In some cases, when you are delaying a discrete change signal in a Discrete Change Delay element, there may be some variability in delay times for signals. This is equivalent to saying that there is a distribution of actual delay times around some mean, and whenever the element receives a signal, the delay time for that signal is sampled from a distribution.
If we conceptualize the Delay as a conveyer belt for the signal, another way to view signal 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 two different ways:
•By specifying the dispersion in terms of an Erlang dispersion factor; or
•By specifying the dispersion in terms of a standard deviation.
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, which corresponds to an exponential distribution of delay times.
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 signal received, the Delay Time for that signal 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.
Note: When dispersion is specified for a delay time, each signal received
by the element is “assigned” an actual delay time by sampling from the
distribution presented above. As a result, when signals in a Discrete Change Delay are dispersed, the signals will
not necessarily be "released" in the order that they were received.