W.D. Grover TRLabs & University of Alberta © Wayne D. Grover 2002, 2003 Mesh-restorable Network Design (2) E E Module 13

E E Module 132 Partly for completeness, and partly because of the “special structure” (unimodularity) of the classic “transportation” problem. This approach also allows formulation without pre-processing to find either cutsets or eligible restoration routes. N.B.: for this we switch to node-based indexing and implicitly directional flow variables. s.t.: Restorability: - source = sink : Spare capacity : - flow transhipment nodes : “Transportation-like” variant of the mesh spare capacity problem

E E Module 133 Technical aspects of the “transportation-like” problem formulation Generates –2 S(S-1) flow variables plus S capacity variables –S sets of { 2 source-sink and (N-2) flow conservation constraints } ~ i.e., O(S  N) = O(S 2 ) –O(S 2 ) spare capacity constraints. Advantages: –compact formulation (in the sense of no pre-processing required) –each failure scenario presents a transportation-like flow sub-problem (however, these are all coupled under a min spare objective) –unimodular nature of transportation problem. Disadvantages: –AMPL / CPLEX memory for ~ O(S 2 ) constraints on O(S 2 ) variables –no direct knowledge of restoration path-sets from solution –no hop or distance-limiting control on restoration –implicitly assumes max-flow restoration mechanism –“blows up” in later “joint” or path-restorable problem formulations

E E Module 134 Henceforth, basing things on Herzberg-like approach... Extensions to Herzberg’s basic formulation: (1) Adding Modularity and economy of scale to the design model (2) Jointly optimizing the routing of working paths

E E Module 135 same Before….To make it modular…. = cost of m th module size on span j = number of modules of size m on span j = capacity of m th module size Ref: J. Doucette, W. D. Grover, “Influence of Modularity and Economy-of-scale Effects on Design of Mesh-Restorable DWDM Networks”, IEEE JSAC Special Issue on Protocols and Architectures for Next Generation Optical WDM Networks, October (1) Adding modularity (and economy of scale) Plus:

E E Module 136 = the set of all (active) O-D pairs = an individual O-D pair (“relation r”) = the set of “eligible working routes” available for working paths on relation r. = the total demand for relation r. = the amount of demand routed over the q th eligible route for relation r. = 1 if the q th “eligible working route” for relation r crosses span j. (2) Additions for “joint” working and spare optimization

E E Module 137 All demands must be routed Working capacity on spans must be adequate Only modular totals are possible All working span capacities must be fully restorable Spare capacity on spans must be adequate Cost of modules of all sizes placed on all spans new Optimizing the working path routes with spare capacity placement modular “joint” capacity (working and spare) placement (MJCP)

E E Module 138 Post-Modularized (PMSC) Working Path Routing Shortest Path Spare Capacity Placement Integer, but non-modular Modular on Totals (Spare + Working) Modularity Rounded Up * On Totals True Modular Design NotesExisting BenchmarkA compromise No approximations Joint Modular IP Formulation * rounding rule = least cost combination of modules such that meets the wi+si requirement, under the same economy-of-scale model as the MSCP and MJCP trial cases. Some recent Research Comparisons on effect of design modularity Modular-aware (MSCP) Joint-modular (MJCP) Ref: J. Doucette, W. D. Grover, “Influence of Modularity and Economy-of-scale Effects on Design of Mesh-Restorable DWDM Networks”, IEEE JSAC Special Issue on Protocols and Architectures for Next Generation Optical WDM Networks, October 2000.

E E Module 139 Each formulation implemented in AMPL Modeling System Solved in CPLEX Linear Optimizer 6.0. Used 9 test networks of various sizes (below). The number of eligible working and restoration routes is controlled by hop-limit strategies. Eligible working routes restricted to 5 to 20 per demand. Eligible restoration routes similarly restricted for each failure scenario. Experimental Design

E E Module 1310 Five module sizes = {12, 24, 48, 96, and 192 wavelengths}. Module costs follow three progressively greater economy-of-scale models notation for economy of scale models : 3x2x --> “3 times capacity for 2 times cost” Cost Model Module Size 12 Module Size 24 Module Size 48 Module Size 96 Module Size 192 3x2x x2x x2x Experimental Design (2)

E E Module 1311 * * Relative to the least-cost post-modularized design (PMSCP) with the same series of module costs Results - “modular aware” spare capacity placement (MSCP)

E E Module 1312 Results - “joint modular” capacity placement (MJCP) * * Relative to the least-cost post-modularized design (PMSCP) with the same series of module costs

E E Module 1313 PMSCP (benchmark)MJCP (joint design) Total Capacity = 504 Total Cost = 2861 Total Used Spans = 17 Total Capacity = 612 Total Cost = 2595 (9.3% savings) Total Used Spans = 13 (23.5% reduction) 9n17s2 - 6x2x Class question: Why is this happening - explanation? Spontaneous Topology Reduction Unexpected finding Happens with strong economy of scale scenarios

E E Module 1314 Post-Modularized Design (PMSCP): 14 % to 37% levels of “excess” (above design working and spare) capacity arises from efficient post-modularization into the {12, 24, 48, 48, 192} set. “Modular-aware” Spare Capacity Design (MSCP): Moderate levels of excess capacity (9% to 30%, average 19.4%). Moderate cost savings (up to 6%). Joint Modular Design (MJCP): Minimal excess capacity (~1 to 4.7%). Highest cost savings (~6 to 21%, average 10.7%). “Spontaneous Topology Reduction” observed (~ 24% of spans for 6x2x) Summary of Results

Routing algorithms and issues related to formulating the mesh design problem files for AMPL / CPLEX

E E Module 1316 Two types of route - sets can be needed: –(1) “all distinct routes” up to a hop or distance limit (or both) for restoration each prospective span failure Needed for both joint and non-joint span-restorable mesh design. These routes are all between nodes that are adjacent in the pre-failure graph. Require S sets of such routes, typically up to hop or distance limit H –(2) distinct route sets for working path routing Needed in addition for the joint working and spare formulations - in addition to routes (1) Require ~ N 2 /2 sets of such routes (each O-D pair) Typically delimited by hop or distance limit in excess of the shortest path distance. –Rationing / budgeting of route-set sizes may be required. Then also need strategies for selecting / sampling which routes to represent Methodology - Generating the “eligible route” sets

E E Module 1317 One practical approach can be to generate a “master database” of all distinct routes up to some “high” hop limit –can take a long time … and produce a large file –but, it is a one-time effort for any number of studies on the same topology. Once the master route-set is available, the route representations for specific problem formulations can be generated by filter programs according to almost any desired specification.. e.g. all routes under 3,000 miles or six hops, except for... all routes that exclude nodes {…} or spans {…} routes up the hop limit that provides at least 15 per span (or OD pair). a set of routes that visit no node more than x times the first 15 routes when sorted by increasing length etc. Possible project idea: statistical “sampling” of master route-sets for practical formulations Overall strategy for generating route-sets needed in formulations