Matrix Structures in Queuing Models

Stationary measures on infinite graphs

“…Proof. First we prove item (3) and the direct implication in (2). Let λ 0 ∈ C \ sp(TSS) be an eigenvalue of T and let u = u S + uS ∈ E be an associated eigenvector.…”

“…All together, we have proved that the operator T − λ 0 I is surjective. On the other hand, by (2), this operator must be injective. Hence T − λ 0 I is invertible, which implies that λ 0 / ∈ sp(T).…”

“…also known as the discrete time birth-and-death process, where the state space can be interpreted as the size of a certain population (for more details see [13]). It is interesting to remark that many problems of the real world can be modeled by birth-and-death processes: frequently cited examples include problems in evolutionary biology, ecology, population genetics, epidemiology and queuing theory (see [2,7,15] and references therein). The most simple example of this class can be defined as follows: the transition probability from the state j to the state i is given by p ij where p ii = δ i , p i+1,i = b i and p i,i+1 = c i+1 (where i ∈ {0, 1, .…”