n computers are trying to connect to a main frame with n ser

n computers are trying to connect to a main frame with n servers. Each of the n servers is randomly picking 3 computers, independently of the other servers, and connects to them.

(a) What is the distribution of number of servers connected to the the first computer?

(b) What is the expected number of computers with which there is no conection?

Solution

There are n servers and n computers.

As each server can connect to 3 computers

we have options like this.

I. Each server is connected to one computer. In this case, all servers will be busy

2. 1 servers handle more than 1 computer say 2 computers.Then any one server will be idle.

3. 1 Server handles 3 computers at a time. Then 2 servers remain idle.

4. 2 servers handle more than 1 computer say 2 computers. Then 2 servers ar idle

5. 2Server handles 3 computers at a time. Then 4 servers remain idle.

Thus if x is the no of servers connected to first computer then x can take values as 1 , 2 or 3

b) As no of servers are equal to no of computers

Expected no of computers with no connection =0