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Comparison of availability results evaluated by numerical and analytical methods for the exponential case
Microelectronics and Reliability, Vol. 16, pp. 567 to 569. Pergamon Pr¢~, 1977. Print~l in C.n'cat Britain
COMPARISON OF AVAILABILITY RESULTS EVALUATED BY NUMERICAL AND ANALYTICAL METHODS FOR THE EXPONENTIAL CASE B. A. BASKER
Department of Engineering Production, University of Technology, Loughborough, Leicestershire LEII 3TU, U.K. Abstract--A numerical method based on simulation was developed by Basker and Martin to predict the Availability of units in a system where the reliability and repairability functions need not be exponentially distributed. In this paper the accuracy of the method is tested by comparing the Availability results obtained by simulation method with the results of the analytical method for the exponential case. 1. INTRODUCTION The definition of Availability is: Uptime Availability = Uptime + Downtime
(1)
where Uptime is the length of time the unit is operating, without a failure and Downtime is that necessary for maintenance. In this paper Downtime is assumed to include the following: (i) the active repair time and (ii) the waiting time for repair due to insufficient number of repairmen. In the case of exponential distribution of the variables, Availability can be expressed as follows: MTBF A = MTBF + MTTR + MTWR"
(2)
where A = Availability, MTBF = Mean time between failures reflecting reliability R, MTTR = Mean time to repair reflecting repairability RM and MTWR = Mean time waiting for repair. The mean time waiting for repair becomes extremely complicated if the distributions qf R and Ru are other than exponential. For example, if R and Ru are normally distributed it becomes almost impossible to evaluate MTWR. A description of the simulation method developed is given in reference [1]. In this paper the accuracy of this simulation method is tested by evaluating Availability for the exponential case and comparing it with that evaluated using eqn (2) when the waiting time for repair is zero.
A flowchart describing the method is given in Fig. 1. The details of this method are given in reference [1]. The Availability results obtained using the simulation method are shown in Table 2. The Availability increases with the number of repairmen. F o r instance, the Availability of unit 1 increases as follows: NRM Availability 1 92?/0 2 96% 3 97% 4 97% The Availability results for 4, 5,..., n repairmen are the same as that for three repairmen for all the units. This implies that there would be no waiting time between repairs if there axe more than three repairmen. Hence the Availability results obtained for four repairmen are for the situation when the waiting time is zero. Table 1. Mean time between failure (MTBF) and mean time to repair (M'FI'R) of eight units considered in the hypothetical system Unit no.
MTBF (hr)
M'I'TR (hr)
1 2 3 4 5 6 7 8
60.0 80.0 95.0 70.0 75.0 88.0 63.0 55.5
2.0 4.0 5.0 4.5 8.0 6.6 8.0 2.5
Table 2. Availability results obtained using the simulation method Availability(~)
2. EVALUATION OF AVAILABILITY FOR THE EXPONENTIAL CASE USING A METHOD OF SIMULATION
NRM
A hypothetical system consisting of eight units is considered. The mean time between failures and mean time to repair are as given in Table 1. 567
2 3 4
1
Unit number 1
2
3
4
5
6
7
8
92 96 97 97
93 96 96 96
90 95 95 95
90 94 94 94
88 91 91 91
92 95 95 95
85 87 88 88
88 95 96 96
568
B. A. BASKER l Read mean time beteen failure OI and mean time to repair e 2
1
Select o random number r I to determine breokdown time ' t ' from the relation t = ( - 8 , ) x Log, ( I - r,)
(
ri= F ( t ) = t - e - * / O , )
1
Breakdown time is added to cumulative breakdown time
1 Select o random number r2 to obtain o corresponding repair time ' T ' from the relation T = ( - O 2 ) x Log, ( I - r 2) ( . " rz = F (T)= I - e - T / O z )
Repeat 'm' m
times
P Repeat for oil the units
Select the unit which has
broken down and awaiting repair
t
Yes
:o
is comp'uted
time + waitin~ time ,~ Cumulate
I
downtime
f Uptime = cumulative breakdown time Downtime = cumulative downtime Evaluate Uptime Availability = Uptime +Downtime
Set
Repeat 'n'
times
Fig. 1. 3. EVALUATION OF AVAILABILITY FOR THE EXPONENTIAL CASE USING THE ANALYTICAL METHOD
and
When reliability R(t) and repairability RM(t) functions are exponentially distributed, Availability can be expressed as shown in eqn (2). Let Reliability R(t) = e -~r
# Availability = ~ .... . Z + ,tt
(3)
The derivation of eqn (3) using the theory of Markov process is given in reference [3]. The Availability results obtained using eqn (3) for the data shown in Table 1 are given in Table 3.
and Repairability
R~(t) =
e-"',
4. C O M P A R I S O N O F T H E T W O R E S U L T S
where 2 and /~ are the failure and repair rates respectively. Then 2
1 MTBF'
~
1 MTTR
The Availability results shown in Table 2 for four repairmen are now compared with the results given in Table 3. The Availability of units 2, 3, 4, 6 and 7 are the same in both the cases. The Availability of units
Table 3. Availability results obtained using the relation: MTBF Availability = MTBF + MTTR Availability (%) Unit number 1
2
3
4
5
6
7
60 x 100 6-2
80 ~ x 100
95 xlO0 1~
70 ~ - - x 100 74.5
75 --xlO0 83
88 xlO0 94.6
63 x tO0 71
=96
=95
=95
=94
=90
=95
=88
8
55.5 --xlO0 58 =95
Comparison of Availability Results 1, 5 and 8 in the two cases differs by only 1~. Thus the Availability evaluated by simulation method agrees correct to 1% with that evaluated by analytical method.
569
Acknowledoements--The author wishes to appreciate the helpful comments made by Mr P. Martin of Liverpool University. during the preparation of this paper. REFERENCES
5. CONCLUSION When the reliability and repairability functions are exponentially distributed, the Availability results evaluated by simulation method agrees correct to 1~ with the results of the analytical method.
1. B. A. Basker and P. Martin, Availability prediction by using a method of simulation. Microelectron. Reliab. 16, (2) (1977). 2. B.A. Basker, Sensitivity analysis of Availability of units in a production/electrical system. Microelectron. Reliab. 16, (3) (1977). 3. A. B. Clarke and R. L. Disney, Probability and Random Processesfor Engineers and Scientists, John Wiley (1970).
COMPARISON OF AVAILABILITY RESULTS EVALUATED BY NUMERICAL AND ANALYTICAL METHODS FOR THE EXPONENTIAL CASE B. A. BASKER
Department of Engineering Production, University of Technology, Loughborough, Leicestershire LEII 3TU, U.K. Abstract--A numerical method based on simulation was developed by Basker and Martin to predict the Availability of units in a system where the reliability and repairability functions need not be exponentially distributed. In this paper the accuracy of the method is tested by comparing the Availability results obtained by simulation method with the results of the analytical method for the exponential case. 1. INTRODUCTION The definition of Availability is: Uptime Availability = Uptime + Downtime
(1)
where Uptime is the length of time the unit is operating, without a failure and Downtime is that necessary for maintenance. In this paper Downtime is assumed to include the following: (i) the active repair time and (ii) the waiting time for repair due to insufficient number of repairmen. In the case of exponential distribution of the variables, Availability can be expressed as follows: MTBF A = MTBF + MTTR + MTWR"
(2)
where A = Availability, MTBF = Mean time between failures reflecting reliability R, MTTR = Mean time to repair reflecting repairability RM and MTWR = Mean time waiting for repair. The mean time waiting for repair becomes extremely complicated if the distributions qf R and Ru are other than exponential. For example, if R and Ru are normally distributed it becomes almost impossible to evaluate MTWR. A description of the simulation method developed is given in reference [1]. In this paper the accuracy of this simulation method is tested by evaluating Availability for the exponential case and comparing it with that evaluated using eqn (2) when the waiting time for repair is zero.
A flowchart describing the method is given in Fig. 1. The details of this method are given in reference [1]. The Availability results obtained using the simulation method are shown in Table 2. The Availability increases with the number of repairmen. F o r instance, the Availability of unit 1 increases as follows: NRM Availability 1 92?/0 2 96% 3 97% 4 97% The Availability results for 4, 5,..., n repairmen are the same as that for three repairmen for all the units. This implies that there would be no waiting time between repairs if there axe more than three repairmen. Hence the Availability results obtained for four repairmen are for the situation when the waiting time is zero. Table 1. Mean time between failure (MTBF) and mean time to repair (M'FI'R) of eight units considered in the hypothetical system Unit no.
MTBF (hr)
M'I'TR (hr)
1 2 3 4 5 6 7 8
60.0 80.0 95.0 70.0 75.0 88.0 63.0 55.5
2.0 4.0 5.0 4.5 8.0 6.6 8.0 2.5
Table 2. Availability results obtained using the simulation method Availability(~)
2. EVALUATION OF AVAILABILITY FOR THE EXPONENTIAL CASE USING A METHOD OF SIMULATION
NRM
A hypothetical system consisting of eight units is considered. The mean time between failures and mean time to repair are as given in Table 1. 567
2 3 4
1
Unit number 1
2
3
4
5
6
7
8
92 96 97 97
93 96 96 96
90 95 95 95
90 94 94 94
88 91 91 91
92 95 95 95
85 87 88 88
88 95 96 96
568
B. A. BASKER l Read mean time beteen failure OI and mean time to repair e 2
1
Select o random number r I to determine breokdown time ' t ' from the relation t = ( - 8 , ) x Log, ( I - r,)
(
ri= F ( t ) = t - e - * / O , )
1
Breakdown time is added to cumulative breakdown time
1 Select o random number r2 to obtain o corresponding repair time ' T ' from the relation T = ( - O 2 ) x Log, ( I - r 2) ( . " rz = F (T)= I - e - T / O z )
Repeat 'm' m
times
P Repeat for oil the units
Select the unit which has
broken down and awaiting repair
t
Yes
:o
is comp'uted
time + waitin~ time ,~ Cumulate
I
downtime
f Uptime = cumulative breakdown time Downtime = cumulative downtime Evaluate Uptime Availability = Uptime +Downtime
Set
Repeat 'n'
times
Fig. 1. 3. EVALUATION OF AVAILABILITY FOR THE EXPONENTIAL CASE USING THE ANALYTICAL METHOD
and
When reliability R(t) and repairability RM(t) functions are exponentially distributed, Availability can be expressed as shown in eqn (2). Let Reliability R(t) = e -~r
# Availability = ~ .... . Z + ,tt
(3)
The derivation of eqn (3) using the theory of Markov process is given in reference [3]. The Availability results obtained using eqn (3) for the data shown in Table 1 are given in Table 3.
and Repairability
R~(t) =
e-"',
4. C O M P A R I S O N O F T H E T W O R E S U L T S
where 2 and /~ are the failure and repair rates respectively. Then 2
1 MTBF'
~
1 MTTR
The Availability results shown in Table 2 for four repairmen are now compared with the results given in Table 3. The Availability of units 2, 3, 4, 6 and 7 are the same in both the cases. The Availability of units
Table 3. Availability results obtained using the relation: MTBF Availability = MTBF + MTTR Availability (%) Unit number 1
2
3
4
5
6
7
60 x 100 6-2
80 ~ x 100
95 xlO0 1~
70 ~ - - x 100 74.5
75 --xlO0 83
88 xlO0 94.6
63 x tO0 71
=96
=95
=95
=94
=90
=95
=88
8
55.5 --xlO0 58 =95
Comparison of Availability Results 1, 5 and 8 in the two cases differs by only 1~. Thus the Availability evaluated by simulation method agrees correct to 1% with that evaluated by analytical method.
569
Acknowledoements--The author wishes to appreciate the helpful comments made by Mr P. Martin of Liverpool University. during the preparation of this paper. REFERENCES
5. CONCLUSION When the reliability and repairability functions are exponentially distributed, the Availability results evaluated by simulation method agrees correct to 1~ with the results of the analytical method.
1. B. A. Basker and P. Martin, Availability prediction by using a method of simulation. Microelectron. Reliab. 16, (2) (1977). 2. B.A. Basker, Sensitivity analysis of Availability of units in a production/electrical system. Microelectron. Reliab. 16, (3) (1977). 3. A. B. Clarke and R. L. Disney, Probability and Random Processesfor Engineers and Scientists, John Wiley (1970).