Is there is some literature for queue with random arrival/service rate?

Question

I have read literature for queuing theory where we have time-varying rates. I am interested in knowing any literature that concerns itself with the service rate or arrival rate as a random variable and then finds the average number of customers waiting in the queue and several other results.

Answer 1

There are a few ways to model this. The problem, as you imagine, is that analysis become quite messy.

One way to do this is assuming Markov modulated arrival/service rates. This means that the rate is changing according to an independent Markov Chain.

See for example (and literature review): https://link.springer.com/article/10.1007/s11134-013-9368-4

Another approach is assuming the arrival rate is a random variable that is generated once before the system starts operating. Here you can actually compute mean stationary values quite easily.

Example:https://www.sciencedirect.com/science/article/pii/S0167637717305321