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Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process

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Brzeźniak Z., Peszat S.
Studia Mathematica
|
1999
|
tom 137
|
nr 3
261-299
EN
Stochastic partial differential equations on $ℝ^d$ are considered. The noise is supposed to be a spatially homogeneous Wiener process. Using the theory of stochastic integration in Banach spaces we show the existence of a Markovian solution in a certain weighted $L^q$-space. Then we obtain the existence of a space continuous solution by means of the Da Prato, Kwapień and Zabczyk factorization identity for stochastic convolutions.
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Analysis of an MMAP/PH₁,PH₂/N/∞ queueing system operating in a random environment

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Kim C., Dudin A., Dudin S., Dudina O.
International Journal of Applied Mathematics and Computer Science
|
2014
|
tom 24
|
nr 3
485-501
EN
A multi-server queueing system with two types of customers and an infinite buffer operating in a random environment as a model of a contact center is investigated. The arrival flow of customers is described by a marked Markovian arrival process. Type 1 customers have a non-preemptive priority over type 2 customers and can leave the buffer due to a lack of service. The service times of different type customers have a phase-type distribution with different parameters. To facilitate the investigation of the system we use a generalized phase-type service time distribution. The criterion of ergodicity for a multi-dimensional Markov chain describing the behavior of the system and the algorithm for computation of its steady-state distribution are outlined. Some key performance measures are calculated. The Laplace-Stieltjes transforms of the sojourn and waiting time distributions of priority and non-priority customers are derived. A numerical example illustrating the importance of taking into account the correlation in the arrival process is presented.
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