Flow Models and Optimal Routing

Flow Models and Optimal Routing

• How can we evaluate the performance of a routing algorithm– quantify how well they do– use arrival rates at nodes and flow on links

• View each link as a queue with some given arrival statistics, try to optimize mean and variance of packet delay – hard to develop analytically

… cont

• Measure average traffic on link Fij

– Measure can be direct (bps) or indirect (#circuits)

– Statistics of entering traffic do not change (much) over time

– Statistics of arrival process on a link– Change only due to routing updates

Some Basics• What should be “optimized”

Dij = link measure =

Cij is link capacity and dij is proc./prop delay

max (link measure)

link measure

These can be viewed as measures of congestion

links all

measurelink

Cij

Fij

ijijijij

ijFd

F - C

F

… cont

• Consider a particular O – D pair in the network W. Input arrival is stationary with rate

• W is set of all OD pairs

• Pw is set of all paths p connection an OD pair

• Xp is the flow on path

W

• The Path flow collection

{ Xp | w W, p PW } must satisfy

The flow Fij on a link is

minimize

0 X W w,Pp ; r X W,w pWW

Pp

p

W

j)(i, containing

p paths all

pX

j)(i,

ijij )(F D

] X [ Dijj)(i, containing

paths all

p subject to

• This cost function optimizes link traffic without regard to other statistics such as variance.

• Also ignores correlations of interarrival and transmission times

• ODs are (1,4), (2,4), (3,4)

• A rate base algorithm would split the traffic 1 2 4 and 1 3 4

• What happens if source at 2 and 3 are non-poisson

4

3

2

1

Link capacity is 2 for all links

Recall that D(x) =Now,

Where the derivative is evaluated at total flows corresponding to X

If D’ij |x is treated as the “length” of link, then

is the length of path p aka first derivative length of p

aka first derivative of length p

] X [ Dijj)(i, containing

paths all

p

p onj)(i, all

ijp

D'X

D(x)

pX

D(x)

• Let X* = {Xp*} be the optimal path flow vector

• We shouldn’t be able to move traffic from p to p’ and still improve the cost !

Xp* > 0

• Optimal path flow is positive only on paths with minimum First Derivative Length

• This condition is necessary. It is also sufficient in certain cases e.g. 2nd derivative of Dij exists and is positive over [0,Cij]

pp' X

D(x*)

X

D(x*)

ii

iii

X-C

X )(XD , r < C1+ C2

minimize D(X) = D1(X1) + D2(X2)

at optimum X1* + X2* = r , X1*, X2* 0

r

1

>

2

>

X2

X1

C2 low capacity

C1 high capacity

X1* = r, X2* = 0

X1* > 0, X2* > 0

The 2 path lengths must be the same

r)-(C

C

0)-(C

C

r)-(C

C

1

1

2

2

1

1

211 CC - C r

2 2 2

2

2

1

1

dX(0)dD

dX

(r)dD

2

22

1

11

dX*)(XdD

dX

*)(XdD

*)X-(C

C

*)X-(C

C

22

2

11

1

21

21211

CC

)]CC - (C - r [ C *X

21

21122

CC

)]CC -(C - r [ C *X

X1* + X2* = r

X1*

X2*

0 r

X1* X2

*

C1+C2211 CC-C

Topology DesignGiven

• Location of “terminals” that need to communicate

• OD Traffic Matrix

Design

• Topology of a Communication Subnet location of nodes, their interconnects / capacity

• The local access network

Topology Design … cont

Constrained by

• Bound on delay per packet or message

• Reliability in face of node / link failure

• Minimization of capital / operating cost

Subnet Design• Given Location of nodes and traffic flow

select capacity of link to meet delay and reliability guarantee

– zero capacity no link

– ignore reliability

– assume liner cost metric

Choose Cij to minimize

j)(i,

ijijCp

Subnet Design … cont

• Assuming M/M/1 model and Kleinrock independence approximation, we can express average delay constraint as

T F - C

F

1j)(i, ijij

ij

is total arrival rate into the network

Subnet Design … cont

• If flows are known, introduce a Lagrange multiplier to get

) F - C

F Cp ( L

ijij

ij

j)(i,ijij

at L = 0

0 )F-(C

F - p

Cij

L

ijij

ijij

2

Subnet Design … contSolving for Cij gives

ij

ijijij

p

F F C

Substituting in constraint equation, we obtain

j)(i,j)(i, ijij

ij pijFij

F - C

F

1 T

Solving for

j)(i,

ijijFp

T

1

A

Subnet Design … contSubstituting in equation A

n)(m,

mnmnFp

pij

Fij

T

1 Fij Cij

Given the capacities, the “optimal” cost is

) pijFij ( T

1 pijFij pijCij

j)(i,j)(i,j)(i,

- So far, we assume Fijs (routes) are known

- One could now solve for Fij by minimizing the cost above w.r.t. Fij (since Cijs are eliminated)

- However this leads to too many local minima with low connectivity that violates reliability

Subnet Design … cont

C1

C2

Cn

.

.

.

.

.

.

.

r

Minimize C1 + C2 + … + Cn while meeting delay constraint

This is a hard problem !!

Some Heuristics

• We know the nodes and OD traffic

• We know our routing approach (minimize cost?)

• We know a delay constraint, a reliability constraint and a cost metric

• Use a “Greedy” approachLoop

Step 1: Start with a topology and assign flows

Step 2: Check the delay and reliability constraints are met

Step 3: Check improvement gradient descent

Step 4: Perturb 1

End Loop

For Step 4

- Lower capacity or remove under utilized links

- Increase capacity of over utilized link

- Branch Exchange Saturated Cut