Finite Queue Analysis. The buffer of a streaming media application (e.g., mp3 player) can hold up to N packets. The processing time for each packet to be played out was found to be exponential with a mean of 25 msec. Degradation in the quality of the playout occurs as a result of buffer over-runs or as a result of a buffer under-runs. Buffer over-runs occur when a packet is dropped due to a filled buffer (causing a "blip" in the playout). Buffer under-runs occur when there are no packets to playout (causing a period of "plop" in the playout). It was measured that the packets arrive to the application as a Poisson process with a mean of 36 packets per second. Answer the following questions:

(1) Assuming an infinite buffer, how many packets do you expect to find in the buffer (i.e., being played out or waiting to be played out)? What is the mean delay between receipt of a packet and the beginning of its playout?

(2) Assuming an infinite buffer and assuming that a streaming media object consists of 10,000 packets, how many "blips" and how many "plops" do you expect to hear for that object?

(3) Assuming N=4 packets, how many packets do you expect to find in the buffer? What is the mean delay between receipt of a packet and the beginning of its playout?

(4) Assuming N=4 packets and that a streaming media object consists of 10,000 packets, how many "blips" and how many "plops" do you expect to hear for that object?

(5) Based on your answers above, explain the following sentence: Figuring out the right size for the buffer of a streaming media application is a tradeoff between tolerance for delay and tolerance for degraded audio quality. In particular, give an example of a streaming media application for which an infinite (or very large) buffer would be preferable to a finite (or small) buffer, and give an example of the opposite.