COM6515 Data Provided: NONE DEPARTMENT OF COMPUTER SCIENCE Spring Semester 2018-2019 NETWORK PERFORMANCE ANALYSIS 2 hours ANSWER ALL QUESTIONS. All questions carry equal weight. Figures in square brackets indicate the percentage of avail- able marks allocated to each part of a question. Registration number from U-Card (9 digits) — to be completed by student COM6515 1 TURN OVER COM6515 THIS PAGE IS BLANK COM6515 2 CONTINUED COM6515 1. a) The Poisson distribution is used for modelling queueing systems. (i) Write down the formula for the Poisson distribution and define all the symbols. [25%] (ii) Calculate the mean and variance of the Poisson distribution. [35%] (iii) N independent Poisson processes whose rates are λ1,λ2, . . . ,λN are combined to form one Poisson process. What is the rate this Poisson process? [10%] b) There are four desks at passport control at London airport, corresponding to passengers arriving from Europe, North and South America, Africa, and Asia (including Australia and New Zealand). Four planes, one from each of these regions, arrive consecutively, and the numbers of passengers on the planes are 150, 250, 200 and 200 respectively. It takes 20 minutes for the passengers to disembark from each plane, such that the passengers arrive randomly and simultaneously for passport control. It is assumed that passengers disembark from the planes at the same time. Derive the formula for the probability that there are 60 arrivals for passport control 2 minutes after the first passenger arrives. Do NOT calculate the probability, but explain how you would compute it. [30%] COM6515 3 TURN OVER COM6515 2. Packets arrive randomly at a switch, according to a Poisson process with an arrival rate of 1000 packets/second. The switch takes 0.8 milliseconds to process each packet. The switch and packets can be modelled as an M/M/1 queue because there is one input port and one output port. a) Show that the average number of packets in the system is E {k}= ρ 1−ρ = λ µ−λ where λ is the arrival rate and µ is the service rate. Calculate the average number of packets in the system. [25%] b) Derive an expression for the average number of packets in the queue. Calculate this average number for the given values of λ and µ. [20%] c) The average number of packets calculated in 2(b) includes the situation when there is no queue. Derive an expression for the average number of packets in the queue by including only those situations for which the queue is not empty. Calculate this average number for the given values of λ and µ. [40%] d) Use Little’s formula to calculate the average time spent by a packet in the system, and the average time by a packet in the queue. [15%] COM6515 4 CONTINUED COM6515 3. Consider a birth-death process with the following arrival rate λk and service rate µk at state k λk = (k+2)λ, k = 0,1,2, . . . , µk = kµ, k = 1,2,3, . . . where λ and µ are constants. The following formulae may be required for this question. ∞ ∑ k=0 kρk = ρ (1−ρ)2 and ∞ ∑ k=0 k2ρk = ρ(1+ρ) (1−ρ)3 where ρ is limited to values for which the infinite sums converge. a) Derive an expression for Pk, the probability that the system is in state k. [30%] b) Derive an expression for the average number of customers in the system. [15%] c) Calculate ¯λ, the average arrival rate. [25%] d) Calculate µ¯ ,the average service rate. [20%] e) Show that the average time spent in the system is 1µ and calculate the ratio ¯λ/µ¯. [10%] END OF QUESTION PAPER COM6515 5