That general, all you can say is "show that it satisfies the definition of "Markov Process". How you would do that, of course, depends upon exactly what the arrival process is.
The process is people arrival at the bus stop. I have time arrivals and now I need to proof that is indeed markovian process and maybe poisson
Surely one would assume a memoryless system to allow for the theory of queues and stochastic processes can be applied to your queueing system? I am sure it is mathematically allowed to assume (wlog) that your system is Markovian.
Perhaps a clear statement of your question is: "I have data for the arrival times of people at a bus stop. What statistical tests can I use to test the hypothesis that the arrival process is Poission?". (Statistical tests aren't "proof".)