Network Design under Demand Uncertainty

1. Network Design under Demand Uncertainty Koonlachat Meesublak National Electronics and Computer Technology Center Thailand

2. Partially known Unknown Known pattern Assumptions • Assumption A: Traffic demand with the uncertainty. Traffic demand matrix

3. Possible Approaches • How to handle the uncertainty? Bandwidth reservation approaches mean-rate based peak-rate based “average cases” Pro: cheap design Con: rejection of demand requests “worst case” Pro: could handle large variation Con: the most expensive design (large safety margin) Statistical approach ?

4. Distribution of Random Demands • Assumption B: • Traffic between a node pair comes from many independent sources • By CLT, the distribution of large aggregate traffic  Normal distribution. J. Kilpi and I.Norros, “Testing the Gaussian approximation of aggregate traffic,” in Proc. Internet Measurement Workshop, 2002, pp. 49-61. R. G. Addie, M. Zukerman, and T.D. Neame, “Broadband traffic modeling: simple solutions to hard problems,” IEEE Commun. Mag., vol. 36,  Issue 8, pp. 88-95,  Aug. 1998. • Measurement experiment T. Telkamp, “Traffic characteristics and network planning,” ISMA Oct 2002.

5. Assumption B • Thus, the traffic from large aggregation is not totally uncharacterized. • Traffic distribution could be useful. Bandwidth reservation approaches mean-rate based peak-rate based “average case” Pro: cheap design Con: rejection of demand requests “worst case” Pro: could handle large variation Con: the most expensive design Based on demand ~N (m, s2) Benefits? How to deal with such demand?

6. Benefits • Why use statistical allocation? • Bandwidth is not unlimited, and is not free  consider the tradeoffs between cost and ability to handle the variation. These are the benefits between mean and peak schemes. How can we handle the uncertainty? E.g., using m+ 3scan cover 99.9% of the area. m s xs peak

7. Applications • Possible applications • A generic network design • A routing/bandwidth allocation scheme at the IP/MPLS layer that considers those tradeoffs or benefits. • A routing design that guarantees the traffic base on its demand statistics, and also based on the resource limitation along the path (as will be explained later).

8. Related optimization models • Deterministic model • Demand is known or easily estimated • Uses mean value or worst-case value • Extension: time dimension, e.g., multi-hour design. • Stochastic model • Demand can be treated as a random variable • Typical Long-term / multi-period planning design: • Stochastic Programming with Recourse [28], [30] • Robust Optimization [29] • involves forecasting of future events.

9. 3 2 1 Example: Scenario-based demand

10. Alternative approach • Chance-constrained programming (CCP) is a SP variation. It uses different probabilistic assumption, and does not assume the future events • Input: statistical information on a random demand • To handle the random demand, levels of probabilistic guarantee can be specified. “Probability that the allocated bandwidth exceeding the volume of random demand is greater than or equal to 0.95.” Level of guarantee Bandwidth allocation Demand volume

11. CCP • Medova [31] studies routing and link bandwidth allocation problem in an ATM network • Level of guarantee = 1 - Probability of blocking ATM connection request • Assumes that this Prob. is very small  the approximation eliminates statistical information of random demands. • Our work • Levels of guarantee are used. Each demand has its own guarantee value. • Aggregate traffic carried on each link is composed of two parts: certain and uncertain parts.

12. CCP • Demand statistics • A random variable x has mean (m) and variance ( s2 ) • Probabilistic guarantee ( ai) • The amount of bandwidth to be allocated, x

13. Subject to: (3.2) (3.3) (3.4) Example: CCP Formulation Minimize total bandwidth cost Chance constraints -guarantee a random demand with some level Bandwidth constraints -set limitation on network resources Non-negativity constraints

14. xis N (m, s2) (.) = cumulative distribution function -1(.) = inverse transform of (.) “certain factor” “uncertain factor” Deterministic Equivalent • Deterministic equivalent of stochastic constraint To guarantee that the link can support the random demand at least a-level, we need to allocate bandwidth at least -1(a)s beyond the mean m of the demand volume.

15. Multiple demands E.g. Let a = 0.95, -1(0.95) = 1.645 Random variables:x1, x2,…,x10 mk = 100, s2 = 100 Sum-part1 = 100*10 = 1000 Sum-part2 =1.645*(10*10) = 164.5 Each flow is guaranteed with a-level.

16. Proposed research • Goal • To develop a methodology for network design under demand uncertainty. • Need to solve a routing and bandwidth allocation problem based on CCP so achieve the benefits from statistical guarantee. • Research Approach • To develop mathematical models for a routing and bandwidth allocation problem with uncertainty constraints. • This is intended for usage in the IP layer, and will not solve the traffic glooming problem in the physical layer.

17. Basic design problem “Given network information and a demand volume matrix, determine routes and the amount of bandwidth to be allocated on such routes so that the total network cost subject to network constraints is minimized.”

18. General network design problem

19. Network Formulation Notation akLevel of guarantee of demand k Fj Fixed cost for routing on link j A cj Variable cost of adding one unit of bandwidth to link j A Input Data Set D Set of random demands A Set of links (arcs) Pk Set of predefined candidate paths Wj Bandwidth bound for total traffic demand on link j djk,p= 1 if path p Pk for flow k uses link j = 0 otherwise Decision and Output Variables f k,p= 1 if flow k selects path p Pk = 0 otherwise yj= 1 if link j is used = 0 otherwise

20. Subject to: (4.3) (4.4) (4.5) (4.6) Mathematical Problem Bandwidth constraints Flow integrity constraints Fixed charge constraints

21. Example: Bandwidth reservation

22. Experimental studies • Three network topologies: Net50 • Pre-calculated candidate path set (8 paths per set): • Max hop: 12 (Net50) • a = {0.90, 0.95, 0.95} • Wj, cj, and Fj are given. • Random demand, s2: 50-100 and m ≥ 2.857s • Use CPLEX 9.1 solver to solve a linear programming part

23. Net 50 (50 nodes, 82 links)

24. Example: Bandwidth reservation

25. Note • The number of demands and network size are crucial factors for an optimization problem. • For this network size and demand input set, computational times are in the order of hundreds of milliseconds, which are still acceptable for these studies. • Parameter a could influence route selection, especially in limited bandwidth environments.

26. Conclusions • Theoretical study • A new interpretation of the Chance-Constrained Programming optimization in the communications networks context, considering both the uncertainty and service guarantees. • A mathematical formulation for network design under traffic uncertainty is developed. This framework is expected to be applied to the virtual network design at the IP layer. • The uncertainty model is based on short-term routing and bandwidth provisioning. • Uses Chance-constraints to capture both the demand variability and levels of uncertainty guarantee.

27. Conclusions • Future work: improvement of accuracy of the model • Simulation studies on the relation between different traffic patterns and the benefit of the Chance-constraint approximation are needed. • Traffic measurement: An investigation on other traffic distributions and their effects on the uncertainty bound. • A study on the benefits of the scheme with real traffic input from measurement.

28. Questions? Thank You