SRPT Applied to
Bandwidth Sharing Networks
(to appear in Annals of Operations Research)
Samuli Aalto
TKK Helsinki University of Technology, Finland
Urtzi Ayesta
LAAS-CNRS, France
BSnetworks.ppt
TKK/ComNet Research Seminar, 7.3.2008
1
Outline
•
•
•
•
•
Introduction: Bandwidth sharing networks
Background: Static setting
Background: Dynamic setting
Theoretical results: Delay reduction by applying SRPT
Numerical results and observations
2
Resource sharing
• Single resource
– loss system (e.g. Erlang model M/G/n/n)
• one customer per server, no buffer
• model for telephone traffic at the connection level
– queueing system (e.g. M/G/1-FCFS)
• customers served one-by-one, infinite buffer
• model for data traffic at the packet level
– sharing system (e.g. M/G/1-PS)
• customers share the server, infinite buffer
• model for (elastic) data traffic at the flow level
• Multiple resources
– loss networks (e.g. Ross)
– queueing networks (e.g. Jackson, BCMP, Kelly, ...)
– bandwidth sharing networks (Massoulié & Roberts)
3
Bandwidth sharing network
• Characterization:
– Flow-level model of a data network loaded with elastic
flows (such as file transfers using TCP)
• Resources:
– Links l with bandwidth cl (bps)
– Routes r (route = collection of consecutive links)
• Traffic:
– Elastic flows i with arrival time (possibly random), size
(bits to be transferred), and route (fixed)
• Control:
– Bandwidth allocation to flows
• inter-route bw allocation
• intra-route bw allocation
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 on route r
– feasible allocation satisfies the capacity constraints:
rR(l ) fr  cl for all l
• Intra-route bandwidth allocation
– tells how the total bandwidth fr is shared among the
flows using route r
– an 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 and observations
7
Static setting
• Characterization:
– Fixed number of saturated flows, n  (nr; r  R)
• 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)
– TM: throughput maximization [a ]; Massoulié &
Roberts (1999)
– PDM: potential delay minimization [a 2]; 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
• alpha-fairness [a ]
n0
f0 
,
1
/
a
a
a
n (n  n )
0
1
• TM [a ]
2
f1  f2 
( n1a  n2a )1 / a
n0  ( n1a  n2a )1 / a
n1n2  0
0,
 n
f0   n  n0  n , n1n2  0, n1  n2  0
1,0 1 2 n  n  0

1
2
• PF [a 1] = BF (in this case, not generally)
n0
n1  n2
f0  n  n  n ,
f1  f2  n  n  n
0 1 2
0 1 2
• MMF [a ]
n0
,

max{
n
,
n
}
0
1 2
f0  n
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 
n1  0 and n2  0
 0,
 n0
f0   n  n  n , either n1  0 or n2  0
1,0 1 2 n  0 and n  0

1
2
• TM* with preemptive priority to routes 1 and 2
0,

f0   0,
1,

n1  0 and n2  0
either n1  0 or n2  0
n1  0 and n2  0
10
Outline
•
•
•
•
•
Introduction: Bandwidth sharing networks
Background: Static setting
Background: Dynamic setting
Theoretical results: Delay reduction by applying SRPT
Numerical results and observations
11
Dynamic setting
• Randomly varying number of flows
– Poisson arrivals, generally distributed flow sizes
• Necessary stability conditions:
rR(l )  r  cl for all l
• Definition:
– Bandwidth allocation policy is (maximally) 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 (1968)
• Non-anticipating mean delay optimization for DHR
(Decreasing Hazard Rate) service times: FB (ForegroudBackground) = LAS (Least-Attained-Service); Yashkov
(1987)
13
Stable bw allocations for multilink networks
• Exponential flow sizes
– PF stable in linear networks; Massoulié & Roberts (1998)
– MMF stable in linear networks; De Veciana et al. (1999)
– alpha-fair bw allocations stable for any topology; Bonald
and Massoulié (2001)
– U-utility maximization bw allocations stable for any
topology; Ye et al. (2003,2005):
• General flow sizes
– BF stable for any topology; Bonald and Proutière (2003)
– MMF stable for any topology; Bramson (2005)
– PF stable for any topology; Massoulié (2005)
– alpha-fair bw allocation stable for tree topologies;
Gromoll and Williams (2006)
• Note: Above, intra-route discipline always PS
14
Stable bw allocations for multilink 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 is 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* not maximally stable in linear network
– TM* stable in symmetric linear network only if
0  (1  1)(1   2 )  min{1  1,1   2}
• Verloop et al. (2005):
– global SRPT not maximally stable in linear network
– global LAS not maximally stable 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 and 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n = 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n
20
Delay reduction by local SRPT’
• Let p Pn 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*
• Proposition 1:
– Let p Pn, r  R and t .
~
For any modification p (including p ),
p
p~
N r (t )  N r (t )
• Theorem 1:
– Let p Pn.
~
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w = family of stable bw allocation policies p for which
Z rp (t )   rp ( Wp (t ))
where
– Zr(t) = total bw allocated to class r at time t
– Wr(t) = total workload on route r at time t
– W(t)  (Wr(t); r  R)
• Note: Policies PR0 and PR12 
Pw
23
Delay reduction by local SRPT*
• Now there is no difference between p’ and p*
• Proposition 2:
– Let p Pw, r  R and t .
~
For any modification p (including p ),
p*
p
p~
N r (t )  N r (t )  N r (t )
• Theorem 2:
– Let p Pw.
~
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 and observations
25
Simulation setup
• Symmetric linear network with L = 2 and unit capacities
– Poisson 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
• Now, PR12 is the optimal non-anticipating allocation policy
• Mean delay comparison between basic policy p and its SRPTmodifications p’ and p* using basic policies
– BF = PF, 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
Increasing variability
5
6
BF
Mean number of flows
Mean number of flows
6
4
3
2
1
0.5
1
1.5
2
2.5
3
3.5
Coefficient of variation
4
5
PR0
4
3
2
1
0.5
1
1.5
2
2.5
3
3.5
Coefficient of variation
4
Mean number of flows
6
5
PR12
4
3
2
1
0.5
1
1.5
2
2.5
3
3.5
Coefficient of variation
4
30
Observations
• Mean delay improved by p’ for each class as Prop 1 predicts
• As Prop 2 implies, for the basic policies PR0 and PR12
N rp *  N rp   N rp
• In all numerical cases, for the 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
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