We consider Jackson queueing networks (JQN) with finite capacity constraints and analyze the temporal computational complexity of sampling from their stationary distribution. In the context of perfect sampling, the monotonicity of JQNs ensures that it is the coupling time of extremal sample-paths. In the context of approximate sampling, it is given by the mixing time. We give a sufficient condition to prove that the coupling time of JQNs with M queues is O(M ∑Mi=1 Ci), where Ci denotes the capacity (buffer size) of queue i. This condition lets us deal with networks having arbitrary topology, for which the best bound known is exponential in the Ci's or in M. Then, we use this result to show that the mixing time of JQNs is log2 1/∈ O(M ∑Mi=1 Ci), where ∈ is a precision threshold. The main idea of our proof relies on a recursive formula on the coupling times of special sample-paths and provides a methodology for analyzing the coupling and mixing times of several monotone Markovian networks.