When a queueing system is modeled as an M/M/1 queue, it is assumed that the arrival time of jobs has Poisson distribution and the service rate has exponential distribution. I am wondering what features a system should have in order to model the arrival rate as Poisson? I known that Poisson is the only distribution that its inter-arrival time of jobs is exponentially distributed which is memoryless. Are there any better and more intuitive features for it?

Also even in a more complex modeling (M/G/1), only the service rate is changed to general which means that the Poisson arrival rate is good enough where G/M/1 or G/G/1 is not as appealing as the previous ones.

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#### Best Answer

I think the main advantage to modeling the arrival distribution as Poisson is the memory-less property. It greatly simplifies the subsequent calculations to be able to assume that the number of arrivals in a particular time interval depends only on the length of the interval, rather than when the interval occurs, how many people came before, etc. Of course, this assumption is not always appropriate. For example, modeling the number of patients arriving to a doctor's office for routine checkups could be considered memory-less, since it would tend to average out to a constant rate over a long period of time. On the other hand, this assumption would likely be inappropriate for modeling the arrival rate of patients to an emergency room, since this would be more likely to follow a boom-and-bust pattern.