1. Chapter 4: Network Layer
4. 1 Introduction
4.2 Virtual circuit and datagram networks
4.3 IP: Internet Protocol
Datagram format
IPv4 addressing
4.4 Routing: Concepts and Algorithms
Introduction
Math Detour – Graph Theory
Routing via Broadcast – PI, PIF
Connectivity Test – CT
Distributed Routing
Bellman – Ford – Distance Vector
Link State
Optimal Routing
Network Model
Math Detour – Convexity
Flow Deviation
Math Detour – Inequality Constraints
Optimal Routing – Necessary and Sufficient Conditions
4.5 Routing in the Internet
Hierarchical Routing
RIP
OSPF
BGP
4.6 Broadcast and multicast routing
Network Layer

2. Network Model Notations: N nodes M links
Ci = capacity of link i ( bits/sec )
Poisson external message arrivals to each node
= msg rate of arrivals at node j with destination k (traffic requirements)
independent exponentially distributed message length, mean (bits/msg)
negligible message processing time at nodes
total external traffic incoming to node j (msg/sec)
total traffic in the network (msg/sec)
= average delay per message in link i
= average message delay in the network

3. Delay calculations Assumption: non-splitting fixed routing Notations:
route from node j to node k
link-route indicator:
Total traffic on link i
namely traffic from j to k is included if the path includes link i
Zjk = Average Delay on path from j to k
namely delay on link i is included only if it is part of the path from j to k

4. Average Network Delay in a general network
example
Path from A to L
by links
by nodes
Examples:
Therefore:
Average Network Delay in a general network

5. Calculation of average delay Ti for a single (separate) link
Messages arrive to the link have exponential length with parameter , namely
therefore service time is random. If the bit rate (capacity) of the link is Ci bits/sec, then transmission time is:
therefore transmission time is exponentially distributed with parameter and
Calculation of network average delay T
Question: is the quantity Ti calculated above the same as Ti used earlier?
Answer: No, since normally the service times at consecutive nodes are not independent and not independent of the arrival process. This is because message length remains the same.
Kleinrock’s assumption: service times at different nodes are independent, good approximation because there are many sources and many destinations.
This allows us to calculate average network delay T
where is the flow in bits/sec on link i.

6. Design Problems
Capacity assignment: Given set of nodes and links, flows fi or , budget D. Find link capacities to minimize network average delay T.
Routing ( flow assignment ): Given network and link capacities. Find flows to minimize average delay T.
Capacity and Flow assignment: Given budget D. Find capacities Ci and flows to minimize average delay T.

7. Chapter 4: Network Layer
4. 1 Introduction
4.2 Virtual circuit and datagram networks
4.3 IP: Internet Protocol
Datagram format
IPv4 addressing
4.4 Routing: Concepts and Algorithms
Introduction
Math Detour – Graph Theory
Routing via Broadcast – PI, PIF
Connectivity Test – CT
Distributed Routing
Bellman – Ford – Distance Vector
Link State
Optimal Routing
Network Model
Math Detour – Convexity
Flow Deviation
Math Detour – Inequality Constraints
Optimal Routing – Necessary and Sufficient Conditions
4.5 Routing in the Internet
Hierarchical Routing
RIP
OSPF
BGP
4.6 Broadcast and multicast routing
Network Layer

8. Convexity Given an N-dimensional space RN
A vector in the space can be represented by its coordinates
is a convex region if
For example, the half space of vectors such that for some holds
is a convex region, since if then .
Similarly the hyperplane of vectors such that for some holds is a convex region.
Theorem: The intersection of two convex regions is convex
Theorem: A linear transformation of a convex region is convex

9. Projections
Normally, more than one point in the original region is mapped into any given point in the projection

10. Convex functions
Given a convex region , a function on is said to be convex if for any
and any , holds
Example: on
Theorem: A twice differentiable function on a convex subset of R1 is convex iff for all x in the subset
Example: The function is convex on
Theorem: A local minimum of a convex function on a convex region is a global minimum.

11. Flow Vectors
Consider the collection of vectors where j is a destination node , is a link and is the flow on link with destination j
This collection of vectors determines an RNM vector space
is the total flow on link
As before is the flow requirement at node k with destination j
set of outgoing links from k
set of incoming links at k
Subspace of flow vectors is defined as the collection of RNM -vectors that satisfy:
Kirkhoff Law
positive
bounded
Subspace is defined as the collection of RNM - vectors that satisfy only the Kirkhoff and positivity conditions
Theorem: The subspace of flow vectors is a convex region. Subspace is also a convex region.

12. Proof of convexity of
Given two flow vectors, need to show that is a flow vector. Explicit proof below. In fact not needed, since are intersections of hyperplanes and half spaces.
Kirkhoff
Positive
Bounded

13. Chapter 4: Network Layer
4. 1 Introduction
4.2 Virtual circuit and datagram networks
4.3 IP: Internet Protocol
Datagram format
IPv4 addressing
4.4 Routing: Concepts and Algorithms
Introduction
Math Detour – Graph Theory
Routing via Broadcast – PI, PIF
Connectivity Test – CT
Distributed Routing
Bellman – Ford – Distance Vector
Link State
Optimal Routing
Queuing & Network Model
Math Detour – Convexity
Flow Deviation
Math Detour – Inequality Constraints
Optimal Routing – Necessary and Sufficient Conditions
4.5 Routing in the Internet
Hierarchical Routing
RIP
OSPF
BGP
4.6 Broadcast and multicast routing
Network Layer

14. Routing as an optimization problem
The optimal routing problem is the following:
where
Fa is the projection of to the RM space of the link flow vectors
Fb is the set of link flows satisfying the “bounded” property
are the link capacities
are the flow requirements
Note: are called flow vectors, while are called link flow vectors
Properties:
is a convex function over the convex region (Fa is convex since it is a linear transformation of a convex region), therefore it has a unique minimum in this region
therefore if we start in Fb and move continuously, we will stay in Fb , thus it is sufficient to minimize over Fa

15. Chapter 4: Network Layer
4. 1 Introduction
4.2 Virtual circuit and datagram networks
4.3 IP: Internet Protocol
Datagram format
IPv4 addressing
4.4 Routing: Concepts and Algorithms
Introduction
Math Detour – Graph Theory
Routing via Broadcast – PI, PIF
Connectivity Test – CT
Distributed Routing
Bellman – Ford – Distance Vector
Link State
Optimal Routing
Network Model
Math Detour – Convexity
Flow Deviation
Math Detour – Inequality Constraints
Optimal Routing – Necessary and Sufficient Conditions
4.5 Routing in the Internet
Hierarchical Routing
RIP
OSPF
BGP
4.6 Broadcast and multicast routing
Network Layer

16. Flow Deviation Method
Iterative method to find optimal routing: at the (n+1)-st iteration we calculate a flow vector that is better than the one we have at the n-th iteration
where we want and are looking for an appropriate
In an unconstrained problem, we would find the gradient of and go in the opposite direction a small amount , then repeat.
Can’t do this in a constrained problem because the resulting vector may not satisfy the constraint.
Solution: let us take where
is some flow vector ( unspecified yet )
our solution at step n
some number
Since , are flow vectors, so is
Problem: selection of and

17. Flow Deviation Method (continued)
Denote by the gradient of T at For small values of holds
Denote the coordinates of the gradient vector. Then the above
equation can be re-written:
Since we are looking for a minimum, we would like to minimize the left-hand side. Thus the right-hand side should be minimized. Therefore we are looking for a flow vector such that is minimized. Solution to this sub-problem:
Assign weights to links in the network
For every source-destination pair , find shortest path according to these weights
Send the entire requirement on shortest path from k to j (disregarding capacities)
Resulting flow vector is
Reason: shifting any amount of flow from shortest path to another path would increase

18. Flow Deviation Method (continued)
In the previous slide, we have decided the direction we should take from to reach The next question is how much should we move.
This is a one parameter optimization problem: minimize and can be solved by simply solving the one unknown equation
Note: if optimal then thus is optimal.
If is not optimal, then as increases from 0, T decreases, thus should be increased until minimum is reached or until

19. Flow Deviation Algorithm
At step n , a flow vector is given.
Calculate
By finding shortest paths with link weights , find a flow vector for which
is minimized.
Substitute
Find for which is minimized
Repeat the above, until
Good convergence properties.

20. Example: Flow Deviation
Start with
holds
then
since with these weights the shortest path from a to b is via link 1, holds
therefore:
and
the minimum is achieved at and the minimum delay is

21. Example: Flow Deviation

22. Flow Deviation ( continued)
Note that in every step, the actual routing of the individual requests that result in the required link flows is not necessarily unique
(same as what was said before: Normally, more
than one point in the original region is mapped
into any given point in the projection )
Example: Suppose in the given network, at a given step we come up with the flows: Then is a solution, but so is

23. Chapter 4: Network Layer
4. 1 Introduction
4.2 Virtual circuit and datagram networks
4.3 IP: Internet Protocol
Datagram format
IPv4 addressing
4.4 Routing: Concepts and Algorithms
Introduction
Math Detour – Graph Theory
Routing via Broadcast – PI, PIF
Connectivity Test – CT
Distributed Routing
Bellman – Ford – Distance Vector
Link State
Optimal Routing
Network Model
Math Detour – Convexity
Flow Deviation
Math Detour – Inequality Constraints
Optimal Routing – Necessary and Sufficient Conditions
4.5 Routing in the Internet
Hierarchical Routing
RIP
OSPF
BGP
4.6 Broadcast and multicast routing
Network Layer

24. Optimal Routing necessary and sufficient conditions
Want to find a set of necessary and sufficient conditions to characterize optimal routing in a network. The conditions are a good tool to verify optimality of a given routing scheme and to develop algorithms to achieve it.
Optimization with equality constraint:
Build Lagrangian by multiplying each constraint by its Lagrange factor
Differentiate with respect to each parameter and equate to 0
result: necessary conditions
if the function and region are convex, this is also a sufficient condition

25. Mathematical Detour - Minimization with inequality constraints
Two cases: minimum inside the region or at the edge of the region
Example:
Lagrangian:
Case 1:
makes sense, since if then a small positive x would have given a higher
Case 2:

26. Minimization with inequality constraints - examples
Case 1: for , we get which satisfies the conditions and thus is minimum. The other possibility , does not satisfy the constraint. Case 2: for , we get which does not satisfy the constraint. Thus and , which do satisfy the conditions and provide the minimum point. Note: In their general form, the conditions on the previous slide are termed the Kuhn-Tucker conditions

27. Minimization with inequality constraint - another example
Suppose a given amount of money is to be invested and there are several possible investment programs. Program gives an income of if shekels are invested in it. The functions are concave. We want to
Necessary and sufficient condition for optimality

28. Minimization with inequality constraint - another example (cont’d)
Obvious, since if not, it is worthwhile to take money out of programs
with low incremental return and reinvest it into programs with high incremental return

29. Chapter 4: Network Layer
4. 1 Introduction
4.2 Virtual circuit and datagram networks
4.3 IP: Internet Protocol
Datagram format
IPv4 addressing
4.4 Routing: Concepts and Algorithms
Introduction
Math Detour – Graph Theory
Routing via Broadcast – PI, PIF
Connectivity Test – CT
Distributed Routing
Bellman – Ford – Distance Vector
Link State
Optimal Routing
Network Model
Math Detour – Convexity
Flow Deviation
Math Detour – Inequality Constraints
Optimal Routing – Necessary and Sufficient Conditions
4.5 Routing in the Internet
Hierarchical Routing
RIP
OSPF
BGP
4.6 Broadcast and multicast routing
Network Layer

30. Optimal Routing Problem
The problem is as before (slightly different notations: are nodes, are the
neighboring nodes of toward/from which there is an outgoing/incoming link):
Subject to
where is a non-negative continuous function, convex, going to when
and whose derivative is strictly positive.
Solution: as before disregard the capacity constraint
where Differentiating w.r.t gives

31. Optimal Routing
Theorem: Set of flows is optimal if and only if there exists a set of Lagrange multipliers such that
Interpretation of equations: The Lagrange multiplier is the sensitivity coefficient of the
delay w.r.t. the requirement: if the requirement is increased by , then the minimum delay is increased by
For a given destination , look at the neighbors of and calculate the sum of their incremental delay coefficient and the incremental delay coefficient on link
Optimality requires that for all neighbors to which sends traffic destined for , this sum will be the same and no larger than the sum for links on which sends no traffic.

32. Optimal Routing
The above interpretation also gives us an idea for distributed routing optimization: if the routing is not optimal, namely the incremental delay sums are not equal for links with traffic and no larger for links without, then one incrementally improve the average delay by transferring traffic from links with high incremental delay sum to links with small incremental delay sum. This can be done if nodes learn the coefficient parameters from their neighbors and can estimate the incremental delay coefficient on their adjacent links.
It can be shown that this distributed algorithm converges to the optimal routing.