# Assumption for an M/M/1 queue

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.

• Another characteristic of the Poisson distribution is that its mean and variance are the same.
– user10525
Nov 9 '12 at 13:23

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.

• This is exactly what I pointed out in the question. Nov 9 '12 at 23:31

There's a theorem in renewal theory related to that. It is roughly equivalent to the Central Limit Theorem for sums of independent random variables. It says that, under certain general conditions, the superposition of a large number of independent arrival processes converges to a Poisson process. That explains why the Poisson process so often arises, and why it is such a common assumption in queueing theory.

I think the theorem is called Cinlar's theorem, but I can't find a reference right now.

EDIT: See this document, section 4.5.B (page 194): http://www.pitt.edu/~super7/19011-20001/19501.pdf

Another intuitive property of Poisson arrivals is this: given a fixed time interval, and conditioned on the fact that the number of arrivals contained in that interval is a fixed number N, those arrivals are uniformly distributed on the interval. See for example here: http://www.netlab.tkk.fi/opetus/s383143/kalvot/E_poisson.pdf

To use an M/M/I queuing model, we must assume Poisson arrivals and exponential service distribution. Then, to derive characteristics for this types of waiting line, we must consider some other assumptions. Most importantly, there must be only one service channel, which arrivals enter one at a time. More so, it is assumed that there is an infinite population from which arrivals originate: we assumed arrivals are served on a first-come-served basis, we assume that there is no room to hold arrival waiting for service.