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### Project SARU

How to situate Access Remote Units and construct a minimal

cost fibre optic cable network

July 2006

Ivana Ljubic University Vienna

Bertram Wassermann Telekom Austria

- Broadband demand increases.
- New products and Services (ADSL TV)
- Number of customers still increasing

- Existing local area access networks are based on copper cables
- Limited with respect to bandwidth and distance
- Will not cover upcoming demand

- Fibre optic technology is the alternative
- nearly unlimited bandwidth
- used for core – net
- rarely for LAN

- Consequence
- Creating a new network -> network design problem

- Terms known in the industry
- FTTH, Fibre To The Home
- FTTC, Fibre To The Curb

1

- Fibre To The Home (FTTH)
- No copper between customer and switching centre anymore
- Customers are directly connected via fibre optic cable
- Probably no multiplexing (or only small scale)
- Passive, no need electricity

- Fibre To The Curb (FTTC)
- A Access Remote Unit (ARU) is placed close (“at the curb”) to several customers
- “Last few meters” still copper
- The ARU functions as a translator between copper (electricity) and optical medium (light)
- Serves also as a multiplexer (Customers share Fibre)

- Solution to these network design problems:
- Steiner Trees and its capacitated variants
- Well studied
- Although NP-hard, fast algorithms do exist

1

- Search for an alternative
- SARU, Situating Access Remote Unit
- Fibre as close to the customer as necessary and as far as possible

- Problems with FTTH and FTTC
- Expensive as a country-wide approach
- Inefficient: Telekom Austria wants to be prepared for any customer, but knows not all customers will come.
- FTTH or FTTC probably suitable for certain LANs or specific parts of LANs

- Key idea
- Within a certain distance (L) of the customer an ARU (Access Remote Unit) has to be placed / situated which houses this customer.
- Copper network still supplies last mile
- At the moment L = 600m

- Distance Metric
- Length of cable is used

1

Switching nodes

Switching centre Root of the Copper Tree

Source node

Customer nodes

Copper cables

Copper tree

Leaves are customers

But customers need not be leaves

Graph structure should be tree-like.

Big pre-processing problem!

Problem Definition Graph2

Potential Fibre Optics Network

Potential Fibre Optics Lines

with intersection nodes

All nodes should be connected to the switching centre

The Fibre Optics net should form a connected graph!

Problem Definition Graph2

*) FON is not shaped like a rectangular grid!

Shape indicates, that FON may be of different form

then copper net. However, nets are superimposed

Potential Fibre Optics Network

Additional Nodes:

Intersection points of FOL and Copper Net

Potential Fibre Optics Lines

with intersection nodes

Problem Definition Graph2

Potential Fibre Optics Network

Potential ARU positions

Additional Nodes: Intersection points of FOL and Copper Net

Potential ARU Positions are chosen in the vicinity of intersections of copper net and fibre optic net

Potential Fibre Optics Lines

with intersection nodes

Problem Definition Graph2

Distance Condition L and Edge Directions

Assignement of Customer to potential ARUs under Distance Condition

Additional Condition:

Never go up the tree, always go down towards root.

But Fibre Optic edges may be used in one of the two directions.

Consequently:

Copper edges are directed.

Problem Definition Graph2

Distance Condition L and Edge Directions

Assignement of Customer to potential ARUs under Distance Condition

Alternative Representation of the Copper Net obeying Distance condition

Problem Definition Graph2

Find Positions for ARUs and create Fibre Optic Network such that

Problem Definition Graph2

- all customers are served

- all ARUs are connected to the root by fibre optic lines

- all other constraints are met (length L)

- all this is done at minimal cost

In FTTH(C) the end-nodes (ARU positions) are given and therefore fixed.

No optimisation of their position is necessary.

This optimisation formulation corresponds to the Steiner Tree Problem.

In our problem the graph consist of two strictly separated layers (copper network, potential FON) and a set of nodes potentially connecting them.

In FTTH and FTTC there is “just” one layer and no set of designated nodes besides customer nodes.

Problem Definition Graph2

Connected Facility Location Problem (ConFL)

- Given a graph G=(V,E), with lengths on the edges, with a subset of facilities, their opening costs and client demands.
- Our goal is to:
- Pick a set of facilities to open
- Assign each demand to an open facility
- Connect all open facilities by a Steiner tree
- Minimize the costs of opening and assigning facilities, plus the cost of the Steiner tree

Our problem reduces to ConFL if edge installation costs are M*length.

- Approximation algorithms:
- Gupta et al. (2003): randomized 3.55-factor algorithm
- (no opening costs)
- Swamy & Kumar (2002): 9-approx. algorithm for general case

3

Capacitated Local Access Network Design (CapLAN)

- Simplifying our problem:
- For already placed access nodes, find minimum-cost capacity installation of the fiber optic network.
- Also known as Network Loading Problem. Edge-cost function depends on capacity and may be piecewise-linear or step function.

- Uniform capacities:
- Edge-cost function the same for all edges Single-sink buy-at-bulk
- Approx. algorithms: Gupta et al. (2003)
- Polyhedral approaches: Magnanti (1995), Günlück (1999)

- Non-uniform capacities:
- Dahl & Stoer (1998): cutting plane approach
- We propose our problem-specific non-uniform ILP formulation

3

Demand in terms of copper lines (twisted pairs of copper lines)

With every customer a certain demand di is associated

Rule:

Demand has to be completely satisfied

Not in the sense of bandwidth

Problem Definition Cost4

d2

d4

d3

d1

d5

d6

di

dn

di+1

Cost related to the Copper Net

The copper network has to be incorporated as it is.

No alteration allowed!

No cost due to copper network.

Problem Definition Cost4

d2

d4

d3

d1

d5

d6

di

dn

di+1

Cost related to the potential ARU locations

costARU( location, demand)

- Location cost factors are:
- Outdoor or indoor
- Electricity
- Rental
- Development

- Demand cost factors are:
- Type of ARU (mainly size = number of copper lines to be served)

Cost function is a step function (also in terms of demand)

Buy at Bulk principle:

Price per unit (=served copper line) decreases with increasing size of ARU

ARUs produce demand. #Fibre Optic Lines depends on type of ARU

Problem Definition Cost4

Cost related to the potential Fibre Optic Network

In general: costFON( length, demand) per edge

But: The potential FON is a union of 3 layers.

Layer 1: Dark Fibre

Existing Fibre Optic Lines which are not in use

Layer 2: Empty Pipes

Empty pipes where fibre optic cables may be inserted

Layer 3: Excavation

Excavating trenches and laying new pipes

None of the layers need to form a connected graph.

New trenches usually follow roadmaps

Two adjacent nodes of the potential FON may be connected by any combination of the 3 edge types!

Problem Definition Cost4

Any combination of the 3 edge types may connect two nodes.

Into both directions

Problem Definition Cost4

Cost related to the potential Fibre Optic Network

Dark Fibre Edge: costDF ( length, demand) = const

The cost for dark fibre may by viewed as being constant.

It is independent of the length of the line.

The work cost resulting from lighting the lines is a constant compared to costs resulting from other layers.

Problem Definition Cost4

Cost related to the potential Fibre Optic Network

Dark Fibre Edge: costDF ( length, demand) = const

Insertion cost for fibre optic cables is linear in terms of edge length

Need to know cost of cables per unit length

Like for ARUs

Cost function is a step function with respect to demand.

Empty Pipes: costEP ( length, demand) =

length*costEP/UL (demand)

Again Buy at Bulk principle:

Price per unit (=optic fibre) decreases with increasing size of fibre optic cables

Problem Definition Cost4

Cost related to the potential Fibre Optic Network

Dark Fibre Edge: costDF ( length, demand) = const

Excavation costs depend linearly on the edge length

- Excavation costs depend on location in two ways:
- regionally cost may differ (big city, small city, country-side)
- surface conditions (concrete, soil, …)

Cost function obeys economies of scale (compare Buy at Bulk principle)

Empty Pipes: costEP ( length, demand) =

length*costEP/UL (demand)

Excavation: costExT ( length, location, demand) =

length*costExT/UL (location, demand)

Simplification:

Costs are based on the assumption,

trenches are filled completely with pipes which are completely filled with fibre cable.

Problem Definition Cost4

Cost related to the potential Fibre Optic Network

Dark Fibre Edge: costDF ( length, demand) = const

Empty Pipes: costEP ( length, demand) =

length*costEP/UL (demand)

Excavation: costExT ( length, location, demand) =

length*costExT/UL (location, demand)

costFON( length, location, demand) per edge =

const + length * [costEP/UL (demand) +costExT/UL (location, demand)]

Cost function is dominated by excavation costs.

Cheapest contribution from Dark Fibre.

With respect to free capacities it will be the other way round.

Problem Definition Cost4

- Phase 1:
- Solving the simplified problem:
- To improve the solution
- To study the cost function

1

- Phase 2:
- Solving the problem
- To find an exact algorithm
- Study the approximation qualities of heuristic solutions

2

- Pre-Phase, Heuristic solution
- A (really) fast algorithm for a first solution
- Finding a feasible solution for a given instance
- Initial upper bound for exact (branch-and-bound based) algorithm

0

Solution Strategy5

Simplified Optimization Problem

Start with copper net

0

Find “optimized” Positions for ARUs heuristically

1

Switch to Fibre Optic Net

2

Phase 0 & 16

Simplified Optimization Problem

Create fibre optic network with:

Phase 0

Heuristic Algorithm

Phase 1

Integer Linear Program (exact)

3

Start with copper net

0

Find “optimized” Positions for ARUs heuristically

1

Switch to Fibre Optic Net

2

Phase 0 & 16

Idea:

Optimal solution will “minimize” the number of ARUs necessary to satisfy all demand.

Hence, a set of ARUs satisfying all demand and minimal in number will approximate the optimal (=cost minimal) solution.

Algorithm

Pick customer furthest away from source.

1

Choose potential ARU node furthest away from this customer still valid under distance condition L

2

Install ARU at this position and serve all customers of sub tree rooted at this node

3

Ignore sub-tree and proceed form step 1

4

Pre-Phase, Heuristic Solution6

Solution is unique

Of all solutions with minimal number of nodes it’s the one where no ARU can be moved closer to the source node without violating the distance condition L for at least one customer.

Dropping this condition gives rise to different solutions

For example:

Nice to have:

We know minimal number of ARU nodes needed to provide complete service.

!

Pre-Phase, Heuristic Solution6

Cost minimized Fibre Optic Net

Simple but fast approach

to connect so found ARUs with source node via FON

Imitation of the Minimal Cost Flow algorithm for linear cost functions

Pick any unconnected ARU and determine shortest path through actual network.

1

- Update network along shortest path:
- cost-functions on used edges
- free capacities
- used capacities

2

Repeat from step 1 until all ARUs are connected.

3

Works for network with “unlimited” capacities on edges.

Pre-Phase, Heuristic Solution6

CapLAN: Notation for ILP formulation

Connectors Type Set … N=N1 U N2 U N3

For every edge type several connectors are possible

Edge Type 3: excavation trenches of different size

and filling … N3

Edge Type 2: different (combination of) cables to

fill empty pipes … N2

Edge Type 1: different (combination of) dark fibres … N1

Different Edge Types:

Edge Type 1: Dark fibre edges

Edge Type 2: Empty pipes edges

Edge Type 3: Excavation edges

Directed graph representing FON

with customer set (ARUs)

and sink (switching centre)

Phase 17

CapLAN: Notation for ILP formulation

length of edge (i,j):

building cost of connector type n:

indicator variable for connector type n being

installed on edge (i,j):

flow on edge (i,j) using connector type n

flow on edge (i,j) using connector type n

for customer k

customers demand (careful! customer=ARU)

capacity limit for edge (i,j) and connector type n

Phase 17

CapLAN: ILP Single-commodity formulation

objective

Flow preservation constraints

Capacity constraints

constraints

Phase 17

CapLAN: ILP Multi-commodity formulation with 0/1 variables

objective

Flow preservation constraints

Capacity constraints

constraints

Phase 17

Additional Connector Type Set

For every edge type several connectors are possible

Edge Type 0: Copper Connectors … N0

(only one element)

Edge Type A: potential ARUs … NA

Additional Edge Types:

Edge Type 0: Copper Connection of Customer and potential ARU node

Edge Type A: potential ARUs represented as edges

(Now real) Customer nodes

ARU nodes in (customer side)

ARU nodes out (sink side)

Phase 28

length of edge (i,j):

building cost of connector type n:

indicator variable for connector type n being

installed on edge (i,j):

flow on edge (i,j) using connector type n

flow on edge (i,j) using connector type n

for customer k

customers demand

ARU demand

capacity limit for edge (i,j) and connector type n

Phase 28

ILP Single-commodity formulation

objective

Fibre Optic Net

Flow preservation constraints

constraints 2

Phase 28

Connected Facility Location in Multi-commodity Networks

How to find an exact solution for the stated optimisation problem?

Phase 28

Connected Facility Location in Multi-commodity Networks

How to find an exact solution for the stated optimisation problem?

Phase 28

Connected Facility Location in Multi-commodity Networks

How to find an exact solution for the stated optimisation problem?

Phase 28

Connected Facility Location in Multi-commodity Networks

How to find an exact solution for the stated optimisation problem?

Phase 28

Connected Facility Location in Multi-commodity Networks

How to find an exact solution for the stated optimisation problem?

Phase 28

Connected Facility Location in Multi-commodity Networks

How to find an exact solution for the stated optimisation problem?

Phase 28

Connected Facility Location in Multi-commodity Networks

How to find an exact solution for the stated optimisation problem?

Phase 28

Preparation of Land for Building

- Difference 1:
- Layer connecting node do not multiplex

- The representation of this problem as a graph is very similar to the presented one:
- Customer demand has to be met through a potential network starting from a source node
- Graph of network consists out of two strictly separated layers (above ground, below ground) and a set of nodes potentially connecting the two layers)

- Difference 2:
- Design of network has to be optimised in both layers not only in one.

9

Marketing Retail – Business & Market Intelligence

Operations Research

Telekom Austria AG

Lassallestrasse 9, A-1020 Wien

Tel: +43 (0)59 059 1 31089

Ivana Ljubic

Mobil: +43 (0)664 629 5527

University of Vienna

E-Mail: [email protected]

E-Mail: [email protected]

Contact:Thank you!

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