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

2. Overview motivation key idea modeling details experimental 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 technology –UDP flows over small networks –TCP flows over single link...

4. Challenge Need to explore systems with a parameter space of: –100s - 1000s network elements –10,000s - 100,000s of flows (TCP, UDP, NG) Belief Fluid based simulation techniques which abstract out and exploit topologies/protocols are key for scalability 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 the SDEs solve the resultant coupled ODEs numerically Differential equation abstraction: computationally highly efficient

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

7. A Single Congested Router TCP flow i AQM router C, p 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) One bottlenecked AQM router –capacity {C (packets/sec) } –queue length q(t) –discard prob. p(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 +  R i (q(t)) ^ W i (t) ^ Incoming traffic Queue length: dq dt = ^ All quantities are average values. Timeouts and slow start ignored

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

11. N+2 coupled equations N flows W i (t) = Window size of flow i R i (t) = RTT of flow i p(t) = Drop probability q(t) = queue length Equations solved numerically using MATLAB dp/dt = f 3 (q)^^ dq/dt =f 2 (W i )^ ^ dW i /dt = f 1 (p,R i, W i ) i =1..N ^^^ ^

12. Extension to Network Networked case: V congested 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. Experimental scenario DE system programmed with RED AQM policy equivalent system programmed in ns transient queuing performance obtained one way, ftp flows used as traffic model 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 capacity 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 Queue length Time

15. Observations on RED RED behavior changes with change in network conditions (load level, packet size, link bandwidth). “Tuning” of RED is difficult, queue length frequently oscillates deterministically. discontinuity of drop function contributes to, but is not the only reason for oscillations. RED uses a variable  (sampling interval). This variable sampling could cause oscillations. averaging mechanism of RED is counter productive from stability viewpoint: introduces a further delay to the existing round trip delay.

16. Future Direction model short lived and non-responsive flows demonstrate applicability to large networks analyze theoretical model to rectify RED shortcomings apply techniques to other “TCP-like” protocols, e.g. equation based TCP-friendly protocols

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

18. Background Sender Loss Probability p i Traditional, Source centric loss model Sender Loss Indications arrival rate New, Network centric loss model New loss model proposed in “Stochastic Differential Equation Modeling and Analysis of TCP Window size behavior”, Misra et. al. Performance 99. Loss model enabled casting of TCP behavior as a Stochastic Differential Equation dw = dt/R-w/2dN td +(1-w)dN to

19. Deficiency of earlier Model B(t) = f(,R) Throughput (B(t)) is a function of loss rate (  and round trip time (R) R Network Network is a (blackbox) source of R and Solution: Express R and as functions of B R

20. t -> - q(t) - x(t) t -> - q(t) - x(t)

21. System of Differential Equations Window size: All quantities are expected values. We ignore timeouts and slowstart in this formulation. Queue length: dq = -1 [q(t) > 0] Cdt +  W i (t)/R i (q(t))dt Average Queue size: dx = ln (1  x(t) - ln (1  q(t) Where  averaging parameter of RED (w th )  sampling interval ~ 1/C

22. Control Theoretic Viewpoint