2What is Simulation? A definition of simulation from Dictionary.com Imitation or representation, as of a potential situation or in experimental testing.The basic idea is to build an experimental device, or simulator, that will “act like” the system of interest … in a quick, cost-effective manner.Representation of the operation or features of one process or system through the use of another: computer simulation of an in-flight emergency.3
3Simulation in Business Analysis Uses mathematical modelsProbabilistic (as opposed to deterministic)Uses the entire range of possible values of a variable in the modelImitates a system or situation (like a coin-flip, or how long a person might have to wait in a line at a restaurant)
4Why Simulate? Safety – flight simulator Cost – easier to simulate adding a new runway and find out effects than to implement in reality and then find outTime – Boeing uses simulated manufacturing before the real thing, with tremendous savings in time and money – can discover parts that do not fit and fix them before actual production
5How does it work? Simulation requires you to know What variable is to be simulatedIs the variable discrete or continuous?The distribution of the variable – values it can take on and the probabilities of those values occurring.Step 1: Generate a variable containing uniformly distributed random variables between 0 and 1 (the rand() function in Excel).Step 2: Create a rule to map the random numbers to values of the variable desired in the right proportion, and apply the rule.
6Types of Variables in Simulation Models DiscreteUsed for simulating specific values or specific points.Example: Number of people in a waiting line (queue).ContinuousUsed for simulating any value (between specific points)Example: The amount of time a person spends in a queue.8
7Discrete Example 1– coin toss Variable to be simulated is “Outcome of a coin toss”. It takes on values “Heads” and “Tails”, each with 0.5 probability.Generate 100 random numbers (100 tosses of coin).Make a rule like – if random number > 0.5, then “Heads”, else “Tails”. This will create the right distribution of outcomes.
8Discrete Example 2: Machine Failures Simulate machine failures based on this historical dataNumber of Failures per monthFrequency(# of months this occurred)1233620Total60
9Discrete Example 2: Machine Failures Create the following cumulative probability table.Number of Failures per monthFrequency(# of months this occurred)ProbabilityCumulative12336200.6000.3330.0500.0160.9330.9831.000Total601.00
10Discrete Example 2: Machine Failures Now map the random numbers between 0 and 1 using the cumulative prob. Column as the cutoffs.Random numbers between 0 and 0.6 represent 0 failures, between 0.6 and represent 1 failure, and so on.0.600.930.980 failures1 failure23 failures
11Discrete Example 2 Solution – Random Number Mapping The random numbers are now mapped to number of failures as follows.Random #Number ofFailures0.3450.0080.9850.87831
12Continuous Example – arrival time Variable to be simulated is arrival time at a restaurant which can literally take on infinite individual valuesFor example someone could arrive at:12:09:3712:09:37:5212:09:37:52:14, etc.
13Continuous Example – arrival time con’t To simulate this situation, we must specify intervalsAt the restaurant the intervals could be all people arriving between 11am and 12pm, 12pm and 1pm, or 1pm and 2pm.As with the coin toss, generate random numbers in Excel ( =RAND() )Make a rule – if random number:<=.333, then =11am-12pm>.333 up to =.666, then 12pm-1pm>.666 up to 1, then 1pm to 2pmeach category is equally likely
14Continuous Example – arrival time con’t If the random number is.47, then this would fall in the 12pm to 1pm category,If the random number is .88, then this would fall in the 1pm to 2pm category, etc.Because each category is equally likely, if we run enough trials, each category will contain about the same number of random numbers, which will tell the restaurant owner that it is equally likely that a person will arrive at any of the three times.
15Continuous Example – arrival time con’t The owner looks at historical information and says that on an average day, 225 people eat lunch at his restaurant , and that typically47 people arrive between 11am and 12pm112 people arrive between 12pm and 1pm66 people arrive between 1pm and 2pmHow do we map these numbers?
16Continuous Example – arrival time con’t To complete the mapping, we need to make a cumulative distribution function (CDF)
17Continuous Example – arrival time con’t Make a new rule – if random number:<=.21, then =11am-12pm>.21 up to =.71, then 12pm-1pm>.71 up to 1, then 1pm to 2pm
18Note on Random Numbers in Excel Spreadsheets Once entered in a spreadsheet, a random number function remains “live.” A new random number is created whenever the spreadsheet is re-calculated. To re-calculate the spreadsheet, use the F9 key. Note, almost any change in the spreadsheet causes the spreadsheet to be recalculated!If you do not want the random number to change, you can freeze it by selecting: tools, options, calculations, and checking “manual.”
19Evaluating ResultsSimulation measures the quality of a solution because it gives the probability of a certain event occurringSimulation also shows the variabilitySimulation does not necessarily give the best possible answer. It gives the most likely answer.6
20AppendixSome useful information on very popular probability distribution function6
21Probability Distributions A probability distribution defines the behavior of a variable by defining its limits, central tendency and natureMeanStandard DeviationUpper and Lower LimitsContinuous or DiscreteExamples are:Normal Distribution (continuous)Binomial (discrete)Poisson (discrete)Uniform (continuous or discrete)Custom (created to suit a specific purpose)
22Uniform DistributionAll values between minimum and maximum occur with equal likelihoodConditionsMinimum Value is FixedMaximum Value is FixedAll values occur with equal likelihoodExcel function: RAND() – returns a uniformly distributed random number in the range (0,1)
23Normal Distribution Conditions: Use when: Uncertain variable is symmetric about the meanUncertain variable is more likely to be in vicinity of the mean than far awayUse when:Distribution of x is normal (for any sample size)Distribution of x is not normal, but the distribution of sample means (x-bar) will be normally distributed with samples of size 30 or more (Central Limit Theorem)Excel function: NORMSDIST() – returns a random number from the cumulative standard normal distribution with a mean of zero and a standard deviation of one [e.g., NORMSDIST(1) = .84]
25DistributionsVariables to be simulated may be normal (e.g. height) or exponential (e.g. service time) or various other distributions.Task is to convert uniform distribution to the required distribution.Freq1Freqinfinity
26Application - Queuing Systems A queuing system is any system where entities (people, trucks, jobs, etc.) wait in line for service (processing of some sort) –retail checkout lines, jobs on a network server, phone switchboard, airport runways, etc.
27Queuing System InputsQueuing (waiting line) systems are characterized by:Number of servers / number of queuesSSSQ – Single Server Single QueueSSMQ – Single Server Multiple QueueMSSQ – Multiple Server Single QueueMSMQ - Multiple Server Multiple QueueArrival Rate (Arrival Intervals)Service Rate (Service Times)
28Performance Variables (outcome) Performance of a queuing system is measured byAverage time waiting in queue/systemAverage number of entities in queue/systemTime in QueueService TimeArrival timeService BeginsService EndsTime in System
29Distributions in Queuing Arrival Intervals (time between two consecutive arrivals) and Service Time (time to serve one customer) are exponentially distributed.Confirm it yourself by watching cars on a street!
30Sample ProblemA loading dock (SSSQ) has trucks arriving every 36 minutes (0.6 hrs) on average, and the average service (loading / unloading) time is 30 minutes (0.5 hrs). A new conveyer belt system can reduce that to 15 minutes (0.25 hours). Simulate the arrival of 200 trucks to see how performance would be affected by the new system.
31Simulating Exponential Distributions To convert the uniform distribution of the random numbers to an exponential distribution, take the negative natural log of the random numbers.This creates an exponential distribution with an average of 1.00.To get an average of 0.6 (to represent average arrival interval in hours), simply multiply result by 0.6.Thus, the conversion formula is:–ln(rand())*µwhere µ is the mean of the exponential distribution desired.
32Sample Conversion Average: 1 infinity ….. ….. Random -ln(rand()) Random-ln(rand())Average:1infinity…..…..