0, ∙∙∙ , ], and the matrix T is written as
where
For the remaining time representation, the vector
1.3.2. The PH/PH/1 system
Consider a PH/PH/1 system with interarrival times and service times that are represented by (α,T) and (β, B), respectively, with means given as a–1 and b–1 . Note that a–1 = α(I – T)–11 and b–1 = β(I – B)–11. At time t ≥ 0, let Xt be the number of items in the system,Yt the phase of interarrival times and Zt the phase of the ongoing service. It is straightforward to see that (Xt, Yt, Zt), t ≥ 0, is a DTMC with transition matrix P given as
where
and where ⊗ is the Kronecker product. For two matrices U of order n × m and V of order r × k, the product U ⊗ W is a matrix of order nr × mk (for more details, see Graham 1981).
It is well-known in the literature that for a stable PH/PH/1 system there is a matrix R, with spectral radius less than 1, which is the minimal non-negative solution to
after appropriately obtaining the boundary solution [x0, x1]. For more details, see Alfa (2016).
The idea of the Geo/Geo/1 system discussed in section 1.2 can be extended to the PH/PH/1, but requires a more involved representation and analysis. Consider the case where (α,T) depends on (β, B). The dependence can be captured by letting
with the condition that
We see that the case of the Geo/Geo/1 is a special case where (β, B) = (1, b), hence
provided
Even though this system is more complicated to analyze compared to the Geo/Geo/1 case, the idea is exactly the same and it can be studied numerically.
Next we consider a model in which there is a general service time distribution; however, there are several interarrival time distributions and the selection of the next interarrival time distribution depends on the service time distribution.
1.4. The model with multiple interarrival time distributions
We consider the case where the service times are independent and the interarrival times depend on the distribution of the service times, and not just one parameter such as the mean:
– let the service time, S, be of a general renewal type in discrete time with finite support. Let Let us write s = [s1, s2, ∙∙∙ , sM];
– let there be K interarrival times with the selection of the interarrival time depending on the service time. Let be the interarrival time between the nth and the st customer if the kth interarrival is selected. Let
– by the results in Alfa (2004), we can represent each of the K interarrival times by discrete PH distribution and in this case chose the remaining time approach. Let (α(k), T(k)) be the PH representation for the kth interarrival time distribution of dimension .
Next we define some parameters that will be used to relate the interarrival times to the service times.
1.4.1. Preliminaries
Define a K × M matrix EKM such that its elements have values of 0 or 1 and in addition
– for K ≥ M, we require that and ;
– for K ≤ M, we require that , and
Next we define a stochastic vector ψ =[ψ1, ψ2, ... ,ψK], which is a function of s, i.e. ψ = f(s), where f : RM → RK. We give some very simple examples of this type of mapping.
– Consider the case when K = M. A simple mapping of f could be ψi = si, ∀i, i.e.ψ = s. Another possible example could be
– There are two possible cases when K ≠ M.- Consider the case where K < M, an example of the mapping here could beas an example only, and- for the case of K > M, an example of the mapping here could beagain just an example.
The K interarrival times can each be written as PH distribution (α(k), T(k)), where k = 1,2, ..., K. Hence the resulting interarrival time, which is a function of the service time, is a PH distribution represented by
