Transformation to Ordered Bidiagonal

3.2 Transformation to Ordered Bidiagonal

Theorem 3.2 does not provide a mechanism for transforming a given APH representa- tion to its ordered bidiagonal representation. In this section, we continue using the running example to illustrate the main idea behind the transformation. This illustra- tion does not only clarify the underlying mechanism of the algorithms later described in Section 3.3, but also provides a better insight into ordered bidiagonal representa- tions.

Let (~ α, A) be the APH representation depicted in Figure 3.1. The LST of the APH distribution is

7s + 42s + 54

f (s) =

4(s + 4)(s + 3)(s + 1)

The underlying CTMC of a PH distribution can be regarded as a transition system, in which each state is associated with some sojourn time, which is governed by some probability distribution. Thus, starting from a particular state in the transition system, the system evolves by making transitions from one state after some sojourn time to another state, and so on, until the absorbing state is reached.

2 , and spend some time there (which is distributed according to an exponential distribution with rate

In our example, the system may start in state 1 1 with probability

4), and then transition to state 1 2 with probability

4 . In state 2 the system sojourns for an exponentially distributed time with rate 3, and then transitions to state 3 with

probability 1

3 . After an exponential delay with rate 1, the system transitions from state

3 to the absorbing state. Such a trajectory from an initial state to the absorbing state is called an elementary series. Each of the elementary series is a hypoexponential representation. Recall that a hypoexponential distribution is the distribution of a convolution of several exponen- tial distributions with possibly different rates. The elementary series described above

occurs with probability 1 1 1 2 1 · 4 · 3 = 24 . Note that the sojourn times are determined by the total outgoing rates of the states in the representation.

40 Chapter 3. Reducing APH Representations

Let the underlying CTMC of the APH representation be M = (S, R, ~π). We observe that each path σ ∈ P aths(M) corresponds to an elementary series, and the probability with which the elementary series occurs corresponds to the occurrence probability P (σ) of the path. Since the APH representation is finite and acyclic, the set P aths( M) is finite, and each path σ ∈ P aths(M) is also of finite length.

Figure 3.4 shows all elementary series of (~ α, A). Each of the elementary series is depicted with its occurrence probability and LST . The representation (~ α, A) is equiv- alent to the convex combination of all its elementary series, in which each series is weighted by its occurrence probability.

4 f ˜ e 1 (s) = (s+1)

12 f ˜ e 3 1 (s) = (s+3) (s+1) (4) 4 4 1

1 8 f ˜ e 4 (s) = (s+4)

(5) 1 4 3 1 1 4 f ˜ e 5 (s) = 4 (s+4) 1 (s+1)

(6) 1 1 4 4 2 3 3 12 f ˜ e 6 (s) = (s+4) (s+3)

(7) 4 3 1 1 4 3 1 1 2 3 24 f ˜ e 7 (s) = (s+4) (s+3) (s+1)

Figure 3.4: All Elementary Series of the APH Representation in Figure 3.1 From the LST of hypoexponential representations (and hence of the elementary

series), it can be observed that exponential distributions forming the convolution can always be reordered. Here, we can define the set of the basic series. It is the set of hypoexponential representations with decreasing rates viewed from the absorbing state to each initial states. Figure 3.5 depicts all basic series of our example together with their LST s.

Remark: We will provide formal definitions of the elementary and the basic series in Chapter 4. Each elementary series is a convex combination of several basic series [Cum82]. In the example, this should be clear for elementary series (4), (6) and (7) (cf. Figure 3.4), but may not be obvious for the rest of them. In the following, we show that the elemen- tary series (3) is a convex combination of the basic series (2) and (3) (cf. Figure 3.5). Consider the LST s of elementary series (3), basic series (2) and (3), and the following holds

s+1 s+3

4 s+3 s+4 4 s+1 s+3 s+4

We rely on the following identity to obtain the right hand side of the previous equation from the left hand side: Given λ 0<λ ≤ µ and p =

In a similar fashion, the other elementary series can be expressed as the convex combinations of several basic series. Figure 3.6 shows the new expressions of all elementary series.

3.3. Transformation Algorithms

1 4 f ˜ b 1 (s) = 4 (s+4)

(2) 2 3 1 4 f ˜ b 3 2 4 (s) = (s+3) (s+4)

1 3 4 ˜ 1 3 3 4 2 1 f b 3 (s) = (s+1) (s+3) (s+4)

Figure 3.5: All Basic Series of the APH Representation in Figure 3.1

Figure 3.6: Elementary Series as Convex Combination of the Basic Series

Now, each of the new expressions of the elementary series is weighted by its occur- rence probability to obtain the initial probabilities of all states forming the elementary series. The elementary series can then be combined to form a single ordered bidiago- nal representation by summing up all the initial probabilities of each state of similar identity. As a result, an APH distribution with an ordered bidiagonal representation

19 14 (~ 15 β, Bi(1, 3, 4)) is obtained, where ~ β=[

48 , 48 , 48 ]. The ordered bidiagonal representa- tion is depicted in Figure 3.2.