Poisson Flows in Single Class Open Networks of Quasireversible Queues

It was shown in [l] and [2] that ‘he flows in open networks of 9 /M/l queues with a single class of customers obe;r the so-called ‘loop criterion’: In equilibrium a flow on a link of such a network is Poisson if and only if the link is not part of a loop. In [3] that criterion was shown to apply to networks of ./M/s queues, i.e., to Jackson networks (see [4]). It was also shown in [3] that this classification of the flows does lnot hold for multiclass networks. The purpose of this note is to provide a simple argument which proves thlat the loop criterion hol3s in a single class open networks of quasireversible queues (see e.g. [7]). A stability condition needed for the argument is shown to be satisfied by the usual networks of quasireversible nodes with bounded service rates (e.g., [S, 61). The paper is organized as follows: In the next section the basic properties of the networks under consideration are briefly reviewed and the necessary notation is introduced, In Section 3 the loop criterion is established under a stability condition. Section 4 is devoted to the discussion of that condition.