Avail-o-nomics – Availability for Fun and Profit

You may or may not wear trousers, but for the sake of argument, imagine you do. Further suppose that you like to use braces to hold up your trousers. But braces can break. Braces cost $60 and a belt costs $40. Is it worth buying both so you can wear braces and a belt?Would you believe that, under certain made-up circumstances, the braces-plus-belt combo beats all other options: braces on their own; belt on its own; two sets of braces; or two belts.

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Just for the sake of illustration, let's use some made-up data. Firstly costs: instead of considering the up-front expense, we will assume that the total annual cost of ownership of a belt is $20 (including maintenance and replacements), and the coresponding figure for braces is $30. We have amortised the CAPEX and included any OPEX to come up with an equivalent annual cost. Also we will assume the following data for reliability and restoration:

- Braces: Mean Time To Fail is 20 months, Mean Time to Restore Service is one day;
- Belt: Mean Time To Fail is 10 months, Mean Time to Restore Service is two days.

Using the Availability Calculator that you downloaded in
**AVAIL**,
you should be able to verify all the numbers in the table below:

Parameters | Belt | Braces |
---|---|---|

Total annual cost including maintenance / replacements | $20 | $30 |

FR, Failure Rate per year | 1.2 | 0.6 |

MSRT, Mean Restoration Time in hours | 24 | 48 |

MSRR, Mean Service Restoration Rate in days ^{-1} |
0.5 | 1.0 |

DT, Average Downtime per year in hours | 57.60 | 14.40 |

Unavailability | 0.0066 | 0.0016 |

Using the Availability Calculator (or manually if you prefer)
you can now verify the parameters for three possible combinations of two components in parallel:
two belts, braces and belt, and two braces.
As usual we assume indpendence, and use the additivity of the Mean Service Restoration Rates and the Nines
(or -log_{10}(U)). Of course we ignore sartorial considerations:

Parameters | Two Belts | Braces and Belt | Two Braces |
---|---|---|---|

Total annual cost including maintenance / replacements | $40 | $50 | $60 |

FR, Failure Rate per year | 0.016 | 0.006 | 0.002 |

MSRT, Mean Restoration Time in hours | 24 | 16 | 12 |

MSRR, Mean Service Restoration Rate in days ^{-1} |
1.0 | 1.5 | 2.0 |

DT, Average Downtime per year in hours | 0.378 | 0.095 | 0.024 |

Unavailability | 4.3 × 10^{-5} |
1.1 × 10^{-5} |
2.7 × 10^{-6} |

So the Availability Calculator provides all the information we require, but how should we make a choice. Clearly the parallel combinations provide better reliability. For example, two braces in parallel will fail about once per five centuries! Clearly, two braces is also an expensive solution. The question arises: does the performance justify the extra cost.

At this point we will invoke an assumption about the preferences of you – the customer. Let's assume that you are willing to pay up to $50 per hour to reduce your “downtime”.

From Table 1 we see that upgrading from belt to braces reduces DT from 57.6 hours per year to 14.4 hours per year – a reduction of 43.2 hours per year at an additional cost of only $10 per year. Clearly that is a bargain at only 23 cents per hour of downtime.

From Table 2, wearing two belts instead of just one set of braces slashes DT by about 14 hours per year for an increased cost of $10 per year. Again, this is a bargain at $0.70 per hour of downtime.

Repeating the calculation for the transition from two belts to the braces-and-belt combination, DT is reduced by about 17 minutes per year for an equivalent cost of about $35 per hour of downtime, which is still worthwhile. However, upgrading to two braces reduces DT by a further 4 minutes per year approximately, at a cost of about $140 per hour of downtime – not cost-justified according to your preferences.

For the given data, the optimal solution is a braces-and-belt combination with DT of about 6 minutes per year and an annual cost of $50. You can expect a 16-hour outage about once every 169 years. The probability of having no failure for 70 years of trousers-wearing (average time span between wearing short pants with an elastic waistband, and wearing pyjamas all day) is about two thirds. A single customer, wearing braces and belt, and enjoying a modicum of good luck, may never experience an outage.

As a more complex example, suppose we are planning to instal a large number of identical items which consist of three components in series called “Power,” “Airconditioning,” and “Equipment”. For each component there are three options: high reliability, medium reliability, and low reliability. As you would expect, the three options come with three different annualised costs. We may choose any of the three options for each of the three components, giving twenty seven possibilities from which we will select one. The one selection will apply to all the items to be instaled.

Suppose we have one further choice to make. The Mean Service Restoration Time may be short, medium, or long. Short restoration time could be achieved by having more repair staff in more locations working more shifts per week, for example. The cost corresponding to each restoration option is not an annualised cost, but a cost per restoration performed. It can be converted to average annualised cost by multilying by the average number of outages per year.

Now we have a total of eighty one possible combinations, based on a one-in-three choice for each of three components, plus a similar choice for restoration. It is not very convenient to use the Availability Calculator for so many separate options, so instead the basic equations will be used in a separate program or spreadsheet. For each of the 27 component choices we simply sum the values of FR and apply each of the possible values of MSRT to find the values of DT for each possibility. We will assume that the value of DT, combined with the average annual cost, will give us the information needed to make a selection. The total average annual cost is comprised of the sum of the three component costs, representing annual amortised CAPEX, plus the sum of the annual Failure Rates times the cost per restoration, corresponding mainly to OPEX (although there may be some part of the restoration cost which is CAPEX, but that is applied when it is accrued rather than being amortised).