1. Assignment Task Definition You task compromises the following.

(1) Produce a report on Traffic Modeling based on the papers provided on Moodle, and other relevant papers you can find.

(2) Part 1 of this report shall compromise four sections (50 marks)

(a) Discuss the background and explain the main issues and problems in network traffic modeling (not more than 400 words).

(b) Discuss the influence of traffic modeling on network performance and design (not more than 800 words).

(c) Discuss the role of simulation in the analysis of complex queueing systems (not more than 800 words).

(d) Explain simulation of random variables such as uniformly distributed random variables, exponentially distributed random variables

and Pareto distributed random variables. Explain simulation of discrete random variables.

(e) Add any other issues relevant to the topic

(f) Include references and use proper citation techniques.

(g) Your report need not be restricted to the material of lecture notes. Please use books, information found on the web. Journal and confer- ence papers are very encouraged.

(3) Part 2 of this report shall compromise the simulation of a queueing system and its analysis (50 marks). You need to provide the following work for simulating an M/M/1 queueing system. (The value of traffic parameter ρ will be suggested individually for each student.)

(a) Simulate the sequence of interarrival times. For this purpose you need first to simulate a sequence of uniformly distributed random variables (denoted Un) in the interval (0,1). Then, the exponentially distributed
random variables are obtained by using the transform En = - 1 ln Un,
where λ is the parameter of the exponential distribution. The sim- ulated number of interarrival times is supposed to be not less than 10,000.

(b) Simulate the sequence of service times. It should be done by the same way as before. You need first to simulate a sequence of uniformly
distributed random variables in the interval (0,1) (denoted by Vn) and then to use the transform Sn = - 1 ln Vn, where µ is the reciprocal to the expected service time. The simulated number of service times is supposed to be not less than 10,000.

(c) On the basis of the two sequences {En} and {Sn} you need to cal- culate the sequence of waiting times. For waiting times we have the recurrence relation:

W1 = 0, Wn+1 = max{0, Wn - En + Sn}.

Here En is the time between the arrivals of the nth and n + 1st cus- tomers and Sn is the service time of the nth customer. Hint: We give some details of the above recurrence relation. The first cus- tomer arrives in the empty system, so its waiting time is set to zero. For the waiting time of the second customer we have the relation

W2 = max{0, S1 - E1}, which means that the waiting time of the sec- ond customer is zero if the service time of the first customer is shorter than interarrival time, i.e. second customer arrives in the empty sys- tem as well. Otherwise, the waiting time is the difference between S1 and E1. For the following calculations of W2, W3 etc. the students are to use the loop.

(d) After calculation of the sequence Wn, estimate the mean waiting time,

EW , by the formula W ≈ 1 .N

Wn and variance σ2

1 .N ≈  (Wn- N n=1 W N n=1W )2 (Here N is the number of variables, say N = 10, 000.) Compare the estimated value W with the theoretical value EW .

(e) Bonus question 1. Calculate the sequence for the number of cus- tomers in the system:
Q1 = 0, Qn+1 = max{0, Qn + 1 - νn},

where νn is the number of served customers between the arrivals of the nth and n + 1st customers. After calculation of the sequence Qn, estimate the mean queue length, EQ, by the formula Q ≈ 1 .N Qn,N n=1 and the variance σ2 1 .N≈ (Qn - Q)2. Compare the  estimated Q Nn=1 value Q with the theoretical value EQ.(f) Bonus

question 2. On the basis of the calculated values of Q and W check the validity of the Little's formula. Estimate the error for this formula compared to the theoretical result.

(4) Your answer must include source code and/or numerical procedure. The choice of programming language is not restricted. It can be C, Matlab or another language or even Microsoft Excel.

(5) Your final report should be 8-10 word processed pages. This page limit excludes the title page, source code listing and references. No additional marks will be given for reports that over the specified length. If the report is more than 50% longer than specified, the total mark will be decreased by 10% for every excess page in the report.

(6) Provide clear references for any theories, arguments, claims, diagrams, equations and any related material that you get from elsewhere.

(7) Each bonus question costs 5 additional marks. However, the total mark must not exceed 100. That is, if you collected 92 marks for the Assignment excluding the bonus questions, then your total mark will be 100, but not 102.

(8) You will then submit a PDF format report in a single file which includes your student number, name, the report and the references. The format is A4 with wide margins from the left and right and 10 point sanf serif font (e.g. Helvetica or Arial).