Question

: A production system has three production machines. Machine A consists one component with constant failure rate of 0.000512 failures per day. Machine B consists of two components; this machine works if one or more of its components work; each one of these components has constant failure rate of 0.000725 and 0.000618 failures per day respectively. The third production machine consists of two components; the production in this machine stops if any of these two components stop; and one of these two components has a constant failure rate of 0.000408 failures per days. If any one of the system's production machine fails, the whole system will come to a complete halt. If the probability that this production system will fail before 100 days of running time is 10%; calculate the following:

1. Draw the system network, and then find the constant failure rate of the second component of the third machine.

2. Mean time to failure of the system.

Answer 1

1. For the system to perform : all the three machine should work.

So , for prodcution as per the given condition

For production system to work
A should work	B should work	C should work
Component a should work	One of Component b1 or b2 should work	Both of component c1 and c2 should work

A	B	C
a	b1 or b2	c1 & c2

a1 = probability of failing A machine

b1 = probability of failing component 1 of B machine,  b2 = probability of failing component 2 of B machine,

c1 = probability of failing component 1 of C machine,  c2 = probability of failing component 2 of C machine,

Also, for A to be running = (1-a)

for B to be running = 1- (b1*b2)

for C to be running = 1- (c1 + c2)

So production running daily = (1-a) * {1- (b1*b2)}* {1- (c1 + c2)}

So production failing daily = 1- [(1-a) * {1- (b1*b2)}* {1- (c1 + c2)}] ..............(equation 1)

Now, probability that this production system will fail before 100 days of running time is 10%.

which is probability that this production system will not fail before 100 days of running time is 90% i.e. 0.9

Lets say probability that this production system running daily without failure be q.

So for 100 days running daily without failure = q ^(100).

Now, q ^(100) = 0.9

q = 0.99894

so, probability that this production system failing per day = p = 1- q = 0.00105........... equation 2

Now, now comparing equation 1 and 2 ,

1- [(1-a) * {1- (b1*b2)}* {1- (c1 + c2)}] = p

c2 = 1- c1- [ (1-p)/{ (1-a) {1- (b1*b2)}}]

putting all the value in Calculator, we have , c2 = 0.00013

Hence, the constant failure rate of the second component of the third machine = 0.00013

2.

Now scene, p = probability that this production system failing per day

Also Mean failure = 1/p = 1/0.00105 = 952 days