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I am studying Markov chains and queuing theory. I was curious about traffic generation models and actually happened to see the Deterministic Traffic Model, referred to as $D$ in Kendall's notation.

So far I only encountered $M$ models (Poisson distributed traffic model and serving model). How does this distribution work? How does it affect a simple queue system like the M/M/1 when we use the $G$ as the input process or the serving process or both?

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In the deterministic traffic arrival model, jobs arrive at fixed instants, for example every 10 seconds. There is no randomness in the model, jobs arrive exactly once every 10 seconds. If you have a model with deterministic services then this means the model takes a fixed amount of time to serve every job, again no randomness.

In contrast, in the M models there is randomness. The time between two consecutive M arrivals has an exponential distribution, so the arrivals can be described by a Poisson process (where there is an exponentially distributed delay between arrivals, sometimes known as 'firings' of the process).

A G (sometimes GI to denote that interarrival times or service times are independent) model is a general model, so allows any kind of interarrival or service time distribution.

Note the mathematical attraction of the M processes is that they have the memoryless property. When arrivals are deterministic, if we observe an arrival at $t=0$ we know exactly when the next arrival will be, so to describe the state of the system we must record how long it was since the most recent arrival. However, in an M/M/1 system we do not need to record this information because of the memoryless property.

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