  We analyze the so-called Shortest Queue First (SQF) queueing discipline
whereby a unique server addresses queues in parallel by serving at any time
that queue with the smallest workload. Considering a stationary system composed
of two parallel queues and assuming Poisson arrivals and general service time
distributions, we first establish the functional equations satisfied by the
Laplace transforms of the workloads in each queue. We further specialize these
equations to the so-called "symmetric case", with same arrival rates and
identical exponential service time distributions at each queue; we then obtain
a functional equation $$ M(z) = q(z) \cdot M \circ h(z) + L(z) $$ for unknown
function $M$, where given functions $q$, $L$ and $h$ are related to one branch
of a cubic polynomial equation. We study the analyticity domain of function $M$
and express it by a series expansion involving all iterates of function $h$.
This allows us to determine empty queue probabilities along with the tail of
the workload distribution in each queue. This tail appears to be identical to
that of the Head-of-Line preemptive priority system, which is the key feature
desired for the SQF discipline.
