On a campus building, there are m offices of similar sizes with identical air conditioners. The electrical grid supplies electric energy to the air conditioners whose thermostats turn on and off in each individual office as needed to maintain each office’s temperature at the desired level of 76◦ Fahrenheit, independent of the other offices. Suppose that a thermostat remains on or off for exponential amounts of times with means 1/μ and 1/λ, respectively.

Find the long-run probability that there will be i, 0 ≤ i ≤ m, thermostats on at the same time.