Minimum and Maximum Operations

5.3.2 Minimum and Maximum Operations

Minimum Let (~δ, D) = min((~ α, Bi(λ 1 , ···,λ m )), (~ β, Bi(µ 1 , ···,µ n ))). Figure 5.11 de- picts a possible representation of the minimum of the two ordered bidiagonal repre-

Figure 5.11: A Representation of the Minimum of Two Ordered Bidiagonal Represen-

tations (~ α, Bi(λ 1 ,λ 2 , ···,λ m )) and (~ β, Bi(µ 1 ,µ 2 , ···,µ n ))

Constructing the core series of (~δ, D), the number of states required to repre- sent PH(~δ, D) is ranging from m + n − 1, namely for the case when (~δ, D) is the minimum of two Erlang distributions, to mn when all states in both representations

(~ α, Bi(λ 1 , ···,λ m )) and (~ β, Bi(µ 1 , ···,µ n )) have distinct total outgoing rates, and all their possible sums are also distinct. From the structure of (~δ, D), we can infer that the number of distinct states with similar total outgoing rates in each ordered bidiagonal representation involved in the operation influences the number of states required in the minimal representation of PH(~δ, D). The more we have such states, the fewer states are needed in the minimal representation.

Maximum Let (~δ, D) = max((~ α, Bi(λ 1 , ···,λ m )), (~ β, Bi(µ 1 , ···,µ n ))). Figure 5.12 de- picts a possible representation of the maximum of the two ordered bidiagonal repre- sentations.

We can construct the core series of (~δ, D) and observe that the number of states required to represent PH(~δ, D) is ranging from m + n + (m + n − 1), namely for the

94 Chapter 5. The Use of APH Reduction

Figure 5.12: A Representation of the Maximum of Two Ordered Bidiagonal Represen-

tations (~ α, Bi(λ 1 ,λ 2 , ···,λ m )) and (~ β, Bi(µ 1 ,µ 2 , ···,µ n ))

case when (~δ, D) is the maximum of two Erlang distributions, to m + n + (mn), namely when all states in both (~ α, Bi(λ 1 , ···,λ m )) and (~ β, Bi(µ 1 , ···,µ n )) have distinct total

outgoing rates, and all their possible sums are also distinct. Similar to the case of the minimum operation, we can infer that the number of dis- tinct states with similar total outgoing rates in each ordered bidiagonal representation involved in the maximum operation influences the number of states required in the minimal representation of PH(~δ, D).

Triangular-Ideal Preservation In the previous subsection, we showed that the convolu- tion operation is not triangular-ideal-preserving. For the minimum and maximum op-

erations, the situation is not as clear. We close this section by proposing the following conjecture. Although we have no strong indication as to the validity of the conjecture, we find that counterexamples are hard to find.

Conjecture 5.12. The minimum and maximum operations are triangular-ideal-preserving. Algorithmic Improvement We mentioned in our characterizations of the structure of

(~δ, D) produced by the minimum and maximum operations that the number of states with similar total outgoing rates in each ordered bidiagonal representation involved in the operation influences the number of states required in the minimal representation of PH(~δ, D). We will clarify and make precise this assertion in the rest of the section.

For k ≤ m and l ≤ n, assume that k states of (~α, Bi(λ 1 , ···,λ m )) have a common total outgoing rate λ, and, similarly, l states of (~ β, Bi(µ 1 , ···,µ n )) have a common

total outgoing rate µ. Since the minimum and maximum operations are basically cross-product operations, somewhere in the underlying Markov chain of (~δ, D), we will find a partial chain like that depicted in Figure 5.13. The partial chain forms the cross product of the k λ-states and the l µ-states.

5.3. Operations on APH Representations

Now, applying the idea of the core series only to the partial chain, we can conclude that we need only k+l − 1 (λ + µ)-states—instead of as many as kl—to represent the partial chain. Moreover, the same is true for (~δ, D): assuming these are the only (λ + µ)-states in the representation, any path in it traverses at most k + l − 1 (λ + µ)- states; the paths that traverse k+l − 1 (λ + µ)-states are those that pass through states (1, 1) and (k, l) in the figure. Therefore, only k + l − 1 (λ + µ)-states are needed in the representation, regardless of the initial probabilities of the partial chain.

Figure 5.13: The Cross-Product between k λ-States and l µ-States The same reasoning can be applied to all other pairs of total outgoing rates. As-

sume that there are ′ m and n distinct total outgoing rates in (~ α, Bi(λ

1 , ···,λ m )) and ′ (~ ′ β, Bi(µ

1 , ···,µ n )), respectively. Then there are m n such pairs. The number of states that can be removed from the cross product of such a pair increases as the number of states in each member of the pair increases.

Further, assume that the number of states required in the representation (~δ, D), once all multiplicities of total outgoing rates in the cross products of all pairs are removed, is p. Then p is no less than the number of states in the core series of (~δ, D), since we assumed that ( λ + µ)-states are confined to the cross product of λ-states and

µ-states. We can build an ordered bidiagonal representation Bi(ν 1 , ···,ν p ) from these p states (total outgoing rates). Now, we can use the spectral polynomial algorithm (S PA ) to transform (~δ, D) into (~κ, Bi(ν 1 , ···,ν p )) by solving the system of equations (from Equation (3.5))

1 ≤ i ≤ p − 1, (5.9)

j =i ν j

where P is a matrix with unit row-sums (i.e., P ~e = ~e), and ~κ = ~δP. By Lemma 4.5 and Lemma 5.8, Bi (ν 1 , ···,ν p ) PH -majorizes D, and therefore the sub-stochasticity of vector ~κ is guaranteed. Once ordered bidiagonal representation (~κ, Bi(ν 1 , ···,ν p )) is obtained, Algorithm 3.13 can be used to further reduce the size of the representation.

96 Chapter 5. The Use of APH Reduction

The pre-processing—namely the removal of the multiplicities in the cross products of all total outgoing rate pairs—we just introduced is advantageous for both S PA and the reduction algorithm. As we will show in several case studies in Chapter 7, S PA con- sumes most of the computation time of the reduction procedure. From Equation (5.9), it is evident that the number of iterations in S PA can be significantly reduced if many states with similar total outgoing rates exist in each ordered bidiagonal representation used in the minimum or maximum operations.

Furthermore, since the number of states in (~κ, Bi(ν 1 , ···,ν p )) is smaller than in (~δ, D), the number of states that have to be checked—whether removable or not—in Algorithm 3.13 also decreases. There can still be, however, many states to remove,

namely those with total outgoing rates whose multiplicities in the core series of (~δ, D) cannot be detected by the pre-processing. In the next section, we will show that given two ordered bidiagonal representations whose APH distributions are triangular ideal, the number of states in the minimal representation of the minimum or the maximum of the two ordered bidiagonal representations will almost always be equal to the number of states in the core series of (~δ, D).