  We study a $d$-dimensional random walk with exponentially distributed
increments conditioned so that the components stay ordered (in the sense of
Doob). We find explicitly a positive harmonic function $h$ for the killed
process and then construct an ordered process using Doob's $h$-transform. Since
these random walks are not nearest-neighbour, the harmonic function is not the
Vandermonde determinant. The ordered process is related to the departure
process of M/M/1 queues in tandem. We find asymptotics for the tail
probabilities of the time until the components in exponential random walks
become disordered and a local limit theorem. We find the distribution of the
processes of smallest and largest particles as Fredholm determinants.
