$(4)(4$pts$)$ How to enable two machines located behind symmetric NATs to communicate with

each other is a difficult problem. One way to tackle this problem is to exploit a behavior of

NATs: they often assign ports in a predicable way by just incrementing the port number when a

new entry is added to their translation tables.

We first deploy a server $S,$ which has a public IP address, to help computers locating behind

symmetric NATs communicate with each other. Any computer which wants to use the service

will periodically open up a TCP connection with $S$ and send in a packet to $S.$ This allows $S$ to

know the current IP addresses and port numbers they are using. Assume their current translated

IP addresses and ports recorded in S are $($IP$\_$a $,$ Port$\_$a$)$ and $($IP$\_$b$,$ Port$\_$b$).$

Two computers A and B are located behind a symmetric NAT and A likes to set up a TCP

connection with computer B$.$ It can use the following steps to do it$.$

$($i$)$ A sends a request to $S$ with both A and $B\text{'}$s name identities $($note: Since A and B do not have

permanent identities at the network layer, we assume computers A and B have other identities at

the application layer, such as DNS names, they can use to identify themselves when they send

requests to $S.$

$($ii$)$ S sends a packet to A and B simultaneously to inform each other's current translated IP

address and port number: $($IP$\_$a $,$ Port$\_$a$)$ and $($IP$\_$b$,$ Port$\_$b$)..$

$($iii$)$ Once A and B receive the message from S$,$ they immediately send out a packet to each other.

Based on the port incrementing behavior of their NATs, Machine A should send its packet to

$($IP$\_$b$,$ Port$\_$b $+1)$ and machine B should send its packet to $($IP$\_$a$,$ Port$\_$a $+1).$

Step $($iii$)$ will open up an entry in A$\text{'}$s and B$\text{'}$s NAT translation tables. The hope is that when the

packet from the other party arrives, the corresponding entry in their NATs' translated tables have

been opened up$.$ If so$,$ the arriving packet will be accepted by the NATs.

But if A$\text{'}$s packet or B$\text{'}$s packet arrive before the corresponding entry in the destination NAT's

entry is opened up$,$ the packet will be dropped and the connection fails.

$($a$)$ Suppose we have the following delay information:

Delay between A and $B=25ms($fixed$)$

Delay between $S$ and $A=40ms($fixed$)$

Delay between $S$ and $B=20ms($fixed$)$

Can the connection between A and $B$ be set up successfully?

$($b$)$ Assume the delay between A and B is a random variable and uniformly distributed within the

range $10ms,25ms.$ What is the probability the above TCP setup is successful?