MINIMIZING LOSS PROBABILITY IN QUEUING SYSTEMS WITH HETEROGENEOUS SERVERS *

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1 Iraia Joural o cic & Tchology, Trasactio A, Vol 3, No A Pritd i Th Islamic Rublic o Ira, 7 hiraz Uivrsity MINIMIZING LO PROBABILITY IN QUEUING YTEM WITH HETEROGENEOU ERVER * V AGLAM ** AND A HAHBAZOV Dartmt o tatistic, Faculty o Arts ad cics, Odouz Mayis Uivrsity, Kurulit, 5539-amsu, Tury, vsaglam@omudutr Abstract Th robability o losig a customr i M/G// ad GI/M// loss quuig systms with htrogous srvrs is miimizd Th irst systm uss a quu discili i which a customr who arrivs wh thr ar r srvrs chooss ay o o thm with qual robability, but is lost othrwis Providd that th sum o th srvrs rats ar ixd, loss robability i this systm attais miimum valu wh all th srvic rats ar qual Th scod systm uss quu discili, i which a customr who trs ito th systm is assigd to th srvr with th lowst umbr Loss robability i this systm tas th miimum valu i th cas wh th astst srvr rul is usd i which a icomig customr is srvd by th r srvr with th shortst ma srvic tim I th ma o th arrival distributio is ixd, th loss robability is miimizd by dtrmiistic arrival distributio Kywords rvic rat, Erlag s loss ormula, htrogous srvrs, loss robability, rcurrt iut, xotial srvr, ovrlow distributio INTRODUCTION I aalyzig may quuig modls, it is usually assumd that all th srvrs (chals i th quuig systm ar idtical (homogous i th ss that thy hav th sam srvic tim (st distributio Howvr, th srvrs o may ral-li systms ar dirt (htrogous uch a situatio aars wh srvrs o th sam mar wr mad at dirt actoris, or bor xloitatio i th systm thy wr usd i at dirt systms, ad thror hav a o-idtical dgr o warig out Th Quuig modls with htrogous srvrs also aris i a umbr o imortat alicatios such as comutr systms, commuicatios systms ad roductio lis Fudamtal loss quuig systms, M / G / / ad GI / M / /, with idtical srvrs, hav b studid almost comltly A vry imortat masur o ctivss or ths systms is th loss robability mat or th statioary robability o losig a customr or th robability that all srvrs o th systm ar busy tatioary robability i which srvrs o th systm M / G / / (Erlag s loss modl ar busy is giv by wll-ow Erlag s ormula ρ ρ!! (, ( whr ρ λ/ is th ord load, / λ is th ma itrarrival tim, ad / is ma T This ormula, irst drivd i [] or th cas wh all srvrs hav th sam xotial st, lays a imortat rol i aalyzig commuicatio systms ad its rortis hav b studid xtsivly ad xhaustivly O Rcivd by th ditor March 7, 5 ad i ial rvisd orm Aril 8, 7 Corrsodig author

2 V aglam / A hahbazov o th surrisig rortis o ormula ( is that th limitig distributio o th umbr o busy srvrs is ivariat to th st distributio G, i, limitig robabilitis,, ar iddt o th orm G ddig o G oly through its ma This rmarabl rsult ad may coctd qustios hav b studid by svral authors For xaml, th validity o ( or absolut cotiuous st distributio has b rovd i [] A xact mathmatical roo o this ormula or arbitrary st distributio with a iit ma has b giv i [3] Erlag s ormula or systm GI / M / / with rcurrt iut ad with th idtical xotial srvrs has b obtaid i [4] I [5] Erlag s ormula has b xtdd to th cas o ddt srvic tims I [6-8], Erlag s loss modl has b studid usig discrt-tim rocss at arrival ad dartur ochs From ormula ( w ca id th loss robability i th systm M / G / / Iraia Joural o cic & Tchology, Tras A, Volum 3, Numbr A rig 7 (/ ρ! (! ( This ormula is calld Erlag s loss ormula ad is xrssd i trms o th ma st ad th ma itrarrival tim I this ar w cosidr th roblm o miimizig loss robability i M / G / / ad GI / M / / quuig modls with dirt srvrs W dot th loss systm with rcurrt iut by GI / G / /, ad with st distributio G at th srvr, whr G ( G,, G symbolizs th htrogity o th srvrs I th cas wh G G or,,, w hav a M / G / / systm with idtical srvrs which w shall call homogous No-homogous quuig systms hav b studid maily or xotial srvrs I [9, ] a limitid distributio o th umbr o customrs i th systm M / M / hav b oud with htrogous xotial srvrs ad a uboudd waitig room I [] th roblm o miimizatio o th loss robability i th M / M / / systm has b solvd, rovidd th sum o th srvic rats (total srvic rat is ixd ad th arrivig customr is assigd to th r srvr with th shortst ma st Most o th o-homogous quus hav b aalyzd or two-srvr cass I [], xlicit xrssios or th stady-stat robabilitis or a M / G / / quu with two classs o Poisso arrivals has b drivd Th rsult o ths authors is or th cas o a arbitrary umbr o arrival classs i [3] Th quuig modls with two o-idtical xotial srvrs hav b aalyzd i [4-7], whr a w quu discili has b itroducd i which a customr who arrivs wh both srvrs ar r, chooss his srvr with som robability Th systms M / G / / ad M / G / with this discili hav b ivstigatd i [8, 9] Th systm GI / M / with rcurrt iut ad a srvic rat ddig o th umbr o busy srvrs was studid i [] LO PROBABILITY IN THE MODEL M / G / / Cosidr th loss quuig systm M / G / / cosistig o htrogous srvrs labld by umbrs,,, Th arrival rocss is Poisso with rat λ, ad th st o ay customr at th th srvr has a distributio uctio G with iit ma or,, For this systm th ollowig discili is usd: A arrivig customr chooss ay o o th r srvrs with qual robability ad is lost i all srvrs ar busy Lt X ( t i th srvr is busy at tim t, ad X ( t othrwis Th limitig robability that srvrs with umbrs i,,i ar busy ca b writt as whr (, ( i, i lim P X ( t,, X ( t, X ( t,, X ( t { }, i i + i i t, (3 i,i is a rmutatio o (,, Not that X ( t X ( t + + X ( t is th umbr o

3 Miimizig loss robability i busy srvrs at tim t Lt dot th limitig robability that srvrs ar busy, i, { X ( t }, lim P t I articular, is th loss robability i th M / G / / quu with dirt srvrs I this sctio w cosidr th roblm o miimizig th loss robability subjct to ++, whr is costat Th solutio o this roblm is basd o th xlicit xrssio or I [] it has b show that th limitig robabilitis (3 xist ad ar giv by whr ρ ( i, i (! ρi ρi (, (4!, λ / Th limitig robabilitis,, ar giv by whr E ( ρ,, ρ as (! E / (! E ( E is th th lmtary symmtric uctio o th ρ,, ρ E, E ρ i ρi,, i < < i, (5, which is did whr th summatio xtds (ovr th C (, all combiatios o distict lmts { i,,i } rom{,,} Lttig E C( i (5, w obtai i th orm,!!, It is a gralizatio o Erlag s ormula ( to th htrogous srvrs cas I articular, rom (6 w coclud that th limitig distributio o th umbr o busy srvrs which ar iddt o th orm G,,G, dds oly thir ma valus,, I articular, i th srvic tims hav th sam ma /, th ρ ad robabilitis,, ar calculatd by Erlag s ormula ( From ormula (6 w ca id th loss robability i th quu M / G / / with dirt srvrs (6! ρ ρ! (7 It is a gralizatio o Erlag s loss ormula ( to th dirt srvrs cas Our mai rsult about th roblm o miimizig th loss robability ca b xrssd by th ollowig thorm Thorm I sum srvic rats ++ is ixd, th loss robability i th systm M / G / / attais its miimum valu or Proo: Rwrit (7 i a way that is mor covit or aalysis Usig th rlatio or ositiv umbr rig 7 a,, a ( a,, a a a ( a,, a ad sttig ( / ρ,, / ρ, w ca writ (7 i orm Iraia Joural o cic & Tchology, Tras A, Volum 3, Numbr A

4 V aglam / A hahbazov! (8! ( Taig accout o th iquality [] w hav From this ad ormula (8 w obtai / /( /, / ( ( / λ + + / λ λ! ( λ (! (9 For th cas wh th loss robability tas th valu that is qual to th xrssio i th right sid o (9, so that tas miimum valu wh Not that th xrssio o th right sid o (9 is th loss robability with ( i th M / G / / systm with homogous srvrs W coclud that a homogous systm is bttr tha th corrsodig htrogous systm, rovidd that th total srvic rat is ixd 3 LO PROBABILITY IN THE MODEL GI / M / / WITH ORDERED ENTRY W cosidr th loss quuig systm cosistig o htrogous xotial srvrs labld by,, ad arragd i sris i that ordr Th ollowig quu discili is usd: Each arrivig customr is srvd by th lowst umbrd srvr that is r Th customr iitially arrivs at th th srvr I this srvr is r, h is srvd ad darts I this srvr is busy, h ovrlows ad arrivs at th d srvr ad so orth Th outut stram rom th th srvr is th iut stram to th (+th srvr, or,, Fially, th customr who ids all srvrs busy is lost rom th systm Th ovrlow rocss rom th th srvr is th sam as that rom th loss systm GI / M / / Itrarrival tims to th systm ar iddt radom variabls ad hav distributio uctio F with ma λ Th st o ay customr at th th srvr is xotial with aramtr or,, uos that F ad,, ar ixd Th loss robability has a varyig valu ddig o th ordr o th srvrs How dos th ordr o th srvrs miimiz? Th solutio o this roblm is basd o th xlicit xrssio or uch a xlicit ormula i th cas that srvrs ar idtical is giv i [4, 3] I [4, 5], th gratig uctio o th irst assag tim rom ay stat to a ull stat i th loss systm with idtical xotial chals has b oud ad hav show that this tim is iddt o th lacmt o call olicy usd Lalac-tiltjs (L trasorm o th itrovrlow tims rom a GI / M // K quuig systm was drivd i [6] It was show i [4] that loss robability i th homogous systm GI / M / / is giv by c, ( whr is th L trasorm o th itrarrival tim ad Iraia Joural o cic & Tchology, Tras A, Volum 3, Numbr A rig 7

5 Miimizig loss robability i 3 ( ( c, c, ( ( ( s dots th L trasorm o th itrovrlow tims distributio rom th irst srvrs or,, Th ( s satisy Palm s dirc quatios ( s ( s + ( s + ( s + (, whr ( s ( s is th L trasorm o F This is th xtsio o Palm s quatio to th htrogous srvrs cas From this quatio w ca id th ma o ovrlow tim rom th irst srvrs ( a ( s ( lim a / ( s s From this rcurrt quatio w obtai a imortat ormula or th ma o th ovrlow tims rom quu GI / M / / a λ ( ( ( Usig this siml rlatioshi a λ, w obtai th loss robability i th htrogous systm GI / M / / ( ( ( ( I articular, rom ( ad (, w ca id th loss robability i quus GI / M // GI / M / / ad (, (3 ( ( + (4 ( + ( + I th cas that th last ormula yilds Palm s loss ormula ( with Thorm Loss robability i th quu GI / M / / systm achivs its miimum valu wh th srvrs ar oratd i th ordr ( i,, i, a arbitrary rmutatio o (,, or which i i i W shall rov th thorm by a itrchag argumt Lt (i dot th valu o th wh th srvrs ar oratd i ordr i ( i,, i Without loss o grality, assum that th otimal ordr o th srvrs is giv by a squc α (,, Itrchagig ad i this squc givs a w squc β (,,,,, +,, which will ot b otimal, i, Usig ( w hav ( α ( β (5 * β ( ( ( ( ( (, (6 ( whr * ( is th L trasorm o th ovrlow distributio rom th irst srvrs with srvic rig 7 Iraia Joural o cic & Tchology, Tras A, Volum 3, Numbr A

6 4 V aglam / A hahbazov rats,, rsctivly This trasorm is obtaid rom Palm s quatio (:, * ( s + ( s (7 ( s + ( s + ubstitutig ( ad (6 ito (5 lads to th iquality ( * ( ( ( * Usig th xrssio o th ( ad ( rom ( ad (7 rsctivly, ad dotig by ϕ (, w hav ( + + ubtractig both sids o this iquality rom givs iquality whr + + b b, (8 ϕ ( b ϕ ( + + ( t t ( df ( t > ad F ( t dot ovrlow distributio rom th irst srvrs Dividig both sids o iquality (8 by actor b ad rormig siml algbra w obtai ( (, (9 ic ( s is o-icrasig, i s w gt,, i, which cocluds th roo Accordig to this thorm th loss robability i th quu G / M / / attais its miimum valu i srvrs ar oratd i th ordr o shortst ma srvic tim I othr words, i ordr to miimiz th loss robability w must us th astst-srvr rul i which ach arrivig customr is srvd by th r srvr with th shortst ma o srvic tim W ow cosidr th abov quuig systm with a giv ma itrarrival tim ad giv srvic rats,, Which itrarrival tim distributio miimizs th loss robability i this systm? This qustio has b solvd i [7] or th cas o idtical srvrs Lt H a dot th class o all itrarrival tim distributios, F havig a ixd ma a, ad (F b loss robability i th abov systm with itrarrival tim distributio F Lt H a, A ( t, t a t > a H a Clarly, A ad -as is th L trasorm o A(t Thorm 3 Loss robability ( F, F H a i th quu GI / M / / is miimizd by FA Iraia Joural o cic & Tchology, Tras A, Volum 3, Numbr A rig 7

7 Miimizig loss robability i 5 as Proo: Usig Js s iquality w hav ( s or ay s, whr a '( is th ma o itrarrival tim Formula (4 or loss robability ( i th quu GI / M / / ca b writt as From Js s iquality w ca id iquality F ( ( F ( ( + ( + ( ( + a( a + Usig this iquality i ( w hav + a a a( + ( F a a a( + a( + as ic th L trasorm o A (t is, w s that th xrssio o th right sid o th last iquality has th valu o ( F or F A Cosqutly, loss robability ( F achivs its miimum valu or F A : + mi ( F ( F H a A REFERENCE Erlag, A K (97 olutio o som robability roblms o sigiicac or automatic tlho xchags Eltrotir, 3, 5-3 Fortt, R M (956 Radom distributio with a alicatio to tlho girig, Proc Brly ymos, Math tat ad Prob (8-88 Los Agls, Brly 3 vastyaov, B A (957 A rgodic thorm or Marov rocsss ad its alicatio to tlho systms with rusals Thor Prob Al (i Russio,, 9-4 Palm, C (943 Itsitätschwaug Frsrchvrhr Ericsso Tchics, 44, Koig, D & Matths, K (963 Wrallgmihrug dr rlagsch ormlu Math Nachr, 6, Taacs, L (969 O Erlag s ormula A Math tat, 4, habhag, D N & Tambouratzis D G (973 Erlag s ormula ad som rsults o th dartur rocss or a loss systm J Al Prob,, Brumll, L (978 A Gralizatio o Erlag s loss systm to stat ddt arrival ad srvic rats Math Orat Rs, 3, -6 9 Gumbl, M (96 Waitig lis with htrogous srvrs Os Rs, 8, 54-5 Blac, J P C (987 A Not o waitig tims i systms with quus i aralll J A Prob, 4, Nath, G B & Es, E B (98 Otimal srvic rats i th multisrvr loss systm with htrogous srvrs J A Prob, 8, Chai, J M & Igall, E (97 A Extsio o Erlag s ormulas which distiguishs idividual srvrs J Al Prob, 9, Wol, R W ad Wrightsoo, C W (976 A xtsio o Erlag s loss ormula J Al Prob, 3, 68- rig 7 Iraia Joural o cic & Tchology, Tras A, Volum 3, Numbr A

8 6 V aglam / A hahbazov 63 4 Mors, P M (958 Quus, Ivtoris ad Maitac Nw Yor, Wily 5 aaty, T L (96 Tim ddd solutio o may-srvr Poisso quu Os Rs, 8, igh, V P & Prasad, J (976 A htrogous systm with iit waitig sac J Egi Math,, igh, V P (97 Two-srvr Marovia quus with balig: Htrogous ad homogous srvrs Os Rs, 8, hahbazov, A A (983 Miimizig o loss robability i quuig systm with two htrogous srvrs Izv Acad ci Azrb R, r Phiz Tch ad Mat ci (i Russio,, Kssl, C, Matowsy, J, chuss, Z & Tir, C (99 O itgral quatio aroach to th M/G/ quu Os Rs, 38, Bhat, U N (966 Th quu G/M/ with srvic rat ddig o th umbr o busy rvrs A Is tat Math, 8, - hahbazov, A A (984 Loss quu with dirt srvrs Zasthosaw Mat, 8, Hardy, G H, Littlwood, J E & Polya, G (954 Iqualitis Lodo, Cambridg Uivrsity Prss 3 Taacs, L (957 O a robability cocrig tlho traic, Acta Mat Acad ci Hug, 7, mith, D R (968 Otimal rair o a sris systm Os Rs, 6, Aatharam, V, Goiath, B & Hajla, D J (988 A gralizatio o th Erlag ormula o traic girig quuig systm Quuig ystms, 3, Cilar, E & Isy, R L (967 trams o ovrlows rom a iit quu, Os Rs, 5, Bs, V E (959 O trus with gativ xotial holdig tims srvig a rwal rocss Bll systm Tch J, 38, -58 Iraia Joural o cic & Tchology, Tras A, Volum 3, Numbr A rig 7

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