Question

There are two machines, one of which is used as a spare. A working machine will function for an exponential time with rate λ and will then fail. Upon failure, it is immediately replaced by the other machine if that one is in working order, and it goes to the repair facility. The repair facility consists of a single person who takes an exponential time with rate μ to repair a failed machine. At the repair facility, the newly failed machine enters service if the repairperson is free. If the repairperson is busy, it waits until the other machine is fixed; at that time, the newly repaired machine is put in service and repair begins on the other one. Starting with both machines in working condition, find

the expected value and

Answer 1

The expected value for the system described in the question can be calculated using the concept of mean time to absorption in a Markov chain.

Step-by-step explanation:

The problem described in the question can be represented as a simple Markov chain, where the states represent the condition of the two machines: one working and the other spare. Let's define the following:

• State 0: Both machines are working.
• State 1: One machine is working and the other is spare.

We also need to define the transition rates:

• λ: Rate at which a machine fails.
• μ: Rate at which a machine is repaired.

Using these definitions, we can construct the transition diagram for this Markov chain.

To find the expected value, we use the concept of mean time to absorption.

In this case, the absorption states are State 0 and State 1. The mean time to absorption from State 0 is given by:

E(X|0) = 1/(μ + λ)

Similarly, the mean time to absorption from State 1 is given by:

E(X|1) = 1/(2μ + λ)

The overall expected value is the weighted sum of these two mean times:

E(X) = (E(X|0) + E(X|1))/2