Results 1 to 2 of 2

Math Help - Inferring transition rates from continuous markov chain question

  1. #1
    Senior Member
    Joined
    Apr 2009
    Posts
    308

    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 $q_{02} = 0$?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Jun 2014
    From
    iraq
    Posts
    5

    Re: Inferring transition rates from continuous markov chain question

    Quote Originally Posted by usagi_killer View Post
    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
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. transition matrix of a Markov Chain
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: November 29th 2013, 05:07 PM
  2. Markov Chain Transition Probabilities Help.
    Posted in the Statistics Forum
    Replies: 3
    Last Post: September 4th 2012, 09:32 AM
  3. Transition matrix for Markov chain
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: May 8th 2009, 04:04 AM
  4. Markov Chain - Transition Matrix
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: April 9th 2009, 03:51 PM
  5. Transition matrix for Markov Chain
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: February 17th 2009, 01:26 AM

Search Tags


/mathhelpforum @mathhelpforum