1. Reliability

2. Reliability
Reliability: The confidence that an emergency system will function as planned when the next large-scale incident or disaster occurs.
“Evaluating the Reliability of Emergency Response Systems for Large-Scale Incident Operations,“ Brian A. Jackson, Kay Sullivan Faith, Henry H. Willis, RAND Corporation, 2010.

3. Reliability
Availability: the proportion of time a system is in a functioning condition.
Resilience: the capability of an asset, system, or network to maintain its function during or to recover from a terrorist attack or other incident.
(DHS, National Infrastructure Protection Plan 2006, p. 104)

4. Emergency Reliability Applications
Human response
Emergency Support Functions (ESFs)
Emergency medicine
Communications systems
Critical Infrastructure
Energy, particularly electrical
Transportation systems
Equipment

5. References
Critical Infrastructure: Systems and assets, whether physical of virtual, so vital to the United States that the incapacity or destruction of such systems and assets would be a debilitating impact on security, national economic security, national public health or safety, or any combination of those matters
Source: USA Patriot Act, Sec. 1016(e).

6. References
Emergency Support Functions (ESFs): Used by the Federal Government and many State governments as the primary mechanism at the operational level to organize and provide assistance. ESFs align categories of resources and provide strategic objectives for their use. ESFs utilize standardized resource management concepts such as typing, inventorying, and tracking to facilitate the dispatch, deployment, and recovery of resources before, during, and after an incident.
Source: NRF Resource Center.

7. Emergency System Requisites
The system must have the necessary components to respond.
The system components must be configured to allow the system to achieve its goals.
The system must operate without error.

8. Economic View of Reliability
Highly reliable systems are expensive
Failures are rare but very expensive
How do we design systems that reduce failure rates and are not expensive?
How much reliability is enough?
How do we improve the reliability of an existing system?

9. Reliability Examples
Redundant Power Supplies/Sources

10. Reliability Assessment
Reliability (R) is the probability that a system will operate in the next instant.
Scientifically, we measure reliability by testing to failure some sufficient number of instances of the system until they fail.

11. Reliability Assessment
Emergency systems are typically so large and diverse that it is economically and logistically infeasible to test them.
It is unethical to create an emergency merely to test an emergency system.
An emergency response system implementation is usually unique. It is designed for a particular application, location, performance and environment. Thus, comparable data from similar emergency systems is unlikely to exist.
Citizens are seldom worried about the reliability of the emergency response system as a whole. They are only concerned with the reliability of that portion of the system which is used to service the their request.
Emergency response systems are loosely coupled combinations of links and processors. Portions of the emergency response system can fail without having a significant impact on the rest of the system.

12. Reliability Assessment
We can, however, practically test components of systems for their reliabilities
Beams, materials, computers, links, etc.
What we need is some way to estimate the reliability of a system R from the reliabilities of its components ri.

13. System Architectures
Series
Parallel
Hybrid

14. Components
Fully Redundant
Partially Redundant

15. Series System Example
Electrical Power System

16. Series System
Two Component System Schematic

17. Series System
How do we estimate the System Reliability from the component reliabilities ra and rb?
R = f(ra, rb)
System Failure Rate
F = 1 - R
Component Failure Rate
fa = 1 - ra

18. Reliability Assumptions
A component is either failed or operating; there are no partial component malfunctions.
A single component failure does not cause other components to fail. In other words, each component operates independently.

19. Two Component System Behavior
System a b
Probability
State
 Working 
ra * rb
Operating
Working
Failed
ra * (1- rb)
Not Operating
( 1 - ra) * rb
 ( 1 - ra) * (1 - rb) 
 Not Operating 

20. Two Component Series System Reliability
The system operates only when each of the components is working
R = ra* rb

21. Series System Example
Two Component System Schematic

22. Two Component Series System Reliability
Two Components Example
R = ra* rb
R = .9 * .8
R= .72

23. Series System
Three Component System Schematic

24. Three Component System Behavior
System a b c
Probability
State
 Working 
ra * rb * rc
Operating
Working
Failed
ra * rb * (1- rc)
Not Operating
ra * ( 1 - rb) * rc
ra * ( 1 - rb) * ( 1 - rc)
 ( 1 - ra) * rb * rc 
 Not Operating 
 ( 1 - ra) * rb * (1 - rc) 
 ( 1 - ra) * (1 - rb) * (1 - rc) 

25. Three Component Series System Reliability
The system operates only when each of the components is working
R = ra* rb * rc

26. Series System Reliability
General
R = ra * rb * rc * rn
For identical reliability components
R = rn

27. Series System Reliability
Number of  Components 
 Reliability  1
0.9 2
0.81 3
0.729 4
0.6561 5
0.5904 6
0.5314 7
0.4782 8 9
0.3874 10
0.3486 11
0.3138 12
0.2824

28. Series System Reliability

29. Series System Reliability
Observations
Adding series components to a system makes that system less reliable. R << Rold * rnew
A series system's reliability is less than the reliability of the system's least reliable component. R << rmin

30. Parallel System Reliability
Two Component System

31. Two Component System Behavior
Probability
State
 Working 
Working
ra * rb
Operating
Failed
ra * (1- rb)
( 1 - ra) * rb
 ( 1 - ra) * (1 - rb) 
 Not Operating 

32. Two Component Parallel System Reliability
Two Components
The system operates when one of the components is working
R = ra* rb + ra* (1- rb ) + (1- ra ) * rb

33. Series System Example
Two Component System Schematic

34. Two Component Series System Reliability
Two Components Example
R = ra* rb + ra* (1- rb ) + (1- ra ) * rb
R = .9 * * * .8
R= .98

35. Three Component Parallel System Behavior
System a b c
Probability
State
 Working 
ra * rb * rc
Operating
Working
Failed
ra * rb * (1- rc)
ra * ( 1 - rb) * rc
ra * ( 1 - rb) * ( 1 - rc)
 ( 1 - ra) * rb * rc 
 Operating 
 ( 1 - ra) * rb * (1 - rc) 
 ( 1 - ra) * (1 - rb) * (1 - rc) 
 Not Operating 

36. Three Component Parallel System Reliability
Three Components
The system operates only when any one of the components is working
R = ra* rb * rc + ra * rb * (1- rc) + ra * ( 1 - rb) * rc + ra * ( 1 - rb) * ( 1 - rc) +  ( 1 - ra) * rb * rc +  ( 1 - ra) * rb * (1 - rc) 

37. Parallel System Reliability
General
R = 1 - Π i ( 1 - ri)
For identical reliability components
R = 1 - ( 1 - r )n

38. Parallel System Reliability
Parallel  Components 
System  Reliability  1
0.9 2
0.99 3
0.999 4
0.9999 5 6 7 8 9 10 11 12

39. Series System Reliability

40. Parallel System Reliability
Observations
Adding parallel components to a system increases the reliability of the system, often referred to as redundancy. R >> 1 - Σ i * ( 1 - ri ) * ( 1 - rnew )
The reliability of a purely parallel system is greater than the reliability of the system's most reliable component. R >> rmax

41. Hybrid Systems Reliability

42. System Reduction

43. Example Emergency Routing
How do I assess the reliability of a person’s evacuating Washington DC to Harrisburg, PA?

44. Example Emergency Routing

45. Routing Schematic

46. Routing Schematic Reduction

47. Routing Schematic Reduction

48. Routing Schematic Reduction

49. n of N Parallel Component Reliability Rn/N
What happens if the components are not completely redundant?
For a system of N parallel components, n or more of them must be working for the system to operate.
We can use components of lower capacity and they ought to be less expensive

50. Three Component Parallel System Behavior
System a b c
Probability
State
 Working 
ra * rb * rc
Operating
Working
Failed
ra * rb * (1- rc)
ra * ( 1 - rb) * rc
ra * ( 1 - rb) * ( 1 - rc)
 ( 1 - ra) * rb * rc 
 Operating 
 ( 1 - ra) * rb * (1 - rc) 
 ( 1 - ra) * (1 - rb) * (1 - rc) 
 Not Operating 

51. n of N Parallel Component Reliability Rn/N
R3/3 = ra * rb * rc
R2/3 = ra * rb * rc +   (3 operate)  
ra * rb * ( 1 - rc ) + ra * ( 1 - rb ) * rc + ( 1 - ra ) * rb * rc   (2 operate)
R1/3 = ra * rb * rc +   (3 operate)  
ra * rb * ( 1 - rc ) + ra * ( 1 - rb ) * rc + ( 1 - ra ) * rb * rc +   (2 operate)  
ra * ( 1 - rb ) * ( 1 - rc ) + ( 1 - ra ) * rb * ( 1 - rc) + ( 1 - ra ) * ( 1 - rb ) * rc   (1 operates)

52. n of N Parallel Component Reliability Rn/N
R3/3 Same as Series System
R1/3 Same as Parallel System
R1/3 = R2/3 = R3/3

53. n of N Parallel Component Reliability Rn/N
Generalization
Rn/N = ΣNnCnNri(1-r)N-I
Where CiK is the number of combinations in K components taken i at a time or: CiK = K!/(i! (K-i)!)        for 0 <= i <= K

54. n of N Parallel Component Reliability Rn/N
Example:
Six component system
Each has a reliability of .95
Four or more must be working for the system to operate at capacity

55. n of N Parallel Component Reliability Rn/N
R4/6 =  C46 * ( .95 )4 * ( .05 )2 +
C56 * ( .95 )5 * ( .05 ) +
C66 * ( .95 )6 * ( .05 )0   15 * ( .95 )4 * ( .05 )2 +
6 * (.95 )5 * (.05 ) + 1 * (.95 )6 * (.05 )0     =

56. n of N Parallel Component Reliability Rn/N
Operating Components (n)
System Parallel Components (N) 2 3 4 5 6 7 8 9 10 1
.99
.999
.9999
.81
.972
.9963
.9985
.729
.9477
.9904
.9987
.9998
.6561
.9175
.9842
.9973
.9996
.5905
.8857
.9743
.9950
.9991
.5314
.8503
.9619
.9917
.9984
.4783
.8131
.9470
.9872
.4305
.7748
.9298
.3874
.7361
.3486

57. n of N Parallel Component Reliability Rn/N

58. Reliability Design Use components that are inherently more reliable.
Add parallel components.

59. Reliability Design How much reliability is enough?
How much reliability can I afford?
What is the optimum (minimum) cost?

60. Reliability Costs

61. Reliability Costs TC TS TF total system reliability cost
costs to make the system reliable TF
the cost associated with the system having a failure

62. Variables n Cc Cf F Number of system components
Cost of an individual component Cf
Cost of a failure F
Failure probability ( F = 1 - R)

63. Reliability Costs
TC = TS + TF
TS = n * Cc
TF = F * Cf

64. Example Parallel System
F = ( 1 - r )n
TC = n * Cc + ( 1 - r )n * Cf

65. Example Parallel System
Component costs = $100
Component Reliability = .9
Failure cost = $5000

66. Example Parallel Systemn  TS F TF TC  1
 100.00
 .1
   
     2
 200.00
 .01
 500.00
 700.00  3
 300.00
 .001
 50.00
 *350.00  4
 400.00
 .0001
 5.00
 405.00  5
 500.00 
 .00001 
 0.50
 500.50
* Optimum components

67. Component Reliability
Factors affecting reliability
Temperature
Pressure
Age
Duty cycles

68. Component Reliability
Reliability has previously been defined as the probability that a system or component will operate in the next instant
r = 1 - Pr( Failure / T )
Pr( Failure / T )
Instantaneous Failure Rate (IFR)
Hazard Rate (HR).

69. Component Reliability
Pr( Failure / T ) = { f(t) } / {Pr( t >= T ) }
f(t) = failure probability density function
f(t) = 1/m e-t/m (e = )
m = Mean Time Before Failure (MTBF)

70. Component Reliability

71. Component Reliability

72. Component Reliability
r = 1 – Pr(Failure/T)
r = 1 – 1/m
r = a constant

73. Component Reliability Example
An LED display has a mean life of 100,000 hours (100000/(365 *24))= 11.5 years.
r = 1 – 1/100000
r =

74. Component Reliability Example
What is the probability that an LED will last longer than 125,000 hours in service?
Pr( t >= 125,000 ) = e-125,000/100,000
= e-1.25  =

75. Component Reliability Example
What is the probability that an LED will fail before its mean life?
Pr( t < 100,000 ) = 1 - e-100,000/100,000
= 1 - e-1  = . 632

76. Weibul Probability Distribution
Pr( Failure / T ) = k * z-k * Tk - 1

77. Weibul
Exponential
k=1, z=m
Linear IFR
k = 2