Advance Waiting Line Theory and Simulation Modeling

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

© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: Supplement 8S, Slide 3 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

© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: Supplement 8S, Slide 4 Alternative Waiting Lines Single-Channel, Single-Phase Multiple-Channel, Single-Phase Single-Channel, Multiple-Phase

© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: Supplement 8S, Slide 5 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

© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: Supplement 8S, Slide 6 Poisson Distribution Probability of n arrivals in T time periods where = arrival rate

© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: Supplement 8S, Slide 7 Waiting Line Formulas

© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: Supplement 8S, Slide 8 P 0 = Probability of 0 Units in Multiple-Channel System

© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: Supplement 8S, Slide 9 Single-Channel, Single-Phase Manual Car Wash Example Arrival rate = 7.5 cars per hour Service rate  = an average of 10 cars per hour Utilization  = /  = 75%

© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: Supplement 8S, Slide 10 Single-Channel, Single-Phase Automated Car Wash Example Arrival rate = 7.5 cars per hour Service rate  = a constant rate of 10 cars per hour Utilization  = /  = 75%

© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: Supplement 8S, Slide 11 Comparisons Manual wash, single server Automated wash, single server Manual wash, two servers Cars waiting Cars in system Time waiting 18 minutes9 minutes1 minute Time in System 24 minutes15 minutes7 minutes

© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: Supplement 8S, Slide 12 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

© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: Supplement 8S, Slide 13 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.

© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: Supplement 8S, Slide 14 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

© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: Supplement 8S, Slide 15 Building a Simulation Model Four basic steps 1)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

© 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: Supplement 8S, Slide 16 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 system 4)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)