The max-plus algebra approach in modelling of queueing 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.

💡 Research Summary

The paper investigates a class of queueing networks composed of single‑server fork‑join nodes with infinite buffers and shows how their dynamics can be expressed compactly using max‑plus algebra. In max‑plus algebra the conventional addition and multiplication are replaced by the operations “maximum” (⊕) and “addition” (⊗). This algebraic framework is particularly suited to systems where synchronization (the join operation) and parallelism (the fork operation) dominate the timing behavior.

The authors first formalize the behavior of a fork‑join node. In a fork, a job splits into several sub‑jobs that are processed simultaneously on different servers; the overall processing time is the sum of the service times of the sub‑jobs (captured by the ⊗ operation). In a join, the job can leave the node only after all sub‑jobs have finished; the departure time is therefore the maximum of the completion times of the incoming streams (captured by the ⊕ operation). By treating each departure epoch as a scalar “time value,” the whole network can be described by an n‑dimensional state vector x(k), where x_i(k) denotes the departure time of the k‑th customer from node i.

The central contribution is the derivation of a common state‑transition equation of the form

  x(k + 1) = A ⊗ x(k)

where A is an n × n max‑plus matrix. Each entry a_ij represents the minimal time needed for a customer to travel from node i to node j, taking into account the service time at node i and any synchronization constraints imposed by downstream joins. Because the buffers are assumed infinite, waiting due to buffer overflow never occurs; the only delays are those inherent to service and synchronization, which are naturally captured by the max‑plus operations.

The paper provides a systematic procedure for constructing A from the known service‑time sequences {s_i(k)} of each node. For a fork node, the service times of the outgoing branches are summed (⊗) to obtain the effective processing time; for a join node, the incoming branches are combined by taking the maximum (⊕). By arranging these relationships in matrix form, the authors obtain explicit expressions for all a_ij. This construction works for arbitrary network topologies, but the authors illustrate it with two concrete examples.

The first example is a small three‑node network that mixes serial and parallel sections. The authors compute the corresponding A matrix, select an initial state x(0), and iterate the max‑plus equation to obtain the exact departure epochs of successive customers. The second example involves a five‑node network with multiple forks and joins, demonstrating how the same methodology scales to more intricate structures. In both cases the max‑plus formulation yields closed‑form recursions that are far more efficient than Monte‑Carlo simulation or traditional Markov‑chain analysis.

Beyond transient analysis, the authors discuss the spectral theory of max‑plus matrices. The max‑plus eigenvalue (the “critical circuit weight”) corresponds to the asymptotic cycle time of the network, while the associated eigenvector indicates the proportion of delay contributed by each node. By examining these spectral quantities, one can identify bottlenecks and assess stability: a finite eigenvalue implies that the network settles into a periodic regime, whereas an unbounded eigenvalue would signal unmanageable growth of delays.

In conclusion, the paper demonstrates that max‑plus algebra provides a powerful, analytically tractable framework for modeling fork‑join queueing networks with infinite buffers. The explicit state‑transition matrix A can be derived directly from service‑time data, enabling exact computation of departure epochs, easy evaluation of long‑run performance metrics, and straightforward identification of critical resources. The authors suggest that future work could extend the approach to finite buffers, stochastic service times, and multi‑server extensions, thereby broadening its applicability to real‑world manufacturing, computer‑system, and communication‑network contexts.