INVESTIGATION OF HM-NETWORK WITH PRIORITY MESSAGES AND DEPENDING ON TIME INCOMES FROM TRANSITIONS BETWEEN ITS STATES

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1 Joral of Ale Maheacs a Coaoal Mechacs 0 - INVETIGATION OF HM-NETWORK WITH PRIORITY MEAGE AND DEPENDING ON TIME INCOME FROM TRANITION BETWEEN IT TATE Olga Kro Mhal Maalys Facly of Maheacs a Coer cece Groo ae Uversy Groo Belars Ise of Maheacs Czesochowa Uversy of Techology Częsochowa Pola syaya_o@al.r.aalys@gal.co Asrac. I he arcle he echqe of fg he exece coe llele qeeg syses close exoeal y HM-ewor wh rory essages a case whe he coe fro rasos ewee s saes s rao varales wh ow average vales s escre. The esy of he servce of essages syses learly ees o her er. The aroache exressos for he exece coe eeg o he average ers of essages syses are receve. For he he syse of he lear ffereal eqaos wh roe rgh ars s ae. The exale s calclae. Keywors: HM-ewor coes rory essages Iroco Marov HM qeeg ewors wh rory essages a case whe he coe fro rasos ewee her saes s he eere fcos eeg o saes a e were cosere arcle []. yses of he ffereceffereal eqaos for he exece coe of qeeg syses Q of a ewor were receve a were rece o syses of he hoogeeos orary ffereal eqaos whch were solve wh he hel of ffereal schees. I hs arcle he exressos for he exece coe of sch HM ewor are receve a case whe he coe fro rasos ewee s saes s rao varales RV wh a gve execao. Le s coser he close exoeal HM Howar-Maalys - ewor wh he rory essages cossg of Q. As a sae of a ewor we wll ersa a vecor ; ;...; where - accorgly he er of rory he frs ye a sal he seco ye essages syse a sa. Prory essages have a asole rory relao o sal essages [ ]. Le K a K - accorgly he er of rory a sal essages serve a ewor

2 O. Kro M. Maalys K K K - he oal er of essages a ewor; - he er of servce les of essages of ye s Q ; - roaly of essages raso afer servce fro Q syse ; s s - esy of essages servce of ye s Q s. Le s asse ha a ay oe are sasfe coos: <. Le s also eer he followg esgaos: ε ε { } < { }. Le s f exressos for he exece coe of ewor syses.. Exece coes of ewor syses Le s coser he yacs of chage of he coe of soe syse of a ewor. Le s esgae hrogh V s coe a sa. Le V 0 v. 0 The coe of hs Q a sa ca e se a loo V V. For fg V we wll wre o cooal roales of eves whch ca occr rg he e a chages of he coe of syses coece wh hese eves. Wh roaly o he essage of he frs ye wll ass afer servce fro syse o syse hs he coe of syse wll crease a a sze r a he coe of syse wll ecrease y hs sze r where r RV wh execao ex.. Wh roaly o he essage of he frs ye wll ass afer servce fro syse o syse hs he coe of syse wll ecrease y sze R a he coe of syse wll crease y hs sze where R RV wh ex.. Wh roaly { } o he essage of he seco ye wll ass afer servce fro syse o syse hs he coe of syse wll crease a a sze r a he coe of sys- e wll ecrease y hs sze where r RV wh ex.. V

3 Ivesgao of HM-ewor wh rory essages a eeg o e coes fro rasos Wh roaly { } o he essage of he seco ye wll ass afer servce fro syse o syse hs he coe of syse wll ecrease y sze R a he coe of syse wll crease y hs sze where R RV wh ex.. O a erval of e sze he chage of a sae of syse wo' hae o roaly [ { } ] [ { }] o. Beses for each sall ero he syse y sze r where r - RV wh ex. c. creases he coe Le s also coser ha he aove-eoe RV are ars eee s ovos ha r R r R wh roaly.e.. Fro he aove follows: r r r r R r R r r wh roaly wh roaly wh roaly wh roaly wh roaly { { V [ }] { { }] [ o o. o } } o o A he fxe realzao of rocess s ossle o wre ow: M { V / }

4 O. Kro M. Maalys 0 } { o c } {. Averagg ag o acco a oralzao sae P for he chage of he exece coe of syse we receve { } { } l V M P V M / P } { o c } {. I s ovos ha s ossle o reove Heavse s fcos owg o her efo a exsece of oher corresog facors hs exresso. Le s eer esgao } { V M v. Le s have he } { V M v v P v } { o c } {. Frher assg o a l a 0 we wll receve hoogeeos lear orary ffereal eqaos ODE of he frs orer.

5 Ivesgao of HM-ewor wh rory essages a eeg o e coes fro rasos v P { } { } c. Havg se al saes v 0 v0 s ossle o f he exece coe of syses of a ewor. Le s coser ha he esy of a servce of essages s s learly ee o he er of essages of ye s syse s. I hs case.e. > < 0 } <. 0 I vew of saes we receve { }. Fco y x s covex a herefore fro Iyese's eqaly follows ha M N a hs eqaly s reache whe s s Ns M{ s } s s. Ths sae s sasfe for exale whe he flows of essages eerg o a ewor are reglar a holg es of essages syses are cosas. Le's also oe ha hs coo s sasfe whe all Q of a ewor fco are coos of sall or hgh loag. As aroxao of average vale of exressos a we wll ae N a N N.e. we eleve ha

6 O. Kro M. Maalys M { } N N where N a N - average of he essages he frs a seco yes resecvely execg a eg serve syse a sa. Tag o acco hs asso we receve he followg eqaos v fro where follows ha N N N N N N c v v 0 f N x N x x 0 where f N x N x - he exresso sag he rgh ar a x.. Ao fg he average er of essages syses Le s esgae hrogh ρ a ρ - a average er of sy servce les y essages of he frs ye a he seco ye resecvely syse a sa. The ρ a ρ - he average er of essages of he frs a seco yes whch have lef syse e a ρ a ρ a seco yes resecvely arrve - average er of essages of he frs fro oher Q e N N ρ ρ N N ρ ρ. Therefore

7 Ivesgao of HM-ewor wh rory essages a eeg o e coes fro rasos fro 0 where a syses he ODE follow for N a N : N N ρ ρ ρ ρ. zes ρ a ρ ca recsely e fo a herefore we aroxae he exressos N N ρ N N > N N ρ N. N > The syses of he eqaos a wll ecoe N N N N N N.. These are syses lear he ODE wh roe rgh ars. I s ecessary o solve he y slg of hase sace o a er of areas a fg he ecso each of he. yses a ca e solve for exale sg a acage of he ale Maheaca rogras. If he ewor fcos so N N N N loo le: N N N N N N.

8 O. Kro M. Maalys The las syses of he eqaos ca e resee as a arx T Ns XsNs 0 where Ns Ns Ns... Ns X s - he sqare arx cossg of elees x f o s. The ecso of syse 0 loos le s s s s Xs N N 0 e where N s 0 - soe se al saes however fg of elees of a arx s a colex as eve for raher sall vales. X s e. Exale Coser he ewor show Fgre. Le Le also: v 0 v 0 v 0 v 0 v 0 v 0 v 0 v 0 v 0 0 v 0 0 v 0 v 0 v 0 v 0 v 0 v 0 v 0 ; c c c0 c c c c c c c c c c c c c c. 0 Fg.. Moel of logsc rasor syse

9 Ivesgao of HM-ewor wh rory essages a eeg o e coes fro rasos Execaos a we wll rece arces: D. B

10 O. Kro M. Maalys olvg syses of a of orary ffereal eqaos of he frs orer wh secfe al vales se y he Rge-Ka forh-orer sol he Maheaca. acage. The we sse he solos oae we f he exece coes ewor syses. v 0 Fg.. Average of essages of syse of he frs ye For exale for syse average er of essages of he frs ye s resee Fgre a he exece coes for he syse s reresee Fgre. v Fg.. Grah of he coe of syse of a ewor Refereces [] Maalys M. Kro O. Czhoraa N. Fg exece coes HM-ewor wh rory reqess a lear e-eee esy of her servce cefc Research of he Ise of Maheacs a Coer cece [] Maalys M. Thoeo O. Kolzaeva E. Qeeg syses a ewors: aalyss a alcaos GrU Groo 0 Rssa.

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