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

Motivations §Evaluate the fundamental impact of inaccuracy in state information, on the performance of QoS routing l Problem tractability l Algorithmic approaches

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

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

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

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

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

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

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 §d l - Static delay value

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

Tractable Solutions for Special Distribution of the Residual Rate §Four special cases: l Deterministic Case l Identical d l ’s l Identical PDF’s l Exponential Distribution

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

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

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

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

An  -Optimal Solution §Assumptions: l p(r)>p min l r l on link l can only take K l different values §Solution Algorithm: l Quantization of pdf’s: Let W l (r)=-logp l (r) l Round up W’ l (r)  (0, ,2 ,…,I  ); l  =(log1/1-  )/N; I=  -logp min /   l QP algorithm for selecting a path §Complexity:O(N 3 M/  )

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

Identical PDF’s §Assumption: l p l (d)  p(d) §Solution Algorithm: l Minimal hop path is an optimal solution to problem D

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

Observations from the Tight Constraints Case §Proportional Windows l Simplified computation of the probability of a success path is still intractable l Pseudopolynomial algorithm of acceptable complexity can be formulated in case of small value of   min l  E  l §Constant Windows l 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

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

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

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

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

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

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

Conclusion