In some cases, when you are delaying a discrete change signal A discrete signal that contains information regarding the response to an event. in a Discrete Change Delay An event element that provides a mechanism for delaying a discrete change signal 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 The square root of the variance of a distribution. The variance is the second moment of the distribution and reflects the amount of spread or dispersion in the distribution..
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 A mathematical representation of the relative likelihood of a variable having certain specific values. It can be expressed as a PDF (or a PMF for discrete variables), CDF or CCDF., which is equivalent to (and a generalization of) the Erlang distribution that is frequently used in simulation models. n is the Erlang value (specified by the user);
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.
- Delaying a Discrete Change Signal
- Discrete Change Delays with Time-Variable Delay Times
- Modeling Discrete Change Delays with Dispersion
- Modeling Discrete Change Delays without Dispersion
- Referencing the Discrete Change Value to Determine the Delay Time
- Specifying Capacities and Modeling Queues for a Discrete Change Delay
- Using Splitter Elements to Route Discrete Changes Based on Their Value