# Math Help - Inferring transition rates from continuous markov chain question

1. ## Inferring transition rates from continuous markov chain question

A house has 2 rooms of similar sizes with identical air conditioners equipped with thermostats which turn on and off as needed to maintain the temperature in each room to a desired level of 22 degrees. Suppose that a thermostat remains on or off for exponential amounts of time with means $1/\mu$ and $1/\lambda$, respectively, independently of other thermostats. Consider the Markov process $\{X(t), t \ge 0\}$ whose state space is the number of active air conditioners. Write down the matrix of transition rates.

I'm not sure how to exactly approach this type of question. My working is as follows but if someone could clarify my confusion that would be good.

Working:

So clearly there are 3 states, 0 for no air conditioners are on, 1 for one air conditioner is on (active), and 2 for two air conditioners are on (active). Now to work out $q_{01}$, i.e., the transition rate from state 0 to 1, assume currently no air conditioners are on. Consider the two independent poisson processes both with parameter $\lambda$, where the interarrival time is the duration of "off", then merging these two processes gives a poisson process with parameter $2\lambda$, so the transition rate from state 0 to state 1 is $2\lambda$.

Now what about the transition rate from state 0 to 2? I am told that it's 0, but why? Isn't it possible for both air conditioners to both go from "off" to "on"? What is the argument that $q_{02} = 0$?

2. ## Re: Inferring transition rates from continuous markov chain question

Originally Posted by usagi_killer
A house has 2 rooms of similar sizes with identical air conditioners equipped with thermostats which turn on and off as needed to maintain the temperature in each room to a desired level of 22 degrees. Suppose that a thermostat remains on or off for exponential amounts of time with means $1/\mu$ and $1/\lambda$, respectively, independently of other thermostats. Consider the Markov process $\{X(t), t \ge 0\}$ whose state space is the number of active air conditioners. Write down the matrix of transition rates.

I'm not sure how to exactly approach this type of question. My working is as follows but if someone could clarify my confusion that would be good.

Working:

So clearly there are 3 states, 0 for no air conditioners are on, 1 for one air conditioner is on (active), and 2 for two air conditioners are on (active). Now to work out $q_{01}$, i.e., the transition rate from state 0 to 1, assume currently no air conditioners are on. Consider the two independent poisson processes both with parameter $\lambda$, where the interarrival time is the duration of "off", then merging these two processes gives a poisson process with parameter $2\lambda$, so the transition rate from state 0 to state 1 is $2\lambda$.

Now what about the transition rate from state 0 to 2? I am told that it's 0, but why? Isn't it possible for both air conditioners to both go from "off" to "on"? What is the argument that $q_{02} = 0$?
well done ..............
Soran University