# Finding a queuing model for waste accumulation

I've been tasked with modeling the accumulation of solid waste in an urban setting. In particular, the objective is to find the steady state distribution describing the amount of waste in a given waste accumulation site.

The situation can be best described as follows; waste is deposited in the accumulation site in discrete steps, whose interarrival times follow an exponential distribution, the amount (mass) of waste deposited in each step follows a normal distribution with known mean and variance (the unit of mass employed is the metric ton).

Waste is removed from the site with the use of a single truck, each time the truck stops by the site it removes all of the waste, it doesn't matter how much waste has accumulated, the truck always removes the entirety of the waste. The truck visits the site with a fixed and known frequency, it can, however, deviate slightly from its schedule (this random deviation can be ignored for simplification, particularly as the mean of this deviation is close to 0).

I've researched the following models, M/D/1 and M/G/1, but I have stumbled onto the following problems:

• Mass is a continuous variable, so in order to use any of the aforementioned queuing models I'd have to "discretize" the waste. I could, for example, consider each ton of waste in the site as a single entity in the queue. My first question is as to how valid (justifiable) this would be, and whether or not it would be a better idea to try a fluid queue model (as this also has a few problems).

• If I consider each ton of waste in the site as an entity in the queue, and the truck as the server, then that would mean that the server can process entities in bulk, so my second question would be, how do I factor this into the model?

• Finally, what would the service time be? Considering that the truck arrives every $${1 \over f}$$ hours ($$f$$ being the truck's arrival frequency), would it be the time required to load the truck?

It's quite possible that queueing theory is simply not the best starting point to model this particular situation, if that is the case, any guidance as to where to start looking would be greatly appreciated.