To me, MATLAB is the best tool for coping to your query. Here are some links for clue to the solution of your query: https://www.mathworks.com/help/simevents/ug/generate-random-numbers-in-simevents.html

Also read about Queuing Theory here!: https://www.mathworks.com/discovery/queuing-theory.html

Article Modeling and simulation of queue waiting time at traffic lig...

I would recommend you to use OMNet++ for your simulation modeling and analyses. In fairness, MATLAB TM is undoubtedly powerful, however, it does not align with your specific requirements. Also, MATLAB TM is not a good tool to replicate Poisson processes as the timing precision is bounded by multi-threading. When you superimpose two or maybe more Poisson processes, you lose the semantics of continuous-time. As an anecdotal remark, you can achieve somewhat similar to real-time in OMNet++.

OMNet++- https://omnetpp.org/download/

Introductory Material-

I hope this answers your queries.