Delay Analysis for Maximal Scheduling in Wireless Networks with Bursty Traffic Michael J. Neely University of Southern California INFOCOM 2008, Phoenix, AZ *Sponsored in part by the DARPA IT-MANET Program, NSF OCE , NSF Career CCF ONOFFONOFFONOFF Capacity Region   -scaled region  ONOFF

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

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

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

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

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

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

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

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

Capacity Region:  = {All rate vectors  = ( 1,…, L ) supportable} Capacity Region  [Tassiulas, Ephremides 92]: Max Weight Match (MWM) Maximize Q l (t)  l (t) Subject to: (Stabilizes Network, Supports all interior to   (t) X(t)

Capacity Region:  = {All rate vectors  = ( 1,…, L ) supportable} Capacity Region  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.  l (t) = 1 iif Q l (t)>0 AND no other active links  S l

Capacity Region:  = {All rate vectors  = ( 1,…, L ) supportable} Capacity Region  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.  l (t) = 1 iif Q l (t)>0 AND no other active links  S l

Capacity Region:  = {All rate vectors  = ( 1,…, L ) supportable} Capacity Region  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.  l (t) = 1 iif Q l (t)>0 AND no other active links  S l

Capacity Region:  = {All rate vectors  = ( 1,…, L ) supportable} Capacity Region  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.  l (t) = 1 iif Q l (t)>0 AND no other active links  S l

Capacity Region:  = {All rate vectors  = ( 1,…, L ) supportable} Capacity Region   -scaled region  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]:  -factor, General Constraint Sets [Wu, Srikant, Perkins 05, 07]:  -factor, General Constraint Sets

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).

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: 1)Define “Reduced Throughput Region”  * 2)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.

Markov Arrival Model: -Arrivals A l (t) modulated by ergodic DTMC Z l (t). -Finite State: Z l = {1, …, M l } p l, m (a) = Pr[A l (t)=a| Z l (t)=m] for a {0, 1, 2, …} l, m = E{ A l (t)| Z l (t)=m}, l = E{ A l (t)} = l, m  l, m Assume E{ A l (t)| Z l (t)=m} < infinity for all states m Example (M = 2 states): 12 ll ll [Possibly ON/OFF process] m

The Reduced Throughput Region  *: Capacity Region   -scaled region  Reduced Region   Define:  * = {( 1, …, L )} such that:  S l  1 for all l L Example: NxN Switch ** **  * = 0.9 2x2: 3x3:

The Reduced Throughput Region  *: Capacity Region   -scaled region  Reduced Region   Example: NxN Switch ** **  * = 0.9 2x2: 3x3:  * is typically within a constant factor  of  [Chaporkar, Kar, Sarkar 05][Lin, Shroff 05] Example: (Bipartite Matching)  * is strictly larger than  /2  S l  1 l L  *:

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

1-step Unconditional Lyapunov Drift  (t):  (t) = E{L( Q (t+1)) - L( Q (t))} Drift Theorem:  S l iff l S   (t) = B - Proof Uses Pair-wise Symmetry Property of the General Interference Sets: l L E{ Q l (t) (1 - A S l (t)) } B = Const Depends on Spatial Correlations E{A l A  } A S l (t) =  S l A  (t) = “group” arrivals for S l

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

Structural Delay Result for General Ergodic Markov Modulated Arrivals (finite state): Theorem: For any maximal scheduling, if  * <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.

Structural Delay Result for General Ergodic Markov Modulated Arrivals (finite state): Theorem: For any maximal scheduling, if  * <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 A l ( t): E{ A l ( t) A l ( t+k)} (details in paper)

More Detailed Analysis for 2-State Markov Modulated Arrivals: 12 ll ll Each A l ( t) has 2-state chain Z l ( t) : Pr[ A l ( t) = a| Z l ( t) = 1] = general dist., rate l Pr[ A l ( t) = a| Z l ( t) = 2] = general dist., rate l (1) (2) Important Special Case: 2-State ON/OFF Processes: ONOFF ll ll

Tight (order-optimal) Delay Analysis for 2-State Markov Modulated Arrivals: 12 ll ll Challenge: Lyapunov Drift term contains: E{ Q l ( t) A l ( t)}, E{ Q l ( t) A  ( t)} These Corellations are Difficult to understand! Solution: Use a combination of Lyapunov Drift, Steady State Markov Chain theory, and Linear Algebra. We can isolate and bound the unknown correlations!

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

Tight Delay Result (2-State Arrival Processes): Theorem: For any maximal scheduling, if  * <1: Example: For independent ON/OFF arrival processes with 1 packet arrival when ON, we have… ONOFF ll ll ON = 1 Packet Arrival OFF = 0 Packet Arrival

Conclusions: ONOFF ll ll ON = 1 Packet Arrival OFF = 0 Packet Arrival  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