Comments on the relative and absolute fairness measures

Joanna Józefowska*)

Łukasz Józefowski*)

Wiesław Kubiak**)

*)Poznań University of Technology**)Memorial University

Presentation outline

Scheduling packets in packet-switched networks Problem formulation Relative fairness bound Absolute fairness bound

Apportionment problem Formulation Properties RFB transformation

PRV problem Formulation AFB transformation

Conclusions

Scheduling packets in a packet switched network

n flows single output from the buffer of packets wi – weight of flow i, i = 1, …, n

Li – size of packet i, i = 1, …, n

Fair sequence?

Relative fairness

SiP(t1, t2) – service obtained by flow i in time interval

(t1, t2) using discipline P

j

Pj

i

Pi

ji w

ttS

w

ttSttRF 2121

21,

,,,

t1 t2

Relative fairness bound

t1 t2

j

Pj

i

Pi

ji w

ttS

w

ttSttRF 2121

21,

,,,

21,21 ,max, ttRFttRF jij

i

2121 ,max, ttRFttRF ii

21),(

,max21

ttRFRFBtt

j

Pj

i

Pi

ttji w

ttS

w

ttSRFB 2121

,,,

,,max

21

Generalized Processor Sharing Policy

j

iGj

Gi

w

w

ttS

ttS

21

21

,

,

Generalized Processor Sharing Policy

W

wCttttS iG

i )(, 1221

C – resource capacity (rate)

n

jjwW

1

Absolute fairness bound

i

Gi

i

Pi

i w

ttS

w

ttSttAF 2121

21

,,,

2121 ,max, ttAFttAF ii

21),(

,max21

ttAFAFBtt

i

Gi

i

Pi

tti w

ttS

w

ttSAFB 2121

,,

,,max

21

Apportionment problemformulation

n – number of states p = [p1, …, pn] – vector of populations h – house size a = [a1, …, an] – vector of apportionment: ha

n

ii

1

j

j

i

i

ji a

p

a

pm

,ax minimize

Apportionment problemproperties

House monotone methods

No method minimizing is house monotone.

Population monotone methods

Every population monotone method is also house monotone.

a'aa'a 1,, hMhM pp

jjii

jjii

j

i

j

i

aaaa

or

aaaa

p

p

p

phMhM

''

''

'

',, p'p a'a

j

j

i

i

ji a

p

a

pm

,ax

Apportionment

number of states

population (pi) of state i

number of seats (ai) assigned to state i in a parliament of size h

n – number of flows

wi – weight of flow i

xi – number of packets of flow i sent in the considered time interval of length h

xi Li /C – number of time units assigned to flow i in the considered time interval

Packet scheduling

RFB transformation

Relation between RFB and the apportionment problem

iipi LxttS *, 21

j

jj

i

ii

jitt w

Lx

w

LxRFB

,),( 2,1

max

j

Pj

i

Pi

jitt w

ttS

w

ttSRFB 2121

,,,

,,max

21

t1 t2

j

j

i

i

h,i,j p

a

p

aCRFB max

Caj

Comments

Theorem

There exists no house monotone method minimizing the RFB measure.

Conclusion

There exists no population monotone method minimizing the RFB measure.

Product Rate Variation

10%

15%

25%

50%

10 pcs

15 pcs

25 pcs

50 pcs

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

100/10=10

100/15=6.67

100/25=4

100/50=2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Product Rate Variation

iik krx

xik – number of copies of product i completed by time k

di – demand for product i in the planning horizon

i – weight of product i

minimize

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

k

iikiki

krx max,

iiki krx

n

ii

ii

d

dr

1

Relation between AFB and the PRV problem

assume Li = L

t1 t2

C

Lktt 12

k packets

PRV

number of products

total number of copies completed in k time units

demand (di) of product i

number of copies (xik) of product i completed in k time units

n – number of flows

k – total number of packets sent

wi – weight of client i

xi – number of packets of flow i sent in the sequence of k packets

Packet scheduling

AFB transformation

Relation between AFB and the PRV problem

LxttS ipi *, 21

i

Gi

i

Pi

tti wttS

wttS

AFB 2121

21

,,max

,,

W

wCttttS iG

i )(, 1221

Ww

LC

ttxwL

AFB ii

itti

1221 ,,

max

Fi

ri

iiiki

krxAFB ,

max

k

C

Lktt 12

W

wCttLx

wAFB i

ii

tti12

,,

1max

21

Comments

AFB with identical packet length can be transformed to the PRV problem in the min-max version.

PRV and thus AFB can be effectively solved as a linear bottleneck assignment problem.

Further research

Transformation of the AFB for the problem with arbitrary packet length.

Analysis of properties of schedules and algorithms minimizing the AFB.