COM6515

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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

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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%]

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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%]

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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.

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