Distributions are useful for optimizing. But the decision whether a queue should generally be empty or have elements comes out of system requirements.

Then you can employ statistics to plan what resources you will allocate to processing the queue.

Though having capacity for Poisson event spikes isn’t nearly the only way to ensure queues trend to zero elements. For example, you can also scale the workload for each queue element depending on the length of the queue.

It’s very popular in video game graphics to allocate a given amount of frame time for ray tracing or anti aliasing, and temporally accumulate the results. So we may process a frame in a real time rendering pipeline for more or less time, depending on the state of the pipeline. Say, if the GPU is still drawing the previous frame, we can accumulate some ray traced lighting info on the CPU for the next frame for longer. If the CPU can’t spit out frames fast enough, we might accumulate ray traced lighting kernels/points for less time each frame and scale their size/impact.

Statistical distributions of frame timings don’t figure here much as workloads are unpredictable and may spike at any time - you have to adjust/scale rendering quality on the fly to maintain stable frame rates. This example assumes that we’re preparing one frame on the CPU while drawing the previous one on the GPU. But there may also be other threads that handle frames in parallel.

Other workload spikes could be predicted by things like Poisson distributions. But once again, the key takeaway is that you can’t generalise this stuff in any practical application. It all depends on system requirements and context.