$1.$ This elementary problem begins to explore propagation delay and transmission delay, two central concepts in data networking. Consider two hosts A and B$,$ connected by a single link of rate R bps$.$ suppose that the two hosts are separated by m meters, and supose the propagation speed along the link is s meters$/$sec$.$ Host A is to send a packet of size L bits to Host B$.$

Express the propagation delay d$($prop$)$ in terms of m and s$.$

Determine the transmission time of the packet dtrans in terms of L and R$.$

Ignoring processing and queuing delays obtain an expression for the end to end delay.

Suppose Host A begins to transmit packet at time t$=0.$ At time t$=$dtrans where is the last bit of the packet?

Suppose dprop is greater than dtrans. At time t$=$dtrans where is the first bit of the packet?

Suppose dprop is less than dtrans. At time t$=$dtrans where is the first bit of the packet?

Suppose s$=2\mathrm{.}5108,$ L$=100$ bits and R$=28$ kbps$.$ find the distance m so that dprop equals dtrans.

$2.$ Suppose users share a $3$ mbps link. Also suppose each user requires $150$kbps when transmitting, but each user only transmits $10$ percent of the time.

When circuit switching is used, how many users can be supported?

Suppose packet switching is used. Find the probability that a given user is transmitting?

Suppose there are $120$ users. Find the probability that at any given time, exactly n users are transmitting simultaneously

find the probability that there are $21$ or more users transmitting simultaneously.

$3.$ Visit the Queuing and Loss interactive animation at the companion Web site. What is the maximum emission rate and the minimum transmission rate? With those rates, what is the traffic intensity? Run the interactive animation with these rates and determine how long it takes for packet loss to occur. Then repeat the experiment a second time and determine again how long it takes for packet loss to occur. Are the values different? Why or why not?

$4.$ Perform a Traceroute between source and destination on the same continent at three different hours of the day.

Find the average and standard deviation of the round$-$trip delays at each of the three hours.

Find the number of routers in the path at each of the three hours. Did the paths change during any of the hours?

Try to identify the number of ISP networks that the Traceroute packets pass through from source to destination. Routers with similar names and$/$or similar IP addresses should be considered as part of the same ISP. In your experiments, do the largest delays occur at the peering interfaces between adjacent ISPs?

Repeat the above for a source and destination on different continents. Compare the intra$-$continent and inter$-$continent results.