QoS Constrained Adaptation for

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QoS Constrained Adaptation for CDMA in the
Wideband Limit
Tim Holliday Andrea Goldsmith Peter Glynn
Stanford University
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Motivation

• Future wireless application will require tight
  constraints on QoS (delay, min. rate, etc.)
• Average metrics are not sufficient
• A dynamic programming approach to adaptation is
  required
• Interactive interference in multi-user systems
• Standard dynamic programming is intractable
• A new set of tools for optimizing systems with
  interacting controllers
• Exploits wideband limit approximations

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WCDMA System Model

• User signals can be decoded by one or more base
  stations
• Assume the base stations coordinate
• Assume linear multi-user receivers

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WCDMA Cross-Layer Model
Receiver
Channel
Traffic Generator
Data Buffer
Source Coder
Channel Coder
Modulator (Power)
Adaptive Control
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WCDMA Problem Formulation

• Optimize adaptation in a multi-user setting
• The users in the system interact through
  interference
• Creates a Chicken and Egg control problem
• We seek an optimal and stable equilibrium state
  and adaptation for the system of users
• The key is the stochastic process describing the
  interference

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Linear Multi-User Receiver

• Assume each of K mobiles is assigned a N-length
  random spreading sequence
• The power of a user affects the SIR of other
  users forcing them to change their power

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Interference Models

• Jointly model the state space of every mobile in
  the system
• Problem The system state space grows
  exponentially
• Assume unresponsive interference
• Avoids the Chicken and Egg control issue
• Problem Unresponsive interference models provide
  misleading results
• Diffusion approximation Stolyar, 2001
• Diffusions do not accurately capture delay
  performance
• We propose a new approach based on wideband limit
  approximations

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Wideband Limit Approximations
Power
Hz
B
B
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Optimization in the Wideband Limit

• We want to find the optimal multi-user
  cross-layer adaptation for a given performance
  metric, subject to QoS constraints
• Approximate the CDMA network dynamics through the
  wideband limit
• Optimize the control in the wideband limit
• We check convergence and uniqueness to ensure the
  solution is a good approximation to a finite
  bandwidth system

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Equilibrium in the Wideband Limit

• At time t, the ith user in the system has a state
  Xi(t) ? S
• For any K, N, the system state vector
  is the fraction of users in each state
• Define as the single user
  transition matrix
• In the wideband limit we have deterministic
  non-linear dynamics for the system state

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Value Function in the Wideband Limit

• The wideband approximation for the system has a
  unique fixed point
• This equilibrium allows us to use a simple form
  for the limiting average value function
• Notice that the fixed point equation is
  non-linear!

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Wideband Optimal Control Problem

• Same size as a single user optimization problem
• The non-linear constraint can introduce
  significant theoretical and computational
  complications
• The non-linear program is not convex
• We show that it can be solved by a sequence of
  linear programs

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Example Power Adaptation With Deadline
Constrained Traffic

• Assume deadline sensitive data (100ms)
• 50 km/h Microcell (COST 207)
• Minimize average transmission power subject to a
  deadline constraint
• What happens as system load increases?
• Let number of users per Hz vary between 0 and 1

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Power vs. System Load vs. Deadline Constraint
Infeasible Region
Average Power
Prob. Of Packet Loss
Users/Hz
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Technical Issues

• We show that we can always find a global optimal
  solution to the non-linear program.
• We have proven convergence and asymptotic
  optimality of the finite user controls
• No discontinuity in the wideband limit
• The nature of the wideband limit
• Technically we have applied an mean-field
  approximation LLN over bandwidth
• The CLT version induces a non-stationary
  Gauss-Markov process

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Summary

• A general framework for adaptation in multi-user
  wireless systems
• Characterization of of QoS constrained capacity
• A rigorous methodology for approximating the
  interaction between users.
• Powerful new machinery for analyzing the optimal
  control of dynamic systems with loosely
  interacting users

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Extensions

• Multi-user wireless networks
• Nonlinear receivers and/or multiple antennas
• Uplink/Downlink duality
• Non-stationary Gauss-Markov approximation for the
  interference (CLT approximation)
• Channel capacity with complex dynamics
• Multiple antenna channels
• Multiple access and broadcast channels

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Future Wireless Networks
Wireless Internet Streaming Audio/video Real-time
Voice Multiplayer Games Sensor Information Automat
ion and Control
Simple connectivity is no longer
sufficient Customers want content and services!
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Challenges

• Provisioning for many types of traffic
• Voice, video, streaming audio, data
• Each traffic type requires different delay
  constraints and rate requirements
• Time varying network quality
• All traffic goes over the same network!
• Introduces the need for cross-layer design.

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Cross-Layer Design

• Application layer
• Adaptive source coding and flexible QoS
• Network Layer
• Adaptive routing
• Medium Access (MAC) layer
• Multi-class queueing and prioritization
• Link layer
• Adaptive techniques (rate, power, coding)

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Future Research

• Cross-layer adaptation
• Multimedia
• Ad-Hoc networks
• Admissions control and routing
• New methodologies for solving these problems
• Intersection of dynamic systems, communications,
  and information theory

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Conclusions

• Cross-layer design is critical for delay
  constrained wireless networks
• Simple average constraints can be misleading!
• Developed a general framework for optimal
  cross-layer adaptation in multi-user networks
• Presented a new method for computing Shannon
  capacity of complex systems
• The techniques we develop can be applied across a
  wide range of problems in communications,
  control, and information theory.

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Capacity, Separation and Cross-Layer Design

• One of Shannons most famous results is a
  separation theorem for a class of channels
• Perhaps this might tell us something about the
  fundamental optimality of cross-layer design
• We consider the fundamental Shannon capacity of
  channels with complex dynamics
• Time-varying channels with memory
• Time-varying input processes with memory
• To obtain these capacity limits we have developed
  a new method of computing capacity using Lyapunov
  exponents

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Outline

• Introduction
• Cross-layer adaptation (single-user)
• Performance results
• Multi-user networks
• Capacity of time-varying systems

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Single User System Model
Receiver
Channel
Traffic Generator
Data Buffer
Source Coder
Channel Coder
Modulator (Power)
Cross-Layer System
Stringent QoS constraints require that the full
dynamics of the system be represented
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State-Space Model

• The interaction between the components of the
  cross-layer system is modeled as a single FSMC
• Judicious modeling choices are required to
  prevent intractable models
• The traffic generator and channel are the
  sources of randomness and are uncontrolled
• The joint adaptation policy defines the
  transition probabilities for the cross-layer
  system

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Problem Formulation

• Find the optimal adaptation for a given
  performance metric
• For example, minimum average transmit power
• Subject to appropriate QoS constraints
• Maximum delay
• Probability of packet loss or consecutive packet
  losses
• Dynamic programming is a natural approach to
  solve this problem
• The complexity of the cross-layer system model
  complicates the DP solution

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Definitions

• State of the cross-layer system
• The adaptive policy at time t is chosen by a
  control g
• The control determines the transition matrix P(g)
  of the cross-layer system.

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Value Function

• The desired performance metric determines a cost
  function r(g)
• The expected cost (value) function is minimized
  over a finite or infinite horizon
• We consider infinite horizon to avoid
  inappropriate time-scale issues

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Optimization Problem

• The optimal value function and optimal control
  can be found via a linear program
• The function f(p) denotes a set of performance
  constraints

The optimization problem is computationally
challenging
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Applications

• We have proposed a general problem formulation
• We illustrate the techniques with a specific
  example
• Power control and source/channel coding
• This example illustrates two key features
• The flaw of averages
• The benefit of cross-layer adaptation

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Power and Joint Source-Channel Coding for EDGE

• Traffic arrives according to an On/Off DTMC
• Source can be coded into 56 byte or 112 byte
  packets with a deadline of 100 milliseconds
• Channel code options are MCS-5 and MCS-7 (Rate
  0.37 and .74 8PSK)
• Power 20mW to 800mW in 2 dB increments
• TU-50 channel model within a microcell shadowing
  environment

First result Minimize average delay subject to
an average power constraint
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Delay Vs. Channel Gain For Different Power
Constraints
Delay in milliseconds
Channel Gain in dB
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Power Vs. Data Rate with a 50ms Constraint on
Conditional Expectation of Delay
Power in milliwatts
Source data rate in bits per second
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Single User Summary

• We develop a general framework for cross-layer
  adaptation.
• We have applied this frame work to a number of
  cross-layer optimization problems
• Permits a wide variety of cross-layer adaptation
  to meet QoS constraints.
• Flaw of Averages
• Cross-layer design provides substantial gains in
  both performance and efficiency

We want to extend previous system model to
multiple users in a cellular context
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Channels with Memory

• We consider the case of no channel state
  information (the transition dynamics are known)
• Time-varying channels with finite memory induce
  infinite memory in the channel output.
• Capacity for time-varying infinite memory
  channels is only known in terms of a limit
• Closed-form capacity solutions are only known in
  a few cases
• Gilbert Elliot Channel and Finite State Markov
  Channels

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A New Characterization of Channel Capacity

• Capacity using Lyapunov exponents
• Similar definitions hold for l(Y) and l(XY)

where the Lyapunov exponent
for Bxi a random matrix whose entries depend on
the input symbol Xi
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Intuition and Connections

• In some cases the Lyapunov exponent is entropy
• The vector pn is the direction associated with
  l(X) and the conditional channel state
  probability
• This vector has a number of interesting
  properties
• It is the standard prediction filter in hidden
  Markov models
• Under certain conditions we can use its
  stationary distribution to directly compute l(X)

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Computing Lyapunov Exponents

• Define p as the stationary distribution of the
  direction vector pn
• We prove that we can compute these Lyapunov
  exponents in closed form as
• This result is a significant advance in the
  theory of Lyapunov exponent computation

p
pn2
pn
pn1
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Computing Capacity

• Closed-form formula for mutual information
• We prove continuity of the Lyapunov exponents
  with respect to input distribution and channel
• Can thus maximize I(XY) relative to p(x), which
  yields the channel capacity
• We also develop a new CLT for sample entropy
• Rigorous confidence interval methodology for
  simulation-based estimates of entropy