Fluidbased Analysis of a Network of AQM Routers

Fluid-based Analysis of a Network of AQM Routers Supporting TCP Flows with an Application to RED Vishal Misra Wei-Bo Gong Don Towsley University of Massachusetts, Amherst MA 01003, USA Beats Zhi-li Zhang’s paper title length by one (hyphenated) word

Overview • • • motivation key idea modeling details validation with ns analysis sheds insights into RED conclusions

Motivation . . • current simulation technology, e. g. ns – appropriate for small networks 10 s - 100 s of network nodes 100 s - 1000 s IP flows – inflexible packet-level granularity • current analysis techniques – UDP flows over small networks – TCP flows over single link

Challenge Explore large scale systems – 100 s - 1000 s network elements – 10, 000 s - 100, 000 s of flows (TCP, UDP, NG) Our Belief Fluid-based techniques that abstract out protocol details are key to scalable network simulation/understanding Contribution of Paper First differential equation based fluid model to enable transient analysis of TCP/AQM networks developed

Key Idea • model traffic as fluid • describe behavior of flows and queues using Stochastic Differential Equations • obtain Ordinary Differential Equations by taking expectations of SDEs • solve resultant coupled ODEs numerically Differential equation abstraction: computationally highly efficient

Loss Model AQM Router B(t) Packet Drop/Mark p(t) Sender Receiver l(t) = B(t)*p(t) Round Trip Delay (t) Loss Rate as seen by Sender: l(t) = B(t-t)*p(t-t)

Start Simple: A Single Congested Router TCP flow i, prop. delay Ai • One bottlenecked AQM router – capacity {C (packets/sec) } – queue length q(t) – drop prob. p(t) • N TCP flows AQM router C, p – window sizes Wi (t) – round trip time Ri (t) = Ai+q (t)/C – throughputs Bi (t) = Wi (t)/Ri (t)

Adding RED to the model Marking probability p RED: Marking/dropping based on average queue length x (t) Marking probability profile has a discontinuity at tmax 1 discontinuity removed in gentle_ variant pmax tmin tmax 2 tmax Average queue length x x (t): smoothed, time averaged q (t) - x (t) t ->

System of Differential Equations All quantities are average values. Timeouts and slow start ignored ^ d. W Window Size: i dt = 1 ^ (q(t)) R i ^ - Additive increase Queue length: ^ dq ^ > 0]C = -1[q(t) dt Outgoing traffic ^ (t-t) ^ W i p(t-t) ^ (q(t-t)) ^ R ^ W i 2 i Mult. decrease Loss arrival rate ^ i(t) W + S ^ ^ R (q(t)) i Incoming traffic

System of Differential Equations (cont. ) ^ Average queue length: dx dt = ln (1 -a) d x(t) ^ - ln (1 -a) d Where a = averaging parameter of RED(wq) d = sampling interval ~ 1/C = dp dx ^ dx dt Where dp is obtained from the marking profile dx p Loss probability: ^ dp dt x q(t) ^

Stepping back: Where are we? W=Window size, R = RTT, q = queue length, p = marking probability d. Wi = f (p, R ) i = 1. . N 1 i dt dp dt = f 3(q) dq = f (W ) 2 i dt N+2 coupled equations solved numerically using MATLAB

Extension to Networked case: V AQM routers queuing delay = aggregate delay q(t) = SV q. V(t) loss probability = cumulative loss probability p(t) = 1 -PV(1 -p. V(t)) Other extensions to the model Timeouts: Leveraged work done in [PFTK Sigcomm 98] to model timeouts Aggregation of flows: Represent flows sharing the same route by a single equation

Validation scenario Topology • DE system programmed with RED AQM policy • equivalent system simulated in ns • transient queuing performance obtained • one way, ftp flows used as traffic sources Flow set 4 Flow set 1 RED router 2 Flow set 3 Flow set 5 5 sets of flows 2 RED routers Set 2 flows through both routers

Performance of SDE method Inst. queue length for router 2 DE method ns simulation Inst. queue length • queue capacities 5 Mb/s • load variation at t=75 and t=150 seconds • 200 flows simulated • DE solver captures transient performance • time taken for DE solver ~ 5 seconds on P 450 Time

Observations on RED • “Tuning” of RED is difficult - sensitive to packet sizes, load levels, round trip delay, etc. • discontinuity of drop function contributes to, but is not the only reason for oscillations. • RED uses variable sampling interval d. This could cause oscillations. Further Work Detailed Control Theoretic Analysis of TCP/RED available at http: //gaia. cs. umass. edu/papers. html

Conclusions • differential equation based model for evaluating. TCP/AQM networks • computation cost of DE method a fraction of the discrete event simulation cost • formal representation and analysis yields better understanding of RED/AQM

Future Directions • model short lived and non-responsive flows • demonstrate applicability to large networks • analyze theoretical model to rectify RED shortcomings • apply techniques to - other protocols, e. g. TCP-friendly protocols - other AQM mechanisms like Diffserv