Simulation_FifthEdition_SheldonM.Ross

Simulation

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Introduction

Consider the following situation faced by a pharmacist who is thinking of setting

up a small pharmacy where he will ﬁll prescriptions. He 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 deviation 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 necessary to ﬁll all the prescriptions ordered that day. Given this

scenario the pharmacist is probably, among other things, interested in the answers

to the following questions:

1. What is the average time that he will depart his store at night?

2. What proportion of days will he still be working at 5:30 p.m.?

3. What is the average time 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)?

4. What proportion of prescriptions will be ﬁlled within 30 minutes?

5. If he changes his policy on 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 to be ﬁlled, how many prescriptions, on

average, will be lost?

6. How would the conditions of limiting orders affect the a nswers to questions

1 through 4?

In order to employ mathematics to analyze this situation and answer the

questions, we ﬁrst construct a probability model. To do this it is necessary to

Simulation. DOI: http://dx.doi.org/10.1016/B978-0-12-415825-2.00001-2

2 1 Introduction

make some reasonably accurate assumptions concerning the preceding scenario.

For instance, we must make some assumptions about the probabilistic mechanism

that describes the arrivals of the daily average of 32 customers. One possible

assumption might be that the arrival rate is, in a probabilistic sense, constant over

the day, whereas a second (probably more realistic) possible assumption is that

the arrival rate depends on the time of day. We must then specify a probability

distribution (having mean 10 and standard deviation 4) for the time it takes to

service a prescription, and we must make assumptions about whether or not the

service time 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 times. 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 constant over time. After these assumptions, and possibly others,

have been speciﬁed, a probability model of our scenario will have been constructed.

Once a probability model has been constructed, the answers to the questions

can, in theory, be analytically determined. However, in practice, these questions

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 days and then utilizes the theory

of statistics to estimate the answers to questions such as those given. In other words,

the computer program utilizes random numbers to generate the values of random

variables having the assumed probability distributions, which represent the arrival

times and the service times of prescriptions. Using these values, it determines over

many days the quantities of interest related to the questions. It then uses statistical

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 so as to decide on certain probability distributions and questions such

as whether appropriate random variables are to be assumed independent or not.

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 how they are

computer generated is presented in Chapter 3. Chapters 4 and 5 show how one can

use random numbers to generate the values of random variables 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 distributions of random variables, are also introduced

in Chapter 6. After completing Chapter 6, the reader should h ave some insight

into the construction 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 use of these generated values to track the system as it evolves

Exercises 3

continuously over time—that is, the actual simulation of the system—is discussed

in Chapter 7, where we present the concept of “discrete events” and indicate how

to utilize these entities to obtain a systematic approach to simulating systems.

The discrete event simulation approach leads to a computer program, which can

be written in whatever language the reader is comfortable in, that simulates the

system a large number of times. Some hints concerning the veriﬁcation of this

program—to ascertain that it is actually doing what is desired—are also given in

Chapter 7. The use of the outputs of a simulation study to answer probabilistic

questions concerning the model necessitates the use of the theory of statistics, and

this subject is introduced in Chapter 8. This chapter starts with the simplest and

most basic concepts 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 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 obtaining new estimators that are improvements

over the raw simulation estimators because they have reduced variances. This

topic of variance reduction is extremely important in a simulation study because

it can substantially improve its efﬁciency. Chapter 11 shows how one can use

the results of a simulation 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 expanded

the class of problems that can be attacked by simulation.

Exercises

1. The following data yield the arrival times and service times that each customer

will require, for the ﬁrst 13 customers at a single server system. Upon arrival,

a customer either enters service if the server is free or joins the waiting line.

When the server completes work on a customer, the next one in line (i.e., the

one who has been waiting the longest) enters service.

Arrival Times: 12 31 63 95 99 154 198 221 304 346 411 455 537

Service Times: 40 32 55 48 18 50 47 18 28 54 40 72 12

(a) Determine the departure times of these 13 customers.

(b) Repeat (a) when there are two servers and a customer can be served by either

one.

service, the next customer to enter service is the one who has been waiting

the least time.

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Simulation_Fifth Edition_Sheldon M. Ross

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Simulation_Fifth Edition_Sheldon M. Ross

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