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17
WideArea Traffic: The Failure of Poisson Modeling
 IEEE/ACM TRANSACTIONS ON NETWORKING
, 1995
"... Network arrivals are often modeled as Poisson processes for analytic simplicity, even though a number of traffic studies have shown that packet interarrivals are not exponentially distributed. We evaluate 24 widearea traces, investigating a number of widearea TCP arrival processes (session and con ..."
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Cited by 1405 (21 self)
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Network arrivals are often modeled as Poisson processes for analytic simplicity, even though a number of traffic studies have shown that packet interarrivals are not exponentially distributed. We evaluate 24 widearea traces, investigating a number of widearea TCP arrival processes (session and connection arrivals, FTP data connection arrivals within FTP sessions, and TELNET packet arrivals) to determine the error introduced by modeling them using Poisson processes. We find that userinitiated TCP session arrivals, such as remotelogin and filetransfer, are wellmodeled as Poisson processes with fixed hourly rates, but that other connection arrivals deviate considerably from Poisson; that modeling TELNET packet interarrivals as exponential grievously underestimates the burstiness of TELNET traffic, but using the empirical Tcplib [Danzig et al, 1992] interarrivals preserves burstiness over many time scales; and that FTP data connection arrivals within FTP sessions come bunched into “connection bursts,” the largest of which are so large that they completely dominate FTP data traffic. Finally, we offer some results regarding how our findings relate to the possible selfsimilarity of widearea traffic.
Task Assignment in a Distributed System: Improving Performance by Unbalancing Load
, 1997
"... We consider the problem of task assignment in a distributed system (such as a distributed Web server) in which task sizes are drawn from a heavytailed distribution. Many task assignment algorithms are based on the heuristic that balancing the load at the server hosts will result in optimal perfo ..."
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Cited by 77 (6 self)
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We consider the problem of task assignment in a distributed system (such as a distributed Web server) in which task sizes are drawn from a heavytailed distribution. Many task assignment algorithms are based on the heuristic that balancing the load at the server hosts will result in optimal performance. We show this conventional wisdom is less true when the task size distribution is heavytailed (as is the case for Web file sizes). We introduce a new task assignment policy, called Size Interval Task Assignment with Variable Load (SITAV). SITAV purposely operates the server hosts at different loads, and directs smaller tasks to the lighterloaded hosts.
Asymptotic results for multiplexing subexponential onoff processes
 Advances in Applied Probability
, 1998
"... Consider an aggregate arrival process AN obtained by multiplexing N OnOff processes with exponential Off periods of rate λ and subexponential On periods τon. As N goes to infinity, with λN → Λ, AN approaches an M/G/ ∞ type process. Both for finite and infinite N, we obtain the asymptotic characteri ..."
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Cited by 70 (19 self)
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Consider an aggregate arrival process AN obtained by multiplexing N OnOff processes with exponential Off periods of rate λ and subexponential On periods τon. As N goes to infinity, with λN → Λ, AN approaches an M/G/ ∞ type process. Both for finite and infinite N, we obtain the asymptotic characterization of the arrival process activity period. Using these results we investigate a fluid queue with the limiting M/G/ ∞ arrival process A ∞ t and capacity c. When On periods are regularly varying (with noninteger exponent), we derive a precise asymptotic behavior of the queue length random variable QP t observed at the beginning of the arrival process activity periods P[Q P t +ρ−c> x] ∼ Λr P[τ c−ρ x/(r+ρ−c) on> u]du x → ∞, where ρ = EA ∞ t < c; r (c ≤ r) is the rate at which the fluid is arriving during an On period. The asymptotic (time average) queuedistributionlower boundis obtained undermoregeneral assumptions on On periods than regular variation. In addition, we analyze a queueing system in which one OnOff process, whose On period belongs to a subclass of subexponential distributions, is multiplexed with independent exponential processes with aggregate expected rate Eet. This system is shown to be asymptotically equivalent to the same queueing system with the exponential arrival processes being replaced by their total mean value Eet.
Waitingtime tail probabilities in queues with longtail servicetime distributions
 QUEUEING SYSTEMS
, 1994
"... We consider the standard GI/G/1 queue with unlimited waiting room and the firstin firstout service discipline. We investigate the steadystate waitingtime tail probabilities P(W> x) when the servicetime distribution has a longtail distribution, i.e., when the servicetime distribution fails to ..."
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Cited by 55 (21 self)
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We consider the standard GI/G/1 queue with unlimited waiting room and the firstin firstout service discipline. We investigate the steadystate waitingtime tail probabilities P(W> x) when the servicetime distribution has a longtail distribution, i.e., when the servicetime distribution fails to have a finite moment generating function. We have developed algorithms for computing the waitingtime distribution by Laplace transform inversion when the Laplace transforms of the interarrivaltime and servicetime distributions are known. One algorithm, exploiting Pollaczek’s classical contourintegral representation of the Laplace transform, does not require that either of these transforms be rational. To facilitate such calculations, we introduce a convenient twoparameter family of longtail distributions on the positive half line with explicit Laplace transforms. This family is a Pareto mixture of exponential (PME) distributions. These PME distributions have monotone densities and Paretolike tails, i.e., are of order x − r for r> 1. We use this family of longtail distributions to investigate the quality of approximations based on asymptotics for P(W> x) as x → ∞. We show that the asymptotic approximations with these longtail servicetime distributions can be remarkably inaccurate for typical x values of interest. We also derive multiterm asymptotic expansions for the waitingtime tail probabilities in the M/G/1 queue. Even three terms of this expansion can be remarkably inaccurate for typical x values of interest. Thus, we evidently must rely on numerical algorithms for determining the waitingtime tail probabilities in this case. When working with servicetime data, we suggest using empirical Laplace transforms.
Subexponential Asymptotics of a MarkovModulated Random Walk with Queueing Applications
, 1996
"... Let f(Xn; Jn)g be a stationary Markovmodulated random walk on R\Theta E (E finite), defined by its probability transition matrix measure F = fF ij g; F ij (B) = P[X 1 2 B; J 1 = jjJ 0 = i]; B 2 B(R); i; j 2 E. If F ij ([x; 1))=(1 \Gamma H(x)) ! W ij 2 [0; 1), as x! 1, for some longtailed distribut ..."
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Cited by 45 (15 self)
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Let f(Xn; Jn)g be a stationary Markovmodulated random walk on R\Theta E (E finite), defined by its probability transition matrix measure F = fF ij g; F ij (B) = P[X 1 2 B; J 1 = jjJ 0 = i]; B 2 B(R); i; j 2 E. If F ij ([x; 1))=(1 \Gamma H(x)) ! W ij 2 [0; 1), as x! 1, for some longtailed distribution function H, then the ascending ladder heights matrix distribution G+ (x) (right WienerHopf factor) has longtailed asymptotics. If EXn! 0, at least one W ij? 0, and H(x) is a subexponential distribution function, then the asymptotic behavior of the supremum of this random walk is the same as in the i.i.d. case, and it is given by P \Theta sup n0 Sn? x
On the M/G/1 Queue with HeavyTailed Service Time Distributions
 IEEE Journal on Selected Areas in Communications
, 1997
"... In present teletraffic applications of queueing theory service time distributions B(t) with a heavy tail occur, i.e. 1 \Gamma B(t) t \Gamma for t !1 with ? 1. For such service time distributions not much explicit information is available concerning the tail probabilities of the inherent waiting ..."
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Cited by 29 (8 self)
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In present teletraffic applications of queueing theory service time distributions B(t) with a heavy tail occur, i.e. 1 \Gamma B(t) t \Gamma for t !1 with ? 1. For such service time distributions not much explicit information is available concerning the tail probabilities of the inherent waiting time distribution W (t). In the present study the waiting time distribution is studied for a stable M=G=1 model for a class of service time distributions with 1 ! ! 2. For = 1 1 2 the explicit expression for Q(t) is derived. For rational with 1 ! ! 2, an asymptotic series for the tail probabilities of W (t) is derived. 1991 Mathematics Subject Classification: 90B22, 60K25 Keywords and Phrases: M=G=1 model, stable, service time distribution, heavytails, waiting time distributions, asymptotic series for tail probabilities. Note: work carried out under project LRD. 1. Introduction In classical applications of teletraffic theory the service time distributions in queueing models are freq...
Subexponential loss rates in a GI/GI/1 queue with applications, Queueing Systems 33
, 1999
"... Consider a single server queue with i.i.d. arrival and service processes, {A, An, n � 0} and {C, Cn, n � 0}, respectively, and a finite buffer B. The queue content process {Q B n, n � 0} is recursively defined as Q B n+1 = min((Q B n + An+1 − Cn+1) +, B), q + = max(0, q). When E(A − C) < 0, and A ha ..."
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Cited by 17 (4 self)
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Consider a single server queue with i.i.d. arrival and service processes, {A, An, n � 0} and {C, Cn, n � 0}, respectively, and a finite buffer B. The queue content process {Q B n, n � 0} is recursively defined as Q B n+1 = min((Q B n + An+1 − Cn+1) +, B), q + = max(0, q). When E(A − C) < 0, and A has a subexponential distribution, we show that the stationary expected loss rate for this queue E(Q B n + An+1 − Cn+1 − B) + has the following explicit asymptotic characterization: E(Q B n + An+1 − Cn+1 − B) + ∼ E(A − B) + as B →∞, independently of the server process Cn. For a fluid queue with capacity c, M/G/ ∞ arrival process At, characterized by intermediately regularly varying on periods τ on, which arrive with Poisson rate Λ, the average loss rate λ B loss satisfies λ B loss ∼ Λ E(τ on η − B) + as B →∞, where η = r + ρ − c, ρ = EAt <c; r (c � r) is the rate at which the fluid is arriving during an on period. Accuracy of the above asymptotic relations is verified with extensive numerical and simulation experiments. These explicit formulas have potential application in designing communication networks that will carry traffic with longtailed characteristics, e.g., Internet data services.
Performance Analysis of ATM Switches with SelfSimilar Input Traffic
 Systems and Computer Engineering J
, 1996
"... In this paper, the performance of ATM switches with selfsimilar input traffic is analyzed. The ATM switch is a nonblocking, outputbuffered switch, and input traffic is approximated by a PoissonZeta process, which is an asymptotically secondorder selfsimilar process. The upper and lower bounds ..."
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Cited by 8 (0 self)
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In this paper, the performance of ATM switches with selfsimilar input traffic is analyzed. The ATM switch is a nonblocking, outputbuffered switch, and input traffic is approximated by a PoissonZeta process, which is an asymptotically secondorder selfsimilar process. The upper and lower bounds of the buffer overflow probability of the switch are obtained by stochastically bounding the number of cells arriving in any interval of time. These upper and lower bounds are very tight and give reliable estimates of the buffer overflow probability. The selfsimilar behavior of traffic has serious implications on the buffer overflow probability of the switches. In contrast to typical ATM traffic models currently considered in the literature, the buffer overflow probability decreases non exponentially with buffer size. This work was supported in part by NSERC Operating grant No. A8450 2 1. Introduction Performance analysis of ATM switches has been extensively studied. From previous studi...
A network multiplexer with multiple time scale and subexponential arrivals,” in: Stochastic Networks: Stability and Rare Events
, 1996
"... ABSTRACT Realtime traffic processes, such as video, exhibit multiple time scale characteristics, as well as subexponential first and second order statistics. We present recent results on evaluating the asymptotic behavior of a network multiplexer that is loaded with such processes. 1 ..."
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Cited by 8 (0 self)
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ABSTRACT Realtime traffic processes, such as video, exhibit multiple time scale characteristics, as well as subexponential first and second order statistics. We present recent results on evaluating the asymptotic behavior of a network multiplexer that is loaded with such processes. 1
To Queue or Not to Queue: When Queueing is Better Than Timesharing in a Distributed System
, 1997
"... We examine the question of whether to employ the firstcomefirstserved (FCFS) discipline or the processorsharing (PS) discipline at the hosts in a distributed server system. We are interested in the case in which service times are drawn from a heavytailed distribution, and so have very high vari ..."
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Cited by 5 (2 self)
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We examine the question of whether to employ the firstcomefirstserved (FCFS) discipline or the processorsharing (PS) discipline at the hosts in a distributed server system. We are interested in the case in which service times are drawn from a heavytailed distribution, and so have very high variability. Traditional wisdom when task sizes are highly variable would prefer the PS discipline, because it allows small tasks to avoid being delayed behind large tasks in a queue. However, we show that system performance can actually be significantly better under FCFS queueing, if each task is assigned to a host based on the task's size. By task assignment, we mean an algorithm that inspects incoming tasks and assigns them to hosts for service. The particular task assignment policy we propose is called SITAE: Size Interval Task Assignment with Equal Load. Surprisingly, under SITAE, FCFS queueing typically outperforms the PS discipline by a factor of about two, as measured by mean waiting t...