Mathematical Challenges in Telecommunication a Tutorial Martin Grötschel IMA workshop on Network Management and Design Minneapolis, MN, April 6, 2003 Martin Grötschel Institute of Mathematics, Technische Universität.
Download ReportTranscript Mathematical Challenges in Telecommunication a Tutorial Martin Grötschel IMA workshop on Network Management and Design Minneapolis, MN, April 6, 2003 Martin Grötschel Institute of Mathematics, Technische Universität.
Mathematical Challenges in Telecommunication a Tutorial Martin Grötschel IMA workshop on Network Management and Design Minneapolis, MN, April 6, 2003 Martin Grötschel Institute of Mathematics, Technische Universität Berlin (TUB) DFG-Research Center “Mathematics for key technologies” (FZT 86) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) [email protected] http://www.zib.de/groetschel 2 Comment This is an abbreviated version of my presentation. The three films I showed are not included. I also deleted many of the pictures and some of the tables: in some cases I do not have permission to publish them since they contain some confidential material or information not for public use, in other cases I am not sure of the copy right status (and do not have the time and energy to check it). Martin Grötschel 3 ZIB Telecom Team The ZIB Telecom Group ZIB Associates Andreas Bley Manfred Brandt Andreas Eisenblätter Sven Krumke Martin Grötschel Thorsten Koch Arie Koster Roland Wessäly Adrian Zymolka Frank Lutz Diana Poensgen Jörg Rambau and more: Clyde Monma (BellCore, ...) Mechthild Opperud (ZIB, Telenor) Dimitris Alevras (ZIB, IBM) Martin Grötschel Christoph Helmberg (Chemnitz) 4 ZIB Partners from Industry Bell Communications Research Telenor (Norwegian Telecom) E-Plus (acquired by KPN in 01/2002) DFN-Verein Bosch Telekom (bought by Marconi) Siemens Austria Telekom (controlled by Italia Telecom) T-Systems Nova (T-Systems, Deutsche Telekom) KPN Telecel-Vodafone Atesio (ZIB spin-off company) Martin Grötschel 5 Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Martin Grötschel Telecommunication: The General Problem The Problem Hierarchy: Cell Phones and Mathematics The Problem Hierarchy: Network Components and Math Network Design: Tasks to be solved Addressing Special Issues: Frequency Assignment Locating the Nodes of a Network Balancing the Load of Signaling Transfer Points Integrated Topology, Capacity, and Routing Optimization as well as Survivability Planning Planning IP Networks Optical Networks Summary and Future 6 Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Martin Grötschel Telecommunication: The General Problem The Problem Hierarchy: Cell Phones and Mathematics The Problem Hierarchy: Network Components and Math Network Design: Tasks to be solved Addressing Special Issues: Frequency Assignment Locating the Nodes of a Network Balancing the Load of Signaling Transfer Points Integrated Topology, Capacity, and Routing Optimization as well as Survivability Planning Planning IP Networks Optical Networks Summary and Future 7 What is the Telecom Problem? Design excellent technical devices and a robust network that survives all kinds of failures and organize the traffic such that high quality telecommunication between very many individual units at many locations is feasible at low cost! Martin Grötschel Speech Data Video Etc. 8 What is the Telecom Problem? Design excellent technical devices and a robust network that survives all kinds of failures and organize the traffic such that high quality telecommunication between very many individual units at many locations is feasible at low cost! Martin Grötschel This problem is too general to be solved in one step. Approach in Practice: • Decompose whenever possible • Look at a hierarchy of problems • Address the individual problems one by one • Recompose to find a good global solution 9 Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Martin Grötschel Telecommunication: The General Problem The Problem Hierarchy: Cell Phones and Mathematics The Problem Hierarchy: Network Components and Math Network Design: Tasks to be solved Addressing Special Issues: Frequency Assignment Locating the Nodes of a Network Balancing the Load of Signaling Transfer Points Integrated Topology, Capacity, and Routing Optimization as well as Survivability Planning Planning IP Networks Optical Networks Summary and Future 10 Cell Phones and Mathematics cell phone picture Designing mobile phones •Computational logic •Combinatorial •Task partitioning optimization •Chip design (VLSI) •Differential algebraic •Component design equations Producing Mobile Phones •Operations research •Production facility layout •Linear and integer programming •Control of CNC machines •Combinatorial optimization •Ordinary differential equations •Control of robots •Lot sizing Marketing and Distributing Mobiles •Scheduling •Financial mathematics •Logistics •Transportation optimization Martin Grötschel Chip Design Schematic for four-transistor static-memory cell CMOS layout for four-transistor static-memory cell Placement Routing Compactification CMOS layout for two four-transistor static-memory cells. Compacted CMOS layout for two four-transistor static-memory cells. ZIB-Film 9:01 – 9:41 12 Design and Production of ICs and PCBs Integrated Circuit (IC) Printed Circuit Board (PCB) Problems: Logic Design, Physical Design Correctness, Simulation, Placement of Components, Routing, Drilling,... Martin Grötschel Production and Mathematics: Examples CNC Machine for 2D and 3D cutting and welding (IXION ULM 804) Sequencing of Tasks and Optimization of Moves Mounting Devices Minimizing Production Time via TSP or IP SMD Printed Circuit Boards Optimization of Manufacturing 14 Drilling 2103 holes into a PCB Significant Improvements via TSP (Padberg & Rinaldi) Martin Grötschel 15 Siemens Problem printed circuit board da1 Martin Grötschel before after 16 Siemens Problem printed circuit board da4 Martin Grötschel before after 17 Mobile Phone Production Line Fujitsu Nasu plant Martin Grötschel 18 Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Martin Grötschel Telecommunication: The General Problem The Problem Hierarchy: Cell Phones and Mathematics The Problem Hierarchy: Network Components and Math Network Design: Tasks to be solved Addressing Special Issues: Frequency Assignment Locating the Nodes of a Network Balancing the Load of Signaling Transfer Points Integrated Topology, Capacity, and Routing Optimization as well as Survivability Planning Planning IP Networks Optical Networks Summary and Future 19 Network Components Design, Production, Marketing, Distribution: Similar math problems as for mobile phones •Fiber (and other) cables •Antennas and Transceivers •Base stations (BTSs) •Base Station Controllers (BSCs) •Mobile Switching Centers (MSCs) •and more... Martin Grötschel 20 Component „Cables“ Martin Grötschel 21 Component „Antennas“ Martin Grötschel 22 Component „Base Station“ Nokia MetroSite Nokia UltraSite Martin Grötschel Component „Mobile Switching Center“: Example of an MSC Plan 24 Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Martin Grötschel Telecommunication: The General Problem The Problem Hierarchy: Cell Phones and Mathematics The Problem Hierarchy: Network Components and Math Network Design: Tasks to be solved Addressing Special Issues: Frequency Assignment Locating the Nodes of a Network Balancing the Load of Signaling Transfer Points Integrated Topology, Capacity, and Routing Optimization as well as Survivability Planning Planning IP Networks Optical Networks Summary and Future 25 Network Design: Tasks to be solved Some Examples Locating the sites for antennas (TRXs) and base transceiver stations (BTSs) Assignment of frequencies to antennas Cryptography and error correcting encoding for wireless communication Clustering BTSs Locating base station controllers (BSCs) Connecting BTSs to BSCs Martin Grötschel 26 Network Design: Tasks to be solved Some Examples (continued) Locating Mobile Switching Centers (MSCs) Clustering BSCs and Connecting BSCs to MSCs Designing the BSC network (BSS) and the MSC network (NSS or core network) Topology of the network Capacity of the links and components Routing of the demand Survivability in failure situations Martin Grötschel Most of these problems turn out to be Combinatorial Optimization or Mixed Integer Programming Problems 27 Connecting Mobiles: What´s up? BSC MSC MSC BSC BSC MSC MSC BSC BSC MSC BSC BTS Martin Grötschel BSC 28 Martin Grötschel Connecting Computers or other Devices The mathematical problems are similar but not identical 29 Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Martin Grötschel Telecommunication: The General Problem The Problem Hierarchy: Cell Phones and Mathematics The Problem Hierarchy: Network Components and Math Network Design: Tasks to be solved Addressing Special Issues: Frequency Assignment Locating the Nodes of a Network Balancing the Load of Signaling Transfer Points Integrated Topology, Capacity, and Routing Optimization as well as Survivability Planning Planning IP Networks Optical Networks Summary and Future 30 F A P F i l m Martin Grötschel 31 Antennas & Interference co- & adjacent channel interference cell antenna x x x x site x x x x cell backbone network Martin Grötschel 32 Interference Level of interference depends on distance between transmitters geographical position power of the signals direction in which signals are transmitted weather conditions assigned frequencies co-channel interference Martin Grötschel adjacent-channel interference 33 Separation Frequencies assigned to the same location (site) have to be separated Site Blocked channels Parts of the spectrum forbidden at some locations: government regulations, agreements with operators in neighboring regions, requirements military forces, etc. Martin Grötschel 34 Frequency Planning Problem Find an assignment of frequencies to transmitters that satisfies all separation constraints all blocked channels requirements and either avoids interference at all or minimizes the (total/maximum) interference level Martin Grötschel 35 Minimum Interference Frequency Assignment Problem Integer Linear Program: min vwE s.t. co co cvw zvw f Fv vwE ad co x vf ad ad cvw zvw 1 xvf xwg 1 vw E d , f g d (vw) co xvf xwf 1 zvw vw E co , f Fv Fw ad xvf xwg 1 zvw vw E ad , f g 1 co ad xvf , zvw , zvw 0,1 Martin Grötschel v V 36 [% mi ni m ] av um era de ge gre de e ma gr ee xim um de dia gr me ee t cli er qu en um be r de ns |V | Ins tan ity ce A Glance at some Instances k 267 56,8 2 151,0 B-0-E-20 1876 13,7 40 257,7 f 2786 4,5 3 135,0 h 4240 5,9 11 249,0 238 779 453 561 E-Plus Project Martin Grötschel 3 69 5 81 12 69 10 130 37 Region Berlin - Dresden 2877 carriers 50 channels Interference reduction: 83.6% Martin Grötschel 38 Region Karlsruhe 2877 Carriers 75 channels Interference Reduction: 83.9 % Martin Grötschel 39 The UMTS Radio Interface Completely new story see talk by Andreas Eisenblätter on Monday Martin Grötschel 40 Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Martin Grötschel Telecommunication: The General Problem The Problem Hierarchy: Cell Phones and Mathematics The Problem Hierarchy: Network Components and Math Network Design: Tasks to be solved Addressing Special Issues: Frequency Assignment Locating the Nodes of a Network Balancing the Load of Signaling Transfer Points Integrated Topology, Capacity, and Routing Optimization as well as Survivability Planning Planning IP Networks Optical Networks Summary and Future 41 G-WiN Data G-WiN = Gigabit-Wissenschafts-Netz of the DFN-Verein Internet access of all German universities and research institutions • Locations to be connected: 750 • Data volume in summer 2000: 220 Terabytes/month • Expected data volume in 2004: 10.000 Terabytes/month Clustering (to design a hierarchical network): • 10 nodes in Level 1a 261 nodes eligible for • 20 nodes in Level 1b Level 1 • All other nodes in Level 2 Martin Grötschel 42 G-WiN Problem • Select the 10 nodes of Level 1a. • Select the 20 nodes of Level 1b. • Each Level 1a node has to be linked to two Level 1b nodes. • Link every Level 2 node to one Level 1 node. • Design a Level 1a Network such that •Topology is survivable (2-node connected) •Edge capacities are sufficient (also in failure situations) •Shortest path routing (OSPF) leads to balanced capacity use (objective in network update) • The whole network should be „stable for the future“. • The overall cost should be as low as possible. Martin Grötschel 43 Potential node locations for the 3-Level Network of the G-WIN Red nodes are potential level 1 nodes Blue nodes are all remaining nodes Cost: Connection between nodes Capacity of the nodes Martin Grötschel 44 Demand distribution The demand scales with the height of each red line Aim Select backbone nodes and connect all non-backbone nodes to a backbone node such that the overall network cost is minimal (access+backbone cost) Martin Grötschel 45 G-WiN Location Problem: Data V set of locations Z set of potential Level 1a locations (subset of V ) K p set of possible configurations at location p in Level 1a For i V , p Z and k K p : wip connection costs from i to p d i traffic demand at location i c kp capacity of location p in configuration k wkp costs at location p in configuration k xip 1 if location i is connected to p (else 0) z kp 1 if configuration k is used at location p (else 0) Martin Grötschel 46 G-WiN Location/Clustering Problem min wip xip pZ iV x ip 1 pZ k K p wkp z kp Each location i must be connected to a Level 1 node p di xip i k z p 1 k k c z p p Capacity at p must be large enough k Only one configuration at each Location 1 node k k z p const p All variables are 0/1. Martin Grötschel # of Level 1a nodes 47 Solution: Hierarchy & Backbone Martin Grötschel 48 G-WiN Location Problem: Solution Statistics The DFN problem leads to ~100.000 0/1-variables. Typical computational experience: Optimal solution via CPLEX in a few seconds! A very related problem at Telekom Austria has ~300.000 0/1-variables plus some continuous variables and capacity constraints. Computational experience (before problem specific fine tuning): 10% gap after 6 h of CPLEX computation, 60% gap after „simplification“ (dropping certain capacities). Martin Grötschel 49 Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Martin Grötschel Telecommunication: The General Problem The Problem Hierarchy: Cell Phones and Mathematics The Problem Hierarchy: Network Components and Math Network Design: Tasks to be solved Addressing Special Issues: Frequency Assignment Locating the Nodes of a Network Balancing the Load of Signaling Transfer Points Integrated Topology, Capacity, and Routing Optimization as well as Survivability Planning Planning IP Networks Optical Networks Summary and Future 50 Re-Optimization of Signaling Transfer Points Telecommunication companies maintain a signaling network (in adition to their communication transport network). This is used for management tasks such as: Basic call setup or tear down Wireless roaming Mobile subscriber authentication Call forwarding Number display SMS messages Martin Grötschel Etc. 51 Signaling Transfer Point (STP) CCD=routing unit, CCLK=interface card CCLK CCD Cluster CCD CCD Link-Sets CCD CCD CCD Martin Grötschel CCD=Common Channel Distributors, CCD Cluster STP CCD Cluster CCD CCLK=Common Channel Link Controllers 52 STP – Problem description Target Assign each link to a CCD/CCLK Constraints At most 50% of the links in a linkset can be assigned to a single cluster Number of CCLKs in a cluster is restricted Objective Balance load of CCDs Martin Grötschel 53 STP – Mathematical model Variables xij 0,1 , i L, j C xij 1 if and only if C set of CCDs j Data L set of links i link i is assigned to CCD j Di demand of link i P set of link-sets Q set of clusters Lp subset of links in link-set p Cq subset of CCDs in cluster q Martin Grötschel cq #CCLKs in cluster q 54 STP – Mathematical model min y z Min load difference x ij 1 iL Assign each link D x ij y j C Upper bound of CCD-load D x z j C Lower bound of CCD-load p P, q Q Diversification q Q CCLK-bound jC iL iL iL p jCq i i ij L xij x iL jCq ij p 2 cq xij 0,1 Martin Grötschel Integrality 55 STP – current (former) solution Minimum: 186 Maximum: 404 Load difference: 218 450 400 Traffic Load 350 300 250 200 150 100 50 0 1 Martin Grötschel 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 CCD 56 STP – „Optimal solution“ Minimum: 280 Maximum: 283 Load difference: 3 450 400 Traffic Load 350 300 250 200 150 100 50 0 1 Martin Grötschel 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 CCD 57 STP – Practical difficulty Problem: 311 rearrangements are necessary to migrate to the optimal solution Reformulation with new objective Find a best solution with a restricted number of changes Martin Grötschel 58 STP – Reformulated Model Min load difference min y z x ij 1 iL Assign each link D x ij y j C Upper bound of CCD-load D x z j C Lower bound of CCD-load x x cq jC iL iL iL p jCq iL iL jCq i i ij ij ij L p 2 xij B p P, q Q Diversification q Q CCLK-bound Restricted number of changes! jC , j j ( i ) * xij 0,1 Martin Grötschel Integrality 59 STP – Alternative Model min iL Min # changes jC , j j ( i ) x ij 1 iL Assign each link D x ij y j C Upper bound of CCD-load D x z j C Lower bound of CCD-load x x cq jC i iL i iL iL p jCq iL jCq ij ij ij L p 2 yz D xij 0,1 Martin Grötschel xij * p P, q Q Diversification q Q CCLK-bound Restricted load difference Integrality 60 STP – New Solutions White: D=50, (alternative) Minimum: 257 D=Load difference: 50 Maximum: 307 Number of changes: 12 Orange: B=8, (reformulated) Minimum: 249 Load difference: 90 Maximum: 339 Number of changes=B: 8 400 350 Traffic Load 300 250 200 150 100 50 0 1 Martin Grötschel 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 CCD 61 STP – Experimental results Max changes 0 5 10 15 20 Load differences 218 129 71 33 14 1 hour application of CPLEX MIP-Solver for each case Martin Grötschel 62 STP - Conclusions It is possible to achieve 85% of the optimal improvement with less than 5% of the changes necessary to obtain a load balance optimal solution ! Martin Grötschel 63 Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Martin Grötschel Telecommunication: The General Problem The Problem Hierarchy: Cell Phones and Mathematics The Problem Hierarchy: Network Components and Math Network Design: Tasks to be solved Addressing Special Issues: Frequency Assignment Locating the Nodes of a Network Balancing the Load of Signaling Transfer Points Integrated Topology, Capacity, and Routing Optimization as well as Survivability Planning Planning IP Networks Optical Networks Summary and Future 64 Network Optimization Film ZIB Film 1994, 9:40 Martin Grötschel 65 Network Optimization Capacities Requirements Networks Cost Martin Grötschel 66 What needs to be planned? Topology Capacities Routing Failure Handling (Survivability) IP Routing Node Equipment Planning Optimizing Optical Links and Switches DISCNET: A Network Planning Tool (Dimensioning Survivable Capacitated NETworks) Atesio ZIB Spin Off Martin Grötschel 67 The Network Design Problem Communication Demands Hamburg 134 Berlin Düsseldorf 42 Berlin Düsseldorf 65 30 Frankfurt 200 Martin Grötschel Hamburg 258 Frankfurt München Potential topology & Capacities München 68 Capacities (P)SDH=(poly)synchronous digital hierarchy PDH 2 SDH Mbit/s 155 Mbit/s 34 Mbit/s 622 Mbit/s 140 Mbit/s 2,4 Gbit/s ... WDM (n x STM-N) Two capacity models : Discrete Finite Capacities Divisible Capacities WDM=Wavelength Division Multiplexer STM-N=Synchronous Transport Modul with N STM-1 Frames Martin Grötschel 69 Survivability Diversification „route node-disjoint“ H B 120 D H B 60 D F 30 F 30 M Reservation „reroute all demands“ M H (or p% of all affected demands) Martin Grötschel H 60 D D 60 F (or p% of all demands) Path restoration „reroute affected demands“ B B 120 F M H M B H 60 D F 60 D 60 F M B 60 M 70 Model: Data & Variables Supply Graph: G=(V,E) Ce0 Z Ce1 CeT Capacity variables: x(e, t ) {0,1} e E Operating states: sS P Puvs e Demand Graph: H=(V,D) D vu , v u d D vu , v u D vu , v u D vu , vul Martin Grötschel Valid Paths: Path variables: f uv ( P) 0 s s S , uv Ds , P uvs 71 Model: Capacities Capacity variables : e E, t = 1, ..., Te x et {0,1} Cost function : Te min k et x et e E t 1 Capacity constraints : 1 x e0 x e1 ye Martin Grötschel Te t t c x e e t 0 eE x eTe 0 72 Model: Routings s Path variables : s S , uv D s , P uv f uvs (P ) 0 Capacity constraints : e E y e f uv0 (P ) 0 uv D P uv :e P Demand constraints : uv D d uv f uv0 (P ) 0 P uv Martin Grötschel Path length restriction 73 Model: Survivability (one example) Path restoration H B D F 60 M Martin Grötschel H 60 120 D „reroute affected demands“ B for all sS, uvDs 60 F 60 M for all sS, eEs 74 Mathematical Model Te min k et x et topology decisison e E t 1 capacity decisions x et {0,1} e E , t 1, ,Te xet 1 xet e E , t 1,, Te Te ye t t c e xe ye f t 0 0 uv 0 uvD Puv :eP d uv f 0 uv ( P) normal operation routing component failure routing eE ( P) eE uv D Pu0v f uvs ( P ) 0 Martin Grötschel s S , uv Ds , P uvs LP-based Methods Feasible integer solutions Objective function Convex hull LP-based relaxation Cutting planes Flow chart LP-based approach: Initialize LP-relaxation Solve LP-relaxation Separation algorithms Augment LP-relaxation Yes Inequalities? Polyhedral combinatorics Valid inequalities (facets) Separation algorithms Heuristics Feasibility of a capacity vector Separation algorithms Run heuristics No No No Solve feasibility problem Feasible routings? Yes x variables Yes integer? Optimal solution 77 Finding a Feasible Solution? Heuristics Manipulation of – Routings – Topology – Capacities Local search Simulated Annealing Genetic algorithms ... Problem Sizes Martin Grötschel Nodes Edges Demands Routing-Paths 15 46 78 > 150 x 10e6 36 107 79 > 500 x 10e9 36 123 123 > 2 x 10e12 78 How much to save? Real scenario • • • 163 nodes 227 edges 561 demands PhD Thesis: http://www.zib.de/wessaely [email protected] 34% potential savings! == > hundred million dollars Martin Grötschel 79 Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Martin Grötschel Telecommunication: The General Problem The Problem Hierarchy: Cell Phones and Mathematics The Problem Hierarchy: Network Components and Math Network Design: Tasks to be solved Addressing Special Issues: Frequency Assignment Locating the Nodes of a Network Balancing the Load of Signaling Transfer Points Integrated Topology, Capacity, and Routing Optimization as well as Survivability Planning Planning IP Networks Optical Networks Summary and Future 80 Comment 9. Planning IP Networks 10. Optical Networks 11. Summary and Future The lecture ended after about 100 minutes. The last three topics above were not covered. Martin Grötschel 81 Summary Telecommunication Problems such as • • • • • • • Frequency Assignment Locating the Nodes of a Network Optimally Balancing the Load of Signaling Transfer Points Integrated Topology, Capacity, and Routing Optimization as well as Survivability Planning Planning IP Networks Optical Network Design and many others can be succesfully attacked with optimization techniques. Martin Grötschel 82 Summary The mathematical programming approach • • • • • • • Helps understanding the problems arising Makes much faster and more reliable planning possible Allows considering variations and scenario analysis Allows the comparision of different technologies Yields feasible solutions Produces much cheaper solutions than traditional planning techniques Helps evaluating the quality of a network. There is still a lot to be done, e.g., for the really important problems, optimal solutions are way out of reach! Martin Grötschel 83 The Mathematical Challenges Finding the right ballance between flexibility and controlability of future networks Controlling such a flexible network Handling the huge complexity Integrating new services easily Guaranteeing quality Finding appropriate Mathematical Models Martin Grötschel Finding appropriate solution techniques (exact, approximate , interactive, quality guaranteed) Mathematical Challenges in Telecommunication The End Martin Grötschel IMA workshop on Network Management and Design Minneapolis, MN, April 6, 2003 Martin Grötschel Institute of Mathematics, Technische Universität Berlin (TUB) DFG-Research Center “Mathematics for key technologies” (FZT 86) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) [email protected] http://www.zib.de/groetschel