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64
Synchronization and linearity: an algebra for discrete event systems
, 2001
"... The first edition of this book was published in 1992 by Wiley (ISBN 0 471 93609 X). Since this book is now out of print, and to answer the request of several colleagues, the authors have decided to make it available freely on the Web, while retaining the copyright, for the benefit of the scientific ..."
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Cited by 249 (10 self)
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The first edition of this book was published in 1992 by Wiley (ISBN 0 471 93609 X). Since this book is now out of print, and to answer the request of several colleagues, the authors have decided to make it available freely on the Web, while retaining the copyright, for the benefit of the scientific community. Copyright Statement This electronic document is in PDF format. One needs Acrobat Reader (available freely for most platforms from the Adobe web site) to benefit from the full interactive machinery: using the package hyperref by Sebastian Rahtz, the table of contents and all LATEX crossreferences are automatically converted into clickable hyperlinks, bookmarks are generated automatically, etc.. So, do not hesitate to click on references to equation or section numbers, on items of thetableofcontents and of the index, etc.. One may freely use and print this document for one’s own purpose or even distribute it freely, but not commercially, provided it is distributed in its entirety and without modifications, including this preface and copyright statement. Any use of thecontents should be acknowledged according to the standard scientific practice. The
Large Deviations and Overflow Probabilities for the General SingleServer Queue, With Applications
, 1994
"... We consider from a thermodynamic viewpoint queueing systems where the workload process is assumed to have an associated large deviation principle with arbitrary scaling: there exist increasing scaling functions (a t ; v t ; t 2 R+ ) and a rate function I such that if (W t ; t 2 R+ ) denotes the wo ..."
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Cited by 180 (18 self)
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We consider from a thermodynamic viewpoint queueing systems where the workload process is assumed to have an associated large deviation principle with arbitrary scaling: there exist increasing scaling functions (a t ; v t ; t 2 R+ ) and a rate function I such that if (W t ; t 2 R+ ) denotes the workload process, then lim t!1 v \Gamma1 t log P (W t =a t ? w) = \GammaI (w) on the continuity set of I . In the case that a t = v t = t it has been argued heuristically, and recently proved in a fairly general context (for discrete time models) by Glynn and Whitt [8], that the queuelength distribution (that is, the distribution of supremum of the workload process Q = sup t0 W t ) decays exponentially: P (Q ? b) ¸ e \Gammaffib and the decay rate ffi is directly related to the rate function I . We establish conditions for a more general result to hold, where the scaling functions are not necessarily linear in t: we find that the queuelength distribution has an exponential tail only if l...
A Stochastic Model of TCP/IP with Stationary Random Losses
 ACM SIGCOMM
, 2000
"... In this paper, we present a model for TCP/IP congestion control mechanism. The rate at which data is transmitted increases linearly in time until a packet loss is detected. At this point, the transmission rate is divided by a constant factor. Losses are generated by some exogenous random process whi ..."
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Cited by 165 (40 self)
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In this paper, we present a model for TCP/IP congestion control mechanism. The rate at which data is transmitted increases linearly in time until a packet loss is detected. At this point, the transmission rate is divided by a constant factor. Losses are generated by some exogenous random process which is assumed to be stationary ergodic. This allows us to account for any correlation and any distribution of interloss times. We obtain an explicit expression for the throughput of a TCP connection and bounds on the throughput when there is a limit on the window size. In addition, we study the effect of the Timeout mechanism on the throughput. A set of experiments is conducted over the real Internet and a comparison is provided with other models that make simple assumptions on the interloss time process. The comparison shows that our model approximates well the throughput of TCP for many distributions of interloss times.
Stochastically recursive sequences and their generalizations
 Siberian Adv. Math
, 1992
"... The paper deals with the stochastically recursive sequences { X ( n) } defined as the solutions of equations X ( n + 1) = f ( X ( n) , ξn) (where ξn is a given random sequence), and with random sequences of a more general nature, named recursive chains. For those the theorems of existence, ergodici ..."
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Cited by 41 (11 self)
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The paper deals with the stochastically recursive sequences { X ( n) } defined as the solutions of equations X ( n + 1) = f ( X ( n) , ξn) (where ξn is a given random sequence), and with random sequences of a more general nature, named recursive chains. For those the theorems of existence, ergodicity, stability are established, the stationary majorants are constructed. Continuoustime processes associated with ones studied here are considered as well. Key words and phrases: stochastically recursive sequence; recursive chain; generalized Markov chain; renovating event; couplingconvergence; ergodicity; stability; rate of convergence; stationary majorants; boundedness in probability; processes admitting embedded stochastically recursive sequences. CHAPTER 1.
Economies of Scale in Queues With Sources Having PowerLaw Large Deviation Scalings.
, 1995
"... We analyse the queue Q L at a multiplexer with L sources which may display longrange dependence. This includes, for example, sources modelled by fractional Brownian Motion (fBM). The workload processes W due to each source are assumed to have large deviation properties of the form P [W t =a(t) ? ..."
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Cited by 39 (10 self)
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We analyse the queue Q L at a multiplexer with L sources which may display longrange dependence. This includes, for example, sources modelled by fractional Brownian Motion (fBM). The workload processes W due to each source are assumed to have large deviation properties of the form P [W t =a(t) ? x] ß e \Gammav(t)K(x) for appropriate scaling functions a and v, and ratefunction K. Under very general conditions, lim L!1 L \Gamma1 log P [Q L ? Lb] = \GammaI (b) provided the offered load is held constant, where the shape function I is expressed in terms of the cumulant generating functions of the input traffic. For powerlaw scalings v(t) = t v , a(t) = t a (such as occur in fBM) we analyse the asymptotics of the shape function: lim b!1 b \Gammau=a i I(b) \Gamma ffi b v=a j = u for some exponent u and constant depending on the sources. This demonstrates the economies of scale available through the multiplexing of a large number of such sources, by comparison with ...
AN INTRODUCTION TO NUMERICAL TRANSFORM INVERSION AND ITS APPLICATION TO PROBABILITY MODELS
, 1999
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Perfect Simulation and Backward Coupling
 Comm. Statist. Stochastic Models
"... Algorithms for perfect or exact simulation of random samples from the invariant measure of a Markov chain have received considerable recent attention following the introduction of the "couplingfromthepast" (CFTP) technique of Propp and Wilson. Here we place such algorithms in the context of backw ..."
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Cited by 30 (2 self)
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Algorithms for perfect or exact simulation of random samples from the invariant measure of a Markov chain have received considerable recent attention following the introduction of the "couplingfromthepast" (CFTP) technique of Propp and Wilson. Here we place such algorithms in the context of backward coupling of stochastically recursive sequences. We show that although general backward couplings can be constructed for chains with finite mean forward coupling times, and can even be thought of as extending the classical "Loynes schemes" from queueing theory, successful "vertical" CFTP algorithms such as those of Propp and Wilson can be constructed if and only if the chain is uniformly geometric ergodic. We also relate the convergence moments for backward coupling methods to those of forward coupling times: the former typically lose at most one moment compared to the latter. Work supported in part by NSF Grant DMS9504561 and by CRDF Grant RM1226 y Postal Address: Institute of Math...
Heavytraffic limits for the G/H∗ 2 /n/m queue
 Math. Oper. Res
, 2005
"... We establish heavytraffic stochasticprocess limits for queuelength, waitingtime and overflow stochastic processes in a class of G/GI/n/m queueing models with n servers and m extra waiting spaces. We let the arrival process be general, only requiring that it satisfy a functional central limit th ..."
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Cited by 28 (12 self)
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We establish heavytraffic stochasticprocess limits for queuelength, waitingtime and overflow stochastic processes in a class of G/GI/n/m queueing models with n servers and m extra waiting spaces. We let the arrival process be general, only requiring that it satisfy a functional central limit theorem. In order to capture the impact of the servicetime distribution beyond its mean within a Markovian framework, we consider a special class of servicetime distributions, denoted by H ∗ 2, which are mixtures of an exponential distribution with probability p and a unit point mass at 0 with probability 1 − p. These servicetime distributions exhibit relatively high variability, having squared coefficients of variation greater than or equal to one. As in Halfin and Whitt (1981), Puhalskii and Reiman (2000) and Garnett, Mandelbaum and Reiman (2000), we consider a sequence of queueing models indexed by the number of servers, n, and let n tend to infinity along with the traffic intensities ρn so that √ n(1 − ρn) → β for − ∞ < β < ∞. To treat finite waiting rooms, we let mn / √ n → κ for 0 < κ ≤ ∞. With the special H ∗ 2 servicetime distribution, the limit processes are onedimensional Markov processes, behaving like diffusion processes with different drift and diffusion functions in two different regions, above and below zero. We also establish a limit for the G/M/n/m + M model, having exponential customer abandonments.
On the Saturation Rule for the Stability of Queues
 J. Appl. Prob
, 1998
"... This paper focuses on the stability of open queueing systems under stationary ergodic assumptions. It defines a set of conditions, the monotone separable framework, ensuring that the stability region is given by the following saturation rule: `saturate' the queues which are fed by the external arriv ..."
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Cited by 28 (9 self)
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This paper focuses on the stability of open queueing systems under stationary ergodic assumptions. It defines a set of conditions, the monotone separable framework, ensuring that the stability region is given by the following saturation rule: `saturate' the queues which are fed by the external arrival stream; look at the `intensity' of the departure stream in this saturated system; then stability holds whenever the intensity of the arrival process, say satisfies the condition ! , whereas the network is unstable if ? .
Products of Irreducible Random Matrices in the (max,+) Algebra  Part I
 Adv. Appl. Prob
, 1993
"... The study of networks with synchronization, and more particularly of Stochastic Event Graphs has raised an interest for products of random matrices in the (Max; +) algebra. We consider a general model of type "x(n + 1) = A(n)x(n)" where x(n+ 1) and x(n) are IR J valued vectors and A(n) is an irredu ..."
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Cited by 25 (5 self)
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The study of networks with synchronization, and more particularly of Stochastic Event Graphs has raised an interest for products of random matrices in the (Max; +) algebra. We consider a general model of type "x(n + 1) = A(n)x(n)" where x(n+ 1) and x(n) are IR J valued vectors and A(n) is an irreducible random matrix of size J \Theta J . The exogeneous sequence fA(n); n 2 INg is i.i.d or more generally stationary and ergodic. The model is completely solved in a sense, as we give necessary and sufficient conditions for the system to couple in finite time to a unique stationary regime. Moreover, we characterize the systems which converge to a unique regime with "\Gammacoupling only. They appear to be limit cases of the former ones.