  The entropy production is one of the most essential features for systems
operating out of equilibrium. The formulation for discrete-state systems goes
back to the celebrated Schnakenberg's work and hitherto can be carried out when
for each transition between two states also the reverse one is allowed.
Nevertheless, several physical systems may exhibit a mixture of both
unidirectional and bidirectional transitions, and how to properly define the
entropy production in this case is still an open question. Here, we present a
solution to such a challenging problem. The average entropy production can be
consistently defined, employing a mapping that preserves the average fluxes,
and its physical interpretation is provided. We describe a class of stochastic
systems composed of unidirectional links forming cycles and detailed-balanced
bidirectional links, showing that they behave in a pseudo-deterministic
fashion. This approach is applied to a system with time-dependent stochastic
resetting. Our framework is consistent with thermodynamics and leads to some
intriguing observations on the relation between the arrow of time and the
average entropy production for resetting events.
