Optimal Delay for Opportunistic Scheduling in Multi-User Wireless Systems

1. S1(t) {ON, OFF} Avg. Delay Order Optimal Delay for Opportunistic Scheduling In Multi-User Wireless Uplinks and Downlinks l1 S2(t) l2 Num. Users N lN SN(t) Michael J. Neely University of Southern California http://www-rcf.usc.edu/~mjneely/ Allerton 2006 *Sponsored in part by NSF OCE Grant 0520324 (DIGITAL OCEAN)

2. The System Model: N Users , 1 Server q1 Uplink l1 q2 l2 N 1 2 lN qN Downlink user 1 user N Discrete Time System: Timeslots t = {0, 1, 2, …} Qi(t) = Current Num. Packets in queue i Ai(t) = Arrivals to Queue i during slot t [ i.i.d over slots , E[Ai(t)] = li ] Si(t) = Current Channel State ({ON, OFF}) [ i.i.d. over slots, Pr[Si(t) = ON] = qi ] mi(t) = Packets Transmitted over link i on slot t

3. mi(t) {0,1} N mi(t) 1 i=1 The System Model: N Users , 1 Server q1 Uplink l1 q2 l2 N 1 2 lN qN Downlink user 1 user N Discrete Time System: Timeslots t = {0, 1, 2, …} mi(t) Ai(t) Qi(t) Qi(t+1) = max[Qi(t) - mi(t), 0] + Ai(t) Scheduling Constraints: Can serve at most one “ON” link per slot: mi(t)=0 if Si(t)=OFF , ,

4. q1 l1 q2 l2 lN qN l2 l1 q1 , l1 q2 l1 + l2 q1 + (1-q1)q2 l1 This model is investigated in [Tassiulas, Ephremides 93]: Results of [Tas, Eph 93]: 1) Capacity Region L 2) LCQ Algorithm (“Largest Connected Queue”) 3) Delay Optimality for Symmetric Systems Model is central to channel-aware (“opportunistic”) scheduling. The Capacity Region L: Set of all rate vectors (l1, .., lN) that can be stabilized. Example: (N=2) L is the set of all (l1, l2) such that:

5. q1 l1 q2 l2 lN qN iI iI (1-qi) 1 - li This model is investigated in [Tassiulas, Ephremides 93]: Results of [Tas, Eph 93]: 1) Capacity Region L 2) LCQ Algorithm (“Largest Connected Queue”) 3) Delay Optimality for Symmetric Systems Model is central to channel-aware (“opportunistic”) scheduling. The Capacity Region L: Set of all rate vectors (l1, .., lN) that can be stabilized. General Case for N: (l1, .., lN) L if and only if for each of the 2N-1 non-empty subsets I of {1, .., N}

6. An isolated set of delay-optimality results: q l q l l q For Symmetric Systems: -Largest Connected Queue (LCQ) [Tassiulas and Ephremides 93]: Proof uses stochastic coupling and exploits symmetry… -Rate Allocation in Gaussian Multiple Access Channels [Yeh 2001 , Yeh and Cohen 2003] -Multi-Server Systems: [Yeh 2001 , Ganti, Modiano, Tsitsiklis 2002]

7. An isolated set of delay-optimality results: q The actual delay that is achieved is unknown (even for these symmetric cases) O(N)? O( N )? O(1)? l q l l q For Symmetric Systems: -Largest Connected Queue (LCQ) [Tassiulas and Ephremides 93]: Proof uses stochastic coupling and exploits symmetry… -Rate Allocation in Gaussian Multiple Access Channels [Yeh 2001 , Yeh and Cohen 2003] -Multi-Server Systems: [Yeh 2001 , Ganti, Modiano, Tsitsiklis 2002]

8. An isolated set of delay-optimality results: q The actual delay that is achieved is unknown (even for these symmetric cases) O(N)? O( N )? O(1)? l q l l q For Heavy Traffic: r = fraction l is away from capacity region boundary Shakkottai, Srikant, Stolyar 2004 r 1 (Heavy Traffic) An exponential Scheduling Rule approaches delay optimality (r 1)

9. Related: Delay for N x N Switch Scheduling: 1 N 3 N 1 2 -[Leonardi, Mellia, Neri, Marsan 2001]: O(N/(1-r)) Delay bound (MWM Sched.) -[Neely, Modiano 2004]: O(log(N)/(1-r)2) Delay bound (Frame Based Sched.)

10. Related: Delay for N x N Switch Scheduling: 1 N 3 N 1 2 Some Interesting Queue Grouping Approaches (mainly to reduce complexity): -Mekkittikul, McKeown (1998) -Shah (2003) -Wu, Srikant (wireless, 2006)

11. Related: Delay for N x N Switch Scheduling: 1 N 3 N 1 2 Some Interesting Queue Grouping Approaches (mainly to reduce complexity): -Mekkittikul, McKeown (1998) -Shah (2003) -Wu, Srikant (wireless, 2006) = + -Leonardi et al. (2001)

12. Related: Delay for N x N Switch Scheduling: 1 N 3 N 1 2 Some Interesting Queue Grouping Approaches (mainly to reduce complexity): -Mekkittikul, McKeown (1998) -Shah (2003) -Wu, Srikant (wireless, 2006) O(1) Delay when r < 1/2 (half loaded) = + -Leonardi et al. (2001)

13. N O( ) 1 O( ) (1-r) (1-r) What is the optimal delay (as a function of N) for the N user wireless problem with varying channels? q l q l l q Time Varying Channels make analysis more complex, cannot use same approaches as switch problems… Previous Upper and Lower Bounds: (N users) E[Delay] “Single-Queue Bound” [Neely, Modiano, Rohrs 03]

14. N 2rN(1-r) What is the optimal delay (as a function of N) for the N user wireless problem with varying channels? q l rN = 1-(1-q)N q l (max possible output rate) l q Our Results: (part 1) If scheduling doesn’t consider queue backlog (such as stationary randomized scheduling) then: E[Delay] is at least linear in N 2) Uniform Poisson Traffic: E[Delay] >

15. What is the optimal delay (as a function of N) for the N user wireless problem with varying channels? q l rN = 1-(1-q)N q l (max possible output rate) l q Our Results: (part 2) For any r such that r < 1 O( ) log(1/(1-r)) Independent of N Av. Delay (1-r) Holds for Symmetric Systems and a large class of Asymmetric ones

16. What is the optimal delay (as a function of N) for the N user wireless problem with varying channels? q l rN = 1-(1-q)N q l (max possible output rate) l q Our Results: (part 2) For any r such that r < 1 O( ) log(1/(1-r)) Independent of N Av. Delay (1-r) We use a form of queue grouping together with Lyapunov drift And statistical multiplexing

17. Intuition about Queue Grouping: N user System, Uniform Poisson inputs: q l rN = 1-(1-q)N q l (max possible output rate) l q Compare to a single-queue system with Pr[ON] = q l (GI/GI/1 queue) Pr[serve]=q l l Can show that any work conserving scheduling policy in multi- queue system yields delay that is stochastically smaller than single- queue system. Leads An easy upper bound on delay…

18. 1 - ltot/2 q - ltot Intuition about Queue Grouping: N user System, Uniform Poisson inputs: q l rN = 1-(1-q)N q l (max possible output rate) l q Compare to a single-queue system with Pr[ON] = q l (GI/GI/1 queue) Pr[serve]=q l Poisson Bernoulli l Single Queue Upper Bound on Avg. Delay: Only works for ltot < q (i.e., r < g where g = q/rN) O( ) 1 = E[Delay] = (1-r/g)

19. Queue Grouping Approach: Form K Groups: {G1, G2, …, GK} i Gk l1 Qsum, k(t) = Qi(t) l2 G1 lM1 G2 lM1+1 GK lN

20. i Gk i Gk Qsum, k(t) = Qi(t) G1 G2 lsum, k = li GK { 1 , if group Gk has at least one non-empty connected queue. 0 , else Define: 1k(t) = The Largest Connected Group (LCG) Algorithm: Every slot t, observe the queue backlogs and channel states, and select the group k {1, …, K} that maximizes 1k(t)Qsum, k(t). Then serve any non-empty connected queue in that group (breaking ties arbitrarily).

21. Actual N-queue System Comparison K-queue System lsum, k= li qmin, k= min {qi} i Gk i Gk q1 1 qmin, 1 lsum, 1 q2 G1 lsum, 2 2 qmin, 2 G2 lsum, N K qmin, K GK qN Define: LK = Capacity region of the K-queue System Theorem: If there is an e > 0 such that: (lsum, 1 + e, lsum, 2 + e, . . . , lsum, K + e) LK Then LCG stabilizes the system and yields average delay:

22. Actual N-queue System Comparison K-queue System lsum, k= li qmin, k= min {qi} i Gk i Gk q1 1 qmin, 1 lsum, 1 q2 G1 lsum, 2 2 qmin, 2 G2 lsum, N K qmin, K GK qN Define: LK = Capacity region of the K-queue System Theorem: If there is an e > 0 such that: (lsum, 1 + e, lsum, 2 + e, . . . , lsum, K + e) LK If arrivals are independent and Poisson, then we have:

23. Theorem: If there is an e > 0 such that: (lsum, 1 + e, lsum, 2 + e, . . . , lsum, K + e) LK If arrivals are independent and Poisson, then we have: Proof Concept: Use the following Lyapunov function: LCG comes within additive constant of minimizing: (Lyapunov drift) 2) (tricky part) Prove there exists another algorithm that yields: (h() linear)

24. l l Q1(t) QN-1(t) l l Q2(t) QN(t) Application to Symmetric Systems: rN = 1-(1-q)N q (max possible output rate) q q ltot = rrN q For any loading r such that 0 < r < 1: For simplicity assume N = MK (K groups of equal size M) log(2/(1-r)) Choose K = log(1/(1-q)) Then e = rN(1-r)/(2K) , … Plug this into the theorem…

25. l l Q1(t) QN-1(t) l l Q2(t) QN(t) Application to Symmetric Systems: rN = 1-(1-q)N q (max possible output rate) q q ltot = rrN q For any loading r such that 0 < r < 1: For simplicity assume N = MK (K groups of equal size M) log(2/(1-r)) Choose K = log(1/(1-q)) 2K O( ) log(1/(1-r)) = Then LCG => E[W] rN(1-r) (1-r)

26. lN-1 QN-1(t) lN QN(t) Application to Asymmetric Systems: N (1-qi) rmax = 1 - q1 l1 Q1(t) i=1 q2 l1 Q2(t) (max possible output rate) ltot = rrmax qN-1 qN ltot = l1 + … + lN Form variable length groups by iteratively packing individual streams until total rate of the group exceeds ltot/N. Then: lsum, k < ltot/N + lmax for all groups k

27. lN-1 QN-1(t) > lN QN(t) Application to Asymmetric Systems: N (1-qi) rmax = 1 - q1 l1 Q1(t) i=1 q2 l1 Q2(t) (max possible output rate) ltot = rrmax qN-1 qN For any loading r such that 0 < r < 1: log(2/(1-r)) Choose K = Assume lmax < (1-r)rmax/(3K) log(1/(1-qmin)) O( ) log(1/(1-r)) For any N K, LCG => E[W] (1-r)

28. Conclusions: Order-Optimal Delay for Opportunistic Scheduling in a Multi-User System (N users) -Backlog-unaware scheduling: Delay grows at least linear with N -Backlog-aware scheduling: It is possible to achieve O(1) delay (independent of N) -The first explicit bound for optimal delay in this setting -Queue Grouping is a useful tool for analysis and design