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Unformatted text preview: 090N618 1. {1 point) You run a newspaper stand. You cannot predict exactly how many copies of
the Daily Bleb newspaper you will be able to sell, but in the past, you have observed that
there are demands for 15 with 10%; 20 with 10%; 35 with 30%; 45 with 40%; and 60 with Each copy of the Daily Blah. costs you $0.5 and sells for $1.00. You must place your order for the papers the night before they are sold, before you know exactly how many copies you will be able to sell. Unsold copies may be returned to the publisher at the end of the 10%.
1 Li” 0. l *3
\ ' 2'0 0‘ l ' day for a credit of $0.1 each.
[.— I; 5 35.5 .J I
5 J 4.9— o‘ Ll. :_. l. (A numerical answer is expected.) (,0 cl {a} (0.5. point) If 25 copies are prepared, what is the expected number of unsold copies? J loxol +ieveeﬁ=+¥§aea§=egﬁ=fgt=e§ than :(g (b) [0.5 point) Calculate the optimal number of copies tonnaxirnize the expected proﬁt. [5— 03’ 'Il‘" of each recurrent state. (a) (M'point) ““77 0.0 0.5 0.6 0.0
0.0 0.7
0.8 0.0 g,“ 1.) —. EM 6%," H 0.0 0.0
I} 0.0 0.0
0.5 0.5 0.0 0.0 S __u itch— {o—D.l off i .ﬁ‘ “x1 '0 CL c “035% C’IWSWI
2. (2‘ point) For each Markov chain with state space {1,13, 4} and the following onestep
transition matrix, (i) classify as recurrent or transient the states and (ii) give the period 0.0 0.5 __  ..
0.4 0.0 All {Etcwgm‘r ..» ,
0.0 0.3 "fa . 0.2 0.0 pg ‘ d2 6‘)? 0.0
0.0
0.0
1.0 1.0 1&5“ {QKLILYmf c g‘“
10 3:3 lfbo‘ocl 3 2 . UL; Faint) (000.09 PotTh (_ L) O‘UYJMYFI— g_ L9 Polaris) 0%? 3. (1.5 point) For the Markov chain with state. space S = {1, 2, 3} and one—step transiticm
matrix 1 _ ' 0.4 0.6 0.0
P21 0.3 0.? 0.0 _ 0.8. 0.1 0.1 calculate Pm. 'l’Cﬂ/D 9%) "I i '. Lil—L . l 1111.;
L’ ‘5 TE 2'. 3 m;:._
), j.“ .3
.— "1‘ :r‘ﬁ 5.“ g
l '’ ‘ \ «x
— : c 4
0°C _, I 3 3 O ”A) f— ) A"; j)
t “— 1 ..J—~ 3: ) l5" xivh
4— 3 % _ D l 0142/
l a“ Gig it o . u
3 4) .._ .. o 9,, ., a
. 3 “— l C Lil—‘1
C. S ": I L H 4. (0. 5 point) A discrete time Markov chain with state Space 5': {1, 2, 3, 4} has the steady state probabilities (0. 2 0.3, 0.4, {1.1) In steady state, how many tlansitions would it take
to 1eturn to. state. 3 given that the system IS currently 111 state 3? /\ enema/Dig
f : _’§i’mn¢n1l7 Ham; Orvagg 5. (2.5 points) For aPoissOn arrival process {Ny(_t_) : t 2 0} with arrival rate Ay = 2 per
hour, and a. second independent Poisson process {.NX(t) : t Z 0} with arrival rate A X = 5
per hour, compute the following: 3 3'
(a) (05 point) )Pr{Ny{9)—Ny(7) )=51Ny(7 )23} {m
I«cl'afg'mci’vd‘ ZWMwﬂS C h) 3
__ 5
Val N492”): E’Lr Ll
5—! 4:1. ?_)1/ fOE/‘fh “—xx—Z ( Of? A
t_———l—~l (b J ((1.5 innt) Pr{Ny(9)— — 12INy( (7)— — 10} Ff£NY (2") :l> #2? ﬁzz—E. a):
EWW :{Ed‘me . (I3) {0.5 pOint) Pr{_Ny(2 )+Nx(2) = 6]Ny_( ) = 2} P5 NYLZHNxUFé; WK”? 5 (W) =1) Cows/Q
5
(cl) (0.5 point) E[Ny(2) +Nx(2)]
M 2+ My: Moo:1 9L (e) (0.5 point) Pr{Ny(2) + NX(2) = 10} leinllTNﬂll ‘V ljois W/ “450114 W5 to
sit{31+ ([ij Gait/‘8 o 2, lg
10/ .
~ Sauna 6. (1.5 point) There is a continuoustime Markov chain with state S : {A,B,C,D}. The corresponding embedded Markov chain has steadystate probabilities [qr:4, «33, «53110) =
(0.2, o.4,0.3,0.1). (a) (1 point) Suppose that sojourn times in states A, B, C, D before making any tran
sition are exponentially distributed with mean 1, 2, 3, 4 minutes, respectively. Ca1
culate the steadystate fraction of time that the system spends in state D. 0“ X [4,. tuYMET megam_ oelXH—O—LI'XZ‘? UApﬁqw Lt 5 , . m 7:; *1 W . ; . ' ____,
L‘L ‘ (Dug I“ % i’bLf’ 2_} 2'g (b) (0.5 point) If 1000 transitions Occurs for this system, how many transitions would.
you expect into state D? (00 ﬁanefivmg OHM/13352 7. (10 points) Draw a transition—rate diagram for each system below. (a) (2.5 points) A small icecream shop competes with several other ice—cream shops in a busy mall. If there are too many customers already in line at the shop, then
potential customers will go elsewhere. Potential customers arrive at a rate of 20 per
hour. They probability that a customer will go elsewhere is j/4 when there are 3' g 4
customers already in the system, and 1I when there are 3' > 4 customers already'in
the system. The server at the shop can serve customers at a rate of 10 customers
per hour. 0
W. 20 l? (b) (2.5 points) The service counter at Southwest Montana Airlines has a single queue for waiting customers and two ticket agents. One of the agents is on duty at all times;
the other agent goes on duty whenever the Queue of customers becomes too long.
Suppose that customer arrivals to the counter are well modeled as a Poisson process
with rate 45 per hour. The agents both 'work at rate 30 customers per hourI and the
Second agent goes on duty if there are 5 or more costomers at the counter (including
the ones being served). Service times are modeled as exponentially distributed. so 2>0 ”’0 30 O to 5'0 (2.5 points) Customers arrive at a single—server facility at a Poisson rate ofﬂ per
hour. When two or fewer customers are present, a single attendant operates the
facility, and the service time for each custOmer is exponentially distributed with a
mean value of two minutes. However, when there. are three or more customers at the
facility, the attendant is joined by an assistant and, working together, they reduce
the mean service time to one minute. WW (d) (2.5 points) A truck company has 7' trucks and its own internal repair shop with two
repairman. Each repairman works on one truck at a time and usually it takes expo—
nential time with mean 1 day. Each truck runs without any problem for exponential
time with mean 7 days. 2 \ customer at a time with exponential time with rate per hour. The system has the
following transition rate diagram. [/Inr Kilg l Inf
IQ/Irw Mimi ﬁll/hr (a) (1 point] Write down the correseonding emb'ecldecl DTMC for this system. 8. Wpoints) A system has two servers with one waitinﬁ'line. Each server processes one (1)) {1 point) Assuming that the system is currently in state 3' for 'i = 0, 1I 2, 3, on average
how long does the system stay in state 1' before making any transition? @ W (D gt,» @311“ 693%” AK 0‘74— 0_)‘§ of“? Oi" CxVLg/é (e) (2 points) Set up equations to solve steadystate probabilities 7r; for the CTMC. 0.5 “t :21n  or
«as ’3? '1 7* TIE+447; Eritr "ii—2.461131} L. (d) (1 point) Suppose a customer arrives and ﬁnds two others in the system. What is
the expected time he spends 1n the systey‘i ECWW g :2) skié; +§3~ Luz I" “£1. K1
9. A production line hasc two station and each static}?K has one serve1. This system has
ThithE following transition rate diagram State space (2' 3') means '5 customers at Station 1
and 3' customers at Station 2 (including those 111 service). For example, {1 2) implies 1 customer at Station 1 and 2 customers at Station 2. Customers can enter as long as the total number of customers Sithe system is (2&9sz 3% (1904/65, kf‘ﬁti/ié §'~/$
@ Guarded/1? has 2. 491 (a) (1 point) What proportion of customers enter the system? Give your answer in terms
of A and steadynstate probab'iiities Eiders (Pi/EM l1’:5<l > king" llo+T> roeMtg To { é... ~05 R Us 71;: Th]  We.) (1;) (1 point) What is the average number of customers in the system? Give your answer
in terms of steadystate probabilities. i, :2 [Lil MD m)"fl(—il_:m+_iiiftmz) 0% (c) {1 point) W hat is the average time an entering customer spends in the system? Give
vour answer in terms of steadystate probabilities. Mg 1% li—rmiiQUizai Twat) Xémvrmi—lm) (d) (1 point) )What is the average number of customer waiting in queue? Give your
answer in terms of steady~state probabilities. i‘j—io +l—HZ'L i{;c,"ﬁ*i[i1tfra ARMflue; ‘hL‘d/L '— 0‘
e pom " a 13 'ie averii n '  : u _rm cus omer spans we ng’irrﬁqneuelnaﬁ
Giv _ .  . w ' a? s 0". e a state probabilities, Fl
2’ 9’ It i. vi ) 10. k2 points) A production line has two stations. Each station has one server whose process
ing times are exponentially distributed with rate 60 per hour .arr'" .
..'" J'— .’ (b) point) If processing times of each station are normally distributed with mean 1 we” 2 9
minute and variance 0.01 what is the average. $0/1mt of time that a joh spendsmL 1 ibissystem? " ‘ QC \_ 1 t
LL}? .1: m —k—_—ﬂ RXSLOK #0 "1L 3 —..___. ...
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 Spring '07
 Billings

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