Question

3. A single channel communication link is used for transmitting data files from one computer toanother in a low-rate data network. The file length can be assumed to be exponentially-distributed with a mean file length of 700 kbytes and files arrive for transmission in a Poissonstream with a mean rate of 1 file/100 seconds. The link is buffered by a FIFO buffer ofsufficient capacity to hold all files awaiting transmission. If the channel transmission rate is64 Kbits/s:a) Determine the probability that a file will not have to wait for transmission.b) Determine the probability that the file will have to wait for more than 10 minutes beforebeing transmitted.c) Would the overall mean waiting time be improved by giving "short" files (e.g. files lessthan 700 kbytes in length) priority over "long" files? A brief discussion will be sufficient.​

Answer 1

The probability that a file will not have to wait for transmission involves analyzing the system as an M/M/1 queue. The probability of waiting more than 10 minutes requires calculating the CDF of the exponential service time conditional on the queue being non-empty. Additionally, prioritizing short files often leads to decreased mean waiting times due to the shorter-job-first principle.

Step-by-step explanation:

The subject is related to data network performance analysis and examines how a communication link behaves using ideas from probability theory. In particular, the system uses a Poisson process to describe file arrival times and an exponential distribution for file lengths. Prior to transmission, files are stored in a FIFO (First In, First Out) queue and the communication channel has a set transmission rate.

1. To determine the probability that a file will not have to wait for transmission, we must consider the arrival rate of files and the service rate of the channel. Because the system is modeled as an M/M/1 queue (a queueing model with Poisson arrival, exponential service times, and one server), the probability that a file will not have to wait is the same as the system being empty, which is P0 = 1 - ρ, where ρ = arrival rate/service rate.
2. The probability that a file will have to wait for more than 10 minutes can be found by considering the cumulative distribution function (CDF) of the exponential service time and calculating P(Wait>10min | Queue not empty).
3. Discussing whether the overall mean waiting time would be improved by prioritizing "short" files over "long" files can involve an analysis of the benefits of a priority queueing system over a FIFO system. Typically, prioritizing short files can lead to a decrease in overall mean waiting time, an effect known as the shorter-job-first principle.