lect0303

# But we do not need the q matrix to solve for the

This preview shows pages 2–4. Sign up to view the full content.

But we do not need the Q matrix to solve for the steady-state distribution. We can use the SMP representation. If we do define Q , then we can alternatively obtain the steady-state probability vector α above by solving the equation αQ = 0, where the elements α i are required to sum to 1. Note that this is just a system of linear equations, just like π = πP . You should work to understand why we here have 0 instead of α for the vector. ———————————————————————- (b) What is the average number of trips Pooh makes per day from tree B to tree A ? ———————————————————————- The long-run fraction of time spent at B is 1 / 4, by part (a). Thus, on average, Pooh spends 6 hours per day at tree B . When at tree B , the rate of trips from B is 1 / 5 per hour (the reciprocal of 5 hours), and thus, on average, Pooh makes 6 / 5 = 1 . 2 trips per day from tree B . However, 3 / 4 of the trips from tree B are to tree A , so the average number of trips per day from B to A is (6 / 5) × (3 / 4) = (18 / 20) = 0 . 9. ———————————————————————- 2

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2. Copier Breakdown and Repair. Consider two copier machines that are maintained by a single repairman. Machine i func- tions for an exponentially distributed amount of time with mean 1 i , and thus rate γ i , before it breaks down. The repair times for copier i are exponential with mean 1 i , and thus rate β i , but the repairman can only work on one machine at a time. Assume that the machines are repaired in the order in which they fail. Suppose that we wish to construct a CTMC model of this system, with the goal of finding the long-run proportions of time that each copier is working and the repairman is busy. How can we proceed? (a) Let { X ( t ) : t 0 } be a stochastic process, where X ( t ) represents the number of working machines at time t . Is { X ( t ) : t 0 } a Markov process? (b) Formulate a CTMC describing the evolution of the system. (c) Suppose that γ 1 = 1, β 1 = 2, γ 2 = 3 and β 2 = 4. Find the stationary distribution. (d) Now suppose, instead, that machine 1 is much more important than machine 2, so that the repairman will always service machine 1 if it is down, regardless of the state of machine 2. Formulate a CTMC for this modified problem and find the stationary distribution. ANSWERS: (a) Let { X ( t ) : t 0 } be a stochastic process, where X ( t ) represents the number of working machines at time t . Is { X ( t ) : t 0 } a Markov process? ———————————————————————- This process is not a Markov process. To be a Markov process, we need the conditional distribution of a future state, given a present state and past states to depend only upon the present state; i.e., we need P ( X ( t ) = j | X ( s ) = i, X ( u ) , 0 u s ) = P ( X ( t ) = j | X ( s ) = i ) for all s and t with 0 s < t , and for all i and j . Here, however, the Markov property does not hold: When both machines are down, the next transition depends on which of the two machines failed first.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern