Project: System reliabilityA telecommunication company aims to construct a new glass fibre network that should cover a large

geographical area. The company has already decided that this network should have a modular design. That is,

the total area to be covered consists of 13 regions. Each region will have its own local network (module). The

local networks will be connected mutually. We will study the reliability of this fibre glass network. Design your

own spreadsheet model for your reliability calculations.

The local networks

Each local network consists of several nodes (connecting stations) that are connected by fibre glass cables.

Various layouts are under consideration. Depending on the region size, a local network may consist of 3, 4, 5 or

6 nodes. We will examine the following layout options:

Layout options with three nodes (3A-3B):

Layout options with four nodes (4A-4D):

Layout options with five nodes (5A-5D):

Layout options with six nodes (6A-6C):

Figure I.1. Layout options for the local networks.

For the local networks, we assume that the nodes are 100% reliable. The reliability of the fibre glass cables are

all identical and equal to p.

The main network

The total network consists of the 13 local networks that are mutually connected by special interregional cables.

Various basic shapes for the main network are being considered:

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Star

Circle

Double circle

Triple circle

Quadruple circle

Figure I.2. Layout options for the main network (each node represents a local network).

The network is “up” if there is at least one path connecting each pair of (local) nodes in the network. So, we

need both the reliability of each local network and the reliability of the interregional cables (denoted by q) to

calculate the system reliability of the full network.

Questions about local network reliability

Hint: it is convenient to use the built-in functions of EXCEL for your calculations, for example the binominal

distribution: BINOMDIST(number_s,trials,probability_s,cumulative); see the EXCEL help function for

details.

Question 1.

What do the layout options 3A, 4A, 5A and 6A for the local networks (Figure I.1) have in common? Calculate

the reliability for these local networks in a spreadsheet for p=0.9, 0.95 and 0.99.

Question 2.

What do the layout options 3B, 4B, 5B and 6B for the local networks (Figure I.1) have in common? Calculate

the reliability for these local networks in a spreadsheet for p=0.9, 0.95 and 0.99.

Question 3.

Give the minimal path vectors and minimal cut vectors for layout option 4C (Figure I.1). Explain how the

system reliability can be calculated using these minimal path vectors and minimal cut vectors (you do not have

to make calculations by now).

Question 4.

Explain how pivotal decomposition can be used to calculate the reliability of local network 4C (Figure I.1).

Question 5.

Add formulas for the reliability of local network 4C (Figure I.1) to your spreadsheet. Calculate the reliability

for p=0.9, 0.95 and 0.99. Check your own work: the reliability of local network 4C should be around 0.977 if

p=0.90

Question 6.

Derive formulas for the reliability of the rest of the local networks (Figure I.1). Add these formulas to your

spreadsheet and calculate the reliability of each local network type for p=0.9, 0.95 and 0.99.

Questions about the overall network reliability

Question 7.

Calculate in your spreadsheet the system reliability for each of the five overall network layouts (Figure I.2),

assuming for the time being that all local networks are 100% reliable and that the interregional connection

reliability equals q=0.99.

Question 8.

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Now suppose that each region may choose its own local network layout (Figure I.1), based on its own local

geographical considerations. The result is as follows:

Local

number of

number of

network

connections in regions preferring

layout

local network

this layout

3A

2

0

3B

3

2

4A

3

0

4B

4

1

4C

5

1

4D

6

2

5A

4

0

5B

5

1

5C

6

2

5D

6

1

6A

5

0

6B

6

2

6C

7

1

TOTAL

13

Table I.1. Overview of local network layout preferences

So, we see from the last column that zero local networks choose layout 3A, two of the local networks choose

layout 3B, etc. Totally, there are 13 nodes, so 13 preferences for the local network layout are given. The second

column just shows auxiliary information, namely the number of connections (links) in the local network (cf.

Figure I.1).

Now the question is to determine the reliability of each of the five network layouts (Figure I.2), taking into

account the local network reliability (related to the layout preferences from Table I.1) and assuming that the

reliability of each local cable equals p=0.95.

Question 9.

The following costs for the connections are given. Each interregional cable costs 5000 units, whereas the local

connections are available in various qualities. The higher the quality results in a higher the reliability, but this

comes at the cost of a higher price. The costs per local cable as function of its reliability r is given by

15

. We assume that the local cables are bought using an overall contract, so only one type of

Costs(r ) =

1− r

cable may be selected for all local networks (recall that we have another cable type for the interregional

network with reliability 0.99 each and costs 5000 each). The company has set the requirement that the overall

network reliability should be at least 0.98.

Now the question is to determine for each of the five layout options for the main network (Figure I.2) which

reliability of the local cables is required to meet the company target of 0.98 overall network reliability (hint: use

the function Goal seek or the Solver from Excel). Calculate the costs involved. Which main network layout can

be used to attain the target reliability at minimal costs?

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