ENM316E Simulation. The output obtained by running the simulation model once is also random.

Transcription

1 ENM 316E Simulation Lesson 6 Output analysis is the analysis of data generated from simulation study. The purpose of the output analysis To estimate the performance of a system To compare the performance of two or more alternative systems. The output obtained by running the simulation model once is also random. Because of the random number generators The values of the input variables For this reason statistical output analysis is needed. 1

2 First step; To obtain te output by running the simulation model once; Let y 1,y 2,..., y m Y i : waiting time of entity i in queue Y i are random variables and are interdependent, But statistical techniques are based on acceptance of independence. So, n independent trials are carried out (Replication number), Each with length of m (Replication length). ***Independence between experiments is achieved using different initial values.*** Now, statistical methods can be used for output analysis after independence between the tests is achieved. 2

3 Types of Simulation The analysis and design of the simulation experiments depends on the type of simulation. Simulations are divided into two groups based on whether the working length can be determined or not. There are several types of parameters for nonterminating simulations. Simulation Terminating Simulation Non-terminating Simulation Steady-state parameters Steady-state cycle parameters Other Parameters Terminating Simulation Estimate the expected values of the interested performance measures for a predetermined simulation time which will be defined with an event E. [O, T E ] T E time when the event E emerged. (T E can be a random variable) The event E often occurs At a time point when the system is cleaned out. At a time point beyond which no useful information is obtained. At a time point specified by management. EXAMPLE: E = {m clients who completed to wait} 3

4 Terminating Simulation EXAMPLE 1: Consider a simulation of a banking system. The bank opens at 9:00 am and closes at 5:00 pm. The aim of simulationstudymaybetoestimateameasure of the quality of customer service for this period. E = {simulation run length for 8 hour and the system is empty} EXAMPLE 2: Suppose an aerospace manufacturer made a contract to produce 100 airplanes. They have to produce these aircraft within 18 months. The company wants to build up a simulation model in order to choose better production alternative that ensure the lowest cost. E = {100 airplanes have been completed} Terminating Simulation EXAMPLE 3: Consider a production system that operates 16 hours a day (2 shifts). Can the simulation of this system be considered as a terminating simulation? E = {16 hour of simulation time}??!!!! It can not be taken because this production system is actually a continuous system.!!!!the terminating conditions for one day is the starting conditions for the next day. EXAMPLE 4: A retailer that sells one type of product wants to decide how much inventory sholuld be kept at stake over a 120-month period. When the initial inventory level is given, determine how much order quantity should be ordered in each month to minimize the monthly expected cost. E = {Follow system for 120 months} Simulation is started with current inventory level 4

5 Estimate the expected values of the interested performance measures for a simulation time which is considered to be approaching to infinity. It occurs when Designing a new system Changing an existing system ***There are no E events that will determine the duration of the simulation run. A performance measure for such a simulation is called the steady-state parameter if this parameter is a characteristic of the output stochastic process Y 1,Y 2,.... Corollary: If random variable y has a steady-state distribution then it is interested with the estimation of the mean of steady-state parameters: Ey 5

6 EXAMPLE 1: Consider a company that is going to build a new production system. This company wants to determine the long-run average production rate of the system (steady-state parameters) after the situation that all workers know their own jobs and the mechanical difficulties to have been worked out. The Assumptions: a) The system will operate 16 hours a day for 5 days a week b) Loss of production at the end of a shift or in the beginning of the next shift is neglected c) There is no break in production at certain times of the day EXAMPLE 1: The system could be simulated by pasting together 16 hour days with ignoring the free time (8 hours idle time in a day) at end of each day and at weekend. Let N i be the number of parts produced in ith hour. N 1, N 2,... : Stochastic process obtained from simulation models If the stochastic process has a steady-state distribution with corresponding random variable N, the we are interested in estimating the mean v=e(n) 6

7 For many real systems, the stochastic process does not have steady-state distribution. Because the characteristics of the system change continuously over time. For example, in a production system, production scheduling rules and factory setup (i.e. number of machines and set-up) may change over time. On the other hand, the simulation model, which is a summary of the truth, may have steady-state distributions, since the characteristics of the model are assumed not to change over time. In EXAMPLE 1, if the management wants to know how long it takes from the start to the normal state (steady-state), the simulation is a terminating simulation. Because there is an E event that ends the simulation model. E = {simulation until system is running normally }!!! A simulation for a particular system might be either terminating or nonterminating, depending on the objectives of the simulation study. 7

8 Steady-state cycle parameter: Let us consider the stochastic process Y 1, Y 2,... for a nonterminating simulation without steady-state distribution. Let assume that the time axis is divided into consecutive time intervals with equal length. For example, in mentioned production system, a cycle might be an 8-hour shift. Let Y c 1 be a random variable defined on the ith cycle. Assume that Y c 1,Y c 2... are comparable. Suppose that the process Y c 1,Y c has an steady-state distribution of F c. c c y F In this case, the performance measure is called the steady-state cycle parameter if it is a characteristic of Y c such as the mean v C =E(Y C ) ***So, a steady-state cycle parameter is just a steady-state parameter of the appropriate cycle process Y c 1,Y c 2. EXAMPLE 2: Suppose that in the production system of Example 1, at the beginning of the fifth hour in each 8-hour shift there is a half-hour lunch. In this case the hourly output process N 1,N 2,... has no steady-state distribution. Let N c 1 be the average hourly production rate in the ith 8-hour shift (That is a cycle!!!) Then we might be interested in estimating the steady state expected average hourly production rate over a cyle. 8

9 Other parameters: Suppose that for nonterminating simulation, the stochastic process Y 1,Y 2,... have no steady-state distribution. At the same time, there is no proper cycle definition Y c 1,Y c 2,... has no steady-state distribution. This is the case when the parameters of the model change over time. For example, if the rate of arrival of telephone calls in a telephone company varies from week to week, the steady-state parameters will not probably be well identified. In this case there will be fixed amount of data describing how the input parameters change over time. In this case there is an E event to simulate and the analysis techniques used for the terminating simulation can be used in the analysis of the simulation output of such systems. Estimating Means of a Single System for Terminating Simulations First step: Perform n independent replications of the terminating simulation model. In each replication, the same initial condition is used. Independence between replications is achieved by using different initial random values. Assume that we are dealing with a performance measure (X) in the simulation model. 9

10 Estimating Means of a Single System for Terminating Simulations : Estimate value of performance measure in jth replication. **Aim is to obtain point estimation of and also the confidence interval. the point estimate of ~t n-1 in situations where the system variance might not be defined. 1 Estimating Means of a Single System for Terminating Simulations At (1 ) level of confidence; Confidence interval for, It is expected that the confidence interval includes at (1 ) level of confidence. 10

11 Estimating Means of a Single System for Terminating Simulations EXAMPLE 1: Let's estimate the point average for a customer's average waiting within one day with the simulation of the M / M / 1 queue system and the confidence interval at 90% confidence/security level. Replication Avg. Waiting in queue , C. I ; 2.35 According to this result, it is expected that the real average is on the interval 1.71; 2.35 at 90% confidence level. Estimating Means of a Single System for Terminating Simulations EXAMPLE 2: For an inventory system, suppose that we want to obtain a point estimatation and an approximate 95 percent confidence interval for the expected average cost over 120-month planning horizon. 10 independent replications were made and ontained following output. Replication Avg. Cost , With 95% confidence level; C. I ;

12 A disadvantage of fixed size procedure based n replications of a model is that the analyst can not control the half-width of the confidence interval (the sensitivity of the X (n) ). The half-length (, ) for a predetermined n value will depend on the variance V(X) of x j. There are 2 ways to measuring the error in confidence interval with the desired half-width. 1- Absolute error 2- Relative error 1- Absolute error: If x, it can be said that the has an absolute error of. If replications are made until the half-width of the confidence interval at the 1 confidence level is less than or equal to ; 1 P X halfwidthμx halfwidth P X μ half width P X μ β 12

13 In other terms, it will have up to as many absolute error as with probability of 1. So, CI will have the desired precision. Let n* be the total number of replications required to construct the CI with the desired error. It is obtained using the following inequality. Half width, β, So, the estimation for n * : The least number satisfied the condition of Initial replication number should be greater than or at least equal to 2 ( 2. Usually at least 4 or 5 replication is selected. Then, if, additional replication has to be done to obtain the desired sensitivity. After that, C.I. at 1 confidence level should be obtained with using all of the observations/outputs from replications. 13

14 EXAMPLE A: It is desired to estimate the average waiting time in the queue by simulating the M / M / 1 queue system. For this reason n = 5 trials were carried out at the beginning and the following C.I. were obtained. We desire that the sensitivity of the C.I. created for the average waiting time in the queue( ) should be 1. We find the number of replications required to achieve this Z S Z *2.46 n* n* n* n* n* n* n Relative error: x If then we can say that has a relative error of. x Assuming that the value obtained by dividing the half-width of CI to is equal to or less than half the length. x half length 1P x x x P x 14

15 Suppose once again that we have constructed a confidence interval for based on a fixed number of replications n. Assume that estimates of both population mean and variance will not change an approximate replication number, required to obtain a relative error is given as;,, so; the least integer number satisfied the following condition If n* > n then it has to be done additional n*-n trials. where is the adjusted relative error needed to get an actual relative error. 1 EXAMPLE: In the simulation run given in earlier Example A, we have the number of replications required to have a relative error of x 15.58, S 2.46 min. (n) s ,82 12 n*-n =12-5 =7 additional trials to obtain CI with desired relative error. 15