a performance model for an asynchronous optical buffer w. rogiest k. laevens d. fiems h. bruneel...

A performance model for an A performance model for an asynchronous optical bufferasynchronous optical buffer

W. Rogiest • K. Laevens • D. Fiems • H. Bruneel

SMACS Research Group

Ghent University

Performance 2005

Juan-Les-Pins, France

Performance 2005 • Juan-Les-Pins • Wouter Rogiest

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MotivationMotivation

core nodes(possibly co-located with the edge nodes)

edgenodes

(legacy) access networks

DWDM channels

optical channels vs. channels vs. electricalelectrical nodes nodes


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AimAim

optical switching (OBS/OPS) all-optical: new transport paradigm still need for contention resolution

a solution: optical buffering (for now) light cannot be stored, only delayed → fibers

aim: analyze model of an asynchronous equidistant fiber delay line (FDL) buffer

set of fibers (N+1 in number) with equidistant fiber lengths → delays 0*D,1*D, ... N*D N is the size, D the granularity, N*D the capacity

example for N=2


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OverviewOverview

ModelModel

ApproachApproach

AnalysisAnalysis

Numerical ResultsNumerical Results

ConclusionConclusion


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ModelModel • system equationsystem equation

for FDL buffers system of infinite size (N=∞)

only delays nD can be realized gives rise to “voids”

scheduling horizon ≠ unfinished work (due to voids) as seen by arrivals

queueing effect [x]+

(max{0,x}) FDL effect x (ceil(x))

valid for both slotted and unslotted systems

interarrival timeTk

"work" being done at

rate 1

kth arrival (k+1)st arrival

k

kk1k T

DH

DBH

burstsize Bk

void

Hk D Hk /D

Hk+1


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Approach (1)Approach (1) • assumptions assumptions

unslotted model for an FDL buffer single wavelength • uncorrelated arrivals • iid burst sizes

conventions slotted = synchronous = discrete time (DT)

unslotted = asynchronous = continuous time (CT)

(N = ∞) : infinite size buffer = infinite system

(N < ∞) : finite size buffer = finite system

strategy three mathematical domains several steps involved


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Approach (2)Approach (2) • domainsdomains

resulting

performance measures

sustainable load tail probabilities moments of the waiting

time

loss probabilitiesCT, N<∞

DT , N=∞

CT, N=∞

mathematical approach

z-domain probability generating functions

Laplace domain Laplace transforms

probability domain probabilities


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Approach (3)Approach (3) • steps steps

z-domain

Laplace domain

probability domain

direct approach

DT , N=∞

CT, N=∞

limit procedure

queueing effectFDL effect

queueing effectFDL effect

CT, N=∞ CT, N<∞

heuristic (1) : dom. pole approx.

heuristic (2) : heuristic approx.

“scratch”


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Analysis (1)Analysis (1) • z-domainz-domain

analysis assuming equilibrium

solution of queueing effect memoryless arrivals, well-known solution (see paper)

analysis of FDL effect in DT

"solve“

yields

where

D’ is DT granularity, an integer multiple of slots

D'H

D'F

k/D'j2πeεk

1D'

0kk

k

D'k )H(zε

1)(zε1)(zε

D'1

F(z)


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Analysis (2)Analysis (2) • to Laplace domainto Laplace domain

starting from results for a slotted model slot length (e.g. in s)

take limit 0 time-related quantities scale accordingly counting-related quantities do not

identity involving comb function

first way: limit procedure

second way: direct approach

k

D/kt2j

k

eD1

kDt


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both ways yield

D is the CT granularity, a real number

sD

k

1 1 eF(s) H(s j2πk/D)

D s j2πk/D

1D'

0kk

k

D'k )H(zε

1)(zε1)(zε

D'1

F(z)

Laplace transform domain

infinite sum

D is real

z-domain

finite sum

D’ is integer

Analysis (3)Analysis (3) • Laplace domainLaplace domain


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special cases for burst size distribution: closed-form formulas exponential deterministic mix of deterministic

heuristic, two parts: (1) dominant pole approximation,

allows to obtain overflow possibilities for infinite system

(2) heuristic approximation,

involving special expressions (see paper),

allows to obtain burst loss probabilities (BLP) for finite system

Analysis (4)Analysis (4) • to probability domainto probability domain

D]NProb[H


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yields

applying steps for each special case yields numerical results

D]NProb[H H(s)

Laplace transform domain

exact

N = ∞

probability domain

approximate

N < ∞

probability domain

approximate

N = ∞

BLP

Analysis (5)Analysis (5) • probability domainprobability domain


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Numerical example (1)Numerical example (1)

BLP as function of D (E[B]=50.0 s, N=20)

exponential burst size distribution

1E-5

1E-4

1E-3

1E-2

1E-1

1E+0

0 20 40 60 80

r = 40%

r= 80%

r = 60%

D

BLP


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Numerical example (2)Numerical example (2)

BLP as function of D (E[B]=50.0 s, N=20)

behaviour is similar to synchronous systems

deterministic burst size distribution

1E-7

1E-6

1E-5

1E-4

1E-3

1E-2

1E-1

1E+0

0 20 40 60 80

BLP

r = 40%

r= 80%

r = 60%

D


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ConclusionsConclusions

performance measures for finite asynchronous optical buffers

derived from infinite synchronous buffer model

asynchronous operation

behaviour is similar to synchronous systems

further research

comparison “synchronous vs. asynchronous” by studying batch arrivals

contact [email protected]

a performance model for an asynchronous optical buffer w. rogiest k. laevens d. fiems h. bruneel...

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