## 3.1 Finite versus Infinite Horizon Simulation Studies

When modeling a system, specific measurement goals for the simulation responses are often required. The goals, coupled with how the system operates, will determine how you execute and analyze the simulation experiments. In planning the experimental analysis, it is useful to think of simulations as consisting of two main categories related to the period of time over which a decision needs to be made:

Finite horizon

In a finite-horizon simulation, a well define ending time or ending condition can be specified which clearly defines the end of the simulation. Finite horizon simulations are often called terminating simulations, since there are clear terminating conditions.

Infinite horizon

In an infinite horizon simulation, there is no well defined ending time or condition. The planning period is over the life of the system, which from a conceptual standpoint lasts forever. Infinite horizon simulations are often called steady state simulations because in an infinite horizon simulation you are often interested in the long-term or steady state behavior of the system.

For a finite horizon simulation, an event or condition associated with the system is present which indicates the end of each simulation replication. This event can be specified in advance or its time of occurrence can be a random variable. If it is specified in advance, it is often because you do not want information past that point in time (e.g. a 3 month planning horizon). It might be a random variable in the case of the system stopping when a condition is met. For example, an ending condition may be specified to stop the simulation when there are no entities left to process. Finite horizon simulations are very common since most planning processes are finite. A few example systems involving a finite horizon include:

• Bank: bank doors open at 9am and close at 5pm

• Military battle: simulate until force strength reaches a critical value

• Filling a customer order: suppose a new contract is accepted to produce 100 products, you might simulate the production of the 100 products to see the cost, delivery time, etc.

For a finite horizon situation, we want to observe the behavior of the system over the specified time horizon and report on its performance. The observations available over one time horizon create a sample of performance. This sample of performance is sometimes referred to as a sample path because it is based on the trajectory of the performance measures over time. When we repeatedly run the simulation over multiple instances of the time horizon, we are observing different samples (sample paths). If each of the generated sample paths are independent and identically distributed, we call an instance a replication, and the set of instances as replications. The length of each sample path or replication corresponds to the finite horizon of interest. For example, in modeling a bank that opens at 9 am and closes at 5 pm, the length of the replication would be 8 hours.

In contrast to a finite horizon simulation, an infinite horizon simulation has no natural beginning point or ending point. In the case of infinite horizon simulation studies, we are almost always interested in reporting on the long-run performance of the system. Thus, the desired starting conditions of interest are sometime in the far off future. In addition, because we want long run performance there is no natural ending point of interest. Thus, we consider the horizon infinite. Of course, when you actually simulate an infinite horizon situation, a finite replication length must be specified. Hopefully, the replication length will be long enough to satisfy the goal of observing long run performance. Examples of infinite horizon simulations include:

• A factory where you are interested in measuring the steady state throughput

• A hospital emergency room which is open 24 hours a day, 7 days of week

• A telecommunications system which is always operational

Infinite horizon simulations are often tied to systems that operate continuously and for which the long-run or steady state behavior needs to be estimated.

Because infinite horizon simulations often model situations where the system is always operational, they often involve the modeling of non-stationary processes. In such situations, care must be taken in defining what is meant by long-run or steady state behavior. For example, in an emergency room that is open 24 hours a day, 365 days per year, the arrival pattern to such a system probably depends on time. Thus, the output associated with the system is also non-stationary. The concept of steady state implies that the system has been running so long that the system’s behavior (in the form of performance measures) no longer depends on time; however, in the case of the emergency room since the inputs depend on time so do the outputs. In such cases it is often possible to find a period of time or cycle over which the non-stationary behavior repeats. For example, the arrival pattern to the emergency room may depend on the day of the week, such that every Monday has the same characteristics, every Tuesday has the same characteristics, and so on for each day of the week. Thus, on a weekly basis the non-stationary behavior repeats. You can then define your performance measure of interest based on the appropriate non-stationary cycle of the system. For example, you can define $$Y$$ as the expected waiting time of patients per week. This random variable may have performance that can be described as long-term. In other words, the long-run weekly performance of the system may be stationary. This type of simulation has been termed steady state cyclical parameter estimation within (Law 2007).

Of the two types of simulations, finite horizon simulations are easier to analyze. Luckily they are the more typical type of simulation found in practice. In fact, when you think that you are faced with an infinite horizon simulation, you should very carefully evaluate the goals of your study to see if they can just as well be met with a finite planning horizon. The analysis of finite horizon simulation studies will be discussed in this chapter while the discussion of infinite horizon simulation is presented in Chapter 5.

In the next section, we will examine in more detail the concepts of samples, sample paths, and replications by defining the types of statistical quantities observed within a simulation.

### References

Law, A. 2007. Simulation Modeling and Analysis. 4th ed. McGraw-Hill.