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# Introduction to modeling_and_simulation

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### Introduction to modeling_and_simulation

1. 1. Proceedings of the 1997 Winter Simulation Conferenceed. S. Andradóttir, K. J. Healy, D. H. Withers, and B. L. Nelson INTRODUCTION TO MODELING AND SIMULATION Anu Maria State University of New York at Binghamton Department of Systems Science and Industrial Engineering Binghamton, NY 13902-6000, U.S.A.ABSTRACT output variables is probabilistic); static (time is not taken into account) or dynamic (time-varying interactionsThis introductory tutorial is an overview of simulation among variables are taken into account). Typically,modeling and analysis. Many critical questions are simulation models are stochastic and dynamic.answered in the paper. What is modeling? What issimulation? What is simulation modeling and analysis? 2 WHAT IS SIMULATION?What types of problems are suitable for simulation? Howto select simulation software? What are the benefits and A simulation of a system is the operation of a model ofpitfalls in modeling and simulation? The intended the system. The model can be reconfigured andaudience is those unfamiliar with the area of discrete experimented with; usually, this is impossible, tooevent simulation as well as beginners looking for an expensive or impractical to do in the system it represents.overview of the area. This includes anyone who is The operation of the model can be studied, and hence,involved in system design and modification - system properties concerning the behavior of the actual systemanalysts, management personnel, engineers, military or its subsystem can be inferred. In its broadest sense,planners, economists, banking analysts, and computer simulation is a tool to evaluate the performance of ascientists. Familiarity with probability and statistics is system, existing or proposed, under differentassumed. configurations of interest and over long periods of real time.1 WHAT IS MODELING? Simulation is used before an existing system is altered or a new system built, to reduce the chances ofModeling is the process of producing a model; a model failure to meet specifications, to eliminate unforeseenis a representation of the construction and working of bottlenecks, to prevent under or over-utilization ofsome system of interest. A model is similar to but resources, and to optimize system performance. Forsimpler than the system it represents. One purpose of a instance, simulation can be used to answer questionsmodel is to enable the analyst to predict the effect of like: What is the best design for a newchanges to the system. On the one hand, a model should telecommunications network? What are the associatedbe a close approximation to the real system and resource requirements? How will a telecommunicationincorporate most of its salient features. On the other network perform when the traffic load increases by 50%?hand, it should not be so complex that it is impossible to How will a new routing algorithm affect itsunderstand and experiment with it. A good model is a performance? Which network protocol optimizesjudicious tradeoff between realism and simplicity. network performance? What will be the impact of a linkSimulation practitioners recommend increasing the failure?complexity of a model iteratively. An important issue in The subject of this tutorial is discrete eventmodeling is model validity. Model validation techniques simulation in which the central assumption is that theinclude simulating the model under known input system changes instantaneously in response to certainconditions and comparing model output with system discrete events. For instance, in an M/M/1 queue - aoutput. single server queuing process in which time between Generally, a model intended for a simulation study arrivals and service time are exponential - an arrivalis a mathematical model developed with the help of causes the system to change instantaneously. On thesimulation software. Mathematical model classifications other hand, continuous simulators, like flight simulatorsinclude deterministic (input and output variables are and weather simulators, attempt to quantify the changesfixed values) or stochastic (at least one of the input or in a system continuously over time in response to 7
2. 2. 8 Mariacontrols. Discrete event simulation is less detailed Step 5. Validate the model.(coarser in its smallest time unit) than continuous Step 6. Document model for future use.simulation but it is much simpler to implement, and Step 7. Select appropriate experimental design.hence, is used in a wide variety of situations. Step 8. Establish experimental conditions for runs. Figure 1 is a schematic of a simulation study. The Step 9. Perform simulation runs.iterative nature of the process is indicated by the system Step 10. Interpret and present results.under study becoming the altered system which then Step 11. Recommend further course of action.becomes the system under study and the cycle repeats. In Although this is a logical ordering of steps in aa simulation study, human decision making is required at simulation study, many iterations at various sub-stagesall stages, namely, model development, experiment may be required before the objectives of a simulationdesign, output analysis, conclusion formulation, and study are achieved. Not all the steps may be possiblemaking decisions to alter the system under study. The and/or required. On the other hand, additional steps mayonly stage where human intervention is not required is have to be performed. The next three sections describethe running of the simulations, which most simulation these steps in detail.software packages perform efficiently. The importantpoint is that powerful simulation software is merely a 3 HOW TO DEVELOP A SIMULATIONhygiene factor - its absence can hurt a simulation study MODEL?but its presence will not ensure success. Experiencedproblem formulators and simulation modelers and Simulation models consist of the following components:analysts are indispensable for a successful simulation system entities, input variables, performance measures,study. and functional relationships. For instance in a simulation model of an M/M/1 queue, the server and the queue are system entities, arrival rate and service rate are input Real Simulation variables, mean wait time and maximum queue length World Study are performance measures, and time in system = wait time + service time is an example of a functional relationship. Almost all simulation software packages System Simulation provide constructs to model each of the above Under Model components. Modeling is arguably the most important Study part of a simulation study. Indeed, a simulation study is as good as the simulation model. Simulation modeling comprises the following steps: Simulation Experiment Step 1. Identify the problem. Enumerate problems with an existing system. Produce requirements for a proposed system. Simulation Analysis Step 2. Formulate the problem. Select the bounds of the system, the problem or a part thereof, to be studied. Define overall objective of the study and a few Conclusions specific issues to be addressed. Define performance measures - quantitative criteria on the basis of which Altered different system configurations will be compared and System ranked. Identify, briefly at this stage, the configurations of interest and formulate hypotheses about system performance. Decide the time frame of the study, i.e., Figure 1: Simulation Study Schematic will the model be used for a one-time decision (e.g., capital expenditure) or over a period of time on a regular The steps involved in developing a simulation basis (e.g., air traffic scheduling). Identify the end usermodel, designing a simulation experiment, and of the simulation model, e.g., corporate managementperforming simulation analysis are: versus a production supervisor. Problems must be Step 1. Identify the problem. formulated as precisely as possible. Step 2. Formulate the problem. Step 3. Collect and process real system data. Step 3. Collect and process real system data. Collect data on system specifications (e.g., bandwidth for Step 4. Formulate and develop a model. a communication network), input variables, as well as
3. 3. Introduction to Modeling and Simulation 9performance of the existing system. Identify sources of variables of a simulation model so that we may observerandomness in the system, i.e., the stochastic input and identify the reasons for changes in the performancevariables. Select an appropriate input probability measures. The number of experiments in a simulationdistribution for each stochastic input variable and study is greater than or equal to the number of questionsestimate corresponding parameter(s). being asked about the model (e.g., Is there a significant Software packages for distribution fitting and difference between the mean delay in communicationselection include ExpertFit, BestFit, and add-ons in some networks A and B?, Which network has the least delay:standard statistical packages. These aids combine A, B, or C? How will a new routing algorithm affect thegoodness-of-fit tests, e.g., χ2 test, Kolmogorov-Smirnov performance of network B?). Design of a simulationtest, and Anderson-Darling test, and parameter experiment involves answering the question: what dataestimation in a user friendly format. need to be obtained, in what form, and how much? The Standard distributions, e.g., exponential, Poisson, following steps illustrate the process of designing anormal, hyperexponential, etc., are easy to model and simulation experiment.simulate. Although most simulation software packagesinclude many distributions as a standard feature, issues Step 7. Select appropriate experimental design.relating to random number generators and generating Select a performance measure, a few input variables thatrandom variates from various distributions are pertinent are likely to influence it, and the levels of each inputand should be looked into. Empirical distributions are variable. When the number of possible configurationsused when standard distributions are not appropriate or (product of the number of input variables and the levelsdo not fit the available system data. Triangular, uniform of each input variable) is large and the simulation modelor normal distribution is used as a first guess when no is complex, common second-order design classesdata are available. For a detailed treatment of probability including central composite, Box-Behnken, and full-distributions see Maria and Zhang (1997). factorial should be considered. Document the experimental design. Step 4. Formulate and develop a model. Developschematics and network diagrams of the system (How do Step 8. Establish experimental conditions for runs.entities flow through the system?). Translate these Address the question of obtaining accurate informationconceptual models to simulation software acceptable and the most information from each run. Determine if theform. Verify that the simulation model executes as system is stationary (performance measure does notintended. Verification techniques include traces, varying change over time) or non-stationary (performanceinput parameters over their acceptable range and measure changes over time). Generally, in stationarychecking the output, substituting constants for random systems, steady-state behavior of the response variable isvariables and manually checking results, and animation. of interest. Ascertain whether a terminating or a non- terminating simulation run is appropriate. Select the run Step 5. Validate the model. Compare the models length. Select appropriate starting conditions (e.g., emptyperformance under known conditions with the and idle, five customers in queue at time 0). Select theperformance of the real system. Perform statistical length of the warm-up period, if required. Decide theinference tests and get the model examined by system number of independent runs - each run uses a differentexperts. Assess the confidence that the end user places random number stream and the same starting conditions -on the model and address problems if any. For major by considering output data sample size. Sample size mustsimulation studies, experienced consultants advocate a be large enough (at least 3-5 runs for each configuration)structured presentation of the model by the simulation to provide the required confidence in the performanceanalyst(s) before an audience of management and system measure estimates. Alternately, use common randomexperts. This not only ensures that the model numbers to compare alternative configurations by using aassumptions are correct, complete and consistent, but separate random number stream for each samplingalso enhances confidence in the model. process in a configuration. Identify output data most likely to be correlated. Step 6. Document model for future use. Documentobjectives, assumptions and input variables in detail. Step 9. Perform simulation runs. Perform runs according to steps 7-8 above.4 HOW TO DESIGN A SIMULATIONEXPERIMENT? 5 HOW TO PERFORM SIMULATION ANALYSIS?A simulation experiment is a test or a series of tests inwhich meaningful changes are made to the input
4. 4. 10 MariaMost simulation packages provide run statistics (mean, 6 AN EXAMPLEstandard deviation, minimum value, maximum value) onthe performance measures, e.g., wait time (non-time A machine shop contains two drills, one straightener, andpersistent statistic), inventory on hand (time persistent one finishing operator. Figure 2 shows a schematic of thestatistic). Let the mean wait time in an M/M/1 queue machine shop. Two types of parts enter the machine shop.observed from n runs be W1 ,W2 ,...,Wn . It is important tounderstand that the mean wait time W is a random Straightenervariable and the objective of output analysis is toestimate the true mean of W and to quantify itsvariability. Notwithstanding the facts that there are no datacollection errors in simulation, the underlying model is Drill #1fully known, and replications and configurations are usercontrolled, simulation results are difficult to interpret. An Finishingobservation may be due to system characteristics or just Drill #2 Operatora random occurrence. Normally, statistical inference canassess the significance of an observed phenomenon, but Legend:most statistical inference techniques assume Type 1 partsindependent, identically distributed (iid) data. Most types Type 2 partsof simulation data are autocorrelated, and hence, do notsatisfy this assumption. Analysis of simulation output Figure 2: Schematic of the Machine Shopdata consists of the following steps. Type 1 parts require drilling, straightening, and finishing in sequence. Type 2 parts require only drilling and Step 10. Interpret and present results. Compute finishing. The frequency of arrival and the time to benumerical estimates (e.g., mean, confidence intervals) of routed to the drilling area are deterministic for both typesthe desired performance measure for each configuration of parts.of interest. To obtain confidence intervals for the meanof autocorrelated data, the technique of batch means canbe used. In batch means, original contiguous data set Step 1. Identify the problem. The utilization offrom a run is replaced with a smaller data set containing drills, straightener, and finishing operator needs to bethe means of contiguous batches of original observations. assessed. In addition, the following modification to theThe assumption that batch means are independent may original system is of interest: the frequency of arrival ofnot always be true; increasing total sample size and both parts is exponential with the same respective meansincreasing the batch length may help. as in the original system. Test hypotheses about system performance.Construct graphical displays (e.g., pie charts, histograms) Step 2. Formulate the problem. The objective is toof the output data. Document results and conclusions. obtain the utilization of drills, straightener, and finishing operator for the original system and the modification. The Step 11. Recommend further course of action. This assumptions include:may include further experiments to increase the precision ♦ The two drills are identicaland reduce the bias of estimators, to perform sensitivityanalyses, etc. ♦ There is no material handling time between the three operations. ♦ Machine availability implies operator availability. ♦ Parts are processed on a FIFO basis. ♦ All times are in minutes. Step 3. Collect and process real system data. At the job shop, a Type 1 part arrives every 30 minutes, and a Type 2 part arrives every 20 minutes. It takes 2 minutes to route a Type 1 part and 10 minutes to route a Type 2 part to the drilling area. Parts wait in a queue till one of the two drilling machines becomes available. After drilling, Type 1 parts are routed to the straightener and Type 2 parts are
5. 5. Introduction to Modeling and Simulation 11routed to the finishing operator. After straightening, Type underutilized. Consequently, the average utilization did not1 parts are routed to the finishing operator. change substantially between the original system and the The operation times for either part were determined to modification; the standard deviation of the drillingbe as follows. Drilling time is normally distributed with operation seems to have increased because of the increasedmean 10.0 and standard deviation 1.0. Straightening time randomness in the modification. The statisticalis exponentially distributed with a mean of 15.0. Finishing significance of these observations can be determined byrequires 5 minutes per part. computing confidence intervals on the mean utilization of the original and modified systems. Step 4. Formulate and develop a model. A modelof the system and the modification was developed using Step 11. Recommend further course of action. Othera simulation package. A trace verified that the parts performance measures of interest may be: throughput offlowed through the job shop as expected. parts for the system, mean time in system for both types of parts, average and maximum queue lengths for each Step 5. Validate the model. The utilization for a operation. Other modifications of interest may be: thesufficiently long run of the original system was judged to flow of parts to the machine shop doubles, the finishingbe reasonable by the machine shop operators. operation will be repeated for 10% of the products on a probabilistic basis. Step 6. Document model for future use. Themodels of the original system and the modification were 7 WHAT MAKES A PROBLEM SUITABLE FORdocumented as thoroughly as possible. SIMULATION MODELING AND ANALYSIS? Step 7. Select appropriate experimental design. In general, whenever there is a need to model andThe original system and the modification described analyze randomness in a system, simulation is the tool ofabove were studied. choice. More specifically, situations in which simulation modeling and analysis is used include the following: Step 8. Establish experimental conditions for runs. ♦ It is impossible or extremely expensive to observeEach model was run three times for 4000 minutes and certain processes in the real world, e.g., next yearsstatistical registers were cleared at time 1000, so the cancer statistics, performance of the next spacestatistics below were collected on the time interval [1000, shuttle, and the effect of Internet advertising on a4000]. At the beginning of a simulation run, there were no companys sales.parts in the machine shop. ♦ Problems in which mathematical model can be formulated but analytic solutions are either Step 9. Perform simulation runs. Runs were impossible (e.g., job shop scheduling problem, high-performed as specified in Step 8 above. order difference equations) or too complicated (e.g., complex systems like the stock market, and large Step 10. Interpret and present results. Table 1 scale queuing models).contains the utilization statistics of the three operations for ♦ It is impossible or extremely expensive to validatethe original system and the modification (in parentheses). the mathematical model describing the system, e.g., due to insufficient data. Table 1: Utilization Statistics Applications of simulation abound in the areas of Drilling Straightening Finishing government, defense, computer and communication Mean Run #1 0.83 (0.78) 0.51 (0.58) 0.42 (0.39) systems, manufacturing, transportation (air traffic Mean Run #2 0.82 (0.90) 0.52 (0.49) 0.41 (0.45) Mean Run #3 0.84 (0.81) 0.42 (0.56) 0.42 (0.40) control), health care, ecology and environment, Std. Dev. Run #1 0.69 (0.75) 0.50 (0.49) 0.49 (0.49) sociological and behavioral studies, biosciences, Std. Dev. Run #2 0.68 (0.78) 0.50 (0.50) 0.49 (0.50) epidemiology, services (bank teller scheduling), Std. Dev. Run #3 0.69 (0.76) 0.49 (0.50) 0.49 (0.49) economics and business analysis.Mean utilization represents the fraction of time a server is 8 HOW TO SELECT SIMULATIONbusy, i.e., busy time/total time. Furthermore, the average SOFTWARE?utilization output for drilling must be divided by thenumber of drills in order to get the utilization per drill. Although a simulation model can be built using generalEach drill is busy about 40% of the time and straightening purpose programming languages which are familiar toand finishing operations are busy about half the time. This the analyst, available over a wide variety of platforms,implies that for the given work load, the system is and less expensive, most simulation studies today are implemented using a simulation package. The
6. 6. 12 Mariaadvantages are reduced programming requirements; interest, and observing the systems operation innatural framework for simulation modeling; conceptual detail over long periods of time.guidance; automated gathering of statistics; graphic ♦ Test hypotheses about the system for feasibility.symbolism for communication; animation; and ♦ Compress time to observe certain phenomena overincreasingly, flexibility to change the model. There are long periods or expand time to observe a complexhundreds of simulation products on the market, many phenomenon in detail.with price tags of \$15,000 or more. Naturally, the ♦ Study the effects of certain informational,question of how to select the best simulation software for organizational, environmental and policy changes onan application arises. Metrics for evaluation include the operation of a system by altering the systemsmodeling flexibility, ease of use, modeling structure model; this can be done without disrupting the real(hierarchical v/s flat; object-oriented v/s nested), code system and significantly reduces the risk ofreusability, graphic user interface, animation, dynamic experimenting with the real system.business graphics, hardware and software requirements, ♦ Experiment with new or unknown situations aboutstatistical capabilities, output reports and graphical plots, which only weak information is available.customer support, and documentation. ♦ Identify the "driving" variables - ones that The two types of simulation packages are simulation performance measures are most sensitive to - and thelanguages and application-oriented simulators (Table 2). inter-relationships among them.Simulation languages offer more flexibility than the ♦ Identify bottlenecks in the flow of entities (material,application-oriented simulators. On the other hand, people, etc.) or information.languages require varying amounts of programming ♦ Use multiple performance metrics for analyzingexpertise. Application-oriented simulators are easier to system configurations.learn and have modeling constructs closely related to the ♦ Employ a systems approach to problem solving.application. Most simulation packages incorporate ♦ Develop well designed and robust systems andanimation which is excellent for communication and can reduce system development time.be used to debug the simulation program; a "correctlooking" animation, however, is not a guarantee of a 10 WHAT ARE SOME PITFALLS TO GUARDvalid model. More importantly, animation is not a AGAINST IN SIMULATION?substitute for output analysis. Simulation can be a time consuming and complex Table 2: Simulation Packages exercise, from modeling through output analysis, that Type Of Examples Simulation necessitates the involvement of resident experts and Package decision makers in the entire process. Following is a Simulation Arena (previously SIMAN), AweSim! (previously checklist of pitfalls to guard against. languages SLAM II), Extend, GPSS, Micro Saint, ♦ Unclear objective. SIMSCRIPT, SLX Object-oriented software: MODSIM III, SIMPLE++ ♦ Using simulation when an analytic solution is Animation software: Proof Animation appropriate. Application Manufacturing: AutoMod, Extend+MFG, ♦ Invalid model. -Oriented FACTOR/AIM, ManSim/X, MP\$IM, ♦ Simulation model too complex or too simple. Simulators ProModel, QUEST, Taylor II, WITNESS Communications/computer: COMNET III, ♦ Erroneous assumptions. NETWORK II.5, OPNET Modeler, OPNET ♦ Undocumented assumptions. This is extremely Planner, SES/Strategizer, SES/workbench important and it is strongly suggested that Business: BP\$IM, Extend+BPR, ProcessModel, ServiceModel, SIMPROCESS, Time machine assumptions made at each stage of the simulation Health Care: MedModel modeling and analysis exercise be documented thoroughly.9 BENEFITS OF SIMULATION MODELING ♦ Using the wrong input probability distribution.AND ANALYSIS ♦ Replacing a distribution (stochastic) by its mean (deterministic).According to practitioners, simulation modeling and ♦ Using the wrong performance measure.analysis is one of the most frequently used operations ♦ Bugs in the simulation program.research techniques. When used judiciously, simulation ♦ Using standard statistical formulas that assumemodeling and analysis makes it possible to: independence in simulation output analysis.♦ Obtain a better understanding of the system by ♦ Initial bias in output data. developing a mathematical model of a system of ♦ Making one simulation run for a configuration.
7. 7. Introduction to Modeling and Simulation 13♦ Poor schedule and budget planning.♦ Poor communication among the personnel involved in the simulation study.REFERENCESBanks, J., J. S. Carson, II, and B. L. Nelson. 1996. Discrete-Event System Simulation, Second Edition, Prentice Hall.Bratley, P., B. L. Fox, and L. E. Schrage. 1987. A Guide to Simulation, Second Edition, Springer-Verlag.Fishwick, P. A. 1995. Simulation Model Design and Execution: Building Digital Worlds, Prentice-Hall.Freund, J. E. 1992. Mathematical Statistics, Fifth Edition, Prentice-Hall.Hogg, R. V., and A. T. Craig. 1995. Introduction to Mathematical Statistics, Fifth Edition, Prentice-Hall.Kleijnen, J. P. C. 1987. Statistical Tools for Simulation Practitioners, Marcel Dekker, New York.Law, A. M., and W. D. Kelton. 1991. Simulation Modeling and Analysis, Second Edition, McGraw-Hill.Law, A. M., and M. G. McComas. 1991. Secrets of Successful Simulation Studies, Proceedings of the 1991 Winter Simulation Conference, ed. J. M. Charnes, D. M. Morrice, D. T. Brunner, and J. J. Swain, 21-27. Institute of Electrical and Electronics Engineers, Piscataway, New Jersey.Maria, A., and L. Zhang. 1997. Probability Distributions, Version 1.0, July 1997, Monograph, Department of Systems Science and Industrial Engineering, SUNY at Binghamton, Binghamton, NY 13902.Montgomery, D. C. 1997. Design and Analysis of Experiments, Third Edition, John Wiley.Naylor, T. H., J. L. Balintfy, D. S. Burdick, and K. Chu. 1966. Computer Simulation Techniques, John Wiley.Nelson, B. L. 1995. Stochastic Modeling: Analysis and Simulation, McGraw-Hill.AUTHOR BIOGRAPHYANU MARIA is an assistant professor in the departmentof Systems Science & Industrial Engineering at the StateUniversity of New York at Binghamton. She receivedher PhD in Industrial Engineering from the University ofOklahoma. Her research interests include optimizing theperformance of materials used in electronic packaging(including solder paste, conductive adhesives, andunderfills), simulation optimization techniques, geneticsbased algorithms for optimization of problems with alarge number of continuous variables, multi criteriaoptimization, simulation, and interior-point methods.