Upload
mechelle-rich
View
36
Download
1
Tags:
Embed Size (px)
DESCRIPTION
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 - PowerPoint PPT Presentation
Citation preview
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