Project SARU How to situate Access Remote Units

Project SARU How to situate Access Remote Units and construct a minimal cost fibre optic cable network July 2006 Ivana Ljubic University Vienna Bertram Wassermann Telekom Austria Business & Market Intelligence / OR 1

1 Problem Definition Overview Introduction Broadband demand increases. • New products and Services (ADSL TV) • Number of customers still increasing Existing local area access networks are based on copper cables • Limited with respect to bandwidth and distance • Will not cover upcoming demand Fibre optic • • • technology is the alternative nearly unlimited bandwidth used for core – net rarely for LAN Consequence • Creating a new network -> network design problem Terms known in the industry • FTTH, Fibre To The Home • FTTC, Fibre To The Curb 2 Business & Market Intelligence / OR

1 Problem Definition Overview Introduction Fibre To The Curb (FTTC) • A Access Remote Unit (ARU) is placed close (“at the curb”) to several customers • “Last few meters” still copper • The ARU functions as a translator between copper (electricity) and optical medium (light) • Serves also as a multiplexer (Customers share Fibre) Fibre To The Home (FTTH) • No copper between customer and switching centre anymore • Customers are directly connected via fibre optic cable • Probably no multiplexing (or only small scale) • Passive, no need electricity Solution to these network design problems: • Steiner Trees and its capacitated variants • Well studied • Although NP-hard, fast algorithms do exist 3 Business & Market Intelligence / OR

1 Problem Definition Overview Introduction Problems with FTTH and FTTC • Expensive as a country-wide approach • Inefficient: Telekom Austria wants to be prepared for any customer, but knows not all customers will come. • FTTH or FTTC probably suitable for certain LANs or specific parts of LANs Search for an alternative • SARU, Situating Access Remote Unit • Fibre as close to the customer as necessary and as far as possible Key idea • • • Within a certain distance (L) of the customer an ARU (Access Remote Unit) has to be placed / situated which houses this customer. Copper network still supplies last mile At the moment L = 600 m Distance Metric • Length of cable is used 4 Business & Market Intelligence / OR

2 Problem Definition Graph Typical LAN structure Switching centre Root of the Copper Tree Source node Customer nodes Switching nodes Copper cables Copper tree Leaves are customers But customers need not be leaves Graph structure should be tree-like. Big pre-processing problem! 5 Business & Market Intelligence / OR

2 Problem Definition Graph Potential Fibre Optics Network Potential Fibre Optics Lines with intersection nodes All nodes should be connected to the switching centre The Fibre Optics net should form a connected graph! *) FON is not shaped like a rectangular grid! Shape indicates, that FON may be of different form then copper net. However, nets are superimposed 6 Business & Market Intelligence / OR

2 Problem Definition Graph Potential Fibre Optics Network Potential Fibre Optics Lines with intersection nodes Additional Nodes: Intersection points of FOL and Copper Net 7 Business & Market Intelligence / OR

2 Problem Definition Graph Potential Fibre Optics Network Potential Fibre Optics Lines with intersection nodes Additional Nodes: Intersection points of FOL and Copper Net Potential ARU positions Potential ARU Positions are chosen in the vicinity of intersections of copper net and fibre optic net 8 Business & Market Intelligence / OR

2 Problem Definition Graph Distance Condition L and Edge Directions Assignement of Customer to potential ARUs under Distance Condition Additional Condition: Never go up the tree, always go down towards root. Consequently: Copper edges are directed. But Fibre Optic edges may be used in one of the two directions. 9 Business & Market Intelligence / OR

2 Problem Definition Graph Distance Condition L and Edge Directions Assignement of Customer to potential ARUs under Distance Condition Alternative Representation of the Copper Net obeying Distance condition 10 Business & Market Intelligence / OR

2 Problem Definition Graph Optimization Problem Find Positions for ARUs and create Fibre Optic Network such that • all customers are served • all ARUs are connected to the root by fibre optic lines • all other constraints are met (length L) • all this is done at minimal cost 11 Business & Market Intelligence / OR

2 Problem Definition Graph Comparison with FTTH and FTTC In FTTH(C) the end-nodes (ARU positions) are given and therefore fixed. No optimisation of their position is necessary. This optimisation formulation corresponds to the Steiner Tree Problem. In our problem the graph consist of two strictly separated layers (copper network, potential FON) and a set of nodes potentially connecting them. In FTTH and FTTC there is “just” one layer and no set of designated nodes besides customer nodes. 12 Business & Market Intelligence / OR

3 Related Problem Connected Facility Location Problem (Con. FL) Given a graph G=(V, E), with lengths on the edges, with a subset of facilities, their opening costs and client demands. Our goal is to: • Pick a set of facilities to open • Assign each demand to an open facility • Connect all open facilities by a Steiner tree • Minimize the costs of opening and assigning facilities, plus the cost of the Steiner tree Our problem reduces to Con. FL if edge installation costs are M*length. Approximation algorithms: • Gupta et al. (2003): randomized 3. 55 -factor algorithm (no opening costs) • Swamy & Kumar (2002): 9 -approx. algorithm for general case 13 Business & Market Intelligence / OR

3 Related Problem Capacitated Local Access Network Design (Cap. LAN) Simplifying our problem: • For already placed access nodes, find minimum-cost capacity installation of the fiber optic network. • Also known as Network Loading Problem. Edge-cost function depends on capacity and may be piecewise-linear or step function. Uniform capacities: • Edge-cost function the same for all edges Single-sink buy-at-bulk • Approx. algorithms: Gupta et al. (2003) • Polyhedral approaches: Magnanti (1995), Günlück (1999) Non-uniform capacities: • Dahl & Stoer (1998): cutting plane approach We propose our problem-specific non-uniform ILP formulation 14 Business & Market Intelligence / OR

4 Problem Definition Cost Customer Demand With every customer a certain demand di is associated Demand in terms of copper lines (twisted pairs of copper lines) d 2 d 1 d 4 d 3 d 5 Not in the sense of bandwidth Rule: Demand has to be completely satisfied d 6 di dn di+1 15 Business & Market Intelligence / OR

4 Problem Definition Cost related to the Copper Net d 2 d 1 d 4 d 3 The copper network has to be incorporated as it is. No alteration allowed! d 5 d 6 No cost due to copper network. di dn di+1 16 Business & Market Intelligence / OR

4 Problem Definition Cost related to the potential ARU locations cost. ARU( location, demand) Location cost factors are: • Outdoor or indoor • Electricity • Rental • Development Demand cost factors are: • Type of ARU (mainly size = number of copper lines to be served) Cost function is a step function (also in terms of demand) Buy at Bulk principle: Price per unit (=served copper line) decreases with increasing size of ARUs produce demand. #Fibre Optic Lines depends on type of ARU 17 Business & Market Intelligence / OR

4 Problem Definition Cost related to the potential Fibre Optic Network In general: cost. FON( length, demand) per edge But: The potential FON is a union of 3 layers. Layer 1: Dark Fibre Existing Fibre Optic Lines which are not in use Layer 2: Empty Pipes Empty pipes where fibre optic cables may be inserted Layer 3: Excavation Excavating trenches and laying new pipes None of the layers need to form a connected graph. New trenches usually follow roadmaps Two adjacent nodes of the potential FON may be connected by any combination of the 3 edge types! 18 Business & Market Intelligence / OR

4 Problem Definition Cost Graph Structure of FON Any combination of the 3 edge types may connect two nodes. Into both directions 19 Business & Market Intelligence / OR

4 Problem Definition Cost related to the potential Fibre Optic Network Dark Fibre Edge: cost. DF ( length, demand) = const The cost for dark fibre may by viewed as being constant. It is independent of the length of the line. The work cost resulting from lighting the lines is a constant compared to costs resulting from other layers. 20 Business & Market Intelligence / OR

4 Problem Definition Cost related to the potential Fibre Optic Network Dark Fibre Edge: cost. DF ( length, demand) = const Empty Pipes: cost. EP ( length, demand) = length*cost. EP/UL (demand) Insertion cost for fibre optic cables is linear in terms of edge length Need to know cost of cables per unit length Like for ARUs Cost function is a step function with respect to demand. Again Buy at Bulk principle: Price per unit (=optic fibre) decreases with increasing size of fibre optic cables 21 Business & Market Intelligence / OR

4 Problem Definition Cost related to the potential Fibre Optic Network Dark Fibre Edge: cost. DF ( length, demand) = const Empty Pipes: cost. EP ( length, demand) = length*cost. EP/UL (demand) Excavation: cost. Ex. T ( length, location, demand) = length*cost. Ex. T/UL (location, demand) Excavation costs depend linearly on the edge length Excavation costs depend on location in two ways: • regionally cost may differ (big city, small city, country -side) • surface conditions (concrete, soil, …) Cost function obeys economies of scale (compare Buy at Bulk principle) Simplification: Costs are based on the assumption, trenches are filled completely with pipes which are completely filled with fibre cable. 22 Business & Market Intelligence / OR

4 Problem Definition Cost related to the potential Fibre Optic Network Dark Fibre Edge: cost. DF ( length, demand) = const Empty Pipes: cost. EP ( length, demand) = length*cost. EP/UL (demand) Excavation: cost. Ex. T ( length, location, demand) = length*cost. Ex. T/UL (location, demand) cost. FON( length, location, demand) per edge = const + length * [cost. EP/UL (demand) +cost. Ex. T/UL (location, demand)] Cost function is dominated by excavation costs. Cheapest contribution from Dark Fibre. With respect to free capacities it will be the other way round. 23 Business & Market Intelligence / OR

Solution Strategy 5 Overview 0 Pre-Phase, • • • Heuristic solution A (really) fast algorithm for a first solution Finding a feasible solution for a given instance Initial upper bound for exact (branch-and-bound based) algorithm 1 Phase 1: Solving the simplified problem: • To improve the solution • To study the cost function 2 Phase 2: Solving the problem • To find an exact algorithm • Study the approximation qualities of heuristic solutions 24 Business & Market Intelligence / OR

6 Phase 0 & 1 Simplified Optimization Problem 0 Start with copper net 1 Find “optimized” Positions for ARUs heuristically 2 Switch to Fibre Optic Net 25 Business & Market Intelligence / OR

6 Phase 0 & 1 Simplified Optimization Problem 0 Start with copper net 1 Find “optimized” Positions for ARUs heuristically 2 Switch to Fibre Optic Net Create fibre optic network with: 3 Phase 0 Heuristic Algorithm Phase 1 Integer Linear Program (exact) 26 Business & Market Intelligence / OR

Pre-Phase, Heuristic Solution 6 Minimal number of ARUs Idea: Optimal solution will “minimize” the number of ARUs necessary to satisfy all demand. Hence, a set of ARUs satisfying all demand minimal in number will approximate the optimal (=cost minimal) solution. Algorithm 1 Pick customer furthest away from source. 2 Choose potential ARU node furthest away from this customer still valid under distance condition L 3 Install ARU at this position and serve all customers of sub tree rooted at this node 4 Ignore sub-tree and proceed form step 1 27 Business & Market Intelligence / OR

Pre-Phase, Heuristic Solution 6 Minimal number of ARUs Solution is unique Of all solutions with minimal number of nodes it’s the one where no ARU can be moved closer to the source node without violating the distance condition L for at least one customer. Dropping this condition gives rise to different solutions For example: ! Nice to have: We know minimal number of ARU nodes needed to provide complete service. 28 Business & Market Intelligence / OR

Pre-Phase, Heuristic Solution 6 Cost minimized Fibre Optic Net Simple but fast approach to connect so found ARUs with source node via FON Imitation of the Minimal Cost Flow algorithm for linear cost functions 1 Pick any unconnected ARU and determine shortest path through actual network. 2 Update network along shortest path: • cost-functions on used edges • free capacities • used capacities 3 Repeat from step 1 until all ARUs are connected. Works for network with “unlimited” capacities on edges. 29 Business & Market Intelligence / OR

7 Phase 1 Cap. LAN: Notation for ILP formulation Different Edge Types: Edge Type 1: Dark fibre edges Edge Type 2: Empty pipes edges Edge Type 3: Excavation edges Connectors Type Set … N=N 1 U N 2 U N 3 For every edge type several connectors are possible Edge Type 3: excavation trenches of different size and filling … N 3 Edge Type 2: different (combination of) cables to fill empty pipes … N 2 Edge Type 1: different (combination of) dark fibres … N 1 Directed graph representing FON with customer set (ARUs) and sink (switching centre) 30 Business & Market Intelligence / OR

7 Phase 1 Cap. LAN: Notation for ILP formulation length of edge (i, j): building cost of connector type n: indicator variable for connector type n being installed on edge (i, j): flow on edge (i, j) using connector type n for customer k customers demand (careful! customer=ARU) capacity limit for edge (i, j) and connector type n 31 Business & Market Intelligence / OR

Phase 1 7 Cap. LAN: ILP Single-commodity formulation objective Flow preservation constraints Capacity constraints 32 Business & Market Intelligence / OR

Phase 1 7 Cap. LAN: ILP Multi-commodity formulation with 0/1 variables objective Flow preservation constraints Capacity constraints 33 Business & Market Intelligence / OR

8 Phase 2 Notation for ILP formulation Additional Edge Types: Edge Type 0: Copper Connection of Customer and potential ARU node Edge Type A: potential ARUs represented as edges Additional Connector Type Set For every edge type several connectors are possible Edge Type 0: Copper Connectors (only one element) Edge Type A: potential ARUs … N 0 … NA (Now real) Customer nodes ARU nodes in (customer side) ARU nodes out (sink side) 34 Business & Market Intelligence / OR

8 Phase 2 Notation for ILP formulation length of edge (i, j): building cost of connector type n: indicator variable for connector type n being installed on edge (i, j): flow on edge (i, j) using connector type n for customer k customers demand ARU demand capacity limit for edge (i, j) and connector type n 35 Business & Market Intelligence / OR

Phase 2 8 ILP Single-commodity formulation objective Copper Net constraints 1 Input ARU 36 Business & Market Intelligence / OR

Phase 2 8 ILP Single-commodity formulation objective Fibre Optic Net Flow preservation constraints 2 37 Business & Market Intelligence / OR

8 Phase 2 Connected Facility Location in Multi-commodity Networks How to find an exact solution for the stated optimisation problem? 38 Business & Market Intelligence / OR

8 Phase 2 Connected Facility Location in Multi-commodity Networks How to find an exact solution for the stated optimisation problem? 39 Business & Market Intelligence / OR

8 Phase 2 Connected Facility Location in Multi-commodity Networks How to find an exact solution for the stated optimisation problem? 40 Business & Market Intelligence / OR

8 Phase 2 Connected Facility Location in Multi-commodity Networks How to find an exact solution for the stated optimisation problem? 41 Business & Market Intelligence / OR

8 Phase 2 Connected Facility Location in Multi-commodity Networks How to find an exact solution for the stated optimisation problem? 42 Business & Market Intelligence / OR

8 Phase 2 Connected Facility Location in Multi-commodity Networks How to find an exact solution for the stated optimisation problem? 43 Business & Market Intelligence / OR

8 Phase 2 Connected Facility Location in Multi-commodity Networks How to find an exact solution for the stated optimisation problem? 44 Business & Market Intelligence / OR

9 Generalisation Preparation of Land for Building The representation of this problem as a graph is very similar to the presented one: • Customer demand has to be met through a potential network starting from a source node • Graph of network consists out of two strictly separated layers (above ground, below ground) and a set of nodes potentially connecting the two layers) Difference 1: • Layer connecting node do not multiplex Difference 2: • Design of network has to be optimised in both layers not only in one. 45 Business & Market Intelligence / OR

Thank you! Contact: Ivana Ljubic University of Vienna Bertram Wassermann Marketing Retail – Business & Market Intelligence Operations Research Telekom Austria AG E-Mail: ivana. ljubic@univie. ac. at Lassallestrasse 9, A-1020 Wien Tel: +43 (0)59 059 1 31089 Mobil: +43 (0)664 629 5527 E-Mail: bertram. wassermann@telekom. at 46 Business & Market Intelligence / OR