This document contains the solutions to a final exam for a communication networks course. It includes multiple problems involving calculating metrics like average transmissions, generating functions, and mean calculations for queueing models. It also contains a multicast routing problem that involves building a minimum cost tree and calculating its cost.

1. EE 333, Communication Networks
Solutions Final Exam (2014-15S)
Maximum Marks = 50 Time = 3 hours
1. (a)
* 2 * 2
1 2 1 2
* 1 2
1 2
1 1 1 1
1 (1 ) 0.5 (1 )
(2 ) 1 (1 0.5 )
N N
N
   
   
 
   
 
   
          
   
 
    
 
* 1 2
1 2
1 1
(2 ) 2
N
 
   
 
   
  
(b)
* 2 * 2
1 1 2
*
1 2
1 1 1
1 (1 ) (1 )
1
(2 ) 1 (1 )
N N
N
   
  

   
 
 
       
 
    
*
1 2
1 1 1 1
(2 ) (2 ) 2
N

     
 
    
   
(c) When both A and B have the packet and both transmit, the average total number of
transmissions required (from both A and B) would be –
1 2 1 2 1 2
2 2
1 (1 )(1 )     

    
Using this,
* 2 * 2
1 2 1 2 1 2
1 1 2
1 (1 ) (1 )N N   
     
 
       
  
* 2
1 2 1 2 1 2
1 1 2
(2 ) 1 (1 )N     
     
 
      
  
Therefore, * 1 2
1 2 1 2 1 2
1 1 2
(2 ) 2 2
N
  
         
      
        
          

2. 2. (a) The State Transition Diagram is given below
(b) Balance Equations
0 1 1 0
1 2 2 1 0 0 1
2
2 1 3 3 1 2 0
( ) (1 ) (1 ) ( )
( ) ( ) (2 )
p p p p
p p p p p p p
p p p p p p p
  
      
    
 
       
     
3 2 4 4 2 3
1 2 1 2
( ) ( )
................................................................................
( ) ( )
..................................................................
n n n n n n
p p p p p p
p p p p p p
  
     
   
   
..............
(c) Summing both sides of all the equations, we get
1 0 0
0 01 2 1 2
k k k
k k k
p p p
p p
 
 
  
  
 
   
  
(d) Generating Function
Multiplying LHS and RHS of the ith
equation by zi
and summing both sides, we get
2 1
1 0 0
i i i
i i i
i i i
p z p z p z 
  
 
  
   
Using
0
( ) i
i
i
P z p z


  and simplifying, we get
2 0
0 2
( ) ( ) ( ) ( )
1
p
P z p z P z zP z P z
z z
 
 
    
 
The Normalization Condition requires that P(1)=1
Therefore, 0 2
1 2
1 2 ( )
1
p P z
z z


 

  
 
(e) Mean
1
( )
z
dP z
N
dz 
 2 2
(1 2 )( 2 )
( )
(1 )
z
P z
z z
  
 
 
 
 
2
3 (1 2 ) 3
(1 2 ) 1 2
N
  
 

 
 
(f) Using Little’s Formula with 2eff  , we get
3
2 (1 2 )eff
N
W
  
 


3. 3(a)
N DB DC DD DE DF DG DH DI DJ
A 2
A-B
3
A-C
7
A-D
∞ 7
A-F
∞ ∞ ∞ ∞
A,B 2
A-B
3
A-C
7
A-D
4
A-B-E
7
A-F
∞ ∞ ∞ ∞
A,B,C 2
A-B
3
A-C
7
A-D
4
A-B-E
5
A-C-F
∞ ∞ ∞ ∞
A,B,C,E 2
A-B
3
A-C
6
A-B-E-D
4
A-B-E
5
A-C-F
5
A-B-E-G
∞ ∞ ∞
A,B,C,E,F 2
A-B
3
A-C
6
A-B-E-D
4
A-B-E
5
A-C-F
5
A-B-E-G
∞ ∞ ∞
A,B,C,E,F,G 2
A-B
3
A-C
6
A-B-E-D
4
A-B-E
5
A-C-F
5
A-B-E-G
7
A-B-E-G-H
9
A-B-E-G-I
∞
ABCEFGD 2
A-B
3
A-C
6
A-B-E-D
4
A-B-E
5
A-C-F
5
A-B-E-G
7
A-B-E-G-H
7
A-B-E-D-I
13
A-D-J
ABCEFGDH 2
A-B
3
A-C
6
A-B-E-D
4
A-B-E
5
A-C-F
5
A-B-E-G
7
A-B-E-G-H
7
A-B-E-D-I
8
ABEGHJ
ABCEFGDHI 2
A-B
3
A-C
6
A-B-E-D
4
A-B-E
5
A-C-F
5
A-B-E-G
7
A-B-E-G-H
7
A-B-E-D-I
8
ABEGHJ
ABCEFGDHIJ 2
A-B
3
A-C
6
A-B-E-D
4
A-B-E
5
A-C-F
5
A-B-E-G
7
A-B-E-G-H
7
A-B-E-D-I
8
ABEGHJ
Forwarding Table at Node A
Destination B C D E F G H I J
Next Node B C B B C B B B B
(b) Multicast Cost = Sum of Link Costs in above tree = 16
(c) Delete all the links of the first path in the network graph and apply Dijkstra’s Algorithm once again
4. Refer to the textbook and the lecture notes for the answers

4. 3(a) ALTERNATE SOLUTION
N DB DC DD DE DF DG DH DI DJ
A 2
A-B
3
A-C
7
A-D
∞ 7
A-F
∞ ∞ ∞ ∞
A,B 2
A-B
3
A-C
7
A-D
4
A-B-E
7
A-F
∞ ∞ ∞ ∞
A,B,C 2
A-B
3
A-C
7
A-D
4
A-B-E
5
A-C-F
∞ ∞ ∞ ∞
A,B,C,E 2
A-B
3
A-C
6
A-B-E-D
4
A-B-E
5
A-C-F
5
A-B-E-G
∞ ∞ ∞
A,B,C,E,G 2
A-B
3
A-C
6
A-B-E-D
4
A-B-E
5
A-C-F
5
A-B-E-G
7
A-B-E-G-H
9
A-B-E-G-I
∞
A,B,C,E,G,F 2
A-B
3
A-C
6
A-B-E-D
4
A-B-E
5
A-C-F
5
A-B-E-G
7
A-B-E-G-H
9
A-B-E-G-I
∞
ABCEGFD 2
A-B
3
A-C
6
A-B-E-D
4
A-B-E
5
A-C-F
5
A-B-E-G
7
A-B-E-G-H
7
A-B-E-D-I
13
A-D-J
ABCEGFDH 2
A-B
3
A-C
6
A-B-E-D
4
A-B-E
5
A-C-F
5
A-B-E-G
7
A-B-E-G-H
7
A-B-E-D-I
8
ABEGHJ
ABCEGFDHI 2
A-B
3
A-C
6
A-B-E-D
4
A-B-E
5
A-C-F
5
A-B-E-G
7
A-B-E-G-H
7
A-B-E-D-I
8
ABEGHJ
ABCEGFDHIJ 2
A-B
3
A-C
6
A-B-E-D
4
A-B-E
5
A-C-F
5
A-B-E-G
7
A-B-E-G-H
7
A-B-E-D-I
8
ABEGHJ
Slightly different sequence if the other “7” is chosen first
Forwarding Table at Node A
Destination B C D E F G H I J
Next Node B C B B C B B B B
(b) Multicast Cost = Sum of Link Costs in above tree = 16
(c) Delete all the links of the first path in the network graph and apply Dijkstra’s Algorithm once again
4. Refer to the textbook and the lecture notes for the answers