APPLICATION OF HM-NETWORKS WITH UNRELIABLE SYSTEMS FOR FINDING THE MEMORY CAPACITY IN THE INFORMATION SYSTEMS

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1 Jourl of Ald Mcs d Couol Mccs 28, 7(2), ISS DOI:.752/c ISS APPLICATIO OF HM-ETWORKS WITH URELIABLE SYSTEMS FOR FIDIG THE MEMORY CAPACITY I THE IFORMATIO SYSTEMS Mkl Mlysk, Pwl Zc 2 3, Dry Kos,2 Isu of Mcs, Czsocow Uvrsy of Tcology Częsocow, Pold 3 Fculy of Mcs d Cour Scc, Grodo S Uvrsy Grodo, Blrus.lysk@gl.co, wl_zc@v.l, dk8395@l.ru Rcvd: 3 ovr 27; Accd: 22 My 28 Asrc. To solv rol of drg ory ccy of foro syss (IS), ws roosd o us socsc odl, sd o us of HM (Howrd- -Mlysk) - quug works w cos. Ts odl ks o ccou srvcg of rquss log w r volus, ly o cg volus of rquss ovr d ossly of dgg IS ods d r rrs, so srvcg of dds c rrud rdoly. T xrssos r grd for (xcd) vlus of ol rquss volus IS ods. MSC 2: 6K2, 6K25, 68M2, 9B22 Kywords: HM-work, foro syss, urll srvc, volus of rquss. Iroduco I IS, ol volu of ory s ld o cr vlu, wc s usully clld ory ccy []. W dsgg syss, sk s o fd xcd volu of ory o k o ccou codos l sr of los foro. O of ods for solvg rols of IS dsg s us of HM-quug works (Q) [2]. Accordg o IS, w wll cou o udrsd syss s foro rsforo ocs, dlvrd for of ssgs (rquss) []. HM-works c usd o dr volu of uffr sorg of syss r rrsg rocssg IS ods d rsfrrg dds. o s rol s crucl, for xl dsg of us or couco crs foro works. T followg odl c lso usd o solv or rol rcly rd IS,.., rol of xcssv uffrg ( rol of drg rqurd

2 52 M. Mlysk, P. Zc, D. Kos ory volu) [3]. Buffrlo - oo occurs ck couco works, w uffrg cuss xcssv crs of cks d ck dly vro, d rsulg dcrs rougu volu of IS. glc ddc of rquss rocssg of r volus c ld o srous rrors fdg uffr ory IS. T soluo grl cs of ov rols c sd o us of HM-works w cos. I suc works, rqus durg rso fro o quug sys (QS) o or rgs so co ls (wc s qul o volu of s rqus), d co (volu) of frs QS s rducd y s ou. o od of fdg o-sory rols of ss d fdg vrg work crcrscs w syss of du urll work syss, org udr g lod w l of uldsol grg fucos r dscrd oogr [4], HM-work w urll srvc [4, 5]. For frs, us of HM-works o s volu IS s dscrd [6]. I [7, 8], roosd od of fdg xcd volu of oogous rquss o work syss w ld wg s quus syss, d [9] - HM-work w ld soour s of rquss. 2. Fdg xcd volus of rquss cs, w s kow oly frs os or cos rsos w work ss L s look Q w urll QS. Suos S dls r rll, d or syss S,...,S ls c dgd. T s of sys S s dscrd y vcor ( k (), d() ), wr k (), d() r ur of rquss d udgd ls sys S, rscvly, d (),,. x, l s scfy ( k (), d(), k2(), d2(),..., k (), d() ) vcor s s of work. Assu srvcg s, l org s, d l rr s r dd rdo vrls (RV). T srvcg for IS rquss s xol dsruo w rr ( k(), d() ); oro of c l wou dg s sys s xol dsruo w rr ( k ( ) ),,. Afr dg of l dly gs rrg, rr lso s xol dsruo w rr ( k(), d() ),,. Assu f l rcvs dg wl srvcg cll, rqus srvcg wll cou fr l s rrd. W ssu dscl of srvcg rquss ll syss s FIFO. T rvl [, ), rqus c dld o vry l of sys S w roly ( k(), d() ), l w roly ( () ) y dgd or corrcd w roly k

3 Alco of HM-works w urll syss for fdg ory ccy 53 ( k(), d () ),,. L us do y - roly of ssg rqus fr srvg S sys o S sys, P rx ( ) ( ) s rx of rso rols of rrducl Mrkov c. L s look dycs of cg xcd volu of rquss so S work syss. L V () volu, d gg of sys s volu V ( ) v. T s volu gv QS c xrssd y forul wr V ( ) rvl [ ) V ( ) V() V( ),, (), - cg volu of rquss sys S,. To fd s vlu, you d o fd rols of vs y occur d volu cgs S sys ssocd w s vs.. W roly λ [ ( k (), d () ) ( k () ) ( k (), d () )] u k () rvl cg sys s S wll o k lc, volu of rquss sys crss y E r c,,. r, wr { } 2. W roly ( k (), d () ) u( k () ) rqus fr srvcg S, gos o xrl vro, volu of rquss sys S dcrss y vlu of R, wr E { R },,. λ o, rqus fro ousd vro rs 3. W roly sys S d crss volu of rquss o sz,. 4. W roly ( k (), d () ) u( k () ) r, E{ r }, rqus, fr srvcg S sys, gos o sys S ; volu of rquss sys S cgs y vlu V( ) R r E R,,,,. 5. W roly k (), d (),, wr { } u( k () ) rqus, fr srvcg sys S gos o sys S ; ol volu of rquss sys S wll crs y vlu of V( ) r r E r,,,,,, wr { }. 6. W roly k () u k o ur of cv ls sys ( ()) S dcrss y, volu of rquss sys S wll crs y vlu V(, ) r,,,.

4 54 M. Mlysk, P. Zc, D. Kos T followg css r ossl. 7. T ur of org ls sys S wll dcrs y, rqus fr srvcg sys S wll go o xrl vro,,,. 8. T srvc l sys S s dgd, rqus rs sys S fro ousd vro,,,. 9. T ur of srvc ls sys S s wll rducd y, rqus fr srvcg sys S wll go o sys S,,,,.. T ur of srvc ls sys S s wll dcrs y, rqus fr srvcg sys S wll go o S,,,,. T rols of corrsodg vs css 7- r qul o ( ), for xl, cs 7 for s roly w v [ ( k () ) u( k ( ) ) ] ( k(), d () ) u( k( ) ) [ ] λ c c c c c c c c c c, ( k (), d () ) ( k () d () ) u( k ) [ ( k (), d () ) ( k () ) ( k (), d () )] u( k ) [ ] (), [ ( k ( ) ) ( k ( ) d ( ) )] u( k ( ) )} ] o( ),. Cgs volu of xcd rquss sys S : V (, ). W roly k (), d () R r cs of 7, r r cs of 8, R r cs of 9, r r cs of. u( k () ), ur of srvc ls sys S wll crs y, d ol volu of rquss sys S y V(, ) r,,,,. 2. W roly ( k(), d() ) u( k() ) o l of srvc s g rrd sys S, ol volu of lcos wll dcrs y V(, ) g r, wr E { g }, g - RV w dsruo fuco H ( x) - rr f for srvc l sys S,,. I ddo, followg css r ossl. 3. Tr wll rr of srvc l sys, rqus fr srvcg sys S wll go o xrl vro;,,,. c

5 Alco of HM-works w urll syss for fdg ory ccy T srvc l wll rrd sys S, sys S wll go o ousd vro;,. 5. T srvc l wll rrd sys S, sys S rs rqus fro ousd vro;,,,. 6. Tr wll rr of srvc l sys S, d s sys s rg rqus fro ousd vro,,. 7. Tr wll rr of srvc l sys S s, rqus fr srvcg sys S wll go o sys S,,, s,, s,. 8. T sys S wll rr srvc l, rqus fr srvcg sys gos o S,,,,. 9. Tr wll rr of srvc l sys S s, rqus fr srvcg sys S gos o S,,, s,, s,. 2. Tr wll rr of srvc l sys S, rqus fr srvcg sys S gos o S,,,,. 2. T sys S c wll rr srvc l, sys wll dgd d sus of or QS wll o cg,, c, s,, c. 22. T srvc l wll rrd sys S, srvc l wll dgd sys S s d wll o cg ss of or QS,, s,. T rols of vs kg lc 3-22 css wll qul ( ) cus, for xl, cs 3 w wll v [ ( k (), d () ) u( k () ) ] ( k(), d () ) u( k() ) [ ] λ q q q q q q q q q, ( k (), d ( ) ) ( k ( ) ) u( k ) [ ( k (), d () ) ( k () ) ( k (), d () )] u k () [ ] () [ ( k ( ) ) ( k ( ) ) d ( ) ] u( k ( ) )} ] o( ),. q q o, Cgs xcd volu of rquss sys S css 3-22 r s follows:

6 56 M. Mlysk, P. Zc, D. Kos V (, ) R r, cs of 3, g R r cs of 4, r r, cs of 5, g r r cs of 6, R r s,, cs of 7, g R r, cs of 8, r r s,, cs of 9, g r r, cs of 2, r c, cs of 2, g r cs of 22. T followg css r ossl: 23. T srvc l wll rrd sys S c, srvc l wll dgd sys S s, wll o cg ss of or QS, d fr oro sys wll go o ousd vro,, c, s,, c. 24. T srvc l wll rrd sys S, srvc l wll dgd sys S s, rqus fr srvcg sys gos o ousd vro,, s,. 25. T srvc l wll rrd sys S c, srvc l wll dgd sys S s, rqus gos o S sys fro ousd vro,, c, s,, c. 26. T srvc l wll rrd sys S, srvc l wll dgd sys S s, rqus gos o S sys fro ousd vros,, s,. 27. T srvc l wll rrd sys S c, srvc l wll dgd sys S s, rqus fr srvcg sys S gos o S,,, c, s,,, c. 28. T srvc l wll rrd sys S, srvc l wll dgd sys S s, rqus fr srvcg sys S s gos o S,,, s,,. 29. T srvc l wll rrd sys S c, srvc l wll dgd sys S s, rqus fr srvcg sys S gos o S,,, c, s,,, c. 3. T srvc l wll rrd sys S d srvc l wll dgd sys S s. T rqus fr srvcg sys gos o S,,, s,,.

7 Alco of HM-works w urll syss for fdg ory ccy 57 2 I ror css, roly of 23-3 vs r qul (( ) ). xl, cs 23, roly wll qul [ c ( kc(), dc() ) u( kc() ) ] [ ( k () ) u( k () ) ] ( k(), d () ) u( k() ) o( ) [ ] s s s o For λ [ q( kq(), dq() ) q( kq() ) q( kq(), dq() )] u( kq() ) q q, c, s ( ) [ ( k (), d () ) ( k () )] u( k () ) [ ( k (), d () ) ( k () d () )] u k () c c c c c c s s s s s, 2 [ ( k() ) ( k(), d() )] u( k() ) o( ) o( ) ). W wll ssu RV R, r, R, r, g, r dd,,,,. T cgs of volus of rquss sys S css 23-3 r qul o: V (, ) R r c, g R r r g R g r g r c, r r R r r r, c, r, c, r cs of 23, cs of 24, cs of 25, cs of 26, cs of 27,, cs of 28, cs of 29,, cs of 3. W cos xcuo of rocss k () w ( ), c wr E{ V (, ) / k( ) } c λ ( k ( ), d ( ) ) u( k ( ) ) ( k(), d( ) ) u( k( ) ) ( k(), d() ) u( k() ),,. V s s

8 58 M. Mlysk, P. Zc, D. Kos k Avrgg fr k (), kg o ccou codo of orlzo P ( k() k) k k K P, w wll o cg xcd volu of sys S { V (, ) } P( k( ) k) E{ V (, ) k( ) } k E / ( k( ) ( k ( ), K, k ( ) )) E{ V (, ) / k( ) ( k ( ), K, k ( ) )} c λ P k ( k( ) k) k () d (), u( k () ) ( k(), d () ) u( k() ) P( k() k) k P( k() k) ( k(), d() ) u( k() ),,. k L sys S v d (), d(). I c of, srvcg, work d rr of dgd ls v xol dsruo w rrs,, d rscvly,,. Assu sy of rqus srvcg d sy of rr of dgd ls sys S dds lrly o ur of rquss d ur of dgd ls. (), k( ) d(), (), k () > d (), k ( k( ), d() ) u( k() ) ( k(), d( ) ), d ( k () d () ) u( k () ) ( d () ),. W wll lso ssu vrgg of xrsso of ( k(), d() ) u( k() ) s ( (), d() ),.. E{ ( k ( ), d ( ) )} ( ( ), d ( ) ). I ddo, E{ d ( ) } d ( ), d () rscvly, wr () ur of rquss (wg d srvcg) d vrg ur of ls org sys S,,. Gv s ssuos, w o rox

9 Alco of HM-works w urll syss for fdg ory ccy 59 E { V (, ) } c λ ( ) d ( ) (, ) ( (), d() ) ( d() ). (2) Bcus org s d rr ls sys S v xol dsruo, d () - vrg ur of srvc ls rrd ovr d () sys S, - vrg ur of srvc ls sys S dgd ovr,, d d d d,. Trfor () () dd(),, wr, w g lr dffrl quo d,, d w l codo d ( ) w v: () ( d() ) d( ), ( ) d(),,. (3) T quo for vrg ur of rquss QS work s for d d () () ρ() λ ρ,,. (4) I s ovd s wy s [4] for works w or fur. T vrg ur of occud srvc ls ρ () sys S c, d, roxd y xrsso ρ () ( (), d() ). d(), () > d(), T sys of quos (4) s for d d () () () () ( (), d () ) (), d () ( ) λ,,, (5) Ts s sys of rgulr lr dffrl quos w dscr rg sds, wc c solvd y dvdg s scs o ur of rs d fdg soluos c of.

10 6 M. Mlysk, P. Zc, D. Kos Fro (), (2) follows v ( ) v ( ) E{ V ( ) }, v() c λ ( (), d () ) ( (), d() ) ( d() ). W, w v rogous frs ordr lr quos dv d () ( (), d () ) (), d () ( d() ) λ c,,. (6) By sg u l codos v ( ) v,,, you c fd xcd volu of work syss. If work s fucog suc wy o vrg r r o quus, d,,, foruls (5) (6) wll v for: (),. () () () d d () dv d v( ) v,, () () λ () () d (),,, (7) 3. Excd volus of rquss crl work sys λ c, L s cosdr closd work w crl QS. L rrl works, d,,, d do o v quus o vrg, () () crl sys ors g lod codos, (), d () T sys (5) s cs c rwr s (8) () d ().

11 Alco of HM-works w urll syss for fdg ory ccy 6 () () () () () ().,,, d d d d d d (9) T grl soluo of sys (9) vg (3) l codos,,,, s qul (),,, () [ ] K, wr,,, () K - ur of rquss work. For os (),,, sys (8) for xcd volu of rquss work syss ks for () [ ] d dv [ ] K c,,, () [ ] K d dv [ ] c. By grg s syss w l codos, v v,,, w v

12 M. Mlysk, P. Zc, D. Kos 62 () v 2 c K 2 v,,, () () v 2 c K 2 v. () 4. Coclusos T rcl rcvd rox xrssos for fdg xcd rvus HM-work syss. Ts syss c usd w fdg xcd ory sc foro syss. Furr rsrc s drco y rld o lyss of rrry (o-mrkov) rdo volu rgsros d Mrkov cs lcos w or culrs, suc s ld quu d fuly srvc syss.

13 Rfrcs Alco of HM-works w urll syss for fdg ory ccy 63 [] Tkoko, O. (26). Mody rsolsycz lzy sysów forcyyc. Akdck Ofcy Wydwcz EXIT. [2] Mlysk, M., Tkoko, O., & Koluzv, E. (2). Syss d quug works: lyss d lco [I Russ: Ssy s ssovogo osluzv: lz r]. GrSU. [3] Buffrlo [Elcroc rsourc]: Wkd: ://.wkd.org/wk/buffrlo. [4] Mlysk, M., & Skvc, S. (24). Socsc work w oudd wg of cls d urll srvc. [I Russ: Soscsk s s ogrcy vr ozd zvok dozy osluzv]. GrSU. [5] Mlysk, M. (25). Alyss d forcsg of xcd cos Mrkov work w oudd wg for cls. Auoo d Ro Corol, 76(6), 5-7. [6] Mlysk, M., & uko V. (25). Zsosow HM-sc kolkowyc dl wyzcz oęośc ęc sysów forcyyc. Sud Iforc, 35(3), [7] Mlysk, M., & Zc, P. (26). Fdg of xcd volus of rqus syss of quug work w ld ou of scs of xcos. Sud Iforc, 36(), [8] Mlysk, M., & Zc, P. (27). Excd volus of rquss syss of quug work w ld ur of wg lcs. Scfc Issus J Długosz Uvrsy Częsocow Mcs XXII, 4-6. [9] Mlysk M., & Zc, P. (27). Alco of HM-works w cls fdg ory ccy foro syss. Jourl of Ald Mcs d Couol Mccs, 6(3),