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

CijFij

ijijijij

ij Fd 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 allp 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 allp

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-CX )(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)-(CC

0)-(CC

r)-(CC

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-(CC

*)X-(CC

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 MatrixDesign• 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 metricChoose 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 ( Lijij

ij

j)(i,ijij

at L = 0

0 )F-(C

F - p CijL

ijij

ijij

2

Subnet Design … contSolving for Cij gives

ij

ijijij

pF 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 T1

A

Subnet Design … contSubstituting in equation A

n)(m,

mnmnFp pij

Fij T1 Fij Cij

Given the capacities, the “optimal” cost is

) pijFij ( T1 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 … contC1

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 flowsStep 2: Check the delay and reliability constraints are metStep 3: Check improvement gradient descentStep 4: Perturb 1

End LoopFor Step 4- Lower capacity or remove under utilized links- Increase capacity of over utilized link- Branch Exchange Saturated Cut