#### Transcript Statistics

```Simulation Statistics


Numerous standard statistics of interest
Some results calculated from parameters


Used to verify the simulation
Most calculated by program
1
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
2
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
3
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
total number of customers
4
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
5
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)
6
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
7
Terms






Event Space
Event
Complement of an Event
Intersection
Union
Mutually Exclusive
8
Examples

Event Space: The set of all possible events
that can occur


Event (E): Any single occurrence


ex: {1,2,3,4,5,6}
ex: E = {4,5}
Complement of E:


Set of all events except E
Ex: Complement of E = {1,2,3,6}
9
Examples

Union: Combination of any 2 event
sets
 A= {1,2,3}
 B = {3,4}
 A U B = {1,2,3,4}
10
Examples

Intersection: Overlap of common
occurrence of 2 event sets



A= {1,2,3}
A Π B = {3}
B = {3,4}
Mutually Exclusive: 2 event sets that
have no events in common


A= {1,2}
AΠB={}
B = {3,4}
11
Random variable
Practical Definition
 a quantity whose value is
determined by the outcome of a
random experiment
12
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
13
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
14
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
15
Probability Function
Example

Consider the rolling of a fair die
P(x) =
1/6
1/6
1/6
1/6
1/6
1/6
0
for
for
for
for
for
for
for
x=1
x=2
x=3
x=4
x=5
x=6
all other x
16
Cumulative Distribution Function



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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
17
Cumulative Distribution
Function - Example

Consider the rolling of a fair die
F(X) =
0
1/6
2/6
3/6
4/6
5/6
1
for
for
for
for
for
for
for
x
x
x
x
x
x
x
<1
<2
<3
<4
<5
<6
>= 6
18
Cumulative Function
1
1/2
1/6
1
2
3
4
5
6
19
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
20
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.
21
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