Post on 11-Jan-2016
description
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