Stochastic monotonicities in Jackson queueing networks

When starting from 0, a standard M/M/k queueing process has a second-order stochastic monotonicity property of a strong kind: its increments are stochastically decreasing (the SDI property). A first attempt to generalize this to the Jackson queueing network fails. This gives us reason to reexamine the underlying theory for stochastic monotonicity of Markov processes starting from a zero-point, in order to find a condition on a function of a Jackson network process to have the SDI property. It turns our that the total number of customers at time t has the desired property, if the network is idle at time 0. We use couplings in our analysis; they are also of value in the comparison of two networks with different parameters.