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Mathematical models of the Internet
Frank Kellywww.statslab.cam.ac.uk/~frank
Hood Fellowship Public Lecture University of Auckland
3 April 2012
Outline
• End-to-end congestion control– square-root formula
• Utilities and fairness – optimization formulation
• Multipath routing– reliability, robustness
End-to-end congestion control
Senders learn (through feedback from receivers) of congestion at queue, and slow down or speed up accordingly. With current TCP, throughput of a flow is proportional to
senders receivers
)/(1 pT
T = round-trip time, p = packet drop probability. (Jacobson 1988, Mathis, Semke, Mahdavi, Ott 1997, Padhye, Firoiu, Towsley, Kurose 1998, Floyd and Fall 1999)
Feedback mechanism
• Now: dropped packets (acknowledgements carry information from receivers to senders)
• Standards track: Explicit Congestion Notification (Ramakrishnan, Floyd and Black RFC3168) - routers mark packets at queues as overload approached
Control of elastic network flows
user
resource
flow
• How should available resources be shared between competing streams of elastic traffic?• Conceptual problem: fairness• Pragmatic problem: stability of rate control algorithm
utility,U(x)
rate, x
utility,U(x)
rate, x
Elastic flows
elastic traffic - prefers to share(Shenker)
inelastic traffic - prefers to randomize
Network structure (J, R, A)
0
1
jr
jr
A
A
R
J - set of resources- set of routes - if resource j is on route r- otherwise
resource
route
Notation
J
R
jr
xr
Ur (xr )
C j
Ax C
- set of resources- set of users, or routes - resource j is on route r- flow rate on route r - utility to user r- capacity of resource j- capacity constraints
The system problem
0
)(
xover
CAxtosubject
xUMaximize rRr
r
Maximize aggregate utility, subject to capacity constraints
SYSTEM(U,A,C):
The user problem
0
r
rr
rr
wover
ww
UMaximize
User r chooses an amount to pay per unit time, wr ,
and receives in return a flow xr =wr /r
USERr(Ur;r):
0
log
xover
CAxtosubject
xwMaximize rRr
r
As if the network maximizes a logarithmic utility function, but
with constants {wr} chosen by the users
NETWORK(A,C;w):
The network problem
Problem decomposition
Theorem: the system problem
may be solved
by solving simultaneously
the network problem and
the user problems
Max-min fairness
Rates {xr} are max-min fair if they are feasible:
CAxx ,0
and if, for any other feasible rates {yr},
rssrr xxysxyr ::
Proportional fairness
Rates {xr} are proportionally fair if they are feasible:
CAxx ,0
and if, for any other feasible rates {yr}, the aggregate of proportional changes is negative:
yr xr
xr
0rR
Weighted proportional fairness
A feasible set of rates {xr} are such that are weighted proportionally fair
if, for any other feasible rates {yr},
w r
yr xr
xr
0rR
Fairness and the network problem
Theorem: a set of rates {xr} solves the network problem,
NETWORK(A,C;w), if and only if the rates are
weighted proportionally fair
Bargaining problem (Nash, 1950)
Solution to NETWORK(A,C;w) with w = 1 is unique point satisfying
• Pareto efficiency
• Symmetry
• Independence of Irrelevant Alternatives
(General w corresponds to a model with unequal bargaining power)
Market clearing equilibrium (Gale, 1960)
Find prices p and an allocation x such that
Rrpxw
AxCp
CAxp
rjjrr
T
,
0)(
,0 (feasibility)(complementary slackness)(endowments spent)
Solution solves NETWORK(A,C;w)
Fairness criteria
We’ve seen three fairness criteria:
• proportional fairness
• max-min fairness
• TCP-fairness (as described by the square-root formula)
Can we unify these fairness criteria within a single framework?
1
1r
rr
xw
subject to
Rrx
JjCxA
r
jrr
jr
0
maximize
Optimization formulation
Suppose the allocation x is chosen to
(weighted -fair allocations, Mo and Walrand 2000)
Rrp
wx
rjj
rr
/1
jp - shadow price for the resource j capacity constraint
Observe alignment with square-root formula when
rj
jrrr ppTw ,/1,2 2
Solution
- maximum flow - proportionally fair- TCP fair - max-min fair)1(
)/1(2
)1(1
)1(0
2
w
Tw
w
w
rr
Examples of -fair allocations
1
1r
rr
xw
subject to
Rrx
JjCxA
r
jrr
jr
0
maximize
Rrp
wx
rjj
rr
/1
Example
0
1 1
JjC
Rrwn
j
rr
1
,1,1
maximum flow:
1/3
2/3 2/3
1/2
1/2 1/2
max-min fairness:
proportional fairness:
0
1
Rate control algorithms
• Can rate control algorithms be interpreted within the optimization framework?
• Several types of algorithm possible, based on– congestion indication using linear increase
and multiplicative decrease rules– explicit rates inversely proportional to
network shadow prices
Notation
J
R
jr
wr
xr (t)
j (t)
- set of resources- set of routes - resource j is on route r- weight of route r- flow rate on route r at time t - shadow price, or rate of congestionindication, at resource j at time t
A primal algorithm
sjssjj
rjjrrrr
txpt
ttxwtxdt
d
:
)()(
)()()(
xr(t) - rate changes by linear increase, multiplicative decreasepj(.) - proportion of packets marked a function of flow through resource
Global stability
Theorem: the above dynamical system has a stable point to which all
trajectories converge. The stable point is proportionally fair with respect to
the weights {wr}, and solves the network problem, when
j
jj
Cx
Cxxp
0)(
TCP
• We’ve seen that TCP’s square root formula is interpretable in terms of an optimization problem
• Next we show that its dynamic behaviour is interpretable as a form of primal algorithm
TCP (Jacobson’s congestion avoidance algorithm)
Source maintains a window of sent, but not yet acknowledged, packets - size cwnd
• cwnd incremented by 1/cwnd on positive ack
• cwnd decremented by cwnd/2 on congestion
• change in rate x per unit time is about
xTcwnd
2
1
/
/2
)1(1
2
2
px
T
p
cwndT
Tpcwnd
pcwnd
Differential equations for TCP
)()(
)(11)(
2
)()()(1)(
:
2
2
txpt
tt
ttx
T
ttx
dt
d
sjssjj
rjjr
rr
r
rr
Equilibrium point
RrT
xr
r
rr
2/11
21
• check: the equilibrium of the dynamical system recovers the inverse square root formula for TCP• note again the round-trip time bias
Consequences of RTT bias
Question: where to place cache?
pTT '
12
long uncongested link, T’
short congested links, T
user
cache?cache?
pT 22
11
If then short RTT preferred under TCP, even though twice the resource usage
TT 122'
Multipath routing Suppose a source-destination pair has access to
several routes across the network:
resource
routesource
destinationsr
S
- set of source-destination pairs- route r serves s-d pair s
Combined multipath routing and congestion control: a robust Internet architecture. Key, Massoulié & Towsley 2005
Routing and optimization formulation
)(nxx
)log( ss
s xn
subject to
Rry
JjCynA
SsxyH
r
jrrr
jr
srr
sr
0
maximize
Suppose is chosen to
( H is an incidence matrix, showing which routes serve a source-destination pair )
Example of multipath routing
1C
2C
3C
3C
1n
2n 3n
Routes, as well as flow rates, are chosen to optimize
over source-destination pairs s)log( ss
s xn
213 ,CCC
Conclusions
Mathematical models of end-to-end congestion control show it to be a distributed algorithm that: solves an optimization problem; computes a fair allocation; and finds a market clearing equilibrium.
The models provide a framework for engineering efforts to improve reliability and robustness.