Algebraic modelling and performance evaluation of acyclic fork-join queueing networks

Simple lower and upper bounds on mean cycle time in stochastic acyclic fork-join queueing networks are derived using a (max,+)-algebra based representation of network dynamics. The behaviour of the bounds under various assumptions concerning the service times in the networks is discussed, and related numerical examples are presented.

💡 Research Summary

The paper tackles the performance analysis of stochastic fork‑join queueing networks that have an acyclic topology. Such networks are common in parallel processing systems, manufacturing lines, and distributed computing, where tasks are split (forked) into multiple parallel branches and later synchronized (joined). Traditional queueing theory struggles to provide tractable results for these systems because the synchronization introduces non‑linear dependencies among service times.

To overcome this difficulty, the authors adopt the max‑plus algebraic framework, which replaces conventional addition with the maximum operator and multiplication with ordinary addition. In this algebra, the evolution of the network can be expressed as a linear‑like recursion: the completion time of a cycle is the max‑plus product of a matrix that encodes the network’s precedence relations and a vector of service times. This representation captures the essential “max” effect of the join operation while preserving a linear structure that is amenable to analytical manipulation.

The core contribution is the derivation of simple, closed‑form lower and upper bounds on the mean cycle time, denoted L and U respectively. The lower bound L is obtained by identifying the most “critical” path in the network—the path whose expected total service time is maximal. Formally, L = max_{paths p} Σ_{i∈p} E