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

copyright © Wayne D. Grover 2000

EE 681 Fall 2000 Lecture 15

Mesh-restorable Network Design (2)

W. D. Grover, October 26, 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


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


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


(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.


(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.


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


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.


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.


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


*

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 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






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









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


*

Results Results - “joint modular” capacity placement - “joint modular” capacity placement (MJCP)(MJCP)









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









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









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


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?


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)


Routing algorithms and issues related to formulating the mesh design problem files

for AMPL / CPLEX


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


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

Documents

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