1. L12. Network optimization
D. Moltchanov, TUT, Fall 2017
D. Moltchanov, TUT, Spring 2008

2. Why do we need to know more?

3. Analysis vs. dimensioning
Forward task
Analyze system
What is the delay, packet loss, throughput, given
Link rate
Buffer space
Routing matrix
Specific protocols
Inverse task
Optimize system
What link rates, buffer space, routing weights, given
Arriving traffic
Packet loss rate
Delay

4. Example: buffer dimensioning
We have output-queuing router
Question: how much buffer is needed at port j
We know internal routing, pij, i=1,2,..,N, j=1,2,…,M
We know arrival traffic on all the links, λi, i=1,2,…,N
We know outgoing rates, Cj, j=1,2,…,M

5. Example: buffer dimensioning
We may consider port j in isolation
M/M/1 is a good model
Performance metric: mean delay
Mean delay for M/M/1 is given by (recalling that ρ = λ/µ)
where µ is the average service rate
where E[P] is the average packet size

6. Example: buffer dimensioning
Performance response
Could analyze M/M/1/K estimating loss
But… how do we know arriving traffic, link rates, routing?

7. Example: buffer dimensioning
Let’s revisit our assumptions…
We know outgoing rates, Cj, j=1,2,…,M???
We only know it when the network is already planned and installed
Should there be a link between two routers?
If yes, what should be its rate?
We know internal routing, pij, i=1,2,..,N, j=1,2,…,M???
Only if someone already assigned weights to OSPF (or IS-IS)
Internal routing = generic OSPF view of the network
We know arrival traffic on all the links , λi, i=1,2,…,N???
If traffic demands are known…
Who is going to do this for us?

8. Generic solution Formulate constraints Capacity constraints
Demand constraints
Specific constraints of protocols/algorithms
Write down objective function
Maximize network throughput
Minimize network cost
Minimize network delay
Solve using optimization techniques
Classify
Solve using well-known methods

9. Optimization problems
Depends on the nature of involved variables
Linear programming
Simplex algorithm
Interior point algorithm
Convex programming
Linearization
Method of Lagrange multipliers
Method of gradients
Integer programming
Continuitization
Branch-and-cut/branch-and-bound algorithms
Heuristic methods, e.g., genetic algorithms

10. Problems Dimensioning problems: minimize network cost given
Set of demands
Each can be routed over different paths
Shortest-path routing: two problems
Given link weights find shortest paths
Given a network with links find the optimal weights
Fair network problem: given greedy traffic
Allocate available resources among demands
Topological design: minimize network cost given
Cost of links (no links are given)
Restoration design: optimize network given
Possible failures
Multi-layer networks

11. Very simple example

12. Toy example Link-demand-path-identifier-based formalization 
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

13. Toy example 2. Continue with links Three nodes example
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

14. Toy example 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

15. Toy example Finish with path-flow variables Demand ID: first index
Path ID for demand: second index

16. Toy example The allocation task now reads as
Minimize routing cost (routing over each link costs 1 unit)
subject to demands constraints
and capacity constraints
and positivity constraints