QoS Routing in Networks with Inaccurate Information: Theory and Algorithms

1. QoS Routing in Networks with Inaccurate Information: Theory and Algorithms Roch A. Guerin and Ariel Orda Presented by: Tiewei Wang Jun Chen July 10, 2000

2. Motivations • Evaluate the fundamental impact of inaccuracy in state information, on the performance of QoS routing • Problem tractability • Algorithmic approaches

3. Contents Table • Sources of Inaccuracy in Network State Information • Flows with Bandwidth Requirements • Flows with End-To-End Delay Requirements: • Advertising of Rate Guarantees • Advertising of Delay Guarantees • Conclusions

4. Sources of Inaccuracy • Communication of updates in resources availability • Infrequently • Imprecisely • Two main components to the cost of timely distribution of changes in network state: • Number of entities generating updates • Frequency at which each entity generates updates

5. Inaccuracy Introduced • Loss of information about the state of individual nodes and links because of aggregation • average guarantee vs.. absolute guarantee • Gap between the actual state and its last advertised value • wait for a large enough change • wait for a minimum amount of time

6. Problem Specification • QoS Routing Environment: • Source-routing model • Link-State model • QoS requirements: • Bandwidth • End-to-end delay • Terms: • Probability distribution function (pdf’s) • Goals: • Find a path that will most likely satisfy the QoS requirement

7. Flow with Bandwidth Requirements • Formal Specification: • Given a bandwidth requirement W, find a path P* such that, for any path P: lP*pl(W) lPpl(W) • Pl(W) -- probability of link l can satisfy W units of bandwidth • Solution Algorithm (Most Reliable Path) • (1) Let Wl= - log pl, for all l  E • (2) Find the shortest path according to the metric{Wl}

8. Flows with End-to-End Delay Requirements • Rate-based service model • The bound of delay is accomplished by ensuring a minimum service rate to the flow • Requires the use of special schedulers • Delay-based service model • End-to-End delay bounds are guaranteed by concatenating local delay guarantees provided at each node/link on the path of a flow

9. End-to-End Delay Requirements with Rate-based Service Model • End-to-End delay bounded by scheduler • n =  +cn •  - Burst Size • r - Minimal guaranteed rate • c - Maximum packet length for the flow • dl - Static delay value

10. R-D Problem • Definition --- Given a maximum delay requirement D, and a path P, find a path that maximizes the probability of satisfying D • Dependency of end-to-end delay bound is only in terms of available bandwidth on each link • Solution Complexity: NP-complete

11. Tractable Solutions for Special Distribution of the Residual Rate • Four special cases: • Deterministic Case • Identical dl’s • Identical PDF’s • Exponential Distribution

12. Deterministic Case • Assumption: • Each link has a deterministic rate rl • Solution Algorithm • Running a shortest-path algorithm for each possible value of r • Time complexity • O(K(NlogN+M)) N=|V|, M=|E| • K is the number of different values for rl

13. Identical dl’s • Assumption: Propagation delay dld • Solution Algorithm • (1)For each 1n  N: Find a path of at most n hops that maximizes pl(r), where r =n/(D-nd), n=+cn • (2) Among the O(N) selected paths choose the one with maximal probability • Complexity: O(N2M)

14. Identical PDF’s • Assumption: Same probability distribution function of rate r , i.e. pl(r)  p(r) • Solution Algorithm: • Maximizes p((n/(D-dl)), i.e. minimize dl • Bellman-Ford shortest-path algorithm

15. Exponential Distribution • Assumption: Exponential distribution of residual rate. i.e. pl(r)=e-r • Solution Idea: • Maximize the probability of success over an n-hop path P which is given by:

16. An -Optimal Solution • Assumptions: • p(r)>pmin • rl on link l can only take Kl different values • Solution Algorithm: • Quantization of pdf’s: Let Wl(r)=-logpl(r) • Round up W’l(r)(0,,2,…,I); • =(log1/1-)/N; I=-logpmin/  • QP algorithm for selecting a path • Complexity:O(N3M/ )

17. End-to-End Delay Requirements with Delay-based Service Model • Specification of problem D: • Find a path P* such that, for any path P: D(P*) D(P). • D(P) - Probability that lPdlD • Pl(d) - probability that link l has at most d units delay • Solution complexity is NP-complete

18. Identical PDF’s • Assumption: • pl(d)p(d) • Solution Algorithm: • Minimal hop path is an optimal solution to problem D

19. Tight Constraints • What are the tight constraints? • End-to-End delay bound is tight • No link can afford to contribute its worst-case delay • Link delays are uniformly distributed • Two cases of uniform delay distribution: • Proportional window, (i (1-/2), i (1+/2)) • Constant window, (i - /2), i + /2)

20. Proportional Windows Simplified computation of the probability of a success path is still intractable Pseudopolynomial algorithm of acceptable complexity can be formulated in case of small value of   minlEl Constant Windows An optimal path can be found by identifying N n-hop( n{1, N}) path that is shortest with respect to the mean values l, and choose the path with the maximum probability Observations from the Tight Constraints Case

21. Split-Constraints Heuristics • Ideas behind the the Split-Constraints Heuristics: • Transform the global delay constraint into local constraints Split D into Dl’s lP • For each link, pl(Dl)=p or pl(Dl) =1

22. Split-Constraints Heuristic-Version 1 (S1) • Assumption: Dl on link l uniformly distributed on (l, l+l) • Heuristic S1: • 1)If shortest distance with respect to(l)>D,Stop • 2)If Shortest distance with respect to (l+l)<D, stop(D(P)=1) • 3) Run algorithm min-CTW(n) to find an n-hop walk P(n) that minimize: • 4) Choose the maximum path

23. Problem with Heuristic S1: • Imposition of same probability on all links does not work for the Heterogeneous inter-network environment • Solution to this drawback: • Assume that l , then the probability of success of path P is:

24. Heuristic SI • 1) If shortest distance with respect to (l) is greater than D,Stop (no solution) • 2)If Shortest distance with respect to (l+l) is less than D, stop(D(P)=1) • 3) Run Bellman-Ford algorithm to find an n-hop path that is shortest with respect to (l) • 4) Choose the maximum path

25. Apply SI in a Hierarchical Network Model (SIH) • Assumption • Link delays dl are uniformly distributed in (l,l+l). • Observation of Hierarchical Network Model • At each layer i, all l’s are identical • For a link l in layer i and for a path P wholly in layer i-1, l= (jP j) • The l of layer i is (m) larger than that of layer (i-1).

26. How SIH Works? • Path is constructed top-down • Recursively choose the best layer-i path: • Choose K layer-i paths and its corresponding layer-(i-1) path. • Identify the best solution for the ith layer by concatenating each layer-i path with corresponding layer-(i-1) path. • For each layer, apply SI algorithm • Higher value of K improve solution quality

27. Conclusion