Delay Analysis for Max Weight Opportunistic Scheduling in Wireless Systems
Download
1 / 19

Delay Analysis for Max Weight Opportunistic Scheduling in Wireless Systems - PowerPoint PPT Presentation


  • 88 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Delay Analysis for Max Weight Opportunistic Scheduling in Wireless Systems' - melora


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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”

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


Delay analysis for max weight opportunistic scheduling in wireless systems

  • Quick Description: N Queues, 1 Server, ON/OFF Channels Wireless Systems

  • 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


Delay analysis for max weight opportunistic scheduling in wireless systems

  • Quick Description: N Queues, 1 Server, ON/OFF Channels Wireless Systems

  • 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


Delay analysis for max weight opportunistic scheduling in wireless systems

  • Quick Description: N Queues, 1 Server, ON/OFF Channels Wireless Systems

  • 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


Delay analysis for max weight opportunistic scheduling in wireless systems

  • Quick Description: N Queues, 1 Server, ON/OFF Channels Wireless Systems

  • 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


Delay analysis for max weight opportunistic scheduling in wireless systems

  • Quick Description: N Queues, 1 Server, ON/OFF Channels Wireless Systems

  • 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


Delay analysis for max weight opportunistic scheduling in wireless systems

  • Notation: Server variables are 0/1 variables. Wireless Systems

  • 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


Delay analysis for max weight opportunistic scheduling in wireless systems

Status Quo #1 – Max-Weight Scheduling Wireless Systems:

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!


Delay analysis for max weight opportunistic scheduling in wireless systems

Status Quo #2 – Queue Grouping and LCG Wireless Systems:

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:


Delay analysis for max weight opportunistic scheduling in wireless systems

  • Our New Results Wireless Systems: 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:


Delay analysis for max weight opportunistic scheduling in wireless systems

  • Lyapunov Function 1 (L Wireless SystemsA): (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


Delay analysis for max weight opportunistic scheduling in wireless systems

  • Lyapunov Function 1 (L Wireless SystemsA): (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


Delay analysis for max weight opportunistic scheduling in wireless systems

Lyap. Drift Wireless SystemsDA(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!


Delay analysis for max weight opportunistic scheduling in wireless systems

Proof Sketch: Wireless SystemsUse Combinatorics to show…

Maximized by any

work-conserving

strategy

Maximized by LCQ

(“max-weight”)

c1 > 0, c2 > 0


Delay analysis for max weight opportunistic scheduling in wireless systems

Thus Wireless Systems: 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


Delay analysis for max weight opportunistic scheduling in wireless systems

  • Lyapunov Function 2 (L Wireless SystemsB): (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


Delay analysis for max weight opportunistic scheduling in wireless systems

  • *Extensions: Wireless Systems(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.


Delay analysis for max weight opportunistic scheduling in wireless systems

Conclusions: Wireless SystemsOrder-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:


Delay analysis for max weight opportunistic scheduling in wireless systems

Conclusions: Wireless SystemsOrder-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: