L11. Link-path formulation D. Moltchanov, TUT, Spring 2015 D. Moltchanov, TUT, Spring 2008

Outline Reminder of link-path formulation Node-identifier notation Link-demand-path-identifier notation Network dimensioning problems Shortest-path routing problems Fair networks Topological design Restoration design

Problem formulation Simplest problem: given Network topology with links and rates Traffic demands between nodes Get the best routing minimizing something (e.g. cost) There are two common problem formulations Link-path formulation Node-link formulation Within one formulation there are different notations Link-path: node-identifier-based Link-path: link-demand-path-identifier-based

Link-path: node-identifier 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

Link-path: node-identifier 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)

Link-path: node-identifier 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:

Link-path: node-identifier 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?

Link-path: node-identifier So the whole problem now looks as Minimize routing cost (routing cost over any link is 1) subject to demand constraints and capacity constraints And positivity constraints

What’s wrong with node-identifier? 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

Link-path: link-demand-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

Link-path: link-demand-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

Link-path: link-demand-path 3. Continue with candidate paths for demand There could be more than one Enumerate from 1 to Pd 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, P1= 2 These are 1-2, 1-3-2 Path 1-2: path ID 1 Path 1-3-2: path ID 2

Link-path: link-demand-path Finish with path-flow variables Demand paid ID: first index Path ID for demand: second index Node-identifier Vs. Link-demand-path Note the following: Previously: we saw which path is taken from indices Now: it is implicitly given

Link-path: link-demand-path The allocation task now reads as Minimize routing cost (routing over any link costs 1) subject to demands constraints and capacity constraints and positivity constraints Paths must be given explicitly prior to formulation

Link-demand-path General procedure Demands d = 1,2,..,D, and their volumes hd Links e = 1,2,…,E, and their rates ce Paths for each demand p = 1,2,…,Pd Flow variables xdp, d = 1,2,…,D, p=1,2,…, Pd Useful for moderate-to-large networks Solution: optimization algorithms

Objective function 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

Optimization software You must get familiar with at least one package Matlab (commercial) Mathematica (commercial) Maple (commercial) MathCad (commercial) CPLEX (reduced version is free) GNU linear programming kit (free) AMPL (reduced version is free) Choose one and get yourself familiar!