Examples
None of the examples in this chapter need to be simulated; they can all be analyzed by mathematical/numerical analysis. But thinking about how to simulate them will help us understand what works, and why, for systems we do need to simulate. These examples
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Examples
None of the examples in this chapter need to be simulated; they can all be analyzed by mathematical/numerical analysis. But thinking about how to simulate them will help us understand what works, and why, for systems we do need to simulate. These examples also provide a tested for evaluating new ideas in simulation design and analysis. You may have encountered these models in other classes, but it is neither necessary to have seen them before, nor to know how the results are derived, to use them throughout the book.
3.1 M(t)/M/∞ Queue This example is based on Nelson (1995, Chap. 8). Example 3.1 (The parking lot). Prior to building a large shopping mall, the mall designers must decide how large a parking garage is needed. The arrival rate of cars will undoubtedly vary over time of day, and even day of year. Some patrons will visit the mall for a brief time, while others may stay all day. Once the garage is built, the mall may open and close floors depending on the load, but the first question is, what should the maximum capacity of the garage be? Since the mall developers would like virtually everyone to be able to have a parking space, a standard modeling trick is to pretend that the parking garage is infinitely large and then evaluate the probability distribution of the number of spaces actually used. If the capacity of the garage is set to a level that would rarely be exceeded in an infinitely large garage, then that should be adequate in practice. The M(t)/M/∞ queue is a service system in which customers arrive according to a nonstationary Poisson arrival process with time-varying arrival rate λ (t) customers/time,1 the service time is exponentially distributed with mean τ , and there 1
A stationary Poisson arrival process has independent times between arrivals that are exponentially distributed with mean 1/λ or equivalently arrival rate λ . The nonstationary Poisson process is a generalization that allows the arrival rate to change over time; it is covered in detail in Chap. 6. B.L. Nelson, Foundations and Methods of Stochastic Simulation: A First Course, International Series in Operations Research & Management Science 187, DOI 10.1007/978-1-4614-6160-9 3, © Springer Science+Business Media New York 2013
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Fig. 3.1 Plot of λ (t) = 1,000 + 100 sin(π t/12) (bottom) and m(t) (top) when τ = 2, and the Poisson distribution of number of cars in the garage with mean m (top left)
are an infinite number of servers. In Example 3.1 the customers are cars arriving at a rate of λ (t) per hour, a “service time” is the time spent occupying a space with mean τ hours, and we pretend that there are infinitely many spaces. A good reference for using queueing models to address the design of service systems like call centers or the parking lot is Whitt (2007). Let N(t) be the number of cars in the parking garage at time t ≥ 0. Then it is known that N(
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