copyright © wayne d. grover 2000 ee 681 fall 2000 lecture 14 mesh-restorable network design (1)...

copyright © Wayne D. Grover 2000

EE 681 Fall 2000 Lecture 14

Mesh-restorable Network Design (1)

Wayne D. Grover, TRLabs / University of Alberta October 24, 2000


Key ideas and vision behind “mesh” restoration

• (1) many real networks are highly mesh-like in their topology

• (2) for restoration, generalized re-routing over the graph can permit greater sharing of spare capacity

– the redundancy will go down in proportion to the average nodal degree

• (3) the network can be its own computer for the real-time solution of the restoration re-routing problem

– the network can self-organize restoration pathsets is a split-second

– without any external control or databases

• (4) if a network is mesh-oriented it is more flexible and adaptable to unforeseen patterns of demand

– the network can continually self-organize its mapping of physical transmission to logical transport configuration to suit time-and-spatially varying demandpatterns


A look at some real network topologies

TORINO

GENOVA

ALESSANDRIA

PISA

MILANOBRESCIA

SAVONA

BOLOGNA

VERONA

VICENZA

VENEZIA

FIRENZEANCONA

PESCARA

PIACENZA

MILANO2

PERUGIA

L’AQUILA

ROMA

ROMA2

NAPOLI SALERNO

CATANZARO

POTENZA

BARI

TARANTO

CAGLIARI

SASSARI

FOGGIA

PALERMOMESSINA

REGGIO C.

32-node Italian backbone transport

network



Belgiannational transport

network

(Belga 39 - 39 nodes, 59 spans)



“COST 239” European Communityproject model

( 19 nodes, 40 spans)



“Bellcore” (New-Jersey LATA)(LATA = local access and

transport area)

0

1 2

3

4

5

6

7

8

910

11

12

13

14



“MCI” North Americancontinental backbone

(homeomorphism oftopology only)

0

1

2

345

6

7

8

9

10

11

12

13

14

15

16

17

18

19

2021

22

23

24

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26

27

28

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31

32

33

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39

40

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48

49

50

5152



Level (3) North American

continental backbone


Representing networks in a “.snif” file

Date: File Name: bellcore.snifNetwork: New Jersey LATA area from 1993 publication ...Program: Jointspandatprep.exe followed by AMPL JointSCP.mod

Node Xcoord Ycoord0 60 751 40 802 70 803 50 704 10 405 10 556 45 507 30 478 45 409 60 3010 30 3011 10 2012 50 1013 15 7014 70 50

Span NodeA NodeB Distance Working Spare1 0 1 9.000000 20 122 0 2 6.000000 8 163 0 14 21.000000 16 44 1 2 14.000000 28 45 1 3 6.000000 32 326 1 13 11.000000 48 167 2 14 16.000000 20 128 3 13 8.000000 24 169 3 14 11.000000 48 1610 4 7 7.000000 28 2811 4 11 7.000000 0 2812 5 6 5.000000 12 2813 5 7 7.000000 76 24

..... to last span entry (28 here)

snif = TRLabs standardnetwork interface file

Notes:

- x,y node co-ordinates are optional to support graphical display applications

- working and spare quantities may or may not be present depending on use or stage of design processing


Our journey will follow this sequence of concepts ....

• Theory and methods for mesh-restorable network capacity design

– span restoration

– path restoration

– non-joint, joint, modular, ..other variations

• “Selfhealing”:

– distributed real-time restoration

• Self-organization:

– continual autonomous adaptation of transport to suit demands


Capacity design of Mesh restorable networks

Start by looking at some actual fiber-optic transmission facilty networks ...


Basics of Mesh-restorable networks

(28 nodes, 31 spans)

30% restoration70% restoration100% restoration

span cut



(28 nodes, 31 spans)

span cut

40% restoration70% restoration100% restoration



Spans where spare capacity was shared over the two failurescenarios ? .....

This sharing efficiency

increases with the degree of

network connectivity

“nodal degree”



a common, but misleading, portrayal of span restoration

the more general concept of span restoration permiting much greater sharing of network spare capacity



Nodal Degree vs. Average Capacity(Network: CSELTNet, Gravity Demand, Least Usage Elimination)

0

100

200

300

400

500

600

700

800

900

2.0 2.5 3.0 3.5 4.0

Average Nodal Degree

Cap

acit

y R

equ

ired

w orking

spare

capacity

Mesh networksrequire lesscapacityas graphconnectivityincreases

(sample result)


A simple lower-bound on achievable redundancy

• Consider two idealizations:– (1) restoration is “end node limited”

• i.e., the min cut governing restoration path number is at one or the other of the custodial nodes

– (2) node has span degree d

– (3) all wi are equal at the node • mesh equivalent of the ‘perfect balance’ notion with rings

then:

. . .

OCX

W

WW

W

d spans in total

if any one span fails, the total sparecapacity on the surviving (d-1) spansmust be >= to w.

hence....

redundancy =(node)

1( 1)( 1)

wd

spare dworking d w d

d


A simple lower-bound on achievable redundancy

• Comment and interpretation:

This just says that in the idealized limit every span has w working capacity

and w/(d-1) spare capacity.

Although the redundancy ratio falls as (d-1)-1 , the absolute working and

spare quantities fall even faster as d increases.

In practise, the lower limit is not achieved primarily because wi values are

not balanced, rather than because (span-) restoration is not end-node

limited (it almost always is above a certain d.)

Class problem (extension of the basic result): show that with unequal wi values at a node, a more general lower bound on spare capacity is given by the relationship:

where the notation s1, s2, w1, w2 denotes the largest and second-largest spare and working capacity quantities at the node.

2..

1 2 2 1 2j d

s s s w wj


Review: Link, Span, Path and Route terminology

A

B

Bandwidth management or cross-connection facility at the link-unit capacity level

Span A-Ba unit-capacity link

G

C

D

E

FA

B

a path

its route


Review: Types of “route”: fully disjoint, span disjoint, and distinct

0

12

3

4

56

7

89

10

1112

1314

0

12

3

4

56

7

89

10

1112

1314

0

12

3

4

56

7

89

10

1112

1314

Paths having (a) fully disjoint, (b) span disjoint and (c) simply distinct routes.

(a) disjoint (a) distinct(a) span-disjoint


Span restoration: what we mean

• The set of working paths severed by a span cut are restored by substitution of a set of local replacement paths between the end nodes of the failed span.

• The restoration path-set is equivalent to single-commodity max-flow routing or k-shortest paths routing between failure end nodes within the surviving portion of the reserve network.

• The number of paths crossing any span must respect the discrete spare capacity on the span.

• Example:


Some notes on span restoration

• A network employing a span restoration mechanism and an optimally designed (e.g., minimal capacity) reserve network that just supports the target level of restorability by that mechanism is what we mean by “a span-restorable mesh network”.

• It has been found that in relatively sparse but non-planar graphs, k-shortest paths > ~ 99% of max-flow.Ref: (D.A. Dunn, W.D. Grover, M.H. MacGregor, “A comparison of k-shortest paths and maximum

flow methods for network facility restoration”, IEEE J. Sel. Areas in Communications, Jan. 1994, vol. 12, no. 1, pp. 88-99.)

• In test cases a distributed restoration algorithm (DRA) must find the same maximal number of simultaneously feasible restoration paths within the discrete-capacitated spare capacity graph as a centralized

reference algorithm. (P.N.E. = 1) .


Overview of three basic * approaches for mesh spare capacity design

•Sakauchi - “cut oriented” formulation

• Herzberg - “arc - path” formulation

• “Transportation-like” problem formulation

Note: In these basic methods, the working paths are first shortest-path routed before solving the spare capacity placement problem. This is what we later refer to

as the “non-joint” problem.


Where:

S = set of all spans

ci = cost of a spare link on span i.

si = number of spare links assigned to span i.

Ci = set of “partial cutsets” relevant to restoration of span i.

= 1 if span j is a member of the cth partial cutset relevant to restoration of span i , 0 otherwise.

c = an individual cutset in the set Ci

wi = the number of working links on span i (working demands are routed prior to solving for the spare capacity.)

jc

Sakauchi’s “min-cut max-flow” approach for spare capacity design

Ref: H. Sakauchi, et al., “A self-healing network with an economical spare-channel assignment”

IEEE GLOBECOM ‘90, pp. 438-443, 1990. min

i

c sii

S

{ }j i

j s wc j is

;c C i Si S.t.


Review of Min-cut max-flow theorem (basis for Sakauchi’s method)

• A cut of a graph G(V,E) is a partitioning of the nodes of the graph into two disjoint sets

• A cutset is the set of edges .

• The capacity of a cut is the sum of the weights on the edges of the cutset.

• The min-cut max-flow theorem is that :

or...

• More intuitively: “regardless of the routing details, the maximum flow

between two nodes is set by the minimum capacity of any cut of the between those two nodes” or…“some combination of spans in parallel will always act as the bottleneck”

P,P : P P = V( , ) : ( ) ( )i j i j E P P

max-flow ( ) min { c( , ) : ( ) ( )}i j G i j P P P P

( , )i j

c sij

P P

P P

( , ) ( ) : ,f c ij i jij

P P P P

( , )c P P

( , )c P P


Recall: the “max-flow = min-cut” concept

A

B

E

C

D

F

20

620

68

6

6

26 1220 32

26

the “min-cut” = 12

Hence --> no routing solution can provide more than 12 units of flow between A -F (i.e., this is the “max-flow”)


How Sakauchi uses this: sizing the cutsets to support restoration

1 2

4

3

5

w1-2

s1-4

s3-4

s1-3s2-5

s4-5

C1

C2

C3 C4

C5

Illustrating the “partial cutsets” relevant to restoration of span (1-2):

Corresponding constraints:

C1: s1-3 + s1-4 + s1-5 w1-2C2: s3-4 + s1-4 + s1-5 w1-2C3: s4-5 + s1-5 w1-2C4: s4-5 + s1-5 + s2-5 w1-2C5: s2-5 w1-2


Sakauchi’s method - technical aspects

• Number of cuts of graph is O(2 S)– Have only S variables but S O(2 S-1) partial cutset constraints in a

complete (fully constrained) formulation– > hence row generation (in the primal) methods have been used.

• Row generation:– Each added row represents a new partial cutset constraint for a

span that is not yet fully restorable.– Uses a separate program to test for full restorability (formulation

itself has no explicit expression of restorability for built-in infeasibility detection.)

– Sakauchi used the N-1 basic cuts of a graph in the initial tableau.– Venables provided composite strategy for incident cutset

constraints in initial tableau plus targeted discovery of “most relevant” additional cutset constraints.

• Other practical aspects:– Can run as LP with rounding up at each iteration– Don’t get (or control) restoration path-sets– Corresponds to perfect max-flow restoration routing


Venables strategy for initializing Sakauchi’s cut- dimensioning approach

“Incident cutset “ constraints:

s2 w0

s3 + s1 w0

s5 + s7 w2

s0 w2

s6 w4

s1 w4

s4 w6

s5 + s8 + s3 w6

s3 + s0 w1

s4 w1

s6 + s5 + s8 w3

s0 + s1 w3

s7 s2 w5

s6 + s8 + s3 w5

s5 + s2 w7

s8 w7

s6 + s5 + s3 w8

s7 w8

Ref: B. Venables, M.Sc.thesis, “Algorithms for near optimal design of mesh-restorable transport networks”, University of Alberta, Fall 1992.


1. Run restoration routing algorithm to find an un-restorable span.

2. Inspect the restoration path-set for that span (available as a result from 1.)

3. Remove saturated edges from the graph.

4. Each combination of disconnected sub-graphs in which two subgraphs remain, one containing s, the other t, defines a new cutset constraint.

5. Done when step 1. Finds all spans fully restorable.

Venables strategy for discovering new cut-set constraints

Next cut-set constraint:

s7 + s3 +s6 >= w5

(also: s0 + s7 >= w5 )

s t

remove edges saturated by the restoration

routing trial (or at zero in the LP)

1 path

1 path

si values after first LP iteration


Where:

• S, ci, si, wi are as before

• Pi is a set of “eligible routes” for restoration of span i

• is an assignment of restoration flow for span i to the pth eligible route

• encodes the eligible restoration routes: = 1 if span j is in the p th eligible route for restoration of span i

,p

i j

pfi

Herzberg’s “arc-path” hop-limited approach

mini

c sii

S

2( , ) .i j i j S

p i

pf wii

P

.i S

,p i

p pf sji ij

P

Subject to:

Restorability :

Spare capacity :

Ref: M. Herzberg, and S. Bye, “An optimal spare-capacity assignment model for survivable networks with hop limits,” Proc. IEEE GLOBECOM ‘94, pp. 1601-1607, 1994.


Understanding the span-restorable mesh spare capacity problem through Herzberg’s approach

failure scen-rios (input)

flows over eli-gible routes (decide)

greatest requirement on span j (result)

flows simulta-neously imposed on span j (result)

all other span spare capacit ies

s (output)

sj

total spare capacity (mini-mize)

Flows over eligible routes

Represented in the eligible route - defining information input

,p

i j

pfi Flows simultaneouslyimposed on any span

si values


Herzberg’s approach - technical aspects

• working paths are routed prior to this formulation --> (provides the wi’s)• A separate program finds “all distinct routes” for each failure scenario (depth-

first search).• DFS may be limited by a hop - count, a distance limit, any other operational

criterion.• These become the “eligible routes” for restoration.

• “Eligible routes” are not apriori decisions about the restoration routes to be taken for each failure…they only represent the routes available for restoration flow assignment.

• There are– S restorability constraints (equalities) and

S(S-1) spare capacity generating inequality constraints. – ~ variables - but controllable via eligible routes.

• Yields restoration path-set details along with reserve network spare capacity.

• The LP relaxation sometimes acts unimodular but this is data-dependent. Solution as IP often solves quickly enough or a “repair” procedure can be devised for fractional outcomes when solved as LP.

S Si P


How hop-limit affects complexity and solution quality in Herzberg’s formulation

500

550

600

650

700

750

800

850

900

2 3 4 5 6 7 8 9

Design Hop Limit, H

To

tal

Sp

are

Cap

acit

y

(864, 51)

(678, 191)

(625, 1351)

(625, 1687)

(642, 476)

(625, 896)

Threshold value( for the network shown )

( Total spare capacity, total number of eligible restoration routes )

Minimum spare

• Below the design threshold hop-limit, solution quality is affected.

• Above the threshold hop limit, computational difficulty grows unnecessarily


Some practical notes re: “hop limit” concept

• In practice,

– hop limits can easily be converted to mileage limits or combined hop / distance limits in generating the eligible route-sets for the formulation.

– the basic idea of a single network-wide hop limit can evolves into approaches such as adapting hop limits per span (and per working route in the joint formulations) so as to assure a minimum representation of route diversity, within a computational “budget” for numbers of variables and constraints.

copyright © wayne d. grover 2000 ee 681 fall 2000 lecture 14 mesh-restorable network design (1)...

Documents

copyright wayne

snif network

real network topologies

real network topologies

meshrestorable network

mesh restoration

transport area slide

real networks