Utility-Optimal Scheduling in Time-Varying Wireless Networks with Delay Constraints

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

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 2 1 AP 3

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, Un(·) 2 1 AP 3

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

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

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

7. Networks with Delay Constraints • Each client periodically generates one packet ever T time slots • τn = prescribed delay bound for client n • tn,c = # of time slots needed for transmitting a packet to client n under channel state c t1,c t2,c t3,c τ1 τ2 τ3 T time slots t1,c t3,c

8. Networks with Delay Constraints • Each client periodically generates one packet ever T time slots • τn = prescribed delay bound for client n • tn,c = # of time slots needed for transmitting a packet to client n under channel state c t1,c t2,c t3,c τ1 τ2 τ3 X T time slots t1,c t2,c

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

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

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 2 4 1 5 3

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 2 1 AP 3

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} = pc {1,2} {1,3} {1} {2,3} {1,2,3} {1,2} {1,3} {1,2} {2,3} {2} {3} 2 1 AP 3

14. Utilities of Clients • Server schedules a feasible set in each interval • Suppose qn = long-term service rate provided to client n • Un(qn) = utility of client n {1,2} {1,3} {1} {2,3} {1,2,3} {1,2} {1,3} {1,2} {2,3} {2} {3} 2 1 AP q2 = 5/6 q1 = 3/6 3 q3 = 4/6

15. NUM in Wireless Max ∑Un(qn) s.t. Network dynamics constraints Network feasibility constraints qn ≥ qn Enhancing fairness or supporting minimum service requirements

16. Server Scheduling Policy • Server adapts λn(k) based on (qn – qn)+ • In each interval, server schedules feasible set S that maximizes • Max-Weight Scheduling Policy • Solves NUM without knowing pc Compensate under-served clients Favor clients that improve total utility most

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

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

19. Optimality of the Auction • Theorem: Let dn(k)≡λn(k). The auction schedules the same set as the Max-Weight Scheduling Policy • This auction design also solves the NUM problem

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

21. Networks with Delay Constraints • Each client generates one packet ever T time slots • τn = prescribed delay bound for client n • tn,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(N2T) τ1 τ2 τ3 T time slots

22. Network with Delay Constraints • 45 clients generate VoIP traffic at 64kbit/sec • An interval = 20 ms • tn,c = 480 μs (under 11 Mb/sec) or 610μs (under 5.5Mb/sec) • wn= 3 + (n mod 3), an = 0.05 + 2n, qn = 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

23. Simulation Results

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

25. Mobile Cellular Networks • 20 clients and one base station with three channels • wn= 1 + (n mod 3), an = 0.2 + 0.1(n mod 7), qn = 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

26. Simulation Results

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 2 4 1 5 3

28. Dynamic Spectrum Allocation • 20 clients randomly deployed in a 1X1 square • wn= 1 + (n mod 3), an = 0.2 + 0.1(n mod 7), qn = 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

29. Simulation Results

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