IEOR 4102, Final Exam
IEOR 4102代写 1. (18 points, 3 each)For each of the following write T (True) or F (False) (noexplanation):a）A continuous-time Markov chain {X(t) : t ≥ 0} is
1. (18 points, 3 each)
For each of the following write T (True) or F (False) (noexplanation):
a）A continuous-time Markov chain {X(t) : t ≥ 0} is irreducible if and only if itsembedded discrete time Markov chain {Xn : n ≥ 0} is irreducible.
b）Anystate i for a continuous-time Markov chain {X(t) : t ≥ 0} is recurrent if and only if it is recurrent for its embedded discrete time Markov chain {Xn : n ≥ 0}.
c）Allcontinuous-time Markov chains are Birth and Death processes.
d）All continuous-time Markov chains have a unique limiting probability distri-bution.
e）If a continuous-time Markov chain is irreducible, and you fifind a probability solution {Pj} to the Global Balance equations, then the chain is positive recurrent and its unique limiting distribution is {Pj}.
f）Little’sLaw (l = λw) only holds for queueing models that have Poisson ar-rivals.
2.(15 points, 5 each) IEOR 4102代写
A light has B ≥ 2 bulbs, and the lifetimes L1, L2, . . . of new bulbs are iid with (continuous) distribution F (x) = P (L ≤ x). For fixed T > 0, a group replacement policy operates as follows: All B bulbs (together as a group) are replaced with new ones at (deterministic) times t1 = T, t2 = 2T, t3 = 3T, . . . , tn = nT, n (All B are replaced wether they are dead or not.)
The cost of each new bulb is CN , and there is a fifixed additional work cost of CN , and there is a fixed additional work cost of CR each time the group is replaced. Moreover, there is an inconvenience cost of CI per bulb that is dead when replaced. Let R(t) = the total (cumulative) cost of using this policy up to time t. (Time is in months.) Our objective is to compute the limit
a）Which of the following does the limit in (1) represent (check one, no explana-tion):
i. The long-run average amount of bulbs that break each month.
ii. The long-run cost rate of using this policy.
iii. The limiting distribution (as t → ∞) of the number of working bulbs.
b）To use the renewal reward theorem (RRT) to evaluate the limit in (1), which of the following serve as the iid cycle lengths {Xn} (check one, no explanation):
i. Xn= Ln, n ≥ 1
ii. Xn= B, n ≥ 1
iii. Xn= T, n ≥
c）To use the renewal reward theorem (RRT) to evaluate the limit in (1), which of the following serves as the reward R over a cycle (check one, no explanation):
3.(25 points, 5 each) IEOR 4102代写
A printer prints (services) jobs one at a time in a FIFO manner. Jobs require an iid amount of time to print { Jn : n ≥ 1 } with a CDF G(x) = P (J ≤ x) such that E(J ) = 5 minutes, and σ2 = V ariance(J ) > At time t = 0 an infinite number of jobs are lined up in a queue for printing. At time t = 0 the first job J1 enters service. Let H(t) denote the total printing time required of the job that is being printed (i.e., in service) at time t. (For example if J6 is in service at time t, then H(t) = J6.)
a）Which of the following is a possible numerical value for
(Check one, no explanation.)
i. 4
ii. 5
iii. 300
b）Supposewe are told that the iid J have an exponential distribution with rate 1/ Which of the following is the exact numerical value of (2) ? (Check one, no explanation.)
i. 7
ii. 10
iii. ∞
c）Suppose we are told that the iid J have CDF, G(x) = P(J ≤ x) = x/10, x ∈ (0, 10). Which of the following is the exact numerical value of (2)? (Check one, no explanation.)
i. 8
ii. 6
iii. 11
d）Suppose that all we are told about the distribution G(x) = P (J ≤ x) is thatE(J ) = 5 and that σ2 = V ar(J ) = Way out in the infinite future you find a job in service. What is the expected age of this job? (how long it already has been in service, on average.) (Check one, no explanation.)
i. 3
ii. 6 ⅔
iii. 7
4.(20points, 5 each) IEOR 4102代写
Consider a single-server FIFO M/M/1 queue with
arrival rate λ = 1
service time rate µ = 1
that has a line (queue) that can hold only two customers—there are two chairs that they sit in to wait; so the system can hold at most 3 (one in service, two in the chairs), the state space is S = { 0, 1, 2, 3 } . Arrivals finding both chairs in use are lost. Additionally assume that any customer waiting in a chair independently becomes impatient after an amount of time with an exponential distribution at rate α = 2: If they have not begun service by that time, then they leave (and never get served.) Let X(t) = the number of customers in the system at time t. (It forms a continuous-time Markov chain.)
a）Whichof the following is the transition matrix P = (Pi,j) of the embedded discrete-time Markov chain? (Check one, no explanation.)
b）Is { X(t) : t ≥ 0} a Birth and Death process? YES or NO (no explanation.)
c）To compute exactly the long-run proportion of arrivals that wait in a chair, which of the following results would you need to use? (Check one, no expla-nation.)
i. l= λw
ii. PASTA
iii. Therenewal reward theorem
d）Suppose that the limiting distribution is given by P0, P1, P2, P3. Which of the following is the long-run proportion of arrivals that wait in a chair? (Check one, no explanation.)
i. P0+ P1
ii. P1
iii. P1+ P2
5.(10points) IEOR 4102代写
Consider a renewal process with iid interarrival times { Xn : n ≥1 }that have a uniform distribution over (0, 1). Let B(t) = t – tN(t), t ≥ 0, denote the age at time t. Compute
6.(10points, 5 each)
Consider a risky asset that has a price per share at time t given by
S(t) = e2e3B(t)−t, t ≥ 0
(where e = 2.71828 ) with {B(t)} a standard Brownian motion.
a）Whichbelow is the probability that S(1) > e3? (Check one, no ) Here, Φ(x) = P (Z ≤ x) where Z is a standard unit normal N (0, 1).
b）Givea simulation algorithm (PSEUDOCODE) for generating one copy of S(10).
其他代写：英国代写 作业代写 report代写 paper代写 code代写 algorithm代写 作业加急 北美代写 CS代写 Data Analysis代写 data代写 澳大利亚代写 essay代写 assignment代写 analysis代写 homework代写 加拿大代写
