Communities

Writing
Writing
Codidact Meta
Codidact Meta
The Great Outdoors
The Great Outdoors
Photography & Video
Photography & Video
Scientific Speculation
Scientific Speculation
Cooking
Cooking
Electrical Engineering
Electrical Engineering
Judaism
Judaism
Languages & Linguistics
Languages & Linguistics
Software Development
Software Development
Mathematics
Mathematics
Christianity
Christianity
Code Golf
Code Golf
Music
Music
Physics
Physics
Linux Systems
Linux Systems
Power Users
Power Users
Tabletop RPGs
Tabletop RPGs
Notifications
Mark all as read
Q&A

A discrete-time queue Markov Chain problem

+0
−0

Hello everyone, I hope somebody can help me with this question, only for parts c and d, because I have already figured out parts a and b. Here is the question:

Items arrive in a queue during time intervals (0, 1), (1, 2), . . . . Let An denote the number of arrivals in the interval (n − 1, n), and assume that (An : n ≥ 1) are mutually independent and identically distributed with pmf aj = P(An = j) = 1 4 , j = 0, 1, 2, 3. The arriving items queue in a buffer with capacity 4: if arrivals exceed the buffer capacity, then the excess is lost. A single server dispatches one item per unit time (if any are waiting), with these dispatches occurring at times n = 1, 2, . . . (by which is meant that the dispatch at time n occurs after the arrivals An, and prior to the next arrivals An+1). Let Xn denote the buffer level at time n, immediately before any dispatch. We know that (Xn) is a Markov chain, with state space S = {0, 1, . . . , 4}. Denote by P = [pij ]i,j the transition probability matrix for (Xn). Suppose that the queue is empty initially, i.e. X0 = 0.

(a) Give the one-step transition probability matrix for (Xn).

(b) Find the mean occupation times (m0j (10) : j ∈ S) for (Xn).

(c) Find the expected time until the buffer reaches its capacity for the first time.

(d) Let Yn denote the number of items that go to waste from arrivals during the interval (n − 1, n). Show that E(Yn+1 | X0 = 0) = (1/4)*P^(n)(subscript 0,3 for P) + (3/4) * P^(n) (subscript 0,4 for P). [Hint: First evaluate E(Yn+1 | Xn = j), for each j ∈ S, and then apply the Law of Total Expectation: E(Yn+1 | X0 = 0) = P j∈S E(Yn+1 | Xn = j, X0 = 0)P(Xn = j | X0 = 0).]

For part a: I got the following matrix (which is correct because it has already been confirmed by my professor):

P =

1/4  1/4  1/4   1/4    0

0    1/4  1/4   1/4    1/4

0    0    1/4   1/4    1/2

0    0    0     1/4    3/4

0    0    0     0      1

For part b: I did the calculations using a statistical package and got the answer by adding up all the entries of the first row of the matrix.

For parts c and d: Well, that's where I need some serious help because I have no idea how to do them, so I would appreciate it any help.

Thanks

Why does this post require moderator attention?
You might want to add some details to your flag.
Why should this post be closed?

2 comment threads

"No idea" (1 comment)
MathJax (1 comment)

0 answers

Sign up to answer this question »

This community is part of the Codidact network. We have other communities too — take a look!

You can also join us in chat!

Want to advertise this community? Use our templates!

Like what we're doing? Support us! Donate