Known Results

5.2.1 Known Results

The first result is for APH distributions whose LST s have numerator polynomials of degree zero or one.

5.2. When Order = Algebraic Degree

Theorem 5.2 ([CC96]). Let the Laplace-Stieltjes transform of a phase-type distribution p be ˜ (s) f(s) =

q (s) . If all of the following conditions hold

1. p(s) and q(s) are co-prime polynomials,

2. q(s) is of degree n and has n real roots, and

3. p(s) is of degree less than or equal to one, then the phase-type distribution has APH representations of size n. It follows that all PH distributions satisfying the conditions are acyclic (cf. Theo-

rem 2.28). Furthermore, the representations described are minimal and triangular minimal, since its algebraic degree is equal to its size (cf. Lemma 2.26). When such representations are transformed into their ordered bidiagonal representations, they start from the first or the first two states. Those that start from the first state cover all hypoexponential representations and hence also all Erlang representations.

Figure 5.3 depicts the associated Cox representation of the ordered bidiagonal rep- resentation that start from the first two states. Since p(s) is of degree 1, it must be of

the form s p(s) = +λ λ , where λ ∈R + and λ 6= µ i , for 1 ≤ i ≤ n, since otherwise q(s) µ n will be of degree n − 1. From the LST , we identify that x= λ , and therefore λ>µ n .

In fact, this condition is necessary so that the LST is of a distribution function, namely that the inverse of the LST is positive for all t ∈R + [BHM87, CC96].

Figure 5.3: Cox Representation with Numerator Polynomial of Degree 1 The second result is for APH distributions whose LST s have numerator polynomials

of degree two. Theorem 5.3 ([CC96]). Let the Laplace-Stieltjes transform of a phase-type distribution

p be ˜ (s) f(s) = q (s) , where p(s) and q(s) are co-prime polynomials with real roots, and

p(s) = L(λ 1 )L(λ 2 ), λ 1 ≥λ 2 > 0, and

q(s) =

L(µ i ), µ 1 ≥µ 2 ≥···≥µ n > 0.

i =1

The phase-type distribution has APH representations of size n if and only if λ 2 >µ n and (λ 1 +λ 2 ) ≥ (µ n −1 +µ n ).

It follows that all PH distributions satisfying the conditions are acyclic, and the representations described must be minimal and triangular minimal. When such rep- resentations are transformed into their ordered bidiagonal representations, the first three states have nonzero initial probability. Figure 5.4 depicts the associated Cox representation of the ordered bidiagonal representation.

Equation (5.3) restricts all zeros of p(s) to be real. Similar to the previous case, λ 1 and λ 2 must be positive and different from any of µ i , for 1 ≤ i ≤ n, since otherwise

84 Chapter 5. The Use of APH Reduction

Figure 5.4: Cox Representation with Numerator Polynomial of Degree 2

q(s) will be of smaller degree. The condition λ 2 >µ n arises from the same reasoning as in the previous case. The condition (λ 1 +λ 2 ) ≥ (µ n −1 +µ n ) is necessary for the PH distribution to have a Cox representation of size n. From the LST , we can compute

For the case where both zeros of p(s) are complex (and hence one is the conjugate of the other), the corresponding APH distribution may or may not have an APH repre- sentation of size n. We refer back to Example 2.29. The numerator polynomial of the

shown in Equation (2.22) has zeros s= − 4 ±ı 4 7. The example showed that the APH distribution associated with this LST has a representation of size 3. On the other hand, consider the following LST

L(1)L(2)L(3)L(5)

√ The numerator polynomial of the LST is of degree 2, and has zeros s= −1 ± ı

2. We will show in Section 5.3.1 that the PH distribution associated with this LST has no APH representation of size 4.

The third result is a generalization of the previous two, namely for APH distributions whose LST s have numerator polynomials that are of any degree less than the degree of the denominator polynomials.

Theorem 5.4 ([CM03]). Let the Laplace-Stieltjes transform of a phase-type distribution p be ˜ (s) f(s) =

q (s) , where p(s) and q(s) are co-prime polynomials with real roots, and

p(s) =

L(λ i ), λ 1 ≥λ 2 ≥···≥λ m > 0, and

i =1

q(s) = L(µ i ), µ 1 ≥µ 2 ≥···≥µ n > 0, n > m.

i =1

If λ m ≥µ n ,λ m −1 ≥µ n −1 , ···,λ 1 ≥µ n −m+1 then the phase-type distribution has APH representations of size n.

Proof. Remark: We present a proof here because there are some ambiguities in the original proof provided in [CM03].

The LST of the PH distribution can be rewritten as

n −m

L(λ i −n+m )

f (s) =

i =1 L(µ i ) i =n−m+1 L(µ i ) | {z } |

{z

(a)

(b)

5.2. When Order = Algebraic Degree

Each term of part (a) in Equation (5.4) corresponds to an exponential distribution with rate µ i . Therefore, part (a) can be represented by a convolution of n − m exponential distributions, producing a hypoexponential representation of size n − m. Each term of part (b), on the other hand, corresponds to the PH distribution of the representation depicted in Figure 5.5.

Figure 5.5: Representation of Each Term of Part (b) in Equation (5.4) Part (b), then, can be represented by a convolution of m such representations,

which produces a bidiagonal representation of size m. The final representation is formed by a convolution of the hypoexponential and the bidiagonal representations, and it is of size n.

Example 5.5. Let the LST of a PH distribution be given by

1 L(5) L(4)

f (s) =

L(3) L(2) L(1)

L (3) , L (2) , and L (1) are depicted in Figure 5.6(a), (b), and (c), respectively. The convolution of the three representations is shown in Figure 5.6(d). The resulting representation is of size 3.

The representations of 1 L (5)

Figure 5.6: Representations Discussed in Example 5.5

A PH distribution satisfying the conditions in Theorem 5.4 must be acyclic, and the representation described is minimal and triangular minimal. In its ordered bidiagonal form, the first m + 1 states have nonzero initial probability. Note that the theorem only provides the necessary conditions for the PH distribution to be of (triangular) order n.

The fourth known result is described in the following theorem. It establishes a connection between the algebraic degree, the size, and the PH -simplicity of the PH - generators of a PH representation and of its dual.

Theorem 5.6 ([CC93]). Let (~ α, A) be a phase-type representation of size n. The associ- ated phase-type distribution has algebraic degree n if and only if the PH -generator of the representation and the PH -generator of its dual representation are both PH -simple.

86 Chapter 5. The Use of APH Reduction

The theorem applies not only to APH distributions. It follows from the theorem that if a PH -generator of a PH representation of a particular size n or the PH -generator of its dual is not PH -simple, then the algebraic degree of the associated PH distribution must

be less than n. It does not follow, however, that the PH distribution has a representation of smaller size, for the representation of size n might already be minimal. This theorem is especially advantageous for APH distributions. The ordered bidi- agonal PH -generator of an APH distribution is always PH -simple (cf. Theorem 3.4). It remains to check whether the PH -generator of its Cox representation, i.e., the dual of the ordered bidiagonal one, is also PH -simple to establish that the size of the ordered bidiagonal representation is equal to the algebraic degree of the APH distribution, and that therefore it is a minimal representation.

We have discussed four existing results in this section. The first three concern several conditions under which an APH distribution is triangular ideal and (therefore) ideal. These conditions are not easy and practical to check, for they work mostly on the LST domain and involve factorizations of polynomials. Nevertheless, the results are interesting in themselves, because they give us insight into the properties of APH distributions and representations. The fourth result provides the most general method to verify that the size of a PH representation is equal to the algebraic degree of its distribution.