Delay Analysis for Maximal Scheduling in

Wireless Networks with Bursty Traffic

Michael J. NeelyUniversity of Southern California

INFOCOM 2008, Phoenix, AZ

*Sponsored in part by the DARPA IT-MANET Program, NSF OCE-0520324, NSF Career CCF-0747525

ON OFF

ON OFF

ON OFF

Capacity Region

-scaled region ON OFF

One-Hop Network Model:

N = Node set = {1, 2…, N}L = Link set = {1, 2, …, L}

Sl = Interference Set for link l 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]

One-Hop Network Model:

N = Node set = {1, 2…, N}L = Link set = {1, 2, …, L}

Sl = Interference Set for link l 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]

Example: Matching, NxN Switch

Link l

One-Hop Network Model:

N = Node set = {1, 2…, N}L = Link set = {1, 2, …, L}

Sl = Interference Set for link l 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]

Example: Matching, NxN Switch

Set Sl

One-Hop Network Model:

N = Node set = {1, 2…, N}L = Link set = {1, 2, …, L}

Sl = Interference Set for link l 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]

Example: Matching, Wireless

Link l

One-Hop Network Model:

N = Node set = {1, 2…, N}L = Link set = {1, 2, …, L}

Sl = Interference Set for link l 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]

Example: Matching, Wireless

Set Sl

One-Hop Network Model:

N = Node set = {1, 2…, N}L = Link set = {1, 2, …, L}

Sl = Interference Set for link l 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]

Example: Arb. Interference Sets

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) l(t) = # packets served from queue l (on slot t)

Al(t) l(t)

Ql(t)

Ql(t+1) = Ql(t) - l(t) + Al(t)

l(t) {0, 1}l(t) = 1 only if Ql(t)>0 AND no other active links Sl

X(t) ={Scheduling Options}

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) l(t) = # packets served from queue l (on slot t)

Al(t) l(t)

Ql(t)

Ql(t+1) = Ql(t) - l(t) + Al(t)

l(t) {0, 1}l(t) = 1 only if Ql(t)>0 AND no other active links Sl

X(t) ={Scheduling Options}

Capacity Region: = {All rate vectors = (1,…, L) supportable}

Capacity Region

[Tassiulas, Ephremides 92]: Max Weight Match (MWM)

Maximize Ql(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 Ql(t)>0 AND no other active links Sl

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 Ql(t)>0 AND no other active links Sl

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 Ql(t)>0 AND no other active links Sl

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 Ql(t)>0 AND no other active links Sl

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 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, …}

l, m = E{Al(t)| Zl(t)=m} , l = E{Al(t)} = l, m l, m

Assume E{Al(t)| Zl(t)=m} < infinity for all states m

Example (M = 2 states):

1 2

l

l

[Possibly ON/OFF process]

m

The Reduced Throughput Region *:

Capacity Region

-scaled region

Reduced Region

Define: * = {(1, …, L)} such that:

Sl

1 for all l L

Example: NxN Switch

.7 .1 .1 .1

.1 .2 0 .3 .2 0 .3 0 .2

*

*

* = 0.9

2x2:

3x3:

The Reduced Throughput Region *:

Capacity Region

-scaled region

Reduced Region

Example: NxN Switch

.7 .1 .1 .1

.1 .2 0 .3 .2 0 .3 0 .2

*

*

* = 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

Sl

1 l L*:

Delay Analysis for Maximal Scheduling (General Interference Sets):

Q(t) = Queue vector = (Q1(t), …, QL(t))

Use concept of Queue Grouping:

Lyapunov Function:

L(Q(t)) =

QSl(t) = Sl

Q(t)

l L

12 Ql(t)QSl(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:

Sl iff l S

(t) = B -

Proof Uses Pair-wise Symmetry Property of the General Interference Sets:

l LE{ Ql(t) (1 - ASl(t)) }

B = Const Depends on Spatial Correlations E{AlA} ASl(t) =

Sl

A(t) = “group” arrivals for Sl

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 ErgodicMarkov 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 techniqueto 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 ErgodicMarkov 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 Al(t):

E{Al(t)Al(t+k)} (details in paper)

More Detailed Analysis for 2-State Markov Modulated Arrivals:

1 2

l

l

Each Al(t) has 2-state chain Zl(t):

Pr[Al(t) = a| Zl(t) = 1] = general dist., rate l

Pr[Al(t) = a| Zl(t) = 2] = general dist., rate l

(1)

(2)

Important Special Case: 2-State ON/OFF Processes:

ON OFF

l

l

Tight (order-optimal) Delay Analysis for 2-State Markov Modulated Arrivals:

1 2

l

l

Challenge: Lyapunov Drift term contains: E{Ql(t)Al(t)}, E{Ql(t)A(t)}

These Corellations are Difficult to understand!

Solution:Use a combination of Lyapunov Drift, Steady StateMarkov Chain theory, and Linear Algebra. We canisolate 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…

ON OFF

l

l

ON = 1 Packet ArrivalOFF = 0 Packet Arrival

Conclusions:

ON OFF

l

l

ON = 1 Packet ArrivalOFF = 0 Packet Arrival

Maximal SchedulingGeneral Interference SetsLog(N) Delay Results for General Markov ArrivalsTight and Order-Optimal (Delay = O(1)) Delay Results for 2-State Chains