Optimal Adaptive Data Transmission over a Fading Channel with Deadline and Power Constraints Murtaza Zafer and Eytan Modiano Laboratory for Information and Decision Systems Massachusetts Institute of Technology

Motivation Big Picture (Main issues) – Deadline constrained data transmission Fading channel Energy limitations Applications – Sensor with time critical data Mobile device communicating multimedia/VoIP data Deep space communication Transmission energy cost is critical – utilize adaptive rate control

Motivation  Convexity convex increasing r data in buffer deadline time data in buffer deadline time Fundamental aspects of the Power-Rate function

Motivation  Convexity  Channel variations convex increasing r c3c3 c1c1 c2c2 improving channel data in buffer deadline timedata in buffer deadline time Fundamental aspects of the Power-Rate function

Problem Setup  B units of data, deadline T  The transmitter can control the rate Transmitter B c(t) receiver Tx. Power,

Problem Setup  B units of data, deadline T  The transmitter can control the rate  Transmission power, g(r) – convex increasing, and, is the channel state Transmitter B c(t) receiver Tx. Power,

Problem Setup  B units of data, deadline T  The transmitter can control the rate  Transmission power, g(r) – convex increasing, and, is the channel state  For this work, (Monomials) Transmitter B c(t) receiver Tx. Power,

Channel model – General Markov process Transition rate, c →ĉ, is, total rate out of c, Simplified representation Define Define Z(c) as, Channel transitions at rate. New state is, c1c1 c2c2 Problem Setup

Problem Summary Transmit B units of data by deadline T over a fading channel Channel state is a Markov process Objective: Minimize transmission energy cost

Problem Setup Problem Summary Transmit B units of data by deadline T over a fading channel Channel state is a Markov process Objective: Minimize transmission energy cost Continuous-time approach Transmitter controls the rate continuously over time Yields closed form solutions

Problem Setup Problem Summary Transmit B units of data by deadline T Channel state is a Markov process Objective: Minimize transmission energy cost Optimal solution - preview Tx. rate at time t = (amount of data left) * (urgency at t) Depends on channel and time

Problem Setup Problem Summary Transmit B units of data by deadline T Channel state is a Markov process Objective: Minimize transmission energy cost Optimal solution - preview Tx. rate at time t = (amount of data left) * (urgency at t) Two settings No power limits Short-term expected power limits

System state is (x,c,t) Transmission policy r(x,c,t) Sample path evolution – PDP process Stochastic Formulation Buffer dynamics time channel x(t) c0c0 c1c1 c2c2 t1t1 t2t2 x – amount of data in the queue at time t c – channel state at time t

Stochastic Formulation Expected energy cost starting in state (x,c,t) is, Minimum cost function J(x,c,t) is, Objective : Obtain J(x,c,t) among policies with x(T)=0 Policy r*(x,c,t) (optimal policy)

Stochastic Formulation Expected energy cost starting in state (x,c,t) is, Minimum cost function J(x,c,t) is, Objective :  Obtain J(x,c,t) for policies with x(T)=0  Policy r*(x,c,t) (optimal policy)

Optimality Conditions Consider a small interval [t,t+h] and apply Bellman’s principle With some algebra and taking limits h → 0, we get

Optimality Conditions Optimality conditions (HJB equation) Boundary conditions

 Optimal rate r*(.) depends on the channel state,, through  Optimal rate r*(.) is linear in x, with slope (“urgency” of transmission at t) Optimal Policy Theorem (Optimal Transmission Policy) amount of data left at t urgency of tx. at t

 ODE solved offline numerically with boundary conds.,  No channel variations, gives, (simple drain policy) Optimal Policy Theorem (Optimal Transmission Policy)

Example – Gilbert-Elliott Channel Good-bad channel model (Gilbert-Elliott channel)  Two states “good” and “bad”  Channel transitions with rate

Example – Constant Drift Channel Constant Drift Channel Optimal Transmission Policy independent of c where, Since, we have,

Problem Setup – Power Limits  The interval [0,T] is partitioned into L partitions 0

Problem Setup – Power Limits  The interval [0,T] is partitioned into L partitions  Let P be the short term expected power limit 0 (k th partition constraint)

Problem Setup – Power Limits  The interval [0,T] is partitioned into L partitions  Let P be the short term expected power limit 0 (k th partition constraint)  Penalty cost at T = transmission in time window (Penalty cost function)

Problem Setup Problem Statement subject to, (objective function) (L constraints) Stochastic optimization Continuous-time – minimization over a functional space Solution Approach – We will take a Lagrangian duality approach

Problem Setup Problem Statement subject to, (objective function) (L constraints)  Stochastic optimization  Continuous-time – minimization over a functional space Solution Approach – We will take a Lagrangian duality approach

Problem Setup Problem Statement subject to, (objective function) (L constraints)  Stochastic optimization  Continuous-time – minimization over a functional space Solution Approach – We will take a Lagrangian duality approach

Duality Approach 1. Form the Lagrangian using Lagrange multipliers 2. Obtain the dual function 3. Strong duality (maximize the dual function) Basic steps in the approach –

Duality Approach 1) Lagrangian function Let be the Lagrange multipliers for the L constraints 1. Form the Lagrangian using Lagrange multipliers 2. Obtain the dual function 3. Strong duality (maximize the dual function) Basic steps in the approach –

Duality Approach 2) Dual function Dual function is the minimum of the Lagrangian over the unconstrained set

Duality Approach 2) Dual function Dual function is the minimum of the Lagrangian over the unconstrained set Consider the minimization term in the equation above, This we know how to solve from the earlier formulation – except two changes 1) Cost function has a multiplicative term, 2) Boundary condition is different

Dual Function Theorem (Dual function and the minimizing r(.) function) 0 B x(t)

Dual Function Theorem (Dual function and the minimizing r(.) function) 0 B x(t)

Dual Function Theorem (Dual function and the minimizing r(.) function) 0 B x(t) where over the k th interval is the solution of the following system of ODE

Optimal Policy 3) Maximizing the dual function (Strong Duality) Theorem – Strong duality holds is the optimal cost of the primal (original constrained) problem is the initial amount of data ( = B) is the initial channel state

Optimal Policy 3) Maximizing the dual function (Strong Duality) Theorem – Strong duality holds If is the optimal policy for the original problem, then, is the maximizing Since the dual function is concave, the maximizing can be easily obtained offline numerically

Simulation Example Simulation setup  Two state (good-bad) channel model  Two policies – Lagrangian optimal and Full power  P is chosen so that for B ≤ 5, Full power Tx. empties the buffer over all sample paths

Summary  Deadline constrained data transmission  Continuous-time formulation – yielded simple optimal solution Future Directions  Multiple deadlines  Extensions to a network setting Summary & Future Work Tx. rate at time t = (amount of data left) * (urgency at t)

Thank you !! Papers can be found at – web.mit.edu/murtaza/www