1. 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

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

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

4. Challenge Explore large scale systems –100s - 1000s network elements –10,000s - 100,000s 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

5. 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

6. Loss Model Sender AQM Router Receiver Loss Rate as seen by Sender: (t  = B(t-  p(t-  Round Trip Delay (  ) B(t) p(t) Packet Drop/Mark (t  = B(t  p(t 

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

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

9. System of Differential Equations Window Size: dW i dt ^ = 1 ^ ^ R i (q(t)) Additive increase Wi2Wi2 ^ - Mult. decrease W i (t-  ) ^ R i (q(t-  )) ^ p(t-  ) Loss arrival rate ^ ^ -1 [q(t) > 0] C ^ Outgoing traffic Queue length: dq dt = ^ All quantities are average values. Timeouts and slow start ignored +  R i (q(t)) ^ W i (t) ^ Incoming traffic ^

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

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

12. Extension to Network Networked case: V AQM routers Other extensions to the model Timeouts: Leveraged work done in [PFTK Sigcomm98] to model timeouts Aggregation of flows: Represent flows sharing the same route by a single equation queuing delay = aggregate delay q(t) =  V q V (t) loss probability = cumulative loss probability p(t) = 1-  V (1-p V (t))

13. Validation scenario 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 1 Flow set 2 Flow set 3 Flow set 4 Flow set 5 RED router 1 RED router 2 Topology 5 sets of flows 2 RED routers Set 2 flows through both routers

14. Performance of SDE method 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 P450 DE method ns simulation Inst. queue length Time Inst. queue length for router 2

15. 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 . This could cause oscillations. Further Work Detailed Control Theoretic Analysis of TCP/RED available at http://gaia.cs.umass.edu/papers/papers.html

16. Conclusions differential equation based model for evaluatingTCP/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

17. 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