本次CS代写的主要涉及如下领域: Python代写,澳洲程序代写,新西兰程序代写,STATS320代写,University of Auckland代写
STATS 320 – Applied stochastic modelling – 2020
Assignment 4
Due: 11 June 2020 at 1pm.
- In the year 2030, the STATS 320 course has fallen into the hands of Jacques Poisson, a much less organized lecturer than your present one. Instead of grouping problems into easily digestible assignments, Jacques hands out a problem whenever the mood so takes him. This happens more or less at random. For smallh, the probability of a problem being handed out between timestandt+his approximatelyλh, whereλ= 2/week, independently of any problems handed out previously. There are no due dates; students hand solutions in whenever they finish writing them. To ensure that no student gains an advantage by procrastinating, there are no model answers, and no marked assignments are ever returned to the students. Bill Battler is a student who responds rather unevenly to pressure. Bill treats his STATS 320 assignment problems as a first-in-first-out queue, always working on the oldest unfinished problem. His rate of working on the problems depends on how many unfinished problems are in his queue, as follows:
Number of unfinished problems 1 2 3 4 5
Work rate (problems/week) 3.0 5.0 2.5 1.5 0.
Bill’s work on the problems is memoryless (one reason why he has such a hard time). The number of unfinished
problems he has can therefore be modelled as a birth-and-death process.
When he has 5 unfinished problems to do, Bill stops all work on STATS 320. This is very unwise. How-
ever, it does mean that no more new problems are added to the queue, since he’s no longer reading Canvas
announcements. Thus, there can never be more than 5 problems in the queue.
(a) Draw the state transition diagram of the system, showing the transition rates.
(b) Find the expected time until Dr. Poisson sets the first problem of the semester.
(c) Find the equilibrium distribution of the system.
The remaining parts of this question assume that the system is in its steady state.
(d) For what fraction of the time is Bill doing STATS 320 work?
(e) What is the mean number of problems in Bill’s queue?
(f) What is the mean number of problems per week handed in by Bill?
(g) If there are currently four problems in the queue, what is the probability that Bill will solve them all
before Dr. Poisson sets any more problems?
(h) Find the mean number of weeks that elapse between Dr. Poisson setting a problem and Bill handing in a
solution.
- Phone calls arrive at my phone as a Poisson process with rateλ= 8 per hour. The time I spend talking to each caller on the phone is a random variable, independent of the time spent talking to other callers and of the call arrival times. Calls which come in while I am already talking on the phone are placed on hold, in a first-come- first-served queue. There is no limit to the number of callers who may be on hold at any one time. I’m interested in several possible distributions for the time spent talking to a caller.
(i) Every call takes exactly 5 minutes.
(ii) Talk times are distributed uniformly between 1 and 9 minutes.
(iii) 10% of calls are wrong numbers (talk time: 15 seconds); 30% of calls require 1 minute to deal with; 40%
require 4 minutes; 20% require 15 minutes.
Foreachof these possible talk-time distributions,
(a) describe the queue inA/S/m/nnotation;
(b) findL, the expected number of callers present in steady state (including those on hold as well as the one
talking);
1
(c) findW, the expected time that a caller takes to complete their call (including time spent on hold);
(d) use the code provided in the fileasst4.Rto do a simulation of the call-waiting queue, and plot a histogram
showing the total times taken by 10000 callers who call while the system is in steady state;
(e) find simulation-based 95% confidence intervals forLandW.
- A Scrabble club may contain members of the following 3 types.
- (E) - Expert players, who consider puns to be the highest form of humour.
- (N) - Novice players who also appreciate puns.
- (D) - Players who do not appreciate puns, and are disinclined to associate with those who do.
The membership is restricted as follows.
- The club’s constitution does not allow more than 4 members in total.
- There cannot be more than 2 expert players. (If there were, the weakest expert player would suffer irreparable damage to the ego, and leave.)
- As pun-hating (type-D) players will not associate with the others, the club must either have no pun-haters, or consist entirely of pun-haters.
- The situation where there are two experts and one novice is an “unstable intermediate”. If this state of affairs should come about, the novice will immediately become too intimidated to continue, and will leave.
- It is possible for the club to have just one member. In this case, the sole member will do crossword puzzles while waiting for someone else to join.
Potential new members arrive as a multi-type Poisson process, with ratesλE= 1/year,λN= 1. 5 /year, and
λD= 2/year for the experts, novices, and pun-haters respectively. They will join the club if they can do so
without violating the above restrictions. A player who joins the club will remain a member for a random length
of time, distributed exponentially with mean 1 year.
(a) Draw the transition diagram for the continuous-time Markov chain which models this process. (Hint:
there are 15 states in all.)
(b) Suppose there are currently 2 novice members and one expert. What is the expected time until the next
change of state?
(c) Find the equilibrium distribution of the system. You may use computer assistance to solve the equations,
or you may solve them by hand.
From now on, you may assume that the system is in its steady state.
(d) You meet a member of the club, and (daringly) make a joke involving a pun. What is the probability of
an unfavourable reaction?
(e) What is the rate at which new members join the club?
(f) Is the process time-reversible? Why, or why not?