comments on the relative and absolute fairness measures joanna józefowska *) Łukasz józefowski *)...
TRANSCRIPT
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.