1. October 28, 2005 Single User Wireless Scheduling Policies: Opportunism and Optimality Brian Smith and Sriram Vishwanath University of Texas at Austin October 28 th, 2005 The 2005 Texas Wireless Symposium

2. October 28, 2005 Overview  Introduction  Wireless Downlink Model  Multi-User Diversity  Single User Scheduling  Gaussian Broadcast Channel Capacity  Ergodic Capacity  Achieving Boundary Points  Summary

3. October 28, 2005 Introduction  Discuss Rate Capacity for Wireless Downlink  Information theoretic viewpoint  Packet scheduling  Max-Rate  Max-Quantile  Simultaneous scheduling in Broadcast Channel  Capacity Region  Achieving maximum rates  Inspired by MIMO systems

4. October 28, 2005  Wireless Base Station with Two Users  Channel gains drawn independently from random distribution  Constant over time-slots, independent between time-slots  Both distribution and realization known to Base Station  Independent Gaussian noise  Transmit power budget P  Single User Rate Capacity:  R 1 ≤ lg (1+  1 P/N) Wireless Downlink Model Base Station P Receiver #1 Receiver #2 22 11

5. October 28, 2005  Channel Randomness Helps  Schedule Better User in each time Slot  Two State Example  Each State occurs with 50% probability Multi-User Diversity Example 6 2 4 8 R1R1 R1R1 R2R2 R2R2 5 5 R1R1 R2R2 State #1 State #2 Ergodic Capacity (4,3)

6. October 28, 2005 Opportunism  Apply Multi-user diversity to Downlink Problem  Fairness can become an issue with max-sum rate  Max Quantile  Schedule user who has best channel, with respect to his own channel distribution  Each user is served equal amount of the time  Many practical strategies to exploit diversity Base Station P Receiver #1 Receiver #2 22 11

7. October 28, 2005 Information Theoretic Broadcast Channel  Transmit messages at reduced rate to both receivers simultaneously  Message intended for other user treated as noise  Better user decodes both messages, discards unintended message  Interesting Feature of this Capacity Region  Max sum-rate always at endpoint  Send message exclusively to better user Base Station P Receiver #1 Receiver #2 22 11 CAPACITY REGION PLOT HERE

8. October 28, 2005 Ergodic Capacity of Fading Broadcast Channel  Assumptions:  Exponential distribution of received powers  In example plot, average powers received are 1 and 3  No power control  Max sum-rate point no longer at endpoint  Consequence of the fact that sometimes, Channel #1 is better than Channel #2 Max Sum- Rate Point

9. October 28, 2005 Optimality: Achieving Boundary Points  Observation:  Already shown how to achieve three boundary points with single-user scheduling  Always User #1, Always User #2, Always best User  Assertion:  No other boundary point can be achieved with a single-user strategy  Simultaneous scheduling on Broadcast channel required

10. October 28, 2005 Convex Region: Boundary Points and Maximization Problem  The boundary points of a convex region can be described by a maximization problem: argmax{R 1 +  R 2 : (R 1,R 2 ) in S} is a boundary point of S  Tangent line with a given slope  To achieve this boundary point in the ergodic capacity region, then we must operate at this maximum in every realization (timeslot)

11. October 28, 2005 Ergodic Capacity: Maximizing at Each Time-Slot  Achieving the corresponding ergodic capacity boundary point requires solving the maximization problem for every realization argmax{R 1 +  R 2 : (R 1,R 2 ) in S} is a boundary point of S  For any parameter  other than 0, 1, infinity (slope of 0º, 45º, 90º) some set of realizations will require simultaneous (multi-user) scheduling  No single-user scheduling can be optimal

12. October 28, 2005 Simulation: Max-Quantile Max Quantile Rate Point What is the capacity region for single-user scheduling policies?

13. October 28, 2005 Summary  Wireless downlink with two or more users  Information theoretic Gaussian broadcast channel  Multi-user diversity valuable  There exist easily implementable single-user scheduling policies  Sometimes very close to optimal  Optimal scheduling requires simultaneous broadcast channel policy unless the goal is one of three specific rate points  Required for MIMO to achieve capacity