Transcript SRPT applied to bandwidth-sharing networks

SRPT Applied to
Bandwidth Sharing Networks
Samuli Aalto
TKK Helsinki University of Technology, Finland
Urtzi Ayesta
LAAS-CNRS, France
aalto.ppt
StoPeRA, EuroNGI Workshop, November 8-10, 2006
1
Outline
•
•
•
•
•
•
Introduction: Bandwidth sharing networks
Background: Static setting
Background: Dynamic setting
Theoretical results: Delay reduction by applying SRPT
Numerical results
Observations
2
Bandwidth sharing network
• BS network = Flow-level model of a data network
• Resources: Links l  L with
– bandwidth = capacity cl (bps)
• Traffic: Routes r  R (set of links) used by
– elastic (TCP) flows i with
• arrival time Ai
• size Si (bits to be transferred)
• route ri
• Control: Link bandwidth allocation to flows
– inter-route bw allocation
– intra-route bw allocation
3
Example: Linear network with L = 4
route 1
route 2
route 3
route 4
link 1
link 2
link 3
link 4
route 0
4
Example: Symmetric linear network with L = 2
route 1
route 2
link 1
link 2
route 0
Symmetric with unit capacities:
c1  c2  1
5
Bandwidth allocation
• Inter-route bandwidth allocation
– fr = total bandwidth allocated to the flows with route r
– feasible allocation satisfies the capacity constraints:
rR(l ) fr  cl
• Intra-route bandwidth allocation
– tells how the total bandwidth fr is shared among the
flows with route r
– for example:
• PS = Processor-Sharing = bandwidth is shared evenly
among the flows with the same route
6
Outline
•
•
•
•
•
•
Introduction: Bandwidth sharing networks
Background: Static setting
Background: Dynamic setting
Theoretical results: Delay reduction by applying SRPT
Numerical results
Observations
7
Static setting
• Fixed number of flows, n  (nr; r  R)
– saturated flows of infinite size
• Problem:
– Fair bandwidth sharing
• Solutions (with PS intra-route discipline):
– MMF: max-min fairness [a ]; trad., Jaffe (1981)
– PF: proportional fairness [a 1]; Kelly (1997)
– PDM: potential delay minimization [a 2]; Massoulié &
Roberts (1999)
– TM: throughput maximization [”a ”]; Massoulié &
Roberts (1999)
– alpha-fairness [a ]; Mo & Walrand (1998,2000)
– U-utility maximization; Ye et al. (2003,2005)
– BF: balanced fairness; Bonald & Proutière (2003)
8
Fairness in symmetric linear network with L = 2
• TM [a ]
n1n2  0
0,
 n0
f0   n  n  n , n1n2  0, n1  n2  0
1,0 1 2 n  n  0

1
2
• PF [a 1] = BF (in this case, not generally)
n
n n
f0  n  n0 n ,
f1  f2  n 1n 2n
0 1 2
0 1 2
• alpha-fairness [a ]
n0
f0 
,
1
/
a
a
a
n (n  n )
0
1
• MMF [a ]
2
n0
,

max{
n
,
n
}
0
1 2
f0  n
f1  f2 
( n1a  n2a )1 / a
n0  ( n1a  n2a )1 / a
max{n1,n2 }
0  max{n1,n2 }
f1  f2  n
9
Fairness in symmetric linear network with L = 2
• Note: Throughput maximization does not specify a
unique bandwidth allocation when n1  or n2 
• TM as a limit a 
n1n2  0
0,
 n0
f0   n  n  n , n1n2  0, n1  n2  0
1,0 1 2 n  n  0

1
2
• TM* with preemptive priority to routes 1 and 2
0,

f0  0,
 1,

n1n2  0
n1n2  0, n1  n2  0
n1  n2  0
10
Outline
•
•
•
•
•
•
Introduction: Bandwidth sharing networks
Background: Static setting
Background: Dynamic setting
Theoretical results: Delay reduction by applying SRPT
Numerical results
Observations
11
Dynamic setting
• Randomly varying number of flows
– Poisson arrivals, generally distributed flow sizes
• Necessary stability conditions:
rR(l )  r  cl
• Definition:
– Bandwidth allocation policy is stable if the necessary
stability conditions are also sufficient
• Primary concern:
– stable bandwidth sharing
• Secondary concern:
– (mean) delay optimization among stable bandwidth
allocations
12
Single (bottleneck) link
• M/G/1 queue
• Fair bandwidth sharing = PS (Processor-Sharing)
• Stability = WC (Work-Conserving disciplines)
• Anticipating mean delay optimization = SRPT (ShortestRemaining-Processing-Time); Schrage (1966)
• Non-anticipating mean delay optimization for DHR service
times = FB (Foregroud-Background) = LAS (Least-AttainedService); Yashkov (1987)
13
Stable bw allocations for multiple link networks
• Massoulié & Roberts (1998,2000):
– PF stable in linear networks
• De Veciana et al. (1999,2000):
– MMF stable in linear networks
• Bonald and Massoulié (2001):
– alpha-fair bw allocations stable for any topology (a )
• Ye et al. (2003,2005):
– U-utility maximization bw allocations stable for any
topology
• Bonald and Proutière (2003):
– BF stable for any topology
• Note: Above, intra-route discipline always PS
14
Stable bw allocations for multiple link networks
• Verloop et al. (2006):
– PR0 and PR12 stable in symm. linear network
• PR0 gives preemptive priority to class 0
whenever nonempty
• PR12 gives preemptive priority to classes 1 and 2
whenever both of them are nonempty; otherwise
preemptive priority is given to class 0
– Intuitive argument:
• Both policies ensure that the whole capacity of each
link used whenever there are flows loading the link
• Note: PR12  TM* which gives preemptive priority to classes
1 and 2 whenever at least one of them is nonempty
15
Unstable bw allocations for multiple link networks
• Bonald and Massoulié (2001):
– TM* (with preemptive priority to routes
1 and 2) unstable in linear network
– TM* stable in symmetric linear network only if
0  (1  1)(1   2 )  1  1
• Verloop et al. (2005):
– global SRPT unstable in linear network
– global LAS unstable in linear network
• Note: In all these cases, the whole capacity of a link is not
necessarily used while there are flows loading the link
16
Delay optimization among stable bw allocations
• Yang & de Veciana (2002,2004):
– optimal allocation: frn or 1 (depending on state n) in
symmetric linear network
• Verloop et al. (2006):
– determined optimal non-anticipating allocation in
symmetric linear network with exponential flow sizes
– if m1 m2 m0, then PR0 optimal
– if m1m2 m0 and m1 m2 m0, then PR12 optimal
• Bonald and Proutière (2003):
– BF insensitive for any topology
17
Outline
•
•
•
•
•
•
Introduction: Bandwidth sharing networks
Background: Static setting
Background: Dynamic setting
Theoretical results: Delay reduction by applying SRPT
Numerical results
Observations
18
Theoretical setup
• Consider a BS network with
– a general topology,
– Poisson arrivals, and
– generally distributed flow sizes
•
P = family of stable bw allocation policies p
19
Delay reduction by local SRPT’
•
P = family of stable bw allocation policies p
for which
Z rp (t )  frp (Np (t ))
where
– Zr(t) = total bw allocated to class r at time t
– Nr(t) = number of flows on route r at time t
– N(t)  (Nr(t); r  R)
• Note: All fair bw allocation policies mentioned above  P
20
Delay reduction by local SRPT’
• Let p P be fixed.
•
p~ = a modified policy
– with the same inter-route bw allocation process,
p~
Z r (t )  Z rp (t )  frp (Np (t ))
– but the intra-route disciplines may be different from the
original ones
•
p’
•
p*
= the modified policy
– that applies SRPT as the intra-route discipline
= the policy
– for which the inter-route bw allocation process is
Z rp * (t )  frp (Np * (t ))
– and that applies SRPT as the intra-route discipline
21
Delay reduction by local SRPT’
• Note the difference between p’ and p*
• Theorem 1:
– Let p P, r  R and t .
~
For any modification p (including p),
p
p~
N r (t )  N r (t )
• Corollary 1:
– Let p P.
~
For any modification p (including p),
p
p~
E[T ]  E[T ]
• Here T refers to delay (= total transfer time)
22
Delay reduction by local SRPT*
•
P b = family of stable bw allocation policies p for which
Z rp (t )   rp (Bp (t ))
where
– Zr(t) = total bw allocated to class r at time t
– Br(t) = 1Nr(t)   = busy period indicator
– B(t)  (Br(t); r  R)
• Note: Policies PR0 and PR12 
Pb
23
Delay reduction by local SRPT*
• Proposition:
– Let p  Pb, r  R and t . Then
Brp (t )  Brp  (t )  Brp * (t )
• Theorem 2:
– Let p  Pb, r  R and t .
~
For any modification p (including p),
p*
p
p~
N r (t )  N r (t )  N r (t )
• Corollary 2:
– Let p  Pb.
~
For any modification p (including p),
E[T
p*
p
p~
]  E[T ]  E[T ]
24
Outline
•
•
•
•
•
•
Introduction: Bandwidth sharing networks
Background: Static setting
Background: Dynamic setting
Theoretical results: Delay reduction by applying SRPT
Numerical results
Observations
25
Simulation setup
• Symmetric linear network with L = 2 and unit capacities
• Poissson arrivals with constant total rate
l 1
• Flow size distribution with mean b .8
– deterministic
– exponential:
m 1/b
– hyperexponential: p1 .9, m1 9/b; p2 .1, m2 1/9b
• Comparison between p, p’ and p* using basic policies
– BF
– PR0
– PR12
26
Deterministic flow sizes
p p’ p*
Mean number of flows
4
3.5
3
BF
2.5
2
1.5
PR12
1
0.5
0.2
0.4
0.6
lambda0
0.8
1
Mean number of flows
4
3.5
3
2.5
2
1.5
PR12
1
0.5
0.2
0.4
0.6
lambda0
0.8
27
1
Exponential flow sizes
p p’ p*
Mean number of flows
4
3.5
3
BF
2.5
2
1.5
PR12
1
0.5
0.2
0.4
0.6
lambda0
0.8
1
Mean number of flows
4
3.5
3
2.5
2
1.5
PR12
1
0.5
0.2
0.4
0.6
lambda0
0.8
28
1
Hyperexponential flow sizes
p p’ p*
Mean number of flows
4
3.5
3
BF
2.5
2
1.5
PR12
1
0.5
0.2
0.4
0.6
lambda0
0.8
1
0.4
0.6
lambda0
0.8
1
Mean number of flows
4
3.5
3
BF
2.5
2
1.5
1
0.5
0.2
29
Outline
•
•
•
•
•
•
Introduction: Bandwidth sharing networks
Background: Static setting
Background: Dynamic setting
Theoretical results: Delay reduction by applying SRPT
Numerical results
Observations
30
Observations
• Mean delay improved by p’ for each class as Thm 1 predicts
• As Thm 2 tells, for basic policies PR0 and PR12
N rp *  N rp   N rp
• In all numerical cases, for basic policy BF
N rp *  N rp
• In all numerical cases, for all basic policies (BF, PR0, PR12)
Np*  Np  Np
• Basic policy PR12 is better than BF for deterministic and
exponential flow sizes but worse for hyperexpontial
• Delay reduction of BF very similar for exponential and
hyperexponential flow sizes
31
THE END
32