Max-plus algebra models of queueing networks
It offers a unified mathematical framework for analyzing queueing networks, which is valuable for researchers in queueing theory and network performance analysis.
This paper derives a max-plus algebra representation for a class of queueing networks with arbitrary topology and single-server fork-join nodes, providing a common dynamic state equation and establishing existence conditions based on network topology.
A class of queueing networks which may have an arbitrary topology, and consist of single-server fork-join nodes with both infinite and finite 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. It is shown how the matrices inherent in particular networks may be calculated from the service times of customers. Since, in general, an explicit dynamic equation may not exist for a network, related existence conditions are established in terms of the network topology.