In Chapter 21, we saw one application for phase-type (PH) distributions: If we need to analyze a system whose workload involves distributions that are non-Exponential (e.g., high-variability workloads), then we can use a PH distribution to at least match 2 or 3 moments of that workload distribution. This allows us to represent the system via a Markov chain, which we can often solve via matrix-analytic methods.

In this chapter we see another application of PH distributions. Here, we are interested in analyzing networks of Processor-Sharing (time-sharing) servers (a.k.a. PS servers). It will turn out that networks of PS servers exhibit product form solutions, even under general service times. This is in contrast to networks of FCFS servers, which require Exponential service times. Our proof of this PS result will rely on phase-type distributions. This result is part of the famous BCMP theorem [16].

Review of Product Form Networks

So far we have seen that all of the following networks have product form:

• Open Jackson networks: These assume probabilistic routing, FCFS servers with Exponential service rates, Poisson arrivals, and unbounded queues.

• Open classed Jackson networks: These are Jackson networks, where the outside arrival rates and routing probabilities can depend on the “class” of the job.

• Closed Jackson networks

• Closed classed Jackson networks

We have also seen (see Exercise 19.3) that Jackson networks with load-dependent service rates have product form. Here the service rate can depend on the number of jobs at the server. This is useful for modeling the effects of parallel processing.