[PDF] - Mo deling TCP Throughput: A Simple Mo del and its Empirical PDF Document

SLIDE 1 Mo deling TCP Throughput: A Simple Mo del and its Empirical V alidation

Jitendra

P adh y e Victor Firoiu Don T

wsley

Jim Kurose f jitu, vroiu, to wsley , kuroseg@cs.umass.edu Departmen t

f

Computer Science Univ ersit y

f

Massac h usetts Amherst, MA 01003 USA Abstract In this pap er w e dev elop a simple analytic c haracterization

f

the steady state throughput, as a function

f

loss rate and round trip time for a bulk transfer TCP

w,

i.e., a

w

with an unlimited amoun t

f

data to send. Unlik e the mo dels in [6 , 7, 10 ],

ur

mo del captures not

nly

the b e- ha vior

f

TCP's fast retransmit mec hanism (whic h is also considered in [6 , 7, 10]) but also the eect

f

TCP's timeout mec hanism

n

throughput. Our measuremen ts suggest that this latter b eha vior is imp

rtan

t from a mo deling p ersp ectiv e, as almost all

f
ur

TCP traces con tained more time-

ut

ev en ts than fast retransmit ev en ts. Our measuremen ts demonstrate that

ur

mo del is able to more accurately predict TCP throughput and is accurate

v

er a wider range

f

loss rates. 1 Intro duction A signican t amoun t

f

to da y's In ternet trac, including WWW (HTTP), le transfer (FTP), email (SMTP), and re- mote access (T elnet) trac, is carried b y the TCP transp

rt

proto col [18 ]. TCP together with UDP form the v ery core

f

to da y's In ternet transp

rt

la y er. T raditionally , sim ulation and implemen tation/measuremen t ha v e b een the to

ls
f

c hoice for examining the p erformance

f

v arious asp ects

f

TCP . Recen tly , ho w ev er, sev eral eorts ha v e b een directed at analytically c haracterizing the throughput

f

TCP's congestion con trol mec hanism, as a function

f

pac k et loss and round trip dela y [6, 10 , 7]. One reason for this recen t in- terest is that a simple quan titativ e c haracterization

f

TCP throughput under giv en

p

erating conditions

ers

the p

s-

sibilit y

f

dening a \fair share"

r

\TCP-friendly" [6 ] throughput for a non-TCP

w

that in teracts with a TCP connection. Indeed, this notion has already b een adopted in the design and dev elopmen t

f

sev eral m ulticast congestion control proto cols [19 , 20 ].

This

material is based up

n

w

rk

supp

rted

b y the National Science F

undation

under gran ts NCR-95-08274, NCR-95-23807 and CD A-95-02639. An y

pinions,

ndings, and conclusions

r

recommen- dations expressed in this material are those

f

the authors and do not necessarily reect the views

f

the National Science F

undation.

In this pap er w e dev elop a simple analytic c haracterization

f

the steady state throughput

f

a bulk transfer TCP

w

(i.e., a

w

with a large amoun t

f

data to send, suc h as FTP transfers) as a function

f

loss rate and round trip time. Unlik e the recen t w

rk
f

[6 , 7, 10 ],

ur

mo del captures not

nly

the b eha vior

f

TCP's fast retransmit mec hanism (whic h is also considered in [6 , 7, 10 ]) but also the eect

f

TCP's timeout mec hanism

n

throughput. The measuremen ts w e presen t in Section 3 indicate that this latter b eha v- ior is imp

rtan

t from a mo deling p ersp ectiv e, as w e

bserv

e more timeout ev en ts than fast retransmit ev en ts in almost all

f
ur

TCP traces. Another imp

rtan

t dierence b et w een

urs

and previous w

rk

is the abilit y

f
ur

mo del to accurately predict throughput

v

er a signican tly wider range

f

loss rates than b efore; measuremen ts presen ted in [7 ] as w ell the measuremen ts presen ted in this pap er, indicate that this to

is

imp

rtan

t. W e also explicitly mo del the eects

f

small receiv er-side windo ws. By comparing

ur

mo del's predictions with a n um b er

f

TCP measuremen ts made b e- t w een v arious In ternet hosts, w e demonstrate that

ur

mo del is able to more accurately predict TCP throughput and is able to do so

v

er a wider range

f

loss rates. The remainder

f

the pap er is

rganized

as follo ws. In Section 2 w e describ e

ur

mo del

f

TCP congestion con trol in detail and deriv e a new analytic c haracterization

f

TCP throughput as a function

f

loss rate and a v erage round trip time. In Section 3 w e compare the predictions

f
ur

mo del with a set

f

measured TCP

ws
v

er the In ternet, ha ving as their endp

in

ts sites in b

th

United States and Europ e. Section 4 discusses the assumptions underlying the mo del and a n um b er

f

related issues in more detail. Section 5 concludes the pap er. 2 A Mo del fo r TCP Congestion Control In this section w e dev elop a sto c hastic mo del

f

TCP congestion con trol that yields a relativ ely simple analytic expression for the throughput

f

a saturated TCP sender, i.e., a

w

with an unlimited amoun t

f

data to send, as a function

f

loss rate and a v erage round trip time (R TT). TCP is a proto col that can exhibit complex b eha vior, esp ecially when considered in the con text

f

the curren t In- ternet, where the trac conditions themselv es can b e quite complicated and subtle [14 ]. In this pap er, w e fo cus

ur

at- ten tion

n

the congestion a v

idance

b eha vior

f

TCP and its impact

n

throughput, taking in to accoun t the dep endence

f

congestion a v

idance
n

A CK b eha vior, the manner in whic h pac k et loss is inferred (e.g., whether b y duplicate A CK detection and fast retransmit,

r

b y timeout), limited

SLIDE 2 receiv er windo w size, and a v erage round trip time (R TT). Our mo del is based

n

the Reno a v

r
f

TCP , as it is b y far the most p

pular

implemen tation in the In ternet to da y [13 , 12]. W e assume that the reader is familiar with TCP Reno congestion con trol (see for example [4 , 17 , 16 ]) and w e adopt most

f
ur

terminology from [4 , 17 , 16 ]. Our mo del fo cuses

n

TCP's congestion a v

idance

mec h- anism, where TCP's congestion con trol windo w size, W ; is increased b y 1=W eac h time an A CK is receiv ed. Con- v ersely , the windo w is decreased whenev er a lost pac k et is detected, with the amoun t

f

the decrease dep ending

n

whether pac k et loss is detected b y duplicate A CKs

r

b y timeout, as discussed shortly . W e mo del TCP's congestion a v

idance

b eha vior in terms

f

\rounds." A round starts with the bac k-to-bac k transmission

f

W pac k ets, where W is the curren t size

f

the TCP congestion windo w. Once all pac k ets falling within the congestion windo w ha v e b een sen t in this bac k-to-bac k manner, no

ther

pac k ets are sen t un til the rst A CK is receiv ed for

ne
f

these W pac k ets. This A CK reception marks the end

f

the curren t round and the b eginning

f

the next round. In this mo del, the duration

f

a round is equal to the round trip time and is assumed to b e indep enden t

f

the windo w size, an assumption also adopted (either implicitly

r

explicitly) in [6, 7 , 10 ]. Note that w e ha v e also assumed here that the time needed to send all the pac k ets in a windo w is smaller than the round trip time; this b eha vior can b e seen in

bserv

ations rep

rted

in [2, 12 ]. A t the b eginning

f

the next round, a group

f

W new pac k ets will b e sen t, where W is the new size

f

the congestion con trol windo w. Let b b e the n um b er

f

pac k ets that are ac kno wledged b y a receiv ed A CK. Man y TCP receiv er implemen tations send

ne

cum ulativ e A CK for t w

consecutiv

e pac k ets receiv ed (i.e., dela y ed A CK, [16 ]), so b is t ypically 2. If W pac k ets are sen t in the rst round and are all receiv ed and ac kno wledged correctly , then W =b ackno wledgmen ts will b e receiv ed. Since eac h ac kno wledgmen t increases the windo w size b y 1=W ; the windo w size at the b eginning

f

the second round is then W = W + 1=b. That is, during congestion a v

idance

and in the absence

f

loss, the windo w size increases linearly in time, with a slop e

f

1=b pac k ets p er round trip time. In the follo wing subsections, w e mo del TCP's b eha vior in the presence

f

pac k et loss. P ac k et loss can b e detected in

ne
f

t w

w

a ys, either b y the reception at the TCP sender

f

\triple-duplicate" ac kno wledgmen ts, i.e., four A CKs with the same sequence n um b er,

r

via time-outs. W e denote the former ev en t as a \TD" (triple-duplicate) loss indication, and the latter as a \TO" loss indication. W e assume that a pac k et is lost in a round indep enden tly

f

an y pac k ets lost in

ther

rounds, a mo deling assumption justied to some exten t b y past studies [1 ] that ha v e sho wn that p erio dic UDP pac k ets that are separated b y as little as 40 msec tend to get lost

nly

in singleton bursts. On the

ther

hand, w e assume that pac k et losses are correlated among the bac k-to-bac k transmissions within a round: if a pac k et is lost, all remaining pac k ets transmitted un til the end

f

that round are also lost. This burst y loss b eha vior, whic h has b een sho wn to arise from the drop-tail queuing discipline (adopted in man y In ternet routers), is discussed in [2 , 3]. W e discuss it further in Section 4. W e dev elop a sto c hastic mo del

f

TCP congestion control in sev eral steps, corresp

nding

to its

p

erating regimes: when loss indications are exclusiv ely TD (Section 2.1), when loss indications are b

th

TD and TO (Section 2.2), and when the congestion windo w size is limited b y the receiv er's adv ertised windo w (Section 2.3). W e note that w e do not mo del certain asp ects

f

TCP's b eha vior (e.g., fast reco v- ery) but b eliev e w e ha v e captured the essen tial elemen ts

f

TCP b eha vior, as indicated b y the generally v ery go

d

ts b et w een mo del predictions and measuremen ts made

n

n u- merous commercial TCP implemen tations, as discussed in Section 3. A more detailed discussion

f

mo del assumptions and related issues is presen ted in Section 4. Also note that in the follo wing, w e measure throughput in terms

f

pac k ets p er unit

f

time, instead

f

b ytes p er unit

f

time. 2.1 Loss indications a re exclusively \triple-duplicate" A CKs In this section w e assume that loss indications are exclusiv ely

f

t yp e \triple-duplicate" A CK (TD), and that the windo w size is not limited b y the receiv er's adv ertised

w

con trol windo w. W e consider a TCP

w

starting at time t = 0, where the sender alw a ys has data to send. F

r

an y giv en time t > 0, w e dene N t to b e the n um b er

f

pac k- ets transmitted in the in terv al [0; t], and B t = N t =t, the throughput

n

that in terv al. Note that B t is the n um b er

f

pac k ets sen t p er unit

f

time regardless

f

their ev en tual fate (i.e., whether they are receiv ed

r

not). Th us, B t represen ts the throughput

f

the connection, rather than its go

dput.

W e dene the long-term steady-state TCP throughput B to b e B = lim t!1 B t = lim t!1 N t t W e ha v e assumed that if a pac k et is lost in a round, all remaining pac k ets transmitted un til the end

f

the round are also lost. Therefore w e dene p to b e the probabilit y that a pac k et is lost, giv en that either it is the rst pac k et in its round

r

the preceding pac k et in its round is not lost. W e are in terested in establishing a relationship B (p) b et w een the throughput

f

the TCP connection and p, the loss probabilit y dened ab

v

e. A sample path

f

the ev

lution
f

congestion windo w size is giv en in Figure 1. Bet w een t w

TD

loss indications, the sender is in congestion a v

idance,

and the windo w increases b y 1=b pac k ets p er round, as discussed earlier. Immediately after the loss indication

ccurs,

the windo w size is reduced b y a factor

f

t w

.

W e dene a TD p erio d (TDP) to b e a p erio d b et w een t w

TD

loss indications (see Figure 1). F

r

the i-th TD p erio d w e dene Y i to b e the n um b er

f

pac k ets sen t in the p erio d, A i the duration

f

the p erio d, and W i the windo w size at the end

f

the p erio d. Considering fW i g i to b e a Mark

v

regenerativ e pro cess with rew ards fY i g i (see for example [15 ]), it can b e sho wn that B = E [Y ] E [A] (1) In

rder

to deriv e an expression for B , the long-term steady- state TCP throughput, w e m ust next deriv e expressions for the mean

f

Y and A. Consider a TD p erio d as in Figure 2. A TD p erio d starts immediately after a TD loss indication, and th us the curren t congestion windo w size is equal to W i1 =2, half the size

f

windo w b efore the TD

ccurred.

A t eac h round the windo w is incremen ted b y 1=b and the n um b er

f

pac k ets sen t p er round is incremen ted b y

ne

ev ery b rounds. W e denote b y

i

the rst pac k et lost in T D P i , and b y X i the round where this loss

ccurs

(see Figure 2). After pac k et

i

, W i

1

more pac k ets are sen t in an additional round b efore a TD loss indication

ccurs

(and the curren t TD p erio d ends), as discussed in more detail in Section 2.2. Th us, a total

f

Y i =

i

+ W i

1

pac k ets are sen t in X i + 1 rounds. It follo ws that: E [Y ] = E [] + E [W ]

1

(2)

SLIDE 3

t A 1 W 1 A 2 A 3 W 2 W 3 W TDP 1 TDP 2 TDP 3

Figure 1: Ev

lution
f

windo w size

v

er time when loss indications are triple duplicate A CKs

no of rounds packets sent ACKed packet lost packet LEGEND W W 1 2 X i

i i-1

..... b b b 3 4 TDP i 1 2

α −1 i

3 4 5 .............

βi αi

TD occurs TDP ends 2 last round penultimate round

Figure 2: P ac k ets sen t during a TD p erio d T

deriv

e E [], consider the random pro cess f i g i , where

i

is the n um b er

f

pac k ets sen t in a TD p erio d up to and including the rst pac k et that is lost. Based

n
ur

assumption that pac k ets are lost in a round indep enden tly

f

an y pac k ets lost in

ther

rounds, f i g i is a sequence

f

indep enden t and iden tically distributed (i.i.d.) random v ariables. Giv en

ur

loss mo del, the probabilit y that

i

= k is equal to the probabilit y that exactly k

1

pac k ets are successfully ac kno wledged b efore a loss

ccurs

P [ = k ] = (1

p)

k 1 p; k = 1; 2; : : : (3) The mean

f
is

th us E [] = 1 X k =1 (1

p)

k 1 pk = 1 p (4) F

rm

(2) and (4) it follo ws that E [Y ] = 1

p

p + E [W ] (5) T

deriv

e E [W ] and E [A], consider again T D P i . W e dene r ij to b e the duration (round trip time)

f

the j

th

round

f

T D P i . Then, the duration

f

T D P i is A i = P X i +1 j =1 r ij . W e consider the round trip times r ij to b e random v ariables, that are assumed to b e indep enden t

f

the size

f

congestion windo w, and th us indep enden t

f

the round n um b er, j . It follo ws that E [A] = (E [X ] + 1)E [r ] (6) Henceforth, w e denote b y R T T = E [r ] the a v erage v alue

f

round trip time. Finally , to deriv e an expression for E [X ], w e consider the ev

lution
f

W i as a function

f

the n um b er

f

rounds, as in Figure 2. T

simplify
ur

exp

sition,

in this deriv ation w e assume that W i1 =2 and X i =b are in tegers. First w e

bserv

e that during the i-th TD p erio d, the windo w size increases b et w een W i1 =2 and W i . Since the increase is linear with slop e 1=b, w e ha v e: W i = W i1 2 + X i b ; i = 1; 2; : : : (7) The fact that Y i pac k ets are transmitted in T D P i is expressed b y Y i = X i =b1 X k =0 ( W i1 2 + k )b +

i

(8) = X i W i1 2 + X i 2 ( X i b

1)

+

i

(9) = X i 2 ( W i1 2 + W i

1)

+

i

using (7) (10) where

i

is the n um b er

f

pac k ets sen t in the last round (see Figure 2). fW i g i is a Mark

v

pro cess for whic h a stationary distribution can b e

btained

n umerically , based

n

(7) and (10) and

n

the probabilit y densit y function

f

f i g giv en in (3). W e can also compute the probabilit y distribution

f

fX i g. Ho w ev er, a simpler appro ximate solution can b e

btained

b y assuming that fX i g and fW i g are m utually indep enden t sequences

f

i.i.d. random v ariables. With this assumption, it follo ws from (7), (10) and (5) that E [W ] = 2 b E [X ] (11)

SLIDE 4 and, 1

p

p + E [W ] = E [X ] 2

E

[W ] 2 + E [W ]

1
+

E [ ] (12) W e consider that

i

, the n um b er

f

pac k ets in the last round, is uniformly distributed b et w een 1 and W i , and th us E [ ] = E [W ]=2. F rom (11) and (12), w e ha v e E [W ] = 2 + b 3b + r 8(1

p)

3bp +

2

+ b 3b

2

(13) Observ e that, E [W ] = r 8 3bp +

(1=

p p) (14) i.e., E [W ]

q

8 3bp for small v alues

f

p. F rom (11), (6) and (13), it follo ws E [X ] = 2 + b 6 + r 2b(1

p)

3p +

2

+ b 6

2

(15) E [A] = RT T 2 + b 6 + r 2b(1

p)

3p +

2

+ b 6

2

+ 1 ! (16) Observ e that, E [X ] = r 2b 3p +

(1=

p p ) (17) F rom (1) and (5) w e ha v e B (p) = 1p p + E [W ] E [A] (18) = 1p p + 2+b 3b + r 8(1p) 3bp +

2+b

3b

2

RT T

2+b

6 + q 2b(1p) 3p + ( 2+b 6 ) 2 + 1

(19)

Whic h can b e expressed as: B (p) = 1 RT T r 3 2bp +

(1=

p p ) (20) Th us, for small v alues

f

p, (20) reduces to the throughput form ula in [6] for b = 1. W e next extend

ur

mo del to include TCP b eha viors (suc h as timeouts and receiv er-limited windo ws) not considered in previous analytic studies

f

TCP congestion con trol. 2.2 Loss indications a re triple-duplicate A CKs and time-

uts

So far, w e ha v e considered TCP

ws

where all loss indications are due to \triple-duplicate" A CKs. Our measuremen ts sho w (see T able 2) that in man y cases the ma jorit y

f

windo w decreases are due to time-outs, rather than fast retransmits. Therefore, a go

d

mo del should capture time-out loss indications. In this section w e extend

ur

mo del to include the case where the TCP sender times-out. This

ccurs

when pac k ets (or A CKs) are lost, and less than three duplicate A CKs are receiv ed. The sender w aits for a p erio d

f

time denoted b y T , and then retransmits non-ac kno wledged pac k ets. F

llo

w- ing a time-out, the congestion windo w is reduced to

ne,

and

ne

pac k et is th us resen t in the rst round after a time

ut.

In the case that another time-out

ccurs

b efore successfully retransmitting the pac k ets lost during the rst time

ut,

the p erio d

f

time

ut

doubles to 2T ; this doubling is rep eated for eac h unsuccessful retransmission un til 64T is reac hed, after whic h the time

ut

p erio d remains constan t at 64T . An example

f

the ev

lution
f

congestion windo w size is giv en in Figure 3. Let Z T O i denote the duration

f

a sequence

f

time-outs and Z T D i the time in terv al b et w een t w

consecutiv

e time-out sequences. Dene S i to b e S i = Z T D i + Z T O i Also, dene M i to b e the n um b er

f

pac k ets sen t during S i . Then, f(S i ; M i )g i is an i.i.d. sequence

f

random v ariables, and w e ha v e B = E [M ] E [S ] W e extend

ur

denition

f

TD p erio ds giv en in Section 2.1 to include p erio ds starting after,

r

ending in, a TO loss indication (b esides p erio ds b et w een t w

TD

loss indications). Let n i b e the n um b er

f

TD p erio ds in in terv al Z T D i . F

r

the j

th

TD p erio d

f

in terv al Z T D i w e dene Y ij to b e the n um b er

f

pac k ets sen t in the p erio d, A ij to b e the duration

f

the p erio d, X ij to b e the n um b er

f

rounds in the p erio d, and W ij to b e the windo w size at the end

f

the p erio d. Also, R i denotes the n um b er

f

pac k ets sen t during time-out sequence Z T O i . Observ e here that R i coun ts the total n um b er

f

pac k et transmissions in Z T O i , and not just the n um b er

f

dieren t pac k ets sen t. This is b ecause, as discussed in Section 2.1, w e are in terested in the throughput

f

a TCP

w,

rather than its go

dput.

W e ha v e M i = n i X j =1 Y ij + R i ; S i = n i X j =1 A ij + Z T O i and, th us, E [M ] = E [ n i X j =1 Y ij ] + E [R]; E [S ] = E [ n i X j =1 A ij ] + E [Z T O ] If w e assume fn i g i to b e an i.i.d. sequence

f

random v ariables, indep enden t

f

fY ij g and fA ij g, then w e ha v e E [( n i X j =1 Y ij ) i ] = E [n]E [Y ]; E [( n i X j =1 A ij ) i ] = E [n]E [A] T

deriv

e E [n]

bserv

e that, during Z T D i , the time b et w een t w

consecutiv

e time-out sequences, there are n i TDPs, where eac h

f

the rst n i

1

end in a TD, and the last TDP ends in a TO. It follo ws that in Z T D i there is

ne

TO

ut
f

n i loss indications. Therefore, if w e denote b y Q the probabilit y that a loss indication ending a TDP is a TO, w e ha v e Q = 1=E [n]. Consequen tly , B = E [Y ] + Q

E

[R] E [A] + Q

E

[Z T O ] (21) Since Y ij and A ij do not dep end

n

time-outs, their means are those deriv ed in (4) and (16). T

compute

TCP throughput using (21) w e m ust still determine Q; E [R ] and E [Z T O ]: W e b egin b y deriving an expression for Q: Consider the round

f

pac k ets where a loss indication

ccurs;

it will b e re- ferred to as the \p en ultimate" round (see Figure 4) 1 . Let w 1 In Figure 4 eac h A CK ac kno wledges individual pac k ets (i.e., A CKs are not dela y ed). W e ha v e c hosen this for simplicit y

f

il- lustration. W e will see that the analysis do es not dep end

n

whether A CKs are dela y ed

r

not.

SLIDE 5

A i1 W i1 A i2 A i3 W i2 W i3 Zi 2T R =2

i

4T T0 W t t i

TD

Zi

TO

S i

Figure 3: Ev

lution
f

windo w size when loss indications are triple-duplicate A CKs and time-outs

RTT

penultimate round last round f1 fk fw fk+1

RTT

time sequence number k w k m s1 sm+1 sk received packet lost packet ACK TD occurs, TDP ends LEGEND

Figure 4: P ac k et and A CK transmissions preceding a loss indication b e the curren t congestion windo w size. Th us pac k ets f 1 ::f w are sen t in the p en ultimate round. P ac k ets f 1 ::f k are ackno wledged, and pac k et f k +1 is the rst

ne

to b e lost (or not A CKed). W e again assume that pac k et losses are correlated within a round: if a pac k et is lost, so are all pac k ets that follo w, till the end

f

the round. Th us, all pac k ets follo wing f k +1 in the p en ultimate round are also lost. Ho w ev er, since pac k ets f 1 ..f k are A CKed, another k pac k ets, s 1 ::s k are sen t in the next round, whic h w e will refer to as the \last" round. This round

f

pac k ets ma y ha v e another loss, sa y pac k et s m+1 . Again,

ur

assumptions

n

pac k et loss correlation mandates that pac k ets s m+2 ::s k are also lost in the last round. The m pac k ets successfully sen t in the last round are resp

nded

to b y A CKs for pac k et f k , whic h are coun ted as duplicate A CKs. These A CKs are not dela y ed ([16 ], p. 312), so the n um b er

f

duplicate A CKs is equal to the n um- b er

f

successfully receiv ed pac k ets in the last round. If the n um b er

f

suc h A CKs is higher than three, then a TD indication

ccurs,
therwise,

a TO

ccurs.

In b

th

cases the curren t p erio d b et w een losses, TDP , ends. W e denote b y A(w ; k ) the probabilit y that the rst k pac k ets are A CKed in a round

f

w pac k ets, giv en there is a sequence

f
ne
r

more losses in the round. Then A(w ; k ) = (1

p)

k p 1

(1
p)

w Also, w e dene C (n; m) to b e the probabilit y that m pac k ets are A CKed in sequence in the last round (where n pac k ets w ere sen t) and the rest

f

the pac k ets in the round, if an y , are lost. Then, C (n; m) = n (1

p)

m p; m

n
1

(1

p)

n ; m = n Then, ^ Q(w ), the probabilit y that a loss in a windo w

f

size w is a TO, is giv en b y ^ Q(w ) = ( 1 if w

3

P 2 k =0 A(w ; k ) + P w k =3 A(w ; k ) P 2 m=0 C (k ; m)

therwise

(22) since a TO

ccurs

if the n um b er

f

pac k ets successfully transmitted in the p en ultimate round, k , is less than three,

r
therwise

if the n um b er

f

pac k ets successfully transmitted in the last round, m is less than three. Also, due to the assumption that pac k et s m+1 is lost indep enden tly

f

pac k et f k +1 (since they

ccur

in dieren t rounds), the probabilit y that there is a loss at f k +1 in the p en ultimate round and a loss at s m+1 in the last round equals A(w ; k )

C

(k ; m), and (22) follo ws.

SLIDE 6 After algebraic manipulations, w e ha v e ^ Q (w ) = min

1;

(1

(1
p)

3 )(1 + (1

p)

3 (1

(1
p)

w 3 )) 1

(1
p)

w

(23)

Observ e (for example, using L'Hopital's rule) that lim p!0 ^ Q (w ) = 3 w : Numerically w e nd that a v ery go

d

appro ximation

f

^ Q is ^ Q(w )

min

(1; 3 w ) (24) Q, the probabilit y that a loss indication is a TO, is Q = 1 X w =1 ^ Q (w )P [W = w ] = E [ ^ Q ] W e appro ximate Q

^

Q (E [W ]) (25) where E [W ] is from (13). W e consider next the deriv ation

f

E [R ] and E [Z T O ]. F

r

this, w e need the probabilit y distribution

f

the n um b er

f

timeouts in a TO sequence, giv en that there is a TO. W e ha v e

bserv

ed in

ur

TCP traces that in most cases,

ne

pac k et is transmitted b et w een t w

time-outs

in sequence. Th us, a sequence

f

k TOs

ccurs

when there are k

1

consecutiv e losses (the rst loss is giv en) follo w ed b y a successfully transmitted pac k et. Consequen tly , the n um b er

f

TOs in a TO sequence has a geometric distribution, and th us P [R = k ] = p k 1 (1

p)

Then w e can compute R 's mean E [R] = 1 X k =1 k P [R = k ] = 1 1

p

(26) Next, w e fo cus

n

E [Z T O ], the a v erage duration

f

a time-

ut

sequence excluding retransmissions, whic h can b e com- puted in a similar w a y . W e kno w that the rst six time-outs in

ne

sequence ha v e length 2 i1 T , i = 1 : : : 6, with all immediately follo wing timeouts ha ving length 64T . Then, the duration

f

a sequence with k time-outs is L k = n (2 k

1)T

for k

6

(63 + 64(k

6))T

for k

7

and the mean

f

Z T O is E [Z T O ] = 1 X k =1 L k P [R = k ] = T 1 + p + 2p 2 + 4p 3 + 8p 4 + 16p 5 + 32p 6 1

p

Armed no w with expressions for Q; E [S ]; E [R ] and E [Z T O ] w e can no w substitute these expressions in to equation (21) to

btain

the follo wing for B (p): B (p) = 1p p + E [W ] + ^ Q(E [W ]) 1 1p RT T (E [X ] + 1) + ^ Q (E [W ])T f (p) 1p (27) where: f (p) = 1 + p + 2p 2 + 4p 3 + 8p 4 + 16p 5 + 32p 6 (28) ^ Q is giv en in (23), E [W ] in (13) and E [X ] in (16). Using (24), (14) and (17), w e ha v e that (27) can b e appro ximated b y B (p)

1

RT T p 2bp 3 + T min

1;

3 p 3bp 8

p(1

+ 32p 2 ) (29) 2.3 The impact

f

windo w limitation So far, w e ha v e not considered an y limitation

n

the congestion windo w size. A t the b eginning

f

TCP

w

estab- lishmen t, ho w ev er, the receiv er adv ertises a maxim um buer size whic h determines a maxim um congestion windo w size, W max . As a consequence, during a p erio d without loss indications, the windo w size can gro w up to W max , but will not gro w further b ey

nd

this v alue. An example

f

the ev

lution
f

windo w size is depicted in Figure 5. T

simplify

the analysis

f

the mo del, w e mak e the follo wing assumption. Let us denote b y W u the unconstrained windo w size, the mean

f

whic h is giv en in (13) E [W u ] = 2 + b 3b + r 8(1

p)

3bp +

2

+ b 3b

2

(30) W e assume that if E [W u ] < W max , w e ha v e the appro ximation E [W ]

E

[W u ]. In

ther

w

rds,

if E [W u ] < W max , the receiv er-windo w limitation has negligible eect

n

the long term a v erage

f

the TCP throughput, and th us the TCP throughput is giv en b y (27). On the

ther

hand, if W max

E

[W u ], w e appro ximate E [W ]

W

max . In this case, consider an in terv al Z T D b e- t w een t w

time-out

sequences consisting

f

a series

f

TD p erio ds as in Figure 6. During the rst TDP , the windo w gro ws linearly up to W max for U 1 rounds, then remains constan t for V 1 rounds, and then a TD indication

ccurs.

The windo w then drops to W max =2, and the pro cess rep eats. Th us, W max = W max 2 + U i b ; 8i

2

whic h implies E [U ] = (b=2)W max . Also, considering the n um b er

f

pac k ets sen t in the i-th TD p erio d, w e ha v e Y i = U i 2 ( W max 2 + W max ) + V i W max and then E [Y ] = 3 4 W max E [U ] + W max E [V ] = 3b 8 W 2 max + W max E [V ] Since Y i , the n um b er

f

pac k ets in the i-th TD p erio d, do es not dep end

n

windo w limitation, E [Y ] is giv en b y (5), E [Y ] = (1

p)=p

+ W max , and th us E [V ] = 1

p

pW max + 1

3b

8 W max Finally , since X i = U i + V i , w e ha v e E [X ] = E [U ] + E [V ] = b 8 W max + 1

p

pW max + 1 By substituting this result in (27), w e

btain

the TCP throughput, B (p), when the windo w is limited B (p) = 1p p + W max + ^ Q (W max ) 1 1p RT T ( b 8 W max + 1p pW max + 2) + ^ Q(W max )T f (p) 1p In conclusion, the complete c haracterization

f

TCP throughput, B (p), is: B (p) = 8 > > < > > : 1p p +E [W ]+ ^ Q (E [W ]) 1 1p RT T ( b 2 E [W u ]+1)+ ^ Q (E [W ])T f (p) 1p if E [W u ] < W max 1p p +W max + ^ Q(W max ) 1 1p RT T ( b 8 W max + 1p pW max +2)+ ^ Q(W max )T f (p) 1p

therwise

(31)

SLIDE 7

A i1 W i1 A i2 A i3 W i2 W i3 Zi Zi 2T R =2

i

4T T0 W t t i Wmax

TD TO

Figure 5: Ev

lution
f

windo w size when limited b y W max

Y1 Y2 Y3

no. of rounds

X1 X2 X3 W

max

U1 U2 U3 V1 V2 V3 W TDP 1 TDP 2 TDP 3

Figure 6: F ast retransmit with windo w limitation where f (p) is giv en in (28), ^ Q is giv en in (23) and E [W u ] in (13). In the follo wing sections w e will refer to (31) as the \full mo del". The follo wing appro ximation

f

B (p) follo ws from (29) and (31): B (p)

min

@ W max RT T ; 1 RT T p 2bp 3 + T min

1;

3 p 3bp 8

p(1

+ 32p 2 ) 1 A (32) In Section 3 w e v erify that equation (32) is indeed a v ery go

d

appro ximation

f

equation 31. Henceforth w e will refer to (32) as the \appro ximate mo del". 3 Measurements and T race Analysis Equations (31) and (32) pro vide an analytic c haracterization

f

TCP as a function

f

pac k et loss indication rate, R TT, and maxim um windo w size. In this section w e empiri- cally v alidate these form ulae, using measuremen t data from 37 TCP connections established b et w een 18 hosts scattered across United States and Europ e. T able 1 lists the domains and

p

erating systems

f

the 18 hosts. All data sets are for unidirectional bulk data transfers. W e gathered the measuremen t data b y running tcpdump at the sender, and analyzing its

utput

with a set

f

analysis programs dev elop ed b y us. These programs accoun t for v arious measuremen t and implemen tation related problems discussed in [13 , 12 ]. F

r

example, when w e an- alyze traces from a Lin ux sender, w e accoun t for the fact that TD ev en ts

ccur

after getting

nly

t w

duplicate

ac ks instead

f

three. Our trace analysis programs w ere further v eried b y c hec king them against tcptrace[9 ] and ns [8 ]. T able 2 summarizes data from 24 data sets, eac h

f

whic h corresp

nds

to a

ne

hour long TCP connection in whic h the sender b eha v es as an \innite source" { it alw a ys has data Receiv er Domain Op erating System ada hofstra.edu Irix 6.2 afer cs.umn.edu Lin ux al cs.wm.edu Lin ux 2.0.31 alps cc.gatec h.edu SunOS 4.1.3 bab el cs.umass.edu SunOS 5.5.1 bask erville cs.arizona.edu SunOS 5.5.1 ganef cs.ucla.edu SunOS 5.5.1 imagine cs.umass.edu win95 manic cs.umass.edu Irix 6.2 mafalda inria.fr SunOS 5.5.1 maria wustl.edu SunOS 4.1.3 mo di4 ncsa.uiuc.edu Irix 6.2 pif inria.fr Solaris 2.5 p

ng

usc.edu HP-UX spi sics.se SunOS 4.1.4 sutton cs.colum bia.edu SunOS 5.5.1 to v e cs.umd.edu SunOS 4.1.3 v

id

US site Lin ux 2.0.30 T able 1: Domains and Op erating Systems

f

Hosts

SLIDE 8 Sender Receiv er P ac k ets Loss TD TO R TT Time Sen t Indic. Out manic alps 54402 722 19 703 0.207 2.505 manic bask erville 58120 735 306 429 0.243 2.495 manic ganef 58924 743 272 471 0.226 2.405 manic mafalda 56283 494 2 492 0.233 2.146 manic maria 68752 649 1 648 0.180 2.416 manic spi 117992 784 47 737 0.211 2.274 manic sutton 81123 1638 988 650 0.204 2.459 manic to v e 7938 264 1 263 0.275 3.597 v

id

alps 37137 838 7 831 0.162 0.489 v

id

bask erville 32042 853 339 514 0.482 1.094 v

id

ganef 60770 1112 414 696 0.254 0.637 v

id

maria 93005 1651 33 1618 0.152 0.417 v

id

spi 65536 671 72 599 0.415 0.749 v

id

sutton 78246 1928 840 1088 0.211 0.601 v

id

to v e 8265 856 5 843 0.272 1.356 bab el alps 13460 1466 1461 0.194 1.359 bab el bask erville 62237 1753 197 1556 0.253 0.429 bab el ganef 86675 2125 398 1727 0.201 0.306 bab el spi 57687 1120 1120 0.331 0.953 bab el sutton 83486 2320 685 1635 0.210 0.705 bab el to v e 83944 1516 1 1514 0.194 0.520 pif alps 83971 762 760 0.168 7.278 pif imagine 44891 1346 15 1329 0.229 0.700 pif manic 34251 1422 43 1377 0.257 1.454 T able 2: Summary data from 1hr traces to send and th us TCP throughput is

nly

limited b y the TCP congestion con trol. The exp erimen ts w ere p erformed at randomly selected times during 1997 and b eginning

f

1998. The third and fourth columns

f

T able 2 indicate the n um b er

f

pac k ets sen t and the n um b er

f

loss indications resp ectiv ely (triple duplicate ac k

r

timeout). Dividing the total n um b er

f

loss indications b y the total n um b er

f

pac k ets sen t, yields an appro ximate v alue

f

p. This appro ximation is similar to the

ne

used in [7 ]. The next t w

columns

sho w a breakdo wn

f

the loss indications b y t yp e: the n um b er

f

TD ev en ts, and the n um b er

f

timeouts. Note that p dep ends

nly
n

the total n um b er

f

loss indications, and not

n

their t yp e. The last t w

columns

rep

rt

the a v erage round trip time, and a v erage duration

f

a \single" timeout T . These v alues ha v e b een a v eraged

v

er the en tire trace. When calculating round trip time v alues, w e follo w Karn's algorithm [5 ], in an attempt to minimize the impact

f

timeouts and retransmissions

n

the R TT estimates. T able 3 rep

rts

summary results from additional 13 data sets. In these cases, eac h data set represen ts 100 serially- initiated TCP connections b et w een a giv en sender-receiv er pair. Eac h connection lasted 100 seconds, and w as follo w ed b y a 50 second gap b efore the next connection w as initiated. These exp erimen ts w ere p erformed at randomly selected times during 1998. The data in columns 3-10

f

T a- ble 3 are cum ulativ e

v

er the set

f

100 traces for the giv en source-destination pair. The last t w

columns

rep

rt

the a v- erage v alue

f

round trip time and \single" timeout. These v alues ha v e b een a v eraged

v

er all

ne

h undred traces for the giv en source-destination pair. An imp

rtan

t

bserv

ation to b e dra wn from the data in these tables is that, in all traces, timeouts constitute the ma jorit y

r

a signican t fraction

f

the total n um b er

f

loss indications. This underscores the imp

rtance
f

including the eects

f

timeouts in the mo del

f

TCP congestion control. W e ha v e also noticed that exp

nen

tial bac k

due

to m ultiple timeouts

ccurs

with signican t frequency . More details are pro vided in [11 ]. Next, w e use the measuremen t data describ ed ab

v

e to v alidate

ur

mo del prop

sed

in Section 2. Figures 7-12 plot the measured throughput in

ur

trace data, the mo del

f

Sender Receiv er P ac k ets Loss TD TO R TT Time Sen t Indic. Out manic ada 531533 6432 4320 2112 0.141 2.223 manic afer 255674 4577 2584 1993 0.180 2.3 manic al 264002 4720 2841 1879 0.188 2.354 manic alps 667296 3797 841 2956 0.112 1.915 manic bask erville 89244 1638 627 1011 0.473 3.226 manic ganef 160152 2470 1048 1422 0.215 2.607 manic mafalda 171308 1332 9 1323 0.250 2.512 manic maria 316498 2476 5 2471 0.116 1.879 manic mo di4 282547 6072 3976 2096 0.174 2.26 manic p

ng

358535 4239 2328 1911 0.176 2.137 manic spi 298465 2035 159 1876 0.253 2.454 manic sutton 348926 6024 3694 2330 0.168 2.185 manic to v e 262365 2603 6 2597 0.115 1.955 T able 3: Summary data from 100s traces [7 ], as w ell as the predicted throughput from

ur

prop

sed

mo del giv en in (31) as describ ed b elo w. The title

f

the trace indicates the a v erage round trip time, the a v erage \single" timeout duration T , and the maxim um windo w size W max adv ertised b y the receiv er (in n um b er

f

pac k ets). The x- axis represen ts the frequency

f

loss indications, p, while y

axis

represen ts the n um b er

f

pac k ets sen t. Eac h

ne-hour

trace w as divided in to 36 consecutiv e 100 second in terv als, and eac h plotted p

in

t

n

a graph represen ts the n um b er

f

pac k ets sen t v ersus the n um b er

f

loss indications during a 100s in terv al. While dividing a con tin u-

us

trace in to xed sized in terv als can lead to some inaccuracies in measuring p, (e.g., the in terv al b

undaries

ma y

ccur

within timeout in terv als, th us p erhaps not attributing a loss ev en t to the in terv al where most

f

its impact is felt), w e b eliev e that b y using in terv al sizes

f

100s, whic h are longer than most timeouts, w e ha v e minimized the impact

f

suc h inaccuracies. Eac h 100 second in terv al is classied in to

ne
f

four categories: in terv als

f

t yp e \TD" did not suer an y timeout (only triple duplicate ac ks), in terv als

f

t yp e \T 0" suered at least

ne

\single" timeout but no exp

nen

tial bac k

,

\T 1" represen ts in terv als that suered a single ex- p

nen

tial bac k

at

least

nce

(i.e a \double" timeout) etc. The line lab eled \TD Only" (stands for T riple-Duplicate ac ks Only) plots the predictions made b y the mo del describ ed in [7], whic h is essen tially the same mo del as describ ed in [6 ], while accoun ting for dela y ed ac ks. The line lab eled \Prop

sed

(F ull)" represen ts the mo del describ ed b y Equation (31). It has b een p

in

ted

ut

in [6 ] that the \TD Only" mo del ma y not b e accurate when the frequency

f

loss indications is higher than 5%. W e

bserv

e that in man y traces the frequency

f

loss indications is higher than 5% and that indeed the \TD Only" mo del predicts v alues for TCP throughput m uc h higher than measured. Also, in sev- eral traces (see for example, Figure 7) w e

bserv

e that TCP throughput is limited b y the receiv er's adv ertised windo w size. This is not accoun ted for in the \TD Only" mo del, and th us \TD Only"

v

erestimates the throughput at lo w p v alues. Figures 13-17 sho w similar graphs, where eac h p

in

t represen ts an individual 100 second TCP connection. T

plot

the mo del predictions, w e used round trip and timeout du- rations that w ere a v eraged

v

er all 100 traces (these v alues also app ear in T able 3). Equation (32) in Section 2 represen ts the simple, but appro ximate form (32)

f

the full mo del giv en in (31). In Figure 18, w e plot the predictions

f

the appro ximate mo del along with the full mo del. The results for

ther

data sets are similar. In

rder

to accurately ev aluate the mo dels, w e compute the a v erage error as follo ws:

SLIDE 9

1 10 100 1000 10000 0.001 0.01 0.1 1 Number of Packets Sent Frequency of Loss Indications (p) manic-baskerville, RTT=0.243, TO=2.495, WMax=6, 1x1hr TD T0 T1 T2 T3 or more TD Only Proposed (Full)

Figure 7: manic to bask erville

1 10 100 1000 10000 0.001 0.01 0.1 1 Number of Packets Sent Frequency of Loss Indications (p) pif-imagine, RTT=0.229, TO=0.700, WMax=8, 1x1hr TD T0 T1 T2 T3 or more TD Only Proposed (Full)

Figure 8: pif to imagine

1 10 100 1000 10000 0.001 0.01 0.1 1 Number of Packets Sent Frequency of Loss Indications (p) pif-manic, RTT=0.257, TO=1.454, WMax=33, 1x1hr TD T0 T1 T2 T3 or more TD Only Proposed (Full)

Figure 9: pif to manic

1 10 100 1000 10000 0.001 0.01 0.1 1 Number of Packets Sent Frequency of Loss Indications (p) void-alps, RTT=0.162, TO=0.489, WMax=48, 1x1hr TD T0 T1 T2 T3 or more TD Only Proposed (Full)

Figure 10: v

id

to alps

1 10 100 1000 10000 0.001 0.01 0.1 1 Number of Packets Sent Frequency of Loss Indications (p) void-tove, RTT=0.272, TO=1.356, WMax=8, 1x1hr TD T0 T1 T2 T3 or more TD Only Proposed (Full)

Figure 11: v

id

to to v e

1 10 100 1000 10000 0.001 0.01 0.1 1 Number of Packets Sent Frequency of Loss Indications (p) babel-alps, RTT=0.194, TO=1.359, WMax=48, 1x1hr TD T0 T1 T2 T3 or more TD Only Proposed (Full)

Figure 12: bab el to alps

Hour-long

traces: W e divide eac h trace in to 100 second in terv als, and compute the n um b er

f

pac k ets sen t during that in terv al (here denoted as N

bser

v ed ) as w ell as the v alue

f

loss frequency (here p

bser

v ed ). W e also calculate the a v erage v alue

f

round trip time and timeout for the en tire trace (these v alues are a v ail- able in T able 2). Then, for eac h 100 second in terv al w e calculate the n um b er

f

pac k ets predicted b y

ur

pro- p

sed

mo del, N pr edicted = B (p

bser

v ed )

100s,

where B is from (31). The a v erage error is giv en b y: P

bser

v ations jN pr edicted N

bser

v ed j N

bser

v ed n um b er

f
bserv

ations The a v erage error

f
ur

appro ximate mo del (using B from (32)) and

f

\TD Only" are calculated in a similar manner. A smaller a v erage error indicates b etter mo del accuracy . In Figure 19 w e plot these error v alues to allo w visual comparison. On the x-axis, the traces

SLIDE 10

1 10 100 1000 10000 0.001 0.01 0.1 1 Number of Packets Sent Frequency of Loss Indications (p) manic-ganef, RTT=0.2150, TO=2.6078, WMax=6.0, 100x100s TD T0 T1 T2 T3 or more TD Only Proposed (Full)

Figure 13: manic to ganef

1 10 100 1000 10000 0.001 0.01 0.1 1 Number of Packets Sent Frequency of Loss Indications (p) manic-mafalda, RTT=0.2501, TO=2.5127, WMax=8.0, 100x100s TD T0 T1 T2 T3 or more TD Only Proposed (Full)

Figure 14: manic to mafalda

1 10 100 1000 10000 0.001 0.01 0.1 1 Number of Packets Sent Frequency of Loss Indications (p) manic-spiff, RTT=0.2539, TO=2.4545, WMax=32.0, 100x100s TD T0 T1 T2 T3 or more TD Only Proposed (Full)

Figure 15: manic to spi

1 10 100 1000 10000 0.001 0.01 0.1 1 Number of Packets Sent Frequency of Loss Indications (p) manic-baskerville, RTT=0.4735, TO=3.2269, WMax=6.0, 100x100s TD T0 T1 T2 T3 or more TD Only Proposed (Full)

Figure 16: manic to bask erville

1 10 100 1000 10000 0.001 0.01 0.1 1 Number of Packets Sent Frequency of Loss Indications (p) manic-sutton, RTT=0.1683, TO=2.1852, WMax=25.0, 100x100s TD T0 T1 T2 T3 or more TD Only Proposed (Full)

Figure 17: manic to sutton

1 10 100 1000 10000 0.001 0.01 0.1 1 Number of Packets Sent Frequency of Loss Indications (p) RTT=0.2539, TO=2.4545, WMax=32.0, 100x100s TD T0 T1 T2 T3 or more TD Only Proposed (Full) Proposed (Approx)

Figure 18: manic to spi, with appro ximate mo del are iden tied b y sender and receiv er names. The

rder

in whic h the traces app ear is suc h that, from left to righ t, the a v erage error for the \TD Only" mo del is increasing. The p

in

ts corresp

nding

to a giv en mo del are joined b y line segmen ts

nly

for b etter visual r ep- r esentation

f

the data.

100

second traces: W e use the v alue

f

round trip time and timeout calculated for eac h 100-second trace. The error v alues are sho wn in Figure 20. It can b e seen from Figures 19 and 20 that in most cases,

ur

prop

sed

mo del is a b etter estimator

f

the

bserv

ed v alues than the \TD Only" mo del. Our appro ximate mo del also generally pro vides more accurate predictions than the \TD Only" mo del, and is quite close to the predictions made b y the full mo del. As

ne

w

uld

exp ect,

ur

mo del do es not matc h all

f

the

bserv

ations. W e sho w an example

f

this in Figure 17. This is probably due to a large n um b er

f

triple duplicate ac ks

bserv

ed for this trace set.

SLIDE 11 Figure 19: Comparison

f

the mo dels for 1hr traces Figure 20: Comparison

f

the mo dels for 100s traces 4 A Discussion

f

the Mo del and the Exp erimental Re- sults In this section, w e discuss v arious simplifying assumptions made while constructing the mo del in Section 2, and their impact

n

the results describ ed in Section 3. Our mo del do es not capture the subtleties

f

the fast r e- c

very

algorithm. W e b eliev e that the impact

f

this

mis-

sion is quite small, and that the results presen ted in Section 3 v alidate this assumption indirectly . W e ha v e also assumed that the time sp en t in slow start is negligible compared to the length

f
ur

traces. Both these assumptions ha v e also b een made in [6 , 7, 10 ]. W e ha v e assumed that pac k et losses within a round are c

rr

elate d. Justication for this assumption comes from the fact that the v ast ma jorit y

f

the routers in In ternet to da y use the drop-tail p

licy

for pac k et discard. Under this p

l-

icy , all pac k ets that arriv e at a full buer are dropp ed. As pac k ets in a round are sen t bac k-to-bac k, if a pac k et arriv es at a full buer, it is lik ely that the same happ ens with the rest

f

the pac k ets in the round. P ac k et loss correlation at drop-tail routers w as also p

in

ted

ut

in [2 , 3]. In addition, w e assume that losses in

ne

round are indep endent

f

losses in

ther

rounds. This is justied b y the fact that pac k ets in dieren t rounds are separated b y

ne

R TT

r

more, and th us they are lik ely to encoun ter buer states that are inde- p enden t

f

eac h

ther.

This is also conrmed b y ndings in [1 ]. Another assumption w e made, that is also implicit in [6 , 7 , 10], is that the round trip time is indep enden t

f

the windo w size. W e ha v e measured the co ecien t

f

correlation b et w een the duration

f

round samples and the n um b er

f

pac k ets in transit during eac h sample. F

r

most traces summarized in T able 2, the co ecien t

f

correlation is in the range

f
0.1

to +0.1, th us lending credence to the sta- tistical indep endence b et w een round trip time and windo w size. Ho w ev er, when w e conducted similar exp erimen ts with receiv ers at the end

f

a mo dem line, w e found the co ecien t

f

correlation to b e as high as 0.97. W e sp eculate that this is a com bined eect

f

a slo w link and a buer dev

ted

exclusiv ely to this connection (probably at the ISP , just b efore the mo dem). As a result,

ur

mo del, as w ell as the mo dels describ ed in [6 , 10 , 7 ] fail to matc h the

bserv

ed data in the case

f

a receiv er at the end

f

a mo dem. In Figure 21, w e

1 10 100 1000 10000 0.001 0.01 0.1 1 Number of Packets Sent Frequency of Loss Indications (p) manic-p5, RTT=4.726, TO=18.407, WMax=22, 1x1hr TD T0 T1 T2 T3 or more TD Only Proposed (Full)

Figure 21: manic to p5 plot results from

ne

suc h exp erimen t. The receiv er w as a P en tium PC, running Lin ux 2.0.27 and w as connected to the In ternet via a commercial service pro vider using a 28.8Kbps mo dem. The results are for a 1 hour connection divided in to 100 second in terv als. W e ha v e also assumed that all

f
ur

senders implemen t TCP-Reno as describ ed in [4, 17 , 16 ]. In [13 , 12 ], it is

b-

serv ed that the implemen tation

f

the proto col stac k in eac h

p

erating system is sligh tly dieren t. While w e ha v e tried to accoun t for the signican t dierences (for example in Lin ux the TD loss indications

ccur

after t w

duplicate

A CKs), w e ha v e not tried to customize

ur

mo del for the n uances

f

eac h

p

erating system. F

r

example, w e ha v e

bserv

ed that the Lin ux exp

nen

tial bac k

do

es not exactly follo w the algorithm describ ed in [4 , 17, 16 ]. Our

bserv

ations also seem to indicate that in the Irix implemen tation, the exp

nen

tial bac k

is

limited to 2 5 , instead

f

2 6 . W e are also a w are

f

the

bserv

ation made in [13 ] that the SunOS TCP implemen tation is deriv ed from T aho e and not Reno. W e ha v e not customized

ur

mo del for these cases.

SLIDE 12 5 Conclusion In this pap er w e ha v e presen ted a simple mo del

f

the TCP- Reno proto col. The mo del captures the essence

f

TCP's congestion a v

idance

b eha vior and expresses throughput as a function

f

loss rate. The mo del tak es in to accoun t the b eha vior

f

the proto col in the presence

f

timeouts, and is v alid

v

er the en tire range

f

loss probabilities. W e ha v e compared

ur

mo del with the b eha vior

f

sev- eral real-w

rld

TCP connections. W e

bserv

ed that most

f

these connections suered from a signican t n um b er

f

timeouts. W e found that

ur

mo del pro vides a v ery go

d

matc h to the

bserv

ed b eha vior in most cases, while mo dels prop

sed

in [6, 7 , 10] signican tly

v

erestimate throughput. Th us, w e conclude that timeouts ha v e a signican t impact

n

the p erformance

f

the TCP proto col, and that

ur

mo del is able to accoun t for this impact. W e ha v e also presen ted a simplied expression for TCP bandwidth in Equation (32), whic h is a go

d

appro ximation for the prop

sed

mo del in most cases. This simple appro ximation can b e used in proto cols suc h as those describ ed in [19 , 20 ] to ensure \TCP-friendliness'. A n um b er

f

a v en ues for future w

rk

remain. First,

ur

mo del can b e enhanced to accoun t for the eects

f

fast reco v ery and fast retransmit. Second, a more precise throughput calculation can b e

btained

if the congestion windo w size is mo deled as a Mark

v

c hain. Third, w e ha v e assumed that

nce

a pac k et in a giv en round is lost, all remaining pac k ets in that round are lost as w ell. This assumption can b e relaxed, and the mo del can b e mo died to incorp

rate

a loss distribution function. Estimating this distribution function for a giv en path in the In ternet is a signican t researc h eort in itself. F

urth,

it is in teresting to further in v estigate the b eha vior

f

TCP

v

er slo w links with dedicated buers (suc h as mo dem lines). W e are curren tly in v estigating more closely the data sets for whic h

ur

mo del is not a go

d

estimator. W e are also w

rking
n

a TCP-friendly proto col to con trol transmission

f

con tin uous media. This proto col will use

ur

mo del to mo dulate its throughput to ensure TCP friendliness. 6 Ackno wledgments W e w

uld

lik e to thank Gary W allace and the CSCF sta at the Departmen t

f

Computer Science, Univ ersit y

f

Mas- sac h usetts, Amherst, for helping us with tcp dump setup and general system administration. W e are grateful to P . Krishnan

f

Hofstra Univ ersit y , Zhi-Li Zhang

f

Univ ersit y

f

Minnesota, Andreas Stathop

ulos
f

College

f

William and Mary , P eter W an

f

Georgia Inst.

f

T ec h., Larry P eterson

f

Univ ersit y

f

Arizona, Lixia Zhnag

f

Univ ersit y

f

California, Los Angeles, John Bolot and Phillip e Nain

f

INRIA, Ch uc k Cranor

f

W ashington Univ ersit y , St. Louis, Grig Gheorhiu

f

Univ ersit y

f

Southern Cali- fornia, Stephen Pink

f

Sw edish Institute

f

Science, Hen- ning Sc h ulzerinne

f

Colum bia Univ ersit y , Satish T ripathi

f

Univ ersit y

f

Maryland, College P ark, and Sneha Kasera

f

Univ ersit y

f

Massac h usetts, Amherst for pro viding us with computer accoun ts that allo w ed us to gather the data presen ted in this pap er. W e also thank Dan Rub enstein and the anon ymous referees for their helpful commen ts

n

earlier drafts

f

this pap er. References [1] J. Bolot and A. V ega-Garcia. Con trol mec hanisms for pac k et audio in the In ternet. In Pr

c

e e dings IEEE Info c

m96,

1996. [2] K. F all and S. Flo yd. Sim ulation-based comparisons

f

T aho e, Reno, and SA CK TCP. Computer Communic ation R eview, 26(3), July 1996. [3] S. Flo yd and V. Jacobson. Random Early Detection gate- w a ys for congestion a v

idance.

IEEE/A CM T r ansactions

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Networking, 1(4), August 1997. [4] V. Jacobson. Mo died TCP congestion a v

idance

algorithm. Note sen t to end2end-in terest mailing list, 1990. [5] P . Karn and C. P artridge. Impro ving Round-T rip time estimates in reliable transp

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proto cols. Computer Communi- c ation R eview, 17(5), August 1987. [6] J. Mahda vi and S. Flo yd. TCP-friendly unicast rate-based

w

con trol. Note sen t to end2end-in terest mailing list, Jan 1997. [7] M. Mathis, J. Semsk e, J. Mahda vi, and T. Ott. The macro- scopic b eha vior

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the TCP congestion a v

idance

algorithm. Computer Communic ation R eview, 27(3), July 1997. [8] S. MCanne and S. Fly

d.

ns-LBL Net w

rk

Sim ulator, 1997. Obtain via h ttp://www-nrg.ee.lbnl.go v/ns/. [9] S. Ostermann. tcptrace: TCP dump le analysis to

l,

1996. h ttp://jarok.cs.ohiou.edu/soft w are/tcptrace/. [10] T. Ott, J. Kemp erman, and M. Mathis. The stationary b e- ha vior

f

ideal TCP congestion a v

idance.

in preprin t. [11] J. P adh y e, V. Firoiu, D. T

wsley

, and J. Kurose. Mo deling TCP throughput: A simple mo del and its empirical v alidation. T ec hnical rep

rt

UMASS-CS-TR-1998-08. [12] V. P axson. Automated pac k et trace analysis

f

TCP implemen tations. In Pr

c

e e dings

f

SIGCOMM 97, 1997. [13] V. P axson. End-to-End In ternet pac k et dynamics. In Pr

c

e e dings

f

SIGCOMM 97, 1997. [14] V. P axson and S. Flo yd. Wh y w e don't kno w ho w to sim ulate the In ternet. In Pr

c

c e dings

f

the 1997 Winter Simulation Confer enc e, 1997. [15] S. Ross. Applie d Pr

b

ability Mo dels with Optimization Ap- plic ations. Do v er, 1970. [16] W. Stev ens. TCP/IP Il lustr ate d, V

l.1

The Pr

to

c

ls.

Addison-W esley , 1994. [17] W. Stev ens. TCP Slo w Start, Congestion Av

idance,

F ast Retransmit, and F ast Reco v ery Algorithms. RF C2001, Jan 1997. [18] K. Thompson, G. Miller, and M. Wilder. Wide-area in ternet trac patterns and c harateristics. IEEE Network, 11(6), No v em b er-Decem b er 1997. [19] T. T urletti, S. P arisis, and J. Bolot. Exp erimen ts with a la y ered transmission sc heme

v

er the In ternet. T ec hnical rep

rt

RR-3296, INRIA, F rance. Obtain via h ttp://www.inria.fr/RRR T/RR-3296.h tml. [20] L. Vicisano, L. Rizzo, and J. Cro w croft. TCP-lik e congestion con trol for la y ered m ulticast data transfer. In Pr

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e e dings

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INF OCOMM'98, 1998.