1. SRPT Applied to Bandwidth Sharing Networks
Samuli Aalto
TKK Helsinki University of Technology, Finland
Urtzi Ayesta
LAAS-CNRS, France

2. Outline Introduction: Bandwidth sharing networks
Background: Static setting
Background: Dynamic setting
Theoretical results: Delay reduction by applying SRPT
Numerical results
Observations

3. Bandwidth sharing network
BS network = Flow-level model of a data network
Resources: Links l Î L with
bandwidth = capacity cl (bps)
Traffic: Routes r Î 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

4. Example: Linear network with L = 4
route 1
route 2
route 3
route 4
link 1
link 2
link 3
link 4
route 0

5. Example: Symmetric linear network with L = 2
route 1
route 2
link 1
link 2
route 0
Symmetric with unit capacities:

6. Bandwidth allocation Inter-route bandwidth allocation
fr = total bandwidth allocated to the flows with route r
feasible allocation satisfies the capacity constraints:
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

7. Outline Introduction: Bandwidth sharing networks
Background: Static setting
Background: Dynamic setting
Theoretical results: Delay reduction by applying SRPT
Numerical results
Observations

8. Static setting Fixed number of flows, n = (nr; r Î 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 ® 0”]; Massoulié & Roberts (1999)
alpha-fairness [a Î (0, ¥)]; Mo & Walrand (1998,2000)
U-utility maximization; Ye et al. (2003,2005)
BF: balanced fairness; Bonald & Proutière (2003)

9. Fairness in symmetric linear network with L = 2
TM [a ® 0]
PF [a = 1] = BF (in this case, not generally)
alpha-fairness [a Î (0, ¥)]
MMF [a ® ¥]

10. Fairness in symmetric linear network with L = 2
Note: Throughput maximization does not specify a unique bandwidth allocation when n1 = 0 or n2 = 0
TM as a limit a ® 0
TM* with preemptive priority to routes 1 and 2

11. Outline Introduction: Bandwidth sharing networks
Background: Static setting
Background: Dynamic setting
Theoretical results: Delay reduction by applying SRPT
Numerical results
Observations

12. Dynamic setting Randomly varying number of flows
Poisson arrivals, generally distributed flow sizes
Necessary stability conditions:
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

13. Single (bottleneck) link
M/G/1 queue
Fair bandwidth sharing = PS (Processor-Sharing)
Stability = WC (Work-Conserving disciplines)
Anticipating mean delay optimization = SRPT (Shortest-Remaining-Processing-Time); Schrage (1966)
Non-anticipating mean delay optimization for DHR service times = FB (Foregroud-Background) = LAS (Least-Attained-Service); Yashkov (1987)

14. 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 > 0)
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

15. 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 ¹ TM* which gives preemptive priority to classes 1 and 2 whenever at least one of them is nonempty

16. 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
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

17. Delay optimization among stable bw allocations
Yang & de Veciana (2002,2004):
optimal allocation: fr(n) = 0 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 ³ m0, then PR12 optimal
Bonald and Proutière (2003):
BF insensitive for any topology

18. Outline Introduction: Bandwidth sharing networks
Background: Static setting
Background: Dynamic setting
Theoretical results: Delay reduction by applying SRPT
Numerical results
Observations

19. 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

20. Delay reduction by local SRPT’
P° = family of stable bw allocation policies p for which 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 Î R)
Note: All fair bw allocation policies mentioned above Î P°

21. Delay reduction by local SRPT’
Let p Î P° be fixed.
p = a modified policy
with the same inter-route bw allocation process,
but the intra-route disciplines may be different from the original ones
p’ = the modified policy
that applies SRPT as the intra-route discipline
p* = the policy
for which the inter-route bw allocation process is
and that applies SRPT as the intra-route discipline ~

22. Delay reduction by local SRPT’
Note the difference between p’ and p*
Theorem 1:
Let p Î P°, r Î R and t ³ 0. For any modification p (including p ),
Corollary 1:
Let p Î P°. For any modification p (including p ),
Here T refers to delay (= total transfer time) ~ ~

23. Delay reduction by local SRPT*
P b = family of stable bw allocation policies p for which where
Zr(t) = total bw allocated to class r at time t
Br(t) = 1(Nr(t) > 0) = busy period indicator
B(t) = (Br(t); r Î R)
Note: Policies PR0 and PR12 Î P b

24. Delay reduction by local SRPT*
Proposition:
Let p Î P b, r Î R and t ³ 0. Then
Theorem 2:
Let p Î P b, r Î R and t ³ 0. For any modification p (including p ),
Corollary 2:
Let p Î P b. For any modification p (including p ), ~ ~

25. Outline Introduction: Bandwidth sharing networks
Background: Static setting
Background: Dynamic setting
Theoretical results: Delay reduction by applying SRPT
Numerical results
Observations

26. 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 = 0.8
deterministic
exponential: m = 1/b
hyperexponential: p1 = 0.9, m1 = 9/b; p2 = 0.1, m2 = 1/9b
Comparison between p, p’ and p* using basic policies BF
PR0
PR12

27. Deterministic flow sizes
p p’ p* BF
PR12
PR12

28. Exponential flow sizes
p p’ p* BF
PR12
PR12

29. Hyperexponential flow sizes
p p’ p* BF
PR12 BF

30. Outline Introduction: Bandwidth sharing networks
Background: Static setting
Background: Dynamic setting
Theoretical results: Delay reduction by applying SRPT
Numerical results
Observations

31. Observations
Mean delay improved by p’ for each class as Thm 1 predicts
As Thm 2 tells, for basic policies PR0 and PR12
In all numerical cases, for basic policy BF
In all numerical cases, for all basic policies (BF, PR0, PR12)
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

32. THE END