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Posted: November 24th, 2022

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Simulation studies are important in investigating various characteristics of a system or process.
Simulation studies are important in investigating various characteristics of a system or process.
Simulation studies are important in investigating various characteristics of a system or process. They are generally employed when the mathematical analysis necessary to answer important questions is too complicated to yield closed form solutions. For example, in a system where the time between successive customer arrivals has a particular pdf and the service time of any particular customer has another particular pdf, simulation can provide information about the probability that the system is empty when a customer arrives, the expected number of customers in the system, and the expected waiting time in queue. Such studies depend on being able to generate observations from a speci ed probability distribution. The rejection method gives a way of generating an observation from a pdf f(# ) when we have a way of generating an observation from g(#) and the ratio f(x)/g(x) is bounded, that is, c for some nite c. The steps are as follows:

1. Use a software package s random number generator to obtain a value u from a uniform distribution on the interval from 0 to 1. 2. Generate a value y from the distribution with pdf g(y). 3. If u f(y)/cg(y), set x y ( accept x); otherwise return to step 1. That is, the procedure is repeated until at some stage u f(y)/cg(y).

a. Argue that c 1. Hint: If c 1, then f(y) g(y) for all y; why is this bad? b. Show that this procedure does result in an observation from the pdf f(); that is, P(accepted value x) F(x). Hint: This probability is P({U f(Y)/cg(Y)} {Y x}); to calculate, rst integrate with respect to u for xed y and then integrate with respect to y. c. Show that the probability of accepting at any particular stage is 1/c. What does this imply about the expected number of stages necessary to obtain an acceptable value? What kind of value of c is desirable? d. Let f(x) 20x(1

x) 3 for 0 x 1, a particular beta distribution. Show that taking g(y) to be a uniform pdf on (0, 1) works. What is the best value of c in this situation?

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Simulation studies are useful for examining numerous aspects of a system or process.
Simulation studies are useful for examining numerous aspects of a system or process.

Simulation studies are useful for examining numerous aspects of a system or process. When the mathematical analysis required to solve significant questions is too hard to give closed form solutions, they are commonly used. Simulation can provide information about the probability that the system is empty when a customer arrives, the expected number of customers in the system, and the expected waiting time in line in a system where the time between successive customer arrivals has a particular pdf and the service time of any particular customer has a different pdf. Such research necessitates the ability to collect data.

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