I am currently modeling an Emergency Department but have a question regarding the use of priorities in queues.

Currently, there are four types of patients with each their own priority level. However, when these patients are in a queue waiting to be processed, their priority level changes linearly with the waiting time in that queue according to four different rates (one rate for each type of patient). It is therefore possible that someone with an initially lower priority level gets priority over another type of patient because the former one has a longer waiting time, which led to an increase in the priority level.

I would appreciate any help on how to model this.