t3fall2008spring2012fall2012

8/12/2019 t3Fall2008Spring2012fall2012

1/12

1

ISYE 3232B Fall 2012 Final - 01

I, , do swear that I abide by the Georgia Tech Honor

Code. I understand that any honor code violations will result in an F.

Signature:

Throughout, you will receive full credit if someone with no understanding ofprobability, set theory, and calculus could simplify your answer to obtain thecorrect numerical solution.

You will have 2 hours.

This exam is closed book and closed notes. Calculators are not allowed. No scrap paperis allowed. Make sure that there is nothing on your desk except pens and erasers.

If you need extra space, use the back of the page and indicate that you have done so.

Do not remove any page from the original staple. Otherwise, there will be 3points off.

Show your work. If you do not show your work for a problem, we will give zero pointfor the problem even if your answer is correct.

We will not select among several answers. Make sure it is clear what part of yourwork you want graded. If two answers are given, zero point will be given for the problem.

Problem 1 (a), (b), (c), (d)

Problem 2Problem 3Problem 4 (a), (b), (c), (d)Problem 5 (a), (b), (c)Problem 6 (a), (b), (c), (d), (e), (f), (g)BonusProblem 7 (a), (b), (c), (d)


2/12

2

1. (17 points) A store stocks a particular item. The demand for the product each week is0 item with probability 0.1, 1 item with probability 0.5, and 2 items with probability0.4. Assume that the weekly demands are independent and identically distributed. Each

Friday evening if the remaining stock is 0 item, the store orders enough to bring the totalstock up to 2 items. These items reach the store before the beginning of the followingMonday. Assume that any demand is lost when the item is out of stock. LetXn be theamount in stock at the end of Friday in the nth week. Then state spaceS={0, 1, 2}andit is found that stationary distribution (0, 1, 2) = (0.5, 0.4, 0.1).

(a) (5 points) This is a Markov chain. Assuming thatX0 = 2, give the initial distributionand transition matrix.

(b) (3 points) If this week (week 0) starts with inventory 2, what is the probability that

week 2 has starting inventory level 2?

(c) (2 points) If this week (week 0) starts with inventory 2, what is the probability thatweek 500 has starting inventory level 2?


3/12

3

(d) (7 points) Suppose that each item sells at $20. Each item costs $5 to order, and eachleftover item by Friday evening has a holding cost of $2. Suppose that each orderhas a fixed cost $10. Find the long-run average profit per week.


4/12

4

2. (10 points) Let X be a Markov chain with state space{a,b,c,d,e} and transition proba-bilities given by

P=

.5 .5 0 0 0

.3 .7 0 0 0

.2 0 0 0 .80 0 .4 .4 .20 0 0 0 1

Compute the limn Pn matrix.


5/12

5

3. (5 points) Consider a bank with two tellers. Alice, Betty, Carol, and David enter thebank at almost the same time and in that order. Alice and Betty go directly into serviceby Teller 1 and Teller 2, respectively, while Carol and David wait for the first available

teller in a single queue. When one of tellers becomes available, Carol will be served beforeDavid. Suppose that the service times for Teller 1 are exponentially distributed withmean 6 minutes and that the service times for Teller 2 are exponentially distributed withmean 9 minutes. What is the expected service time for David?

4. (15 points) A basketball player takes 3-point shots and the number of 3-point shots hetakes by time t is a Poisson arrival process with rate 10 per hour. He also takes 2-point

shots and the number of 2-point shots he takes is an independent Poisson process withrate 30 per hour. Compute the following:

(a) (3 points) What is the probability that he takes five 2-point shots in the first 30minutes and takes seven 2-point shots in the first one hour?


6/12

6

(b) (6 points) What is the probability that the player takes two 2-point shots in the first30 minutes given that the total number of shots (2-point and 3-point shots) is ninethe first 30 minutes?

(c) (3 points) The player can make any particular 2-point shot with probability 0.4(independently of all other shots). What is the probability that he makes exactlyfour 2-point shots in the first 30 minutes?

(d) (3 points) What is the expected amount of time until he takes his fourth shot (foreither 2 or 3 points)?


7/12

7

5. (8 points) A continuous-time Markov chain has the following generator matrix:

G=

2 2 0 0

7 9 2 03 0 10 75 0 0 5

.

(a) (2 points) Draw a transition-rate diagram for state spaceS= {0, 1, 2, 3}

(b) (3 points) Find the roadmap matrixR.

(c) (3 points) Set up calculations to get steady-state probabilities (i). (Do not attemptto solve your equations to get numerical answers for i. Just set up calculations.)


8/12

8

6. (25 points) Below is a transition-rate diagram for a queueing system with system capacity3 and 2 identical servers. Both inter-arrival times and service times are exponentiallydistributed. Steady-state probabilities for this system (i.e., long-run fractions of time the

system spends in each state) are (0, 1, 2, 3) = (0.2, 0.4, 0.3, 0.1).

(a) (2 point) Define the generator matrix G.

(b) (2 points) Give the expected holding time in state 1 until it makes a transition outto any other state.

(c) (5 points) When both servers are idle, an arriving customer chooses a server randomlywith 50% probability. What is the utilization of one server?

(d) (5 points) What proportion of potential customers enter the system (i.e., what is theeffective arrival rate or average arrival rate eff)?


9/12

9

(e) (5 points) What is the long-run expected number of customers in the system?

(f) (3 points) What is the long-run expected amount of time in units of hours a customerspends in the system from arrival until departure?

(g) (3 points) What is the expected waiting time in queue in units of hours?

Bonus: True or False? If your answer is correct, you get 2 points. If you leave it blank,you get 0 point. If you answer is wrong, you lose 1 point. Let N(t) represent the numberof arrivals from time 0 to time t.

(a) Even if N(t) has independent and stationary increments properties,Pr(N(3) N(1) = 0, N(1) = 0)= Pr(N(2) = 0, N(1) = 0).

(b) A demand-arrival process for school backpacks throughout a year does notsatisfy the stationary increments property.

(c) A demand-arrival process for swimsuits in June does satisfy the stationaryincrements property.

Move to next page for Problem 7


10/12

10

7. (20 points) For each system, (i) define system states and (ii) draw a transition-rate dia-gram using the system states you defined.

(a) (5 points) A truck company runs 5 trucks and its own internal repair shop with

three repairmen. Each repairman works on one truck at a time and usually it takesexponential time with mean 2 days to repair a truck. Each truck runs without anyproblem for exponential time with mean 7 days.

(b) (5 points) Customers arrive at a small bank with one banker at a Poisson rate of 40per hour. The banker is on duty at all time and his service times are exponentiallydistributed with rate 45 per hour. A study shows that customers waiting in the firstand second position in the queue line do not leave and wait for their turn regardless

of waiting times in queue. However, customers in the third or higher position in thequeue leave the queue line without receiving service if their waiting times exceedexponential time with mean 15 minutes.

Move to next page for Part(c) and (d)


11/12

11

(c) (5 points) GT students arrive at a Bank of America ATM machine in the studentcenter. Students arrive following a Poisson process with rate 10 per hour but a stu-dent may not join a queue if the queue is long. More specifically, a student joins the

queue with probability 1i+1 where i is the number of students at the ATM machine.

For example, if there is no one at the ATM machine, then a new arriving studentjoins the system with probability 1. Service times are exponentially distributed withmean 5 minutes.

(d) (5 points) A call center has two agents (Amy and Ben) and three phone lines (i.e.,hold only up to three calls). Calls arrive at the call center according to a Poissonprocess with rate = 10 per hour. The service time of Amy is exponential with rate8 per hour and the service time of Ben is exponential with rate 4 per hour. A call willbe served by next available agent. However, when both agents are free, an arriving

call always prefers Amy. Once a call is served, the call departs the call center.

END!


12/12

12

Discrete-Time Markov Chain (DTMC)

Pnij = Pr{Xn= j | X0 = i}.

A Markov chain is irreducible if there is only one class.

State i is recurrent if

n=1 Pnii = and transient if

n=1 Pnii s + t| X > t) = Pr{X > s}.

Suppose that X1 exp with 1 and X2 exp with 2 and X1 and X2 are independent.Then (i) Pr(X1 < X2) = 1/(1 +2); (ii) min(X1, X2) expo with 1 +2; (iii)max(X1, X2) =X1+ X2 min(X1, X2).

Poisson Process

A Poisson arrival process N(t) is a Poisson distributed with Pr{N(t) =n}= et(t)n

n! .

Nonhomogeneous (or non-stationary) Poisson process with (t) has = E[N(t+ s)

N(s)] =t+ss

(t)dtand Pr{N(t + s) N(s) =n}= en

n!

.

CTMC/Queueing/Queueing Networks

steady-state probabilities for a CTMC: G= 0 and 1= 1.

Littles Law: L= effW andLq =effWq.

For M/M/1, L =

, W = 1

, Lq = 2

() , and Wq = () .

t3fall2008spring2012fall2012

Documents