The Erlang model as a rival model for the PE model

In the literature, a wide variety of probability distributions are proposed to account for RT data. Among these, the gamma distribution was the favorite (see e.g., Bush & Mosteller, 1955; Christie, 1952; Luce, 1960; Restle, 1961). The gamma distribution can be considered as a special case of the Poisson-Erlang distribution, arising if the number of distractions is assumed to be constant. In that particular case the gamma distribution is referred to as the Erlang distribution. The Erlang distribution has three parameters: the parameter $\delta$, which again represents the transition rate to switch from distraction to work, the number of distraction N and real total working time A. Note, that N is assumed to be constant across RTs. According to the Erlang model, expectation and variance can be written as

$$E(T) = A + \frac{N}{\delta}$$ (10)

and

$$Var(T) = \frac{N}{\delta^2}$$ (11)

Taking the natural logarithm of both sides one obtains:

$$\ln E(T) = \ln(A+\frac{N}{\delta})$$ (12)

and

$$\ln Var(T) = \ln(\frac{N}{\delta^2}) = \ln N - 2\ln \delta$$ (13)

As was the case in the Poisson-Erlang model, in the Erlang model the logarithm of the variance is also a linear function of the logarithm of the parameters, that is the parameters N and $\delta$. And also similar to the Poisson-Erlang model, in the Erlang model this is not the case with respect to the logarithm of the expectation. But, again, in the Erlang model the difference between the expectation E(T) and the real total working time A: E(T)-A is a linear function of the logarithm of the Erlang parameters:

$$\ln(E(T)-A) = \ln N - \ln \delta$$ (14)

So, according to the simpler Erlang model, the correlation across subjects between $\ln(E(T)-A)$ and $\ln Var(T)$ is also linearly dependent on the log parameters $\ln A$, $\ln N$ and $\ln \delta$.