and bounded delay for customers, and (ii) conditions for queueing networks in which the queue length distribution of ..."
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Cited by 171 (20 self)
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Motivated by recent development in high speed networks, in this paper we study two types of stability problems: (i) conditions for queueing networks that render bounded queue lengths and bounded delay for customers, and (ii) conditions for queueing networks in which the queue length distribution of a queue has an exponential tail with rate `. To answer these two types of stability problems, we introduce two new notions of traffic characterization: minimum envelope rate (MER) and minimum envelope rate with respect to `. Based on these two new notions of traffic characterization, we develop a set of rules for network operations such as superposition, inputoutput relation of a single queue, and routing. Specifically, we show that (i) the MER of a superposition process is less than or equal to the sum of the MER of each process, (ii) a queue is stable in the sense of bounded queue length if the MER of the input traffic is smaller than the capacity, (iii) the MER of a departure process from a stable queue is less than or equal to that of the input process (iv) the MER of a routed process from a departure process is less than or equal to the MER of the departure process multiplied by the MER of the routing process. Similar results hold for MER with respect to ` under a further assumption of independence. These rules provide a natural way to analyze feedforward networks with multiple classes of customers. For single class networks with nonfeedforward routing, we provide a new method to show that similar stability results hold for such networks under the FCFS policy. Moreover, when restricting to the family of twostate Markov modulated arrival processes, the notion of MER with respect to ` is shown to be
Fitting Mixtures Of Exponentials To LongTail Distributions To Analyze Network Performance Models
, 1997
"... Traffic measurements from communication networks have shown that many quantities characterizing network performance have longtail probability distributions, i.e., with tails that decay more slowly than exponentially. File lengths, call holding times, scene lengths in MPEG video streams, and interva ..."
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Cited by 144 (13 self)
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Traffic measurements from communication networks have shown that many quantities characterizing network performance have longtail probability distributions, i.e., with tails that decay more slowly than exponentially. File lengths, call holding times, scene lengths in MPEG video streams, and intervals between connection requests in Internet traffic all have been found to have longtail distributions, being well described by distributions such as the Pareto and Weibull. It is known that longtail distributions can have a dramatic effect upon performance, e.g., longtail servicetime distributions cause longtail waitingtime distributions in queues, but it is often difficult to describe this effect in detail, because performance models with component longtail distributions tend to be difficult to analyze. We address this problem by developing an algorithm for approximating a longtail distribution by a hyperexponential distribution (a finite mixture of exponentials). We first prove tha...
Local search algorithms for SAT: An empirical evaluation
 JOURNAL OF AUTOMATED REASONING
, 2000
"... Local search algorithms are among the standard methods for solving hard combinatorial problems from various areas of Artificial Intelligence and Operations Research. For SAT, some of the most successful and powerful algorithms are based on stochastic local search and in the past 10 years a large num ..."
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Cited by 62 (18 self)
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Local search algorithms are among the standard methods for solving hard combinatorial problems from various areas of Artificial Intelligence and Operations Research. For SAT, some of the most successful and powerful algorithms are based on stochastic local search and in the past 10 years a large number of such algorithms have been proposed and investigated. In this article, we focus on two particularly wellknown families of local search algorithms for SAT, the GSAT and WalkSAT architectures. We present a detailed comparative analysis of these algorithms' performance using a benchmark set which contains instances from randomised distributions as well as SATencoded problems from various domains. We also investigate the robustness of the observed performance characteristics as algorithmdependent and problemdependent parameters are changed. Our empirical analysis gives a very detailed picture of the algorithms' performance for various domains of SAT problems; it also reveals a fundamental weakness in some of the bestperforming algorithms and shows how this can be overcome.
Modelbased evaluation: From dependability to security
 IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING
, 2004
"... The development of techniques for quantitative, modelbased evaluation of computer system dependability has a long and rich history. A wide array of modelbased evaluation techniques are now available, ranging from combinatorial methods, which are useful for quick, roughcut analyses, to statebased ..."
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Cited by 56 (2 self)
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The development of techniques for quantitative, modelbased evaluation of computer system dependability has a long and rich history. A wide array of modelbased evaluation techniques are now available, ranging from combinatorial methods, which are useful for quick, roughcut analyses, to statebased methods, such as Markov reward models, and detailed, discreteevent simulation. The use of quantitative techniques for security evaluation is much less common, and has typically taken the form of formal analysis of small parts of an overall design, or experimental red teambased approaches. Alone, neither of these approaches is fully satisfactory, and we argue that there is much to be gained through the development of a sound modelbased methodology for quantifying the security one can expect from a particular design. In this work, we survey existing modelbased techniques for evaluating system dependability, and summarize how they are now being extended to evaluate system security. We find that many techniques from dependability evaluation can be applied in the security domain, but that significant challenges remain, largely due to fundamental differences between the accidental nature of the faults commonly assumed in dependability evaluation, and the intentional, human nature of cyber attacks.
The load and availability of byzantine quorum systems
 SIAM Journal of Computing
, 1997
"... Abstract. Replicated services accessed via quorums enable each access tobe performed at only a subset (quorum) of the servers and achieve consistency across accesses by requiring any two quorums to intersect. Recently, bmasking quorum systems, whose intersections contain at least 2b+1 servers, have ..."
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Cited by 46 (17 self)
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Abstract. Replicated services accessed via quorums enable each access tobe performed at only a subset (quorum) of the servers and achieve consistency across accesses by requiring any two quorums to intersect. Recently, bmasking quorum systems, whose intersections contain at least 2b+1 servers, have been proposed to construct replicated services tolerant of barbitrary (Byzantine) server failures. In this paper we consider a hybrid fault model allowing benign failures in addition to the Byzantine ones. We present four novel constructions for bmasking quorum systems in this model, each of which has optimal load (the probability of access of the busiest server) or optimal availability (probability of some quorum surviving failures). To show optimality we also prove lower bounds on the load and availability of any bmasking quorum system in this model.
Departures from Many Queues in Series
, 1990
"... We consider a series of n singleserver queues, each with unlimited waiting space and the firstin firstout service discipline. Initially, the system is empty; then k customers are placed in the first queue. The service times of all the customers at all the queues are i.i.d. with a general distribu ..."
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Cited by 44 (5 self)
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We consider a series of n singleserver queues, each with unlimited waiting space and the firstin firstout service discipline. Initially, the system is empty; then k customers are placed in the first queue. The service times of all the customers at all the queues are i.i.d. with a general distribution. We are interested in the time D(k, n) required for all k customers to complete service from all n queues. In particular, we investigate the limiting behavior of D(k, n) as n and/or k . There is a duality implying that D(k, n) is distributed the same as D(n , k) so that results for large n are equivalent to results for large k. A previous heavytraffic limit theorem implies that D(k, n) satisfies an invariance principle as n , converging after normalization to a functional of kdimensional Brownian motion. We use the subadditive ergodic theorem and a strong approximation to describe the limiting behavior of D(k n , n) where k n as n . The case of k n = xn corresponds to a hydrodyna...
Towards a characterisation of the behaviour of stochastic local search algorithms for SAT
 ARTIFICIAL INTELLIGENCE
, 1999
"... Stochastic local search (SLS) algorithms have been successfully applied to hard combinatorial problems from different domains. Due to their inherent randomness, the runtime behaviour of these algorithms is characterised by a random variable. The detailed knowledge of the runtime distribution provi ..."
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Cited by 44 (16 self)
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Stochastic local search (SLS) algorithms have been successfully applied to hard combinatorial problems from different domains. Due to their inherent randomness, the runtime behaviour of these algorithms is characterised by a random variable. The detailed knowledge of the runtime distribution provides important information about the behaviour of SLS algorithms. In this paper we investigate the empirical runtime distributions for Walksat, one of the most powerful SLS algorithms for the Propositional Satisfiability Problem (SAT). Using statistical analysis techniques, we show that on hard Random3SAT problems, Walksat's runtime behaviour can be characterised by exponential distributions. This characterisation can be generalised to various SLS algorithms for SAT and to encoded problems from other domains. This result also has a number of consequences which are of theoretical as well as practical interest. One of these is the fact that these algorithms can be easily parallelised such that optimal speedup is achieved for hard problem instances.
A New Ordering for Stochastic Majorization: Theory and Applications
 Advances in Applied Probability
, 1991
"... In this paper, we develop a unified approach for stochastic load balancing on various multiserver systems. We expand the four partial orderings defined in Marshall and Olkin, by defining a new ordering based on the set of functions that are symmetric, Lsubadditive and convex in each variable. This ..."
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Cited by 19 (7 self)
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In this paper, we develop a unified approach for stochastic load balancing on various multiserver systems. We expand the four partial orderings defined in Marshall and Olkin, by defining a new ordering based on the set of functions that are symmetric, Lsubadditive and convex in each variable. This new partial ordering is shown to be equivalent to the previous four orderings for comparing deterministic vectors but differ for random vectors. Sample path criteria and a probability enumeration method for the new stochastic ordering are established and the ordering is applied to various forkjoin queues, routing and scheduling problems. Our results generalize previous work and can be extended to multivariate stochastic majorization which includes tandem queues and queues with finite buffers. Keywords: load balancing, stochastic convexity, majorization ordering, routing, scheduling, forkjoin queues 0 1 Introduction Recently, stochastic comparison has received a great deal of attention. ...
Limiting exit location distributions in the stochastic exit problem
 SIAM J. Appl. Math
, 1997
"... Abstract. Consider a twodimensional continuoustime dynamical system, with an attracting fixed point. If the deterministic dynamics are perturbed by white noise (random perturbations) of strength, the system state will eventually leave the domain of attraction of. We analyse the case when, as, the ..."
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Cited by 18 (1 self)
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Abstract. Consider a twodimensional continuoustime dynamical system, with an attracting fixed point. If the deterministic dynamics are perturbed by white noise (random perturbations) of strength, the system state will eventually leave the domain of attraction of. We analyse the case when, as, the exit location on the boundary is increasingly concentrated near a saddle point of the deterministic dynamics. We show using formal methods that the asymptotic form of the exit location distribution on is generically nonGaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter, equal to the ratio of the stable and unstable eigenvalues of the linearized deterministic flow at. If then the exit location distribution is generically asymptotic as! " to a Weibull distribution with shape parameter #$ % , on the &'(*) +,. lengthscale near. If 0/1 it is generically asymptotic to a distribution on the &'(23+, lengthscale, whose moments we compute. Our treatment employs both matched asymptotic expansions and stochastic analysis. As a byproduct of our treatment, we clarify the limitations of the traditional Eyring formula for the weaknoise exit time asymptotics. Key words. Stochastic exit problem, large fluctuations, large deviations, WentzellFreidlin theory, exit location, saddle point avoidance, first passage time, matched asymptotic expansions, singular perturbation theory, stochastic analysis, AckerbergO’Malley resonance. AMS subject classifications. 60J60, 35B25, 34E20 1. Introduction. We
Importance Sampling For Monte Carlo Estimation Of Quantiles
 Publishing House of Saint Petersburg University
, 1996
"... This paper is concerned with applying importance sampling as a variance reduction tool for computing extreme quantiles. A central limit theorem is derived for each of four proposed importance sampling quantile estimators. Efficiency comparisons are provided in a certain asymptotic setting, using ide ..."
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Cited by 16 (0 self)
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This paper is concerned with applying importance sampling as a variance reduction tool for computing extreme quantiles. A central limit theorem is derived for each of four proposed importance sampling quantile estimators. Efficiency comparisons are provided in a certain asymptotic setting, using ideas from large deviation theory. Keywords: quantiles, importance sampling, large deviations. 1 Introduction Let X be a realvalued random variable with distribution function F (\Delta). For 0 ! p ! 1, the quantity F \Gamma1 (p) 4 = inffx : F (x) pg is called the p'th quantile of X. The quantile estimation problem is concerned with computing such quantities efficiently. Quantile estimation is of interest in many applications settings. For example, in computing tables of critical values associated with complicated hypothesis tests in which F cannot be computed analytically in closed form, it may be necessary to resort to Monte Carlo methodology as a means of calculating such critical v...