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39
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 168 (41 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.
Characterizing the Variability of Arrival Processes with Indices of Dispersion
 IEEE Journal on Selected Areas in Communications
, 1990
"... We propose to characterize the burstiness of packet arrival processes with indices of dispersion for intervals and for counts. These indices, which are functions of the variance of intervals and counts, are relatively straightforward to estimate and convey much more information than simpler indic ..."
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Cited by 62 (0 self)
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We propose to characterize the burstiness of packet arrival processes with indices of dispersion for intervals and for counts. These indices, which are functions of the variance of intervals and counts, are relatively straightforward to estimate and convey much more information than simpler indices, such as the coefficient of variation, that are often used to describe burstiness quantitatively.
Asymptotics for steadystate tail probabilities in structured Markov queueing models
 Commun. Statist.Stoch. Mod
, 1994
"... In this paper we establish asymptotics for the basic steadystate distributions in a large class of singleserver queues. We consider the waiting time, the workload (virtual waiting time) and the steadystate queue lengths at an arbitrary time, just before an arrival and just after a departure. We s ..."
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Cited by 38 (10 self)
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In this paper we establish asymptotics for the basic steadystate distributions in a large class of singleserver queues. We consider the waiting time, the workload (virtual waiting time) and the steadystate queue lengths at an arbitrary time, just before an arrival and just after a departure. We start by establishing asymptotics for steadystate distributions of Markov chains of M/GI/1 type. Then we treat steadystate distributions in the BMAP/GI/1 queue, which has a batch Markovian arrival process (BMAP). The BMAP is equivalent to the versatile Markovian point process or Neuts (N) process; it generalizes the Markovian arrival process (MAP) by allowing batch arrivals. The MAP includes the Markovmodulated Poisson process (MMPP), the phasetype renewal process (PH) and independent superpositions of these as special cases. We also establish asymptotics for steadystate distributions in the MAP/MSP/1 queue, which has a Markovian service process (MSP). The MSP is a MAP independent of the arrival process generating service completions during the time the server is busy. In great generality (but not always), the basic steadystate distributions have asymptotically exponential tails in all these models. When they do, the asymptotic parameters of the different distributions are closely related. 1.
Approximation Techniques for Computing Packet Loss in FiniteBuffered Voice Multiplexers
 IEEE Journal on Selected Areas in Communications
, 1991
"... In this paper we examine three different approximation techniques for modeling packet loss in finitebuffer voice multiplexers. The performance models studied differ primarily in the manner in which the superposition of the voice sources (i.e., the arrival process) is modeled. The first approach ..."
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Cited by 37 (5 self)
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In this paper we examine three different approximation techniques for modeling packet loss in finitebuffer voice multiplexers. The performance models studied differ primarily in the manner in which the superposition of the voice sources (i.e., the arrival process) is modeled. The first approach models the superimposed voice sources as a renewal process and performance calculations are based only on the first two moments of the renewal process. The second approach is based on modeling the superimposed voice sources as a Markov Modulated Poisson Process (MMPP). Our choice of parameters for the MMPP attempts to capture aspects of the arrival process in an alternate, more intuitive, manner than previously proposed approaches for determining the MMPP parameters and is shown to compute loss more accurately. Finally, we also evaluate a fluid flow approximation for computing packet loss. For all three approaches, we consider as a unifying example, the case of multiplexing voice sou...
Parallel rollout for online solution of partially observable Markov decision processes,” Discrete Event Dynamic Systems: Theory and Application
, 2004
"... Abstract We propose a novel approach, called parallel rollout, to solving (partially observable) Markov decision processes. Our approach generalizes the rollout algorithm of Bertsekas and Castanon (1999) by rolling out a set of multiple heuristic policies rather than a single policy. In particula ..."
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Cited by 16 (6 self)
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Abstract We propose a novel approach, called parallel rollout, to solving (partially observable) Markov decision processes. Our approach generalizes the rollout algorithm of Bertsekas and Castanon (1999) by rolling out a set of multiple heuristic policies rather than a single policy. In particular, the parallel rollout approach aims at the class of problems where we have multiple heuristic policies available such that each policy performs nearoptimal for a different set of system paths. Parallel rollout automatically combines the given multiple policies to create a new policy that adapts to the different system paths and improves the performance of each policy in the set. We formally prove this claim for two criteria: total expected reward and infinite horizon discounted reward. The parallel rollout approach also resolves the key issue of selecting which policy to roll out among multiple heuristic policies whose performances cannot be predicted in advance. We present two example problems to illustrate the effectiveness of the parallel rollout approach: a buffer management problem and a multiclass scheduling problem. I.
Heavytraffic asymptotic expansions for the asymptotic decay rates
 in the BMAP/G/1 queue. Stochastic Models
, 1994
"... versatile Markovian point process, tail probabilities in queues, asymptotic decay rate, PerronFrobenius eigenvalue, asymptotic expansion, caudal characteristic curve, heavy traffic In great generality, the basic steadystate distributions in the BMAP / G /1 queue have asymptotically exponential tai ..."
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Cited by 15 (10 self)
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versatile Markovian point process, tail probabilities in queues, asymptotic decay rate, PerronFrobenius eigenvalue, asymptotic expansion, caudal characteristic curve, heavy traffic In great generality, the basic steadystate distributions in the BMAP / G /1 queue have asymptotically exponential tails. Here we develop asymptotic expansions for the asymptotic decay rates of these tail probabilities in powers of one minus the traffic intensity. The first term coincides with the decay rate of the exponential distribution arising in the standard heavytraffic limit. The coefficients of these heavytraffic expansions depend on the moments of the servicetime distribution and the derivatives of the PerronFrobenius eigenvalue δ(z) of the BMAP matrix generating function D(z) at z = 1. We give recursive formulas for the derivatives δ (k) ( 1). The asymptotic expansions provide the basis for efficiently computing the asymptotic decay rates as functions of the traffic intensity, i.e., the caudal characteristic curves. The asymptotic expansions also reveal what features of the model the asymptotic decay rates primarily depend upon. In particular, δ(z) coincides with the limiting timeaverage of the factorial cumulant generating function (the logarithm of the generating function) of the arrival counting process, and the derivatives δ (k) ( 1) coincide with the asymptotic factorial cumulants of the arrival counting process. This insight is important for admission control schemes in multiservice networks based in part on asymptotic decay rates. The interpretation helps identify appropriate statistics to compute from network traffic data in order to predict performance. 1.
A MultiDimensional Martingale for Markov Additive Processes and its Applications
, 1998
"... We establish new multidimensional martingales for Markov additive processes and certain modifications of such processes (e.g., such processes with reflecting barriers). These results generalize corresponding one dimensional martingale results for L'evy processes. This martingale is then applied to v ..."
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Cited by 15 (8 self)
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We establish new multidimensional martingales for Markov additive processes and certain modifications of such processes (e.g., such processes with reflecting barriers). These results generalize corresponding one dimensional martingale results for L'evy processes. This martingale is then applied to various storage processes, queues and Brownian motion models. AMS 1991 Subject Classification 60J30; 60K25 Keywords L' evy process, EOQ model, Markov modulation, multivariate martingale, reflected Brownian motion, storage process Department of Mathematical Statistics; University of Lund; Box 118; S221 00 Lund; Sweden (asmus@maths.lth.se) y Department of Statistics; The Hebrew University of Jerusalem; Mount Scopus, Jerusalem 91905; Israel (mskella@olive.mscc.huji.ac.il) 1 Introduction With X t being a L'evy process with exponent ' (see Section 2), e ffX t \Gammat'(ff) is the well known Wald martingale. From a recent paper [14], it follows that under certain conditions '(ff) Z t 0...
The Extended Alternating Fractal Renewal Process for Modeling Traffic in HighSpeed Communication Networks
, 2001
"... Extensive studies indicate that traffic in highspeed communication networks exhibits longrange dependence (LRD) and impulsiveness, thus posing new challenges in network engineering. While many models have recently appeared for capturing the traffic LRD, far less models exist that account for impul ..."
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Cited by 12 (9 self)
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Extensive studies indicate that traffic in highspeed communication networks exhibits longrange dependence (LRD) and impulsiveness, thus posing new challenges in network engineering. While many models have recently appeared for capturing the traffic LRD, far less models exist that account for impulsiveness as well as LRD. One of the few existing constructive models for network traffic is the celebrated On/Off model, or Alternating Fractal Renewal Process (AFRP). However, while the AFRP results in aggregated traffic with LRD, it fails to capture impulsiveness, yielding traffic with Gaussian marginal distribution. A new constructive model, namely the Extended AFRP (EAFRP), is proposed here, which overcomes the limitations of the AFRP model. We show that, for both singleuser and aggregated traffic, it results in impulsiveness and longrange dependence, the LRD being defined here in a generalized sense. We provide queueing analysis of the proposed model, which clearly demonstrates the implications of the impulsiveness in traffic engineering, and validate all theoretical findings based on real traffic data.
The Transient BMAP/G/1 Queue
, 2000
"... We derive the twodimensional transforms of the transient workload and queuelength distributions in the singleserver queue with general service times and a batch Markovian arrival process (BMAP). This arrival process includes the familiar phasetype renewal process and the Markov modulated Poisson ..."
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Cited by 9 (2 self)
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We derive the twodimensional transforms of the transient workload and queuelength distributions in the singleserver queue with general service times and a batch Markovian arrival process (BMAP). This arrival process includes the familiar phasetype renewal process and the Markov modulated Poisson process as special cases, and allows correlated interarrival times and batch sizes. Numerical results are obtained via twodimensional transform inversion algorithms based on the Fourierseries method. From the numerical examples we see that predictions of system performance based on transient and stationary performance measures can be quite different. n
Calculation of the steady state waiting time distribution in GI/PH/c and MAP/PH/c queues
, 2000
"... This paper is concerned with the explicit evaluation of the steadystate distribution of the waiting time W in a manyserver queue with (possibly heterogeneous) servers each having a phasetype service time distribution. Consider first the most classical case GI=PH het =c of renewal arrivals (we ..."
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Cited by 7 (0 self)
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This paper is concerned with the explicit evaluation of the steadystate distribution of the waiting time W in a manyserver queue with (possibly heterogeneous) servers each having a phasetype service time distribution. Consider first the most classical case GI=PH het =c of renewal arrivals (we later generalize to MAP=PH het =c, i.e. a Markovian arrival process as in Neuts [15]). Denote the interarrival distribution by H and the service time distribution of server i by F i . It is assumed that any F i is phasetype, say with representation (fi i ; S i ) where fi i is a m i dimensional row vector and S i is a m i \Theta m i matrix. 2 S. Asmussen, J.R. Mller / Calculation of the waiting time That is, the density is \Gammafi i e S i x S i e (throughout, e denotes the column vector of ones with the appropriate dimension depending on the context). In the homogeneous case F 1 = : : : = F c = F , m i = m, we write GI=PH=c. The service time is FCFS in the sense that the customers form a single line in the order of arrival and joins the next server to become available. In the homogeneous case, this suffices for a complete model description, whereas in the heterogenous case, we also need to describe the rule for the allocation of a server to a customer arriving in a notallbusy period, that is, a period in which some or all servers are idle. We will consider two situations, one where any of s ! c idle servers is selected with equal probability 1=s, and one where the servers have priorities in the sense that the customer selects server i before server j when i ! j (other Markovian rules can easily be treated with small modifications of the analysis given for these two cases). The waiting time of a customer is the time from his arrival until service starts; it can b...