  Network Utility Maximisation (NUM) addresses the problem of allocating
resources fairly within a network and explores the ways to achieve optimal
allocation in real-world networks. Although extensively studied in classical
networks, NUM is an emerging area of research in the context of quantum
networks. In this work, we consider the quantum network utility maximisation
(QNUM) problem in a static setting, where a user's utility takes into account
the assigned quantum quality (fidelity) via a generic entanglement measure as
well as the corresponding rate of entanglement generation. Under certain
assumptions, we demonstrate that the QNUM problem can be formulated as an
optimisation problem with the rate allocation vector as the only decision
variable. Using a change of variable technique known in the field of geometric
programming, we then establish sufficient conditions under which this
formulation can be reduced to a convex problem, a class of optimisation
problems that can be solved efficiently and with certainty even in high
dimensions. We further show that this technique preserves convexity, enabling
us to formulate convex QNUM problems in networks where some routes have certain
entanglement measures that do not readily admit convex formulation, while
others do. This allows us to compute the optimal resource allocation in
networks where heterogeneous applications run over different routes.
