1. Lecture 3. Notations and examples D. Moltchanov, TUT, Spring 2008 D. Moltchanov, TUT, Spring 2015

2. Outline Simple example Link-path formulations Common notation for link-path formulation Addon: node-link formulations

3. Why this lecture? We will be solving mathematical models of networks Abstractions … that are close to reality classified as multi-commodity network flow problems Introduce mathematical notation You’ll see it is common for all the problems There are in fact several equivalent ones Introduce important notions Again common for all the problems Consider an example in detail Motivating you…

4. Link-path formulation

5. Simple example Simple example: three nodes Triangle topology Common term node Nodes could be: routers, telco switches, SDH cross-connects Traffic between pair of nodes Common term demand volume Different meaning for different networks Could be averaged traffic volume between two nodes Could be required bandwidth Could be: number of wavelengths Pair of nodes: demand pair, demand Sometimes O-D (originator-destination) Can be directional Links can also be directional

6. Simple example: link-path Demands: bidirectional 1-2: 5 1-3: 7 2-3: 8 Routing of demands Two path for each demand Demand paths 1-2 and 1-3-2 We see that flows over paths For other demands Another concern: how much to put over those paths

7. Simple example: link-path Are there limitation on how much we put over paths? Yes! Link bandwidth! (we assume bidirectional links) We will denote links by 1-2, 2-3, 1-3 and capacities First important note: Demand: between any two nodes Link: connects two nodes directly Second important note: Same units must be used for demands and link rates You need to convert: pps/pps, Mbps/Mbps E.g. to get pps: link rate in Mbps / average packet size Link capacity unit (LCU) Demand volume unit (DVU)

8. Simple example: link-path So which demands are using links, what are implications For link 1-2 Similarly for links 1-3 and 2-3 we have What we got? (demands constraints + capacity constraints) and of course non-negativity of allocations:

9. Simple example: link-path So what we got so far? A system of equalities/inequalities Gives feasible solutions to allocations Possibly no solutions exist, possibly infinitely many Which of these solutions are of interest Depends on our objective! Question: what is the goal of your network design? Minimize the routing cost? Minimize congestion? Something else? Objective function! (AKA Utility function) Example: minimizing the total routing cost Let the cost of transmission of a unit flow over any link be 1 Can you tell me why coefficient 2?

10. Simple example: link-path So the whole problem now looks as Minimize subject to demand constraints and capacity constraints And positivity constraints

11. Simple example: link-path What is known about the problems like this? A lot… Special case of multi-commodity network flow problem Multi-commodity: multiple demands Single-commodity is way more simple We provided link-path (arc-path) formulation Optimization theory tells us This is linear programming problem As all the constraints and objective function are linear Solution? Just try to use the common-sense Assign everything on the shortest-paths to get Minimum cost is (how it is computed?)

12. Simple example: link-path Wasn’t it simple? Yes as the choice of demands and link rates favored shortest paths It is not always like this What is about the following objective function two times expensive to go over shortest paths a bit artificial but not that unrealistic sometimes especially, in non-networking applications (see aviation) Solving the problem? Trying to route everything over long paths? Capacity constraints strikes! Not feasible! No solutions? No… we have one for ! Best allocation is somewhere in-between… How to find? Linear programming!

13. Simple example: link-path One more look at objective function Sum of links costs Link cost: link load link cost unit Link load: sum of all flows crossing it One alternative: minimize delay on the most congested link Changing objective function May affect the optimal solution May dramatically affect how we find the optimal solution Sometimes the most complex thing We considered link-path formulation It also valid for directed links/demands

14. Better link-path formulation

15. New notation for link-path Why? Recall link-path notation Demands volumes:, where i,j are nodes… we are OK Links capacities :, where i,j are nodes… we are OK Paths for demand (1->2): we used nodes as indices, e.g. Paths for demands It was OK for three nodes What’s about paths for networks having N nodes? What we used is called node-identifier-based notation Additional shortcomings Some nodes may not have demands: we still need to Not all nodes are directly connected Flow variables have indices of different length More than one link between two nodes is also a problem Simply put: we have a problem modeling large networks

16. Notation for link-path Link-demand-path-identifier-based notation Compact Allows to list only necessary objects Good for moderate-to-large networks Seem strange for small networks at the first glance though… 1. Start with demands Enumerate from 1 to D Only those that are non-zero Three nodes example Demand (1->2): demand ID 1 Demand (1->3): demand ID 2 Demand (2->3): demand ID 3 We have d=1,2,3 demands In general: d=1,2,..,D demands

17. Notation for link-path 2. Continue with links Enumerate from 1 to E Only those that exist Three nodes example Link 1->2: link ID 1 Link 1->3: link ID 2 Link 2->3: link ID 3 We have e=1,2,3 links In general e=1,2,…,E links Can perform mapping of Demand volumes Link capacities

18. Notation for link-path 3. Continue with candidate paths for demand There could be more than one Enumerate from 1 to P d for demand d Note! paths have to be found prior to the solution of the task Example: demand pair (1->2) ID 1 There exist two paths, P 1 = 2 These are 1-2, 1-3-2 Path 1-2: path ID 1 Path 1-3-2: path ID 2

19. Notation for link-path Finish with path-flow variables (former ) Demand paid ID: first index Path ID for demand: second index Note the following: Previously: we saw which path is taken from indices Now: it is implicitly given

20. Simple example The allocation task now reads as Minimize (routing cost) subject to demands constraints and capaacity constraints and positivity constraints Paths must be given explicitly prior to formulation

21. Addon: node-link formulation

22. Simple example: node-link Let links and demands be directed Consider a demand pair Consider a node which is not originator nor destination Differences in formulations Link-path: considers path flows realizing a demand Node-link: considers flows passing through nodes Single node: flow in = flow out Flow conservation low (recall electricity!) Holds for end nodes assuming that sources are external

23. Simple example: node-link Get back to the three nodes example Bidirected link 1-2: now two links 1-2, 2-1 Demands should also be directinal Now sufficient to use one direction only 1->2 Consider demand (1->2) Demand (1->2) starts at 1 terminates at 2 Has volume Has two outgoing arcs 1->2, 1->3 to realize its demand via Over these arcs we send first index: arc second index: demand

24. Simple example: node-link Flow conservation: originator node 1, demand (1->2) out is positive, in is negative Flow conservation: transit node 3, demand (1->2) Flow conservation: destination node 3, demand (1->2)

25. Simple example: node-link Let’s merge what we did for demand (1->2)! Is there some redundancy? Network is symmetric with respect to links/demands Why not to set backward flows to zero?

26. Simple example: node-link

27. Let’s merge what we did for demand (1->2)! Equations we get for demand (1->2)

28. Simple example: node-link For demand (1->3) by setting we get equations For demand (2->3) by setting we get equations

29. Simple example: node-link Now consider links Each link consist of two arcs For each demand its flow on one of arcs is zero The sum of arc flows should be less than capacity Capacity constraint for link 1->2 is recalling that Capacity constraint for link 1->3 is Similarly for other links… We got: demands constraints + capacity constraints

30. Simple example: node-link Minimize Here we minimize routing costs (1 unit cost per link) Subject to demands constraints:

31. Simple example: node-link and capacity constraints and non-negativity of all Important notes Can you even get an idea what the solution could be? Usually node-link is that complex Still it is linear programming Starting from now we will use link-path notation