computer networks

(1) (a) Write a computer program to generate random numbers between [0,1]. Such a random number generator simulates the values generated by a uniform random variable U[0,1]. (b) Write another progam (using the program in (a)) to estimate P(U > x). Plot P(U > x) for values of x ∈ (0.5,1). (2) (a) Write a computer program to simulate the values generated by Exponential (X) and Poisson random (Y) variables using the program you developed in (1). (b) Provide plots for P(X > x) and P(Y > x), for E(Y) = E(X) = 2. It may be necessary to show your result on a plot where the vertical axis is logarithmic. (3) (a) Write a computer program that simulates an M/M/1 queue. (b) Based on this program plot Pn against n when λ = 5 and μ = 6. (c) Again, from your program, find the expected number and expected delay in your M/M/1 queueing system when ρ= 5/6. (4) (a) Write a computer program that simulates an M/Ek/1 queue. Here, Ek is an Erlang random variable with k phases. (b) Based on this program, plot Pn against n when k = 4, λ = 5 and μ = 6. Also, find the expected number in the system. How do these results compare with your M/M/1 results in (3)? (c) Plot the expected number in the system for different values of the utilization when k = 40. Also plot the expected number in the system in an M/D/1 queue from the analysis in class and compare the results with your simulation. What does this tell you and why?