CCC Processes and PH Distributions 111

6.2. CCC Processes and PH Distributions 111

Now that we are certain that the absorbing CTMC semantics of any process P ∈L always represents an APH distribution, we formalize their relationship in the following definition.

Definition 6.10. For a CCC process P ∈ L, let PH(P ) be the PH distribution associated with M P =( S, R, ~π).

In the following lemma, we prove a stronger assertion than the fact that the ab- sorbing CTMC semantics of any process P ∈ L always represents an APH distribution. The lemma establishes that for any APH distribution having no mass at t = 0, we can generate a corresponding CCC process only by using the disabling operator.

Lemma 6.11. Let PH be an acyclic phase-type distribution having no mass at t = 0. There is a process P , which is generated only by using the disabling operator, such that PH = PH(P ).

Proof. PH must have an APH representation. We can transform this representation to

a Cox distribution by using the spectral polynomial algorithm (see Section 3.3.3) and Theorem 3.5. Let (~e 1 , Cx([λ 1 ,p 1 ], [λ 2 ,p 2 ], ···,λ n )) be the obtained Cox representation, then

PH = PH(~e 1 , Cx([λ 1 ,p 1 ], [λ 2 ,p 2 ], ···,λ n )). Let process P=P 1 be defined as follows:

1. P n = (λ n ),

2. For 1 ≤ i < n,

P i = (µ) ⊳ (λ)P i +1 ,

where µ = (1 −p i )λ i and λ=p i λ i . From the form of the Cox representation described in Section 3.1.2, M P is the

underlying CTMC of Cox representation (~e 1 , Cx([λ 1 ,p 1 ], [λ 2 , p2], ···,λ n )). Therefore

PH = PH(~e 1 , Cx([λ 1 ,p 1 ], [λ 2 ,p 2 ], ···,λ n )) = PH(P ).

In the beginning of Section 6.1, we started the development of our simple calculus by laying out the intuitive interpretation of each operator of language L. This interpre- tation is actually a goal, and we have worked out the semantics to achieve it. In the following, we verify whether the intuitive interpretation is matched by the semantics.

Lemma 6.12. For all processes P, Q ∈ L:

1. con(PH(P ), PH(Q)) = PH(P.Q),

2. min(PH(P ), PH(Q)) = PH(P + Q), and

3. max(PH(P ), PH(Q)) = PH(P kQ).

A proof sketch of this lemma is available in Appendix B.3. For the case of (µ) ⊳ (λ)P , assume that the APH representation associated with M P is (~e 1 , A). By following SOS rules (2.a) and (2.b), we conclude that the representation associated with the absorbing CTMC semantics of (µ) ⊳ (λ)P is given by (~e ′ 1 ,A ′ ), where

112 Chapter 6. A Simple Stochastic Calculus

and ~e ′ 1 is a row vector whose first component is equal to 1 of appropriate dimension. We close this section by offering the following remarks. In the calculus, we pro- posed the disabling operator instead of the traditional and more general choice oper- ator. The traditional choice operator usually proceeds by selecting one from several processes based on the set of outgoing transitions from the processes. Thus the oper- ator provides a mechanism to branch to the starting points of several processes. The reason we proposed one but not the other is so that the PH -equivalence notion—an equivalence notion on CCC processes that we will introduce in the next section—is a congruence with respect to all operators of language L. With respect to the traditional choice operator, on the other hand, PH -equivalence is a not a congruence. We will show this in the end of Section 6.3.