1. 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.

2. 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

3. 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 between Exponential distribution ↔ Poisson distribution X i : Continuous random variable, time between arrivals, has Exponential distribution with mean = 1/4 Y i : 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

4. 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

5. 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

6. Scenario 1: Constant interarrival time (2 min) and service time (1 min) Scenario 2: Variable interarrival time and service time

7. 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 IE 429, Parisay, January 2010

8. 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

9. 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

10. Analysis of Basic Queuing System Based on the theoretical M/M/1 IE 429, Parisay, January 2010

11. 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

12. Example 2 Logical Model IE 429, Parisay, January 2010

13. 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