Group Round Robin: Improving the Fairness and Complexity of Packet Scheduling

October 27, 2005 ANCS 2005

Group Round Robin: Improving the Fairness and

Complexity of Packet Scheduling

Bogdan Caprita, Jason Nieh, and Wong Chun Chan

Columbia University


GRR

Hierarchical framework for proportional sharing of network links with good fairness and overhead


GRR


• Proportional sharing

• Related work• GRR scheduling• Results


Scheduling

Time-multiplexing a shared resource among a set of competing entities


Scheduling

Time-multiplexing a shared link among a set of competing flows


Scheduling

Time-multiplexing a shared link among a set of competing clients


Scheduling

Time-multiplexing a shared link among a set of competing clients

Goals:metric of interest (throughput, delay, fairness …)speed, simplicity


Proportional Scheduling

Allocate to each client resource share in proportion to its weight

Goals:

fairness: is the allocation really ~ weight?

speed, simplicity



Allocate to each client resource time (WC) in proportion to its weight (φC)

WC ≈ (φC/ΦT)WT

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WC ≈ (φC/ΦT)WT

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Client work




WC ≈ (φC/ΦT)WT

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Client weight




WC ≈ (φC/ΦT)WT

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Total weight




WC ≈ (φC/ΦT)WT

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Total work




WC ≈ (φC/ΦT)WT

Goals:fairness: is the allocation really ~ weight?

Ideal: GPS (Parekh, Gallager)speed, simplicity



Issues:

skewed weights –

unfair

many clients –

slow, complicated



Formalize:

fairness service error EC=|WC – (φC/ΦT)WT|

Issues:

skewed weights –

unfair

many clients –

slow, complicated



Issues:

skewed weights –

unfair

many clients –

slow, complicated

Formalize:

fairness service error EC=|WC – (φC/ΦT)WT|

complexity

running time / decision


Related Work

Round-Robin

Virtual Time


Round Robin

• Weighted Round Robin

• Deficit Round Robin

• Virtual Time Round Robin

• Smoothed Round Robin

• Stratified Round Robin

Fast ( usually O(1) ), but unfair ( Ω(N) )


Virtual Time

• Weighted Fair Queuing• Start-Time Fair Queuing• Self-Clocked Fair Queuing • Worst-case Fair Weighted Fair Queuing• Earliest Eligible Virtual Deadline First• Hierarchical Stride

Slow (usually Ω(log N) ), but fair ( O(N), O(1) )


GRR





GRR Scheduling

• Grouping Strategy• Scheduling groups using intergroup

scheduler• Scheduling clients within groups using

intragroup scheduler


GRR Scheduling

Our approach: Exponential grouping by weight

G = { C : 2σ ≤ φC < 2σ+1 } σG = “group order”

1 2,3 4..7 8..15 16..31 32..63 …

σ=0 σ=1 σ=2 σ=3 σ=4 σ=5

number of groups: g = constant


GRR Scheduling

Our approach: Exponential grouping by weight

G = { C : 2σ ≤ φC < 2σ+1 } σG = “group order”

why? Decrease:

number of schedulable entities (few groups)

weight skew(clients within groups have similar weights)


Example:

C1 C2 C3 C4 C5

φ1=12 φ2=3 φ3=3 φ4=2 φ5=4

Grouping Strategy


G1 G2 G3

C1 C4 C5

Example:

C1 C2 C3 C4 C5

φ1=12 φ2=3 φ3=3 φ4=2 φ5=4

G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Grouping Strategy


GRR Scheduling

Data structures

Group G: unordered list of clients

group weight: Φ = ∑CG φC

Group list:

Φ1 , Φ2 , Φ3 ,… Φg g ≤ 32


GRR Scheduling

Two-level scheduler

1. Intergroup scheduling

2. Intragroup scheduling


GRR Scheduling

Two-level scheduler

1. Intergroup scheduling

idea: use an existing scheduler to schedule groups


GRR SchedulingExamples (1 - Intergroup):

G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

Φ 12 8 4

G1 G2 G3



Φ 12 8 4

G1 G2 G3



G1 G2 G3

12 8 4



G1 G2 G3

12 8 4



G1 G2 G3

12 8 4

Scheduling algorithm



G1 G2 G3

12 8 4

Scheduling algorithmWFQ HS

SCFQ WF2Q

SFQ SRR



G1 G2 G3

12 8 4

Scheduling algorithm

G1 G2 G3G1 G1 G2 G1 G2 G3G1 G1 G2

Schedule:


GRR Scheduling

Two-level scheduler

2. Intragroup scheduling

idea: round robin with deficit tracking

“current client” index i < |G| :

if ( Di < 1 ) {

i = i + 1; Di = Di + φC/2σ;

}

Di = Di – 1;


G1 G2 G3

C1 C4 C5

Examples (2 - Intergroup):

G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Intragroup Scheduling


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 00 00 0



C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 00 00 0


C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 00.5 00 0

C1


C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0.50.5 00 0

C1 C2


C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0.5 1 00 0

C1 C2 C1


C3C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C2

Deficit: 0.5 0 00 0

C1 C2 C1 C1


C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0.5 0 00.5 0

C1 C2 C1 C1 C3


C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0.5 0 00.5 0

C1 C2 C1 C1 C3 C5


C4C4C1 C5C3C2C1


G1 G2 G3

C1 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0.5 0.5 00.5 0

C1 C2 C1 C1 C3 C5 C1


C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0.5 0 0

C1 C2 C1 C1 C3 C5 C1 C4

0.5 0.5


C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 1 0 0

C1 C2 C1 C1 C3 C5 C1 C4 C1

0.5 0.5


C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0 0 0

C1 C2 C1 C1 C3 C5 C1 C4 C1 C1

0.5 0.5


C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0 0 0

C1 C2 C1 C1 C3 C5 C1 C4 C1 C1

1 0.5

C2


C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0 0 0

C1 C2 C1 C1 C3 C5 C1 C4 C1 C1

1 0.5

C2 C5


C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0.5 0 0

C1 C2 C1 C1 C3 C5 C1 C4 C1 C1

1 0.5

C2 C5

C1


C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0.5 0 0

C1 C2 C1 C1 C3 C5 C1 C4 C1 C1

0 0.5

C2 C5

C1 C2


C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 1 0 0

C1 C2 C1 C1 C3 C5 C1 C4 C1 C1

0 0.5

C2 C5

C1 C2 C1


C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0 0 0

C1 C2 C1 C1 C3 C5 C1 C4 C1 C1

0 0.5

C2 C5

C1 C2 C1 C1


C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0 0 0

C1 C2 C1 C1 C3 C5 C1 C4 C1 C1

0 1

C2 C5

C1 C2 C1 C1 C3


C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0 0 0

C1 C2 C1 C1 C3 C5 C1 C4 C1 C1

0 1

C2 C5

C1 C2 C1 C1 C3 C5


C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0.5 0 0

C1 C2 C1 C1 C3 C5 C1 C4 C1 C1

0 1

C2 C5

C1 C2 C1 C1 C3 C5 C1


C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0.5 0 0

C1 C2 C1 C1 C3 C5 C1 C4 C1 C1

0 0

C2 C5

C1 C2 C1 C1 C3 C5 C1 C3


C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 1 0 0

C1 C2 C1 C1 C3 C5 C1 C4 C1 C1

0 0

C2 C5

C1 C2 C1 C1 C3 C5 C1 C3 C1


C4C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0 0 0

C1 C2 C1 C1 C3 C5 C1 C4 C1 C1

0 0

C2 C5

C1 C2 C1 C1 C3 C5 C1 C3 C1 C1


C4C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0 0 0

C1 C2 C1 C1 C3 C5 C1 C4 C1 C1

0 0

C2 C5

C1 C2 C1 C1 C3 C5 C1 C3 C1 C1 C4


C4C1 C5C3C2C1


G1 G2 G3

C1 C4 C5


G1={C1} Φ1=12 ; G2={C2, C3, C4} Φ1=8 ; G3={C5} Φ3=4

φ1=12 ; φ2=3, φ3=3, φ4=2 ; φ5=4

C3C2

Deficit: 0 0 0

C1 C2 C1 C1 C3 C5 C1 C4 C1 C1

0 0

C2 C5

C1 C2 C1 C1 C3 C5 C1 C3 C1 C1 C4 C5


GRR





GRR Properties

Theoretical

Worst-case bounds

Experimental

Simulations


GRR Theoretical Properties

Error: EC < EG + O(1)

Time complexity: TCG + O(1)


Intergroup Schedulers

Scheduler Error – Θ() Time – Θ()

WFQ N Log(N)

SCFQ N Log(N)

SFQ N Log(N)

HS Log(N) Log(N)

WF2Q 1 Log(N)

SRR N*Log(φC) 1


Simulation Studies

0.1

1

10

100

1000

32 64 128 256 512 1024 2048 4096 8192

Number of Clients

Neg

ativ

e S

ervi

ce E

rro

rGRR WFQ SRR WF2Q


Simulation Studies

0.1

1

10

100

1000

32 64 128 256 512 1024 2048 4096 8192

Number of Clients

Po

sit

ive

Se

rvic

e E

rro

rGRR WFQ SRR WF2Q


GRR C & FW


• Scheduler

Integration• Analysis• Implementation


Thank You!

http://ncl.cs.columbia.edu/

Documents

Group Round Robin: Improving the Fairness and Complexity of Packet Scheduling