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STATS 325/721 Assignment 4
发布时间：2022-10-11
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STATS 325/721
Assignment 4
1. Consider the continuous time Markov chain X = (Xt)t≥0 on statespace S = {A, B, C}
|
whose transition rates are shown in the following diagram:
2
(a) Write down the Q-matrix for X. [2]
(b) Find the equilibrium distribution of X. [2]
(c) Using resolvents, find PA(X(t) = C) for t ≥ 0. Explain intuitively what happens as t → ∞. [4]
(d) Using resolvents, find the distribution of the first hitting time of state B given
X starts in state A at time 0. Explain your answer intuitively. [4]
(e) An observer looks at the state of the Markov chain X at a random time T
which is exponentially distributed with rate λ and independent of X.
i. What is the probability distribution for the state the observer finds X in at time T given that the chain started in state A? Explain intuitively what happens to your answer as λ → 0. [4]
ii. Starting in state C, find the chance that X manages to make a return trip to state A before the observer looks (ie. X first hits A and then returns back to C, all before time T elapses). [*4]
2. Let Xt represent the number of people in a single server queue at time t. Service times are independent exponentials of rate α > 0. Arrivals of new customers occurs at rate β > 0 whenever the queue is empty, and at rate β/i whenever there are i ≥ 1 customers in the queue (in particular, the longer the queue the slower new customers will join). Suppose X = (Xt)t≥0 forms an irreducible continuous time Markov Chain
with state-space S = {0, 1, 2, . . . } and Q-matrix Q = (qi,j)i,j∈I specified by
β
q0,1 = β, qi,i+1 =
and qi,j = 0 for all |i − j| > 1.
i (i ≥ 1), qi+1,i = α (i ≥ 0),
Recall, if πQ = 0, πi ≥ 0 (∀i ∈ S) and Σi∈S πi = 1, then π is the equilibrium
distribution of X. Also recall, if ν is in detailed balance with Q, that is
νiqi,j = νjqj,i (∀i, j ∈ S), then ν is also invariant for Q, that is νQ = 0.
(a) Write out the structure of Q. [2]
(b) By solving the detailed balance equations, πiqi,j = πjqj,i (∀i, j), show that the Markov chain X is positive recurrent and find the equilibrium distribution π = (πn)n∈Z+ , simplifying your answer. [6]
(c) If initially the queue is empty, explain intuitively why the average time until
first returning back to an empty queue is β−1(1 + ρeρ) where ρ := β/α. [2]
(d) Deduce the average length of a busy period for the server (ie. the average length of time from the arrival of the first customer into an empty queue until the next time the queue empties). [*2]
3. Consider a queue with two servers where the times between arrivals of customers are independent exponentially distributed random variables of parameter 6. There is an independent probability of 1/6 that an arrival consists of a pair of customers, otherwise the arrival consists of a single customer. Customers must be served indi- vidually by one of the two servers, the service times of individuals being independent exponentially distributed random variables of parameter 8.
Let Xt be the number of people in the queue at time t, including anyone currently being served. Then X := (Xt)t≥0 is a continuous-time Markov chain with state- space I = {0, 1, 2, . . . } and Q-matrix Q.
(a) Briefly explain why the first three lines of Q are
|
−6 |
5 |
1 |
0 |
0 |
0 |
. . . |
|
8 |
−14 |
5 |
1 |
0 |
0 |
. . . |
|
0 |
16 |
−22 |
5 |
1 |
0 |
. . . |
and give the whole structure of the Q-matrix. [3]
(b) Let π = (πn : n ≥ 0) be the equilibrium distribution for the number of people in the queue. Define Π˜(s) := Σn≥0 πnsn for s ∈ [0, 1]. Show that
Π˜(s) = (16π0 + 8sπ1)(1 − s). s3 + 5s2 − 22s + 16
[7]
(c) Find the long term proportion of time that the queue is empty, and the long time average queue length. [4]
(d) If one of the servers is taken ill, use intuitive arguments to show why the remaining server can still cope with the customer arrivals and to find the long term proportion of time that the single server is busy. [*4]
