Problem
Suppose that you have $N$ servers, $\{s_1, s_2, \ldots, s_N\}$. Let $T_i$ represent the random variable for service time on server $s_i$. $T_i \sim \text{Gamma}(k_i, \lambda_i)$. Suppose jobs arrive at a rate $\lambda_a$. Suppose that they become impatient and leave at rate $\sigma_a$. What is the stationary distribution for the queue?
Simplifications I'd Still Be Interested In:
- There is a common parameter $k$ such that $k_i = k$ for all $i \in \{1, 2, \ldots, N\}$.
- $\sigma_a = 0$.
- There is a common parameter $\lambda$ such that $\lambda_i = \lambda$ for all $i \in \{1, 2, \ldots, N\}$.
Simplifications I Am Not Interested In:
- $k = 1$.
Context
I work at a mathematics tutoring lab. We have 7 years or so of data regarding when students signed up for help. We recorded when the tutor started helping the student, as well as when they finished (they would write both down when they helped the student). We recently have implemented an electronic queue, which would allow us to estimate $\sigma_a$.
By plotting a histogram of help duration, it's very clear that tutor help time is not exponentially distributed.
As a class project, I'm trying to figure out if we can use this data to help inform scheduling decisions.
I am VERY inexperienced with queueing theory. I've mostly just skimmed the Wikipedia article. I'm aware there's a closed-form solution for M/M/c at https://en.wikipedia.org/wiki/M/M/c_queue#Stationary_analysis, but this assumes the job service time is exponentially distributed.
Other Questions
I'm also interested in how tractable these problems are. I don't want to fall down a rabbit hole of queueing theory and make zero progress on my project.