  A simple analytical solution is proposed for the stationary loss system of
two parallel queues with finite capacity $K$, in which new customers join the
shortest queue, or one of the two with equal probability if their lengths are
equal. The arrival process is Poisson, service times at each queue have
exponential distributions with the same parameter, and both queues have equal
capacity. Using standard generating function arguments, a simple expression for
the blocking probability is derived, which as far as we know is original. Using
coupling arguments and explicit formulas, comparisons with related loss systems
are then provided. Bounds are similarly obtained for the average total number
of customers, with the stationary distribution explicitly determined on $\{K,
\dots, 2K \}$, and elsewhere upper bounded. Furthermore, from the balance
equations, all stationary probabilities are obtained as explicit combinations
of their values at states $(0,k)$ for $0 \le k \le K$. These expressions extend
to the infinite capacity and asymmetric cases, i.e., when the queues have
different service rates. For the initial symmetric finite capacity model, the
stationary probabilities of states $(0,k)$ can be obtained recursively from the
blocking probability. In the other cases, they are implicitly determined
through a functional equation that characterizes their generating function. The
whole approach shows that the stationary distribution of the infinite capacity
symmetric process is the limit of the corresponding finite capacity
distributions. For the infinite capacity symmetric model, we provide an
elementary proof of a result by Cohen which gives the solution of the
functional equation in terms of an infinite product with explicit zeroes and
poles.
