Cover sheet for Assignment 3

Transcription

1 Faculty of Arts and Science University of Toronto CSC Introduction to Computer Networks, Winter 2018, LEC0101 Cover sheet for Assignment 3 Due Monday March 5, 10:00am. Complete this page and attach it to the front of your assignment. Name: (Underline your last name) Student number: I declare that this assignment is solely my own work, and is in accordance with the University of Toronto Code of Behavior on Academic Matters. Signature: 1

2 Question 1 : In class, we discussed queues with infinite buffer space. However, in reality, buffers are not infinite but finite. We consider this more realistic situation in this question, where we analyze the M/M/1/m queue. The M/M/1/m queueing system is the same as the M/M/1 system, except that there can be no more than m packets in the system (that is waiting in the buffer or in service), and packets arriving when the system is full are dropped and lost. Packets arrive according to a Poisson process with rate λ and are served at rate µ. (a) Draw the state-transition diagram for the M/M/1/m queue. (b) Derive the steady-state probabilities p n, n = 0,1,...,m, that there are n packets in the queue. (c) Find the probability that a new packet is lost and the rate at which packets are dropped. (d) Compute the throughput of the system. (e) Assume that ρ m << 1 (when is this the case?), and redo parts (b) and (c). (f) Using Little s formula, find the expected delay (queuing plus transmission delay) of a packet that enters the system. (g) Find the expected delay (queuing plus transmission delay) of a packet that enters the system directly by using the same approach as in Question 1 of Tutorial 5. Show all steps of your derivation. 2

3 Question 2 In this question, we analyze the M/M/m queueing system which is identical to the M/M/1 system, except that there m servers. Packets arrive according to a Poisson process with rate λ, and each server serves packets at rate µ, independently from the other servers. A packet at the head of the buffer is routed to any server that is currently not busy, or to the first server that becomes available. (a) Draw the state-transition diagram for the M/M/m queue. (b) Find the steady-state probabilities p n, n = 0,1,2,..., that there are n packets in the system. 3

4 Question 3 Consider a M/M/ queue (i.e. a queue with an infinite number of servers) with packet arrival rate λ and service rate µ for each server. (a) What is the range of values of λ for which the system is stable, i.e. the expected number of packets in the system will stay bounded (as time increases)? (b) Draw the state transition diagram for the system. (c) Compute the average E[N] and E[T], i.e. the average number of packets in the system and the total expected delay of a packet 4

5 Question 4 Consider the a buffer with two servers, server 1 and 2. Service times at each server are exponentially distributed with rate µ 1 for server 1 and µ 2 for server 2. Suppose that server 1currentlyservespacketp 1 andserver2servespacketp 2. Furthermore, thereisonepacket, packet p 3, waiting in the buffer. There are no new packets (in addition to the three packets already in the system) arriving to the system. As long as there is at least one packet in the buffer, we have that as soon as one of the two server has finished serving a packet then the first packet in the buffer enters service at that server. For this situation, answer the following questions (hint: to answer these questions, first draw a picture of the current situation and the sequences of events that have to happen for the situations in the questions below to occur). (a) What is the probability that packet p 1 leaves the system before packet p 2? (b) What is the expected queueing delay of packet p 3, i.e. what is the expected time packet p 3 has to wait until it gets served by one of the servers? (c) What is the expected time until packet p 3 leaves the system, i.e. what is the expected delay (queueing plus transmission delay) of packet p 3? (d) What is the probability that packet p 3 leaves the system before packet p 1? 5