Our Reduction Algorithm

5.2.2 Our Reduction Algorithm

The results described in the previous subsection establish several conditions under which an APH distribution is triangular ideal. We may encounter such an APH distri- bution, however, in a representation having strictly larger size than the order of its distribution. In this case, verifying the conditions will be difficult. Even more so if we wish to build a representation of the same size as the order of the distribution. Obtain- ing such minimal representation is of high interest in the field of stochastic modelling and analysis. In this section, we prove that, given an APH representation whose APH distribution is triangular ideal—no matter how large the size of the representation is—our reduction algorithm is certain to produce a minimal representation.

Definition 5.7. Let Red(~ α, A) denote the reduced representation of acyclic phase-type representation (~ α, A) obtained by applying Algorithm 3.13.

From the algorithm, Red(~ α, A) is an ordered bidiagonal representation. We need the following lemma in our main proof.

Lemma 5.8. Let (~ β, Bi(λ 1 ,λ 2 , ···,λ n )) be an ordered bidiagonal representation. For an arbitrary λ x ≥λ i ,

1 ′ ≤ i ≤ n, there is a unique sub-stochastic vector ~β such that PH(~ ′ β, Bi(λ

1 ,λ 2 , ···,λ n )) = PH(~ β , Bi(λ 1 , ···,λ i ,λ x ,λ i +1 , ···,λ n )). Proof. The LST of (~ β, Bi(λ 1 ,λ 2 , ···,λ n )) expressed in L-terms (from Equation (3.8)) is

β ~ 1 +~ β 2 L(λ 1 )+ · · · + ~β n L(λ 1 )L(λ 2 ) · · · L(λ n −1 ) , L(λ 1 )L(λ 2 ) · · · L(λ n )

which can be rewritten as β ~ 1 1 β ~ 2 L(λ 1 )+ · · · + ~β n L(λ 1 )L(λ 2 ) · · · L(λ n −1 )

. (5.5) L(λ 1 ) L(λ 2 ) · · · L(λ n )

L(λ 1 )L(λ 2 ) · · · L(λ n )

5.2. When Order = Algebraic Degree

Recall the identity in Equation (3.2), which, when expressed in L-terms, can be written as

(1 − p)

L(λ)

L(µ) L(λ)L(µ)

where λ λ ≤ µ and p = µ . Let w= 1 λ x . Then, by using the identity, the first term of Equation (5.5) can be expressed in a different way, and Equation (5.5) then becomes

1 (1 − w)

+ L(λ 1 )L(λ x ) L(λ x ) L(λ 2 ) · · · L(λ n )

We can construct an APH representation of Equation (5.6). A possible representation is depicted in Figure 5.7.

Figure 5.7: A Possible Representation of the LST in Equation (5.6) We can now build the core series of the obtained representation by following the

procedure described in Definition 4.2. Consider the multisets of the total outgoing rates in the first branch (which starts in state (1, 2)) and the second branch (which starts in state (2, 1)) in Figure 5.7. Each of these multisets is a subset of the the multiset of the total outgoing rates in the third branch (which starts in state (3, 0)). Therefore, the longest core series is formed by ordering the total outgoing rates in this third branch ascendingly, and it is of size (length) n + 1.

By using Lemma 4.5 on the resulting core series, we obtain an ordered bidiagonal representation (~ β ′ , Bi(λ 1 , ···,λ i ,λ x ,λ i +1 , ···,λ n )) of size n + 1 for some sub-stochastic vector ~ ′ β . Since throughout the procedure the LST remains the same, the two ordered

bidiagonal representations represent the same PH distribution. Now, PH -generator Bi (λ 1 , ···,λ i ,λ x ,λ i +1 , ···,λ n ) is PH -simple (cf. Theorem 3.4). By Definition 2.36, the obtained vector ~ β ′ is unique for this ordered bidiagonal representation, in the sense that this is the only sub-stochastic vector such that

1 ,λ 2 , ···,λ n )) = PH(~ β , Bi(λ 1 , ···,λ i ,λ x ,λ i +1 , ···,λ n )) holds. We can then conclude that there is a unique way of incorporating a proper new

PH(~ ′ β, Bi(λ

state with a particular total outgoing rate to an existing ordered bidiagonal represen- tation. Furthermore, since the ordered bidiagonal representation is a canonical form,

88 Chapter 5. The Use of APH Reduction

for the particular sets of total outgoing rates the obtained ordered bidiagonal represen- tation is unique. Note that λ x must not be less than λ 1 , since, in that case, the resulting representation may be of a different distribution, because −λ 1 is the largest pole of the LST of the distribution. According to Theorem 2.20, a PH distribution is characterized by the unique largest real pole of its LST .

Lemma 5.9. Let (~ α, A) be an acyclic phase-type representation. If PH(~ α, A) is triangu- lar ideal, then the size of representation Red(~ α, A) is equal to the triangular order of

PH(~ α, A). Proof. Let m be the dimension of PH -generator A, and let the ordered bidiagonal

representation of (~ α, A) be (~ β, Bi(λ 1 ,λ 2 , ···,λ m )). Let n be the algebraic degree of PH(~ α, A). Since n is also the triangular order of the distribution, PH(~ α, A) must have at least one acyclic representation of size n. Let (~γ, Bi(µ 1 ,µ 2 , ···,µ n )) be the ordered bidiagonal representation of this acyclic representation. Let P = {λ 1 ,λ 2 , ···,λ m } and Q = {µ 1 ,µ 2 , ···,µ m } be the multisets of the total

outgoing rates of states of ordered bidiagonal representations (~ β, Bi(λ 1 ,λ 2 , ···,λ m )) and (~γ, Bi(µ 1 ,µ 2 , ···,µ n )), respectively. Based on Lemma 5.1, we have Q ⊆ P. For each λ ∈ P − Q, according to Lemma 5.8, a state with total outgoing rate λ can be incorporated into (~γ, Bi(µ 1 ,µ 2 , ···,µ n )), resulting in a new ordered bidiagonal representation, say (~γ ′ , Bi), of size n + 1 having the same APH distribution, where ~γ ′ is unique for the PH -generator Bi. It follows that all λ ∈ P − Q can be incorporated into

(~γ, Bi(µ 1 ,µ 2 , ···,µ n )) in any arbitrary order, and by the uniqueness of the resulting

initial probability vectors, we must obtain (~ β, Bi(λ 1 ,λ 2 , ···,λ m )).

Therefore, each state associated with an L-term L(λ), where λ ∈ P − Q, is remov- able from (~ β, Bi(λ 1 ,λ 2 , ···,λ m )), namely (1) both the numerator and denominator polynomials of the LST of (~ β, Bi(λ 1 ,λ 2 , ···,λ m )) expressed in L-term contain L(λ); and (2) removing L(λ) from both polynomials results in a sub-stochastic initial prob- ability vector. Furthermore, the removal of all such states can be performed in any

arbitrary order. Algorithm 3.13 eliminates all removable states in a particular order: from the state with the smallest total outgoing rate to the state with the largest total outgoing rate. (For a particular L-term L(λ), the procedure described in Lemma 5.8 reverses the pro-

cedure in line 7 of the algorithm). From the uniqueness of the initial probability vec- tor for a particular ordered bidiagonal representation, eliminating all removable states

must result in (~γ, Bi(µ 1 ,µ 2 , ···,µ n )). Therefore, Red(~ α, A) = (~γ, Bi(µ 1 ,µ 2 , ···,µ n )), and the size of Red(~ α, A) is equal to the triangular order of PH(~ α, A).

In conclusion, Lemma 5.9 establishes that applying Algorithm 3.13 to any APH rep- resentation whose APH distribution is triangular ideal is certain to result in a minimal representation. The algorithm can also be applied to APH representations whose APH distribution is not triangular ideal, although in this case it is not guaranteed to result in a minimal representation.

In the following, we summarize the inner mechanism and the effects of the reduc- tion algorithm. First, we clarify some notions. For an ordered bidiagonal representa- tion (~ β, Bi(λ 1 ,λ 2 , ···,λ n )), a common factor is an L-term that exists both in the numer- ator and denominator polynomials of the LST of the ordered bidiagonal representation when it is expressed in L-terms. A common factor is removable if removing the fac-

5.2. When Order = Algebraic Degree

tor results in a valid initial probability distribution. Conversely, a common factor is irremovable if removing the factor results in a non-valid initial probability distribution.

The reduction algorithm removes all unnecessary common factors from the numer- ator and denominator polynomials of (the LST of the form of) Equation (3.8). These removable common factors must have real zeros, since all factors of the denominator polynomial have real zeros. At every step of the removals, the resulting form of the LST has a straightforward associated APH representation. Once the reduction algorithm fin- ishes, we obtain a certain expression of the LST and its associated representation. We distinguish the following possible results:

(1) If the numerator and denominator polynomials have no common factors, then the associated representation is minimal, because the triangular order (and there- fore the order) of its distribution is equal to its algebraic degree, i.e., its distribu- tion is (triangular) ideal.

(2) From result (1), it follows that if the numerator polynomial only has complex zeros, then the associated representation is minimal, because the (triangular) order of its distribution is equal to its algebraic degree, i.e., its distribution is

(triangular) ideal. (3) If the numerator and denominator polynomials have a single irremovable com-

mon factor, then the associated representation is triangular minimal, and the triangular order of its distribution is not equal to its algebraic degree, i.e., its distribution is not triangular ideal.

(4) If the numerator and denominator polynomials have more than one irremovable common factor, then the associated representation may or may not be triangular minimal, and the triangular order of its distribution is certainly not equal to its algebraic degree, i.e., its distribution is not triangular ideal.

Results (1) and (2) are straightforward. For result (3), assume that the algebraic degree of the distribution of the ordered

bidiagonal representation is n. The LST of the ordered bidiagonal representation once all of its removable common factors are removed is then of the form

L(λ) i =1 L(λ i )

p ′ Since the common factor (s) L (λ) is irremovable, the LST fragment Q n i =1 L (λ i ) has no or- dered bidiagonal representation of size n, namely there is no sub-stochastic vector ~ β

L (λ)

of dimension n such that p ′ (s) = ~ β

1 +~ β 2 L(λ 1 )+ · · · + ~β n L(λ 1 )L(λ 2 ) · · · L(λ n −1 ). Hence the PH distribution of the ordered bidiagonal representation must be of trian-

gular order more than n. But the ordered bidiagonal representation is itself of size n + 1. Therefore, the triangular order of the PH distribution is n + 1, and the ordered bidiagonal representation is indeed triangular minimal. As an example, assume that given an APH representation, Algorithm 3.13 reduces it to the representation depicted in Figure 5.1(b). This representation has one irremovable common factor, namely L(24), and therefore it is triangular minimal.

90 Chapter 5. The Use of APH Reduction

As for result (4), we may be given another APH representation, which Algorithm 3.13 then reduces to the representation depicted in Figure 5.1(a). This representation has more than one irremovable common factor, namely L(22) and L(32). Since, as we have shown, the associated distribution has a representation of size 4, this representation is not triangular minimal.