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  1. Getting Started with Algorithm
    What is an Algorithm?
  2. Characteristics of Algorithm
    1 Topic
  3. Analysis Framework
  4. Performance Analysis
    3 Topics
  5. Mathematical Analysis
    2 Topics
  6. Sorting Algorithm
    Sorting Algorithm
    10 Topics
  7. Searching Algorithm
    6 Topics
  8. Fundamental of Data Structures
  9. Queues
  10. Graphs
  11. Trees
  12. Sets
  13. Dictionaries
  14. Divide and Conquer
    General Method
  15. Binary Search
  16. Recurrence Equation for Divide and Conquer
  17. Finding the Maximum and Minimum
  18. Merge Sort
  19. Quick Sort
  20. Stassen’s Matrix Multiplication
  21. Advantages and Disadvantages of Divide and Conquer
  22. Decrease and Conquer
    Insertion Sort
  23. Topological Sort
  24. Greedy Method
    General Method
  25. Coin Change Problem
  26. Knapsack Problem
  27. Job Sequencing with Deadlines
  28. Minimum Cost Spanning Trees
    2 Topics
  29. Single Source Shortest Paths
    1 Topic
  30. Optimal Tree Problem
    1 Topic
  31. Transform and Conquer Approach
    1 Topic
  32. Dynamic Programming
    General Method with Examples
  33. Multistage Graphs
  34. Transitive Closure
    1 Topic
  35. All Pairs Shortest Paths
    6 Topics
  36. Backtracking
    General Method
  37. N-Queens Problem
  38. Sum of Subsets problem
  39. Graph Coloring
  40. Hamiltonian Cycles
  41. Branch and Bound
    2 Topics
  42. 0/1 Knapsack problem
    2 Topics
  43. NP-Complete and NP-Hard Problems
    1 Topic
Lesson 35, Topic 6
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Reliability Design

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In reliability design, the problem is to design a system that is composed of several devices connected in series.

Reliability design problem

If we imagine that r1 is the reliability of the device.

Then the reliability of the function can be given by πr1.

If r1 = 0.99 and n = 10 that n devices are set in a series, 1 <= i <= 10, then reliability of the whole system πri can be given as: Πri = 0.904

So, if we duplicate the devices at each stage then the reliability of the system can be increased.

It can be said that multiple copies of the same device type are connected in parallel through the use of switching circuits. Here, switching circuit determines which devices in any given group are functioning properly. Then they make use of such devices at each stage, that result is increase in reliability at each stage. If at each stage, there are mi similar types of devices Di, then the probability that all mi have a malfunction is (1 – ri)^mi, which is very less.

And the reliability of the stage I becomes (1 – (1 – ri) ^mi). Thus, if ri = 0.99 and mi = 2, then the stage reliability becomes 0.9999 which is almost equal to 1. Which is much better than that of the previous case or we can say the reliability is little less than 1 – (1 – ri) ^mi because of less reliability of switching circuits.

Reliability design problem

In reliability design, we try to use device duplication to maximize reliability. But this maximization should be considered along with the cost.

Let c is the maximum allowable cost and ci be the cost of each unit of device i. Then the maximization problem can be given as follows:

Maximize π Øi (mi) for 1 <= I <= n

Subject to:

Reliability design problem

mi>= 1 and integer 1 <= i <= n

Here, Øi (mi) denotes the reliability of the stage i.

The reliability of the system can be given as follows:

Π Øi (mi) for 1 <= i <= n

If we increase the number of devices at any stage beyond the certain limit, then also only the cost will increase but the reliability could not increase.

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