1. مدلسازی شبکه ای و ارزیابی سیستم های ساده و پیچیده
فصل دوم
مدلسازی شبکه ای و ارزیابی سیستم های ساده و پیچیده

2. در فصل قبل کاربرد توزیع دو جمله ای را در محاسبات قابلیت اطمینان دیدیم که در اغلب مسائل، کافی است
در عمل یک سیستم به صورت شبکه ای سری، موازی، حلقوی و ... عمل می کند که در این فصل به آن می پردازیم.
مدلسازی پیش از تحلیل
عدم الزام در تشابه ساختار توپولوژیکی سیستم و مدل

3. تعریف سیستمهای سری و موازی
Series systems
The components in a set are said to be in series from a reliability point of view if they must all work for system success or only one needs to fail for system failure.
Parallel systems
The components in a set are said to be in parallel from a reliability point of view if only one needs to be working for system success or all must fail for system failure.

4. مثال: انواع سیستمها سری موازی non-redundant system
fully redundant system

5. Parallel Redundant Systems
دو المان
n المان

6. سیستمهای سری دو سیستم n سیستم
R=probability of successful operation of a component
Q=probability of failure of a component

7. مثال
A system consists of 10 identical components, all of which must work for system success. What is the system reliability if each component has a reliability of 0.9?

8. partially redundant system
identical components
Non-identical components

9. مثال
A system consists of 3 identical components, one must work for system success. What is the system reliability if each component has a reliability of 0.7?
A system is to be designed with an overall reliability of using components having individual reliabilities of 0.7. What is the minimum number of components that must be connected in parallel?

10. مثال: سیستمهای سری/موازی
Derive a general expression for the unreliability of the model shown below, and hence evaluate the unreliability of the system if all components have a reliability of 0.8.

11. مثال

12. Standby redundant systems
در برخی سیستمها، یک یا چند شاخه از المانهای اضافه، به طور پیوسته در شرایط نرمال در حال بهره برداری نیستند
و تنها وقتی که یک المان خراب شود مورد بهره برداری قرار می گیرند.
In some applications it is physically not possible for both branches to be operating.

13. Standby redundant systems
Perfect switching
Therefore, the failure of this system is given by failure of A and failure of B, given A has failed.
If A and B are independent

14. Standby redundant systems
Imperfect switching

15. Standby redundant systems
Imperfect switching
switch can fail in its initial operating position
Ps=probability of successful changeover
Rs=reliability of normal operating mode

16. Example

17. مدلسازی شبکه ای و ارزیابی سیستم های پیچیده

18. تکنیکهای ارزیابی Conditional probability approach
(with Network reduction method)
Cut set method
Tie set method
Tree diagrams (Event Trees, Fault Trees)
logic diagrams
connection matrix techniques
several of the methods are very similar in concept
main difference is
in the formal presentation or logic of the method and not the essential underlying concept

19. Conditional Probability Approach
E is bad
E is good

20. مثال
قابلیت اطمینان سیستم زیر را با فرض اینکه قابلیت اطمینان هریک از المانها 0.98 باشد محاسبه کنید.

21. حل

22. Cut Set Method کات ست Minimal cutset
set of system components which, when failed, causes system failure, or
set of components which if removed from the network separate the input from the output
Minimal cutset
any cut set which does not contain any other cut set as a subset
all components of a minimal cut set must fail in order to cause system failure

23. …Cut Set Method Minimal cutsets:
A and B in parallel, since both must fail for system failure
C1, C2, C3 and C4 in series since all 4 must be successful for system success

24. Advantages of Cut Set Method
cut sets identify ways in which a system may fail
approximation can be used to simplify evaluation
can be easily programmed on a computer

25. مثال: سیستم تغذیه ایستگاه تکرار رادیویی
با روش network reduction

26. مثال: سیستم تغذیه ایستگاه تکرار رادیویی
با روش مینیمال کات ست

27. Cut-Set Method for Large Complex Systems
1st step: Creating Connection Matrix 2nd step: Building Incidence Matrix 3rd step: Determine Minimal Cutsets from Incidence Matrix column operations

28. 1st step: Connection Matrix Techniques to find minimal paths
روش اول: حذف گره (kام)
روش دوم: ضرب ماتریس اتصال

29. 2nd and 3rd steps: Finding minimal cutsets based on Incidence Matrix column operations
Minimal Paths:
1. AC 2. BD 3. AED 4. BEC

30. Tie Set Method
The tie set method is essentially the complement of the cut set method
It is used less frequently, in practice
it does not directly identify the failure modes of the system.
A tie set is a minimal path of the system
a set of system components connected in series
a tie set fails if anyone of the components in it fails
For system failure, all of the tie sets must fail
Therefore, all tie sets are effectively connected in parallel.

31. Tie Sets:
AC, BD - 2nd order
AED, BEC – 3rd order

32. …

33. Event Trees
Pictorial representation of all the events that can occur in a system.
Event Trees can be applied to:
Continuously operated systems (mainly independent events)
Standby & sequential logic systems (dependent events)
e.g. safety and mission oriented systems (widely used )
Events:
success – represented by vertical line upwards
failure – vertical line downwards
partial failure can also be considered

34. مثال: یک سیستم متشکل از دو المان

35. Event Trees: Continuously operated systems
mainly independent events
components can be taken in any order while creating the event tree
starting point is usually the normal operating condition of the system

37. Reduced event tree

38. Reduced Event Tree:Reduced Data

39. Event Trees: Standby & Sequential Logic Systems
dependent events
events must be taken in sequential order while creating the event tree
starting point is the initiating event

42. Fault Trees Evaluation process:
widely used in standby & sequential logic systems
to evaluate the probability of a particular failure condition of a system
Evaluation process:
Deduce the fault tree (top-down approach)
start from the Top Event (system failure condition of interest)
identify immediate events that cause the top event, and how they interact(represented by logic gates) to cause the top event
identify further events that cause the events above, and proceed downwards until Basic Events are reached
Calculate the failure probability
Direct numerical approach (bottom-up approach)
(applicable when all events are independent: no duplicate basic events)
Boolean algebra approach (top-down) - complex for large systems
Minimal cutset method (top-down)
approximation can be used to simplify evaluation

43. Example1: A computer Power Supply

45. Example2: Direct Numerical approach

46. Example3: Boolean algebra approach
اگر قابلیت اطمینان المانهای مربوط به E1، E2 , و
E3به ترتیب 0.933، 0.925، باشد

47. Example4:
Consider the fault tree shown in Figure Evaluate the probability of occurrence of the top event T when:
a) all basic events are independent of each other
b) basic event E3 is the same as basic event E6 , i.e., these events represent the same failure mode of the same component.

48. solution a)
Boolean algebra approach
numerical approach

49. solution
b) it is virtually impossible to account for this in the numerical approach.

50. Minimal Cut Set Method Start from the top logic gate the cutsets:
Replace the gate by its inputs (i.e. gates or basic events)
OR gate - take the events below as separate items
AND gate - take the events below as a combined item
the cutsets:
Separate or combined items finally obtained include only the basic events

51. Example

52. Multi-failure modes
Example: Derive the expression for the success probability of the system if system success is defined as a unidirectional path existing between X and Y. Assume that both diodes have the same probabilities
Pn = probability of normal operation
Po = probability of failure due to an open circuit
Ps = probability of failure due to a short circuit

53. Solution: Numerical

54. تمرین:

55. Solution:conditional probability approach

56. کوئیز
Deduce an expression for the reliability of the system