1. Optimal Partition of QoS Requirements on Unicast Paths and Multicast Trees Dean H.Lorenz and Ariel Orda Department of Electrical Engineering Technion – Israel Institute of Technology INFOCOM ’ 99.Eighteenth Annual Joint Conference of the IEEE Computer and Communication Societies.Proceedings.IEEE Volume:1,1999,Page(s)246~253,vol.1

2. Outline Introduction Model and problems QoS requirements Cost functions Solution to problem OPQ Solution to problem MOPQ Conclusions

3. Abstract Investigate the problem of optimal resource allocation for end-to-end QoS requirements given unicast path or multicast tree Based on local QoS requirement on each link Each link has a cost function How to partition end-to-end QoS requirements into local requirement such that overall cost is minimal

4. Introduction How to provide resources in order to meet the requirements of each connection Supporting QoS connections requires several mechanisms : QoS routing mechanism Provide QoS guarantee by allocating resources given QoS requirements and topology Cost of establishing a connection is a major consideration for connection admission

5. Introduction(cont.) Network optimization problem is how to establish connections that minimizes their cost Routing process End-to-end QoS partition Associate with each link a cost function that increases severity of local QoS requirement Provide efficient solutions for unicast and multicast connections by imposing some assumptions on link costs

6. Model and problems: QoS requirements A QoS partition with end-to-end QoS Q and a path p is a vector: Two fundamental QoS parameter classes: Bottleneck parameter Additive parameter For bottleneck parameter: for i.e:

7. QoS requirements Optimal partition for bottleneck parameter: For additive QoS requirements,a feasible partition,X p,must satisfy: Some QoS parameters are multiplicative like loss rate:, This study focuses on general partition of additive QoS requirements

8. Cost functions Associate with each local QoS requirement x l,a cost c l (x l ) Make natural assumption that c l (x l ) is higher as x l is tighter X l is delay,then c l (x l ) is non-increasing X l is bandwidth,then c l (x l ) is non-decreasing The overall cost of a partition is:

9. Meaning of cost Cost reflects the resources needed to guarantee the QoS requirement The cost is the price that user must pay to guarantee a specific QoS Associated with set-up and run-time of a connection Cost may be used for network management Assigning higher costs to those congested links

10. Problem formulation The optimal QoS partition problem is defined as follows Problem OPQ(optimal partition of QoS) Given a path P,end-to-end QoS requirement Q, find a QoS partition such that for any QoS partition

11. Solution to problem OPQ Investigate optimal solution to problem OPQ for additive QoS parameters and present efficient algorithms Further assumption QoS parameter is end-to-end delay All parameters are integers Link cost functions are non-increasing with delay and convex

12. Notations X P (D) is a feasible partition with End-to-end delay requirement D Path P If it satisfies denotes the optimal partition Denote the norm,hence X is feasible if

13. Notations(cont.) increment gain: move gain:

14. Pseudo-polynomial solution GREEDY-ADD(D,,c(.),P) algorithm 1. 2. while do 3. e 4. 5. return  augmenting the link with minimal gain,where it most affects the cost  Overall complexity is

15. Greedy-Move algorithm Starts from any feasible allocation and modifies it until it reaches optimal partition GREEDY-MOVE(X,,c(.),P) 1. Loop 2. 3. if return 4. 5.

16. Greedy-Move algorithm(cont.) Let be the distance of a given partition from optimal one Lemma Each iteration of algorithm decreases at least, unless X is a -optimal partition Theorem Algorithm greedy-move solves problem OPQ in

17. Polynomial solution Algorithm Binary-OPQ finds optimal solutions for different values of It consecutively considers smaller values of,until minimal possible value is reached BINARY-OPQ(D,c(.),P) 1. 2. start from the partition X={D,0, ……,0} 3. repeat 4. 5. 6. Until

18. Binary-OPQ algorithm The number of iteration is order of Each iteration starts from 2 -optimal partition and employs greedy-move until reach - optimal partition Algorithm binary-OPQ solves problem OPQ in

19. Solution to multicast OPQ Allocate delay on each link such that the end- to-end delay bound of each O-D pair is satisfied and cost of whole tree is minimized Notation : Source s Multicast group M A multicast tree is a set of edges For all, there exists a path from s to v that belongs to tree T The only one outgoing link from s is denoted by

20. Notation(cont.) N(l=(u,v)) denotes all of the outgoing link from v, i.e.,all of l ’ s neighbors If N(l) is an empty set,then l is a leaf T l is whole sub-tree originating from l (including l) Branches of T are denoted by Observe that

21. Problem formulation A feasible partition for a multicast tree T A set of link requirements Satisfy for all Problem MOPQ (Multicast OPQ) Given a multicast tree T,end-to-end delay requirement D,find a partition, such that for every feasible partition

22. Problem MOPQ(cont.) MOPQ(T,d) means the set of optimal partitions on a tree T with delay d C T (d) denotes tree cost function C T (d) also means the cost of optimally allocating delay d on the tree T.,where

23. Greedy properties At each iteration we augment the sub-tree that most improves overall cost,than an optimal solution is achieved Lemma If are convex,then so is C T (d) By lemma,T can be replaced by an equivalent convex link,and so can T l and However,this applies only if allocation on every sub-tree is optimal

24. Greedy properties(cont.) Lemma Let.,and let sub- partition, where, then Lemma implies for any optimal partitioned trees,we can apply greedy properties of problem OPQ Partition on and is a solution to problem OPQ on two link path (, ) Employing greedy moves between and will solve problem MOPQ

25. Tree-add algorithm Tree-add performs a augmentation on a tree T Tree-Add(X,,T) 1. If then 2. 3. else 4. for each do 5. Tree-Add(X,,T l ) 6. 7.

26. Algorithm balance Balance (X,,T) 1. If T is a leaf then 2. 3. 4. return 5. else for each do 6. Balance(X,, T l ) 7. While 8. 9. for each do 10. Tree-Add(X,, T l )

27. Algorithm balance(cont.) 11. while 12. 13. for each do 14. Tree-Add(X, -, T l ) 15. 16.

28. Conclusions QoS provisioning is characterized through cost functions,providing a powerful method for dealing with QoS networking Actual implementation of the solution in practical network architecture A sub-optimal solution that runs substantially faster might be preferable in practice Consider the impact of the chosen solution for QoS partitioning on routing process Simpler partitions should result in simpler routing