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 Problem tractability Algorithmic approaches

Contents Table

Sources of Inaccuracy in Network State Information

Flows with Bandwidth RequirementsFlows with End-To-End Delay Requirements:

Advertising of Rate Guarantees Advertising of Delay Guarantees

Conclusions

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

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

Problem SpecificationQoS 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

Flow with Bandwidth Requirements

Formal Specification: Given a bandwidth requirement W, find a path P*

such that, for any path P:lP*pl(W) lP pl(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}

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

End-to-End Delay Requirements with Rate-based Service Model

End-to-End delay bounded by scheduler

n = +cn

- Burst Size r - Minimal guaranteed ratec - Maximum packet length for the flowdl - Static delay value

Pl

ln dr

Pd )(

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: Deterministic Case Identical dl’s Identical PDF’s Exponential Distribution

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

Identical dl’s

Assumption: Propagation delay dld

Solution Algorithm (1)For each 1n 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)

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

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:

e

epj

jln

pj

jnl

dD

dD

plDp

)/()(

)/()()(

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/ )

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 lPdlD

Pl(d) - probability that link l has at most d units delay

Solution complexity is NP-complete

Identical PDF’s

Assumption: pl(d)p(d)

Solution Algorithm: Minimal hop path is an optimal solution to

problem D

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)

Observations from the Tight Constraints Case

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 minlEl

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

Split-Constraints Heuristics

Ideas behind the the Split-Constraints Heuristics: Transform the global delay constraint into local

constraints Split D into Dl’s lP

For each link, pl(Dl)=p or pl(Dl) =1

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

Pj j

Pj j

D

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:

)(

n

D Pj jn

l

n

DD

Pj jl

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

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= (jP j)

The l of layer i is (m) larger than that of layer (i-1).

How SIH Works?

Path is constructed top-downRecursively 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

Conclusion