Problem Set 2 (Due: Friday, October 19, 2018)

Transcription

1 Electrical and Computer Engineering Memorial Univerity of Newfoundland ENGI Advanced Data Network Fall 2018 Problem Set 2 (Due: Friday, October 19, 2018) Quetion 1 Conider an M/M/1 model of a queue in a packet witched network, for which = 50 and = 60. (a) Determine the probability that between 5 and 7, incluive, packet arrive at the queue in a 100 m period. (b) Determine the probability that a packet i longer than 5000 bit, given that the output link rate i 4 Mbp. Quetion 2 Conider the model of an output buffer in a packet witch for a 200 Mbp link. he arrival rate of packet to the queue i given by = 400,000 packet per econd. Determine the average delay through the ytem and the average number of packet in the queue given: (a) exponential packet ize with an average length of 53 byte (b) fixed packet ize of length 53 byte (c) packet length with a known mean of 53 byte and a variance of Quetion 3 A queue i utilized in an AM witch to control the packet uing a link of rate 622 Mbp. Arrival to the queue are Poion and arrive at an average rate of = 1.2 packet/ ec. (a) Given that AM cell are fixed length of 53 byte, determine the average time pent in the queuing ytem (i.e., including waiting time and ervice time). (b) Aume that the deigner of the ytem want to enure that the buffer allocated to the queue i capable of toring at leat 5 time a many packet in the queue a the average number in the queue. What i the ize of the buffer allocated in term of the number of bit required to tore the neceary packet? 1

2 Quetion 4 Conider the following queueing ytem. All queue are M/M/1 and traffic tream plit with the probabilitie a given. Determine the expected delay through the entire ytem. Arrival rate are given a arrival per econd and ervice rate are given a departure per econd. q1=.25 2=1.2 p2=.6 =4 q2=.4 1=5 4=4 5=5 p1=.75 3=3.5 Quetion 5 Conider the following queuing ytem. A indicated, 70% of the incoming packet enter Q1 and 30% enter Q2. 70% Q1 1=16 Q3 Q4 =20 30% Q2 2=8 3=25 4=21 (a) Auming that all queue are M/M/1, determine the average delay through the queuing ytem. (b) Repeat part (a) auming that Q1, Q2, and Q3 are M/M/1 and Q4 i an M/D/1 queue. (c) Repeat part (a) auming that Q1, Q2, and Q3 are M/M/1 and Q4 i an M/G/1 queue with the mean ervice time being 1/21 econd and the variance of the ervice time being econd 2. 2

3 Quetion 6 Conider that a queue in a communication network i modeled a an M/M/1/N queue where the ize of the queuing ytem i N = 3. (a) Auming that the arrival rate i 10,000 packet per econd and the average ervice time i 80 ec: (i) Draw the tate tranition diagram that repreent the behaviour of the queuing ytem. (ii) Determine the probability that a packet arriving at the queue mut be dicarded becaue the queue i full. (iii) Determine the average number of packet in the queue. (iv) Determine the average delay for packet that are not dicarded. (b) Aume now that N = 2. Given that the arrival rate i 10,000 packet per econd, determine the maximum poible average ervice time o that the probability of a packet being dicarded i le than Quetion 7 A queuing ytem ha Poion arrival with an arrival rate of 50 packet/ ec and two erver with an exponential ervice time of t =.015 ec. In addition to the two erver, the ytem ha one waiting poition. Hence, the ytem i a M/M/2/3 queue. (a) Draw the tate tranition diagram and determine the tate probabilitie. (b) Determine the average number of cutomer in the queuing ytem. (c) What i the probability of blocking? Quetion 8 Conider a queuing ytem that behave in the following manner: - here are never more than two cutomer in the queuing ytem. - he erver mut erve either two cutomer at a time or no cutomer. - If a cutomer arrive and there are no cutomer in the queuing ytem, the cutomer will wait and will not be erved immediately. - If a cutomer arrive and there i one cutomer waiting and no cutomer being erved, both cutomer will immediately begin to be erved by the erver. - If a cutomer arrive and two cutomer are being erved, the cutomer i dicarded. (Hence, there can never be cutomer waiting while cutomer are being erved.) - he erver erve two cutomer at once for the ame length of time and the ervice time i exponentially ditributed. (a) Draw the tate tranition diagram for the ytem. (b) Auming that cutomer arrival are Poion at a rate of 20 cutomer per minute and the average time cutomer will take while being erved (not including waiting time) i 2 econd, what i the probability that a cutomer i blocked from being erved (i.e., the probability that a cutomer i dicarded)? 3

4 Quetion 9 A ynchronou packet witch contain only input queue, with one firt-come-firt-erve queue at each input. During a witching cycle, one packet may placed in the queue and/or one packet may leave the queue. Packet at the head of the queue are kept in the queue until they can be uccefully tranmitted to the output link. hat i, if the packet at the head of the queue i not blocked, it i removed from the queue. In any given witching cycle, the probability of a packet arriving at an input i 20% and the probability of blocking occurring for a packet at the head of the queue i 30%. he input queue can tore up to 4 packet. Arriving packet are queue at the beginning of the witching cycle and departing packet are dequeued at the end of the witching cycle. Hence, an incoming packet mut be queued before it i poible to dequeue the packet at the head of the queue. A packet mut be placed in the queue for at leat one witching cycle before it i tranmitted. (For example, if a packet arrive in witching cycle i to an empty queue, it will be removed from the input queue no earlier than witching cycle i+1.) (a) Draw the tate tranition diagram of the Markov chain formulation of the ytem. (b) Determine the tate probabilitie. (c) What i the probability that a packet will be dicarded? Note that an arriving packet mut be queued before the packet at the head of the queue can be dequeued. 4

5 ANSWERS: Q1. (a) (b) Q2. (a) (b) (c) Q3. (a) (b) 5936 bit Q4. (a) 6.17 Q5. (a) 1.7 (b) (c) Q6. (a) (ii) PB = (iii) (b) k (iv) Q7. (a) P0 = 0.468, P1 = 0.351, P2 = 0.132, P3 = (b) average # packet = (c) PB = Q8. (a) P0 = 3/8, P1 = 3/8, P2 = 1/4 (b) PB = 1/4 Q9. (b) P0 = , P1 = , P2 = , P3 = , P4 = (c) PB =