Fig. 1. A simple network where (x, y) represents the QoS metrics of delay and bandwidth, respectively.

Fig. 2. Parameters on delay-bandwidth plane.

Fig. 3. A simple two-level hierarchical network.

Fig. 4. Joining QoS line l and point p. (a) p.w  l.lp.w, (b) p.w  l.up.w, (c) l.lp.w < p.w < l.up.w.

Fig. 5. Join operation of p and stair_l when rp₁.w < p.w < rp_n.w. (a) stair′ = (stair₁  p) where rp_k.w  p.w, (b) point p′ = (stair₂  p) where p.w  rp_k+1.w, (c) resulted staircase stair_new = p′  stair′.

Fig. 6. Two possible cases for stair₁ and stair₂. (a) stair₁ and stair₂ in disjoint bandwidth ranges, ; (b) stair₁ and stair₂ in overlap bandwidth ranges, .

Fig. 7. Joining two lines. (a) Disjoint bandwidth ranges, (b) overlap bandwidth ranges.

Fig. 8. Illustrations of lemmas.

Fig. 9. Service supported by A.2 to T.

Fig. 10. Service outline.

Fig. 11. Finding the next point(s) on the service outline. (a) Last point is identified, (b) there is another line on the same height, (c) there is no line on the same height.

Fig. 12. Illustration of lp.d, lp.w, up.d and up.w checking for TC and AHC.

Fig. 13. Routing loop illustration.

Fig. 14. Line segments in a loop.

Fig. 15. Delay deviation of a line at different bandwidth values.

Fig. 16. Delay deviation of a line at different bandwidth values. (a) Shaded area for accepted feasible requests, (b) shaded area for crankbacked requests.

Fig. 17. Accepted feasible requests and crankbacked requests. (a) Delay over-estimation, (b) delay under-estimation.

Fig. 18. Comparison between average actual shortest path delay, estimated delay and absolute delay deviation for simulations with advertisement history checking (AHC). (a) Brite topologies with 50 domains, (b) random topologies with 50 domains, (c) Brite topologies with 10 domains, (d) random topologies with 10 domains.

Fig. 19. Comparison between average actual shortest path delay, estimated delay and absolute delay deviation for simulations with threshold checking (TC). (a) Brite topologies with 50 domains, (b) random topologies with 50 domains, (c) Brite topologies with 10 domains, (d) random topologies with 10 domains.

Fig. 20. Success and crankback ratios. (a) Brite topologies with 50 domains, (b) random topologies with 50 domains, (c) Brite topologies with 10 domains, (d) random topologies with 10 domains.

Initial distance table of A.2

Routing entry for T of A.2

Summary on simulation parameters

Convergence time (time units)

Routing loop encounter rate