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I ISS-2()01) 1 ol4

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I, Attempt any !hoe. sub-palls

PAPER · IV SEC.TlON-A

(j) The matn~ of transilion p(obabllities P= l'~l of the

Markov cham ;. with stales I and 2 IS defined by

111 = 1- rl. fl1 =a, /'21 = /1.1', =1- J!

find the probabnities P0(1) of transition duriug the time and the suuinmll) probnbili lies ~r; ,

8 l11) Consider a pure death process \\here p, = I•Jl for n; I. 2 .... 1 .• 1'1-\' (I +II) = if X (I) = q =4l lbr

J > ~and I and~ pQSJ tlve. Assume un milia! population of srl.e 1

find l~ (!)=I'{X(I )= Ill . li[.X(I))"'"l ~\11:\'(1)

I iii) Consider n discrete time blllllcbiog process IX,. I with probabtlily gcncrruing function 1-(hc) bs

\I (.() a 1 --,CI <<<h <t>< l 1- < 1- <'>

Where ( l - b- r;,)/c (1 - c) > I Assume Xu = I Deter·rmne the coudi~ouallu!llt d:stribution Un:Pr(X, = lei:<, > 0} ·-~

M

(iv) If the pr[tual problem has o unite optimum soluHon. pro1•e that, the dual problem must have a finlle optimum solution and they are equal

8 (v) Explain the nuuri~ minim~ me~:od of (feterminlng 1he initial basic f~ible solution of a

trnnsponat•on problem

3 ( 11) A bake~· keeps stock of n popular hrnnd of tal. e. Pre11ous expenence sho"s the dai I)

demand pall em [or jhe item with assocmled probabililies, as given helo11 Daily demand (Nos.) U 10 2tl 3tJ ~0 50 Prooob•lit~· o.rH 0.20 tJ, rs o.so tU2 um

Use the foUo11 mg sequence of random numbers 10 simulate d1e demand for oex:L I 0 days Random numbers 25, 39, fl6. 76, 12. ()5, 7'3. R9, 19, 49 Also esumate th<>druly nl'erage. demand for the cal.es on the basts ol' s11nul a ted dmn.

2. For rbe M/M/l r"'JfCFS) queueing model (a) Obtain the steady slate solmion for the number ol' unlt~ in the system

Ill (b) In astalionury slate. sho11 1haL the dislnbution of time between successive departures has the

exponential distributi ons as the iotemrri val time distribuuon

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3.

4.

~.

(~) Find the ~xpected voluc:nf queue length

(d) Find the probability distribution oftlte time·•pcud5 io lite •Ysl<m.

(A) P..X11IAin Ute Hungarian mett.ltod of~olvins au as ~cgmd1tiS problam.

"lor I 1{)

10

10

10 (b) Develop tb~ a:onomic lot $ize model fqr the inventory sinmlion where R 1~ the consbnl rate

of demand p<>r unit lim~ and Uu: production is insltlnlancous whott (i) ~bortag"" arc ool •ll<twed und Cii) •hor1fig~• ""' aU•)Wcd wh= c, o l nnu c1 ore the unit lt.oldin!; cool, unit shor1ltge coM ond unit set.up cnst n!llpi>CI ivcly I (I

(c) Explain how the optim>ll\vo·pcnod .inventory poltcy is obtained by staling lite ••sumptioru made.

I (I

I d) Explain the problem ol' replJJcoment or hems that deteriorate with lime by ignormg the chang., tn lhe valne of money during Ute penod.

ta) 10

t>orticles aro pbced succussively and mdcpendcrttl~· mlo N cells. wilh equal probability. l.et IIV(n} equ•l the num!)cr of cells remaining empl)' after dte distrlbution of n p:u1icles. ShOvr that the Kcquenc<: ~~ n). n = 1.2.., •• i~ • i'vl:trk.ov chaio. Frnd tho transition probabilitie~<.

10

(h) Sho~>th•L ii'(t.x.)= m( l m)

Where m = Bf,\'"11< I in a branching proces~.

(c) Usc M·tocltni~ua

Minllniz" : ~ ·h; ~ x.

Snbj""l to 3.r, ~ . .-, =3

'l.r; ,.. .3.1:, .., 3

X,+ 2.r, s,J

.r; .? ''· ,., ~ And cbc-ck your OLL~\\crby lilliog the graphic• I method.

HI

tO (I') Obt~in the np1i·malsolution urthe ll'tlluwins lrnn>l)Ortation prohlcm wh()se unit co~t matri)( is

ghtc:n a.'f under! 10

Mark"l I II Ill 1\" Supply

A G 3 s ~ 22 \Vnrehou.<es B 5 9 2 7 15

c 5 7 ~ 6 8 Rcqurroment 7 12 7 9

SECfJON - P AUempllllly live sub· parts

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3 Ill I (i) Given lh3tl~e c.omplete CXpeCi<lllOU of life 01 ages;>() :ond " .1 ror. particular gTOilp ure

r¢>1'o:~tively 2U9 <t11d '2Q.(}I ye3rll and lh>t. the numher living at age .iO i.~ 41, i 76, ru1d.(a) the number$ t)l3l4tltl.i.ns the age 3lond (\>) lbo ttlltnbcr th>l \I iU die \\ithout. n(bJWlll!lbe nge 31,

8 (ii) Exploin \T<anvill" '$ mt thod to "<msttuct t.heabridgccllifc: table.

(iii I Explain the lugi.~ti'c: cnrve1'iuing to project the pnpulat!on •.

(iv) Dillcu<S bnetly the c.ootputcnystcrn cump11t1enl• walo a dingr:utt.

tV) 13ncfl~·- explain ibe concept of systom suppori proy anune with an t-'Cample.

!vlJ Write shoot notes on sofiw.are for; (a) multi-tast.:lng

(bl multi-progromm•ns 6, (a) Dillcus$ briefly, about the following;

7.

(i) 1.+••• oflife table in demography

(ii ) Tolnl fertility rltle

(w) Gro•• rcproduclionJ·at<> (b) Dc.fifte [nfanl Moo'lll.lity Rllto (Thffi.). Di>cuss thll nualysis of llvffi.by biometric mothod.

]()

(c ) Explain how the Crude Death Rntcs ~'" ' lllndardized directly and indireelly to cs-timlito- ll\c: levels ur IIHJdnlil~ lj,c fforn the dcmognophic iniJuttlte of varyiul! '"I!"" ond sell ' " udur<l.

10 (d) Dntlly c.'<plain the tlH:mry ur ~iablc llOI>ul4tittn mud<:t and "''l'lfiin Ut~ relation~hip bet\\ c:en

the scltcdulcs of fertility, mortality and Uoe ogo di~tJibution of u t10pulalioln O~~> ul\tllinod.

10 (4) Write ~hoot notes on (My 1\\ol) .

Ill (i} CRT ( ii) I\IODF.M I iii) Non-unpact priutetl'

(\1) \\-rite ~hori no~ om (any lWO)l

111 ( i) I\1ICR (u) Opttcal disk

(iii) lndo.'l:d sequential life

(c) O~;~cuss how the database and file organization •nd proces~Jng are made in dM,1hase manage:menl.

Ill (i) l~tch proces•ing mode and 'l'ime ~haring mode ( ii) Bitond Byte (ill) Queue and S tack

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8. 4 oft

(n ) Explain clearly about 1he following: )()

(i) lnlc.mal. Inrcrnnl[onuland Net migrorions (ii) lnlcr·<:<."Jl~lll Md l'osl-ccnijuleslimalcs

(\)) Write o note oo decaUtial population census in India. 1(1

(c) Describe brieJly 01e cxlxtms sollware packages on word processing and spread sheell!. 10 (d) f:..xplain ~1e fundamentals of data traosmi~sion and proce<*ing by providing an e)(ample for

each. H)