1. A production process contains a machine that deteriorates rapidly in both quality and output under heavy usage, so that it is inspected at the end of each day. Immediately after inspection, the condition of the machine is noted and classified into one of four possible states: 0: Good as new, 1: Operable - Operableminimum deterioration, 2: Operablemajor deterioration, 3: Inoperable and replaced by a good-as-new machine.

The process can be modeled as a Markov chain with its (one-step) transition matrix P given by

0 7/8 1/16 1/16

0 3/4 1/8 1/8

0 0 1/2 1/2

0 0 0 1

Find the expected length of time a machine can be used before it must be replaced.