1. Modeling TCP in Small-Buffer Networks
Mark Shifrin and Isaac Keslassy
Technion (Israel)

2. How much buffering do routers need?
Problem
How much buffering
do routers need?

3. Why Does Buffer Size Matter?
Buffers are costly.
Today’s buffers:
1/2 board space
1/3 power consumption
Small buffers:
On chip buffers
Higher density
Lower cost
Scalability
[N. McKeown, Stanford]

4. How much Buffer does a Router need?
Universally applied rule-of-thumb:
A router needs a buffer size:
2T is the round-trip propagation time (or just 250ms)
C is the capacity of the outgoing link
Background
Mandated in backbone and edge routers.
Appears in RFPs and IETF architectural guidelines.
Has major consequences for router design.
Comes from dynamics of TCP congestion control.
Villamizar and Song: “High Performance TCP in ANSNET”, CCR, 1994.
Based on 2 to 16 TCP flows at speeds of up to 40 Mb/s.

5. Synchronized Flows Aggregate window has same dynamics
Therefore buffer occupancy has same dynamics
Rule-of-thumb still holds. t

6. Many TCP Flows
Probability
Distribution B
Buffer Size

7. Stanford Model
[Appenzeller et al., ’04 | McKeown and Wischik, ’05 | McKeown et al., ‘06]
Assumption 1: TCP Flows modeled as i.i.d.  total window W has Gaussian distribution
Assumption 2: Queue is the only variable part  queue has Gaussian distribution: Q=W-CONST  use smaller buffer than in rule of thumb

8. Impact on Router Design
40Gb/s linecard with 1,000,000 flows
Rule of thumb: Buffer = 10Gbits
Requires external, slow DRAM
Stanford model: Buffer = 10Mbits
Can use on-chip, fast SRAM
Delays halved for short-flows

9. Motivation In a small-buffer world…
Assumption 1: TCP Flows modeled as i.i.d.  total window W has Gaussian distribution
Assumption 2: Queue is the only variable part  queue has Gaussian distribution: Q=W-CONST OK
Queue is small  negligible?
Gaussian part is… on the lines!

10. Contributions Distribution models for:
Lines
Arrival rates to queues
Queue sizes
Packet loss rates
General closed-loop model for small-buffer networks
Result: queues are not the only variable part in the network

11. Model Development Flow
li1 pdf
cwnd
total arrival rate
l1i arrival rates
Q pdf
packet loss

12. Model of li1
li1 pdf
cwnd
packet loss

13. Bursty Model of Window Distribution
dest.
source
li5
li1
li3
li2
li4
li6
rtti=tp1i+tp2i+tp3i+tp4i+tp5i+tp6i
Common Approach 1: Uniform packet distribution. l1(t)i = wi(t)* tpi1/rtti.
Approach 2: Bursty Packet Distribution

14. Bursty Model of Window Distribution
Assumption: All packets in a flow move in a single burst
Conclusion 1: All packets belonging to an arbitrary flow i are present almost always on the same link.
Conclusion 2: The probability of burst of flow i being present on the certain link is equal to the ratio of its propagation latency to the total rtti.

15. Simulation Results

16. Model Development Flow
li1 pdf
cwnd
total arrival rate
l1i arrival rates

17. Rate Transmission Derivation
The objective is to find the pdf of the number of packets sent on some link i, ri, in a time unit δt.
Assumptions:
The rate on each one of the links in L1 is statistically independent
We assume that the transmissions are bursty.
We assume that the rate is proportional to the distribution of l1i and to the ratio δt/tp1i.

18. Arrival rate of a single flowB
source
li2
li1
li1 *δt/tp1i δt
tp1i
We find the arrival distribution for every flow in δt msec.

19. Total Rate Result: Proof based on the Lindeberg condition.
Generalizes Central Limit Theorem for non-identically distributed components
Holds if the share of each flow comparatively to the sum is negligible as the number of the flows grows.
Argument for the proof: cwnd is limited by maximum value  same for l1i and ri.

20. Instantaneous Rate Model - Results
Probability
Total arrival rate – number of packets per δt

21. Model Development Flow
li1 pdf
cwnd
total arrival rate
l1i arrival rates
Q pdf
packet loss

22. PDF for Q To find the queue size distribution:
Run Markov Chain simulation (compared with [Tran-Gia and Ahmadi, ‘88])
Use samples of R for the transitions
Packet loss p is derived from the queue size distribution.

23. PDF of Q
Probability
Q state – 0 to 585 packets

24. Fixed Point Solution: p=f(p)
li1 pdf
cwnd
total arrival rate
l1i arrival rates
Q pdf
packet loss

25. Packet loss - results Model gives about 10%-25% of discrepancy.
Case 1: Measured: p=2.7%, Model: p=3%
Case 2: Measured: p=0.8%, Model: p=0.98%
Case 3: Measured: p=1.4%, Model: p=1.82%
Case 4: Measured: p=0.452%, Model: p=0.56%

26. The Gaussian distributionsL4 L2 L5 L6 L1 W
NS2 simulation of 500 flows with different propagation times

27. Are these lines really Gaussian?

28. Summary Introduced general closed-loop model for small-buffer networks
Proved wrong the usual assumption that queues are the only variable part in the network

29. Thank you.