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Information Theory for Mobile Ad-Hoc Networks
(ITMANET): The FLoWS Project
Competitive Scheduling in Wireless Networks with Correlated
Channel State
Ozan Candogan, Ishai Menache, Asu Ozdaglar, Pablo Parrilo
Competitive Scheduling in Wireless Collision Channels with
Correlated Channel State (Candogan, Menache, Ozdaglar, Parrilo)
Achievement:
• Convergent dynamics and equilibria characterization for
hard to analyze centralized schemes
with single performance objective
• Competitive scheduling models allow
the flexibility to incorporate different user
objectives, but focus mainly on users
with independent channel models
• Correlated channels are more realistic
extensions of the current model as they
incorporate joint fading effects
competitive scheduling in collision channels
• Bounds on price of stability (efficiency losses due to
selfishness)
• Equilibrium paradoxes: Bad-quality states can be used
more frequent than good states.
How it works:
• Best response dynamics (or fictitious play) and potential
game property imply convergence
p1
NEW INSIGHTS
+1
0
+1
-1, -1
4, 0
0
0, 4
-1, -1
A similar game with 2 users and a single
state. Our problem is more complex, with
continuous action spaces and multiple
fading states
• Selfish utility maximization framework
for collision channels
• Use tools from optimization theory to
analyze wireless network games
• Potential games and convergence
algorithms
p3
• Aggregate utility maximization
problem is non-convex and
difficult to solve. Instead, a
simple distributed framework
is suggested.
• Robust system design in the
presence of non-cooperative
users is achieved and
efficiency loss due to
selfishness of users is
bounded.
p1
p2
Combine tools from optimization
and game theory
IMPACT
• Existing work on scheduling focuses on
Central component
p2
BR(p-3)
Principle: When user 3 updates its strategy
through best-response, the potential function
increases after each update
• Characterization of the social optimum in order to find
efficiency losses due to selfishness.
• Bounds on price of stability is achieved by bounding the
change in aggregate utility when a Nash equilibrium is
obtained by perturbing the social optimum
Assumptions and limitations:
• Perfect correlation across channel state processes of users
• Symmetric-rate assumption for a more tractable analysis
NEXT-PHASE GOALS
STATUS QUO
FLOWS ACHIEVEMENT
Local components (buildings)
• Extend the results to partial channel
state-correlation (local & central
fading components)
• Additional channel models (e.g.,
CDMA)
• Convergence of dynamics with
asynchronous updates and with
limited information
• Extending the results to general
networks models which also include
routing
Robust scheduling and power allocation in wireless networks with selfish users
Motivation
• Centralized resource allocation in wireless networks is usually
– Hard to implement and sustain
– not robust to selfishness of users
• Hence, there is an increasing interest in distributed resource
allocation in wireless networks under the presence of self-interested
users.
• One of the most distinctive features of wireless networks is the timevarying nature of the channel quality, an effect known as fading.
• Users may base their transmission decision on the current channel
quality. This decision model naturally leads to a non-cooperative
game, since each user’s decision affects other users through the
commonly shared channel.
Motivation
• Unlike existing work in the area which assumes
independent state-processes for different users, we
consider the case where the channel state is correlated
across users.
• This correlated-fading model allows to capture global
network elements that affect all mobiles.
Goals and Relevance


Goals:
– Study the equilibria of the competitive scheduling game
– Establish tight bounds on efficiency loss due to noncooperative behavior
of the users
– Suggest network dynamics that converge to desired equilibrium points.
Relevant examples:
– Satellite networks
– Any uplink wireless network in which global noise effects are possible.
The Model
•
We consider a finite set of mobiles
transmitting to a common basestation.
•
Time is slotted, and the channel
quality is revealed to the user prior to
each transmission.
•
Each user may base its transmission
decision on the current channel state.
•
In the present work, we assume
perfect correlation, meaning that all
users observe the same channel state,
yet may obtain different rates for a
given state.
User 1, p1
User 2, p2
User 3, p3
Due to perfect correlation, all users
observe the same fading state at a given
time slot
•Let 1,2, ,hbe the state space. We denote by R j the rate that user m
will obtain in state j, provided that there are no simultaneous transmissions
•Monotonicity assumption: If j<i then
for every user m.
m
The Model
• Underlying fading process is assumed to be stationary-ergodic. We
denote by
the state-state probability for channel state i.
• Reception Model: We assume a collision channel: Simultaneous
transmissions are lost. A single transmission is always successful.
• Users are assumed to adopt stationary
strategies,
: policy of user m, where
: transmission probability of user m in state j
The Noncooperative Game
•
Performance measures: Average power and average throughput
•
Average power:
•
Average throughput:
•
Each user is interested in maximizing its throughput with minimal
power investment. We capture this tradeoff via the following utility
where
•
is a positive constant.
Each user is interested in maximizing the above utility subject to an
individual power constraint, i.e.,  i pim  P m
i
•
Nash equilibrium:
p m  arg max u m  p m , p  m 
pm
Centrally Optimal Scheduler
• Central optimization problem
• The central optimization problem is non-convex and hard to
characterize.
• Partial characterization: There exists an optimal solution of the central
optimization problem where users utilize threshold policies - meaning
that for each user m there exist at most a single state j so that 0  pim  1
Equilibrium Characterization
• A Nash equilibrium point always exists.
• There can be infinitely many Nash equilibria.
• For the case where P m  1, best equilibrium coincides
with the social optimum (i.e., price of stability is one).
• Price of anarchy (upper bound on the performance ratio
between the social optimum and the worst Nash
equilibrium) can be arbitrary large.
• Bad quality states might be utilized more frequently
than good states, where utilization is defined as the
overall transmission probability at a given state
Convergence to Equilibrium
• Additional assumption: Assume that rates are user independent, i.e.,
• Potential games are games in which a change in a single user’s
payoff is equivalent to a change in a system’s potential function. An
important property of such games is the convergence of sequential
best-response dynamics.
• Thus, under the above assumption, our game convergences to an
equilibrium in case that users update their strategies in a sequential
manner.
• Convergence properties without the above assumption are under
current study.
Future Work
• Further characterization of equilibrium points. Explicit bounds on
price of stability/anarchy for special cases of interest.
• Partial state-correlation models.
• Asynchronous best response dynamics and their properties
• Multi-hop networks.
• Additional reception models (such as CDMA based networks).