  We are concerned with an $M/M$-type join the shortest queue ($M/M$-JSQ for
short) with $k$ parallel queues for an arbitrary positive integer $k$, where
the servers may be heterogeneous. We are interested in the tail asymptotic of
the stationary distribution of this queueing model, provided the system is
stable. We prove that this asymptotic for the minimum queue length is exactly
geometric, and its decay rate is the $k$-th power of the traffic intensity of
the corresponding $k$ server queues with a single waiting line. For this, we
use two formulations, a quasi-birth-and-death (QBD for short) process and a
reflecting random walk on the boundary of the $k+1$-dimensional orthant. The
QBD process is typically used in the literature for studying the JSQ with 2
parallel queues, but the random walk also plays a key roll in our arguments,
which enables us to use the existing results on tail asymptotics for the QBD
process.
