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  1. 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

  2. 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 Problem Definition Overview 1

  3. Introduction • 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 • 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) • Solution to these network design problems: • Steiner Trees and its capacitated variants • Well studied • Although NP-hard, fast algorithms do exist Problem Definition Overview 1

  4. Introduction • Search for an alternative • SARU, Situating Access Remote Unit • Fibre as close to the customer as necessary and as far as possible • 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 • 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 = 600m • Distance Metric • Length of cable is used Problem Definition Overview 1

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

  6. 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! Problem Definition Graph 2 *) FON is not shaped like a rectangular grid! Shape indicates, that FON may be of different form then copper net. However, nets are superimposed

  7. Potential Fibre Optics Network Additional Nodes: Intersection points of FOL and Copper Net Potential Fibre Optics Lines with intersection nodes Problem Definition Graph 2

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

  9. 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. But Fibre Optic edges may be used in one of the two directions. Consequently: Copper edges are directed. Problem Definition Graph 2

  10. Distance Condition L and Edge Directions Assignement of Customer to potential ARUs under Distance Condition Alternative Representation of the Copper Net obeying Distance condition Problem Definition Graph 2

  11. Optimization Problem Find Positions for ARUs and create Fibre Optic Network such that Problem Definition Graph 2 • 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

  12. 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. Problem Definition Graph 2

  13. Connected Facility Location Problem (ConFL) • 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 ConFL 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 Related Problem 3

  14. Capacitated Local Access Network Design (CapLAN) • 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 Related Problem 3

  15. Customer Demand Demand in terms of copper lines (twisted pairs of copper lines) With every customer a certain demand di is associated Rule: Demand has to be completely satisfied Not in the sense of bandwidth Problem Definition Cost 4 d2 d4 d3 d1 d5 d6 di dn di+1

  16. Cost related to the Copper Net The copper network has to be incorporated as it is. No alteration allowed! No cost due to copper network. Problem Definition Cost 4 d2 d4 d3 d1 d5 d6 di dn di+1

  17. Cost related to the potential ARU locations costARU( 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 ARU ARUs produce demand. #Fibre Optic Lines depends on type of ARU Problem Definition Cost 4

  18. Cost related to the potential Fibre Optic Network In general: costFON( 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! Problem Definition Cost 4

  19. Graph Structure of FON Any combination of the 3 edge types may connect two nodes. Into both directions Problem Definition Cost 4

  20. Cost related to the potential Fibre Optic Network Dark Fibre Edge: costDF ( 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. Problem Definition Cost 4

  21. Cost related to the potential Fibre Optic Network Dark Fibre Edge: costDF ( length, demand) = const 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. Empty Pipes: costEP ( length, demand) = length*costEP/UL (demand) Again Buy at Bulk principle: Price per unit (=optic fibre) decreases with increasing size of fibre optic cables Problem Definition Cost 4

  22. Cost related to the potential Fibre Optic Network Dark Fibre Edge: costDF ( length, demand) = const 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) Empty Pipes: costEP ( length, demand) = length*costEP/UL (demand) Excavation: costExT ( length, location, demand) = length*costExT/UL (location, demand) Simplification: Costs are based on the assumption, trenches are filled completely with pipes which are completely filled with fibre cable. Problem Definition Cost 4

  23. Cost related to the potential Fibre Optic Network Dark Fibre Edge: costDF ( length, demand) = const Empty Pipes: costEP ( length, demand) = length*costEP/UL (demand) Excavation: costExT ( length, location, demand) = length*costExT/UL (location, demand) costFON( length, location, demand) per edge = const + length * [costEP/UL (demand) +costExT/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. Problem Definition Cost 4

  24. Overview • Phase 1: • Solving the simplified problem: • To improve the solution • To study the cost function 1 • Phase 2: • Solving the problem • To find an exact algorithm • Study the approximation qualities of heuristic solutions 2 • 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 0 Solution Strategy 5

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

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

  27. 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 and minimal in number will approximate the optimal (=cost minimal) solution. Algorithm Pick customer furthest away from source. 1 Choose potential ARU node furthest away from this customer still valid under distance condition L 2 Install ARU at this position and serve all customers of sub tree rooted at this node 3 Ignore sub-tree and proceed form step 1 4 Pre-Phase, Heuristic Solution 6

  28. 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. ! Pre-Phase, Heuristic Solution 6

  29. 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 Pick any unconnected ARU and determine shortest path through actual network. 1 • Update network along shortest path: • cost-functions on used edges • free capacities • used capacities 2 Repeat from step 1 until all ARUs are connected. 3 Works for network with “unlimited” capacities on edges. Pre-Phase, Heuristic Solution 6

  30. CapLAN: Notation for ILP formulation Connectors Type Set … N=N1 U N2 U N3 For every edge type several connectors are possible Edge Type 3: excavation trenches of different size and filling … N3 Edge Type 2: different (combination of) cables to fill empty pipes … N2 Edge Type 1: different (combination of) dark fibres … N1 Different Edge Types: Edge Type 1: Dark fibre edges Edge Type 2: Empty pipes edges Edge Type 3: Excavation edges Directed graph representing FON with customer set (ARUs) and sink (switching centre) Phase 1 7

  31. CapLAN: 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 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 Phase 1 7

  32. CapLAN: ILP Single-commodity formulation objective Flow preservation constraints Capacity constraints constraints Phase 1 7

  33. CapLAN: ILP Multi-commodity formulation with 0/1 variables objective Flow preservation constraints Capacity constraints constraints Phase 1 7

  34. Notation for ILP formulation Additional Connector Type Set For every edge type several connectors are possible Edge Type 0: Copper Connectors … N0 (only one element) Edge Type A: potential ARUs … NA Additional Edge Types: Edge Type 0: Copper Connection of Customer and potential ARU node Edge Type A: potential ARUs represented as edges (Now real) Customer nodes ARU nodes in (customer side) ARU nodes out (sink side) Phase 2 8

  35. 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 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 Phase 2 8

  36. ILP Single-commodity formulation objective Copper Net Input ARU constraints 1 Phase 2 8

  37. ILP Single-commodity formulation objective Fibre Optic Net Flow preservation constraints constraints 2 Phase 2 8

  38. Connected Facility Location in Multi-commodity Networks How to find an exact solution for the stated optimisation problem? Phase 2 8

  39. Connected Facility Location in Multi-commodity Networks How to find an exact solution for the stated optimisation problem? Phase 2 8

  40. Connected Facility Location in Multi-commodity Networks How to find an exact solution for the stated optimisation problem? Phase 2 8

  41. Connected Facility Location in Multi-commodity Networks How to find an exact solution for the stated optimisation problem? Phase 2 8

  42. Connected Facility Location in Multi-commodity Networks How to find an exact solution for the stated optimisation problem? Phase 2 8

  43. Connected Facility Location in Multi-commodity Networks How to find an exact solution for the stated optimisation problem? Phase 2 8

  44. Connected Facility Location in Multi-commodity Networks How to find an exact solution for the stated optimisation problem? Phase 2 8

  45. Preparation of Land for Building • Difference 1: • Layer connecting node do not multiplex • 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 2: • Design of network has to be optimised in both layers not only in one. Generalisation 9

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