I'm attempting to generate output similar to the queueview program offered by ICMI (but can't use that program). For a given number of staff (computed using the Erlang-C probability in iterative process), ASA (computed and target), AHT, arrival rate, it outputs a list of the numbers of arrivals who are predicted to wait beyond a certain length of time. It then has a table of particular times of interest, distributed above and below the target ASA.
We're modelling our system as M/M/c at this stage as a first-order attempt, i.e. our arrival distribution doesn't quite fit the Markov distribution normally assumed for this kind of work. We're just ignoring that for now.
I've seen plenty of places which allow calculation of average waiting time, but I need to somehow convert the average to a set of predicted wait lengths distributed across the set of arrivals to calculate the wait times.
I know there's the Erlang distribution function $D(x) = 1-(\Gamma(h, x \lambda)/\Gamma(h))$ but I'm at a loss on how to use this (or any other distribution function) to get what I'm after. In particular, I'm not sure how to relate any of my inputs to the $\lambda$, $h$ and $x$ used in the equation. Once I'm able to solve for the different delays of each arrival, I know that I just need to iterate over that set to find if the delay is at or above any of the specified times of interest.
I'd appreciate some help with making the (conceptual) connections between the values used in the function and my inputs, or something similar for a different distribution function if the Erlang one is not the right one to use. And please be gentle; it's been a very long time since I've done any of this kind of math!
I eventually figured out the solution. Given the Erlang-C probability function produces the projected ratio of calls that will be held in the queue, subtracting that figure from 1 and multiplying by the call volume gives the number of calls that have 0 delay.
The iterative process of solving for the optimum staff number gives you the ratio of calls with the delay at or higher than the target ASA. Multiplying that by the call volume gives you a second point.
Using those 2 points, you can establish an exponential decay formula... into which you can then plug the other delays you're interested in.