Minimal and Non-Minimal Representations

5.1 Minimal and Non-Minimal Representations

In the example in Section 3.5, the reduction algorithm produces a minimal representa- tion for the given APH representation. This can be verified by the fact that the size of the produced representation—depicted in Figure 3.12—is equal to its algebraic degree and hence the representation is minimal (cf. Theorem 2.26). The algebraic degree, in this case, is precisely the number of L-terms that are not the divisor of the numerator polynomial, which is equal to the number of states in the representation.

To arrive at a minimal representation is not always possible in general. To shed some light on this, consider the representations depicted in Figure 5.1. Figure 5.1 shows two distinct APH representations of size 5 and 4, respectively. Both representations in the figure, however, have the same PH distribution; their dis- tribution function is given by

648 −2t 3 −16t 1024 −21t

F (t) = 1 −

e , t ∈R ≥0 .

5.1. Minimal and Non-Minimal Representations

Figure 5.1: Non-Minimal and Minimal Representations

The LST of the PH distribution is

672(s 2 + 16s + 55)

f(s) = ˜

, s ∈R ≥0 .

55(s + 2)(s + 16)(s + 21)

The algebraic degree of the PH distribution is 3, and the poles of its LST are s= −2, s= −16, and s = −21.

If we insist on obtaining an ordered bidiagonal “representation” of the distribu- tion by only using states with total outgoing rates 2, 16, and 21 (therefore, of size 3), we obtain a matrix-exponential distribution whose representation is depicted in Figure 5.2. Obviously, this is no longer a PH representation, but an ME representation.

1 2 3 Figure 5.2: A Matrix-Exponential Representation of those in Figure 5.1

Now a question arises: what is the relationship between the poles of the LST of a particular APH distributions and each of its APH representations? The following lemma provides the answer.

Lemma 5.1. If −λ ∈ R is a pole of the Laplace-Stieltjes transform of an acyclic phase- type distribution, then every acyclic phase-type representation of the distribution has at least one state with total outgoing rate λ.

Proof. Every APH representation must have a unique ordered bidiagonal representa- tion of the form (~ β, Bi(λ 1 ,λ 2 , ···,λ n )). The LST of the ordered bidiagonal representa- tion expressed in L-terms is (from Equation (3.8))

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

f (s) = . (5.2)

L(λ 1 )L(λ 2 ) · · · L(λ n )

s Recall that an +λ L-term L(λ) =

The LST of the APH distribution (in irreducible ratio form) can be obtained by re- moving all common factors from the numerator and denominator polynomials in Equa- tion (5.2). Therefore, the poles of the LST of the distribution are among the poles of Equation (5.2), namely they are among the zeros of the L-terms in the denominator polynomial. But then each L-term L(λ i ) in the denominator polynomial represents a

state with total outgoing rate λ i in (~ β, Bi(λ 1 ,λ 2 , ···,λ n )), and thus in the original APH representation.

82 Chapter 5. The Use of APH Reduction

Every APH representation contains states that represent poles of the LST of the associated APH distribution. Hence, by Lemma 5.1 and Equation (5.1), any APH repre- sentation of the PH distribution depicted in Figure 5.1 must contain states with total rates 2, 16, and 21. It is clear then that this PH distribution has no APH , representation of size 3, because, as shown in Figure 5.2, the “representation” that consists only of the three states is no longer APH but ME representation. It follows that the representation depicted in Figure 5.1(b) is the smallest APH representation of the PH distribution. The

PH distribution is then of triangular order 4. It is not clear, however, whether the order of the PH distribution is also 4: it may have a cyclic PH representation of size 3.

Algorithm 3.13 cannot reduce the size of the representation depicted in Figure 5.1(a), because none of its states is removable in the sense of Section 3.4.3 (or Algorithm 3.13). However, the representation depicted in Figure 5.1(b) is of smaller size. Therefore, the algorithm is not guaranteed to produce the smallest or minimal representations. The reason behind this deficiency lies in the fact that the algorithm is bound to the set of present total outgoing rates (as L-terms), while in reality the representation depends on the interplay of total outgoing rates and the initial probability distribution. These dependencies are in full generality difficult to detect, because we are then left with the problem of finding matches over a continuous domain of candidates, akin to the non-linearity of the problem encountered in [HZ07a].

Recall that H E and Z HANG in [HZ07a] proposed an algorithm for computing the minimal ordered bidiagonal representations of APH distributions. Initially, their algo- rithm transforms the given APH representation to the ordered bidiagonal representa- tion that contains only states that represent the poles of the LST of the distribution. As shown in Figure 5.2, the resulting ordered bidiagonal representation is not necessarily of a PH distribution, but is certainly of an ME distribution. If the representation is not of a PH distribution, a new state is appended to the representation, and its total outgo- ing rate is determined by solving a system of non-linear equations. This is performed one by one until the obtained representation represents a PH distribution. The first such representation found is a minimal APH representation.

Given the representation depicted in Figure 5.1(a), for instance, the algorithm of

H E and Z HANG can produce the minimal representation depicted in Figure 5.1(b). However, even this minimal representation is not unique. As reported by the authors in [HZ07a], valid total outgoing rates—namely those that correspond to valid initial probability vectors—of an appended state form an interval of real values.