1. Simulation Statistics
Numerous standard statistics of interest
Some results calculated from parameters
Used to verify the simulation
Most calculated by program

2. Some Statistics Average Wait time for a customer
= total time customers wait in queue
total number of customers
Average wait time of those who wait
= total time of customers who wait in queue
number of customers who wait

3. More Statistics Proportion of server busy time
= number of time units server busy
total time units of simulation
Average service Time
= total service time
number of customers serviced

4. More Statistics Average time customer spends in system
= total time customers spend in system
total number of customers
Probability a customer has to wait in queue
= number of customers who wait

5. Traffic Intensity
A measure of the ability of the server to keep up with the number of the arrivals
TI= (service mean)/(inter-arrival mean)
If TI > 1 then system is unstable & queue grows without bound

6. Server Utilization % of time the server is busy serving customers
If there is 1 server
SU = TI = (service mean)/(inter-arrival mean)
If there are N servers
SU = 1/N * (service mean)/(inter-arrival mean)

7. Statistical Models
Probability: a quantitive measure of the chance or likelihood of an event occurring.
Random: unable to be predicted exactly
In an experiment where events randomly occur but in which we have assigned to each possible outcome a probability, we have determined a probability or stochastic model

8. Terms Event Space Event Complement of an Event Intersection Union
Mutually Exclusive

9. Examples Event Space: The set of all possible events that can occur
Event (E): Any single occurrence
ex: E = {4,5}
Complement of E:
Set of all events except E
Ex: Complement of E = {1,2,3,6}

10. Examples Union: Combination of any 2 event sets A= {1,2,3} B = {3,4}
A U B = {1,2,3,4}

11. Examples Intersection: Overlap of common occurrence of 2 event sets
A= {1,2,3} B = {3,4}
A Π B = {3}
Mutually Exclusive: 2 event sets that have no events in common
A= {1,2} B = {3,4}
A Π B = { }

12. Random variable Practical Definition
a quantity whose value is determined by the outcome of a random experiment

13. Random Variable Examples
X = the number of 4's that occur in 10 rolls
Y = the number of customers that arrive in 1 hour
Z = the number of services that are completed in 5 minutes

14. Discrete vs. Continuous RV
EXAMPLE
Discrete: X = number of customers that arrive in 1 hour
Continuous: Y = gallons that flow into the pool in 1 hour
????: Z = the average age of the customers that arrive in an hour

15. Discrete: Probability Function
Let X be a discrete R.V. with possible values x1, x2,…xn. Let P be the probability function
P(xi) = (X = xi) such that
(a) P(xi) >= 0 for i = 1,2,…n
(b) Σ P(xi) = 1

16. Probability Function Example
Consider the rolling of a fair die
1/6 for x = 1
P(x) = 1/6 for x = 2
1/6 for x = 3
1/6 for x = 4
1/6 for x = 5
1/6 for x = 6
0 for all other x

17. Cumulative Distribution Function
CDF of a random variable X is F such that F(x) = P (X <= x)
F(X) is continuous
Discrete: sum of probabilities
Continuous: area under the curve

18. Cumulative Distribution Function - Example
Consider the rolling of a fair die
0 for x < 1
1/6 for x < 2
F(X) = 2/6 for x <3
3/6 for x < 4
4/6 for x < 5
5/6 for x < 6
1 for x >= 6

19. Cumulative Function 1
1/2
1/6

20. Discrete vs. Continuous R.V.
Cumulative Distribution Function (CDF)
The CDF of a discrete R.V. X is F such that F(x)= P (X<= x)
Continuous: The CDF of a continuous RV has the properties:
F(x) is continuous, at least piecewise
F(x) exists except in at most a finite number of points

21. Discrete vs. Continuous Random Variables
Random variable: a function whose domain is the event space & whose range is some subset of real numbers
If a random variable assumes a discrete (finite or countably infinite) number of values, it is called a discrete random variable. Otherwise, it is called a continuous random variable.