The Leaky Bucket As a Policing Device: Transient Analysis and Dimensioning 1 Introduction

The Leaky Bucket as a Policing Device Transient

Analysis and Dimensioning

Dimitris Logothetis Kishor Trivedi

Center for Advanced Computing Comm

GTE Lab oratories Inc Department of Electrical Engineering

Sylvan Rd DukeUniversityBox

Waltham MA Durham NC

Abstract

Policing the mechanism that the network uses to ensure that the user conforms

with his negotiated contract is an imp ortant asp ect of the forthcoming broadband

networks based on ATM The leaky bucket algorithm has b een adopted by ITU for

merly CCITT as the means to p olice ATM connections The p erformance of the leaky

bucket ie howwell can the leaky bucket algorithm enforce the parameters of the

negotiated contract is a topic of ma jor interest In this pap er we derive expressions

for the timedep endent state probabilities and the timeaveraged stateprobabilities for

the leaky bucket rate control scheme Our mo del is based on the theory of Markov

regenerative pro cesses In particular we show that the leaky bucket state probabil

ities are p erio dic when studied in continuous time We also compute timeaveraged

measures and show that our results sp ecialize to those obtained bySidi et al for

the longrun b ehavior of leaky bucket Our results are more general however in that

they apply to the transient regime and to more general arrival pro cesses Finallyusing

our timedep endent analysis we dimension the leaky bucket parameters and quantify

tradeos on these

Intro duction

Due to ATMs inherent prop erty to supp ort any bandwidth requirement an access control

scheme is needed to ensure that a user do es not exceed his negotiated parameters One

asp ect of Usage Network Parameter Control UPCNPC is p olicing The role of p olicing

Part of this pap er has app eared in the Pro ceedings of INFOCOM

This work was supp orted in part by the National Science Foundation under Grant EEC

is to ensure that nonconforming cells are marked as low priority cells or they are lost ITU

formerly CCITT has standardized two equivalentmechanisms for p olicing one of them

b eing the leaky bucket In this pap er we will present the timedep endent analysis of the

leaky bucket for general p oint arrival pro cesses In contrast to previous works we allowfor

nonstationary arrival pro cesses ie pro cesses where their parameters change over time since

such pro cesses can capture overload b ehavior Wewillshow that the state distribution of

the leaky bucket when studied in continuous time and assuming a p oint pro cess as general as

the BatchMarkovian Arrival Pro cess BMAP is p erio dic This b ehavior although rep orted

earlier is not widely known

We then compute the timeaveraged state distribution and show that its limits exist Since

we treat the leaky bucket as a p olicing mechanism ie as a mechanism to lter incoming

trac we concentrate on loss measures Weidentify twotyp es of loss measures The rst

one that we will refer to as tagged cell loss is dened as the probability that an incoming

cell is lost or marked with low priority As we shall see later this measure p ossesses a

p erio dic limiting b ehavior The other one that we will refer to as fractional loss is dened

as ratio of the exp ected numb er of lost cells over the exp ected numb er of arrived cells in

a given interval of length tWe shall see that this measure has a limiting distribution

Despite the fact that ITU has standardized only p eak rate as a trac descriptor the

ATM Forum considers two other optional trac descriptors namely burst tolerance or

p eak rate duration and sustainable cell rate or mean rate The second part of the pap er

will b e devoted to dimensioning the leaky bucket enforcing trac parameters such as p eak

rate mean rate and p eak rate duration The mathematical to ol that we use is called the

Markov regenerative or semiregenerative pro cess

The remaining of the pap er is organized as follows In Section we describ e the leaky

bucket rate control scheme and present the necessary background on Markov regenerative

pro cesses and the BMAP pro cess In Section wedevelop the mo del based on the BMAP

input pro cess while in Section we present the mo del for nonstationary input pro cesses

and study the overload b ehavior of the leaky bucket In Section we study the abilityof

the leaky bucket to control the desired trac parameters Finally in Section we present

our conclusions

Background

In this section we review existing work in the analysis of the leaky bucket rate control

scheme and wepresent some mathematical preliminaries needed to study the timedep endent

b ehavior of the leaky bucket namely the delayed Markov regenerative theory and the batch

Markovian arrival pro cess

Fractional loss directly related to the cell loss ratio CLR Quality of Service QoS parameter of ATM

is probably more suitable for dimensioning but more dicult to compute when compared with the tagged

cell loss

The Leaky Bucket Rate Control Scheme

In high sp eed networks conventional windowow control is not feasible since propagation

delays are the dominating factor in overall delay rather than transmission or queueing delays

In simple words if we force the transmitter to wait for an acknowledgmentwe will force it to

remain idle for a long p erio d of time since a time equal to the roundtrip delay elapses b efore

a new message can b e transmitted For this reason alternativeschemes for owcontrol need

to b e found One of these is the so called ratecontrol scheme where the arrival rate to a

buer is ltered through a p olicing device One such device is the leaky bucket rate control

scheme

The leaky bucket see Figure consists of a token generator a p o ol of tokens and an

optional buer Tokens are generated every seconds and stored in the p o ol if the p o ol

contains less than L tokens Otherwise the token is lost Packetscells arrive to the buer

according to a trac pro cess that needs to b e controlled The arrival pro cess can b e as

general as the BatchMarkovian Arrival Pro cess BMAP although most of our numerical

investigations consider sp ecial cases of the BMAP suchasMarkov Mo dulated Poisson Pro cess

MMPP the Interrupted Poisson Pro cess IPP and the Poisson Pro cess PP Furthermore

the input pro cess can b e timeinhomogeneous ie its parameters maychange with time

as we shall see in Section An arriving packetcell that nds the token p o ol nonempty

departs the system immediately while removing one token from the token p o ol An arriving

packetcell that nds the token pool empty joins the queue if the buer is not full Now

when the queue is not emptywhich means that the token pool is empty and a token arrives

one packetcell departs immediately taking the token with it

Arriving cells Departing cells M 321

Buffer 1 2 Token pool

Token generator

Figure The leaky bucket rate control scheme

Existing mo dels in the op en literature are concerned with steadystate measures Wecan

characterize these mo dels as follows uidow mo del for arrival sources are considered in

p oint pro cess mo dels with continuoustime in p oint pro cesses

It should b e noted here that this denition of the leaky bucket is slightly dierent from the one adopted

by ITU and ATM Forum

with discretetime in Brownian motion mo dels and analysis based on sample

paths in

In this pap er we will describ e the continuous timedep endent b ehavior of the leaky bucket

adopting a p oint pro cess to characterize cell arrivals

Delayed Markov regenerative pro cesses

In the development of BMAPG and GIBMAP typ e of queueing mo dels it is

commonly assumed that the time origin coincides with a service completion or an arrival

instant When studying overload b ehavior of queueing systems as we shall see in this pap er

we need mo dels where the time origin do es not necessarily coincide with service completion

or arrival instant For this reason in this section we generalize the wellknown Markov

regenerative pro cess MRGP to the delayed Markov regenerative pro cess DMRGP

and develop the underlying equations that govern the conditional state probabilities Before

we formally dene the delayed Markov regenerative pro cess wegive some short of lo ose

denition of the Markov regenerative pro cess An MRGP can b e thought of as a pro cess

where there exist time ep o chs where the pro cess regenerates in the Markovian sense ie the

sto chastic evolution of the marking pro cess do es not dep end on the history of the pro cess

but rather on the current state The time ep o chs that the pro cess regenerates together

with the value that pro cess has at that p oint form what we call a Markov renewal sequence

MRS The sto chastic b ehavior of the Markov regenerative pro cess can b e describ ed bytwo

matrices namely Kt global kernel that captures the b ehavior of the pro cess assuming

the regeneration has o ccurred and Etlocalkernel that considers the b ehavior b etween two

regenerations It has b een shown that matrix V twithentries V t the time dep endent

state probabilities of an MRGP P Z tj jZ i satises a matrix generalized

Markov renewal equation V t Et dKxV t x Now for the developmentofthe

delayed Markov regenerative pro cess we rst dene the delayed Markov renewal sequence

the generalization of the Markov renewal sequence in Markov regenerative theory

Denition Asequence of bivariate random variables fY T g is cal led a time homoge

n n

neous Delayed Markov Renewal Sequence DMRS if

T T T Y

n n n

P fY j T x j Y ig F x

P fY j T T x j Y i T Y T Y T g

n n n n n n n

P fY j T T x j Y ig

n n n n

P fY j T T x j Y ig K x

Then the DMRGP is dened as follows

Denition A stochastic process fZ tt g is cal led a Delayed Markov Regenerative

Process DMRGP if there exists a T such that the process fZ t T t g is a

Markov Regenerative process

D D

Theorem Let V t V t be the matrices with entries V t and V t the condi

tional state probabilities of a Delayed Markov Regenerative ProcessDMRGP ie P Z t

j jZ i and a standard Markov regenerative process ie the process fZ t T t g

respectively Then V t satises the fol lowing integral equation

Z Z Z

t t ts

V tMt dFsEt s dFs dKxV t s x

where Mt represents the local kernel for the rst regeneration instant with entries

M t P fZ t j T tjY ig

Et the local kernel for al l other regeneration instants with entries

E t P fZ t j T T tjY ig

Proof Conditioning on T s the rst Markovian regeneration instant it is obvious

that for any t s matrix V t satises the following equation

dKxV t s x t s V t j T sEt s

Unconditioning we obtain Equation

Note that Equation is not a generalized Markovrenewal Equation since on the left hand

side wehave V t and on the righthandsidewehave V t Lo osely sp eaking in a delayed

Markov pro cess we allow for the statistics of the rst regeneration to b e dierent that the

remaining ones That is the reason whywe dene two lo cal kernels Etand Mt and two

global kernels KtandFt

The BatchMarkovian Arrival Pro cess

The Batch Markovian Arrival Pro cess BMAP is a p oint pro cess where times b etween

arrivals are governed by a Continuous Time Markov Chain CTMC with innitesimal gen

erator D over nite state space S For every state i S we asso ciate an exit rate and

a set of probabilities p k j k and j S fig

The BMAP pro cess determines arrivals based on state transitions given that the CTMC

was in state i after an exp onentially distributed so journ time with parameter when a

transition from state i to state j o ccurs it will giveriseto k arrivals and the corresp onding

probabilityisp k j

The evolution of the BMAP can b e describ ed by a sequence of matrices fD k g with

entries D i m D p j i j m j i and D p k j

i i i k i i

ii ij ij

k i j m

Let N t b e the numb er of arrivals in tand J t represent the state of the arrival pro cess

at time t The pair N tJt is a CTMC on the state space fi j i j mg

and innitesimal generator Q As the state space is innite along the rst dimension we

truncate it at N t n for the purp ose of representation The corresp onding generator

matrix Q is given by

D D D D

k n

D D D

k n

D D

k n

Q t

Note that Q lim Q Matrix e is given by

n n

At At At Ak t

k n

At At Ak t

k n

Q t

At Ak t

k n

Ak t

Ai tisan m m matrix with entries a i t the probabilityofhaving i arrivals in t

time units given that the auxiliary pro cess was in state j at time and is in state k at

Q t

time t The entries of e with few exceptions cannot b e computed in closedform In the

App endix we develop a metho d to compute Ai t for the Markovian arrival pro cess MAP

ie D k For large size problems however uniformization should then b e

used to numerically evaluate the elements Ai t

The BMAP contains many widely used pro cesses such as the Poisson pro cess the Batch

Poisson pro cess phasetyp e renewal pro cess the Markovian arrival pro cess MAP and the

Markov Mo dulated Poisson pro cess MMPP as sp ecial cases

The Mo del with a BMAP input source

Let X t denote the numb er of tokens in the token p o ol at time t Y t denote the number

of cells in the buer at time t and J t denote the phase of the auxiliary pro cess at time

t By observing the fact that token p o ol and the cell buer cannot b e simultaneously non

empty ie X t Y t andY t X t we dene the pro cess

Z tL X tY t ie wekeep track of the sum of the empty token places in the p o ol

and the numb er of cells in the cell buer It is easy to see that

X t L Z t

Y t Z t L

where a maxa and L the token p o ol size We will therefore study the two

dimensional sto chastic pro cess Z tJt

Theorem Consider a leaky bucket fed by a BMAP as an input and x the time origin at s

time units relative to the previous next token generation instant The twodimensional

stochastic process Z tJt is a delayed Markov regenerative process

Proof It is easy to see that the sequence F T where T is the time of the next token

n n n

generation instantand F Z T is a delayed Markov renewal sequence

Theorem Let Et and Kt given by

Aj i tut ut i

i jM L

i h

Al t ut ut i E t

lM Li

j M L

otherwise

A Aut i

Aj ut i

jM L

Aj i ut i

K t

i jM L

h i

Al ut i

lM Li

j M L

otherwise

with Ai t as denedearlier then the local and global kernels for the rst regeneration

periodare given by Et and Kt while for any other regeneration periods are given

by Et and Kt respectively

Proof Straightforward from the denitions of matrices Ktand Et

Note that matrix Kt Equation dep ends on t only through the unit step function

ut Hence Kt can b e written as

Kt Kut P ut

where P is the transition probability matrix of the DTMC emb edded immediately after

the token generation instants Observing the fact that the derivativeof Kt can b e

written as P t where t is the Diracs delta function also called the unit

impulse The recursive relationship b etween conditional state probabilities at time t and

state probabilities at time t is

Et t

P Et t

V t

P V t t

Figure shows the transient state probabilities obtained from the solution of a three state

system ie L M with Observe the p erio dic b ehavior of V t for large values

ij 1.0

P0(t)

P1(t) 0.8 P2(t)

0.6

0.4 Transient probabilities

0.2

0.0 0.0 2.0 4.0 6.0 8.0

Time (sec)

Figure Transient probabilities for state system

of t and the discontinuities that o ccur at regeneration p oints For any time t Equation

can b e written as

ut ut n ut

V t Et P P Es

with n b c and s t n The ab ove equation is useful when studying the leaky

bucket b ehavior under nonstationary conditions overload in a later section For an initial

system state probabilityvector p the system state probabilityvector at time t is given

ut ut n ut

pt pEt P P Es

where ut is the unit step function shifted units from the origin In the limit as t

approaches innity pt b ecomes p erio dic with p erio d

lim p n s Es

where lim pP P This p erio dicity has b een rep orted in Note that

satises the equation P The elements of represent the stationary probability

vector of a DTMC emb edded immediately after the token generation instants

We can also quantify the discontinuities in state probabilities as follows Let Rn

V n V n the discontinuities of the elements of matrix V tattime

n We can easily see that

ut n

RnP P P E

As n approaches innity R lim Rnisgiven by

R I E

ut n

and lim P P One can also see that

V n V n i M L j M L

ij ij

V n i M L

iM L

V n V n V n i M L

i i i

For single arrivals we dene the loss probabilityattimet as follows

p t Prob fp o ol empty and buer full j arrival o ccurs at time tg

loss

The ab ove measure is basically the probability that an incoming customer will nd the buer

full and the token pool empty at the time of the arrival Relating the outside observers

distribution pt with the arriving customers distribution and for single arrivals we can show

see Co op er that

p t

LM LM

p t

loss

p t

j j

For the Poisson pro cess arrivals are not statedep endentandtheabove equation b ecomes

p t a p t a

loss LM

Figure shows the loss probability obtained as V t assuming that we start with a

full token p o ol For MAP arrivals tagged cell loss can b e expressed as a vector

p t aD

p t a

loss

p tD e

with D as dened in Section and p tisavector with entries

loss

p tP fZ t L M tj g

loss

It is interesting to note is that the denominator of Equation do es not dep end on

For example for an IPP source Equation b ecomes

p t

loss

p t

with p t the probability that the source is ON at time t ON 1.0

M+L=10 M+L=20 0.8

0.6

Loss probability 0.4

0.2

0.0 0.0 10.0 20.0 30.0 40.0

Time (secs)

Figure Transient loss probabilities for dierent leaky bucket parameters

Generalizing this measure of loss for batch arrivals we dene the probability that a batch

of size n will suer m n losses as

p tD

j n

LM nm

p n m t

loss

p tD e

j n

Another measure of loss whichisvery closely related to the Cell Loss Ratio CLR dened

in ATM Forum is the exp ected fraction of cells lost up to time tThiscanbeevaluated

Exp ected numb er of losses int

f t

loss

Exp ected numb er of arrivals int

For Poisson arrivals the ab ove measure is simply

f t p xdx

loss loss

The ab ove measure dictates the need to compute timeaverages of the state probabilities

p t Dene st to b e the time average of vector ptie

st pxdx

Indeed it is not dicult to show after integrating Equation for any t n s

Exdx t

R R

Exdx P P Exdx

Exdx t P P

As t approaches innity st approaches a limit

s limst Exdx

Figure shows the time averages corresp onding to the probabilities of Figure Poisson

arrivals Notice that no discontinuities are present It is interesting to see that for Poisson

1.0

0.8

l00(t)

l01(t)

l02(t)

0.6

Time averages 0.4

0.2

0.0 0.0 2.0 4.0 6.0 8.0

Time (sec)

Figure Time averages for a state system

arrivals as t tends to innity f t approaches a limit

M Ln M L i

X X X

f limf t a

n j

i n j

Bucket t

size

Steadystate

Table Steadystate vs transient measures

where are the elements of vector limiting probabilityvector of the emb edded DTMC just

after token generation instants and a e After appropriate change in notation

and relating limiting state probabilityvectors just prior and just after token generation

instants Equation wecanshow that Equation is in agreement with the steady

state measure of loss probability dened in Fractional losses can b e computed for more

complex arrival pro cesses For example for MAP arrivals using Equation fractional

loss in the interval t is given by

s tD

f t

s tD e

Vectors s t are computed using Equation

In Table we compare transient and steadystate fractional loss computed using Equa

tions and resp ectively The mission time t was chosen as time units that

corresp onds to a connection of duration of minutes Note that transient and steadystate

measures are very close This indicates the fact during the duration of the connection the

leaky bucket practically reaches steadystate Time intervals where transient analysis is im

p ortant should therefore b e much smaller We consider such short intervals in subsequent

examples In Figure we show the timeaveraged loss probabilities for dierentbucket sizes

We can also dene the throughput at time t as

T t ip t p xdx

i loss

where the summation term represents the mean queue length of the buer As t tends to

innity

T limT t f

Once again this matches with Formula in 1.0

0.8

0.6 M+L = 10 M+L = 20

Time averages 0.4

0.2

0.0 0.0 20.0 40.0 60.0 80.0

Time (sec)

Figure Time averaged loss probabilities for dierent leakybucket parameters

In Figure the loss probabilityofa leakybucket with L M and an IPP input

with and isshown for dierentvalues of J A leaky bucket with Poisson

arrivals and the same eectivearrival rate is also shown for comparison

ploss(t), Poisson arrivals

p loss(t|J(0)=OFF), IPP p (t|J(0)=ON), IPP 0.50 loss

0.40

0.30 Cell loss probability

0.20

0.10

0.00 0.0 5.0 10.0 15.0 20.0

Time (sec)

Figure Loss probabilities for various input sources

In Figure we plot fractional losses that corresp ond to the loss probabilities of Figure

Note that in spite of the fact wematched the mean arrival rates in b oth arrival streams

the IPP mo del gives much larger loss measures that the Poisson mo del This strengthens

the fact that loss probability strongly dep ends on higher moments not only the mean of

interarrival times In Figure we plot fractional loss for various initial bucket sizes

0.40

Poisson arrivals ON-OFF source, J(0)=OFF ON-OFF source, J(0)=ON

0.30

0.20 Expected fractional loss

0.10

0.00 0.0 20.0 40.0 60.0 80.0

Time (sec)

Figure Time averaged loss probabilities for various input sources

1.0

0.8 Z(0) = 10 Z(0) = 7 Z(0) = 5 Z(0) = 0

0.6

0.4 Expected fractional loss

0.2

0.0 0.0 20.0 40.0 60.0 80.0

Time (sec)

Figure Time averaged loss probabilities for various bucket initial states

Resp onse to an overload

In this section we will consider the leaky bucket resp onse to an overload In particular we

will assume that the leaky bucket is observed when its limiting state probabilityvector is

i Here i is a vector whose elements are the set of parameters used to describ e the

arrival pro cess For an ONOFF source this vector consists of three elements a b

At s s time units after a token generation an overload o ccurs so that an arrival

ii010i τ τ s0 0 v

τ sv

τ = τ − 0 s0 tv Overload Overload

occurs disappears

Figure Relevant times for overload b ehavior

pro cess with a parameter vector i is activated For instance in the case of an ONOFF

We seek an expression for system state probabilities source i a b with

a a

t time units after the overload o ccurs Using Equation and i as the initial state

probabilityvector we obtain

i Ei s Ei t t

ps i i t

i Ei s Pi P i Ei s t

where n b c s and s t n Let us now assume that t time units

after the o ccurrence of the overload it disapp ears Then generalizing Equation wecan

write down for any time t

i Ei s Ei t t

i Ei s Pi P i Ei s t t

ps i i t t

ps i i t t Ei t t t t t

v v v v v v

i Ei s t t ps i i t t Pi P

v v v v v

v v

c s t n t Expressions with n s same as in Equation and n b

v v

for fractional losses can also b e obtained byintegrating Equation as

i Ei s I i t t

i Ei s Pi P i I i

i Ei s Pi P i I i s t t

f s i i t t

v v

f s i i t t

ps i i t t I i t t t t t

v v v v v v

f s i i t t

v v

ps i i t t Pi P i I i

v v v

ps i i t t Pi P i I i t t

v v v v

where I i t Ei xdx In Figures and we assume a nonhomogeneous Poisson

arrival pro cess with rate

The ab ove dened nonhomogeneous pro cess represents an overload with i b

i b andt We assign

Figures shows the loss probability while Figure shows the fractional loss Note

that standard transient resp onse measures suchasoversho ot and relaxation times can b e

observed in Figure

Dimensioning the Leaky Bucket

In the previous sections we fo cused on the derivation of the leaky bucket state probability

vector at an arbitrary time p oint t assuming a given input parameter vector i and leaky

bucket parameters token p o ol size L buer size M and intertoken generation time In

this section we consider the reverse problem for an input trac with sp ecic input trac

characteristics what are the leaky bucket parameters to achievegiven system state distri

bution In particular we will fo cus on measures of loss probability The p erio dic b ehavior

of the leaky bucket state distribution makes this problem very interesting M+L=10 M+L=20 0.4

Loss probability 0.2

0.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0

Time (secs)

Figure Overload b ehavior loss probabilities for dierent LB parameters

M+L = 10 Overload M+L = 20 0.20 duration

Fractional loss 0.10

Relaxation time Overshoot

0.00 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0

Time (sec)

Figure Overload b ehavior fractional loss probabilities for dierent LB parameters

It was observed in this pap er and also noted previously that the loss probability

dep ends on the sum of the sizes of the two buers cell and token L M LetT denote

this sum The role of the cell buer is to smo oth out the input trac at the exp ense of

larger delays The cell buer b ecomes imp ortant when the leaky bucket is used as trac

shap er and its b ehavior is characterized using the cell departure pro cess When the leaky

bucket is used as a p olicing device ie as a means to control and protect network resources

from malicious as well as unintentional misb ehavior loss probability is the primary measure

of interest and as mentioned earlier the partition of the total buer space into token buer

and cell buer is not imp ortant We will therefore dimension the leaky bucket considering

the parameters T and

The role of the leaky bucket as p olicing mechanism is to enforce negotiated trac param

eters at connection setup The numb er of parameters needed to characterize trac and QoS

in BISDN is still an op en question So far ITU and ATM Forum have standardized

p eak cell rate B cell delayvariation tolerance v sustainable or mean cell rate and

burst tolerance Note that burst tolerance denes the maximum burst length that should

b e accommo dated by the network Many authors however such as consider the mean

burst size as opp osed to the worstcase burst size as the trac parameter In any case

the parameters can b e mapp ed to an ONOFF source trac mo del ONOFF source mo dels

have b een used extensively to mo del ATM trac due to their simplicity and their abilityto

capture source burstiness Figure shows the ONOFF source mo del in the ON p erio d

1/B

Figure The ONOFF source mo del

of duration D a constant p eak rateB is oered to the leaky bucket whereas in the OFF

p erio d of duration F the rate is The mean rate is then given by B B When the

m p

D F

duration of the p eak rate has the interpretation of burst tolerance the ON and OFF p erio ds

are deterministic numbers If however mean burst sizes are considered then ON and OFF

p erio ds random variables For most of our results we will consider an IPP as our ONOFF

source mo del and therefore our burst size interpretation is that of a mean value

Dimensioning based on Peak Rate enforcement

In this section we will fo cus on p eak rate dimensioning ie successivecellinterarrival

Let us assume that the arrival source can b e times should exceed a presp ecied value

characterized as a renewal pro cess with an interarrival distribution F t P At We

will say that the source complies if two successive arrivals o ccur at least b time units

apart Let C denote the event that the source complies with the imp osed constraint on the

p eak rate Then

P C F b

and

C F b P

Using the theorem of total probabilitywe can express cell loss probability as

P p p P C p C

loss lossjC

C lossj

In cases where p has a large value the contribution to loss probability can b e due

lossjC

and therefore dimensioning for a small p may actually mean minimizing loss to p

loss

C lossj

when the source is not complying and therefore the leaky bucket is not actually rejecting or

marking cells as low priority cells when it is supp osed to do so The waywe conceive p eak

rate enforcement is as follows

Given an interarrival distribution F t P At and a peak rate constraint B ie

A p

successive cel ls should bespacedatleast b time units apart nd the values of leaky

bucket parameters T L M and such that p is minimized ideal ly and p is

lossjC

lossj C

maximized ideal ly

Wenow turn our attention to computing p and p We will also assume that

lossjC

lossj C

b Let Es denote the state probabilityvector at time s s Using the

arrival theorem we can nd the state distribution as seen by arrivals s If an

arriving customer sees the system in state L M or in state L M with corresp onding

a a

probabilities sand s then immediately after the arrival the state of the

LM LM

a a

s Our goal is to s system is L M with the corresp onding probability

LM LM

compute the loss probability conditional on the next arrival Referring to Figure wecan

as express P and P

lossC

C loss

τ > s+b

0 s s+b τ

0 s τ s+b

Figure Timing relations b etween two arrivalsinaleakybucket

a a

s s F s F b b s

A A

LM LM

lossC

b s

a a

s sF b b s

LM LM

a a

loss C

s sF s s b s

LM LM

p and p are commonly referred to as probability of false alarm and probability of detection

lossjC

lossj C

resp ectively

Noting the fact that an arrival instant is uniformly distributed in the interval and

taking time averages for the interval of length from Equation an expression for loss

probability that do es not dep end on the observation instant can b e written down as

P f f P C f C

loss

lossjC

C lossj

In Figure weplotf vs f as a function of and dierentvalues of T The

lossjC

arrival pro cess is assumed to b e a Poisson pro cess with parameter b is chosen as

is also shown As time units The straight line that corresp onds to f f

lossjC

C lossj

mentioned earlier the ideal b ehavior is obtained at the p ointf f Note

lossjC

lossj C

that by decreasing we actually decrease b oth fractional losses and for b f becomes

lossjC

zero This means that the parameter by itself cannot eectively control the loss of non

increase complying cells Note that by decreasing the buer size b oth f and f

lossjC

C lossj

therefore the choice b and T will at least guarantee no loss for compliantcells

and simultaneously will reject a large fraction of but not all the noncompliant cells The

ab ove suggests that deterministic token generation times cannot guarantee the rejection of

all noncompliant cells Nevertheless there exists a waytomake this happ en if one token is

generated b time units after the last compliant cell In fact this trivial onebuer leaky

bucket with this nonp erio dic token generation times is equivalent to the continuousstate

leaky bucket standardized by ITU for p eak rate enforcement with the assumption that

the cell delayvariation tolerance is zero An example that illustrates why p erio dic token

generations cannot eectively enforce p eak rate is shown in Figure Scheme will often

let noncompliant arrivals enter the network while scheme will not Note that b oth will

let all compliant arrivals in

τ Leaky bucket with regular token generation instants (1)

Non compliant arrival

Compliant arrival

Lost arrival

Leaky bucket with irregular token generation instants (2)

Figure A sample arrival stream to the two leaky bucket algorithms 1.0

Increasing τ

0.8

- f = f 0.6 l | C l | C τ = 0.5 L+M=1 L+M=3 | L+M=10 l | C f

0.4

0.2

0.0 0.0 0.1 0.2 0.3 0.4 0.5

fl | C

Figure f vs f as a function of

lossjC

Dimensioning based on Mean Rate enforcement

Wenowtake a dierent angle and dimension based on mean rate The leaky bucket is now

required to enforce a given mean rate We rst dene two parameters that are imp ortanton

our p olicing mechanisms

Probability of false alarm is the probability the leaky bucket will reject a compliant

cell For mean rate enforcement f mayhave the interpretation as fraction of cells

loss

rejected due to false alarms when the mean rate is compliant

Detection time is elapsed time b efore the leaky bucket takes action in the o ccurrence

of an overload

The problem is stated as follows

Given a desiredmean rate what are the values of and T L M that guarantee

minimum loss ie probability of false alarm is minimized while the detection time is smal l

In order to answer this problem in Figure welookathow the fractional loss f b ehaves

loss

as a function of the normalized mean arrival rate for dierentvalues of and T The

p oint on the x axis represent a compliant source eg its mean rate is the presp ecied

one Ideally wewould liketohave f equal to zero at that p oint Also as the mean rate

loss

deviates from the presp ecied one loss probability should increase such that the throughput

is reduced to the presp ecied one ie

f f

m loss loss

for

m 0.8

0.6

Ideal τ = 1 τ = 0.9 τ = 0.8

0.4 Fractional loss

0.2

0.0 0.0 1.0 2.0 3.0

Normalized Arrival Rate

Figure Fractional loss vs normalized arrival rate

gives results very close to the ideal b ehavior From Figure we see that the choice

except when is very close to Now in order to minimize the probability of false alarm

we need to cho ose T L M as large as p ossible Nevertheless we also minimize the

quick resp onse to an overload or short term congestion ie we make the leaky bucket

less sensitive to short term congestions In order to quantify the detection time we use our

previously develop ed mo del for overload b ehavior We consider the following nonstationary

arrival pro cess

The detection time can b e interpreted as the time of the fractional loss to rise from the

value to a value f for the arrival pro cess dened in Equation A go o d

loss

b ehavior would imply that f will achievea value close to in a short p erio d of time

loss

In Figure we illustrate the eect of the total buer size on the detection time assuming

and We note that t t t and we conclude as we

increase the total buer size detection time increases where t denotes detection time for a

leaky bucket with total buer space equal to i

Controlling the p eak rate duration

In this section wewillquantify the eectiveness of the leaky bucket to control the duration

of the p eak rate Assume a source that emits cells at constant p eak rate B and let token

generation rate b e r Let I T denote the sum of the number of tokens in the token buer

and the number of empty places in the cell buer Then for a given p eak duration D D B

p λ2λ Ideal behavior

0.50

t5 0.40 t10

0.30 Fractional loss

0.20 L+M=5 L+M=10 L+M=20

0.10

0.00 -5.0 5.0 15.0 25.0 35.0

Time

Figure Tradeo b etween probability of false alarm and detection time

cells and D r tokens are generated We require that no more than lost cells for the

duration of the burst ie

maxfD B D r I T g

D B

We can nd the maximum allowable burst that satises the given loss requirements as

I T

B r

Now since I T is prop ortional to T wecansay that the burst tolerance is directly prop or

tional to T and is inversely prop ortional for a given to the dierence b etween p eak rate

and token rate

If the burst size parameter is dened as a mean value an ONOFF source mo del suchas

IPP can b e used to dimension the leaky bucket We will then nd the value of T L M

suchthatatypical burst will suer a loss less than a given threshold As a typical

burst Figure we consider an ON p erio d that lasts for time units the mean so journ

time in the ON state and that o ccurs time units b efore or after a token generation

instant The limiting probability that the b eginning of a burst that o ccurs relatively to

a token generation instant will nd the leaky bucket in a given state is given by

PrfState i source is ONg

PrfState i j source is ONg

Prfsource is ONg

The ab ove suggests that Now the longrun probability that the source is ON is simply

the limiting distribution at the b eginning of ON p erio ds is simply found by normalizing ON Period τ/2

1/a

Figure A typical burst for controlling the duration of the p eak rate

according to the state probabilities that the source is ON Fractional loss will then b e given

f S

loss ON

a a

In Figure we plot fractional loss for an ONOFF source with parameters a b

and Observethatifwe require the fractional loss for the typical burst to b e

0.25

0.20

L+M = 5 L+M=10 L+M=15

0.15 Fractional loss

0.10 fractional loss 9%

0.05 0.0 1.0 2.0

Time

Figure Controlling the p eak rate duration

less than we should cho ose a bucket size equal to

To summarize we observe that for mean rate enforcement the choice is an essential

one It should b e noted that this choice leads to large loss probabilities false alarm for

reasonable values of T L M Additionally the sizes of the buer and the token pool

control the detection time or equivalently resp onsiveness to shortterm congestions the

burst size and the probability of false alarm

Conclusions

In this pap er wepresented the timedep endent b ehavior of the leaky bucket rate control

scheme under general p ointarrival pro cesses Using the theory of Markov regenerative pro

cesses we showed that system state distribution has no p ointwise limits but its time averages

have limits Due to its p opularity as a p olicing device we dened two timedep endent loss

related measures We used these measures to quantify the overload resp onse of the leaky

bucket and to study the eectiveness of the leaky bucket to control standard trac param

eters such as p eak rate mean rate and p eak rate duration Transient analysis provides us

with additional insight ab out the dynamic b ehavior of the leaky bucket and its resp onsive

ness to overload conditions Our results suggest the tradeos for the dierent quantities to

b e controlled

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App endix

Evaluation of An t for a MAP pro cess

The i j th element of matrix An t denotes the probabilityof n arrivals in t time units

given that the initial state is i and the state at time t is j Matrices An t satisfy the

dierential equations

dAn t

An tD An tD n

with the initial condition A I The entries of matrices D and D are dened in

Section The system of dierential equations can b e solved in s domain to give us the

Laplace transform of matrix An t

An ssI D D

Inverting the Laplace transform weobtain

n n

k k

X X

t t

An t B e C e

k k

Calculating matrices B C is not a trivial task Here we will use an alternative approach

k k

By taking Laplace transforms in b oth sides of Equation weobtain

An ssI D An sD

The ab ove equation involves a matrix inversion for the sp ecial case of a state MAP we

have

sI D Ls

s s

where

s p

p s

are the real ro ots of the p olynomial s s p p and

p p as dened in Section

n n

Let B C k n denote the co ecients of An t Wewould like to express

k k

n n n n

the co ecients B C k n in terms of B C k n

k k k k

In sdomain wehave

n n

X X

An s B C

k k

s s

k k

Now An s can b e written as

An s LsB D

s s

LsC D

s s

Using partial fraction expansion the ab ove equation b ecomes

k n

X X

B An s B

kl k

s s

l k

n k

X X

C C

k kl

s s

k l

The co ecients B and C are given by

kl kl

B L D l

k l

B L D lk

k l

l k B L D

and

C L D l

k k

k l

C L D l k

k l

C L D l k

The ab ove equation after some algebraic manipulation b ecomes

n n n n

X X X X

B C B

k k kl

s s

k k l k l

n n n n

X X X X

C B C

k k kl

s s

k k l k l

We can then write

n n n

B C B

k k k

n n

l n B B

kl k l l

n n n

C C B

k k k

n n

C l n C

k l kl l

Lets also dene the matrix An t as

An t Ak t

k n

Matrix An t for n represents the probabilityofhaving more than n arrivals up

to time t given that wewere in state i at time and we are in state j at time t This can b e

computed as

An tAn t An t

given the initial condition At

ab ab

n n e n e

ab ab

n e n n e

a b

with n n a p p and b p p A similar

ab ab

approach can b e applied for an mstate MAP Nevertheless this metho d requires the ro ots

of a p olynomial of degree m Uniformization could then b e applied to compute numerically

the entries An t