tcp

Performance models of TCP



 can simulate (ns-2)

+ faithful to operation of TCP

- expensive, time consuming

 deterministic approximations

+ quick

- ignore some TCP details, steady state

 fluid models

+ transient behavior

- ignore some TCP details

TCP behavior

TCP runs at end-hosts

 congestion control:

 decrease sending rate

when loss detected,

increase when no loss

 routers

 discard, mark packets

when congestion occurs

 interaction between end

systems (TCP) and

routers?

 want to understand congested router drops packets

(quantify) this interaction

Generic TCP behavior

 window algorithm (window W)

 up to W packets in network

 return of ACK allows sender to send another

packet

 cumulative ACKS



 increase window by one per RTT

W window

reduced to one, W p = p(x)



1





Marking probability p

 More generally:

active queue

management pmax



(AQM) 0 tmin tmax

2tmax

Average queue length x

Bottleneck behavior

bottleneck router:

 capacity fully utilized

 all interfering sessions

see same loss prob.

 do all sessions see

same thruput?





i Bi (RTTi ,p) = C

C - router bandwidth

Bi - throughput of flow i

Single bottleneck: infinite flows



 N infinite TCP sessions

 two way propagation

delay Ai, i = 1,…,N

 throughput Bi(p,RTTi)





 one bottleneck router

 RED queue management

• avg. queue length x ; dropping probability p(x)

 to discover

 Bi: TCP sessions’ throughput,

 router behavior, e.g., drop prob. avg. queue len.

Model and solution

 model



p = p(x) (AQM)

RTTi = Ai + x /C (round trip time)

i Bi (x) = C, for i =1 ,…,N

i B (p , RTT i) = C, for i =1 ,…,N

 solve a fixed point problem for x

 unique solution provided B is monotonic and

continuous on x

 resulting x can be used to obtain RTTi and p

Model versus simulation: single

bottleneck, infinite flows

• fixed router capacity 4 Mbps and RED parameters

• 10-120 TCP flows

• two-way prop. delay 20+2i ms, i = 1,…,N









throughput router loss

Bottleneck principle: a qualitative

result





Bnew(p)

 new/improved,

Bnew(p)









thruput

 TCP, BTCP(p)

BTCP(p)





Bnew(p) > BTCP(p)

p

Sharing bottleneck with TCP



C

Nnew

NTCP





Nnew Bnew(p) + NTCP Bni(p) = C

 Bnew(p) > BTCP(p)



a win! friendly? p

Replacing TCP with TCP-new



N Bnew(pnew) = C C

vs N



N BTCP(pTCP) = C

 pnew > pTCP



 a loss!



pTCP pnew

 simple model for TCP B c , c ≈ 1.2

T p



 bottleneck principle









 multiple bottlenecks

 fluid models

Multiple Bottleneck: infinite flows

 N TCP flows

 throughputs B =

 V congested AQM routers

 capacities C =

 avg. queue lengths x =

 discard prob. p =







bottleneck router model



i Bi (x ) = Cv , v =1,…,V

V equations, V unknowns

Results: multiple bottleneck, infinite flows



• tandem network core, 5 -

10 routers

• 2-way propagation delay

20-120 ms

• bandwidth, 2-6 Mbps

throughput

• PFTK model error

• throughput

System of Differential Equations

Timeouts and slow start ignored



Window Size: dWk  1 - p ( x (t - t k )) - Wk (t ) Wk (t - t k ) p ( x (t - t k ))

dt Rk (t ) 2 Rk (t - t k )

Additive Mult. Loss arrival

increase decrease rate









dq Wk (t )

Queue length:  -C 1{q > 0} + k

dt Rk (t )

Outgoing Incoming

traffic traffic

System of Differential Equations

(cont.)

Average smoothed dx ln(1 - a ) ln a

queue length:  x (t ) - q (t )

dt d d

Where

a = averaging parameter of RED(wth)

d = sampling interval ~ 1/C



dp dp dx

Loss probability: 

dt dx dt

Where dp is obtained from the marking profile

dx

N+2 coupled equations

N flows

dWi dt  f1  p, Ri , Wi ), i  1,  , N

Wi(t) = Window size

of flow i



Ri(t) = RTT of flow i dp dt  f 3 q ) dq dt  f 2 Wi )



p(t) = Drop probability



q(t) = queue length



Equations solved numerically using MATLAB

Steady slate behavior

 let t → ∞

dWk

 0, p(t )  p, W (t )  W , Rk (t )  Rk

dt

 this yields



1 - p Wk Wk 2(1 - p)

0 - p or Wk 

Rk 2 Rk p

 the throughput is



2(1 - p) 2

Bk   for small p

Rk p Rk p

A Queue is not a Network

Network - set of AQM routers, V

sequence Vi for session i



Round trip time - aggregate delay

Ri(t) = Ai + vVi qv(t)





Loss/marking probability - cumulative prob

pi (t) = 1-Pv Vi (1 - pv(qv(t)))



Link bandwidth constraints



Queue equations

How well does it work?

 OC-12 – OC-48 links

 RED with target delay

5msec OC-12

 2600 TCP flows

OC-48







 decrease to 1300 at

30 sec. 2600  j 2600  j

1300  j

 increase to 2600 at 90

sec.

t=30 t=90

simulation

fluid model

instantaneous delay









time (sec)



Good queue length match

average window size

simulation

fluid model

window size









simulation

fluid model







time (sec)



time (sec)

matches average window size

Scaling Properties



OC-12 OC-12  j





OC-48 OC-48  j









2600 1300 2600 2600  j 2600  j

1300  j





t=30 t=90 t=30 t=90





Wk(t) = Wj k(t)



qv(t) = qjv(t)/100

Summary: TCP flows as fluids

What have we seen?

 model TCP as constant rate fluid flows

 rate sensitive to congestion via:

 capacities C =

 avg. queue lengths x =

 discard prob. p =

 dynamic (transient) behavior of TCP modeled as

system of differential equations





ability to predict performance of

system of TCP flows using fluid models