span-restorable mesh network design (1) w. d. grover trlabs & university of alberta © wayne d....

Span-restorable Mesh Span-restorable Mesh NetworkNetwork

Design (1)Design (1)W. D. Grover

TRLabs & University of Alberta© Wayne D. Grover 2002, 2003

E E 681 - Module 11

( Version for book website Dec. 2003)

E E 681 - Module 11 © Wayne D. Grover 2002, 2003 2

Concepts to be covered

• Basics of Mesh-restorable networks

• The concept of span restoration

• The lower-bound on redundancy in mesh restorable network

• Three basic approaches for mesh Spare Capacity Placement (SCP) Design

- Sakauchi “cut oriented” formulation

- Herzberg “arc-path” formulation

- “Transportation-like” problem formulation (to be covered in next lecture)


• (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

Key ideas and vision behind “mesh” restoration


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

A look at some real network topologies


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

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2021

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5152



Level (3) North American

continental backbone



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”



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 networks require less capacity as graph connectivity increases (sample result)


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.

• 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”.



a common, but misleading, portrayal of span restoration

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


• 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) .

Some notes on span restoration


Design Objective: Span Restorable Mesh Networks

• Given a transport network with known working capacities, how much spare capacities should we place on each span so than the network is 100% restorable to a single span failure?

Working capacities

of each span are known

How much spare

capacities do we need

for each span?

4

2

5

3

11

9

15

4

4

2

25

78


• 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


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

Overview of three basic Overview of three basic approaches for mesh spare approaches for mesh spare

capacity designcapacity design

• Sakauchi - “cut oriented” formulation

• Herzberg - “arc - path” formulation

• “Transportation-like” problem formulation

Note: In these 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

(Ref: H. Sakauchi, et al., “A self-healing network with an economical spare-channel assignment”, IEEE GLOBECOM ‘90, pp. 438-443, 1990.)

mini

c sii

S

{ }j i

j s wc j is

;c C i Si S.t.

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


• 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

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


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”)

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


1 2

4

3

5

w1-2

s1-4

s3-4

s1-3

s2-5

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

How Sakauchi uses this: sizing the cutsets to support restoration


• 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

Sakauchi’s method - technical aspects


“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. )

Venables strategy for initializing Sakauchi’s cut- dimensioning approach


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.

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

Venables strategy for discovering new cut-set constraints


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

,pi j

pfi

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.)

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


Understanding the span-restorable mesh spare capacity problem

Net

wor

k st

ruct

ure

Failure scenarios

Failure scenarios

Failure scenarios

Greatest requirement on

all spans

Total sparecapacity

(minimize)

Failure scenarios

Flows overeligible routes Flows simulta-

neously imposed on any span

All other spare capacities

si values

(Based on Herzberg’s approach)

pfi

Represented in the

eligible route - defining

information input ,pi j


Solving Herzberg’s SCP ...

Input Parameters to Solvers

Output (Variables) from Solvers

Working capacities of each span i

Set of eligible restorationroutes for each span failure i

wi

Pi, i,j p

Unit cost for each span i c i

Spare capacities of each span i

Restoration Flow p

s i

f i p

Restoration routes finder for all network spans

Shortest path routing

LPsolve

LPsolve


• working paths are routed prior to this formulation --> (provides the wi’s)

• A separate program finds “all distinct routes” for each failure scenario based on the depth-first search algorithm (DFS).

• 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 a priori 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

Herzberg’s approach - technical aspects


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

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


• 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.

Some practical notes re: “hop limit” concept

span-restorable mesh network design (1) w. d. grover trlabs & university of alberta © wayne d....

Documents

mesh restoration slide

mesh restorable network

real network topologies

basics of mesh

restoration span

lecture slide

real networks

restoration pathsets