GATE Overflow Test Series | Mixed Subjects | Test 4 | Question: 13

Question

In a packet switched network, packets are routed from source to destination along a single path having $3$ intermediate nodes. If the message size is $24$ bytes and each packet contains a header of $8$ bytes, then the optimum packet size in bytes is _________

Answer 1

We can consider the total transmission time as propagation delay remains same irrespective of the packet size.
    
If each packet (excluding header)  is $x$ we will have $n$ packets where $n = \left \lceil\dfrac{24}{x}\right \rceil$
    
Total transmission time $ = 4 \times  \dfrac{\text{Packet Size}}{BW}\;(1$ transmission from source followed by $3$ transmissions from $3$ intermediate nodes $) + (n-1) \times$ transmission time.
    
$\qquad =\dfrac{1}{BW} (4 \times  \text{Packet Size} + (n-1) \times \text{Packet Size} $
    
$\qquad =\dfrac{1}{BW} (4 \times  (x+8) + (n-1) \times (x+8) $
    
$\qquad =\dfrac{1}{BW} (4 \times  (x+8) + \left(\left \lceil \dfrac{24}{x}\right \rceil-1\right) \times (x+8) $

$\qquad =\dfrac{1}{BW} (4 \times  x + 32 + 24 -x +192/x -8 )  $

$\qquad =\dfrac{1}{BW} (3 x + 48  +192/x  )  $

For optimal packet size, differentiating w.r.t. $x,$ time should be $0.$

$\implies 3 - 192/x^2 = 0 \implies x^2 = 64.$

$\implies x = 8.$

So, optimal packet size including header $ = 8 + 8 = 16 $ bytes.