Lecture 19
Nov10, 2010
Discrete event simulation (Ross)
• discrete and continuous distributions
• computationally generating random variable following
various distributions.
• repair problem
Exercise 2: (coupon collector’s problem) When you buy
a can of soda, it contains a coupon with a label in 1:n.
The labels on the can are random in the sense that, at
any point during simulation, the probability that the next
can you buy will contain label j is 1/n. The goal is the
determine the average number of cans you need to
buy before you have collected all the n coupons.
Idea: create an array A of size n, all set to 0. Create
another array to keep track of the number of cans to
buy before all n coupons are collected. The size of this
array B is m = the number of trials. Repeat generating a
random number j in 1:n, and increment A(j) until all A(j)
are > 0. At this point, set B(k) to sum(A) and repeat fro k
= 1, 2, …, m.
Probability distributions
• In a Discrete Event Simulation, you need to decide what
probability distribution functions best model the events.
in most situations, uniform distribution does not work.
• Pseudorandom number generators generate numbers in a
uniform distribution
• One basic trick is to transform that uniform distribution into
other distributions.
• Some standard probability distributions are convenient to
represent mathematically.
• They may or may not represent reality, but can be useful
simplification.
Mean and Variance of a Discrete Random Variable
Example:
Mean and Variance of a Discrete Random
Variable
Expected Value of a Function of a Discrete Random
Variable
Discrete Uniform Distribution
Definition
Discrete Uniform Distribution
Example:
Discrete Uniform Distribution
Probability mass function for a discrete uniform random
variable.
Discrete Uniform Distribution
Mean and Variance
Binomial Distribution
Random experiments and random variables
Binomial Distribution
Definition
Binomial Distribution
Binomial distributions for selected values of n and p.
Binomial Distribution
Example:
Binomial Distribution
Example
Binomial Distribution
Mean and Variance
Geometric and Negative Binomial Distributions
Example
Geometric Distribution
Definition
Geometric Distribution
Definition
Poisson distribution
X follows a Poisson distribution if:
P{ X  i}  e

i
i!
Normal or Gaussian distribution
 Ubiquitous in statistics
 Many phenomena follow
this distribution
 When an experiment is
repeated, the sum of the
outcomes tend to be
normally distributed.
 We can test this
experimentally using a
Matlab simulation.
Normal Distribution
Normal probability density functions for selected values of
the parameters  and 2.
Normal Distribution
Some useful results concerning the normal distribution
Normal Distribution
Definition : Standard Normal
Exponential Distribution
Simulating a Probability distribution
Sampling values from an observational distribution with a given set
of probabilities (“discrete inverse transform method”).
Suppose the distribution we want to simulate is X where
p(X = x1) = p0, p(X = x2) = p1, …, p(X = xn) = pn-1
Generate a random number U
If U < p0 return X1
If U < p0 + p1 return X2
If U < p0 + p1 + p2 return X3
etc.
This can be speeded up by sorting p so that the larger intervals are
processed first, reducing the number of steps.
Poisson distribution
• Example of algorithm to sample from a distribution.
• X follows a Poisson distribution if:
P{ X  i}  e

i
i!
An algorithm for sampling from a Poisson distribution:
1.
2.
3.
4.
Generate a random number U
If i=0, p=e-l, F=p
If U < F, return I
P = l * p / (i + 1),
F = F + p, i = i + 1
5. Go to 3
There are similar tricks to sampling from other
probability distributions. Some of the distributions (e.g. Poisson,
Normal etc.) can be generated using Matlab’s built-in
functions.
A repair problem
Question: What is the expected time for the system to
crash? System crashes when fewer than n machines
are available.
Input:
N = the number of machines needed to run the system
S = the number of spare machines
P1 = the distribution of failure time of a machine
P2 = the probability distribution of the time to service a faulty
machine
Output:
The time at which the system crashes.
To get the average time between successive crashes, we should
repeat the simulation many times and sum the times, divide by the
number of trials.
Algorithm for simulation
Variables used: time – t, system variable – r: the
number of machines down at time t.
An event occurs when a machine fails or a machine
has been repaired.
Event list:
t1  t2  t3  ...tn , t *
tj is the time at which the j-th machine currently in use
will fail.
t* is the time at which the present machine being
repaired will be ready for use, t* =  if no such machine.
Simulation function – based on algorithm from
Ross’s book (Chapter 6)
function T = simulate_repair(n, s, l1, l2)
% simulation of a repair problem
% F = prob distribution defining failure of machine, i.e.,
% F(t) = prob(time taken for machine to break down = t)
% G = prob distribution defining the time taken to repair
% ie., G(t) = prob(time taken to repair = t)
% n = number of machines, s = additional number of machines
% goal is to compute time T at which the sytem fails
% system fails when more than s machines are in repair
% failure rate is a uniform distribution with values 1, 2,
..., l1
% repair time follows a uniform distribution in [1, l2]
t = 0; r = 0; tstar = Inf;
mqueue = [];
for j=1:n
mqueue(end+1) = uniform(l1);
end;
mqueue(end+1)=tstar;
mqueue = sort(mqueue);
while 1
t1 = mqueue(1);
if t1 < tstar
t = mqueue(1);
r = r + 1;
if r == s+1
T = t; return;
else
x = uniform(l1);
mqueue(end+1)=t+x;
mqueue = mqueue(2:end);
mqueue = sort(mqueue);
end;
if r == 1
y = uniform(l2);
tstar = t+y;
end;
else
t = tstar;
r = r - 1;
if r == 0
tstar = Inf;
else
y =
uniform(l2);
tstar = t + y;
end;
end;
end;
It is easy to modify the code so that the
distributions (for failure time and repair time)
are not uniform, but say Poisson, exponential
or normal etc.
Exercise: Try replacing uniform by Poisson or
exponential and compute the expected
time for crash.
Does it increase the expected time before
the system crashes (given the same mean)?
Service queue simulation
Problem: Each customer joins a queue serviced by a
single server. The arrival time of the customer is a
random number given by a known distribution. Time to
service obeys a known (possibly different) probability
distribution. Assume that there are a total of N
customers. (N is known.) Goal is to calculate the
average time a customer has to wait in the service line?
We may also want to calculate the average waiting
time of the k-th customer.
Variations: 1) assume that after some time T, no
additional customer allowed.
2) There are two service providers.
Simulation of a car wash system