The Minimal of the Minimum

4.3.3 The Minimal of the Minimum

We now introduce a representation of the minimum of Erlang distributions that is minimal.

Let hC 1 , ···,C l i be the core series of the standard representation of the minimum of n Erlang distributions Erl(λ i ,k i ), for 1 ≤ i ≤ n. Let Bi be the ordered bidiagonal

PH -generator associated with C 1 .

Lemma 4.10. For some stochastic vector ~ β, MinErl := (~ β, Bi) is a representation of the P minimum of n n Erlang distributions Erl(λ

−n+1 states, each with total outgoing rate P

i ,k i ), for 1 ≤ i ≤ n, and it has i =1 k i

i =1 λ i .

Proof. That MinErl is a representation of the minimum of Erlang distributions Erl(λ i ,k i ), for 1 ≤ i ≤ n, is straightforward from the construction of the core series and Lemma 4.5. The stochastic vector ~ β provides the parameters of the convex combination. The vector can be obtained by using the spectral polynomial algorithm of [HZ06b]. Property 4.9 and the way we construct the core series as described in Definition 4.2 assure us that

P the representation has n

i =1 k i −n+1 states, each with total outgoing rate i =1 λ i . P The number of states in representation n MinErl is

i =1 k i − n + 1. This means that this representation of the minimum of Erlang distributions grows linearly in n. The

main question right now is whether this representation is minimal. Lemma 4.11. The representation MinErl is minimal.

P Proof. n MinErl is an ordered bidiagonal representation of size l =

i =1 k i − n + 1. The LST of this representation—following Equation (3.8)—can be written as

1 +~ β 2 L(λ) + · · · + ~β l

−1 L(λ) +~ β l L(λ)

l −2

, P where n λ=

f (s) =

L(λ) l

i =1 λ i . Now, for MinErl to be of size l, 0 < ~ β 1 ≤ 1 must hold. We will show that none of the states of the representation is removable. Assume that for 1 < i < l, state i is removable, then

i β −1

1 +~ β 2 L(λ) + · · · + ~β i L(λ)

must be divisible by L(λ). However this cannot be true since ~ β 1 6= 0. Since none of the states in the representation is removable, and the distribution is of algebraic degree l, we conclude that the algebraic degree of the distribution is equal to the size of the representation. This proves that the representation is minimal.

68 Chapter 4. Operations on Erlang Distributions

By virtue of Lemma 4.10 and Lemma 4.11, we can conclude that the size of the minimal representation of the minimum of n Erlang distributions grows linearly in the number of involved Erlang distributions n.

Example 4.12. In this example, we demonstrate the steps described in the proof of Lemma 4.10 to obtain the minimal representation of the minimum of Erlang distri-

butions with the help of Figure 4.6. Figure 4.6(a) depicts three Erlang distributions: Erl(1, 2), Erl(2, 2), and Erl(4, 2). The standard representation of their minimum—i.e., min {Erl(1, 2), Erl(2, 2), Erl(4, 2)}—is shown in Figure 4.6(b). The standard representa- tion consists of eight transient states, each with total outgoing rate 7. Note that in the figure, a label on a state indicates the rate of the transition from the state to the absorbing state. Let (~ α, A) denote this standard representation.

We can apply the spectral polynomial algorithm (cf. Section 3.3.3) directly on (~ α, A), and we obtain that AP = PBi(7, 7, 7, 7, 7, 7, 7, 7), where

Hence, the ordered bidiagonal representation of (~ α, A) is given by (~ β, Bi(7, 7, 7, 7, 7, 7, 7, 7)), where ~

48 148 β=~ 3 αP = [0, 0, 0, 0, 343 , 343 , 7 , 0]. This ordered bidiagonal representation is de- picted in Figure 4.6(c).

Instead of directly applying S PA on the standard representation (~ α, A), Lemma 4.10 maintains that min

{Erl(1, 2), Erl(2, 2), Erl(4, 2)} can be represented by (~β ′ , Bi(7, 7, 7, 7)) as well, for some stochastic vector ~ β ′ . This is because Bi (7, 7, 7, 7) is the ordered bidi-

agonal PH -generator associated with the longest core series of the (~ α, A). According to Property 4.9, there are at most 2+2+2 − 3 + 1 = 4 states with total outgoing rate

7, and at least two states with total outgoing rate 7 traversed in any path in the un- derlying CTMC of (~ α, A). Hence, the core series of the underlying CTMC are given by

(~ e 1 , Bi(7)),( ~ e 1 , Bi(7, 7)), ( ~ e 1 , Bi(7, 7, 7)) and ( ~ e 1 , Bi(7, 7, 7, 7)). As described in the lemma, S PA can be used to obtain vector ~ β ′ . This is carried out by solving AQ = QBi(7, 7, 7, 7), from which we obtain

4.4. Maximum Operation

Hence, by Lemma 4.10 and Lemma 4.11, the minimal ordered bidiagonal representation of

48 148 (~ 3 α, A) is given by (~ β , Bi(7, 7, 7, 7)), where ~ β =~ αQ = [ 343 , 343 , 7 , 0]. This ordered bidiagonal representation is depicted in Figure 4.6(d).

Figure 4.6: (a) Three Erlang representations, (b) The standard representation of the minimum of the three Erlang distributions: a label on a state indicates its rate to the absorbing state, (c) The ordered bidiagonal representation of the standard represen- tation of the minimum produced by the spectral polynomial algorithm (S PA ), and (d) The minimal representation of the minimum.

Note that for the case of the minimum of Erlang distributions, applying S PA on the standard representation by using the basic series (cf. Figure 4.6(c)) or by using the core series (cf. Figure 4.6(d)) yields the same result. This is because all states in the standard representation have the same total outgoing rate.