Delay Analysis for Maximal Scheduling in Wireless Networks with Bursty Traffic Michael J. Neely...

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]






Example: Matching, NxN Switch

Link l






Example: Matching, NxN Switch

Set Sl






Example: Matching, Wireless

Link l






Example: Matching, Wireless

Set Sl






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}

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

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

-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

Delay Analysis for Maximal Scheduling in Wireless Networks with Bursty Traffic Michael J. Neely...

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