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Delay Analysis for Maximal Scheduling in Wireless Networks with Bursty Traffic

This study, presented at INFOCOM 2008 in Phoenix, delves into the analysis of maximal scheduling in wireless networks with bursty traffic. The capacity region of such networks is explored in depth, focusing on interference sets and queueing dynamics to maximize scheduling efficiency. Sponsored by DARPA IT-MANET Program and NSF grants, the research provides insights into optimal scheduling strategies for wireless communication systems.

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Delay Analysis for Maximal Scheduling in Wireless Networks with Bursty Traffic

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  1. ON ON ON ON OFF OFF OFF OFF Delay Analysis for Maximal Scheduling inWireless Networks with Bursty Traffic Capacity Region L Michael J. Neely University of Southern California INFOCOM 2008, Phoenix, AZ g-scaled region gL *Sponsored in part by the DARPA IT-MANET Program, NSF OCE-0520324, NSF Career CCF-0747525

  2. Sl = Interference Set for link lL One-Hop Network Model: N = Node set = {1, 2…, N} L = Link set = {1, 2, …, L} General Interference Set Model: Sl = l U {links that interfere with link l transmission} [Chaporkar, Kar, Sarkar Allerton 2005] [Wu, Srikant, Perkins, Trans. Mobile Comput. June 2007]

  3. Sl = Interference Set for link lL One-Hop Network Model: N = Node set = {1, 2…, N} L = Link set = {1, 2, …, L} Example: Matching, NxN Switch Link l General Interference Set Model: Sl = l U {links that interfere with link l transmission} [Chaporkar, Kar, Sarkar Allerton 2005] [Wu, Srikant, Perkins, Trans. Mobile Comput. June 2007]

  4. Sl = Interference Set for link lL One-Hop Network Model: N = Node set = {1, 2…, N} L = Link set = {1, 2, …, L} Example: Matching, NxN Switch Set Sl General Interference Set Model: Sl = l U {links that interfere with link l transmission} [Chaporkar, Kar, Sarkar Allerton 2005] [Wu, Srikant, Perkins, Trans. Mobile Comput. June 2007]

  5. Sl = Interference Set for link lL One-Hop Network Model: N = Node set = {1, 2…, N} L = Link set = {1, 2, …, L} Example: Matching, Wireless Link l General Interference Set Model: Sl = l U {links that interfere with link l transmission} [Chaporkar, Kar, Sarkar Allerton 2005] [Wu, Srikant, Perkins, Trans. Mobile Comput. June 2007]

  6. Sl = Interference Set for link lL One-Hop Network Model: N = Node set = {1, 2…, N} L = Link set = {1, 2, …, L} Example: Matching, Wireless Set Sl General Interference Set Model: Sl = l U {links that interfere with link l transmission} [Chaporkar, Kar, Sarkar Allerton 2005] [Wu, Srikant, Perkins, Trans. Mobile Comput. June 2007]

  7. Sl = Interference Set for link lL One-Hop Network Model: N = Node set = {1, 2…, N} L = Link set = {1, 2, …, L} Example: Arb. Interference Sets General Interference Set Model: Sl = l U {links that interfere with link l transmission} [Chaporkar, Kar, Sarkar Allerton 2005] [Wu, Srikant, Perkins, Trans. Mobile Comput. June 2007]

  8. Queueing Dynamics: -Slotted System: t = {0, 1, 2, 3, …} -One Queue for each link l: Ql(t) = # packets in currently in queue l (on slot t) Al(t) = # new packet arrivals to queue l (on slot t) ml(t) = # packets served from queue l (on slot t) Al(t) ml(t) Ql(t) Ql(t+1) = Ql(t) - ml(t) + Al(t) X(t) ={Scheduling Options} ml(t) {0, 1} ml(t) = 1 only if Ql(t)>0 AND no other active links wSl

  9. Queueing Dynamics: -Slotted System: t = {0, 1, 2, 3, …} -One Queue for each link l: Ql(t) = # packets in currently in queue l (on slot t) Al(t) = # new packet arrivals to queue l (on slot t) ml(t) = # packets served from queue l (on slot t) Al(t) ml(t) Ql(t) Ql(t+1) = Ql(t) - ml(t) + Al(t) X(t) ={Scheduling Options} ml(t) {0, 1} ml(t) = 1 only if Ql(t)>0 AND no other active links wSl

  10. m(t) X(t) Capacity Region: L = {All rate vectors l = (l1,…, lL) supportable} Capacity Region L [Tassiulas, Ephremides 92]: Max Weight Match (MWM) Maximize Ql(t)ml(t) Subject to: (Stabilizes Network, Supports all linterior to L)

  11. Capacity Region: L = {All rate vectors l = (l1,…, lL) supportable} Capacity Region L Simpler “Greedy” Scheduling: Maximal Scheduling -Activate any non-empty link that does not conflict. -Keep going until we cannot activate any more links. -Non-unique solution. Easy for distributed implementation. ml(t) = 1 iif Ql(t)>0 AND no other active links wSl

  12. Capacity Region: L = {All rate vectors l = (l1,…, lL) supportable} Capacity Region L Simpler “Greedy” Scheduling: Maximal Scheduling -Activate any non-empty link that does not conflict. -Keep going until we cannot activate any more links. -Non-unique solution. Easy for distributed implementation. ml(t) = 1 iif Ql(t)>0 AND no other active links wSl

  13. Capacity Region: L = {All rate vectors l = (l1,…, lL) supportable} Capacity Region L Simpler “Greedy” Scheduling: Maximal Scheduling -Activate any non-empty link that does not conflict. -Keep going until we cannot activate any more links. -Non-unique solution. Easy for distributed implementation. ml(t) = 1 iif Ql(t)>0 AND no other active links wSl

  14. Capacity Region: L = {All rate vectors l = (l1,…, lL) supportable} Capacity Region L Simpler “Greedy” Scheduling: Maximal Scheduling -Activate any non-empty link that does not conflict. -Keep going until we cannot activate any more links. -Non-unique solution. Easy for distributed implementation. ml(t) = 1 iif Ql(t)>0 AND no other active links wSl

  15. Capacity Region L g-scaled region gL Capacity Region: L = {All rate vectors l = (l1,…, lL) supportable} Constant-Factor Throughput Results for Maximal Scheduling: [Shah 2003]: 1/2-factor, Matching on NxN Switches [Lin, Shroff 2005]: 1/2-factor, Matching on Graphs [Chaporkar, Kar, Sarkar 2005]: g-factor, General Constraint Sets [Wu, Srikant, Perkins 05, 07]: g-factor, General Constraint Sets

  16. Prior Delay Results: Network of Size N nodes [Leonardi, Mellia, Neri, Marsan Infocom 2001]: NxN Packet Switch, full thruput, MWM, iid arrivals Delay = O(N). [Neely, Modiano, Cheng HPSR 04, TON 07]: NxN Packet Switch, full thruput, MSM-variation, iid arrivals, Delay = O(log(N)). [Deb, Shah, Shakkottai CISS 06]: NxN Packet Switch, 1/2 thruput, iid arrivals Maximal Matching, Delay = O(1).

  17. Goals of this paper: Develop a unified treatment of throughput/delay for maximal scheduling with bursty arrivals -Develop Order-Optimal Delay Results -Treat General Interference Sets -Treat Time-Correllated “Bursty” (non-iid) Arrivals We will: Define “Reduced Throughput Region” L* Get Structural Result for General Markovian Traffic: Delay =O(log(# interferers)) 3) Tight and order-optimal (Delay = O(1)) results for 2-state Markov arrivals (such as ON/OFF processes) 4) Get Delay Bounds as a function of spatio-temporal corellations in arrival processes.

  18. 1 2 Markov Arrival Model: -Arrivals Al(t) modulated by ergodic DTMC Zl(t). -Finite State: Zl = {1, …, Ml} pl, m(a) = Pr[Al(t)=a| Zl(t)=m] for a {0, 1, 2, …} ll, m = E{Al(t)| Zl(t)=m} , ll= E{Al(t)} = ll, m pl, m m AssumeE{Al(t)| Zl(t)=m} < infinity for all states m dl Example (M = 2 states): [Possibly ON/OFF process] bl

  19. for all lL wSl The Reduced Throughput Region L*: Capacity Region L Example: NxN Switch .7 .1 .1 .1 .1 .2 0 .3 .2 0 .3 0 .2 L* 2x2: Reduced Region L* L* 3x3: g-scaled region gL r* = 0.9 Define: L* = {(l1, …, lL)} such that: 1 lw

  20. wSl The Reduced Throughput Region L*: Capacity Region L Example: NxN Switch .7 .1 .1 .1 .1 .2 0 .3 .2 0 .3 0 .2 L* 2x2: Reduced Region L* L* 3x3: g-scaled region gL r* = 0.9 lL 1 L*: lw L* is typically within a constant factor g of L [Chaporkar, Kar, Sarkar 05][Lin, Shroff 05] Example:(Bipartite Matching) L* is strictly larger than L/2

  21. 1 2 wSl l L Delay Analysis for Maximal Scheduling (General Interference Sets): Q(t) = Queue vector = (Q1(t), …, QL(t)) Use concept of Queue Grouping: Qw(t) QSl(t) = Lyapunov Function: L(Q(t)) = Ql(t)QSl(t) Similar Lyapunov Functions used for stability analysis in: [Dai, Prabhakar 2000] , [Wu, Srikant, Perkins 07]

  22. E{ Ql(t) (1 - ASl(t)) } wSl wSl l L 1-step Unconditional Lyapunov Drift D(t): D(t) = E{L(Q(t+1)) - L(Q(t))} Drift Theorem: D(t) = B - B = Const Depends on Spatial Correlations E{AlAw} Aw(t) = “group” arrivals for Sl ASl(t) = Proof Uses Pair-wise Symmetry Property of the General Interference Sets: lSw iff

  23. Quick Delay Result for Arrivals iid over slots: Suppose there is a value r* (0 < r* < 1) s.t.: r* = “relative network loading” (relative to L*) Under any maximal scheduling… Example: Simple Delay Bound for independent Bernoulli or Poisson Inputs: (independent of network size!)

  24. Structural Delay Result for General Ergodic Markov Modulated Arrivals (finite state): Theorem: For any maximal scheduling, if r* <1 then: where |S| = 1 + Largest # interferers at any link (< N). Proof: Uses a Delayed Lyapunov Analysis technique to couple sufficiently fast to the stationary distribution. The technique is different from the T-Slot Lyapunov technique of [Georgiadis, Neely, Tassiulas NOW F&T 2006], which would yield looser (O(N)) delay results for bursty arrivals.

  25. Structural Delay Result for General Ergodic Markov Modulated Arrivals (finite state): Theorem: For any maximal scheduling, if r* <1 then: where |S| = 1 + Largest # interferers at any link (< N). The coefficient multiplier in the numerator depends on the auto-correlation of the arrival processes Al(t): E{Al(t)Al(t+k)} (details in paper)

  26. 1 ON 2 OFF More Detailed Analysis for 2-State Markov Modulated Arrivals: dl Each Al(t) has 2-state chain Zl(t): bl Pr[Al(t) = a| Zl(t) = 1] = general dist., rate ll Pr[Al(t) = a| Zl(t) = 2] = general dist., rate ll (1) (2) Important Special Case: 2-State ON/OFF Processes: dl bl

  27. 1 2 Tight (order-optimal) Delay Analysis for 2-State Markov Modulated Arrivals: dl These Corellations are Difficult to understand! bl Challenge: Lyapunov Drift term contains: E{Ql(t)Al(t)}, E{Ql(t)Aw(t)} Solution: Use a combination of Lyapunov Drift, Steady State Markov Chain theory, and Linear Algebra. We can isolate and bound the unknown correlations!

  28. Tight Delay Result (2-State Arrival Processes): Theorem: For any maximal scheduling, if r* <1: Where: Example: For independent ON/OFF arrival processes, we have…

  29. ON OFF Tight Delay Result (2-State Arrival Processes): Theorem: For any maximal scheduling, if r* <1: Example: For independent ON/OFF arrival processes with 1 packet arrival when ON, we have… dl ON = 1 Packet Arrival OFF = 0 Packet Arrival bl

  30. ON OFF Conclusions: dl ON = 1 Packet Arrival OFF = 0 Packet Arrival bl • Maximal Scheduling • General Interference Sets • Log(N) Delay Results for General Markov Arrivals • Tight and Order-Optimal (Delay = O(1)) Delay Results for 2-State Chains

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