1. L11. Link-path formulation
D. Moltchanov, TUT, Spring 2015
D. Moltchanov, TUT, Spring 2008

2. Outline Reminder of link-path formulation Node-identifier notation
Link-demand-path-identifier notation
Network dimensioning problems
Shortest-path routing problems
Fair networks
Topological design
Restoration design

3. Problem formulation Simplest problem: given
Network topology with links and rates
Traffic demands between nodes
Get the best routing minimizing something (e.g. cost)
There are two common problem formulations
Link-path formulation
Node-link formulation
Within one formulation there are different notations
Link-path: node-identifier-based
Link-path: link-demand-path-identifier-based

4. Link-path: node-identifier
Demands: bidirectional
1-2: 5
1-3: 7
2-3: 8
Routing of demands
Two path for each demand
Demand paths 1-2 and 1-3-2
We see that flows over paths
For other demands
Another concern: how much to put over those paths

5. Link-path: node-identifier
Are there limitation on how much we put over paths?
Yes! Link bandwidth! (we assume bidirectional links)
We will denote links by 1-2, 2-3, 1-3 and capacities
First important note:
Demand: between any two nodes
Link: connects two nodes directly
Second important note:
Same units must be used for demands and link rates
You need to convert: pps/pps, Mbps/Mbps
E.g. to get pps: link rate in Mbps / average packet size
Link capacity unit (LCU)
Demand volume unit (DVU)

6. Link-path: node-identifier
So which demands are using links, what are implications
For link 1-2
Similarly for links 1-3 and 2-3 we have
What we got? (demands constraints + capacity constraints)
and of course non-negativity of allocations:

7. Link-path: node-identifier
So what we got so far?
A system of equalities/inequalities
Gives feasible solutions to allocations
Possibly no solutions exist, possibly infinitely many
Which of these solutions are of interest
Depends on our objective!
Question: what is the goal of your network design?
Minimize the routing cost? Minimize congestion? Something else?
Objective function! (AKA Utility function)
Example: minimizing the total routing cost
Let the cost of transmission of a unit flow over any link be 1
Can you tell me why coefficient 2?

8. Link-path: node-identifier
So the whole problem now looks as
Minimize routing cost (routing cost over any link is 1)
subject to demand constraints
and capacity constraints
And positivity constraints

9. What’s wrong with node-identifier?
Why? Recall link-path notation
Demands volumes: , where i,j are nodes… we are OK
Links capacities : , where i,j are nodes… we are OK
Paths for demand (1->2): we used nodes as indices, e.g.
Paths for demands
It was OK for three nodes
What’s about paths for networks having N nodes?
What we used is called node-identifier-based notation
Additional shortcomings
Some nodes may not have demands: we still need to
Not all nodes are directly connected
Flow variables have indices of different length
More than one link between two nodes is also a problem
Simply put: we have a problem modeling large networks

10. Link-path: link-demand-path
Link-demand-path-identifier-based notation 
Compact
Allows to list only necessary objects
Good for moderate-to-large networks
Seem strange for small networks at the first glance though…
1. Start with demands
Enumerate from 1 to D
Only those that are non-zero
Three nodes example
Demand (1->2): demand ID 1
Demand (1->3): demand ID 2
Demand (2->3): demand ID 3
We have d=1,2,3 demands
In general: d=1,2,..,D demands

11. Link-path: link-demand-path
2. Continue with links
Enumerate from 1 to E
Only those that exist
Three nodes example
Link 1->2: link ID 1
Link 1->3: link ID 2
Link 2->3: link ID 3
We have e=1,2,3 links
In general e=1,2,…,E links
Can perform mapping of
Demand volumes
Link capacities

12. Link-path: link-demand-path
3. Continue with candidate paths for demand
There could be more than one
Enumerate from 1 to Pd for demand d
Note! paths have to be found prior to the solution of the task
Example: demand pair (1->2) ID 1
There exist two paths, P1= 2
These are 1-2, 1-3-2
Path 1-2: path ID 1
Path 1-3-2: path ID 2

13. Link-path: link-demand-path
Finish with path-flow variables
Demand paid ID: first index
Path ID for demand: second index
Node-identifier Vs. Link-demand-path
Note the following:
Previously: we saw which path is taken from indices
Now: it is implicitly given

14. Link-path: link-demand-path
The allocation task now reads as
Minimize routing cost (routing over any link costs 1)
subject to demands constraints
and capacity constraints
and positivity constraints
Paths must be given explicitly prior to formulation

15. Link-demand-path General procedure
Demands d = 1,2,..,D, and their volumes hd
Links e = 1,2,…,E, and their rates ce
Paths for each demand p = 1,2,…,Pd
Flow variables xdp, d = 1,2,…,D, p=1,2,…, Pd
Useful for moderate-to-large networks
Solution: optimization algorithms

16. Objective function One more look at objective function
Sum of links costs
Link cost: link load link cost unit
Link load: sum of all flows crossing it
One alternative: minimize delay on the most congested link
Changing objective function
May affect the optimal solution
May dramatically affect how we find the optimal solution
Sometimes the most complex thing
We considered link-path formulation
It also valid for directed links/demands

17. Optimization software
You must get familiar with at least one package
Matlab (commercial)
Mathematica (commercial)
Maple (commercial)
MathCad (commercial)
CPLEX (reduced version is free)
GNU linear programming kit (free)
AMPL (reduced version is free)
Choose one and get yourself familiar!