My home uses two light bulbs. A light bulb lasts for an exponentially distributed length of time with mean 22 days. When a light bulb burns out, it takes an exponentially distributed length of time (with an average of 2 days) before I replace the bulb.
a) Model this situation as a three-state birth–death process in which the state is defined as the number of working light bulbs. Clearly draw the states and transitions, and includes the rates on each arc.
b) Determine the fraction of the time that there are two working light bulbs. To determine this, you will need to use the “death rate at i = birth rate at i – 1” equations, and the normalization equation.