copyright © wayne d. grover 2000 ee 681 fall 2000 lecture 15 mesh-restorable network design (2) w....
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copyright © Wayne D. Grover 2000
EE 681 Fall 2000 Lecture 15
Mesh-restorable Network Design (2)
W. D. Grover, October 26, 2000
copyright © Wayne D. Grover 2000
copyright © Wayne D. Grover 2000
“Transportation-like” variant of the mesh spare capacity problem
• 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.
minij
c si jij
SS.t.:
Restorability:
- source = sink :
Spare capacity :
j t j s
tjsjx x wstst st
( , )s t S
- flow conservation @ transhipment nodes :
0j i j i
jijix xst st
{ { } { }}i s t V( , )s t S
( , )s t
ijx sijst
S
{( , ) ( , )}i j s t S
copyright © Wayne D. Grover 2000
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
copyright © Wayne D. Grover 2000
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
copyright © Wayne D. Grover 2000
(1) Adding modularity (and economy of scale)
1
Mm mj j
m j
Minimize C n
S
M
m
mmjjj Znws
1
j S
i S
Minimize c sii
2( , )i j
i j
S
p Pi
pf wii
i S
,p Pi
p pf sii ij
same
same
Before…. To make it modular….
mjC
mjn
mZ
= cost of mth module size on span j
= number of modules of size m on span j
= capacity of mth 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 2000.
copyright © Wayne D. Grover 2000
(2) Additions for “joint” working and spare optimization
,r qg
rd
rQ
,r qj
r
D = 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 for routed over the qth eligible route for relation r.
= 1 if the qth “eligible working route” for relation r crosses span j.
copyright © Wayne D. Grover 2000
Optimizing the working path routes with spare capacity placement
modular “joint” capacity (working and spare) placement (MJCP)
1
Mm mj j
m j
Minimize C n
S
M
m
mmjjj Znws
1
j S
2( , )i j
i j
S
p Pi
pf wii
i S
rQ
q
rqr dg1
, r D
, ,r q r qj j
r q
g w
rD Q
j S
,p Pi
p pf sii ij
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
copyright © Wayne D. Grover 2000
Some recent Research Some recent Research Comparisons on effect of design Comparisons on effect of design
modularitymodularityPost-Modularized
(PMSCP)
“Modular-aware” Spare(MSCP)
Joint Modular (MJCP)
Working Path
RoutingShortest Path Shortest Path
Spare Capacity
Placement
Integer, but non-modular
Modular on Totals (Spare + Working)
Modularity Rounded Up * On Totals
True Modular Design
Notes Existing Benchmark A compromise No approximations
Joint Modular IP Formulation
Ref: Doucette, Grover JSAC 2000 * 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.
copyright © Wayne D. Grover 2000
Experimental DesignExperimental Design• Each formulation implemented in AMPL Modeling System 6.0.2.• Solved in CPLEX Linear Optimizer 6.0.• Used 9 test networks of various sizes (below).
Network Nodes SpansDemand
PairsTotal
Demand
JED9807b 6 14 15 75Bellcore1 11 23 55 341Bellcore2 15 28 105 4659n17s1 9 17 36 1679n17s2 9 17 36 18310n19s1 10 19 45 25110n19s2 10 19 45 24411n21s1 11 21 55 28511n21s2 11 21 55 282
• 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.
copyright © Wayne D. Grover 2000
Experimental Design Experimental Design (2)(2)
• 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 120 186 288 446 6904x2x 120 170 240 339 4806x2x 120 157 205 268 351
copyright © Wayne D. Grover 2000
*
Results Results - “modular aware” spare capacity - “modular aware” spare capacity placement (MSCP)placement (MSCP)
N e t w o r kR e q ' d
C a p a c i t yS i z e 1 2
M o d u l e sS i z e 2 4
M o d u l e sS i z e 4 8
M o d u l e sS i z e 9 6
M o d u l e s C o s tS p a n E l i m .
T o t . M o d . C a p a c i t y
% C o s t I m p r o v e m e n t
9 n 1 7 s 1 3 7 0 4 1 1 2 0 3 1 0 2 N / A 4 0 8 1 0 . 6 %9 n 1 7 s 2 4 0 4 3 1 1 3 0 3 2 7 0 N / A 4 4 4 8 . 4 %
1 0 n 1 9 s 1 5 6 5 1 1 2 7 0 4 3 6 8 N / A 6 3 6 1 2 . 3 %1 0 n 1 9 s 2 5 6 9 2 1 0 6 1 4 2 7 4 N / A 6 4 8 9 . 3 %1 1 n 2 1 s 1 7 0 4 0 1 2 7 2 5 1 4 0 N / A 8 1 6 7 . 3 %1 1 n 2 1 s 2 6 5 5 2 1 2 9 0 5 0 6 4 N / A 7 4 4 9 . 7 %A v e r a g e 9 . 6 %
M S C P - 3 x 2 x
N e t w o r kR e q ' d
C a p a c i t yS i z e 1 2
M o d u l e sS i z e 2 4
M o d u l e sS i z e 4 8
M o d u l e sS i z e 9 6
M o d u l e s C o s tS p a nE l i m .
T o t . M o d .C a p a c i t y
% C o s tI m p r o v e m e n t
9 n 1 7 s 1 3 7 7 3 1 1 3 0 2 9 5 0 N / A 4 4 4 4 . 8 %9 n 1 7 s 2 4 0 1 3 1 1 3 0 2 9 5 0 N / A 4 4 4 6 . 9 %
1 0 n 1 9 s 1 5 7 0 0 1 2 6 1 3 8 1 9 N / A 6 7 2 9 . 9 %1 0 n 1 9 s 2 5 6 5 2 1 0 6 1 3 7 1 9 N / A 6 4 8 7 . 7 %1 1 n 2 1 s 1 7 1 0 0 1 2 7 2 4 3 9 8 N / A 8 1 6 7 . 1 %1 1 n 2 1 s 2 7 1 7 0 1 5 2 4 4 3 8 6 N / A 8 4 0 8 . 7 %A v e r a g e 7 . 5 %
M S C P - 4 x 2 x
N e t w o r kR e q ' d
C a p a c i t yS i z e 1 2
M o d u l e sS i z e 2 4
M o d u l e sS i z e 4 8
M o d u l e sS i z e 9 6
M o d u l e s C o s tS p a n E l i m .
T o t . M o d . C a p a c i t y
% C o s t I m p r o v e m e n t
9 n 1 7 s 1 3 7 4 4 1 1 2 0 2 6 1 7 N / A 4 0 8 7 . 0 %9 n 1 7 s 2 3 9 3 3 1 1 3 0 2 7 0 2 N / A 4 4 4 5 . 6 %
1 0 n 1 9 s 1 5 6 7 0 1 2 6 1 3 3 8 2 N / A 6 7 2 7 . 8 %1 0 n 1 9 s 2 5 7 5 2 1 0 6 1 3 3 0 8 N / A 6 4 8 6 . 2 %1 1 n 2 1 s 1 7 1 2 0 1 2 7 2 3 8 5 5 N / A 8 1 6 5 . 4 %1 1 n 2 1 s 2 7 1 7 0 1 5 2 4 3 8 3 7 N / A 8 4 0 7 . 0 %A v e r a g e 6 . 5 %
M S C P - 6 x 2 x
* Relative to the least-cost post-modularized design (PMSCP) with the same series of module costs
copyright © Wayne D. Grover 2000
*
Results Results - “joint modular” capacity placement - “joint modular” capacity placement (MJCP)(MJCP)
N e t w o r kR e q ' d
C a p a c i t yS i z e 1 2
M o d u l e sS i z e 2 4
M o d u l e sS i z e 4 8
M o d u l e sS i z e 9 6
M o d u l e s C o s tS p a n E l i m .
T o t . M o d . C a p a c i t y
% C o s t I m p r o v e m e n t
9 n 1 7 s 1 3 4 4 5 1 2 0 0 2 8 3 2 0 3 4 8 1 8 . 3 %9 n 1 7 s 2 4 0 4 6 8 3 0 3 0 7 2 0 4 0 8 1 3 . 9 %
1 0 n 1 9 s 1 5 7 3 0 1 3 6 0 4 1 4 6 0 6 0 0 1 6 . 7 %1 0 n 1 9 s 2 5 6 8 4 1 2 5 0 4 1 5 2 0 5 7 6 1 1 . 9 %1 1 n 2 1 s 1 7 9 0 0 1 4 2 4 4 9 6 4 1 8 1 6 1 0 . 5 %1 1 n 2 1 s 2 7 1 1 0 1 0 1 0 0 4 7 4 0 1 7 2 0 1 5 . 5 %A v e r a g e 0 . 3 1 4 . 5 %
M J C P - 3 x 2 x
N e t w o r kR e q ' d
C a p a c i t yS i z e 1 2
M o d u l e sS i z e 2 4
M o d u l e sS i z e 4 8
M o d u l e sS i z e 9 6
M o d u l e s C o s tS p a n E l i m .
T o t . M o d . C a p a c i t y
% C o s t I m p r o v e m e n t
9 n 1 7 s 1 3 4 0 5 1 2 0 0 2 6 4 0 0 3 4 8 1 4 . 8 %9 n 1 7 s 2 4 3 8 1 1 2 3 0 2 8 8 0 1 4 4 4 9 . 1 %
1 0 n 1 9 s 1 5 8 4 0 1 3 6 0 3 6 5 0 0 6 0 0 1 3 . 9 %1 0 n 1 9 s 2 6 8 1 0 4 1 1 1 3 6 5 9 3 7 2 0 9 . 2 %1 1 n 2 1 s 1 8 5 1 1 2 1 3 2 4 2 5 8 3 8 7 6 1 0 . 1 %1 1 n 2 1 s 2 8 1 3 2 1 1 6 0 4 2 5 0 2 8 1 6 1 1 . 6 %A v e r a g e 1 . 5 1 1 . 5 %
M J C P - 4 x 2 x
N e t w o r kR e q ' d
C a p a c i t yS i z e 1 2
M o d u l e sS i z e 2 4
M o d u l e sS i z e 4 8
M o d u l e sS i z e 9 6
M o d u l e s C o s tS p a n E l i m .
T o t . M o d . C a p a c i t y
% C o s t I m p r o v e m e n t
9 n 1 7 s 1 4 2 7 1 1 0 4 0 2 5 1 0 2 4 4 4 1 0 . 8 %9 n 1 7 s 2 5 2 5 3 2 8 1 2 5 8 2 3 5 6 4 9 . 8 %
1 0 n 1 9 s 1 8 0 5 0 2 9 4 3 2 3 1 4 8 6 4 1 2 . 0 %1 0 n 1 9 s 2 6 6 3 0 5 1 0 1 3 1 0 3 3 6 9 6 1 2 . 0 %1 1 n 2 1 s 1 1 0 2 7 0 1 8 7 3 6 7 3 5 1 0 8 0 9 . 9 %1 1 n 2 1 s 2 1 0 3 1 0 0 5 1 0 3 7 0 5 6 1 2 0 0 1 0 . 2 %A v e r a g e 3 . 8 1 0 . 8 %
M J C P - 6 x 2 x
* Relative to the least-cost post-modularized design (PMSCP) with the same series of module costs
copyright © Wayne D. Grover 2000
Unexpected finding: Spontaneous Topology Unexpected finding: Spontaneous Topology ReductionReduction under strong economy-of-scale under strong economy-of-scale
scenarioscenario
PMSCP (benchmark) MJCP (joint design)
24
24
48
1224
24
48 96
24
24
24
24
48
48
24 48
48
24
24
48
48
48
48
48
48
48
4824
24
48
24
Total Capacity = 504Total Cost = 2861Total Used Spans = 17
Total Capacity = 612Total Cost = 2595 (9.3% savings)Total Used Spans = 13 (23.5% reduction)
9n17s2 - 6x2x
Class question: Why is this happening - explanation?
copyright © Wayne D. Grover 2000
Summary of ResultsSummary of Results
• 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)
copyright © Wayne D. Grover 2000
Routing algorithms and issues related to formulating the mesh design problem files
for AMPL / CPLEX
copyright © Wayne D. Grover 2000
Methodology - Generating the “eligible route” sets
• 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 ~ N2/2 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
copyright © Wayne D. Grover 2000
An overall strategy for generating route-sets needed in the formulations
• 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 a 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