1. Delay Analysis for Max Weight Opportunistic Scheduling in Wireless Systems Michael J. Neely --- University of Southern California http://www-rcf.usc.edu/~mjneely Proc. Allerton Conference on Communication, Control, and Computing, Sept. 2008 *Sponsored in part by NSF Career CCF-0747525 and DARPA IT-MANET Program  1 (t)  2 (t)  N (t) 1 2 N ON/OFF New Max-Weight bound, O(1) Prior Max-Weight Bound, O(N) Network Size N Avg. Delay or: “A Tale of Two Lyapunov Functions”

2. 1 4 N Quick Description: N Queues, 1 Server, ON/OFF Channels Slotted Time, t {0, 1, 2, 3, …}. A i (t) = # packets arriving to queue i on slot t (integer). S i (t) = 0/1 Channel State (ON or OFF) for queue i on slot t. Can serve 1 packet over a non-empty connected queue per slot. (Scheduling: Which non-empty ON queue to serve??) ? 2 3 ON OFF ON Assume: {A i (t)} and {S i (t)} processes are independent. A i (t) i.i.d. over slots: E{A i (t)} = i S i (t) i.i.d. over slots: Pr[S i (t) = ON] = p i

3. 1 4 N Quick Description: N Queues, 1 Server, ON/OFF Channels Slotted Time, t {0, 1, 2, 3, …}. A i (t) = # packets arriving to queue i on slot t (integer). S i (t) = 0/1 Channel State (ON or OFF) for queue i on slot t. Can serve 1 packet over a non-empty connected queue per slot. (Scheduling: Which non-empty ON queue to serve??) ? 2 3 OFF ON OFF ON Assume: {A i (t)} and {S i (t)} processes are independent. A i (t) i.i.d. over slots: E{A i (t)} = i S i (t) i.i.d. over slots: Pr[S i (t) = ON] = p i

4. 1 4 N Quick Description: N Queues, 1 Server, ON/OFF Channels Slotted Time, t {0, 1, 2, 3, …}. A i (t) = # packets arriving to queue i on slot t (integer). S i (t) = 0/1 Channel State (ON or OFF) for queue i on slot t. Can serve 1 packet over a non-empty connected queue per slot. (Scheduling: Which non-empty ON queue to serve??) ? 2 3 OFF ON OFF ON Assume: {A i (t)} and {S i (t)} processes are independent. A i (t) i.i.d. over slots: E{A i (t)} = i S i (t) i.i.d. over slots: Pr[S i (t) = ON] = p i

5. 1 4 N Quick Description: N Queues, 1 Server, ON/OFF Channels Slotted Time, t {0, 1, 2, 3, …}. A i (t) = # packets arriving to queue i on slot t (integer). S i (t) = 0/1 Channel State (ON or OFF) for queue i on slot t. Can serve 1 packet over a non-empty connected queue per slot. (Scheduling: Which non-empty ON queue to serve??) ? 2 3 OFF ON OFF Assume: {A i (t)} and {S i (t)} processes are independent. A i (t) i.i.d. over slots: E{A i (t)} = i S i (t) i.i.d. over slots: Pr[S i (t) = ON] = p i

6. 1 4 N Quick Description: N Queues, 1 Server, ON/OFF Channels Slotted Time, t {0, 1, 2, 3, …}. A i (t) = # packets arriving to queue i on slot t (integer). S i (t) = 0/1 Channel State (ON or OFF) for queue i on slot t. Can serve 1 packet over a non-empty connected queue per slot. (Scheduling: Which non-empty ON queue to serve??) ? 2 3 OFF ON OFF ON Assume: {A i (t)} and {S i (t)} processes are independent. A i (t) i.i.d. over slots: E{A i (t)} = i S i (t) i.i.d. over slots: Pr[S i (t) = ON] = p i

7. Notation: Server variables are 0/1 variables. Q i (t) = # packets in queue i on slot t (integer).  i (t) = server decision (rate allocated to queue i) = 1 if we allocate a server to queue i and S i (t) = ON. (0 else)  i (t) = min[  i (t), Q i (t)] = actual # packets served over channel i 1 4 N ? 2 3 Q i (t+1) = max[Q i (t) –  i (t), 0] + A i (t) equivalently: Q i (t+1) = Q i (t) –  i (t) + A i (t) New Max-Weight (LCQ) bound, O(1) Prior Max-Weight (LCQ) Bound, O(N) Network Size N Avg. Delay

8. Status Quo #1 – Max-Weight Scheduling: Well known algorithm [Tassiulas-Ephremides 93] Gives full throughput region (0 <  < 1). Generalizes to multi-rate channels and multi-hop nets with backpressure, performance opt. [NOW F&T 06] Simple and Adaptive: No prior traffic rates or channel probabilities are required for implementation. Capacity Region    Example  Previous Delay Bound: (1-  ) cN ≤ E{Delay} N = Network Size (# of queues)  = Fraction away from capacity region boundary (0 <  c = constant Advantages: Disadvantages: Max-Weight has no tight delay analysis!

9. Status Quo #2 – Queue Grouping and LCG: “Largest Connected Group algorithm” (LCG) [Neely06,08] gives O(1) Average Delay, for any 0 <  < 1 in the “f-balanced region”: no individual arrival rate is more than a constant above the average rate. Capacity Region    Largest Connected Group (LCG) Delay Bound: (1-  ) c log(1/(1-  ) ≤ E{Delay} Advantages: Disadvantages: [Neely, Allerton 2006, TON 2008] Delay is O(1), independent of N “f-balanced” region More Restrictive “balanced” Throughput Region. Requires pre-organized queue group structure based on knowledge of  and p min = min i {Pr[S i (t)=ON]}. Less Adaptive, not clearly connected to backpressure.

10. Our New Results: For ON/OFF channel… We analyze delay of Max-Weight! (use queue group concepts) Max-Weight gives O(1) delay (anywhere in  ). We develop 2 new Lyapunov functions (“L A ” & “L B ”). These tools may be useful for more general networks *(see end slide for extensions to multi-rate models). “f-Balanced” Rates in    Ex  (1-  ) c log(1/(1-  )) ≤ E{Delay} LA:LA: Anywhere in    Ex  (1-  ) 2 c log(1/(1-  )) ≤ E{Delay} LB:LB:

11. Lyapunov Function 1 (L A ): (ON/OFF channel) We sum over all possible partitions of N into K disjoint groups, where K is same as in LCG algorithm: “f-Balanced” Rates in    Ex   1 (t) 1  2 (t) 2  3 (t) 3  4 (t) 4  5 (t) 5  6 (t) 6  7 (t) 7  N (t) N (1-  ) c log(1/(1-  )) ≤ E{Delay} LA:LA: 0

12. Lyapunov Function 1 (L A ): (ON/OFF channel) We sum over all possible partitions of N into K disjoint groups, where K is same as in LCG algorithm: “f-Balanced” Rates in    Ex  (1-  ) c log(1/(1-  )) ≤ E{Delay} LA:LA:  1 (t) 1  2 (t) 2  3 (t) 3  4 (t) 4  5 (t) 5  6 (t) 6  7 (t) 7  N (t) N 0

13. Lyap. Drift   (t) of L A (Q(t)) : (ON/OFF channel) Theorem: Scheduling to minimize drift involves maximizing: where: Further, this is maximized by the Max-Weight (LCQ) Policy!

14. Proof Sketch: Use Combinatorics to show… Maximized by LCQ (“max-weight”) Maximized by any work-conserving strategy c 1 > 0, c 2 > 0

15. “f-Balanced” Rates in    Ex   1 (t) 1  2 (t) 2  3 (t) 3  4 (t) 4  5 (t) 5  6 (t) 6  7 (t) 7  N (t) N (1-  ) c log(1/(1-  )) ≤ E{Delay} LA:LA: 0 Thus: The first Lyapunov function (L A ) gives:

16. Lyapunov Function 2 (L B ): (ON/OFF channel) A 2-part Lyapunov function, inspired by similar function in [Wu, Srikant, Perkins 2007] for different context. Stabilizes full  Low delay when # non-empty queues is large (via multi-user diversity) Anywhere in    Ex  (1-  ) 2 c log(1/(1-  )) ≤ E{Delay} LB:LB:

17. *Extensions: (Multi-Rate Channels)  S i (t) in {0, 0.1, 0.2, …,  max }  ***We note that this slide originally contained an incorrect claim that multi-rate channels can also achieve O(1) average delay. This claim was not in the Allerton paper, but unfortunately was in our original Arxiv pre-print (v1). We have made a new Arxiv report (v2, Dec. 08) with the corrections and discussion of issues involved: ***  M. J. Neely, “Delay Analysis for Max Weight Opportunistic Scheduling in Wireless Systems,” arXiv:0806.2345v2, Dec. 2008.

18. Paper: available on web: http://www-rcf.usc.edu/~mjneely/http://www-rcf.usc.edu/~mjneely/ Extended version with the multi-rate analysis (also on web): M. J. Neely, “Delay analysis for max-weight opportunistic scheduling in wireless systems,” arXiv: 0806.2345v2, Dec. 2008. Conclusions: Order-Optimal (i.e., O(1)) Delay Analysis for the thruput-optimal Max-Weight (LCQ) Algorithm! “f-Balanced” Rates in    Ex  (1-  ) c log(1/(1-  )) ≤ E{Delay} LA:LA: Anywhere in    Ex  (1-  ) 2 c log(1/(1-  )) ≤ E{Delay} LB:LB:

19. Brief Advertisement: Stochastic Network Optimization Homepage: http://www- rcf.usc.edu/~mjneely/stochastic/http://www- rcf.usc.edu/~mjneely/stochastic/ Contains list of papers, descriptions, other web resources, and an editable wiki board. Conclusions: Order-Optimal (i.e., O(1)) Delay Analysis for the thruput-optimal Max-Weight (LCQ) Algorithm! “f-Balanced” Rates in    Ex  (1-  ) c log(1/(1-  )) ≤ E{Delay} LA:LA: Anywhere in    Ex  (1-  ) 2 c log(1/(1-  )) ≤ E{Delay} LB:LB: