Network Bandwidth Allocation
(and Stability)
In Three Acts
Problem Statement
How to allocate bandwidth to users?
How to model the network?
What criteria to use?
Act I
Modeling
A Physical View
C6
C1
Host
1
C2
Host
2
C7
C3
C4
C5
Host
3
Host
4
Host
5
Router : interconnect, where links meet.
Host : multi-user, endpoint of communication.
Link / Resource : bottleneck, each has finite capacity Cj.
System Usage
N 3 (t )
L 3 (N(t ))
Host
1
Host
2
N1 (t )
N 2 (t )
L1 (N(t ))
L 2 (N(t ))
Host
3
Host
4
Host
5
Route : static path through network, supporting Ni(t)
flows with Li(N(t)) allocated bandwidth.
Flows / Users : transfer documents of different sizes,
evenly split allocated bandwidth along route.
Dynamic. Not directed.
Simplification
C7
C6
C1
C3
C2
C4
N 3 (t )
N1 (t )
N 2 (t )
L 3 (N(t ))
L1 (N(t ))
L 2 (N(t ))
Extraneous elements have been removed.
C5
Abstraction
C1
N 3 (t )
L 3 (N(t ))
C7
C6
C3
C2
C5
C4
N1 (t )
N 2 (t )
L1 (N(t ))
L 2 (N(t ))
Routes are just subsets of links / resources.
Represented by [Aji] : whether resource j is used by route i.
Capacity constraint:
 A L (N(t ))  C
i:Ni ( t ) 0
ji
i
j
Stochastic Behavior
020
1L1 (120)
2
110
1
1
2
120
3
2L2 (120)
3L 3 (121)
220
1L1 (220)
2L2 (130)
130
121
Model N(t) as a Markov process with countable state space.
Poisson user arrivals at rate i.
Exponential document sizes with parameter i.
Define traffic intensity ri = i / i.
Act II
Performance
Criteria
Allocation Efficiency
• An allocation L is feasible if capacity constraint
satisfied.
• A feasible allocation L is efficient if we don’t have
  L for any other feasible .
• Defined at a point in time, regardless of usage.
10
10
10
10
L3  5
L1  5
10
10
L2  2
10
Stability
• Stable  Markov chain positive recurrent.
• Returns to each state with probability 1 in finite
mean time.
• Necessary, but not sufficient condition:
A
ji
ri  C j for all j
i
• How tight this is gives us an idea of utilization.
• Does not uniquely specify allocation.
Maximize Overall Throughput
L
• That is, max
i:N i ( t )  0
i
• No unique allocation.
• Could get unexpected results.
L4
10
10
10
L1
L2
L3
Max-Min Fairness
• Increase allocation for each user, unless doing
so requires a corresponding decrease for a
user of equal or lower bandwidth to satisfy the
capacity constraints.
• Uniquely determined.
• Greedy algorithm. Not distributed.
L3
12
12
L1
L2
Proportional Fairness
• L is proportionally fair if for any other
feasible allocation L* we have:
L*i  L i
i Ni (t ) L  0
i
• Same as maximizing:
 Ni (t ) log L i
i
• Interpret as utility function.
• Distributed algorithms known.
a-Fair Allocations
1a

L
a
i
Maximize

N
(
t
)
if a  (0, ), a  1
  i i
1a
G ( Λ , t )   i: N i ( t )  0
if a  1
   i N i log L i
i:N i (t ) 0
Subject to
A L
i:Ni ( t ) 0
With i=1,
a 
a=1
a 
With i = 1 / RTTi2,
a=2
ji
i
 Cj
Li  0
:maximize throughput
: proportional fairness
:max-min fairness
: TCP
TCP Bias
RTT
timeout
• Congestion window based on
additive increase /
multiplicative decrease
mechanism.
• Increase for each ACK
received, once every
Round Trip Time.
• Timeouts based on RTT.
• Bias against long RTT.
Properties of a-Fair Allocations
Assume Ni(t) > 0.
Let L(N(t)) be a solution to the a-fair optimization.
• The optimal L exists and is unique.
• It’s positive: L > 0.
• Scale invariance: L(rN) = L(N), for r > 0.
• Continuity: L is continuous in N.
•
System is stable when
A
i
ji
ri  C j for all j
Act III
Fluids &
Formalities
Fluid Models
Decompose into non-decreasing processes:
Ni (t )  Ni (0)  Ei (t )  Si (Ti (t ))
t
Ti (t )   Li (N(s)) ds
0
Ni(0) : initial condition
Ei(t)
: new arrival process
Ti(t)
: cumulative bandwidth allocated
Si(t)
: service process
Consider a sequence indexed by r > 0:
N r (rt )
N (t ) 
r
r
1000
9
900
8
800
7
700
6
600
E(t)
10
5
500
4
400
3
300
2
200
1
100
0
0
2
r 1
4
6
8
r  10
10
t
0
0
200
80
70
60
50
40
30
20
10
0
20
40
60
t
600
800
r  100
90
0
400
t
100
E(t)
E(t)
Fluid Limit : Visual
80
100
1000
Fluid Limit : Math
Look at slope:
r
r
E (t  t )  E (t ) E (rt  rt )  E (rt )

t
rt
By strong law of large numbers for renewal processes:
E (rt )
  as r  
rt
Thus
r
E (t )  t with probability 1.
Fluid Model Solution
A fluid model solution is an absolutely continuous
function N(t )
so that at each regular point t and each route i
d
dt
 
 i  i L i N(t ) if N i (t )  0
N i (t )  
if N i (t )  0
0
and for each resource j
 A L N(t )  A
ji
i:N i ( t ) 0
i
i:N i ( t ) 0
ji
ri  C j
Fluid Analysis is Easier
Definition
A complex function f is absolutely continuous on I=[a,b]
if for every e > 0 there is a d > 0 such that
n

i 1
f ( b i )  f (a i )  e
for any n and any disjoint collection of segments
(a1,b1),…,(an,bn) in I whose lengths satisfy
n
 b
i 1
i
 ai   d
Theorem
If f is AC on I, the f is differentiable a.e. on I, and
x
f ( x)  f (a)   f ' (t ) dt
a
Visualizing Fluid Flow
3
1
1L1 (N)  2
2 L 2 (N)
 3 L 3 ( N)
For Stability
• If fluid system empties in finite time, then system is
stable.
N(t )  0 for all t  T
• Show that
d
dt
N(t )  0
• In general, what happens as t when some of the
resources are saturated?
A
ji
ri  C j
i
• We approach the invariant manifold, aka the set of
invariant states
M a  N : Li (N)  ri for all i where N i 0
Towards a Formal Framework
• Interested in stochastic processes with samples
paths in D[0, , the space of right continuous real
functions having left limits.
x(t )
lim x( s )  x(t )
s t
t
lim x(s)  x(t  )
s t
• Well behaved. At most countably many points of
discontinuity.
Why we need a better metric.
x1 (t )
…
t
…
t
x2 (t )
What goes wrong in Lp ? L ?
Skorohod Topology
Let L be the set of strictly increasing Lipschitz
continuous functions  mapping [0,) onto [0,),
such that
'
 ( )  log  (t )   
Put
r  r 1
(standard bounded metric)
~
d ( x, y,  , u)  sup r x(t  u), y( (t )  u)
t 0

u ~

d ( x, y )  inf  ( )   e d ( x, y,  , u ) du 

0
L 

For functions mapping to any Polish (complete,
separable, metric) space.
Prohorov Metric
Let (S,d) be a metric space,
B(S) the s-algebra of Borel subsets of S,
P(S) family of Borel probability measures on S.
Define
r ( P, Q)  inf e  0 : P( F )  Q( F e )  e for all closed F  S 

F e  x  S : inf d ( x, y)  e
yF

The resulting metric space is Polish.
Fluid Limit Theorem
from Gromoll & Williams
Outline of Proof
• Apply functional law of large numbers to load
processes.
• Derive dynamic equations for state and bounds.
• State contained in compact set with probability 1
in limit.
• State oscillations die down with probability 1 in
limit.
• Sequence is C-tight.
• Weak limit points are fluid solutions with
probability 1.
Papers
1995
2000
Dai
Kelly
Kelly,
Maulloo,
Tan
2005
Bonald,
Massoulié
Gromoll,
Williams
Kelly,
Williams
Mo,
Walrand
Massoulié