Simulation

Abstract

The previous chapter dealt with one of the tools for performance analysis—queueing theory. This chapter concentrates on another tool—simulation. In this chapter, we provide an overview of simulation: its historical background, importance, characteristics, and stages of development.

Science without religion is lame, religion without science is blind.

—Albert Einstein

Problems

• 5.1 Define simulation and list five attractive reasons for it?

• 5.2 Generate 10,000 random numbers uniformly distributed between 0 and 1. Find the percentage of numbers between 0 and 0.1, between 0.1 and 0.2, etc., and compare your results with the expected distribution of 10 % in each interval.

• 5.3 (a) Using the linear congruential scheme, generate ten pseudorandom numbers with a = 1573, c = 19, m = 1000, and seed value X₀ = 89.

  (b) Repeat the generation with c = 0.

• 5.4 Uniformly distributed random integers between 11 and 30, inclusive, are to be generated from the random numbers U shown below. How many of the integers are odd numbers?

  0.2311	0.7919
  0.2312	0.9218
  0.6068	0.7382
  0.4860	0.1763
  0.8913	0.4057
  0.7621	0.9355
  0.4565	0.9169
  0.0185	0.4103
  0.8214	0.8936
  0.4447	0.0579

• 5.5 Generate 500 random numbers, exponentially distributed with mean 4, using uniformly distributed random numbers U. Estimate the mean and the variance of the variate.

• 5.6 Using the rejection method, generate a random variable from

  $$ f(x)=5{x}^2,\begin{array}{cc}\hfill \hfill & \hfill 0\le x\le 1\hfill \end{array} $$

• 5.7 (a) Using the idea presented in this chapter, generate 100 Gaussian variates with mean 3 and variance 2.

  (b) Repeat part (a) using MATLAB command randn.

  (c) By estimating the mean and variance, which procedure is more accurate?

• 5.8 The probability density function of Erlang distribution is

  $$ f(x)=\frac{\alpha^k{x}^{k-1}}{\Gamma (k)}{e}^{-\alpha x},\begin{array}{cc}\hfill \hfill & \hfill x>0,\alpha >0\hfill \end{array} $$

  where Γ(k) = (k − 1)! and k is an integer. Take k = 2 and α = 1. Use the rejection method to describe a procedure for generating random variates from Erlang distribution.

• 5.9 Write a computer program to produce variates that follow hyperexponential distribution, i.e.

  $$ f(x)= p\lambda {e}^{-\lambda x}+\left(1-p\right)\mu {e}^{-\mu x} $$

  Take p = 0.6, λ = 10, μ = 5.

• 5.10 Write a program to simulate the M/E_k/1 queueing system. Take k = 2. Compare the results of the simulation with those predicted by queueing theory.

• 5.11 A random sample of 50 variables taken from a normal population has a mean of 20 and standard deviation of 8. Calculate the error with 95 % confidence limits.

• 5.12 In a simulation model of a queueing system, an analyst obtained the mean waiting time for four simulation runs as 42.80, 41.60, 42.48, and 41.80 μs. Calculate the 98 % confidence interval for the waiting time.

• 5.13 Discuss the OPNET simulation results of Fig. 5.29 results?

• 5.14 Discuss the OPNET simulation comparison results of Figs. 5.30 and 5.31?

• 5.15 Discuss the OPNET simulation comparison results Figs. 5.32 through 5.35?

• 5.16 What are different purposes for C++ and OTcl languages in NS2?

• 5.17 What are the limitations of NS2?