IE 429, Parisay, January 2010

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.

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

X1=1/4 X2=1/2 X3=1/4 X4=1/8 X5=1/8 X6=1/2 X7=1/4 X8=1/4 X9=1/8 X10=1/8 X11=3/8 X12=1/8

0 1:00 2:00 3:00 Y1=3 Y2=4 Y3=5

Relation betweenExponential distribution ↔ Poisson distribution

Xi : Continuous random variable, time between arrivals, has Exponential distribution with mean = 1/4

Yi : Discrete random variable, number of arrivals per unit of time, has Poisson distribution with mean = 4. (rate=4) Y ~ Poisson (4)

4

1

12

12

1 i iXX

43

321

YYY

Y

IE 429, Parisay, January 2010

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 timeScenario 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

Scenario 1: Constant interarrival time (2 min) and

service time (1 min)

Scenario 2: Variable interarrival time and service time

= Indicates Arrival of Customer

a) Constant Interarrival Times & Service Times

ArrivalTime

Status of Queue

ArrivalTime

t2 t3 t4 t5 t6 t7

0.5 1 0.5

4 5

2.5

9

2 3 4 5

3Arrival Time

Service Time

1

0

0.5

6

11

1.5

7

12

0.5

Entity Number

= Indicates Arrival of Customer

a) Constant Interarrival Times & Service Times

ArrivalTime

Status of Queue

ArrivalTime

t2 t3 t4 t5 t6 t7

139 10 11 123 5 60 1 2 4 7 8

= Indicates Arrival of Customer

a) Constant Interarrival Times & Service Times

ArrivalTime

Status of Queue

ArrivalTime

t2 t3 t4 t5 t6 t7

6 90 1 2 3 4 5 10 11 12 137 8

Analysis of Basic Queuing SystemBased on the field dataRefer to handout on web page.

T = study period

Lq = average number of customers in line

Wq = average waiting time in line

m

jjTT

1

m

jjjq TL

TL

1

1

n

iiq W

nW

1

1

IE 429, Parisay, January 2010

Queuing TheoryBasic queuing system: Customers arrive to a bank, they will wait if the teller is busy, then are served and leave. Assume: Interarrival times ~ exponentialService times ~ exponentialE(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 lineLs = Number of customers in serviceW = Time a customer spends in the systemWq = Time a customer spends in the lineWs = Time a customer spends in service IE 429

Analysis of Basic Queuing System Based on the theoretical M/M/1

1

)(

2

qL

L

sL

)(

qW

1W

1

sW

IE 429, Parisay, January 2010

Example 2: Packing Station with break and cartsRefer 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