Sheldon m ross (eds ) simulation academic press (2012)
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Sheldon m ross (eds ) simulation academic press (2012)
IntroductionConsider the following situation faced by a pharmacist who is thinking of setting up a small pharmacy where he will fill prescriptions. He Sheldon m ross (eds ) simulation academic press (2012) e plans on opening up at 9 A.M. every weekday and expects that, on average, there will be about 32 prescriptions called in daily before 5 P.M. experience that the time that it will take him to fill a prescription, once he begins working on it. is a random quantity having a mean and standard deviatio Sheldon m ross (eds ) simulation academic press (2012) n of 10 and 4 minutes, respectively. He plans on accepting no new prescriptions after 5 P.M.. although he will remain in the shop past this time if neSheldon m ross (eds ) simulation academic press (2012)
cessary to till all the prescriptions ordered that day. Given this scenario the pharmacist is probably, among other things, interested in the answers IntroductionConsider the following situation faced by a pharmacist who is thinking of setting up a small pharmacy where he will fill prescriptions. He Sheldon m ross (eds ) simulation academic press (2012) P.M.?3What is the average lime it will take him to fill a prescription (taking into account that he cannot begin working on a newly arrived prescription until all earlier arriving ones have been filled)?4What proportion of prescriptions will be filled within 30 minutes?5If he changes his policy on Sheldon m ross (eds ) simulation academic press (2012) accepting all prescriptions between 9 A.M. and 5 P.M., but rather only accepts new ones when there are fewer than five prescriptions still needing toSheldon m ross (eds ) simulation academic press (2012)
be filled, how many prescriptions, on average, will be lost?6I low would the conditions of limiting orders affect the answers to questions I through 4IntroductionConsider the following situation faced by a pharmacist who is thinking of setting up a small pharmacy where he will fill prescriptions. He Sheldon m ross (eds ) simulation academic press (2012) toSimulation. DOI: http://dx.doi.org/10.I016/B978-0-12-415825-2.00001-2© 2013 Elsevier Inc. All rights reserved.121 Introductionmake some reasonably accurate assumptions concerning the preceding scenario. For instance, we must make some assumptions about the probabilistic mechanism that describes t Sheldon m ross (eds ) simulation academic press (2012) he arrivals of the daily average of 32 customers. One possible assumption might be that the arrival rale is. in a probabilistic sense, constant over tSheldon m ross (eds ) simulation academic press (2012)
he day. whereas a second (probably more realistic) possible assumption is that the arrival rale depends on the time of day. We must then specify a proIntroductionConsider the following situation faced by a pharmacist who is thinking of setting up a small pharmacy where he will fill prescriptions. He Sheldon m ross (eds ) simulation academic press (2012) ether or not the service lime of a given prescription always has this distribution or whether it changes as a function of other variables (e.g., the number of waiting prescriptions to be filled or the time of day). That is. we must make probabilistic assumptions about the daily arrival and service t Sheldon m ross (eds ) simulation academic press (2012) imes. We must also decide if the probability law describing a given day changes as a function of the day of the week or whether it remains basically cSheldon m ross (eds ) simulation academic press (2012)
onstant overtime. After these assumptions, and possibly others, have been specified, a probability model of our scenario will have been constructed.OnIntroductionConsider the following situation faced by a pharmacist who is thinking of setting up a small pharmacy where he will fill prescriptions. He Sheldon m ross (eds ) simulation academic press (2012) tions are much too difficult to determine analytically, and so to answer them we usually have to perform a simulation study. Such a study programs the probabilistic mechanism on a computer, and by utilizing “random numbers” it simulates possible occurrences from this model over a large number of day Sheldon m ross (eds ) simulation academic press (2012) s and then utilizes the theory of statistics to estimate the answers to questions such as those given. In other words, the computer program utilizes rSheldon m ross (eds ) simulation academic press (2012)
andom numbers to generate the values of random variables having the assumed probability distributions, which represent the arrival times and the serviIntroductionConsider the following situation faced by a pharmacist who is thinking of setting up a small pharmacy where he will fill prescriptions. He Sheldon m ross (eds ) simulation academic press (2012) cal techniques to provide estimated answers—for example, if out of 1000 simulated days there are 122 in which the pharmacist is still working at 5:30. we would estimate that the answer to question 2 is 0.122.In order to be able to execute such an analysis, one must have some knowledge of probability Sheldon m ross (eds ) simulation academic press (2012) so as to decide on certain probability distributions and questions such as whether appropriate random variables are to be assumed independent or not.Sheldon m ross (eds ) simulation academic press (2012)
A review of probability is provided in Chapter 2. The bases of a simulation study are so-called random numbers. A discussion of these quantities and IntroductionConsider the following situation faced by a pharmacist who is thinking of setting up a small pharmacy where he will fill prescriptions. He Sheldon m ross (eds ) simulation academic press (2012) bles having arbitrary distributions. Discrete distributions are considered in Chapter 4 and continuous ones in Chapter 5. Chapter 6 introduces the multivariate normal distribution, and shows how to generate random variables having this joint distribution. Copulas, useful for modeling the joint distr Sheldon m ross (eds ) simulation academic press (2012) ibutions of random variables, are also introduced in Chapter 6. After completing Chapter 6. the reader should have some insight into the constructionSheldon m ross (eds ) simulation academic press (2012)
of a probability model for a given system and also how to use random numbers to generate the values of random quantities related to this model. The usIntroductionConsider the following situation faced by a pharmacist who is thinking of setting up a small pharmacy where he will fill prescriptions. He Sheldon m ross (eds ) simulation academic press (2012) d in Chapter 7, where we present (he concept of “discrete events" and indicate how to utilize these entities to obtain a systematic approach lo simulating systems. The discrete event simulation approach leads to a computer program, which can be written in whatever language the reader is comfortable Sheldon m ross (eds ) simulation academic press (2012) in. that simulates the system a large number of limes. Some hints concerning the verification of this program—to ascertain lhal il is actually doing wSheldon m ross (eds ) simulation academic press (2012)
hat is desired—arc also given in Chapter 7. The use of the outputs of a simulation study to answer probabilistic questions concerning the model necessIntroductionConsider the following situation faced by a pharmacist who is thinking of setting up a small pharmacy where he will fill prescriptions. He Sheldon m ross (eds ) simulation academic press (2012) s in statistics and continues toward "bootstrap statistics.” which is quite useful in simulation. Our study of statistics indicates the importance of the variance of the estimators obtained from a simulation study as an indication of the efficiency of the simulation. In particular, the smaller this Sheldon m ross (eds ) simulation academic press (2012) variance is. the smaller is the amount of simulation needed to obtain a fixed precision. As a result we are led, in Chapters 9 and 10. to ways of obtaSheldon m ross (eds ) simulation academic press (2012)
ining new estimators that are improvements over the raw simulation estimators because they have reduced variances. This topic of variance reduction isIntroductionConsider the following situation faced by a pharmacist who is thinking of setting up a small pharmacy where he will fill prescriptions. He Sheldon m ross (eds ) simulation academic press (2012) lation to verify, when some real-life data are available, the appropriateness of the probability model ( which we have simulated) to the real-world situation. Chapter 12 introduces the important topic of Markov chain Monte Carlo methods. The use of these methods has. in recent years, greatly expande Sheldon m ross (eds ) simulation academic press (2012) d the class of problems that can be attacked by simulation.Exercises1The following data yield the arrival limes and service limes lhal each customer wSheldon m ross (eds ) simulation academic press (2012)
ill require, for the first 13 customers al a single server system. Upon arrival, a customer either enters service if the server is free or joins the IntroductionConsider the following situation faced by a pharmacist who is thinking of setting up a small pharmacy where he will fill prescriptions. He Sheldon m ross (eds ) simulation academic press (2012) val Times:12 31 63 95 99 154 198 221 304 346 411 455 537Service Times: 40 32 55 48 18 50 47 18 28 54 40 72 12(a)Determine the departure limes of lhese 13 customers.(b)Repeal (a) when there arc two servers and a customer can be served by cither one.(c)Repeal (a) under the new assumption lhal when the Sheldon m ross (eds ) simulation academic press (2012) server completes a service, the next customer to enter service is the one who has been w ailing the least lime.41 Introduction2Consider a service staSheldon m ross (eds ) simulation academic press (2012)
tion where customers arrive and are served in their order of arrival. Let „. s„, and Dn denote, respectively, the arrival time, the service lime, andIntroductionConsider the following situation faced by a pharmacist who is thinking of setting up a small pharmacy where he will fill prescriptions. He Sheldon m ross (eds ) simulation academic press (2012) or II > 0Dn — s„ = Maximum {/V, D„_)}(b)Determine the corresponding recursion formula when there arc two servers.(c)Determine the corresponding recursion formula when there arc k servers.(d)Write a computer program to determine the departure times as a function of the arrival and sen ice times and u Sheldon m ross (eds ) simulation academic press (2012) se it to check your answers in parts (a) and (b) of Exercise 1.Elements of Probability2.1 Sample Space and EventsGọi ngay
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