The Maximum of More Erlang Distributions

4.4.2 The Maximum of More Erlang Distributions

In the previous subsection we discussed the maximum of two Erlang distributions. In this section, we generalize the concept to more than two Erlang distributions. Assume for simplicity that we have n Erlang distributions Erl(λ i ,k i ), for 1 ≤ i ≤ n, where λ i ’s and all their possible sums are pair-wise distinct. The maximum of the n distri- butions can be obtained by multiplying their distribution functions. In terms of their representations, Theorem 2.23(c) can be applied n − 1 times to obtained the stan- dard representation of the maximum. Based on the obtained representation, we can generalize Property 4.13–4.14, as follows.

Property 4.16. For all Q ψ ⊆ {1, 2, · · · , n}, there are

i ∈ψ k i states with total outgoing rate P

i ∈ψ λ i . Since the maximum operation described in Theorem 2.23(c) is a cross-product

operation, this property is straightforward. As an example, observe the graph in Fig- ure 4.9, which represents the maximum of three Erlang distributions. The states la- belled with ➀ , ➁ , and ➂ correspond to the case where ψ is {1}, {2}, and {3}, re- spectively. The states labelled with ➃ , ➄ , and ➅ correspond to the cases where ψ is {1, 2}, {1, 3}, and {2, 3}, respectively, while the states labelled with ➆ correspond to ψ= {1, 2, 3}.

72 Chapter 4. Operations on Erlang Distributions

Figure 4.9: A representation of the maximum of Erl(λ 1 ,k 1 ), Erl(λ 2 ,k 2 ) and Erl(λ 3 ,k 3 ), showing two opposite sides. The second cube is obtained by rotating the first cube 180 ◦ about the straight line from the leftmost state labelled with ➁ to the rightmost state labelled with ➂ .

4.4. Maximum Operation

Accumulating those states for all different sets ψ, we can conclude that the size of P the representation is Q

i ∈ψ k i .

We can think of ψ, in this case, as the set of Erlang distributions that have not yet finished as one traverses the representation from the starting state (i.e., when ψ= {1, 2, · · · , n}) to the absorbing state (i.e., when ψ = ∅).

k j Property 4.17. There are )!

j =1 k j ! distinct paths from the starting state to the absorbing state.

Q n j =1

Reasoning inductively, assume that we have the representation of the maximum of n − 1 Erlang distributions. Building the cross product of this representation with the n-th Erlang representation of length k n means that each path in the n − 1 Erlang representation is extended by k n . Furthermore, for each one of the paths we have to

P place n k

n λ n -states within a path of length i =1 k i . Hence the number of paths is

j =1 k j ! P Property 4.18. Each of these paths is of length n

(k 1 +k 2 )!k 3 !

( j =1 k j )!k n !

i =1 k i . For ψ ⊆ {1, 2, · · · , n}, at most P

i ∈ψ k i − |ψ| + 1 states with total outgoing rate i ∈ψ λ i are traversed in any path, and P P the paths containing i ∈ψ k i − |ψ| + 1 states with total outgoing rate i ∈ψ λ i exist.

The length of the paths is straightforward. According to Property 4.16, there are Q

i ∈ψ λ i . These states form a region in the representation. The regions are distinguished by their state labels in Figure 4.9. The region has a similar structure to the representation of the maximum of Erl(λ i ,k i − 1) for all i ∈ ψ. The special paths described in Property 4.18 are the paths that contain

i ∈ψ k i states with total outgoing rate

the longest sequence of P (

i ∈ψ λ i )-states while traversing within the ( i ∈ψ λ i )-region. These are the paths in the representation of the maximum of Erl(λ i ,k i

P −1) for all i ∈ ψ

whose length is i P

∈ψ (k i − 1). However, the last state in the smaller representation is also a ( i ∈ψ λ i )-state, hence they are of length

Our region construction proves the existence of such special paths.