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Mcepco opaoa Poccco eepa Taoc ocyapce exec yepce A. B. COOAXO MATEMATA B OHOME ac I eo-paecoe ocoe Tao aeco TT 2001 330.105(075) B173-5 M34 Peee:

aa ooecx ay, aey aepo "yxye ay" TT . B. apxoeo oop o-aeaecx ay, poeccop T .. P. epaa C. M. a Maeaa ooe. eo-paecoe ocoe / A.-coc. A. B.

M34 Cooaxo. Tao: -o Ta. oc. ex. y-a, 2001. . 1. 71 c.

ISBN 5-8265-0136-7 oe ocoe eopeece cee paopa oe aa o epo ac ypca "ooo-aeaece eo". Paccapac opoc aa eop cce accooo ocya, eoo popapoa, oee oa poocex ao, eop pao eop p.

poc apa opox aa cyeo opoo ypca aooo oee (ceaoce 0604, 0605, 0608).

330.105(075) B173-5 Taoc ocyapce exec yepce (TT), 2001 ISBN 5-8265-0136-7 Cooaxo A. B.

BBEEHE Meoece papao o ypcy "ooo-aeaece eo" coepa aepa cooecy popae peeo ceecpa cyeo aooo oee. pao aac cee o cey paea: "Teop cce accooo ocya", "eoe popapoae", "Moe oa poocex ao", "Teop pao" "Teop p". A ae poc pep ox aa opoo paccapac x pee, o ooe cyea caocoeo cpac c oee opox aa.

opooe aae coco aa. Bapa aa opeeec y oce pa aeo . opo opeee apaa poc aa.

Tea 1 CCTEM MACCOBOO OCBAH Teop cce accooo ocya (CMO) [1, 2] ec o apae pao aea. B e yac oooc cooa aeaecx oee pacea paoopax cce, oopx pocxo ocyee ocyae eoopoo ooa ao (peoa). p o oo peoaaec, o a ocya cyae oe pee pe eoxooe ocya ao a ae cyao. pepo oox cya eca oo. o ocyae accpo cyaoo ooa oyaee, ocyee a eeoy ca ao a coee c aoeo, oepe oe a pe pay (eop CMO aco ae aa eope oepee), .. Bo ox ax cyax ec oe coca poecco ocye ocya ao, o ooe cooa cooecyy eop.

1.1 Ocoe o eop apocx cyax poecco p oepoa ox poecco pocxox ppoe (aye, exe, oepeco eeoc) x oo paccapa a e co, oopx a ao ae cya opao ocyecec epexo ooo coco pyoe, pe eca epooc oo oo epexoa. Coaa poo coyec aeaeca eop yaa ae cyae poecc. B acoc a eop coyec p ye CMO, ooy ee ecoo cooecyx opeee ao.

Cya npoecc o cy oo paccapa a eoopy cyay yu (t) (.e. y, ae oopo c cyae e), p o apye t oo paccapa a pe (oa a pocpacey epeey). Apye t oe c epep cpe, o opo cyae eec oceoaeoc oeo {ti} oop o poecce c cpe peee cyao oceoaeoc (ti).

pocmpacmo cocmou aaec oeco cex oox coco, oope oe pa poecc. o oeco oe oe ce (oa oop o ucpemo cyao npoecce) epep (oa cya poecc aaec enpep).

aee ye paccapa oo cyae poecc c oe pocpaco coco. Oa s1,..., sN - coco aao pocpace N coco. ac si sj, ye oaa, o poecc epexo i-o coco j-e.

Cya poecc aaec apocu en Mapoa, ec oeo aoe ycoe P{ (tn+1) = (tn) = si,..., (t0) = si }= P{(tn+1 ) = si n + 1 n = si (tn) = si }, n +1 n .e. epooc oaa poecca aoe o cocoe a ceye ae ac oo o eyeo coco poecca e ac o ooo coco, ocyx paee.

Ec epooc e ec c eee pee, .e. e ac o n, o ao poecc aaec cmauoap. caoapx apocx poecco oo opee epooc epexoa coco si sj, a o a, e acy o oepa aa pij = P{si sj}.

p o oeo oc yco:

N pij = 1, i = 1,..., N; pij 0, i = 1,..., N.

j = coooe paa o, o ec poecc a eoopo ae axoc coco si, o a ceye ae o c epooc 1 epexo eoopoe pyoe oooe cocoe pocpaca coco ( acoc poecc oe e e coeo coco, ec pii 0).

Haoe apy P = {pij} paepa N N ape epexox epooce.

epoe cocoe, oopoo poecc aaec, oeo oo a ae aao. o eaec c oo eoopoo eopa a(0) = (a1(0),..., aN(0)), e aj(0) ec epooc oo, o aa oe, poecc axoc coco sj: aj(0) = P{(t0) = s }. Hapep, ec ocoepo (o ec c epooc 1) eco, o epe j ep ao poecc axoc coco sm, e 1 m n, o am(0) = 1, a pye ooe oo eopa pa y. aae aaoo coco poecca c oo eopa epooce ooe ya pacca cya, oa epe aao oceoaeoc oeo, oopo paccapae poecc, eco, o poecc c ec epooc oe axoc oo ecox coco. Taa cya ee eco aco.



aee co, o ocoy ca epexo cya, o cocoe poecca oce k ao oe ocac oo eopo epooce oo eopy a(0).

Ooa epe a(k) - n-ep eop, oopa oopoo paa epooc, o poecc axoc coco si oce k ao. Oaaec eop a(k) oe ae o opye k a(k) = a(0)P. (1.1) a, caoapa oea apoca e ooc opeeea, ec aa:

oeoe pocpaco coco;

aa eop epooce a(0);

apa epexox epooce P.

pep 1.1 B aece pepa oe a apocy e cey cya poecc. a opa cpe py pya oopeeo epe pae poey pee. p ao oee cpea opa A opaae opa B c epooc 1/2, a opa B opaae opa A c epooc 3/8. peoaaec, o p o oaa opa xo cpo. Paccapac peya cep cpeo. Cocae pocpaco coco apy epexoa, ec coco e yy oa opae, ocaxc cpo.

pocpaco coco ye a ceye 4 coco:

si = {e oa opa}, s2 = {A e, B opae}, s3 = {A opae, B e}, s4 = {oa opa opae}.

ocoy cocox s1, s2, s3 cpe e pooc, o, oa oo x coco, c epooc 1 e ocaec o e coco. Ceoaeo, p22 = p= p44= 1, a pij = 0, i = 2, 3, 4, j i. Opee epooc epexoa s1 s1, s2, s4.

Bepooc oo, o p cpeax oa opa yee p11 = (1 - 1/2)(1 - 3/8) = 5/16.

Bepooc oo, o p cpeax ye opae opa B, a opa A ocaec e p12 = 1/2(1 - 3/8) = 5/16; epooc oo, o A ye opae, a B e p13 = (11/2)3/= 3/16; epooc oo, o yy opae oa opa p14 = 1/23/8 = 3/16. Toa apa epexoa P aec e 5/16 5/16 3/16 3/ 0 1 0 P =.

0 0 1 0 0 0 Hae, apep, cocoe paccapaeoo poecca oce ooo oea cpea a(1) = a(0)P = (1, 0, 0, 0)P =(5/16, 5/16, 3/16, 3/16), ec po a(0)P = (1, 0, 0, 0) ocoy o oea cpea oa opa e.

aee oce yx oeo cpea a(1) = a(0)P2 = a(1)P = (5/16, 5/16, 3/16, 3/16)P = = (25/256, 105/256, 63/256, 63/256), o oaae, o oce yx cpeo oa opa ocayc e c epooc 25/256, oo ep - 105/256, oo opo - 63/256, oa yy opae c epooc 63/256. Aaoo oo pacca epooc coco oce oo oeoo oeca cpeo.

1.2 Caoapoe pacpeeee peypo apoco e Bo ox cyax, acoc ex cceoa CMO, oco epec pecae cocoe cyaoo poecca oce ecoeo ooo oeca ao. ocoy eceoc ecoeoo oeca ao e oe poceeo oa, ec e pe o eoopo eope, oop ec peeo oceoaeoc eopo a(i), p i. Ec ao pee cyecye, o o eop aaec cmauoap pacnpeeeue cyaoo poecca. Ooa eo epe b, oa oo aca lim aPi = b.

i Caoapoe pacpeeee cyecye e cea, xapaepc poecca eoxoe ocaoe eo cyecoa oca e.

Mapoca e aaec poueco, ec aoe cocoe pocpaca coco S oe ocyo oo pyoo coco S. poece e a yx o:

uueco aaec e, oopo aoe cocoe oe pac epe opeeee epoece epa;

peypo aaec poeca e, e ac eco.

Peypa e ( peyp apoc poecc) ec ey cex coco pocpace coco, oaa oe aoe cocoe epe eopeeeoe co eepoecx epao pee. Mapa epexoo peypoo apocoo poecca aaec peypo ape epexoo.

Hapep, poecc oca peye pepe e ec peyp.

ec o e coo cxeaec opa oeco eo oox coco epexoe epooc e pc. 1.1. Ha o pcye e oaa cpea epexo c ye epooc. Ta opao, a oo o o e pcya, coco s1, s2, s3 pye coco ye oac eooo, a a a poecc e ec poec , ceoaeo, - peyp.

S5/16 5/3/SS3/SPc. 1.yaeaa eopea peypx apocx ee yepae, o ec P ec peypo ape epexoo, o:

1) cyecye pee lim Pk = B ;





k 2) aa cpoa ap B p o ec o o e eop b = (b1,..., b2), ooe oopoo yoeop yco N bi 0, bi =1.

i = peea eopea, acoc, oaae, o cyae peypo e oe epooc oaa pae coco p ocaoo oo ce ao e aucm om aaoo cocmou.

axoe caoapoo pacpeee b peypo e yoo ooac oe coco, o b - ece epooc eop ao, o oec eopoe paeco bP = b. (1.2) B oopao ope paeco (1.2) pe :

b1p11 + b2 p21 +...+ bN pN1 = b1, b1p12 + b2 p22 +...+ bN pN 2 = b2,...................................................

+ b2 p2N +...+ bN pNN = bN.

b1p1N pep 1.2 Hae, apep caoapoe pacpeeee peypo e, ee apy epexoa 0 1/ 2 1/ P = 3 0 2/ 3.

1/ 1/ 4 1/ 4 1/ yc b = (b1, b2,b3) cooe caoapoe pacpeeee. Be ypaee (1.2) e cce ex ypae:

1/3 b2 + 1/4 b3 = b1, 1/2 b1 + 1/4 b3 = b2, 1/2b1 + 2/3b2 + 1/2b3 = b3.

oa ycoe oo, o eop b ec epooc:

b1 + b2 + b3 = 1.

Pea epoe, opoe eepoe ypae (a a pee ypaee ec eo oae epoo opoo), oyae b1 = = (b1, b2, b3) = (8/37, 9/37, 20/37).

Paccop caoapoe pacpeeee cyaoo poecca cxeaec opaeoo a pc. 1.2. B, o p cex eyex epoocx epexoa o ec peyp.

P0,P1,2 P2,Pk-2, k-1 Pk-1, k ssssk-sk P1,P2,P2,1 Pk-2, k-1 Pk, k-P0,0 P1,P2,Pk-1, k-1 Pk, k Pc. 1.Ooa epe b = (b0,..., bk) eop eo caoapoo pacpeee, oa ccea ypae (1.2) aec e b0 p0,0 + b1p1,0 = b0, bi-1pi-1,i + bipi,i + bi+1pi+1, i = bi, i = 1,..., k - 1, bk-1 pk-1,k + bkpk,k = bk, k.

=bi i=He coo poep, o peee o cce c e - 1+ p0,1 + p0,1p1,2 +... + pk -1,k....p1,2 p0,b0 =, p1,0 p1,0 p2,1 pk,k -1.....p2,1p1, (1.3) p0,1 p1,2 p0,1 pk -1,k...p1,2 p0,b1 = b0, b2 = b0,...., bk = b0.

p1,0 p2,1p1,0 pk,k -1...p2,1p1,1.3 Ocoe o eop accooo ocya epee eep eocpeceo CMO. opee xapaepca CMO oocc:

oeco ocyax popo. Paa ooaae ooaae CMO;

opaa oepe. Ec o ocyaee peoae e a coooo ocyaeo popa oae CMO, o aa CMO aaec CMO c oaa, ec caoc oepe, o - CMO c oae. Moe e eco opaee a y oepe. Toa ec o ocyaee peoae pxo ccey oe, oa a oepe ao oax ocya oca acao oyco e, o aa yxo e ocyeo;

ca ocya. To ec opo opa oepe a ocya. Moy cooac p: "ep pe - ep ocyc", "ep pe - oce ocyc", pye.

epa oe yaeo eop ec oe nomoa comu, o oop oaec oceoaeoc oopox co pocxox cyae e cyae oe pee [1]. oo aaec cmauoap, ec eo xapaepc e ec co peee. Baee xapaepco ooa ec eo umecuocm, o oopo oaec cpeee oeco ao ocyax ey pee. oop, o eec nomo e noceecmu, ec c eac cyae e oeca co pooex a x yx eepeceaxc poeyax pee. oo aaec opuap, ec ooee epooc, o a pe t pooo a oee co, epooc, o pooo oo coe cpec y p t 0. ae oop peoa ccey pxo o ooe, a e pya. oo aaec npocmeu, ec o caoape opape e ee oceec.

Moo oaa, o poceeo ooa ecoc epooc oe m co ( o oe caoe - oe peoa) a poeyo pee o paaec opyo ()m Pm() = e-, (1.4) m! oopa aaec opyo yaccoocoo pacpeee. B cooec c (1.4) aeaecoe oae cyao e m pao. To ec ecoc ec ece apaepo ooa. Tae ao, o eep oe acaa epooc oe xo ooo peoa a poeyo pee t:

P(t) =1- e-t, (1.5) e ee ea e oee ceyee pee t, P(t) =1- e-t t. (1.6) Ec ca, o oo ocyex ao a e ec poce c ecoc , o aaoo opye (1.5) oo aca epooc, o a pe t ye ocyea xo oa aa ( o oe caoe: epooc, o pe ocya oo a T oaec ee t) P(T < t) =1- e-t. (1.7) To ec opya (1.7) ae y pacpeee pee ocya a. Pacpeeee cyao e a (1.7) aaec cnoeua.

B ayo epaype aco coyc ocoe ycoe ooae a CMO. Hapep, M/M/1 oaae CMO c yaccooc ooo peoa, coea peee ocya o ocya popo.

Ec p o oo peoa ec e cya c ocoo ecoc (peyp), o oa ccea ooaaec /M/1.

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