# Discrete event simulation

Discrete event simulation

In discrete-event simulation, the operation of a system is represented as a chronological sequence of events. Each event occurs at an instant in time and marks a change of state in the system [1]. For example, if an elevator is simulated, an event could be "level 6 button pressed", with the resulting system state of "lift moving" and eventually (unless one chooses to simulate the failure of the lift) "lift at level 6".

A common exercise in learning how to build discrete-event simulations is to model a queue, such as customers arriving at a bank to be served by a teller. In this example, the system entities are CUSTOMER-QUEUE and TELLERS. The system events are CUSTOMER-ARRIVAL and CUSTOMER-DEPARTURE. (The event of TELLER-BEGINS-SERVICE can be part of the logic of the arrival and departure events.) The system states, which are changed by these events, are NUMBER-OF-CUSTOMERS-IN-THE-QUEUE (an integer from 0 to n) and TELLER-STATUS (busy or idle). The random variables that need to be characterized to model this system stochastically are CUSTOMER-INTERARRIVAL-TIME and TELLER-SERVICE-TIME.

A number of mechanisms have been proposed for carrying out discrete-event simulation, among them are the event-based, activity-based, process-based and three-phase approaches (Pidd, 1998). The three-phase approach is used by a number of commercial simulation software packages, but from the user's point of view, the specifics of the underlying simulation method are generally hidden.

## Components of a Discrete-Event Simulation

In addition to the representation of system state variables and the logic of what happens when system events occur, discrete event simulations include the following:

### Clock

The simulation must keep track of the current simulation time, in whatever measurement units are suitable for the system being modeled. In discrete-event simulations, as opposed to real time simulations, time ‘hops’ because events are instantaneous – the clock skips to the next event start time as the simulation proceeds.

### Events List

The simulation maintains at least one list of simulation events. This is sometimes called the pending event set because it lists events that are pending as a result of previously simulated event but have yet to be simulated themselves. An event is described by the time at which it occurs and a type, indicating the code that will be used to simulate that event. It is common for the event code to be parameterised, in which case, the event description also contains parameters to the event code.

When events are instantaneous, activities that extend over time are modeled as sequences of events. Some simulation frameworks allow the time of an event to be specified as an interval, giving the start time and the end time of each event.

Single-threaded simulation engines based on instantaneous events have just one current event. In contrast, multi-threaded simulation engines and simulation engines supporting an interval-based event model may have multiple current events. In both cases, there are significant problems with synchronization between current events.

The pending event set is typically organized as a priority queue, sorted by event time.[2] That is, regardless of the order in which events are added to the event set, they are removed in strictly chronological order. Several general-purpose priority queue algorithms have proven effective for discrete-event simulation,[3] most notably, the splay tree. More recent alternatives include skip lists and calendar queues.[4]

Typically, events are scheduled dynamically as the simulation proceeds. For example, in the bank example noted above, the event CUSTOMER-ARRIVAL at time t would, if the CUSTOMER_QUEUE was empty and TELLER was idle, include the creation of the subsequent event CUSTOMER-DEPARTURE to occur at time t+s, where s is a number generated from the SERVICE-TIME distribution.

### Random-Number Generators

The simulation needs to generate random variables of various kinds, depending on the system model. This is accomplished by one or more Pseudorandom number generators. The use of pseudorandom numbers as opposed to true random numbers is a benefit should a simulation need a rerun with exactly the same behaviour.

One of the problems with the random number distributions used in discrete-event simulation is that the steady-state distributions of event times may not be known in advance. As a result, the initial set of events placed into the pending event set will not have arrival times representative of the steady-state distribution. This problem is typically solved by bootstrapping the simulation model. Only a limited effort is made to assign realistic times to the initial set of pending events. These events, however, schedule additional events, and with time, the distribution of event times approaches its steady state. This is called bootstrapping the simulation model. In gathering statistics from the running model, it is important to either disregard events that occur before the steady state is reached or to run the simulation for long enough that the bootstrapping behavior is overwhelmed by steady-state behavior. (This use of the term bootstrapping can be contrasted with its use in both statistics and computing.)

### Statistics

The simulation typically keeps track of the system's statistics, which quantify the aspects of interest. In the bank example, it is of interest to track the mean waiting times.

### Ending Condition

Because events are bootstrapped, theoretically a discrete-event simulation could run forever. So the simulation designer must decide when the simulation will end. Typical choices are “at time t” or “after processing n number of events” or, more generally, “when statistical measure X reaches the value x”.

## Simulation Engine Logic

The main loop of a discrete-event simulation is something like this:

### Start

• Initialize Ending Condition to FALSE.
• Initialize system state variables.
• Initialize Clock (usually starts at simulation time zero).
• Schedule an initial event (i.e., put some initial event into the Events List).

### “Do loop” or “While loop”

While (Ending Condition is FALSE) then do the following:

• Set clock to next event time.
• Do next event and remove from the Events List.
• Update statistics.

### End

• Generate statistical report.

## Common Uses

### Diagnosing process issues

Simulation approaches are particularly well equipped to help users diagnose issues in complex environments. The Goal (Theory of Constraints) illustrates the importance of understanding bottlenecks in a system. Only process ‘improvements’ at the bottlenecks will actually improve the overall system. In many organizations bottlenecks become hidden by excess inventory, overproduction, variability in processes and variability in routing or sequencing. By accurately documenting the system inside a simulation model it is possible to gain a bird’s eye view of the entire system.

A working model of a system allows management to understand performance drivers. A simulation can be built to include any number of performance indicators such as worker utilization, on-time delivery rate, scrap rate, cash cycles, and so on.

#### Hospital applications

An operating theater is generally shared between several surgical disciplines. Through better understanding the nature of these procedures it may be possible to increase the patient throughput. Example: If a heart surgery takes on average four hours, changing an operating room schedule from eight available hours to nine will not increase patient throughput. On the other hand, if a hernia procedure takes on average twenty minutes providing an extra hour may also not yield any increased throughput if the capacity and average time spent in the recovery room is not considered.

#### Custom order environments

Many systems show very different characteristics from day to day depending on the order mix. Many small orders may cause bottle-necks due to excess changeovers. Large custom orders may require extra processing at a point where the system has particularly low capacity. Simulation modeling allows management to understand what changes ‘on average’ would have the largest impact and greatest return-on-investment.

### Lab test performance improvement ideas

Many systems improvement ideas are built on sound principles, proven methodologies (Lean, Six Sigma, TQM, etc.) yet fail to improve the overall system. A simulation model allows the user to understand and test a performance improvement idea in the context of the overall system.

### Evaluating capital investment decisions

Simulation modeling is commonly used to model potential investments. Through modeling investments decision-makers can make informed decisions and evaluate potential alternatives.

Often these decisions look at altering existing operations. Typically, a model of the current state is constructed. This ‘current state’ model is tested and validated against historical data. Once the model is operating correctly, the simulation is altered to reflect the proposed capital investments. This 'future state' model is then stress-tested to ensure the alterations perform as desired.

Occasionally, organizations take on entirely new operations processes. These could be new Lean facilities, designed around new products or using new technology. In these cases only a ‘future state’ model is constructed. The testing and validation may require more analysis. There are companies and experts that specialize in simulation building who may be brought in to help.

### Stress test a system

Models can be used to understand how a system will be able to weather extraordinary conditions. A simulation can help management understand: large increases in orders, significant swings in product mix, new client delivery demands (e.g. 1 week lead times), and economic events (e.g. a multinational with operations in South America and Asia sees significant swings in currencies).

System Modeling approaches:

Computational techniques:

Software:

Disciplines:

## References

1. ^ Stewart Robinson (2004). Simulation - The practice of model development and use. Wiley.
2. ^ Douglas W. Jones, ed. Implementations of Time, Proceedings of the 18th Winter Simulation Conference, 1986.
3. ^ Douglas W. Jones, Empirical Comparison of Priority Queue and Event Set Implementations, Communications of the ACM, 29, April 1986, pages 300-311.
4. ^ Kah Leong Tan and Li-Jin Thng, SNOOPy Calendar Queue, Proceedings of the 32nd Winter Simulation Conference, 2000

• Michael Pidd (1998). Computer simulation in management science - fourth edition. Wiley.
• Jerry Banks, John Carson, Barry Nelson and David Nicol (2005). Discrete-event system simulation - fourth edition. Pearson.
• Averill M. Law and W. David Kelton (2000). Simulation modeling and analysis - third edition. McGraw-Hill.
• Bernard P. Zeigler, Herbert Praehofer and Tag Gon Kim (2000). Theory of modeling and simulation: Integrating discrete event and continuous complex dynamic systems - second edition. Academic Press.
• Roger W. McHaney (1991). Computer Simulation: A Practical Perspective. Academic Press.
• William Delaney, Erminia Vaccari (1988). Dynamic Models and Discrete Event Simulation. Dekker INC.
• A, Alan Pritsker, Jean J. O'Reilly (1999). Simulation with Visual SLAM and AweSim. Wiley.
• James J. Nutaro (2010). Building software for simulation: theory and algorithms, with applications in C++. Wiley.

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