5.8: Summary

This chapter discusses the basic operations of a simulation engine. While these operations are performed transparently to the modeler, an understanding of them helps clarify how simulation experiments work. Events are organized and processed in time sequence. Entities waiting for resources are sorted and maintained. Random samples from distribution functions are generated and pseudo-random number streams are managed.

Problems

1. State a procedure for generating a random sample from each of the following distributions using the inverse transformation method. Use the procedure in section 5.6 as a guide.
   1. Uniform distribution: \(\ F(x)=\frac{x-m i n i m u m}{m a x i m u m-m i n i m u m}\)
   2. Exponential distribution: \(\ F(x)=1-e^{x / m e a n}\)
   3. Weibull distribution: \(\ F(x)=1-e^{(-x / c)^{m}}\) where c and m are the scale and shape parameters of the distribution respectively.
   4. Triangular distribution:

      \(\ F(x)=\left\{\begin{aligned}\left.\frac{(x-\text { minimum } }{(\text { mode }-\text { minimum }) *( \text { maximum - minimum } }\right)^{2} \text {, minimum }\leq x \leq \text { mode } & \\ 1-\frac{(\text { maximum }-x)^{2}}{(\text { maximum }-\text { mode }) *( \text { maximum - minimum)} }) \text { , mode }<x \leq \text { maximum } \end{aligned}\right.\)

   5. Discrete distribution:

      \(\ \begin{aligned}
      F(x) &=0.1, x=1 \\
      &=0.4, x=2 \\
      &=0.6, x=3 \\
      &=0.9, x=4 \\
      &=1.0, x=5
      \end{aligned}\)

2. Create a new trace based on the one shown in Table 5-2 by adding a entity with ID number 4 that arrives at time 2.0 with a processing time at workstation A of 6.4.
3. Consider the properties of pseudo-random number generators presented in section 5-8. Does property four imply property two?
4. Consider the two workstation in a series model and the last event list shown in section 5- 5.

   Current Simulation Time: 8.0

   Next Simulation Time = Time of first event occurrence in list = 8.0

   \[
   \begin{array}{l|cl}
   \text { Event } & \text { Time of Occurrence } & \text { Entity ID } \\
   \hline \text { Entity Arrives to B } & 8.0 & 1 \\
   \text { Start Service at A } & 8.0 & 2 \\
   \text { Entity Arrives to A } & 32.5 & 3
   \end{array}
   \nonumber\]

   Use the event graph shown in Figure 5-1 as well as the trace shown in Table 5-2 as a guide.

   1. Show the event list after the processing of the entity arrives to B event for the entity with ID number 1. What single event occurrence was removed from the list? What event occurrences remain on the list? What event occurrences are added to the list?
   2. Show the event list after the first event on the list resulting from 2a is processed. What single event occurrence was removed from the list? What event occurrences remain on the list? What event occurrences are added to the list?
5. Implement a (bad) LCG generator in Excel with the following parameters: a = 5; m = 16; c = 3; Z₀ = 0.

   Generate the first 20 samples from the generator. Assess its behavior using the four properties in section 5.7.

6. Compute the following table using a spreadsheet.
   1. Generate the two random number streams, corresponding to interarrival time and operation time at station A, for the first ten arriving entities in the two workstation in a series model. Do this by using the random number generator built into your spreadsheet program. In Excel, this would be accomplished by entering the function rand() into each cell of the Pseudo-random Number / Bet. Arrivals and the Pseudo- random Number / Service Time columns.
   2. Use the inverse-transformation method to generate the time between arrivals and service time samples. This means entering equation 5-1 into each cell in the Sample / Bet. Arrivals column and entering equation 5-2 into each cell in the Sample / Service Time column. The corresponding pseudo-random number in the columns should be referenced for each cell.

   Table for Problem 6
   Entity ID	Pseudo-random Number		Sample
   	Bet. Arrivals	Service Time	Bet. Arrivals	Service Time
   1
   2
   3
   4
   5
   6
   7
   8
   9
   10

7. Considering only the first two entities from the data generated in the solution to number 6, create a trace similar to Table 5-2 for the two workstations in a series model.