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

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.