Modeling Event Delays with Dispersion

In some cases, there may be some variability in when a signal is emitted once the element is triggered. This is equivalent to saying that there is a distribution of actual delay times around some mean, and whenever the element is triggered, the delay time for that event is sampled from a distribution.

If we conceptualize the Delay as a conveyer belt for the signal, another way to view event dispersion is that as the signal moves along the belt, it slips randomly forward or backward on the belt, with the amount of movement proportional to the degree of dispersion.

You can specify such dispersion in the delay time in three different ways:

•   By specifying the dispersion in terms of an Erlang dispersion factor;

•   By specifying the dispersion in terms of a standard deviation; or

•   By specifying a custom distribution of delay times (by referencing a Stochastic element).

If “Defined Delay Time + Erlang Dispersion” is selected, you must enter an Erlang n-value, which is a dimensionless value greater than or equal to 1. As n increases, the degree of dispersion decreases. As n goes to infinity, the dispersion goes to zero. The maximum amount of dispersion allowed is represented by n = 1.

If “Defined Delay Time + Std. Deviation” is selected, you must enter a Std. Deviation, which is a value with dimensions of time. The value must be greater than or equal to zero and less than or equal to the Mean Time. As the Std. Deviation decreases, the degree of dispersion decreases. When the Std. Deviation goes to zero, the dispersion goes to zero. The maximum amount of dispersion allowed is represented by Std. Deviation = Delay Time.

The Erlang n and the Std. Deviation are related by the following equation:

If the signal is dispersed, for every event, the Delay Time for that event is sampled from the following distribution:

where:

n is the Erlang value (specified by the user);

β = D/n;

D is the mean delay time; and

Г is the gamma function (not the Gamma distribution).

f(t) is the gamma probability distribution, which is equivalent to (and a generalization of) the Erlang distribution that is frequently used in simulation models.

The gamma distribution represents the time until the occurrence of n sequential Poisson-process events. Each event’s random time is represented by an exponential distribution with mean D/n. The gamma distribution does not require n to be an integer. Note that for n=1, the distribution is exponential, and for increasing values of n it becomes less skewed, approaching normality for large n.

The “Stochastic Delay Time Definition” option provides a way to specify a custom distribution for the delay time (rather than the gamma used for the Erlang and Std. Deviation options). If this option is selected, you must specify a Stochastic element. The Stochastic element that is specified represents the distribution of the delay time.  As such, it must have dimensions of time. The Distribution output of the Stochastic must be referenced:

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