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Delay Analysis for Max Weight Opportunistic Scheduling in Wireless Systems

Delay Analysis for Max Weight Opportunistic Scheduling in Wireless Systems. ON/OFF. or: “ A Tale of Two Lyapunov Functions ”. m 1 (t). l 1. m 2 (t). l 2. Prior Max-Weight Bound, O(N). Avg. Delay. New Max-Weight bound, O(1). l N. m N (t). Network Size N.

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Delay Analysis for Max Weight Opportunistic Scheduling in Wireless Systems

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  1. Delay Analysis for Max Weight Opportunistic Scheduling in Wireless Systems ON/OFF or: “A Tale of Two Lyapunov Functions” m1(t) l1 m2(t) l2 Prior Max-Weight Bound, O(N) Avg. Delay New Max-Weight bound, O(1) lN mN(t) Network Size N 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

  2. Quick Description: N Queues, 1 Server, ON/OFF Channels • Slotted Time, t {0, 1, 2, 3, …}. • Ai(t) = # packets arriving to queue i on slot t (integer). • Si(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??) ON l1 • Assume: • {Ai(t)} and {Si (t)} processes • are independent. • Ai (t) i.i.d. over slots: • E{Ai (t)} = li • Si(t) i.i.d. over slots: • Pr[Si(t) = ON] = pi ON l2 ? OFF l3 OFF l4 ON lN

  3. Quick Description: N Queues, 1 Server, ON/OFF Channels • Slotted Time, t {0, 1, 2, 3, …}. • Ai(t) = # packets arriving to queue i on slot t (integer). • Si(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??) OFF l1 • Assume: • {Ai(t)} and {Si (t)} processes • are independent. • Ai (t) i.i.d. over slots: • E{Ai (t)} = li • Si(t) i.i.d. over slots: • Pr[Si(t) = ON] = pi ON l2 ? OFF l3 OFF l4 ON lN

  4. Quick Description: N Queues, 1 Server, ON/OFF Channels • Slotted Time, t {0, 1, 2, 3, …}. • Ai(t) = # packets arriving to queue i on slot t (integer). • Si(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??) OFF l1 • Assume: • {Ai(t)} and {Si (t)} processes • are independent. • Ai (t) i.i.d. over slots: • E{Ai (t)} = li • Si(t) i.i.d. over slots: • Pr[Si(t) = ON] = pi ON l2 ? OFF l3 ON l4 ON lN

  5. Quick Description: N Queues, 1 Server, ON/OFF Channels • Slotted Time, t {0, 1, 2, 3, …}. • Ai(t) = # packets arriving to queue i on slot t (integer). • Si(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??) OFF l1 • Assume: • {Ai(t)} and {Si (t)} processes • are independent. • Ai (t) i.i.d. over slots: • E{Ai (t)} = li • Si(t) i.i.d. over slots: • Pr[Si(t) = ON] = pi ON l2 ? ON l3 ON l4 OFF lN

  6. Quick Description: N Queues, 1 Server, ON/OFF Channels • Slotted Time, t {0, 1, 2, 3, …}. • Ai(t) = # packets arriving to queue i on slot t (integer). • Si(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??) OFF l1 • Assume: • {Ai(t)} and {Si (t)} processes • are independent. • Ai (t) i.i.d. over slots: • E{Ai (t)} = li • Si(t) i.i.d. over slots: • Pr[Si(t) = ON] = pi ON l2 ? OFF l3 OFF l4 ON lN

  7. Notation: Server variables are 0/1 variables. • Qi(t) = # packets in queue i on slot t (integer). • mi(t) = server decision (rate allocated to queue i) • = 1 if we allocate a server to queue i and Si(t) = ON. (0 else) • mi(t) = min[mi(t), Qi(t)] = actual # packets served over channel i Qi (t+1) = max[Qi(t) – mi (t), 0] + Ai (t) equivalently: Qi (t+1) = Qi(t) – mi (t) + Ai (t) l1 ? l2 Prior Max-Weight (LCQ) Bound, O(N) Avg. Delay l3 New Max-Weight (LCQ) bound, O(1) l4 Network Size N lN

  8. Status Quo #1 – Max-Weight Scheduling: Previous Delay Bound: Capacity Region L l2 Example: r= 0.95 • N = Network Size (# of queues) • r = Fraction away from capacity • region boundary (0 < r <1) • c = constant l1 cN ≤ E{Delay} (1-r) Advantages: • Well known algorithm [Tassiulas-Ephremides 93] • Gives full throughput region (0 < r < 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. Disadvantages: Max-Weight has no tight delay analysis!

  9. Status Quo #2 – Queue Grouping and LCG: Largest Connected Group (LCG) Delay Bound: Capacity Region L l2 “f-balanced” region [Neely, Allerton 2006, TON 2008] l1 Delay is O(1), independent of N c log(1/(1-r)) ≤ E{Delay} (1-r) Advantages: “Largest Connected Group algorithm” (LCG)[Neely06,08] gives O(1) Average Delay, for any 0 < r < 1 in the “f-balanced region”: no individual arrival rate is more than a constant above the average rate. • More Restrictive “balanced” Throughput Region. • Requires pre-organized queue group structure based on knowledge of r and pmin = mini{Pr[Si(t)=ON]}. • Less Adaptive, not clearly connected to backpressure. Disadvantages:

  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 L). • We develop 2 new Lyapunov functions (“LA” & “LB”). • These tools may be useful for more general networks *(see end slide for extensions to multi-rate models). c log(1/(1-r)) c log(1/(1-r)) ≤ E{Delay} E{Delay} ≤ “f-Balanced” Rates in L Anywhere in L (1-r)2 (1-r) l2 l2 Ex: r= 0.95 Ex: r= 0.95 l1 l1 LA: LB:

  11. Lyapunov Function 1 (LA): (ON/OFF channel) • We sum over all possible partitions of N into K disjoint groups, where K is same as in LCG algorithm: c log(1/(1-r)) E{Delay} ≤ “f-Balanced” Rates in L l1 m1(t) (1-r) l2 l2 m2(t) Ex: r= 0.95 l3 m3(t) l4 m4(t) l5 m5(t) l6 m6(t) l1 l7 m7(t) lN mN(t) LA: 0

  12. Lyapunov Function 1 (LA): (ON/OFF channel) • We sum over all possible partitions of N into K disjoint groups, where K is same as in LCG algorithm: c log(1/(1-r)) E{Delay} ≤ “f-Balanced” Rates in L l1 m1(t) (1-r) l2 l2 m2(t) Ex: r= 0.95 l3 m3(t) l4 m4(t) l5 m5(t) l6 m6(t) l1 l7 m7(t) lN mN(t) LA: 0

  13. Lyap. Drift DA(t) of LA(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 any work-conserving strategy Maximized by LCQ (“max-weight”) c1 > 0, c2 > 0

  15. Thus: The first Lyapunov function (LA) gives: “f-Balanced” Rates in L l1 m1(t) l2 l2 m2(t) Ex: r= 0.95 l3 m3(t) c log(1/(1-r)) l4 m4(t) E{Delay} ≤ (1-r) l5 m5(t) l6 m6(t) l1 l7 m7(t) lN mN(t) LA: 0

  16. Lyapunov Function 2 (LB): (ON/OFF channel) • A 2-part Lyapunov function, inspired by similar function in [Wu, Srikant, Perkins 2007] for different context. c log(1/(1-r)) ≤ E{Delay} Low delay when # non-empty queues is large (via multi-user diversity) (1-r)2 Stabilizes full L Anywhere in L l2 Ex: r= 0.95 LB: l1

  17. *Extensions: (Multi-Rate Channels) • Si(t) in {0, 0.1, 0.2, …, mmax} • ***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. Conclusions: Order-Optimal (i.e., O(1)) Delay Analysis for the thruput-optimal Max-Weight (LCQ) Algorithm! c log(1/(1-r)) c log(1/(1-r)) ≤ E{Delay} E{Delay} ≤ “f-Balanced” Rates in L Anywhere in L (1-r)2 (1-r) l2 l2 Ex: r= 0.95 Ex: r= 0.95 • Paper:available on web: 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. l1 l1 LA: LB:

  19. Conclusions: Order-Optimal (i.e., O(1)) Delay Analysis for the thruput-optimal Max-Weight (LCQ) Algorithm! c log(1/(1-r)) c log(1/(1-r)) ≤ E{Delay} E{Delay} ≤ “f-Balanced” Rates in L Anywhere in L (1-r)2 (1-r) l2 l2 Ex: r= 0.95 Ex: r= 0.95 • Brief Advertisement: • Stochastic Network Optimization Homepage: http://www-rcf.usc.edu/~mjneely/stochastic/ • Contains list of papers, descriptions, other web resources, and an editable wiki board. l1 l1 LA: LB:

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