Buer overow probability at handovers

4.2 Measuring high-speed TCP performane during mobile handovers

4.2.4 Buer overow probability at handovers

solving for

α

^,^we ^obtain ^the ^following ^hain inequalities.

α =

1 − T ₀ T ₀ ^′ + Q ^′

W ^∗ >

1 − T ₀ T ₀ ^′ + Q ^′

µ ^′ T ₀ ^′ >

1 − T ₀

T ₀ ^′

µ ^′ T ₀ ^′ = (T ₀ ^′ − T ₀ )µ ^′

^(4.3)

Theseinequalitiesinturnyieldalowerbound for

α

^using^linkharateristisonstantsonly.

α > (T ₀ ^′ − T ₀ )µ ^′

Itmeansthatforasingleowif

α

^is^set^below^this^level,^the^protool^will^be^unable^to^utilize^the

newlink. Forexample,FASTinourdown-swithsenariowith1500bytepaketswillneverreah

full link utilization in the equilibrium state for

α < (250

^ms

− 120

^ms

) · 10

^Mb/s

≈ 110

^pakets.

Thisfat isreeted inFigure4.2(o); for

α = 200

^F^AST ^utilizes^the^full ^apaity ^of^the^Type-1

linkbut for

α = 40

^it^does^not.

Measurements summary

We observed that loss-based TCP variants adapt slowly to apaity up-swithes. Although

urrentTCPvariantsdoprobeinreasedapaity(e.g.,duringongestionavoidane inNewReno

or during max inrease in BIC), the time these probes take limit the time resolution of the

detetion. A possible solution ould be to expliitly detet inreased link apaity and initiate

an aggressive window-growing phaseon suh events.

FASTTCP handlesapaityup-swithesimmediately,but mayfailto utilizethelinkaftera

down-swith. ThereasonliesinFAST'swaytoalulatetheamountofpaketsinthebottlenek

buerif

α

^is ^too ^low ^for ^a ^give ^down-swith ^senario, ^F^AST ^will ^never ^utilize ^the ^new ^link.

Inreasingthe default value of

α

^wouldunneessarilyllthebuer too,therefore webelieve the dynami setting of

α

^based ^on ^available ^bandwidth measurements would be the real solution here.

S

C

∆

RTT base

link outage

Figure 4.4: A simplied TCP ontrol loop as seen by the sender. Data pakets leaving the

sender S rstgetqueued (andpossiblydropped)atthebottleneklink,thenarriveat thelient

C. There, ACK pakets are triggered that travel bak to the sender. Base propagation delay

(

RT T

base

) is aumulated at some point during theyle.

In Figure 4.5 we illustrate the eet of short linkoutages enountered e.g., during wireless

handovers. Assuming that the wireless aess link is the bottlenek of the system we would

see wireless buer oupany growing up to the maximum buer size then dropping to some

non-zerovalueandgrowing tomaximumbuersizeagain, periodially. Whenalinkoutagehits,

link bandwidth past the buer drops to zero for a short timeausing an abrupt peak inbuer

oupany. If at this point there is no free buer spae to aommodate the peak, an overrun

ours.

Themoment whenthebuerusagepeakhits,isindependent oftheTCPsender'sstate,thus

the probability of buer overruns equals the ratio of the buer growth period when a peak of

given size ould not be aommodated. In partiular, if

T _p

^denotes ^the ^buer ^growth ^yle's

period timeand

T _o (B _p )

^is^the^time^during^this ^period^when^a^usage^peak^of^size

B _p

^would^ause

an overrun, theprobability ofthe overrun is

P(B _p ) = T o (B p )

T _p .

^(4.4)

Wewillalulate

T _p

^and

T _o

^for^a^single^AIMD^TCP ^ow^with^the^followingassumptionsand notation.

•

^Thepropagationround-tripdelay

T

^,^the^maximum^buer^size

B

max

andthelinkbandwidth

µ

^past^the^queue^are^onstants. ^(Note: ^The^BDP ^produt^of^the^link^is^thus ^noted^as

µT

^.)

•

^Let

W

min

and

W

max

denote the extremes of the ongestion window

W (t)

orresponding to minimumand maximumbuer oupany,respetively,whenthere isnolinkoutagein

A

B

C

PSfragreplaements

B(t) B

max

time

T _p T _o

Figure 4.5: Buer oupany dynamis withpeaks due to link outages (AIMD ongestion

on-trol). Dottedline indiates the maximumbuer size above whih pakets aredropped fromthe

buer. (Note: Therelativewidthofthepeaks isillustration;inatypialsenarioitisnegligible

ompared to

T _p

^.)

•

^Let

B

max

> µT

^and

W

max

> µT + B

max

, otherwise the buer would empty during the

dereasephaseofthe well-knownsawtooth patternof

W (t)

^,^and^TCP ^ould^not ^utilize^the

fullavailablebandwidth. Thisalsoimpliesthatthebuerisneverempty,thatis

B(t) > 0

for all

t

^. ^As ^the ^buer ^is ^never^empty^,^the ^TCP ^sender's ^window

W (t)

^grows ^and ^shrinks

periodially with

B (t)

•

^W^e ^onsider Appropriate ByteCounting to beimplemented or delayed ACK-s turnedo,

thus windowgrowth isreally 1paket/

T

•

^As ^we ^onsider ^only ^a ^single ^ow, ^buer ^oupany ^at ^any ^time

t

^is ^the^dierene ^of ^the

atualongestionwindowofthatowandthedatanotinthebuerat

t

B(t) = W (t) − µT

Letthen

W ^∗ (B _p )

^denote^the^ritial ^window ^size ^that^solves

B

max

= W ^∗ (B _p ) + B _p − µT

thatis at whihwindowsize a buerusage peak

B _p

^auses^a ^buer ^overrun.

µT

^is^onstant,

T o (B p )

^equals^the^time^needed^to^grow^the^window^from

W ^∗ (B p )

^to

W

^max^.

In general, the time

t(W ₁ , W ₂ )

^to ^grow ^the ^window ^from

W ₁

^to

W ₂

^an ^be ^alulated ^as ^given

in[116℄,provided theTCP isinongestionavoidanephaseandthequeueisnon-empty,thatis

both

W ₁

^and

W ₂

^are

> µT

^(for^more ^details^please ^refer ^to ^the^original^publiation ^[116℄):

t(W ₁ , W ₂ ) = W ₂ ² − W ₁ ² 2µ .

From here

T _o (B _p ) = t(W ^∗ (B _p ), W

max

)

with

W

max

= B

max

+ µT W

min

= B

max

+ µT

2 W ^∗ (B _p ) = B

max

− B _p + µT

beomes

T _o (B _p ) = W

max

² − W ^∗ (B _p ) ²

2µ = B _p (2(B

^max

+ µT ) − B _p )

2µ .

^(4.5)

T _p

^on ^the^other ^hand^is ^the ^time^it^takes ^to^grow ^the^window^from

W

min

W

max

,thatis

T _p = t(W

min

, W

max

) = W

max

² − W ²

min

2µ =

3 4 (B

max

+ µT ) ²

2µ .

^(4.6)

Substituting (4.5) and(4.6) into (4.4) we obtainthe probability ofbuer overrunsas

P (B p ) = T _o (B _p )

T _p = 4B _p (2(B

max

+ µT ) − B _p )

3(B

^max

+ µT ) ² .

^(4.7)

Weontinue withestimationof thebuer usagepeak

B p

^. ^Suh ^peaks ^are^aused^by^the^fat

thatduringlinkoutagethebuerisstillbeinglleduntileitheroutageendsorthesendersstops

sending, due to lak of returning ACK-s. The latter happens when the ACK triggered by the

last paket served before the outage reahes the sender, that is exatly a base round-trip time

(

T

⁾ ^after ^the^start ^of^the ^outage. ^Clearly,

B _p

^depends^on^the^duration ^(width)^of ^the^peak,^and

therefore weexamine

B (t)

^around^the ^peak ⁱⁿ^more^detail.

We note theduration of link outageby

τ

^,^and ⁱⁿ ^Figure ^4.6^we ^show ^the^two ^ases ^of

B(t)

evolution depending on the relation of

τ

^and

T

^. ^In ^both ^ases ^prior ^to ^the ^outage ^the ^buer

grows at the dierene of the TCP sending rate

λ

^and ^the ^queue's ^servie ^rate

µ

^. ^During ^the

outage theservierate iszero, and thebuer lls at therate of

λ

^. ^This^inreased ^lling^period

lasts for the shorter of

τ

^and

T

^, ^but ^the ^sender ^does ^not ^ease ^sending ^until ^after

T

^(the ^rst

non-reeption ofan ACK). Whenlinkoutageends,thebuerresumesto growwithrate

λ − µ

if pakets are still oming from the sender. Otherwise the buer starts to deplete at rate

µ

Depletion ends when the sender reeives the rst ACK after the outage, and this duration is

min(τ, T )

In both ases in Figure 4.6,

B _p

^is ^the^maximum ^growth ^of

B(t)

^ompared ^to ^the ^referene

ase. For thetwo ases thengrowth is

B _p =







λτ

^if

τ < T ,

λT − (λ − µ)T = µT

^if

τ > T .

(4.8)

Letus also assumehere thatthe periodtimeof TCP windowgrowth is

T _p ≫ max(τ, T )

PSfragreplaements

B (t)

− µ λ − µ

λ − µ

λ B _p

τ τ

T

t

(a)

τ < T

PSfragreplaements

B (t)

− µ

λ − µ

λ B _p

τ

T T

t

(b)

τ > T

Figure4.6: Bueroupationgrowtharoundthepeaks. Dashedlinesmarkbuergrowthinase

of nolinkoutages. Non-zerogrowthrates arealso noted.

from whih itfollows that

λ − µ ≪ λ

during the time-frame of an outage. It means thatwe an approximate

λ ≈ µ

ⁱⁿ^{(4.8) ,} ^thus ^we

express buer oupany growth as

B _p ≈ µ min(τ, T ).

^(4.9)

We show two examples of our ndings. First, we plot the probability

P (B p )

ⁱⁿ ^(4.7) ^using

(4.9) in Figure 4.7 with various interruption times and maximum buer sizes (

µ = 10

^Mbit/s,

T = 200

^ms). ^It ^an ^be ^seen^that ^theprobabilityof bueroverruns an be reduedifthebuer is sized appropriately for handover outages. Seondly, in Figure 4.8 we show an example

B(t)

peak as measured in our emulation testbed. A NewReno TCP in ongestion avoidane phase

was measured with parameters

µ = 10

^Mbit/s,

B

^max

= 250

^kbyte,

T = 200

^ms ^and

τ = 50

^ms.

The emulated handover started at

t ≈ 101.45

^s, ^and ^as ^expeted, ^the^buer ^grew ^suddenly^and

the sender stopped due to ACK shortage after

T

^. ^The ^alulation ⁱⁿ^(4.9) ^gives ^an ^estimated

buer growth of

B _p = 62.5

^kbyte, ^whih ^mathes^the ^testbed ^results. ^Using ^the^analyti ^model

theprobabilityof bueroverrun inthisase is

P (B _p ) ≈ 0.25

Theabove ndings were derived for AIMDTCPs, but basedon these we an formulate the

following about non-AIMDvariantsaswell. First,FAST behaves dierently from AIMDTCPs

beause of its buer lling behavior. If there is only a single FAST ow in the equilibrium

state, it would use

α

^pakets ^from ^the ^buer,^sine ⁱⁿ equilibriumthe window is

W = µT + α

for FAST (no sawtooth). It means that if

B _p > B

^max

− α

^, ^a ^buer ^overrun ^will ^our ^with ^a

probabilityof 1,otherwise

P (B _p ) = 0

^. ^Seondly^,^other ^non-AIMD^TCP ^variants^like ^BIC ^result

0.01 0.1 1

100 50 200 150

300 250

250 500 750 1000 0.01

0.1 1 probability

outage [ms]

buffer size [kbyte]

probability

Figure 4.7: Buer overow probability of NewReno. (Note: Lowest value for

B

^max ^is

µT = 2

^Mbit

= 250

^kbyte,^below^whih

P (B _p ) = 1

^.)

infundamentallydierent,moreaggressivequeuedynamis. Asaresulttheprobabilityofbuer

overruns isexpetedto belargerthan inase ofAIMDprotools.

Inreasing buer sizes to overome the problem presented in this setion is not feasible for

many reasons, therefore we propose a dierent solution, provided that the bottlenek buer is

in the mobile network. Our basi onept is to temporarily inrease the maximum buer size

until the transient eets of the handover wear o. The buer size inrease an be given and

revoked per user for those that need it, and this way we do not impose large requirements on

total system buer apaity. We propose that on handover the buer is inreased by

B _p

^as ⁱⁿ

(4.9) , as all parameters of this expression are known to the mobile system. It isexpeted that

TCPs in steady state will derease their buer usage aording to Figure 4.6, thus the extra

buer spaean be laimedbak. Thiswaythe probability ofbuer overruns ouldberedued

or eliminated.

In document B daeUiveiyfTe h (Pldal 107-112)

4.2 Measuring high-speed TCP performane during mobile handovers

4.2.4 Buer overow probability at handovers

α

α =

1 − T 0 T 0 ′ + Q ′

W ∗ >

1 − T 0 T 0 ′ + Q ′

µ ′ T 0 ′ >

1 − T 0

T 0 ′

µ ′ T 0 ′ = (T 0 ′ − T 0 )µ ′

α

α > (T 0 ′ − T 0 )µ ′

α

α < (250

− 120

) · 10

≈ 110

α = 200

α = 40

α

α

α

S

C

∆

RTT base

link outage

RT T

T p

T o (B p )

B p

P(B p ) = T o (B p )

T p .

T p

T o

•

T

B

µ

µT

•

W

W

W (t)

A

B

C

B(t) B

T p T o

T p

•

B

> µT

W

> µT + B

W (t)

B(t) > 0

t

W (t)

B (t)

•

T

•

t

t

B(t) = W (t) − µT

W ∗ (B p )

B

= W ∗ (B p ) + B p − µT

B p

µT

T o (B p )

W ∗ (B p )

W

t(W 1 , W 2 )

W 1

W 2

W 1

1 − T ₀ T ₀ ^′ + Q ^′

W ^∗ >

1 − T ₀ T ₀ ^′ + Q ^′

µ ^′ T ₀ ^′ >

1 − T ₀

T ₀ ^′

µ ^′ T ₀ ^′ = (T ₀ ^′ − T ₀ )µ ^′

α > (T ₀ ^′ − T ₀ )µ ^′

T _p

T _o (B _p )

B _p

P(B _p ) = T o (B p )

T _p .

T _p

T _o

T _p T _o

T _p

W ^∗ (B _p )

= W ^∗ (B _p ) + B _p − µT

B _p

W ^∗ (B p )

t(W ₁ , W ₂ )

W ₁

W ₂

W ₁

W ₂

t(W ₁ , W ₂ ) = W ₂ ² − W ₁ ² 2µ .

T _o (B _p ) = t(W ^∗ (B _p ), W

W ^∗ (B _p ) = B

− B _p + µT

T _o (B _p ) = W

² − W ^∗ (B _p ) ²

2µ = B _p (2(B

+ µT ) − B _p )

T _p

T _p = t(W

² − W ²

+ µT ) ²

P (B p ) = T _o (B _p )

T _p = 4B _p (2(B

+ µT ) − B _p )

+ µT ) ² .

B _p

B _p

B _p =

T _p ≫ max(τ, T )

λ B _p