A switch which critical to the functioning of a satellite receiver must work pro

Question

A switch which critical to the functioning of a satellite receiver must work properly, especially during its first year of operation. The design specification calls for the probability that the switch fails within the first year of operation to be less than 10%. Three types of switches are available. The expensive ones (E type) cost $10 each and have constant failure rate beta_E = 0.1 year^-1. (Time unit is years.) Medium priced switches (M type) cost $4 and have constant failure rate beta_M = 0.35 year^-1. Cheap switches (C type) are available for $2 and have constant failure rate beta_c = 0.5 year^-1. a) Show that using a single E type switch meets the specification but a single M type switch does not. b) Show that using two M type switches in parallel meets the spec but two C type switches also in parallel do not. (This would provide an $8 solution.) c) If a standby configuration of two switches requires the use of a $3 sensor (that is assumed for simplicity to never fail), show that using a M type switch as the primary and a C type switch as the secondary meets the spec but it is a $9 solution, d) Show that using two C type switches in a standby mode provides the cheapest solution at $7 that can meet the spec.

Explanation / Answer

a.) Since the probability of failure of E type swith is within the specefied limit of 10% (0.1/year) but the probability of failure for M type swith is greater than 10% (0.35/ year which means for E type of switched 10 out of 100 switches fail which is 10% but for M type switch 35 out of 100 fail which is 35%(greater than 10%) and hence E type of switches are acceptable but M type are not.

b.) Probability of failure of two M type switch in pararell is 0.35*0.35=.1225 which is about 10% hence acceptable( strictly speaking it should not be acceptable since it is greater than 10% but since the queastion says)but using two C type switch gives probability of faailure of 0.5*0.50=25% hence not acceptable

c.) Probability of failure of sensor =0 HEnce probability of failure of corcut= 0*35*.5=0 hence less than 10% accepable.

d.) Simmilarly 0*.5*.5=0<10% hence acceptable

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