# Homework 6 (FCN) [Actual since I messed up the numbering]

# 1
No. Because network congestion often has short time-scale dynamics, the routing protocols may take longer to converge than the duration of the changes they try to react to. Since congestions and the convergence timing are different in way larger scales, network congestion can be ignored and routing protocols would not take them into account.

Also, route instability is possible as the routes adapt in concert.

# 2

At each step, each updating of a node’s distance vectors is based on the Bellman-Ford equation, i.e., only decreasing those values in its distance vector. There is no increasing in values. If no updating, then no message will be sent out. Thus, D(x) is non-increasing. Since those costs are finite, then eventually distance vectors will be stabilized in finite steps.

# 3

Textbook 7th edition, Chapter 5, Problem P7

## a)

$D_{x}(w) = 2$
$D_{x}(y) = D_{x}(w) + D_{w}(y) = 4$ # Shortest
$D_{x}(u) = D_{x}(w) + D_{w}(u) = 2+5 = 7$

## b)

First consider what happens if c(x,y) changes. If c(x,y) becomes larger or smaller (as long as c(x,y) >= 1) , the least cost path from x to u will still have cost at least 7. Thus a change in c(x,y) (if c(x,y)>=1) will not cause x to inform its neighbors of any changes. If c(x,y)= d<1, then the least cost path now passes through y and has cost $δ+6$.

Now consider if c(x,w) changes. If c(x,w) = $ε <= 1$, then the least-cost path to u continues to pass through w and its cost changes to $5 + ε$; $x$ will inform its neighbors of this new cost. If c(x,w) = $δ > 6$, then the least cost path now passes through y and has cost 11; again x will inform its neighbors of this new cost.

## c)
Any change in link cost c(x,y) (and as long as c(x,y) >=1) will not cause x to inform its neighbors of a new minimum-cost path to u .