Bozarth_ch08S

Advance Waiting Line Theory and

Simulation Modeling

© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036

Supplement 8S, Slide 2

Supplement Objectives

Be able to: Describe different types of waiting line systems. Use statistics-based formulas to estimate waiting

line lengths and waiting times for three different types of waiting line systems.

Explain the purpose, advantages and disadvantages, and steps of simulation modeling.

Develop a simple Monte Carlo simulation using Microsoft Excel.

Develop and analyze a system using SimQuick.



Alternative Waiting Lines

• Single-Channel, Single-Phase– Ticket window at theater,

• Multiple-Channel, Single-Phase– Tellers at the bank, windows at post office

• Single-Channel, Multiple-Phase– Line at the Laundromat, DMV



Alternative Waiting Lines

Single-Channel, Single-Phase

Multiple-Channel, Single-Phase

Single-Channel, Multiple-Phase



Assumptions

• Arrivals– At random (Poisson, exponential distributions)– Fixed (appointments, service intervals)

• Service times– Variable (exponential, normal distributions)– Fixed (constant service time)

• Other– Size of arrival population, priority rules,

balking, reneging



Poisson Distribution

Probability of n arrivals in T time periods

where = arrival rate

Tn

n enT

P !

)(



Waiting Line Formulas



P0 = Probability of 0 Units in Multiple-Channel System

MM

Mn

PM

M

n

n

!1

!1

1

1

0

0



Single-Channel, Single-PhaseManual Car Wash Example

• Arrival rate = 7.5 cars per hour• Service rate = an average of 10 cars per hour• Utilization = / = 75%



Single-Channel, Single-PhaseAutomated Car Wash Example

• Arrival rate = 7.5 cars per hour• Service rate = a constant rate of 10 cars per hour• Utilization = / = 75%



Comparisons

Manual wash, single server

Automated wash, single server

Manual wash, two servers

Cars waiting

2.25 1.125 0.1227

Cars in system

3 1.875 1.517

Time waiting

18 minutes 9 minutes 1 minute

Time in System

24 minutes 15 minutes 7 minutes



Simulation Modeling

Advantages• Off-line evaluation of

new processes or process changes

• Time compression• “What-if” analysis• Provides variance

estimates in addition to averages

Disadvantages• Does not provide

optimal solution• More realistic the

more costly and more difficult to interpret

• Still just a simulation



Monte Carlo Simulation

• Maps random numbers to cumulative probability distributions of variables

• Probability distributions can be either discrete (coin flip, roll of a die) or continuous (exponential service time or time between arrivals)

• Random numbers 0 to 99 supplied by computer functions such as = INT(100*RAND()) in Excel.



Monte Carlo Simulation Examples

• Coin toss: Random numbers 0 to 49 for ‘heads’, 50 to 99 for ‘tails’

• Dice throw: Use Excel function = RANDBETWEEN(1,6) for throws

• Service time: Use Excel function = –(avg service time)*ln(RAND()) for exponential service time



Building a Simulation Model

Four basic steps1) Develop a picture of system to be modeled (process

mapping)2) Identify objects, elements, and probability distributions

that define the system Objects = items moving through system Elements = pieces of the system

3) Determine experiment conditions (constraints) and desired outputs

4) Build and test model, capture the output data



Simulation Example(Using single-channel, single-phase waiting line)

1) Process map

2) Time between arrivals (exponential distribution), service time (exponential distribution), objects = cars, elements = line and wash station

3) Maximum length for line, time spent in the system4) Run model for a total of 100 cars entering the car

wash, average the results for waiting time, cars in line, etc.

‘SimQuick’ Simulation

An Excel-based application for simulating processes that allows use of constraints

(see text example 8S.5)

Documents

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