# Relationship Between The Failure Rate And The Mean Time Between Failures

**Relationship between the failure rate and the mean time between failures (MTBF), MTTR (Mean time to repair), MTTF (Mean time to failure)**

MTBF (Repairable systems only) is defined as the mean time interval between successive failures. It is denoted by θ. Related to the MTBF is the failure rate, which is denoted by λ. The failure rate is the reciprocal of MTBF.

If a large no. of terms are placed in the test of the same type and operated until each one fails, the mean time to failure (irrepairable system), i.e., MTTF is given by,

where n = No. of items failed, T = tested duration.

**Prove that the Failure Rate is the Reciprocal of the MTBF**

Proof. Let us test ‘n’ items each t hours and the items which fail are repairable. Suppose there are ‘r’ failures,

The failure rate,

r = nt λ...................(1)

.................from (1)

...................Hence proved.

The reliability function of any system is given by,

R = Prob. of survival ; t = Tested duration.