The max-plus algebra approach in modelling of queueing networks
This work offers a new mathematical framework for analyzing queueing networks, which is incremental as it extends existing max-plus methods to a specific class of networks.
The paper derives a max-plus algebra representation for queueing networks with fork-join nodes, providing a state equation that relates customer departure epochs via a state transition matrix. It shows how to compute this matrix from service times and gives examples for specific networks.
A class of queueing networks which consist of single-server fork-join nodes with infinite buffers is examined to derive a representation of the network dynamics in terms of max-plus algebra. For the networks, we present a common dynamic state equation which relates the departure epochs of customers from the network nodes in an explicit vector form determined by a state transition matrix. We show how the matrix may be calculated from the service time of customers in the general case, and give examples of matrices inherent in particular networks.