Almost Surely Minimal

5.4 Almost Surely Minimal

It is shown in [CM02] that in the set of all size n PH representations of some pre- specified structure, the set of all parameter values giving rise to PH distributions of algebraic degree less than n has measure zero. Stated differently, PH distributions of algebraic degree n are almost everywhere in the set of all size n PH representations of some pre-specified structure. Recall that a property holds almost everywhere if the set of points the property fails has measure zero. Therefore a PH representation whose size is greater than its algebraic degree arise not from the structure of the representation, but rather from the particular parameter values of the representations [Fac03].

In the following we will prove a somewhat stronger but more restricted result.

Lemma 5.13. Let Bi (λ 1 , ···,λ n ) be an ordered bidiagonal PH -generator. In the polytope PH(Bi(λ n 1 , ···,λ

n )), the set of all PH(~ α, Bi(λ 1 , ···,λ n )), where ~ α ∈R ≥0 and α~e ~ ≤ 1, whose algebraic degree is less than n has measure zero.

Proof. Consider the polytope of PH -generator Bi (λ 1 , ···,λ n ). Since Bi(λ 1 , ···,λ n ) is an ordered bidiagonal PH -generator, it is PH -simple (cf. Theorem 3.4). Therefore, the polytope is n-dimensional, i.e., it resides in an n-dimensional affine subspace. Let ψ ⊆ {1, 2, · · · , n} and ψ 6= ∅. With each ψ we associate a bidiagonal representation

q ψ := (~e 1 , Bi ψ ), where the PH -generator Bi ψ is built by all λ j ’s such that j ∈ ψ. Let Ψ denote the collection of all such ψ. By Lemma 2.45, the associated PH distribution of each ψ ∈ Ψ is on the boundary of the polytope.

We have shown in the proof of Lemma 4.20 that each of the n polytopes

conv( {δ, q {n} ,q {n−1,n} , · · ·, q {3,··· ,n−1,n} ,q {2,3,··· ,n−1,n} }), conv( {δ, q {n} ,q {n−1,n} , · · ·, q {3,··· ,n−1,n} ,q {1,3,··· ,n−1,n} }),

. .. conv( {δ, q {n} ,q {n−2,n} , · · ·, q {2,··· ,n−2,n} ,q {1,2,··· ,n−2,n} }), conv( {δ, q {n−1} ,q {n−2,n−1} , · · ·, q {2,··· ,n−2,n−1} ,q {1,2,··· ,n−2,n−1} }).

5.4. Almost Surely Minimal

is an (n − 1)-dimensional polytope, i.e., it resides in an (n − 1)-dimensional affine subspace.

The intersection of all of these (n − 1)-dimensional affine subspaces and the poly- tope PH(Bi(λ 1 , ···,λ n )) is exactly the region containing APH distributions of algebraic degree n − 1 or less with poles taken from {−λ 1 , −λ 2 , · · · , −λ n }. We refer to this region as Q. Thus, Q is the union of countably many (in this case n) subsets of (n − 1)- dimensional affine subspaces. But then in the n-dimensional affine subspace on which

Q and the polytope PH(Bi(λ 1 , ···,λ n )) reside, the region Q has measure zero. Remark: The lemma can also be proved using Theorem 5.6 and the fact that all

ordered bidiagonal PH -generators are PH -simple. The PH -generator of the dual of an or- dered bidiagonal representation, however, is not always PH -simple, but can be shown to be generically PH -simple in a similar manner as the proof of Theorem 5.1 in [CM02].

The lemma shows that even when we fix a particular size n representation structure (namely ordered bidiagonal) and particular parameter values for the total outgoing rates of the states of the representation, PH distributions of algebraic degree n are still almost everywhere. In this sense, the result described in Lemma 5.13 is stronger than that of [CM02] described above. On the other hand, the result is also more restricted, because it applies solely to ordered bidiagonal representations, and hence to only APH representations.

We showed in the previous lemma that almost all of the set of initial probability distributions of any ordered bidiagonal representation of size n gives rise to APH dis- tributions of algebraic degree n. Since the results of the convolution, minimum, and maximum operations can always be transformed into ordered bidiagonal representa- tions, we expect the same result can be established for them. In the rest of the section, we will prove that this is indeed the case. First, we require the following lemma.

Lemma 5.14. Let (~e 1 , Bi(λ 1 , ···,λ m )) and (~e 1 , Bi(µ 1 , ···,µ n )) be two arbitrary hypoex- ponential representations. Then the number of states in the longest core series of

1. con((~e 1 , Bi(λ 1 , ···,λ m )), (~e 1 , Bi(µ 1 , ···,µ n ))),

2. min((~e 1 , Bi(λ 1 , ···,λ m )), (~e 1 , Bi(µ 1 , ···,µ n ))), or

3. max((~e 1 , Bi(λ 1 , ···,λ m )), (~e 1 , Bi(µ 1 , ···,µ n ))),

is equal to the algebraic degree of its corresponding APH distribution.

A proof sketch of this lemma is available in Appendix B.2. Now, we can prove the main theorem.

Theorem 5.15. Convolution, minimum and maximum operations are triangular-ideal- preserving almost everywhere.

Proof. To prove the lemma, we show that given two arbitrary ordered bidiagonal PH - generators Bi 1 and Bi 2 , the set of all possible initial probability vectors ~ α and ~ β such that either con((~ α, Bi 1 ), (~ β, Bi 2 )), min((~ α, Bi 1 ), (~ β, Bi 2 )) or max((~ α, Bi 1 ), (~ β, Bi 2 )) rep-

resents a triangular-ideal APH distribution has measure zero in the set of all possible initial probability vectors ~ α and ~ β.

We accomplish this task as follows: A core series is constructed for the representa- tion produced by the operation. Our knowledge of the structure of this representation

98 Chapter 5. The Use of APH Reduction

and the total outgoing rates of all states in it allow us to do the construction. Let p

be the size (length) of the longest core series in this representation. Now, p is a new tighter lower bound for the number of states required to represent the APH distribution produced by the operation.

By using the obtained core series, we build the ordered bidiagonal PH -generator Bi (ν 1 , ···,ν p ) for the produced representation. PH -generator Bi (ν 1 , ···,ν p ) is PH -

simple (cf. Theorem 3.4). For the given initial probability vectors ~ α and ~ β, let the ordered bidiagonal representation of the representation produced by the operation be

(~γ, Bi(ν 1 , ···,ν p )). In the rest of the proof, we will show that the PH -generator of the dual represen- tation of (~γ, Bi(ν 1 , ···,ν p )) is PH -simple almost everywhere in the set of all possible α and ~ ~ β. By Theorem 5.6, if the PH -generator of the dual representation is PH -simple almost everywhere, then the algebraic degree of PH(~γ, Bi(ν 1 , ···,ν p )) is p for almost all α and ~ ~ β. Therefore, the set of ~ α and ~ β such that the representation produced by the operation represents an APH distribution whose order is not equal to its algebraic degree (i.e., not a triangular-ideal APH distribution) has measure zero.

To show that the PH -generator of the dual of (~γ, Bi(ν 1 , ···,ν p )) is PH -simple al- most everywhere in the set of all possible α and ~ ~ β, we use a method similar to that of [CM02] we mentioned above. Let B be the PH -generator of the dual representation

of (~γ, Bi(ν 1 , ···,ν p )). From Equations (2.16) and (2.17), we obtain

where matrix M −1 = diag( ~ m), and vector ~ m= −~γBi(ν

1 , ···,ν p ) . From Equation (2.24), PH -generator B is PH -simple if and only if matrix

n R −1 =[~ B B~ B ···B B], ~

· · · (Bi(ν 1 , ···,ν p ) ) ~γ ] (5.10) has rank p.

For matrix R to have full rank p (and hence for B to be PH -simple), det(R) must

be nonzero. Matrix M is nonsingular, and the determinant of M −1 is a nontrivial

polynomial (namely det(M −1 ) 6= 0) in the parameters ~γ 1 , · · · , ~γ p . It follows that the determinant is a nontrivial polynomial in the m + n parameters ~ α 1 , · · · , ~α m ,~ β 1 , · · · , ~β n . On the other hand, the determinant of matrix

· · · (Bi(ν 1 , ···,ν p ) ) ~γ ] is also a polynomial in the mentioned m + n parameters. In conclusion, det(R) is a

polynomial in the m + n parameters ~ α 1 , · · · , ~α m ,~ β 1 , · · · , ~β n .

Now, m +n det(R) = 0 defines an algebraic variety in R . An algebraic variety in R q is defined as the set of common zeros of a finite number of polynomials in q vari-

ables [CM02, Lin74]. We call an algebraic variety proper if at least one of the poly- nomials is nontrivial. A proper algebraic variety, furthermore, has measure zero in the parameter set R q . Therefore, if we prove that this algebraic variety det(R) = 0

· · · , ~α ′ m ,~ β 1 , · · · , ~β n results in a PH - generator B that is PH -simple, then det(R) 6= 0 almost everywhere. The particular realizations for the operations are given in Lemma 5.14. They are

is proper, namely that a particular realization ~ α ′ 1 ,

given by ′ α ~

1 = 1, ~ α i = 0 for 2 ≤ i ≤ m, ~β 1 = 1 and ~ β i = 0 for 2 ≤ i ≤ n.

99

5.5. Conclusion

We established that the three operations are triangular-ideal-preserving almost ev- erywhere. Hence, given two arbitrary triangular-ideal APH distributions, regardless of the size of their representations, we are almost certain that the APH distribution of their convolution, minimum, or maximum is also triangular ideal, and hence its APH representation is reducible to minimal size by applying our reduction algorithm. In other words, if we restrict ourselves to use only triangular-ideal APH distributions in

a stochastic modelling formalism that is equipped with the three operations, we will only deal with models that are almost surely minimal.