  We consider multi--hop wireless networks carrying unicast flows for multiple
users. Each flow has a specified delay deadline, and the lossy wireless links
are modelled as binary symmetric channels (BSCs). Since transmission time, also
called airtime, on the links is shared amongst flows, increasing the airtime
for one flow comes at the cost of reducing the airtime available to other flows
sharing the same link. We derive the joint allocation of flow airtimes and
coding rates that achieves the proportionally fair throughput allocation. This
utility optimisation problem is non--convex, and one of the technical
contributions of this paper is to show that the proportional fair utility
optimisation can nevertheless be decomposed into a sequence of convex
optimisation problems. The solution to this sequence of convex problems is the
unique solution to the original non--convex optimisation. Surprisingly, this
solution can be written in an explicit form that yields considerable insight
into the nature of the proportional fair joint airtime/coding rate allocation.
To our knowledge, this is the first time that the utility fair joint allocation
of airtime/coding rate has been analysed, and also, one of the first times that
utility fairness with delay deadlines has been considered.
