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1.4 cooae apocx ee oepoa CMO ae ceo cpeac cceyc CMO, oopx o xoo, o xoo oo ec yaccaoc. Oao oa oo peoo, o eec eoop (ocaoo a) poeyo pee t, oopoo epooc, a pxoa, a ocya yx oee peoa oo ca pa y. Toa, ec aa epooc ocye oo a p1 ocya p2 a poeyo pee t, o oepoa oaeo poecca accooo ocya oo cooa apocy e.

yc oepe e oe oee k peoa. Toa oyae e paec pecaey a pcye 1.2 (a ye oopoc oa ec peypo). pocpaco coco ae k + 1 eeo {si, i = 0, k }.

ee ap epexox epooce e :

pi,j = 0, i - j > 1, pi,i+1 = p1(1 - p2), i 0, k, pi,i-1 = (1- p1) p2, i0, pi,i = p1 p2 + (1 - p1)(1 - p2), i 0, k, p0,0 = (1 - p1), p0,1 = p1, pk,k = p1 + (1 - p1)(1 - p2).

A ee eopa caoapoo pacpeee aac opya (1.3).

pep 1.3 B aece pepa aoo oxoa pacey CMO paccop ceyy cya. yc apae acepco o peoy aooe oe paecc e oee yx a. B cpee a peo aoo apaaec aco; a ac c epooc 1/4 oe ocy oa aa a peo aoo. p op acepco oepe a yca. Opee epooc oo, o:

a) epe a aca oce aaa pao acepco e ye aooe;

) oepe epe 12 aco oce op acepco ye 2 peoa.

yco aa oo opee e p1 = 1/4 p2 = 1/5. coy opy, oy apy epexox epooce 0,75 0,25 P = 0,15 0,65 0,2.

0 0,15 0, a) Bepooc oo, o epe a aa oepe co 0, 1 2 peoa aac ooea eopa a(2). B cooec c ycoe aa eop aaoo coco a(0) = (1, 0, 0). Coaco (1) a(2) = a(0)P2, oa 0,6 0,35 0, a(2) = (1, 0, 0) 0,21 0,49 0,3 = (0,6, 0,35, 0,05), 0,0225 0,225 0, .e. epe 2 aca oce op oepe e ye peoa c epooc 0.6.

) oce eaa ao oo peo ca, o CMO oe caoap pe epooc oo, o oepe yy 0, 1 2 peoa pa cooecy ooea epoocoo eopa b. oca (1.3) ee ap P = (pij) axo b0 = 9/44, b1 =15/44, b2 = 5/11.

Ceoaeo, peo oo ca, o epe 12 aco oce aaa pao acepco 2 peoa oepe yy co c epooc 5/11.

1.5 CMO a M/M/1 c oae CMO a M/M/1, o ec c o ocya popo, aoee aco coy a oe peax cce ocya ooa ao. Paccop aae cya oepe opaeo (e oee k ao).

yc eco, o ecoc cooeceo ooa ao poecca ocya aac ea . Ec cooecyee o oe epepoe pe pa a oceoaeoc peex epao ocaoo ao e t, o epooc ocye oo a a pe t peo opeeec opyo aaoo (1.6):

P1(t) t, (1.8) a epooc ocya a o pe oo a P2(t) t. (1.9) p o e ee ea t, e oee ae pe.

Bco eep cc yco opapoc ooa, oo peo aca epexox epooce poecca pi,j = 0, i - j > 1, pi,i+1 = t(1 - t), i 0, k, pi,i-1 = (1 - t) t, i 0, pi,i = t t + (1 - t)(1 - t), i 0, k, p0,0 = (1 - t), p0,1 = t, pk,k = t +(1 - t)(1 - t).

B x opyax peepe caae epexox epooce cooecy oe a pe t yx oee peoa, o, ooe oop ooo. Oao, cooec c ycoe opapoc poecca p t yaaoe oyee oocoao.

B e aoe cocoe oe ocyo oo pyoo coco.

Ceoaeo, p aae ao cce oo ooac peyaa, oe peypx apocx ee. Bae oopx ee ye CMO ec eopea ocaoapo pacpeee.

Ec poce yopoae CMO c oea ocye epo a, o oeo oo aa e cocoe a eeec o pee, o pace oepe, o p ecox ocyax popax yeaec o ax popo. Cooeceo ec ce apaep, oope a ae xapaepy cocoe CMO. Hapep, aee opae c epooc oo, o ccee poo i peoa, ao i popo, o ec epooc pax oox coco CMO p(si). B eop CMO paccapac ypae ocae ae e p(si), ypae c epea. Oao p ye CMO ac pee ceo epecy xapaepc poecca, oa o ye oc coe ycaoec a. oop o cyax poecco ac epecye eo caoapoe pacpeeee. o oo a xapaepc poecca caoapo coco oo cy o aece ocya ao eo oCMO.

Ooa epooc p(si) ycaoec coco poecca epe bi, o c ae apaepa, a a a e eo oo pacca ce pye apaep. Paccae x ocao oe CMO. ee ap epexox epooce ye ca e. coy opy (1.3) oyae:



- k +2(1- t)k + (1- t) k +2(1- t)k +b0 =, 1+ (1- t) - (1- t) 1- 1- (1- t) (1- t) (1.10) i i- 1 1- t bi = b0, i = 1,k 1- t t 1- ae pae peo opee epooc coco epepoo poecca. Tooc opeeec oc e (1.8) - (1.9) c epooc. epexo (1.10) peey p t 0 oyae oe pae epooce coco cce caoapo pee 1- i b0 =, bi= b0, i = 1,k 1- k + ec cooao ooaee =, y ey aa apyo cce.

Teep oyc oye epooc ecoo oy pae pax ax apaepo CMO caoapo pee.

Bepooc oaa, .e. epooc, o aa ocy oe, oa ye eec oepe k ao POT = k+1 b0.

Oocea poyca cocooc, .e. o ocyex ao ocyx ccey Q = 1 - POT.

Acoa poyca cocooc CMO, .e. cpeee oeco ao ocyaex ey pee:

A = (1 - POT).

Cpeee co a oepe (e yea ocyaeo a), .e.

aeaecoe oae o cyao e:

1- k (k +1- k).

LO = 2 +(1- k )(1- ) Cpeee pe oa ocya ao (e a pee caoo ocya):

M(n) = LO/. (1.11) o pocoe ocyaeo popa paaec eo b0, o ec epooc oo, o pop cooe.

Paccop eep cya, oa e opae a y oepe. O eo oyaec peyx opy, a pee epexo p k. Aaoe xapaepc ao CMO e ecey .

Bepooc coco:

b0 = 1 -, bi =i b0, i = 1,.

Cpe a oepe, a peoae, axoeec a ocya M(n) =. (1.12) 1- Cpeee pe oa ocya peoa ( - pe oa peoa) M() =. (1.13) (1- ) Cpe a oepe, oae ocya. (1.14) 1- Cpeee pe, poeeoe peoae ccee. (1.15) (1- ) Paccop cey pep.

pep 1.4 yc peoa ocya a ocyaee ycpoco cyepapea cyao, pe cpe poeyo pee ey ocye peoa pae 0,5 . Bpe ocya pacpeeeo coeao co cpe aee, pa 0,25 . Opeee: a) cpeee co peoa ccee; ) cpeee pe oa ocya peoa; ) cpeee pe, poeeoe peoae ccee; ) o pee, eee oopoo ocya pop pocaae.

Cpe copoc xooo ooa pecae coo ey, opay cpee e poeya ey ocye peoa = 1:0,5 = 2 (peoa yy). Cpe copoc ocya peoa paa = 1:0,25 = 4 (peoa yy). Toa = / = 1/2.

Ta a < 1, o a ocaoo ooe pe ccea ocya oe caoap pe.

a) Coaco (1.12), cpeee co peoa ccee = 1 (peoae).

1- ) Coaco (1.13), cpeee pe oa ocya =1/4 ().

(1- ) ) Coaco (1.15), cpeee pe, poeeoe peoae ccee = 1/2.

(1- ) ) o pocoe ocyaeo popa P0 = 1 - = 1/2.

Tea 2 HEHOE POPAMMPOBAHE Bo ox cyax, cpeaxc poeoc, ceco xoce, ooeco eeoc .., aaa oa aa eoopx ooopoocex ec oe acaa e ex ypae epaec c e e, ooceo cox, opeex o a epeex ee yoao. aaa oo e a coc oe oe aa oa p pee eoopx pyx caocoex apae pao aea.

Cooeceo oae opeoc aeaeco eop ooe pea ae aa. Taa eop cyecye aaec ue npopaupoaue [3, 4]. a oeaec o, aoe aae e ec ya. Oo oo 30-e o, oa pecae o popapoa a oepe e e cyecoao. B ao cyae o popapoae, aec opayeaec apoae. Oao, o ep ye yopec, e oo eo cyae. ec a e ae aa aeaecx eop pee aa oa, a eueoe npopaupoaue uauecoe npopaupoaue.

2.1 Oa ocaoa aa eoo popapoa B oe e aaa eoo popapoa () aaec oca ax eopaex ce x1, x2,..., xn, oope acpy ay ey y n C = c1x1 + c2x2 +...+ cnxn = ci xi (2.1) i=p yco oe cce epaec:

a11x1 + a12x2 +...+ a1nxn b1, a21x1 + a22x2 +...+ a2nxn b2,.....................................................

ak1x1 + ak2x2 +...+ aknxn bk, a(k +1)1x1 + a(k +1)2x2 +...+ a(k +1)nxn bk +1, bi0, (i=1,m )......................................................

(2.2) a + a +...+ a bp p1x1 p2x2 pnxn a( p +1)1x1 + a( p +1)2x2 +...+ a( p +1)nxn = bp +1,........................................................

am1x1 + am2x2 +...+ amnxn = bm, xi0, (i=1,n ) (2.3) coy ceye ocoe o ep:

y (2.1) aaec eeo ye ( ee yoao, yoao aeca) aa, epaeca paeca (2.2) - (2.3) aac opae aa, o aop ce {xi}, yoeop (2.2) - (2.3), aaec oyc peee oyc ao;





oycoe peee, oopoe acpye eey y, aaec oa peee;

acaoe ae eeo y Cmax aa aee oaoo pee aa.

Opaee-epaeco cxoo aa , eee "", oo peopaoa opaee-paeco oaee eo eo ac eoopo oo eopaeo epeeo, a opaee-epaeco a "" - opaeepaeco ae eo eo ac eopaeo epeeo. Ta opao, opaee-epaeco ai1x1 + ai2x2 +....+ ainxn bi oo peopaoa opaee-paeco ai1x1 + ai2x2 +.... + ainxn + xn+1 = bi, (xn+1 0), a opaee-epaeco ai1x1 + ai2x2 +.... + ainxn bi opaee-paeco ai1x1 + ai2x2 +.... + ainxn - xn+1 = bi, (xn+1 0).

epeee oe peopaoa opae-epaec opaepaeca aa onoumeu. co, o x co pao cy peopayex epaec.

Boe ooee epeee e oe opeee ooec cc. Ta, ec opaex cxoo aa eoo popapoa opaac pacxo ae poocex pecypco, o ceoe aee eoopo ooeo epeeo pao oey ecoyeoo cooecyeo pecypca.

Ec epeea xk e oea yco eopaeoc, o ee ceye ae y cooae eopae epee uk vk, p xk = uk - vk.

2.2 Caapa ocoa aa eoo popapoa Ec ce opae (2.2) e epaec "", o aa aaa aaec cmaapmo ( cuempuo). Ee yoee aca , opao oee oao, apo ope C = max, (2.4) Ax b, (2.5) xi 0, (i=1, n ), (2.6) ec - capoe poeee eopo (c1, c2, , cn) (x1, x2,, xn); A - apa paepoc (m n) cocaea oeo cce opae (2.2) ( oopo oao e, a cae, oo epaeca a ""); b - eop-coe (b1, b2, , bm), cocae pax ace cce opae (2.2).

Ec ce opae (2.2) e paec "=", o aa aaa aaec ocoo ( aoueco). Ee ae yoee aca apo ope C = max, (2.7) Ax = b, (2.8) xi 0, (i = 1, n ). (2.9) 2.3 paec eo pee aa eoo popapoa EC B HEOTOPO AAE TOO BE EPEMEHHE X1 X2, TO EE MOHO EO PET, TA HABAEMM PAECM COCOOM. TOT COCO OCHOBAH HA TOM ATE, TO B TOM CAE MHOECTBO OCTMX PEEH MOHO OCTPOT HA BXMEPHO OCOCT. PACCMOTPM CE PMEP.

pep 2.1 yc peyec acpoa yoa C = 11x1 + 7x2 max, p opaex 3x1 + 2,4x2 12, 5x + 2,5x2 15, x2 3, x1 0, x2 0.

Peee ocpo ococ x00x2 cooecye opae aa pe 3x1 + 2,4x2= 12, 5x1 + 2,5x2= 15, x2 = 3, x1 = 0, x2 = 0.

aa x opeee oyococ, oopo oec cooecyee ycoe. pe pecae a pc. 2.1.

x3x1 + 2,4x2 = x1 = 5x1 + 2,5x2 = x2 = 11x1 + 7x2 = h Pc. 2.HA TOM E PCHE ATPXOBAHA OACT BAC EPECEEHEM TX OOCOCTE, TO ECTTO ATPXOBAHHO OACT MET OOPHAT OBETBOPE BCEM OPAHEHM AA. EC TEEP OPAT COOTBETCTB HOHA PM 11X1 + 7X2 = H, E H - HEOTOPA BEHA, BEBA H EPEMEAT T PM BBEPX, MCEHHO EPECTPABA EE EOMETPEC, TO MOHO OPEET PAH OCEH TO EPECEEH TO PMO C ATPXOBAHHO OACT.

TA TOA ET COOTBETCTBOBAT PEEH AA.

BHAEM CAE COMA TOA OMEEHA HA PCHE, A C.

TEEP OOHATEHOO OEH PEEH OCTATOHO TOHEE PACCTAT OOPHAT C, PEB CCTEM PABHEH 3x1 + 2,4x2 = 12, + 2,5x2 = 15, 5xoyae ae x1 = 4/3, x2 = 10/3. Toa acaoe aee yoaa Cmax = 114/3+710/3 = 114/3 = 38.

Oca eo oa yaec pe cyae, ec cxoo aae oee yx epeex. e oopoc e o epexoe o opae-epaec opae-paeca, p o opacae oeco epeex. Aaoo eoopx cyax eo oo ye (apep, o yx) ye epexoa o opae-paec epaeca. Oao, a ae ooo ocy pex cyax.

2.4 Cec-eo pee aa eoo popapoa Papaoa poo peec yepca eo pee o aa , aae cec-eoo. Paccop eo cooae a pepe pee opeo aa.

pep 2.2 ooe pax e A, B C pepe coye p a oopyoa. Hop apa cao-aco aoo a a opaoy ooo e A, B C, ea ooo e, a ae oee eeec oeco caoaco pee a. 2.1.

Taa 2.Hop apa Oee B oopyoa cao-aco a oeco oo ee cao-aco A B C peepoe 18 15 12 Toapoe 6 4 8 oaoe 5 3 3 ea ooo 9 10 e Caec, o c oecee. Coca a pooca e, p oopo cooc poy ye acao.

Peee Coca aeaecy oe aa. co yc e A ooa epe x1, e B - epe x2, e C - epe x3. epeee o yoeop ceye ccee epaec:

18x1 + 15x2 + 12x3 360, 6x1 + 4x2 + 8x3 192, 5x1 + 3x2 + 3x3 180.

Oa cooc poeeo poy coca C = 9x1 + 10x2 + 16x3.

co, o oo oc:

x1 0, x2 0, x3 0.

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