Consider sending a file of 2Mbits from Host A to Host B. There are three links and two store-and-forward routers (or switches) between A and B. Assume that no link is congested (that is, no queueing delays), and the one-way propagation delay of each link is 10ms. Host A divides this file into packets of S bits each and adds 100 bits of header to each packet, forming packets of L = 100 + S bits overall. Each link has a transmission rate of R =1Mbps. Find the value of S (in bits) that minimizes the delay of moving this file from Host A to Host B (end-to-end delay).
To find the value of S (in bits) that minimizes the end-to-end delay of moving the file from Host A to Host B, we need to consider both the transmission delay and the propagation delay.
Let's break down the different components of the end-to-end delay:
1. Transmission Delay (T):
The transmission delay is the time it takes to transmit a packet over a link. It can be calculated using the formula:
T = L / R
where L is the packet size (including the header) and R is the transmission rate of the link.
2. Propagation Delay (P):
The propagation delay is the time it takes for a packet to travel from one end of a link to the other. Since there are three links between Host A and Host B, the total propagation delay is 3 * 10ms = 30ms.
3. Queueing Delay (Q):
In this scenario, we assume there are no queueing delays, so Q = 0.
Now, let's analyze the end-to-end delay (D) for moving the file from Host A to Host B:
D = N * (T + P)
where N is the number of packets required to transmit the file.
Given that the file size is 2 Mbits and each packet is S bits, we have:
N = 2 Mbits / S
Note: 1 Mbit = 1,000,000 bits.
Substituting the values of T, P, and N, we get:
D = (2 Mbits / S) * (L / R + P)
To minimize the end-to-end delay, we need to find the value of S that minimizes D. We can achieve this by taking the derivative of D with respect to S and setting it to zero.
Differentiating D with respect to S:
dD/dS = - (2 Mbits / S^2) * (L / R + P) + (2 Mbits / S) * (L / R^2)
Setting dD/dS = 0:
-(2 Mbits / S^2) * (L / R + P) + (2 Mbits / S) * (L / R^2) = 0
Simplifying and solving for S:
-(L / R + P) + (L / R^2) = 0
Solve for S.
Once you solve the equation, you will find the value of S (in bits) that minimizes the end-to-end delay of moving the file from Host A to Host B.