1. T305: Digital Communications
Block II – Modelling Activities
Probability & Reliability
Arab Open University Tutorial 6

2. Arab Open University Tutorial 6
Introduction:
Probability and Reliability used in communication Technology (e.g. Network Traffic)
Topics be covered
Topic 1: Probability
Topic 2: Reliability
Topic 3: Preparation for Next Tutorial
Arab Open University Tutorial 6

3. Arab Open University Tutorial 6
Topic 1: Probability
If a process or measurement can have more than one outcome, then the frequency of the ith possible outcome, referred to as the occurrence of event Ei, is the number of times, Ni, that Ei occurs out of a total number, N, of events. The value of Ni may vary each time the experiment is repeated.
Example of Ei are bits in error arrived at communicatioon channel
The frequency of occurrence of an event may be expressed as a fraction of the total number of events. This is called a relative frequency, and the relative frequency of event Ei is given by Ni /N.
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Example Relative freq. (Probability)
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A set of events which includes every possible outcome of a process or measurement is said to be exhaustive. The events are mutually exclusive if no two of them can occur simultaneously. An exhaustive list of all the events that may occur is known as the event space.
The relative frequency can be used to estimate the probability, P(Ei), of event Ei , provided that N is sufficiently large such that any further increases in the value of N do not significantly affect the value of the relative frequency of event Ei.
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If the result of an experiment or measurement can be one of a set of r exhaustive and mutually exclusive events, E1, E2, E3, …, Ei,…, Er, and if, as the result of a large number, N, of measurements, the frequencies are found to be N1, N2, N3, …, Ni , …, Nr , respectively, then the probability of event Ei is:
P(Ei ) = Ni / N
where N = N1 + N2 + N3 + … + Ni + … + Nr
It follows directly from the above that:
P( E1 ) + P( E2 ) + … + P( Ei ) + … + P( Er ) = 1
and that the value of the probability of event Ei occurring is in the range from zero to one:
0 ≤ P( Ei ) ≤ 1
Probabilities can be represented by the
use of a Venn diagram as shown below.
Fig. Venn diagram
for event E.
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Example : All pattern of 4 bits would be 16 patterns Probability of any one of those patterns would be 1/16= or 6.25%
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Question
16/32
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Question
8/32
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Question
2/32
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The total area covered by a Venn diagram (the rectangular area labelled S) represents all the events which can occur. The frequency of a specific event corresponds to a given area of the diagram (such as area E). The ratio (area of E)/(area of S) is the probability of the event E occurring. Instead of representing frequencies, areas on a Venn diagram can represent probabilities.
The sum of the probabilities of a set of exhaustive and mutually exclusive events is 1. If P(E) is the probability that event E occurs, then P( E ) is used to represent the probability that the event does not occur, and: P( E ) + P( E ) = 1
Arab Open University Tutorial 6

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The notation P(AB) is used for the joint probability that events A and B will both occur. The notation P(A OR B) is used for the probability that event A, or event B, or both event A and event B will occur. The probabilities are related by the equation:
P( A OR B) = P( A) + P( B) - P(AB)
P(AB) = 0 for mutually exclusive events, in which case:
P(A OR B)= P(A) + P(B)
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Question
0.44
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= =0.75
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The notation P(A | B) is used for the conditional probability (a method to check if two events are dependent) that A should occur, given that B has definitely occurred. For any two events, A and B:
P(AB) = P( A | B) P( B) = P( B | A) P( A)
P(T|C)=P(CT)/P(C) = 0.08/0.4
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Independent events are ones that have no influence on each other –that is, the occurrence of one event does not affect the probability of occurrence of the other.
P( A | B) = P( A) for independent events A and B
If A and B are independent events, the equation for the joint probability of the two events reduces to the multiplication rule:
P(AB) = P( A) P( B)
If A and B are independent, the expression for P(A OR B) reduces to the addition rule:
P( A OR B) = P( A) + P( B) − P( A) P( B)
Generally, if the occurrence of an event A affects the probability of occurrence of an event B, then they are said to be dependent events.
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19. Arab Open University Tutorial 6
Question
Arab Open University Tutorial 6

20. Binomial distribution
if a joint event can be represented as the outcome of n identical elementary events, each capable of having only one of two outcomes (outcome X, say, with probability p and outcome Y with probability q = 1 - p) then the probability, P(r), that the outcome of the joint event will contain r occurrences of outcome X, and therefore n - r occurrences of outcome Y, is given by the binomial distribution:
where is called a binomial coefficient and is given by:
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The binomial coefficients can be conveniently tabulated using Pascal’s triangle as is shown below. The top row of Pascal’s triangle corresponds to the binomial coefficient 0C0. Row 2 corresponds to 1C0 and 1C1. Row 3 corresponds to 2C0, 2C1 and 2C2, and so on.
Fig. Pascal’s triangle.
Arab Open University Tutorial 6

23. binomial distribution:
The binomial distribution is an example of a probability distribution, which is in general a function that expresses the probability of an event occurring.
A probability distribution may be for either discrete events (as in the binomial distribution) or for continuous events (such as the probability of a continuous variable taking a value within a given range).
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24. Arab Open University Tutorial 6
Topic 2: Reliability
Basic concepts
The reliability function, R(t), is the probability that a component will survive to time t. This can be estimated by a test of N0 components:
where Ns(t) is the number of components surviving at time t.
Arab Open University Tutorial 6

25. survivors at start of day accumulated failures at start of day
failures in day
survivors at start of day
accumulated failures at start of day
reliability R(t) 1
2214
10000 2
1422
7786
0.7786 3
976
6364
3636
0.6364 4
812
5388
4612
0.5388 5
686
4576
5424
0.4576 6
576
3890
6110
0.389 7
497
3314
6686
0.3314 8
410
2817
7183
0.2817 9
358
2407
7593
0.2407 10
300
2049
7951
0.2049
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27. failure function (Unreliability)
The failure function, Q(t), is the probability that a component will have failed by time t. The failure function, in terms of the results of a test on N0 components, is defined as:
where Nf (t) is the number of components which have failed by time t.
Because survival and failure are mutually exclusive and exhaustive events:
R(t ) + Q(t ) = 1
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28. survivors at start of day accumulated failures at start of day
failures in day
survivors at start of day
accumulated failures at start of day
failure function Q(t) 1
2214
10000 2
1422
7786
0.2214 3
976
6364
3636
0.3636 4
812
5388
4612
0.4612 5
686
4576
5424
0.5424 6
576
3890
6110
0.611 7
497
3314
6686
0.6686 8
410
2817
7183
0.7183 9
358
2407
7593
0.7593 10
300
2049
7951
0.7951
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30. failure density function
It is often useful to know the probability of failure during a small interval around time t. This can be found from the failure density function, ƒ(t). The probability of failing, Q(t), during a very small interval t at t can be expressed as:
Q(t) = ƒ(t)  t
Fig. Failure densityFunction.
Arab Open University Tutorial 6

31. What is the probability of failure at day 10?
failures in day
survivors at start of day
accumulated failures at start of day
probability of failure in day 1
2214
10000
0.2214 2
1422
7786
0.1422 3
976
6364
3636
0.0976 4
812
5388
4612
0.0812 5
686
4576
5424
0.0686 6
576
3890
6110
0.0576 7
497
3314
6686
0.0497 8
410
2817
7183
0.041 9
358
2407
7593
0.0358 10
300
2049
7951
0.03 11
262
1749
8251
0.0262
Prob of failure by day 10
Prob of failure by day 11
Q(t)=
=0.03=Prob of failure ON day 10
Arab Open University Tutorial 6

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Note: prob of failure on each day decreases with time reaching zero on day 32
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For a continuous reliability function R(t) and failure function Q(t), the relationship to the failure density function ƒ(t) can be found by differentiation:
Equivalently, Q(t) and R(t) can be found by integrating ƒ(t):
Arab Open University Tutorial 6

34. Relationship between failure density function,ƒ(t), reliability.
The integrals can be thought of as areas under the curve of ƒ(t). The relationship between the failure density function, ƒ(t), and the reliability function, R(t), and failure function, Q(t), is shown below.
Fig. Relationship between failure
density function,ƒ(t), reliability
function, R(t), and
failure function, Q(t).
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The assumption that all components will eventually fail implies that:
This is known as the normalization condition.
The reliability function, R(t), and failure function, Q(t), are known as cumulative probability distributions because they give the probability of something happening up to a specified time.
The failure density function, ƒ(t), is a probability density function because it is used to give the probability of failure during an interval of time.
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Hazard rate
The hazard rate, h(t), or failure rate, is the ratio of the number of failures per unit time, at time t, to the number of components exposed to failure at that time:
Arab Open University Tutorial 6

37. survivors at start of day
failures in day
survivors at start of day 1
2214
10000 2
1422
7786 3
976
6364 4
812
5388 5
686
4576 6
576
3890 7
497
3314 8
410
2817 9
358
2407 10
300
2049 11
262
1749 12
218
1487 13
184
1269 14
157
1085 15
137
928 16
114
791
Question How many components survived to start of day 10?___________ How many failed during day 10?________ Given a component survived until day 10. (i.e. this is conditional probability) What is the probability it might fail DURING day 10?_____ Calculate the same for day 16?__________
300/2049, 114/791: Similar Results in this section. In practice it is a bath tub shape.
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Plots of hazard rate against time often take the form of a ‘bath-tub’ curve shown below, with a steep fall in the burn-in period (early life and it is when faulty components are weeded out), a level useful life period (or normal operating period) when h(t) is effectively constant, and wear-out (or end-of-life period), when the hazard rate rises steeply.
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hazard rate
The hazard rate is also related to the failure density function and the reliability function by the equation:
During the useful life period, the hazard rate, h, is independent of time.
The failure process during the useful life period is known as memoryless, because the probability of failure is independent of previous history.
Arab Open University Tutorial 6

40. survivors at start of day
failures in day
survivors at start of day 1
2214
10000 2
1422
7786 3
976
6364 4
812
5388 5
686
4576 6
576
3890 7
497
3314 8
410
2817 9
358
2407 10
300
2049 11
262
1749 12
218
1487 13
184
1269 14
157
1085 15
137
928 16
114
791
Question ON Day 14 What is hazard rate h(t)=_________ What is the failure density Function f(t)=___________
H(t)=157/1085=0.144; f(t)=157/10,000= Day 14
Day H(t)=0.147
Arab Open University Tutorial 6

41. Exponential probability distribution
During useful life, when the hazard rate, h, is constant, the reliability function and failure density function are simple exponential functions of time:
R(t ) = exp(-ht )
and
f (t ) = h exp(−ht )
These functions represent examples of the exponential probability distribution. Figure below shows three examples of reliability functions with constant hazard rates.
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Fig. Plots of reliability function for constant hazard rate h, where the units of h are months-1.
Arab Open University Tutorial 6

43. survivors at start of day
failures in day
survivors at start of day
reliability R(t) 1
200
1000
1.000 2
160
800
0.800 3
128
640
0.640 4
102
512
0.512 5 82
410
0.410 6 66
328
0.328 7 52
262
0.262 8 42
210
0.210 9 34
168
0.168 10 27
134
0.134 11 21
107
0.107 12 17 86
0.086 13 14 69
0.069 55
0.055 15 44
0.044 16 35
0.035
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44. Figure below shows the corresponding failure density functions.
Fig. Plots of failure density function for constant hazard rate h, where the units of h are months-1.
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45. survivors at start of day failure density function f(t)
failures in day
survivors at start of day
reliability R(t)
exp(-ht)
failure density function f(t) 1
200
1000
1.000
0.200 2
160
800
0.800
0.819
0.160 3
128
640
0.640
0.670
0.128 4
102
512
0.512
0.549
0.102 5 82
410
0.410
0.449
0.082 6 66
328
0.328
0.368
0.066 7 52
262
0.262
0.301
0.052 8 42
210
0.210
0.247
0.042 9 34
168
0.168
0.202
0.034 10 27
134
0.134
0.165
0.027 11 21
107
0.107
0.135
0.021 12 17 86
0.086
0.111
0.017 13 14 69
0.069
0.091
0.014 55
0.055
0.074
0.011 15 44
0.044
0.061
0.009 16 35
0.035
0.050
0.007
Note that beginning of the test, the value of failure density function is equal to the hazard rate h= This is because at the start of 1st interval, the number of survivals is the number of components at the start of the test.
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46. Arab Open University Tutorial 6
Average lifetimes
The average lifetime of a batch of components is called the mean time to failure (MTTF). The MTTF of a batch of components can be found by recording the time at which each component failed. For instance, if N1 components failed at time t1, N2 components at time t2, and so on until all the components have failed, the MTTF is given by the equation:
where  is the summation sign and the summations extend over all the relevant values of i.
In the case of an exponential distribution with constant hazard rate h, the MTTF is simply:
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47. 15 components tested in 5 days What is the MTTF?
Failure time(days
Number of components 1 2 3 4 5 6
61/15=4 days approx
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Confidence levels
The degree to which an MTTF estimate for a type of component can be trusted depends on the number of components used in obtaining it.
This degree of trust is often expressed in terms of confidence levels.
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49. MTBF, MTTR and availability
For repairable components, the mean time between failures (MTBF) is used instead of the mean time to failure (MTTF). It is defined as the average time from when a component starts to be operational until it fails– that is, it does not include the time when the component is waiting to be repaired.
The mean time to repair (MTTR) of a system is the mean time taken to repair a fault. The system availability, A, is the proportion of time for which a system is operating correctly:
Arab Open University Tutorial 6

50. Topic 3: Preparation for Next Tutorial
Please inform the students to do the following activities before coming to the next tutorial:
Overview the Contents of Block
Arab Open University Tutorial 6