Reliability Properties of Series and Parallel Systems

Series system and parallel system properties provide valuable business and operational implications.

Reliability Properties of Series Systems

Property 1

Series system reliability property 1 is a simple mathematical rule. When multiplying decimal places, the result will always be lower than the numbers in the formula. As you can see from the equations below, the system reliability can never be higher than the least reliable component in the system.

When a “Bad Actor” item of plant has low reliability, it prevents the plant from achieving its best performance Series System Reliability Property 1 tells you to improve the reliability of the Bad Actor. Until its reliability is raised the operating plant’s reliability is always going to be lower than that of the Bad Actor equipment.

This property has a valuable implication for businesses looking to improve their overall reliability for the least expenditure.

1. The reliability of a series system can be no higher than the least reliable component.

Worked Example:

The cost to increase a system’s reliability from 0.72 to 0.81 by improving a single piece of equipment reliability from 0.8 to 0.9 is $100,000. Alternatively, it costs $10,000 to improve the reliability of neighboring piece of equipment from 0.9 to 0.95. This improvement would increase the system’s reliability from 0.72 to 0.76.

Which is going to achieve the greatest return on investment?

If $100,000 was to be spent, then the increase in reliability would be 12.5%.

However, if the neighboring equipment were improved for only $10,000 the increase in reliability would be 5.5%.

This means, that for 10% of the cost of the first option, the reliability of the system would increase by nearly half the predicted increase. As such, it produces a greater return on investment to spend less for the reliability improvement.

Property 2

When you add an item into a series system the system reliability falls. This is illustrated in the equations below, where the more items added to a series reduces the overall system reliability.

For every part added to a series system, there is a greater chance of an issue occurring and more work tasks are required to maintain each part. The only way to keep system reliability the same when adding items to a series system is for each new item to be perfectly reliable, i.e. R = 1. Currently this impossible for equipment to forever be perfect and reliable. Additionally, when work is done on the extra items more opportunities arise for human error to create defects.

Property 2 of series systems implies that smart businesses should seek to simplify their operations wherever possible. Simplified designs with fewer components automatically achieve far better reliability than complex ones. Consequentially there is also far less maintenance cost and fewer operational problems.

2. If "k" more items are added into a series system of items, the reliability of other items must rise to maintain the original system reliability.

Rsystem= R1 x R2 x R3 …Rn




 R3 Rsystem 
0.90.9=0.81Before adding one more item
-10% adding one item
0.9320.9320.932=0.81After improving the system


Property 3

By applying best practices throughout an operation that produces a small increase in reliability causes a much larger rise in system reliability. As shown below, an increase of reliability from 0.93 to 0.95 for each component in a four-item series system produces as system-wide reliability increase by 8.9%.

This Property indicates that there are system-wide, repeatable processes that can improve the level of reliability of equipment. Examples of this include Lubrication Management, Maintainer and Operator up-skilling, and the use of Standard Operating Procedures. For these processes they should be conducted to meet best practices for assets across their lifetime.

3. A small rise in reliability of all items causes a much larger rise in system reliability.

Rsystem= R1 x R2 x R3 …Rn




 R3 Rsystem

System-wide reliability improvements improve system reliability by a vastly greater amount.


R1 R2 R3 R4 Rsystem
0.930.90.930.93=0.805Reliability before
2.1%2.1%2.1%8.9%Percentage improved
0.950.590.950.95=0.857Reliability after

Reliability Properties for Parallel Systems

Parallel system arrangements, in either fully active or duty-and-standby configuration, have two key properties. The first is, the more components in parallel the greater the system reliability. As more items are added in parallel, there are more ways the output can be sustained when one item fails.

The second is, the reliability of a parallel system is higher than the most reliable part of the system. Putting things in parallel delivers the greatest system reliability possible.

This produces three important business implications:

  1. Parallel systems should be used when the risk of failure from a series arrangement is unacceptable.
  2. Intentionally provide options for a series process to continue operation if an equipment part fails.
  3. Build redundancy into all aspects as you go to give yourself as many options to be successful as possible.

Rsystem= 1 – [ (1 – R1) x (1 – R2) x (1 – R3) …(1 – Rn) ]

Series and parallel systems properties provide valuable business and operational implications.

  • By improving the least reliable equipment, the reliability of the whole system increases,
  • Systems should be simplified as much as possible,
  • Making system-wide reliability improvements brings greater benefit than improving individual equipment reliability,
  • Spare parts for equipment in series systems should be immediately available for quick repairs if a breakdown occurs
  • Standardize equipment as much as possible
  • Eliminate component failure modes wherever possible
  • Provide connections into series systems to temporary replacements can be utilized if breakdowns occur.