Chapter 21 SimulationDescription21.1 Basic TerminologySlide 5Slide 621.2 An Example of a Discrete-Event SimulationSlide 8Slide 9Slide 10Slide 11Slide 12Slide 1321.3 Random Numbers and Monte Carol SimulationSlide 15Slide 16Random Number GeneratorsSlide 18Slide 19Computer Generation of Random NumbersSlide 2121.4 An Example of Monte Carlo Simulation21.5 Simulations with Continuous Random VariablesSlide 24Inverse Transformation MethodSlide 26Slide 27Slide 28Slide 29Acceptance – Rejection MethodSlide 31Slide 32Slide 33Direct and Convolution Methods for the Normal DistributionThe Convolution AlgorithmThe Direct MethodSlide 3721.6 An Example of a Stochastic SimulationSlide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 5321.7 Statistical Analysis of SimulationsSlide 55Slide 56Slide 57Simulation TypesSlide 59Slide 60Slide 61Slide 6221.8 Simulation LanguagesSlide 64Slide 65Slide 66Slide 6721.9 The Simulation ProcessSlide 69Slide 70Slide 71Chapter 21Simulationto accompanyOperations Research: Applications and Algorithms 4th edition by Wayne L. WinstonCopyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.2DescriptionSimulation is a very powerful and widely used management science technique for the analysis and study of complex systems.Simulation may be defined as a technique that imitates the operation of a real-world system as it evolves over time. This is normally done by developing a simulation model. A simulation model usually takes the form of a set of assumptions about the operation of the system, expressed as mathematical or logical relations between the objects of interest in the system.Simulation has its advantages and disadvantages. We will focus our attention on simulation models and the simulation technique.321.1 Basic TerminologyIn most simulation studies, we are concerned with the simulation of some system.Thus, in order to model a system, we must understand the concept of a system.Definition: A system is a collection of entities that act and interact toward the accomplishment of some logical end.Systems generally tend to be dynamic – their status changes over time. To describe this status, we use the concept of the state of a system.4 Definition: The state of a system is the collection of variables necessary to describe the status of the system at any given time.Customers arrive and depart, the status of the system changes. To describe these changes in status, we require a set of variables called the state variables.In a system, an object of interest is called an entity, and any properties of an entity are called attributes.Systems may be classified as discrete or continuous.5 Definition: A discrete system is one in which the state variables change continuously over time. A bank is an example of a discrete system.Definition: A continuous system is one in which the state variables change continuously over time.There are two types of simulation models, static and dynamic.Definition: A static simulation model is a representation of a system at a particular point in time.6 We usually refer to a static simulation as a Monte Carlo simulation.Definition: A dynamic simulation is a representation of a system as it evolves over time.Within these two classifications, a simulation may be deterministic or stochastic.A deterministic simulation model is one that contains no random variables; a stochastic simulation model contains one or more random variables.721.2 An Example of a Discrete-Event SimulationTo simulate a queuing system, we first have to describe it.We assume arrivals are drawn from an infinite calling population.There is unlimited waiting room capacity, and customers will be serve in the order of their arrival (FCFS).Arrivals occur one at a time in a random fashion.All arrivals are eventually served with the distribution of service teams as shown in the book.8 Service times are also assumed to be random. After service, all customers return to the calling population.For this example, we use the following variables to define the state of the system: (1) the number of customers in the system; (2) the status of the server – that is, whether the server is busy or idle; and (3)the time of the next arrival.An event is defined as a situation that causes the state of the system to change instantaneously.9 All the information about them is maintained in a list called the event list.Time in a simulation is maintained using a variable called the clock time.We begin this simulation with an empty system and arbitrarily assume that our first event, an arrival, takes place at clock time 0.Next we schedule the departure time of the first customer.Departure time = clock time now + generated service time10 Also, we now schedule the next arrival into the system by randomly generating an interarrival time from the interarrival time distribution and setting the arrival time asArrival time = clock time now + generated interarrival timeBoth these events are their scheduled times are maintained on the event list.This approach of simulation is called the next-event time-advance mechanism, because of the way the clock time is updated. We advance the simulation clock to the time of the most imminent event.11 As we move from event to event, we carry out the appropriate actions for each event, including any scheduling of future events.The jump to the next event in the next-event mechanism may be a large one or a small one; that is, the jumps in this method are variable in size.We contrast this approach with the fixed-increment time-advance method.With this method, we advance the simulation clock in increments of Δt time units, where Δt is some appropriate time unit, usually 1 time unit.12 For most models, however, the next event mechanism tends to be more efficient computationally.Consequently, we use only the next-event approach for the development of the models for the rest of the chapter.To demonstrate the simulation model, we need to define several variables:TM = clock time of the simulationAT = scheduled time of the next arrival13 DT = scheduled time of the next departureSS = status of the server (1=busy, 0=idle)WL = length of the waiting lineMX = length (in time units) of a simulation runWe now begin the simulation by initializing all the variables. Details of the example are found