Article: Stepwise explicit solution for the joint distribution of queue length of a MAP single-server service queueing system with splitting and varying batch size delayed-feedback Journal: International Journal of Mathematics in Operational Research (IJMOR) 2016 Vol.9 No.1 pp.39 - 64 Abstract: Applying truncation, augmentation and tridiagonalisation on infinite block matrices with infinite block matrix elements and duality properties of G/M/1 and M/G/1, and considering two Poisson arrivals as a MAP queueing network, we develop a stepwise algorithm to explicitly compute the joint distribution of the number of tasks in a system (queue length). We believe it is the first time such a development is offered in the literature. The system consists of an infinite-buffer single-server service-station, a splitter and an infinite-buffer single-mover delay-station. Tasks arrive from two sources: singly from outside and by batch from inside, the delay-station to the service-station. Both types of tasks arrive according to a Poisson process with two different parameters. Batch sizes vary between a minimum and a maximum number. A numerical example that demonstrates when the algorithm works and how the parameters must be chosen to reduce the approximation error together with an error analysis is included. Inderscience Publishers - linking academia, business and industry through research

Title: Stepwise explicit solution for the joint distribution of queue length of a MAP single-server service queueing system with splitting and varying batch size delayed-feedback

Authors: Aliakbar Montazer Haghighi; Dimitar P. Mishev

Addresses: Department of Mathematics, Prairie View A&M University, P.O. Box 519, MS2225, Prairie View, Texas 77446 0519, USA ' Department of Mathematics, Prairie View A&M University, P.O. Box 519, MS2225, Prairie View, Texas 77446 0519, USA

Abstract: Applying truncation, augmentation and tridiagonalisation on infinite block matrices with infinite block matrix elements and duality properties of G/M/1 and M/G/1, and considering two Poisson arrivals as a MAP queueing network, we develop a stepwise algorithm to explicitly compute the joint distribution of the number of tasks in a system (queue length). We believe it is the first time such a development is offered in the literature. The system consists of an infinite-buffer single-server service-station, a splitter and an infinite-buffer single-mover delay-station. Tasks arrive from two sources: singly from outside and by batch from inside, the delay-station to the service-station. Both types of tasks arrive according to a Poisson process with two different parameters. Batch sizes vary between a minimum and a maximum number. A numerical example that demonstrates when the algorithm works and how the parameters must be chosen to reduce the approximation error together with an error analysis is included.

Keywords: Poisson arrivals; splitting; queue length; MAP queueing networks; single server queueing; batch sizes; delayed feedback; truncation; augmentation; tridiagonalisation; infinite block matrices; stepwise algorithm; joint task distribution.

DOI: 10.1504/IJMOR.2016.077557

International Journal of Mathematics in Operational Research, 2016 Vol.9 No.1, pp.39 - 64

Received: 22 Sep 2014
Accepted: 11 Oct 2014
Published online: 06 Jul 2016 *

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Title: Stepwise explicit solution for the joint distribution of queue length of a MAP single-server service queueing system with splitting and varying batch size delayed-feedback

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