## 1.5 Randomness in Simulation

In most real-life situations, the arrival process and the service process occur in a random fashion. Even though the processes may be random, it does not mean that you cannot describe or model the randomness. To have any hope of simulating the situation, you must be able to model the randomness. One of the ways to model this randomness is to describe the phenomenon as a random variable governed by a particular probability distribution. For example, if the arrivals to the bank occur according to a Poisson process, then from probability theory it is known that the distribution of inter-arrival times is an exponential distribution. In general, information about how the customers arrive must be secured either through direct observation of the system or by using historical data. If neither source of information is available, then some plausible assumptions must be made to describe the random process by a probability model.

If historical data is available, there are two basic choices for how to handle the modeling. The first choice is to develop a probability model given the data. The second choice is to try to drive the simulation directly from the historical data. The latter approach is not recommended. First of all, it is extremely unlikely that the captured data will be in a directly usable form. Secondly, it is even more unlikely that the data will be able to adequately represent all the modeling scenarios that you will need through the course of experimenting with the model. For example, suppose that you only have 1 day’s worth of arrival data, but you need to simulate a month’s worth of system operation. If you simply re-drive your simulation using the 1 day’s worth of data, you are not simulating different days! It is much more advisable to develop probability models either from historical data or from data that you capture in developing your model. Appendix B discusses some of the tools and techniques for modeling probability distributions.

Once a probability model has been developed, statistical theory provides the means for obtaining random samples based on the use of uniformly distributed random numbers on the interval (0,1). These random samples are then used to map the future occurrence of an event on the time scale. For example, if the inter-arrival time is exponential then a random sample drawn from that distribution would represent the time interval until the occurrence of the next arrival. The process of generating random numbers and random variables within simulation is presented in Appendix A.