1. Utility-Optimal Scheduling in Time- Varying Wireless Networks with Delay Constraints I-Hong Hou P.R. Kumar University of Illinois, Urbana-Champaign 1/30

2. Wireless Networks  A system with one server and N clients  Links can fade  Links interfere with each other  Clients have strict per-packet delay bounds for their packets  Impossible to deliver all packets on-time AP 1 2 3 2 /30

3. Wireless Networks  Each client needs a minimum throughput of on- time packets  Additional throughput for each client n increases its utility through its utility function, U n (·) AP 1 2 3 3 /30

4. Conflict of Interests  Server’s goal: maximize TOTAL utility while supporting minimum throughput Server is in charge of scheduling clients Support minimum throughput of each client Offer additional throughput to maximize total utility  Each client’s goal: maximize its OWN utility Can lie about its utility function to gain more throughput 4 /30

5. Overview of Results  An on-line scheduling policy for the server that achieves maximum total utility while respecting all minimum throughput requirements  A truthful auction conducted by the server that makes all clients report their true utility functions  Three applications Networks with Delay Constraints Mobile Cellular Networks Dynamic Spectrum Allocation 5 /30

6. Networks with Delay Constraints  Each client periodically generates one packet ever T time slots τ n = prescribed delay bound for client n t c,n = # of time slots needed for transmitting a packet to client n under channel state c T time slots 6 /30

7. Networks with Delay Constraints  Each client periodically generates one packet ever T time slots  τ n = prescribed delay bound for client n  t n,c = # of time slots needed for transmitting a packet to client n under channel state c τ1τ1 τ2τ2 τ3τ3 T time slots t 2,c t 3,c t 1,c t 3,c t 1,c 7 /30

8. Networks with Delay Constraints  Each client periodically generates one packet ever T time slots  τ n = prescribed delay bound for client n  t n,c = # of time slots needed for transmitting a packet to client n under channel state c τ1τ1 τ2τ2 τ3τ3 T time slots t 2,c t 3,c t 1,c t 2,c X 8 /30

9. Mobile Cellular Network  α channels  Each channel between the base station and mobile fades ON or OFF X 9 /30

10. Mobile Cellular Network  α channels  Each channel between the base station and mobile fades ON or OFF X X 10 /30

11. Dynamic Spectrum Allocation  One primary user and many secondary users  Channel unused by the primary user can be used by secondary users  However, secondary users can interfere with each other  Schedule an interference-free allocation 1 2 3 5 4 11 /30

12. General Model  A system with one server and N clients  Time is divided into time intervals An interval may consist of multiple time slots  Server schedules a feasible set of clients in each interval Feasibility depends on network constraints AP 1 2 3 12 /30

13. Network Feasibility Model  c ( k ) = network “state” at interval k  State = sets of feasible clients  { c (1), c (2), c (3),…} are i.i.d. random variables Prob{ c ( k )= c } = p c AP 1 2 3 {1,2} {1,3} {1} {2,3}{1,2,3} {1,2} {1,3} {1,2} {2,3} {2} {3} 13 /30

14. Utilities of Clients  Server schedules a feasible set in each interval  Suppose q n = long-term service rate provided to client n  U n ( q n ) = utility of client n AP 1 2 3 {1,2} {1,3} {1} {2,3}{1,2,3} {1,2} {1,3} {1,2} {2,3} {2} {3} q 1 = 3/6 q 2 = 5/6 q 3 = 4/6 14 /30

15. NUM in Wireless Max ∑U n ( q n ) s.t. Network dynamics constraints Network feasibility constraints q n ≥ q n Enhancing fairness or supporting minimum service requirements 15 /30

16. Server Scheduling Policy  Server adapts λ n (k) based on (q n – q n ) +  In each interval, server schedules feasible set S that maximizes  Max-Weight Scheduling Policy  Solves NUM without knowing p c Favor clients that improve total utility most Compensate under-served clients 16 /30

17. Concepts of Truthful Auction  Clients may lie about their utility functions  In each interval, each client n receives a reward r n proportional to U n ( q n )  e n = amount that n has to pay  Each client n greedily maximizes its net reward = r n -e n  Marginal utility of client n = { r n if it is served} – { r n if it is not served}  An auction is truthful if all clients report their true marginal utility 17 /30

18. Design of a Truthful Auction  The server announces a discount d n ( k ) in each interval k  Each client n offers a bid b n ( k )  The server schedules the set S that maximizes  Each scheduled client n is charged  Theorem: For each client n, choosing b n ( k ) to be its marginal utility is optimal 18 /30

19. Optimality of the Auction  Theorem: Let d n (k)≡λ n (k). The auction schedules the same set as the Max-Weight Scheduling Policy  This auction design also solves the NUM problem 19 /30

20. Simulation Overview  Compare with one state-of-the-art technique and a random policy  Utility functions  Metrics: total utility and total penalty 20 /30

21. Networks with Delay Constraints  Each client generates one packet ever T time slots  τ n = prescribed delay bound for client n  t n,c = # of time slots needed for transmitting a packet to client n under channel state c  A variation of knapsack problem  Solved by dynamic programming in O(N 2 T) τ1τ1 τ2τ2 τ3τ3 T time slots 21 /30

22. Network with Delay Constraints  45 clients generate VoIP traffic at 64 kbit/sec  An interval = 20 ms  t n,c = 480 μs (under 11 Mb/sec ) or 610 μs (under 5.5 Mb/sec )  w n = 3 + (n mod 3), a n = 0.05 + 2n, q n = 0.5+0.01(20n mod 300)  Compared against the modified-knapsack policy of [Hou and Kumar] Modified-knapsack focuses on satisfying minimum service rate requirements only 22 /30

23. Simulation Results 23 /30

24. Mobile Cellular Network  α channels  Each channel between the base station and mobile fades ON or OFF  Schedule the α ON clients with largest X 24 /30

25. Mobile Cellular Networks  20 clients and one base station with three channels  w n = 1 + (n mod 3), a n = 0.2 + 0.1(n mod 7), q n = 0.05(n mod 5), Prob(n is ON) = 0.6+0.02(n mod 10)  Compared against the WNUM policy in [O’Neil, Goldsmith, and Boyd] WNUM optimizes utility on a per-interval basis without considering long-term average 25 /30

26. Simulation Results 26 /30

27. Dynamic Spectrum Allocation  One primary user and many secondary users  Channel unused by the primary user can be used by secondary users  Secondary users can interfere with each other  Schedules a maximum weight independent set with weights 1 2 3 5 4 27 /30

28. Dynamic Spectrum Allocation  20 clients randomly deployed in a 1X1 square  w n = 1 + (n mod 3), a n = 0.2 + 0.1(n mod 7), q n = 0.05(n mod 8)  Compared against the VERITAS policy of [Zhou, Gandhi, Suri, and Zheng] VERITAS optimizes utility on a per-interval basis without considering long-term average behavior 28 /30

29. Simulation Results 29 /30

30. Conclusions  Network Utility Maximization (NUM) in wireless Client utilities depend on long-term average throughput of on-time packets Network constraints are dynamic with unknown distribution Clients may lie about utility functions to gain more service Solutions of the NUM problem:  An on-line scheduling policy for the server  A truthful auction design  Applied the solutions to three applications 30 /30