8.7 Application: Computer Simulations

Computers have long been used to model and simulate real-world systems and phenomena. These simulations are simply computer applications that have been designed to represent and appropriately react to the significant events occurring in the system. Simulations can allow humans to study certain behaviors or experiment with certain changes and events in a system to determine the appropriate strategy.

Some of the more common simulations include weather forecasting and flight simulators. A flight simulator, which is a mock-up of a real cockpit and controlled by software, helps train pilots to deal with real situations without having to risk the life of the pilot or loss of the aircraft. Weather forecasts today are much more reliable due to the aide of computer simulations. Mathematical models have been developed to simulate weather patterns and atmospheric conditions. These models can be solved using computer applications, which then provide information to meteorologists for use in predicting the weather.

Computer simulations are also used for less glamorous applications. Businesses can use a computer simulation to determine the number of employees needed to provide a service to its customers. For example, an airline may want to know how many ticket agents are needed at certain times of the day in order to provide timely service. Having too many agents will cost the airline money, but too few will result in angry customers. The company could simply study the customer habits at one airport and experiment with a different number of agents at different times. But this can be costly and time consuming. In addition, the results may only be valid for that one airport. To reduce the cost and allow for events that may occur at various airports, a computer simulation can be developed to model the real system.

Airline Ticket Counter

Simulating an airline ticket counter, or any other queuing system where customers stand in line awaiting service, is very common. A queue structure is used to model the queuing system in order to study certain behaviors or outcomes. Some of the typical results studied include average waiting time and average queue length. Queuing systems that use a single queue are easier to model. More complex systems like those representing a grocery store that use multiple queues, require more complex models. In this text, we limit our discussion to single-queue systems.

Queuing System Model

We can model a queuing system by constructing a discrete event simulation. The simulation is a sequence of significant events that cause a change in the system. For example, in our airline ticket counter simulation, these events would include customer arrival, the start or conclusion of a transaction, or customer departure.

The simulation is time driven and performed over a preset time period. The passing of time is represented by a loop, which increments a discrete time variable once for each tick of the clock. The events can only occur at discrete time intervals. Thus, the time units must be small enough such that no event can occur between units. A simulation is commonly designed to allow the user to supply parameters that define the conditions of the system. For a discrete event simulation modeling a queuing system, these parameters include:

• The length of the simulation given in number of time units. The simulation typically begins at time unit zero.
• The number of servers providing the service to the customers. We must have at least one server.
• The expected service time to complete a transaction.
• The distribution of arrival times used to determine when customers arrive.

By adjusting these parameters, the user can change the conditions under which the simulation is performed. We can change the number of servers, for example, to determine the optimal number required to provide satisfactory service under the given conditions.

Finally, a set of rules are defined for handling the events during each tick of the clock. The specific rules depends on what results are being studied. To determine the average time customers must wait in line before being served, there are three rules:

Rule 1:

If a customer arrives, he is added to the queue. At most, one customer can arrive during each time step.

Rule 2:

If there are customers waiting, for each free server, the next customer in line begins her transaction.

Rule 3:

For each server handling a transaction, if the transaction is complete, the customer departs and the server becomes free.

When the simulation completes, the average waiting time can be computed by dividing the total waiting time for all customers by the total number of customers.

Random Events

To correctly model a queuing system, some events must occur at random. One such event is customer arrival. In the first rule outlined earlier, we need to determine if a customer arrives during the current tick of the clock. In a real-world system, this event cannot be directly controlled but is a true random act. We need to model this action as close as possible in our simulation.

A simple approach would be to flip a coin and let "heads" represent a customer arrival. But this would indicate that there is a 50/50 chance a customer arrives every time unit. This may be true for some systems, but not necessarily the one we are modeling. We could change and use a six-sided die and let one of the sides represent a customer arrival. But this only changes the odds to 1 in 6 that a customer arrives.

A better approach is to allow the user to specify the odds of a customer arriving at each time step. This can be done in one of two ways. The user can enter the odds a customer arrives during the current time step as a real value between $0.0$ (no chance) and $1.0$ (a sure thing). If $0.2$ is entered, then this would indicate there is a $1$ in $5$ chance a customer arrives. Instead of directly entering the odds, we can have the user enter the average time between customer arrivals. We then compute the odds within the program. If the user enters an average time of $8.0$, then on average a customer arrives every 8 minutes. But customers can arrive during any minute of the simulation. The average time between arrivals simply provides the average over the entire simulation. We use the average time to compute the odds of a customer arriving as $\frac{1}{8}$, or $0.125$.

Given the odds either directly by the user or computing them based on the average arrival time, how is this value used to simulate the random act of a customer arriving? We use the random number generator provided by Python to generate a number between $0.0$ and $1.0$. We compare this result to the probability (prob) of an arrival. If the generated random number is between $0.0$ and prob inclusive, the event occurs and we signal a customer arrival. On the other hand, if the random value is greater than prob, then no customer arrives during the current time step and no action is taken. The arrival probability can be changed to alter the number of customers served in the simulation.