What you need to know from
Probability and Statistics:
•Experiment outcome: constant, random variable
•Random variable: discrete, continuous
•Sampling: size, randomness, replication
•Data summary: mean, variance (standard deviation),
median, mode
•Histogram: how to draw, effect of cell size
Refer to handout on web page.
IE 429, Parisay, January 2010
What you need to know from
Probability and Statistics (cont):
•Probability distribution: how to draw, mass function,
density function
•Relationship of histogram and probability distribution
•Cumulative probability function: discrete and
continuous
•Standard distributions: parameters, other specifications
Read Appendix C and D of your textbook.
IE 429, Parisay, January 2010
Relation between
Exponential distribution ↔ Poisson distribution
Xi : Continuous random variable, time between arrivals,
has Exponential distribution with mean = 1/4
X 
X1=1/4
X2=1/2
0
X3=1/4
X4=1/8 X5=1/8
X6=1/2
1:00
Y1=3
X7=1/4
X8=1/4
i 1
12
X9=1/8 X10=1/8
2:00
Y2=4
Yi : Discrete random variable, number of arrivals per unit
of time, has Poisson distribution with mean = 4. (rate=4)
Y ~ Poisson (4)
IE 429, Parisay, January 2010

12
Xi

X11=3/8
1
4
X12=1/8
3:00
Y3=5
Y 
Y1  Y2  Y3
4
3
What you need to know from
Probability and Statistics (cont):
•Confidence level, significance level, confidence interval,
half width
•Goodness-of-fit test
Refer to handout on web page.
IE 429, Parisay, January 2010
Demo on Queuing Concepts
Refer to handout on web page.
Basic queuing system: Customers arrive to a bank, they
will wait if the teller is busy, then are served and leave.
Scenario 1: Constant interarrival time and service time
Scenario 2: Variable interarrival time and service time
Objective: To understand concept of average waiting
time, average number in line, utilization, and the effect
of variability.
IE 429, Parisay, January 2010
= Indicates Arrival of Customer
Scenario
1:3 Constant
interarrival
time
(2 10min)11and12
0
1
2
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
service time (1 min)
a) Constant Interarrival Times & Service Times
1
0
1
1
2
2
3
2 3
4
5
6
43
5
7
8
9
4
6
10
11
13
9
Arrival
Time
10
Status of Que
57
12
13
Arrival
Status of Serv
6
Time
Status of Queu
b) Variable
Interarrival
Times & Service
Times
b) Variable
Interarrival
Times
& Service Times
Scenario 2: Variable interarrival time and service time
EntityEntity
NumberNumber
1
2 1 3
1 Time
Arrival
02
3 0 43
Arrival Time
Service
Time Time
0.5
2.5 0.5 0.5
Service
24
35
1
2.5
53 6 4 7
5
94 4 11 5 12
0.50.5 1.5 1 0.5
5
9
0.5
6
6
11
1.5
7
7
12
0.5
Status of Serve
b) Variable Interarrival Times & Service Times
Entity Number 1
2
3
4
5
6
7
0 0 1
21
3 2 4
5
6
7
8
9
5
Arrival Time
0
3
43
5 4 9
11
12 6
Service Time 0.5 2.5 0.5 4 1
0.5 1.54 0.5
3
4
3
4
107
11 8 12
913
7
Arrival
Time10
Status of Que
Analysis of Basic Queuing System
Based on the field data
Refer to handout on web page.
T = study period
Lq = average number of customers in line
Wq = average waiting time in line
m
T  Tj
j 1
IE 429, Parisay, January 2010
1 m
Lq   L jT j
T j 1
1 n
Wq   Wi
n i 1
Queuing Theory
Basic queuing system: Customers arrive to a bank, they
will wait if the teller is busy, then are served and leave.
Assume:
Interarrival times ~ exponential
Service times ~ exponential
E(service times) < E(interarrival times)
Then the model is represented as M/M/1

IE 429, Parisay, January 2010

Notations used for QUEUING SYSTEM in steady
state (AVERAGES)
 = Arrival rate approaching the system
 e = Arrival rate (effective) entering the system
 = Maximum (possible) service rate
 e = Practical (effective) service rate
L = Number of customers present in the system
Lq = Number of customers waiting in the line
Ls = Number of customers in service
W = Time a customer spends in the system
Wq = Time a customer spends in the line
Ws = Time a customer spends in service
IE 429
Analysis of Basic Queuing System
Based on the theoretical M/M/1

  1

2
Lq 
 (   )

L
 

Wq 
 (   )
1
W
 
IE 429, Parisay, January 2010

Ls 

Ws 
1

Example 2: Packing Station with break and carts
Refer to handout on web page.
Objectives:
•
Relationship of different goals to their simulation
model
•
Preparation of input information for model creation
•
Input to and output from simulation software
(Arena)
•
Creation of summary tables based on statistical
output for final analysis
IE 429, Parisay, January 2010
Example 2 Logical Model
IE 429, Parisay, January 2010
You should have some idea by now about the answer of
these questions.
* What is a “queuing system”?
* Why is that important to study queuing system?
* Why do we have waiting lines?
* What are performance measures of a queuing system?
* How do we decide if a queuing system needs improvement?
* How do we decide on acceptable values for performance measures?
* When/why do we perform simulation study?
* What are the “input” to a simulation study?
* What are the “output” from a simulation study?
* How do we use output from a simulation study for practical
applications?
* How should simulation model match the goal (problem statement)
of study?
IE 429, Parisay, January 2010