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MatrixGeometric Invariant Vectors
Probability Distributions of Phase Type
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A2 Ai A0 algorithmic Appl arrival process arrival rate behavior blocks buffer busy period clearly column computation conditional probability consider corresponding defined denote density differential equations eigenvalue embedded Markov chain Erlang distributions evaluate example exponential servers exponentially distributed finite follows formula GI/PH/l queue given input queue interarrival invariant probability vector invariant vector irreducible iterative Laplace-Stieltjes transform left eigenvector left transitions Lemma M. F. Neuts M/G/l type M/M/l queue Markov process Markov renewal process matrix Q methods minimal nonnegative solution nonnegative matrix nonsingular number of customers obtained Opns parameters partitioned PH-distribution phase type Poisson process positive recurrent Prob probability distributions process Q Proof QBD process queueing models queueing systems queueing theory random readily representation satisfies service rate service time distribution single server spectral radius stationary probability vector steady-state stochastic matrix stochastic models structure theorem transition probability matrix Unit values waiting time distributions zero