Advance Waiting Line Theory and Simulation Modeling

Advance Waiting Line Theory and Simulation Modeling

Supplement Objectives Be able to: q Describe different types of waiting line systems. q Use statistics-based formulas to estimate waiting line lengths and waiting times for three different types of waiting line systems. q Explain the purpose, advantages and disadvantages, and steps of simulation modeling. q Develop a simple Monte Carlo simulation using Microsoft Excel. q Develop and analyze a system using Sim. Quick. © 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036 2

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 3

Alternative Waiting Lines 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: 0131791036 4

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 5

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 6

Waiting Line Formulas © 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036 7

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: 0131791036 8

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: 0131791036 9

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: 0131791036 10

Comparisons Cars waiting Manual Automated wash, single server 2. 25 1. 125 Manual wash, two servers 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 © 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036 11

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: 0131791036 12

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 13

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 14

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: 0131791036 15

Simulation Example (Usingle-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 © 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036 results for waiting time, cars in line, etc. 16

‘Sim. Quick’ Simulation An Excel-based application for simulating processes that allows use of constraints (see text example 8 S. 5) © 2008 Pearson Prentice Hall --- Introduction to Operations and Supply Chain Management, 2/e --- Bozarth and Handfield, ISBN: 0131791036 Chapter 8, Slide