T305: Digital Communications

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

2. 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. Topic 1: Probability • If a process or measurement can have more than one outcome, then the frequencyof the ith possible outcome, referred to as the occurrence of event Ei, is the number of times, Ni, that Eioccurs out of a total number, N, of events. The value of Nimay 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 Eiis given by Ni/N. Arab Open University Tutorial 6

4. Example Relative freq. (Probability) Arab Open University Tutorial 6

5. A set of events which includes every possible outcome of a process or measurement is said to be exhaustive. The events are mutually exclusiveif 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. Arab Open University Tutorial 6

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7. 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 arefound to be N1, N2, N3, …, Ni, …, Nr, respectively, then the probability ofevent Eiis: • 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. Arab Open University Tutorial 6

8. Example : All pattern of 4 bits would be 16 patternsProbability of any one of those patterns would be 1/16=0.0625 or 6.25% Arab Open University Tutorial 6

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12. 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 doesnotoccur, and: P( E ) + P( E ) = 1 Arab Open University Tutorial 6

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14. The notation P(AB) is used for the joint probabilitythat 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) Arab Open University Tutorial 6

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17. 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) Arab Open University Tutorial 6

18. Independent eventsare 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. Arab Open University Tutorial 6

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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 coefficientand is given by: Arab Open University Tutorial 6

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22. The binomial coefficients can be conveniently tabulated using Pascal’s triangleas is shown below. The top row of Pascal’s trianglecorresponds 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). Arab Open University Tutorial 6

24. 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

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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 Arab Open University Tutorial 6

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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? Prob of failure by day 10 Prob of failure by day 11 • Q(t)=0.8251-0.7951 =0.03=Prob of failure ON day 10 Arab Open University Tutorial 6

32. Note: prob of failure on each day decreases with time reaching zero on day 32 Arab Open University Tutorial 6

33. 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). Arab Open University Tutorial 6

35. 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 probabilitydistributionsbecause they give the probability of something happening up to a specified time. • The failure density function, ƒ(t), is a probability density functionbecause it is used to give the probability of failure during an interval of time. Arab Open University Tutorial 6

36. 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. QuestionHow 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?__________ Arab Open University Tutorial 6

38. Plots of hazard rate against time often take the form of a ‘bath-tub’ curve shown below, with a steep fall in the burn-inperiod (early lifeand it is when faulty components are weeded out), a leveluseful 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. Arab Open University Tutorial 6

39. 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. QuestionON Day 14What is hazard rate h(t)=_________What is the failure density Function f(t)=___________ 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 withconstant hazard rates. Arab Open University Tutorial 6

42. Fig. Plots of reliability function for constant hazard rate h, where the units of h are months-1. Arab Open University Tutorial 6

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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. Arab Open University Tutorial 6

45. Note that beginning of the test, the value of failure density function is equal to the hazard rate h=0.2. This is because at the start of 1st interval, the number of survivals is the number of components at the start of the test. Arab Open University Tutorial 6

46. 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 byrecording the time at which each component failed. For instance, if N1components failed at time t1, N2 components at time t2, and so on until allthe 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: Arab Open University Tutorial 6

47. 15 components tested in 5 days What is the MTTF? Arab Open University Tutorial 6

48. 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. Arab Open University Tutorial 6

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