Acyclic Phase-Type Distributions

2.6 Acyclic Phase-Type Distributions

An interesting subset of PH distributions is the family of acyclic phase-type ( APH ) dis- tributions. The family can be identified by the form of their representation matrices.

24 Chapter 2. Preliminaries

An APH distribution must have at least one representation that, under some permuta- tion of its state space, has a triangular representation matrix. In other words, an APH distribution must have at least one representation whose associated graph contains no cycle or is acyclic. Such representations are called APH representations.

A triangular minimal representation of an APH distribution is an APH representation with the least number of states. In a similar way we defined the order of a PH distri- bution, we can also define the triangular order of an APH distribution [O’C93]. The triangular order of an APH distribution is the size of its triangular minimal representa- tion. As shown in [BHM87], the triangular order and the order of an APH distribution are not always the same. The triangular order of an APH distribution may exceed, but cannot be smaller than, its order.

Lemma 2.26. An acyclic phase-type representation whose size is equal to the algebraic degree of its phase-type distribution is a triangular minimal representation.

In this case, the triangular order of the distribution is then simply given by the size of the representation. Also, the order of the distribution is then equal to the triangular order of the distribution.

The following definition simplifies the way we refer to APH distributions whose triangular order is equal to their algebraic degree.

Definition 2.27. An acyclic phase-type distribution is called triangular ideal if and only if its triangular order is equal to its algebraic degree.

2.6.1 Characterization

Similar to the case of the general PH distributions, APH distributions can be character- ized by the properties of their density functions and LST s.

Theorem 2.28 ([Cum82, O’C91]). A probability distribution defined on R ≥0 is an acyclic phase-type distribution if and only if it is the point mass at zero ( δ), or it satisfies the following conditions:

1. its density function is strictly positive on R + , and

2. its Laplace-Stieltjes transform is a rational function having only real poles. Actually, [Cum82] and [O’C91] proved a stronger result than Theorem 2.28, namely

that PH distributions satisfying those conditions are not only acyclic but also have bidi- agonal PH representations. This will be discussed in more detail in Section 3.2.

Example 2.29 (An acyclic phase-type distribution). Consider a PH distribution with representation

1 (~γ, G), where ~γ = [ 1

PH(~γ, G) is an APH distribution, because its LST

s + 5.5s + 8

f (s) =

(s + 4)(s + 2)(s + 1)

2.6. Acyclic Phase-Type Distributions

is a rational function, and has only real poles. The poles are s= −1, s = −2 and s = −4. One of the APH representations of the APH distribution is ′ ′

4 , 16 , 16 ], and

1 5 (~γ 7 ,G ), where ~γ =[

2.6.2 Closure Properties

The following theorem describes the closure characterization of the set of APH distri- butions.

Theorem 2.30 ([AL82]). The family of acyclic phase-type distributions is the smallest family of distributions on R ≥0 that:

1. contains the point mass at zero ( δ) and all exponential distributions,

2. is closed under finite mixture and convolution. Therefore, the family of APH distributions can be generated by the point mass at

zero and exponential distributions together with the finite mixture and convolution operations. This result will be clearer in the next chapter when we introduce several canonical forms of APH representations. The family of APH distributions is also closed under the minimum and maximum operations.

2.6.3 Erlang and Hypoexponential Distributions

Two of the most frequently used APH distributions in this thesis are Erlang distributions and hypoexponential distributions.

Erlang Distributions Erlang distributions are formed by convolutions of several expo- nential distributions with the same rate. They are named after A GNER K. E RLANG

(1878-1929), a Danish mathematician and a pioneer in traffic engineering. Definition 2.31. A random variable

X is distributed according to an Erlang distribution with rate λ ∈R + and phase k ∈Z + if its distribution function is given by

F (t) = Pr(X ∈R ≥0 ,

P k −1 (λt) i 1 −λt −

otherwise . The probability density function of the Erlang distribution is then

An Erlang distribution with rate λ and phase k is denoted by Erl(λ, k).

A representation of Erl(λ, k) is simply a concatenation of k states, each with total outgoing rate λ as depicted in Figure 2.7(a). Such a representation is called an Erlang representation .

26 Chapter 2. Preliminaries

(a)

(b)

Figure 2.7: Erlang and Hypoexponential Representations

Hypoexponential Distributions Hypoexponential distributions are formed by convolu- tions of several exponential distributions with possibly different rates. Hence the set of Erlang distributions is a subset of hypoexponential distributions.

Let λ i , for 1 ≤ i ≤ k and k ∈ Z + , be the rates of exponential distributions forming

a hypoexponential distribution. The rates are not necessarily distinct from each other.

A representation of the hypoexponential distribution is simply a concatenation of k states, each having total outgoing rate λ i , for 1 ≤ i ≤ k, as depicted in Figure 2.7(b). Such a representation is called a hypoexponential representation. A hypoexponential representation is a bidiagonal representation that starts only from the first state, i.e.,

(~e 1 , Bi(λ 1 ,λ 2 , ···,λ k )). Lemma 2.32. Every hypoexponential representation (hence, also every Erlang represen-

tation) is a minimal representation. Proof. Consider the representation in Figure 2.7(b). The LST of the distribution of the

hypoexponential representation is

namely it is the multiplication of the LST s of all exponential distributions. The LST is in irreducible ratio form. We can then conclude that the algebraic degree of the hypo- exponential representation is k. This means that the hypoexponential representation is minimal, because its size is equal to its algebraic degree.