Phase-type (PH) and matrix–exponential (ME) distributions both have densities of the form
\(\varvec{\alpha }\mathrm {e}^{\varvec{T}x}\varvec{v}\),
\(x>0\), where
\(\varvec{\alpha }\) is
\(1\times p\),
\(\varvec{T}\) \(p\times p\) and
\(\varvec{v}\) \(p\times 1\). The PH class is a subset of the class of ME distributions in which the parameters
\(\varvec{\alpha },\varvec{T},\varvec{v}\) are linked to a Markov process
\(J=\) \(\{ J(t)\}_{t\ge 0}\) with almost surely finite lifetime
Y and a finite number
p of (transient) states (phases). A point process analogue of a PH distribution is the Markovian arrival process (MAP), driven by a finite Markov process
\(X=\) \(\{ X(t)\}_{t\ge 0}\) such that there are Poisson arrivals with rates varying according to the states of
X and possibly additional arrivals at state changes of
X. A point process extending the MAP in a somewhat similar vein as ME distributions extend PH distributions was introduced in [
5] and termed the RAP (rational arrival process). For example, interarrival times in a MAP are PH; in a RAP, they may also be ME. For more detail and background, see [
12]. …
