**Advance Waiting Line Theory and Simulation Modeling**

**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: 0131791036

**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: 0131791036

**Alternative Waiting Lines** Multiple-Channel, Single-Phase Single-Channel, Single-Phase Single-Channel, Multiple-Phase © 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036

**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: 0131791036

**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: 0131791036

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

**P0 = 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: 0131791036

**Single-Channel, Single-PhaseManual Car Wash Example** • Arrival rate = 7.5 cars per hour • Service rate = an average of10 cars per hour • Utilization = / = 75% © 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036

**Single-Channel, Single-PhaseAutomated Car Wash Example** • Arrival rate = 7.5 cars per hour • Service rate = a constant rate of10 cars per hour • Utilization = / = 75% © 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036

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

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

**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: 0131791036

**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: 0131791036

**Building a Simulation Model** Four basic steps • Develop a picture of system to be modeled (process mapping) • Identify objects, elements, and probability distributions that define the system • Objects = items moving through system • Elements = pieces of the system • Determine experiment conditions (constraints) and desired outputs • 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: 0131791036

**Simulation Example(Using single-channel, single-phase** waiting line) • Process map • Time between arrivals (exponential distribution), service time (exponential distribution), objects = cars, elements = line and wash station • Maximum length for line, time spent in the system • Run model for a total of 100 cars entering the car wash, average the results for waiting time, cars in line, etc. © 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036

**‘SimQuick’ SimulationAn Excel-based application for** simulating processes that allows use of constraints (see text example 8S.5)