I am currently trying to model an uniform arrival process within my simulation model. However, I can only model it by means of an interarrival time (I can let the model wait for a certain amount seconds).

I want to have:

$$G(x) = {1 \over F(x)}$$

where $ F(x) \sim U(a,b) $. I started by using inverse transform sampling on the CDF of the inverse continious distribution (I should use a discrete distribution, but since the numbers are generally quite big, this is acceptable):

$$ G(y) = { b - y^{-1} \over b - a }$$

$$ x = { 1 \over b - y(b-a) } $$

where $Y \sim U(0,1)$. However this gives me the wrong results. Lets take some number to clarify this. Let say I want to model an arrival process between 600 and 1800 units per hour. On average (1200/hr), the model has to wait 3 seconds. The boundaries of the waiting time are between 2 and 6 seconds. This means that 50% of the samples should lie in the interval $[2,3]$ and 50% in the interval $[3,6]$. This is exactly what $G(y)$ does, so I was expecting good results. However, if I calculate the average from this function, it does not match with what I want.

$$g(y) = y^-2 { 1 \over b^* - a^*}$$ $$ E\left[G\right] = \int_{b^*}^{a^*} g(y) y dy = { b^* -a^* \over \ln \left| b^* \right| - \ln \left| a^* \right| } $$ where $a^* = {1 \over a}$ and $b^* = {1 \over b}$. When I plug in the numbers I get an average of 3.64, while I would like to have an average of 3.

Do I make a mistake somewhere? Or, does somesome have a better approach in generating a random arrival process?


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