Post on 23-Jun-2020
transcript
Networks: Design, Analysis andOptimization
Urtzi Ayesta
NETWORKS Grouphttp://www.bcamath.org/ayesta
BCAM-INRIA WorkshopSeptember 23, 2010
Motivation
• Internet is today the fundamental component of the
worldwide communication infrastructure
• Critical role in education, entertainment, business, social
life and tourism
• The use of both the Internet and wireless services has
experienced an explosive growth. Anticipate further
expansion in future, boosted by convergence of wireless and
internet access
• Need for the development and analysis of mathematical
models to predict and control the QoS in communication
systems
Objectives
We aim at contributing with both fundamental and applied
results:
• To develop mathematical methods and software tools
for the performance evaluation, optimization and control of
communication networks.
• Optimization and enhancement of communication
networks (protocols, architectures, applications)
Methodology
Stochastic modeling and analysis.
• Randomness is inherent in the network, for example the
behavior of Internet users when they connect, what they
download, when they disconnect is random
• Specifically: Queueing theory, stochastic scheduling
theory, optimal stochastic control, and asymptotic regimes
Game theory. The study of optimal decentralized decision
making
European context
• INRIA: CQFD (F. Dufour), TREC (F. Baccelli),
MAESTRO (Ph. Nain), RAP (Ph. Robert), PLANETE
(W. Dabbous), RESO (P. Goncalves), DYONISOS (G.
Rubino), MESCAL (B. Gaujal)
European context
• INRIA: CQFD (F. Dufour), TREC (F. Baccelli),
MAESTRO (Ph. Nain), RAP (Ph. Robert), PLANETE
(W. Dabbous), RESO (P. Goncalves), DYONISOS (G.
Rubino), MESCAL (B. Gaujal)
• CNRS: LAAS, LIX (Polytechnique), LIAFA, LaBRI
European context
• INRIA: CQFD (F. Dufour), TREC (F. Baccelli),
MAESTRO (Ph. Nain), RAP (Ph. Robert), PLANETE
(W. Dabbous), RESO (P. Goncalves), DYONISOS (G.
Rubino), MESCAL (B. Gaujal)
• CNRS: LAAS, LIX (Polytechnique), LIAFA, LaBRI
• Europe: EURANDOM (Eindhoven), CWI (Probability
and Stochastic Networks), EPFL, University of Cambridge,
UPMC (LIP6), ...
European context
• INRIA: CQFD (F. Dufour), TREC (F. Baccelli),
MAESTRO (Ph. Nain), RAP (Ph. Robert), PLANETE
(W. Dabbous), RESO (P. Goncalves), DYONISOS (G.
Rubino), MESCAL (B. Gaujal)
• CNRS: LAAS, LIX (Polytechnique), LIAFA, LaBRI
• Europe: EURANDOM (Eindhoven), CWI (Probability
and Stochastic Networks), EPFL, University of Cambridge,
UPMC (LIP6), ...
• Industry: Technicolor (Paris), TNO (Delft), Microsoft
Lab (Cambridge), Telefonica (Barcelona), France Telecom
R&D (Issy-les-Moulineaux and Sophia Antipolis), T-LABS
(Berlin)
Research directions
• (Nearly) Optimal Stochastic Control
Research directions
• (Nearly) Optimal Stochastic Control
• Asymptotic Analysis: Heavy-Traffic, Fluid limit
Research directions
• (Nearly) Optimal Stochastic Control
• Asymptotic Analysis: Heavy-Traffic, Fluid limit
• Queueing Games
Research directions
• (Nearly) Optimal Stochastic Control
• Asymptotic Analysis: Heavy-Traffic, Fluid limit
• Queueing Games
• Stochastic coupling and sample-path techniques
Research directions
• (Nearly) Optimal Stochastic Control
• Asymptotic Analysis: Heavy-Traffic, Fluid limit
• Queueing Games
• Stochastic coupling and sample-path techniques
and other techniques like
• Mean Field Limit, Large deviations and Differential
Traffic Theory
Scheduling in Wireless
• In each time slot, the base station
selects a customer to serve
• Channel conditions vary due to
fading and interference effects
• In each channel condition, the prob-
ability of completing the job in one
time slot is different
Which user to choose?
Channel quality varies over multiple time scales
Which user to choose?
Channel quality varies over multiple time scales
=⇒ Capacity scales with number of users
=⇒ The greedy control that selects always the user with
instantaneous higher capacity performs very poorly
Multi armed bandit problem
• A sequential decision problem where at each time slot the agent
must choose one of K available options
• Depending on the chosen action, the agent receives a payoff at
the end of the time slot
• Goal: maximize the present value of the future payoffs, choosing
the right sequence of actions
Opportunistic Scheduling
Serve a user who has a “good” channel condition with
respect to its own statistics
Proportional Fair Algorithm: At time t selects the user
with highest:feasible current rate
average rate
Opportunistic Scheduling
Serve a user who has a “good” channel condition with
respect to its own statistics
Proportional Fair Algorithm: At time t selects the user
with highest:feasible current rate
average rate
=⇒ Little mathematical understanding concerning the
properties of opportunistic schedulers
Problem Formulation
• Time is slotted.
• K classes of jobs, a new class- k user arrives with
probability λk
• Nk = {1, 2, ..., Nk} set of possible states for customer of
type k
• ∀n ∈ Nk, qk,n is the probability for customer k of being at
state n and µk,n is departure probability for customer k,
if served, when it is at state n
• 0 ≤ µk,1 ≤ µk,2 ≤ · · · ≤ µk,Nk≤ 1
• Independence in the state evolution history, independence
between different customer’s current states
MDP formulation
• A = {0, 1}: Action space.
• The expected one-period reward earned by customer k at state n,
depending if it is served or not, will be given by
R1k,n = −(1− µk,n) R0
k,n = −1
• Xk(·): state process of customer k and ak(·): action process of
customer k
Objective:
lim supT→∞
1
T
T−1∑
t=0
Eπ0
[
∑
k∈K
Ra(t)k,X(t)
]
subject to∑
k∈K
ak(t) = 1, for all t ∈ T
Relaxation
• We relax the constraint: serve one customer on average.
∑
k∈K
ak(t) = 1 =⇒ lim supT→∞
1
T
T−1∑
t=0
Eπ
0
[
∑
k∈K
ak(t)
]
= 1
We obtain the next relaxed problem:
lim supT→∞
1
T
T−1∑
t=0
Eπ0
[
∑
k∈K
Ra(t)k,X(t)
]
subject to lim supT→∞
1
T
T−1∑
t=0
Eπ
0
[
∑
k∈K
ak(t)
]
= 1
Solution: Potential Improvement rule
Relaxed problem can be approached using Lagrangian methods.
∑
k∈K
lim supT→∞
1
T
T−1∑
t=0
Eπ0
[
Ra(t)k,X(t)
− νak(t)]
− ν
Solution: Potential Improvement rule
Relaxed problem can be approached using Lagrangian methods.
∑
k∈K
lim supT→∞
1
T
T−1∑
t=0
Eπ0
[
Ra(t)k,X(t)
− νak(t)]
− ν
Theorem: Let
νk,n =µk,n
∑
m>n
qk,m(µk,m − µk,n)for n 6= Nk, νk,Nk
= ∞
Then:
• If ν ≤ νk,n, it is optimal to serve customer k under state n ∈ Nk;
• If ν ≥ νk,n, it is optimal not to serve customer k under state
n ∈ Nk
Solution: Potential Improvement rule
• We construct a feasible policy for the original problem,
using the optimal solution of the relaxed problem:
• Potential Improvement rule: Give service at time t to
job k∗ (t) such that:
k∗(t) := argmaxk∈K
µk,n∑
m>n
qk,m(µk,m − µk,n)
• Not necessarily optimal for the original problem
Scheduling disciplines
• µ-index: νµk,n := µk,n for n ∈ Nk
• Score Based index (T.Bonald, 2004):
νSBk,n :=∑n
m=1 qk,m, for n ∈ Nk
• Relatively Best index (Qualcomm 3G standard, 2000):
νRBk,n :=
µk,n
Nk∑
m=1
qk,mµk,m
, for n ∈ Nk.
• Potential Improvement index:
νPIk,n =µk,n
∑
m>n
qk,m(µk,m − µk,n)for n 6= Nk, νPIk,Nk
= ∞
Simulations
0.5 0.60 0.70 0.80 0.90 10
20
40
60
80
ρ
Mea
n nu
mbe
r of
jobs
in th
e sy
stem
CµRBSBPI
How good is the heuristic?
Questions:
• Stability
• Optimal in some sense?
How good is the heuristic?
Questions:
• Stability
• Optimal in some sense?
Fluid limit: Y r(t) := Xr(rt)r
=⇒ we characterize the Maximum stability condition and
the set of policies that achieve maximum stability
Fluid Limit
Theorem: For a given policy f inducing a partially increasing vector
field with uniform limits drift, we have:
limr→∞
P( sup0≤s≤t
|Y f,r(s)− yf (s)| ≥ ε) = 0, for all ε > 0,
with yf (t) a piece-wise linear function
0 20 40 60 80 100 1200
0.5
1
1.5
2
2.5
t
Y 1 r
( t
) +
Y 2 r
( t
)
PIcµRBSB, PB
Stability
Theorem: A policy f inducing a partially increasing drift
with uniform limits is stable if Tfl < ∞ for all l, where
Tfl = inf{t ≥ 0, s.t. yl(t) = 0}
Stability
Theorem: A policy f inducing a partially increasing drift
with uniform limits is stable if Tfl < ∞ for all l, where
Tfl = inf{t ≥ 0, s.t. yl(t) = 0}
Theorem: The maximum stability condition is
K∑
k=1
λk
µk,Nk
< 1.
Any policy that prefers a customer in its best state over any
other customer is maximum stable
=⇒ PI has maximum stability region
Asymptotic Fluid Optimality
min
∫ D
0
K∑
k=1
ckxk(t),
xk(t) = xk(0) + λkt−
Nk∑
n=1
µk,n
∫ t
0uk,n(v)dv,
xk(t) ≥ 0, k = 1, . . . ,K
such that for all v ≥ 0,
K∑
k=1
Nk∑
n=1
uk,n(v) ≤ 1, uk,n(v) ≥ 0, for all k, n
Asymptotic Fluid Optimality
Lemma. Assume µ1,N1≥ µ2,N2
≥ . . . ≥ µK,NK. The fluid
control that solves the fluid control problem is as follows: Let
l = argmin{k : xk(t) > 0}. Then
u∗k,Nk(t) =
λk
µk,Nk
, for k < l, u∗l,Nl(t) = 1−
l−1∑
i=1
λi
µi,Ni
Asymptotic Fluid Optimality
Lemma. Assume µ1,N1≥ µ2,N2
≥ . . . ≥ µK,NK. The fluid
control that solves the fluid control problem is as follows: Let
l = argmin{k : xk(t) > 0}. Then
u∗k,Nk(t) =
λk
µk,Nk
, for k < l, u∗l,Nl(t) = 1−
l−1∑
i=1
λi
µi,Ni
Lemma. For any policy f and D > 0 we have
lim infr→∞
E
(
∫ D
0
K∑
k=1
Yf,rk (t)dt
)
≥
∫ D
0
K∑
k=1
x∗k(t)dt
Theorem. The PI policy is asymptotically fluid optimal
Conclusions
• PI: A heuristic simple to implement
• Characterize the stability region, the set of stable policies, and set
of asymptotic fluid optimal policies
• PI: Maximum stability region and is asymptotically fluid optimal
Conclusions
• PI: A heuristic simple to implement
• Characterize the stability region, the set of stable policies, and set
of asymptotic fluid optimal policies
• PI: Maximum stability region and is asymptotically fluid optimal
U. Ayesta, M. Erausquin, P. Jacko, A Modeling Framework for Optimizing
the Flow-Level Scheduling with Time-Varying Channels. to appear
Performance Evaluation 2010
U. Ayesta, M. Erausquin, M. Jonckheere, I.M. Verloop, Opportunistic
scheduling in wireless systems: stability and asymptotic optimality, submitted
for publication
U. Ayesta, P. Jacko, Method for selecting a transmission channel within a
time division multiple access (TDMA) communication system, BCAM,
Application Number 10380060.3, April 2010
Perspectives
• Combination of Deterministic and Stochastic techniques
Perspectives
• Combination of Deterministic and Stochastic techniques
• Performance Evaluation+Game Theory = Queueing
Games
Perspectives
• Combination of Deterministic and Stochastic techniques
• Performance Evaluation+Game Theory = Queueing
Games
• Bio-Inspired Networks, Swarm Intelligence
Perspectives
• Combination of Deterministic and Stochastic techniques
• Performance Evaluation+Game Theory = Queueing
Games
• Bio-Inspired Networks, Swarm Intelligence
• New applications: Clean-Slate Internet, Energy
minimization, electrical networks, power grid
Networks: Design, Analysis andOptimization
Urtzi Ayesta
NETWORKS Grouphttp://www.bcamath.org/ayesta
BCAM-INRIA WorkshopSeptember 23, 2010
NET group at BCAM
• U. Ayesta (Ikerbasque)
• J. Anselmi (postdoc)
• P. Jacko (postdoc)
• I.M. Verloop (postdoc)
• M. Erauskin (PhD, co-supervised with E. Ferreira)
• A. Izagirre (internship)
Solution to Relaxed Formulation
Relaxed problem can be approached using Lagrangian
methods.
lim supT→∞
1
T
T−1∑
t=0
(
Eπ0
[
Ra(t)X(t)
]
− ν Eπ
0
∑
k∈K
ak(t)
)
− ν
We decompose this problem in K subproblems:
lim supT→∞
1
T
T−1∑
t=0
(
Eπ0
[
Rak(t)k,X(t)
− νak(t)])
− ν
We solve K subproblems, and we obtain the joint optimal
policy for the relaxed problem combining them.
Wireless Channel
Channel quality varies over multiple time scales
Class of policies
Best Rate (BR) The BR policies are such that whenever
there are users present that are currently in their best channel
condition, i.e., in state Nk, only such users are served.
Best Rate Priority (BRB) A BRP policy is a BR policy
where ties are broken using the cµ-rule, that is, based on
ckµk,Nk.
Capacity scales with number of users
Simulations
0 1000 2000 3000 40000
50
100
150
200
250
300
350
Time
Num
ber
of jo
bs in
the
syst
ems
CµRBSBPI
Fluid Limit
Theorem: For a given policy f inducing a partially increasing vector
field with uniform limits drift, we have:
limr→∞
P( sup0≤s≤t
|Y f,r(s)− yf (s)| ≥ ε) = 0, for all ε > 0,
with yf (t) a piece-wise linear function
y1
y2
(0, 0)
(y1(0), y2(0))
Performance in overload
0 2000 4000 6000 8000 10000 120000
10
20
30
40
t
Y1r (t
)+Y
1r (t)
PI
Cµ
RB
SB