Fig. 1. We consider a (σ,ρ) constrained flow which traverses a succession of M nodes in a network of GR servers. The flow can be interpreted as a differentiated services aggregate flow, in which case (σ,ρ) is the sum of the parameters of the constituent microflows.

Fig. 2. Example of non-FIFO behavior of two GR nodes, traversed by a (σ,ρ) constrained flow, with σ=l, ρ=r=1, and where the propagation delay at all links is zero. a_k and b_k are, respectively, the arrival times of packets at nodes 1 and 2, and c_k are the departure times of packets from node 2. At all nodes only packet 1 takes its maximum possible delay at the node (equal to its GR clock value at the node), whereas all other packets get no delay from the nodes. The end-to-end delay of packet 1 is of 3l−2ε time units: if , packet 1 gets a larger delay than the delay bound in [11], which is of 2ltime units.

Fig. 3. Evolution of a sequence of packets, at the input to each of three GR nodes on its path, and at the output of node 3. At the input to node 1, there is a sequence with the characteristics described in the proof of Theorem 4.4, and which is (σ,ρ) constrained, with σ=4l, time units, time unit. As the propagation delay at all links is of 1 time unit, the delay experimented by the packet marked in black at nodes 1, 2 and 3 (taking into account propagation delay of the link at the output of each node) is, respectively, of 7, 10 and 13 time units (for an end-to-end delay of 30 time units), and the burstiness of the output flow at each node is, respectively, of 7l, 10l and 13l, as predicted by Theorem 4.4.

Symbols used in formulas

Comparison between the values assumed by the FIFO delay bound in Eq. (2) from [11], and by the non-FIFO delay bound in Eq. (9)^a