Perspectives on Traffic Modeling in Networks . . . . . Peter W. - PowerPoint PPT Presentation

. . Perspectives on Traffic Modeling in Networks . . . . . Peter W. Glynn Stanford University 2012 Stochastic Networks Conference, June 21, 2012 . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 1 / 21

25 years of stochastic network conferences since Tom Kurtz and I organized the first one in Madison, WI Madison, June, 1987 Minneapolis, March 1994 Edinburgh, August 1995 Madison, June 2000 Stanford, June 2002 Montreal, July 2004 Urbana-Champaign, June 2006 Paris, Ecole Normale Superieure, June 2008 Cambridge, Newton Institute, March 2010 MIT, June 2012 . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 2 / 21

These conferences have significantly impacted the field: Queueing theory in 1987: Early days of heavy-traffic theory, loss networks, etc Much work focused on very detailed "closed form" calculation for various variants of the single-server queue Transforms, recursive methods, etc were key tools . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 3 / 21

Today: Appreciation of multi-scale phenomena is central Stochastic geometry, random graphs, etc play a key role in many models Rich connections to probability, theoretical computer science, etc Modeling tends to be more ambitious → bigger questions to be answered Algorithm design as an endpoint; modeling as a vehicle to design algorithms More focus on qualitative insights; use of asymptotic regimes to study optimality issues . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 4 / 21

What has made these conferences successful? Brings together a diverse and active collection of researchers, cutting across multiple communities having an interest in such models Has been highly responsive to the changing applications landscape Circuit-switched networks ATM networks Wafer fabs Internet/TCP Call centers Mobile networks Sensor networks Peer-to-peer networks Data centers . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 5 / 21

Many important ideas and methods have been promoted at this conference Multi-class stability issues Connecting stability to fluid models and LPs Understanding heavy traffic in single-server and many-server settings State space collapse/snapshot principle Use of economic principles to guide the construction of distributed algorithms Use of randomized algorithms to lower computational/communications costs Large deviations, random graphs, etc . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 6 / 21

The rest of this talk... Perspectives on traffic modeling: A view or vista A mental view or outlook The relationship of aspects of a subject to each other and to a whole Subjective evaluation of relative significance; a point of view Something fun to discuss over lunch today... . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 7 / 21

What is a traffic model? The exogenous input to a network exogenous traffic Three main uses of models: Descriptive Predictive Prescriptive The choice of traffic model can impact conclusions in all 3 settings... . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 8 / 21

The Simplest Traffic Model: Poisson Arrivals Justified by "Superposition Theorem" Large number n of independent sources, none of which contributes a significant percentage of the overall traffic Describes traffic at the time scale of individual interarrival times Does NOT describe traffic at larger time scales ("Poisson breakdown phenomenon") Analog to superposition theorem establishes that for many-server queues in heavy-traffic, abandonment process is "doubly stochastic" Poisson . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 9 / 21

The Renewal Model: Other than Poisson case, difficult to conceive of a mechanistic explanation for renewal arrival epochs Correlated Arrival Stream: Often realistic as a means of incorporating "burstiness" in traffic Markov-modulated Poisson processes Long-range dependence . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 10 / 21

To the degree that we build "Markov-modulated burstiness" into our models, the "standard asymptotics" suppress the presence of the burstiness: If N ( · ) is the associated counting process, √ N ( t ) ≈ λ t + η tN ( 0 , 1 ) Fluid limits, diffusion limits, etc look qualitatively identical to what is seen in the renewal case Describes a very "statistically regular" world Explanation: The "decorrelation time" for the underlying Markov modulation is O ( 1 ) ; the Markov modulation describes bursts that are O ( 1 ) . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 11 / 21

Now, feed the traffic into a network in "heavy traffic": "Relaxation time/time to equilibrium" of order 1 / ( 1 − ρ ) 2 but bursts are O ( 1 ) in duration This asymptotic is relevant to a model with (very) short-lived bursts; same applies to large deviation paths to overflow in large buffer regime But there are problem settings where burstiness persists over long time scales... . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 12 / 21

Six Different Tuesdays 18 80 12 70 16 10 60 14 8 50 12 40 6 10 30 4 8 20 2 6 10 4 0 0 14 14.5 15 15.5 16 14 14.5 15 15.5 16 14 14.5 15 15.5 16 20 18 14 18 16 12 16 14 10 14 12 8 12 10 6 10 8 4 8 6 2 6 4 0 14 14.5 15 15.5 16 14 14.5 15 15.5 16 14 14.5 15 15.5 16 . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 13 / 21

In call center setting: Relaxation time/time to equilibrium for many-server systems is O ( 1 ) But bursts persist over time scales that are O ( 1 ) And time-of-day effects kick in over time scales that are O ( 1 ) . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 14 / 21

A Statistical Analysis (joint work with J. Hong, X. Zhang): We want a process that is "locally Poisson" Our choice: (∫ t ) N ( t ) = � X ( s ) ds N 0 where � N is a unit rate Poisson process independent of X , where X satisfies √ dX ( t ) = κ ( t )( µ ( t ) − X ( t )) dt + σ ( t ) X ( t ) dB ( t ) with κ ( · ) , µ ( · ) , and σ ( · ) piecewise constant Low parameter model Exhibits mean reversion Analytically tractable (affine process) . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 15 / 21

! 200 150 100 50 0 6 8 10 12 14 16 18 20 22 24 Clock TIme £ 0.18 0.17 0.16 0.15 0.14 0.13 6 8 10 12 14 16 18 20 22 24 Clock TIme « 2.2 2 1.8 1.6 1.4 6 8 10 12 14 16 18 20 22 24 Clock TIme Mean reversion time is of order over which time-of-day effects manifest (fit by "generalized method of moments") . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 16 / 21

At some level, the "dominant randomness" in this model comes from unpredictable bursts that are not captured in "standard asymptotics" Question: How does presence of long time-scale burstiness impact the qualitative conclusions we reach from traditional models? (e.g. "square-root staffing") . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 17 / 21

Another example to illustrate this point... Application of large deviations to traditional traffic models predicts very specific "conditional dynamics" for rare events (e.g. buffer overflow) Q ( t ) b t 1 t 2 . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 18 / 21

4 more paths to overflow 4 more paths to overflow 5000 5000 6000 4500 4500 5000 4000 4000 3500 3500 4000 3000 3000 2500 2500 3000 2000 2000 2000 1500 1500 1000 1000 1000 500 500 0 0 0 −350 −300 −250 −200 −150 −100 −50 0 50 −350 −300 −250 −200 −150 −100 −50 0 50 −400 −350 −300 −250 −200 −150 −100 −50 0 50 time before o.f. [10ms] time before o.f. [10ms] time before o.f. [10ms] 4 more paths to overflow 4500 6000 4000 5000 3500 3000 4000 2500 3000 2000 1500 2000 1000 1000 500 0 0 −350 −300 −250 −200 −150 −100 −50 0 50 −400 −350 −300 −250 −200 −150 −100 −50 0 50 time before o.f. [10ms] time before o.f. [10ms] • Traffic source: UNC Network Data Analysis Study Group (2003). • Buffer size = 4,000 packets (about 60 ms worth of traffic). . . . . . . Peter W. Glynn (Stanford University) Perspectives on Traffic Modeling June 11, 2012 19 / 21