Refining the Basic Series

4.1 Refining the Basic Series

In Section 3.2, we described a method to obtain the ordered bidiagonal representation of a given APH representation (~ α, A). The method proceeds via the set P aths( M), where M = (S, R, ~π) is the underlying absorbing Markov chain of the APH represen- tation. Since the CTMC M is finite and acyclic, the set P aths(M) is finite in size, and each path σ ∈ P aths(M) is also of finite length. The elementary series we described in that section correspond to the elements of P aths( M).

Assume that a PH representation (~ α, A) is acyclic and of size n, and has total out- going rates λ n ≥λ n −1 ≥···≥λ 1 > 0, so ordered by their magnitudes. Now consider the family hB 1 , ···,B n i of n hypoexponential representations

B i = (~e 1 , Bi(λ i , ···,λ n )), for 1 ≤ i ≤ n − 1, and B n = (~e 1 , Bi(λ n )). These hypoexponential representations

correspond to the basic series we described in Section 3.2.

4.1. Refining the Basic Series

Lemma 4.1. For each path σ ∈ P aths(M), there are α 1 ,α 2 ,

···,α n ∈R ≥0 , where

0 P ≤α

i ≤ 1, for 1 ≤ i ≤ n, and i =1 α i = 1, such that

PH(σ) =

α i PH(B i ).

i =1

The lemma is a restatement of Lemma 1 in [Cum82], and thus of Theorem 3.2. As described in Lemma 2.11, APH representation (~ α, A) is a convex combination of

its paths. Hence this lemma implies that the representation is, furthermore, a convex combination of its basic series.

Therefore, a particular convex combination of the basic series of M constitutes a canonical representation of the APH representation. In the following, we show that the basic series as defined in [Cum82] are sometimes redundant. We again assume that the given PH representation is acyclic and of size n. Furthermore, the representation has m ≤ n distinct total outgoing rates λ (m) >λ (m−1) > ···>λ (1) > 0, so strictly ordered by their magnitudes.

Let c(σ, λ) denote the number of occurrences of states with total outgoing rate λ in path σ. We extend this to P aths( M) by defining

c( M, λ) =

max c(σ, λ).

σ ∈P aths(M)

Thus c( M, λ) denotes the maximum number of occurrences of states with total out- going rate λ on any path in CTMC M. Our main observation is that c(M, λ) can, for some particular rate λ, be considerably smaller than the number of states with total outgoing rate λ in CTMC M.

We refine the definition of the basic series in the following, and then prove that each of the elementary series is still a convex combination of the thus refined series.

h =1 c( M, λ (h) ), k 0 = 0, and l = k m . Thus, k i denotes the maximum number of occurrences of states with total outgoing rates λ (h) , for all

P For i 1 ≤ i ≤ m, let k

1 ≤ h ≤ i, in any path of CTMC M. Obviously m ≤ l. Furthermore, l ≤ n, since for each λ occurring in M, the chain contains at least c( M, λ) distinct states with total outgoing rate λ. The acyclicity of the chain is important in this observation, which gives us, ranging over all m distinct rates, the bound l for the number of states.

Definition 4.2 (Core Series). Consider the family hC 1 , ···,C l i of l hypoexponential rep-

resentations: C l = (~e 1 , Bi(λ (m) )), and for 1 ≤i≤l−1

C i = (~e 1 , Bi(λ i , ···,λ l )),

where λ j =λ (h) such that k h −1 <j ≤k h , for i ≤ j ≤ l. This family of hypoexponential representations is called the core series of CTMC M.

In the definition, we duplicate λ (h) in the series of the total outgoing rates c( M, λ (h) )- times, where c( M, λ (h) ) is the maximum number of occurrences of states with total outgoing rate λ (h) in any path of M. For instance, the core series C c (M,λ (m) )+1 looks like

λ s (m)

l −c(M,λ (m) ) −−−−−→ s l −c(M,λ (m) )+1 −−−→ s l −c(M,λ (m) )+2 −−−→ · · · −−−→ s l +1 ,

c (M,λ (m) )−times

62 Chapter 4. Operations on Erlang Distributions

while the corresponding hypoexponential representation looks like

(~e 1 , Bi(λ (m−1) ,λ (m) , ···,λ (m) )).

{z

c (M,λ (m) )−times

Example 4.3. Figure 4.1 depicts an APH representation and its elementary series. Each of the elementary series is weighted by its occurrence probability. The basic series of this representation are

B 1 = (~e 1 , Bi(1, 2, 2, 3, 4, 5, 5)),

B 2 = (~e 1 , Bi(2, 2, 3, 4, 5, 5)),

B 3 = (~e 1 , Bi(2, 3, 4, 5, 5)),

B 4 = (~e 1 , Bi(3, 4, 5, 5)),

B 5 = (~e 1 , Bi(4, 5, 5)),

B 6 = (~e 1 , Bi(5, 5)),

B 7 = (~e 1 , Bi(5)).

The elementary series, in terms of their distribution functions, can be expressed as the convex combinations of the basic series as follows

Figure 4.1: Acyclic Phase-Type Representation and its Elementary Series Consider again the APH representation in Figure 4.1. Even though there are two states

with total outgoing rate 2, and similarly with total outgoing rate 5, the maximum number of their occurrences in any path is c( M, 2) = 1 and c(M, 5) = 1. Therefore, the core series of this representation are

C 1 = (~e 1 , Bi(1, 2, 3, 4, 5)),

C 2 = (~e 1 , Bi(2, 3, 4, 5)),

C 3 = (~e 1 , Bi(3, 4, 5)),

C 4 = (~e 1 , Bi(4, 5)),

C 5 = (~e 1 , Bi(5)).

Furthermore, in terms of their distribution functions, the elementary series can be ex- pressed as the convex combinations of the core series as follows

σ 1 =C 2 and

In the following we show that each path is also a convex combination of several core series. To do that we require several concepts and a lemma. Let Bi (λ 1 ,λ 2 , ···,λ n ) be a PH -generator, and λ n ≥λ n −1 ≥···≥λ 1 > 0. The polytope of this PH -generator is a simplex (see Section 3.1.1). Let ψ ⊆ {1, 2, · · · , n} and ψ 6= ∅. With each ψ we associate a hypoexponential representation (~e 1 , Bi ψ ), where the PH -generator Bi ψ is built by all λ i ’s such that i ∈ ψ. Let Ψ denote the collection of all such ψ’s.

4.2. Convolution Operation

Lemma 4.4 (Restatement of Lemma 2.45). For each ψ ∈ Ψ, the PH distribution as- sociated with ψ is on the boundary of polytope PH(Bi(λ 1 ,λ 2 , ···,λ n )), except for that associated with ψ= {1}, which resides inside the polytope.

We can now establish the desired canonicity result. Lemma 4.5. For each path σ ∈ P aths(M), there are α 1 ,α 2 , ···,α l ∈R ≥0 , where

P 0 l ≤α

i ≤ 1, for 1 ≤ i ≤ l, and i =1 α i = 1, such that

PH(σ) =

α i PH(C i ).

i =1

Proof. Let Bi := Bi(µ 1 ,µ 2 , ···,µ l ) be the PH -generator of C 1 . Consider the polytope of PH -generator Bi. Since Bi is an ordered bidiagonal PH -generator, it is PH -simple (cf. Theorem 3.4), and the point mass at zero δ and (~e i , Bi), for 1 ≤ i ≤ l, are therefore

vertices of the polytope. However, notice that (~e i , Bi) = C i , for 1 ≤ i ≤ l. Hence to prove the lemma we just have to show that each PH distribution associated with each path σ ∈ P aths(M) (PH(σ)) resides in the polytope.

Construct the set Ψ as described in Lemma 4.4. We have to show that for each σ ∈ P aths(M), we can find ψ ∈ Ψ such that the associated representation of ψ is equivalent to the representation of σ.

Let σ ∈ P aths(M) be an arbitrary path of length k (which is at most equal to l) given by σ = (~e 1 , Bi(λ 1 ,λ 2 , ···,λ k )). First of all, by the definition of the core series, each λ i , for 1 ≤ i ≤ k, must occur in the core series. This means that we can define a mapping

f: {1, · · · , k} → {1, · · · , l}

such that λ i =µ f (i) , and thus σ = (~e 1 , Bi(µ f (1) ,µ f (2) , ···,µ f (k) )). If we, in addition, establish that f can be an injective mapping, then range(f ) is a subset ψ of the form

required above, and σ = (~e 1 , Bi ψ ), and we can conclude the proof.

The mapping f can be injective, because for each distinct λ occurring in σ, the number of its occurrences in σ is at most c( M, λ). Since

{j ∈ {1, · · · , l} | µ j =λ }

is of size c( M, λ), we can map the index subset

{j ∈ {1, · · · , k} | λ j =λ }

injectively onto that set, and this holds for each distinct λ.