Some queue models with different service rates. Július REBO, Žilinská univerzita, DP Prievidza

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1 Some queue models wth dfferet servce rates Júlus REBO, Žlsá uverzta, DP Prevdza Itroducto: They are well ow models the queue theory Kedall s classfcato deoted as M/M//N wth equal rates of servce each servce ot The frst M descrbes a arrval assumto of requremet for servce ad follows the Posso dstrbuto wth a fte rate λ > Each servce ot of dsoses of a exoetal dstrbuted servce tme wth robablty desty fucto f ( t t e, > ( the secod M A teger N deotes a umber of commo laces reserved for requremets the queue ( arallel servce ots for ther servce ad a watg le ( N laces I the ext arts we shall cosder that umber N ca tae coutable value ( ether fte or fte Queues wth dfferet rates of servce are studed as secal cases of the models wth oe servce ot ad a value of rate usually deeds o some actvty dcators of the systems I ths aer we shall be terested a geeral method to derve characterstcs of those systems whe a assumto of the dfferet rates reresets a soltary roerty of system We shall aly above assumtos detal for a closed model mared M/M//m ad accomlshed results we shall use for a geeralsato of two models M/M//N ad M/M// As a commo foudato of solvg method for cosdered models s a alcato of a geeral brth-death rocess Geeral brth-death rocess: For the geeral brth-death rocess, eg [], we assume that rate of chages betwee states deeds o the state of system The states of system are usually meat as a umber of requremet the system ad robabltes of those trastos durg tme dt we defe the as: from to : λ dt,,,,, N, from to -: dt,,, 3,, N from to : - ( λ dt, whe λ N,,,,, N ( We ca exress a state robablty P{ S } accordg to [] the form λ λ λ ad N λ λ λ,,,,, N ( 3 Rates of servce: Let us have a queue wth servce ots ad a umber of laces the queue s a ostve teger N whch ca tae a fte or fte coutable value The umber of

2 Some queue models wth dfferet servce rates 45 requremets for servce has the Posso dstrbuto wth fte rate λ > ad a ser-vce each ser- t vce ot of has a exoetal desty fucto f ( t e Thus, f the state S s equal to, N, the a total rate of servce wth rates >,,,, that state s ot deedet o the uderstood state ad s defed by the sum of servce rates over each occued servce ot, so we have It s clear to see that for states S equal to, <, the total rate of servce deeds o the umber of occued servce ots ad also o ther ow servce rates Otherwse sad, the rate of servce for the state S s gve by a combato of servce ots from wthout ther reeatg Those combatos for every state {,,, } s ad each rate belogs to each other exactly tmesif we are assumg that etered re- quremet s served wth equal robablty ( ( ( for each combato C (,, of servce rates state S, we shall defe the rate of the servce the state S as whe ( ( ( ( ( ( C (,, C,, ( (? ˆ (, ( 3 Cosder ow, every combato of servce rates ( C (,, the state S wll oc- (, for,, {,,, ( } Set ( ( s whe cur wth a robablty ( so that ( we are summg over all dexes of robabltes ( corresodg to combatos ob-tag a servce rate The robablty ( s dcates a total robablty for a aearace of the rate the gve combatos of rates For a total servce rate the state S the we have (,, ( ( C ( ( ( s ( 3 4 Closed model M/M//m: A closed model of queue wth watg le s called a mo-del wth a fte source of requremets for servce It reflects real systems whch get a essetal weght wth alcatos The frst of all s a roblem of the servce of the several mache A rearma ( or team of rearme teds m ( < m of maches The maches are rug utl they dro out A role of rearma ( rearme s to elmate those accdets

3 Some queue models wth dfferet servce rates 46 We assume that umber of requremets follows the Posso dstrbuto wth arameter λ > ad t each servce ot has a exoecal desty fucto f ( t e of servce tme wth >, for,,, States of system {,,, m} mea that exactly maches uderle servce or they are watg le Always m - maches occur out of the system ad they rereset actve maches Itesty of the requremets for servce s comarable to the umber of maches out of the system ad t ca exress as (m - λ The total rates of servce the state S reresets exactly occued servce ots ad we ca deote t as, for If s > the the rate of beg servce s deoted Accordg to ( ad a revous aalyss trasto robabltes durg tme dt wll be: from to : λ dt ( m λdt, m -, from to -: dt,, from to -: dt, m, from to : - ( λ dt, whe λ, m m I geeral there s ow a soluto of the revous roblem eg [], [], f all servce rates are equal to We shall call that model as a basc model ad f we deote m m! states hold: ψ, <, ψ, m,! λ ψ the ts robabltes of m m m! ψ ψ ( 4! It s clear to see that the above model s the secfc brth-death rocess By meas of that mo-del we ca exress robablty of state S for a closed queue model wth dfferet rates accordg to ( as m! λ!,, ( m m! λ ( m! (, m m, so that Probabltes of states wll be ext arraged f we deose ˆ from ( 3 ad λ We shall ˆ ( c m! λ ( c m λ ( c m ( c get! ˆ ψ ˆ,, ( m! ( ( c m! λ ( c m! λ ( c m!! ( c! ˆ ( m! ( (,!

4 Some queue models wth dfferet servce rates 47 for m, so as ( c exress wth codto m ( c ( 4 5 Oe model M/M//N: O the other had f we shall assume a fte source of requremet to the queues they shall be called oe models the Cosder ow so-called oe system M/M//N wth arallel servce ots ad watg le whch our revous assumtos stad Thus we assume dfferet servce rates wth a exoetal desty fucto accor-dg to d ad 3rd chaters Let λ > be the testy of comg stream of requremet to the queue A umber of laces queue taes a gve value ad t s deoted as N If we cosder the state of system as a umber of requremets the system we ca exress trasto robabltes durg dt as: from to : λ dt λdt, N, from to -: dt,, from to -: dt, N, from to : - ( λ dt, N, λ N Substtutg corresodg rates to the ( ad ext modfyg we shall get roba-bltes for a queue wth a fte watg le ad dfferet servce rates the form: ( ˆ N ( N, for <, ψ, for N, ( 5!! ( N ( N subect to ( N N, where λ ad!! ˆ ˆ Usg a well-ow techque uder revous assumtos for dfferet rates of servce ad equal robablty of the servce we ca derve for N state robabltes of the system M/M// So we have! ( ( N ( lm N ( ( ˆ N ψ (, <, lm N!,, ( 5 ad ( ( s determed by a codto!!! (! ( λ λ If we relace wth ψ ˆ ( 5 ad ( 5, for, we shall get the ˆ models called aga basc models For a queue M/M// we eed to remd also a codto for ts stablty treatmet I the basc model that codto has a form λ < It dcates that a total caacty of servce has exceed a comg rateλ to the queue I the case we reaso dfferet servce rates we shall get a aalogous request for a comg rate ad servce rates λ <

5 Some queue models wth dfferet servce rates 48 6 Urelable servce ots: We ca tae advatage of a revous techque from 3rd chater used for dfferet servces rates for solvg queues wth urelable servce ots Ve-cetel troduces [3] a very smle aroach to the queue model wth urelable servce ots whch cossts a correcto of the model arameters for a sgle servce ot We mea a queue wth a gve robablty of the successful servce whch dcates a robablty to termate successfully of servce for attedat requremet the queue Moreover, a robablty dcates a robablty of falure servce for attedat requremet the queue Let deote a rate of the servce for the basc model each servce ot Let ext r exress a robablty of the successful servce the th servce ot ad - r s the robablty of the falure of the servce The r yelds the rate of the successful servce for th servce ot Assumg equal robabltes of access to a arb- ( trary servce ot, {,,, } we shall get models wth dfferet servce rates from d ad 3rd chaters We derved robabltes of states formulas ( 4, ( 5, ( 5 for the meat queues where s r r ˆ Thus the queue model wth dfferet rates of servce seems to be a secal case of the basc model wth urelable servce ots We ca cosder that those dfferet rates of servce rereset a fudametal codtos for fuctog of the queue Cosder that a effcecy of every ot ad ther relablty are dfferet That ot of vew leads to a very geeral model wth the dfferet rates for the urelable servce ots Its solvg method s assemblg both revous methods Let a every servce ot oerate wth the servce rate υ ad corresodg robablty of the successful servce wll be r The r υ exresses the rate of the successful servce for th servce ot Uder codto of the equal robablty of the access to servce ( 3, we get aga a model wth the dfferet servce rates wth robabltes of the states ( 4, ( 5, ( 5, settg r ˆ υ r υ r υ 7 Otmal rate of servce: Let us loo at a roblem of otmsg rate of servce for the queue M/M// The we ca use a submtted techque for the other uderstood models wth the dfferet servce rates after mmal modfcatos t Let us have a queue wth servce ots wth a exoetal desty fucto f ( t e of

6 Some queue models wth dfferet servce rates 49 the servce tme, wth >, Let c be average servce costs er tme ut ad let c be average store costs er tme ut both reduced er oe requremet Next let d S (,ˆ be the meavalue of requremets the queue deeded o the umber of the servce ots ad the average rate of servce ˆ from ( 3 By course of that we have doe ( (, ˆ d S ( ˆ ( ψ, whe (! (! A total costs fucto C(, ˆ c ˆ c d (, ˆ S (! ( ( 7 cludes the servce costs wth the costs for watg le Moreover we assume that a umber of servce ots s gve ad the rate λ > of the Posso comg stream of requremet we shall tae as a costat value to the otmsg varable? Valuatos of the crtera fucto C (,ˆ ad (,ˆ varable ˆ ad t wll be deote C( ˆ c ˆ c d ( ˆ ( 7 S d S wll be deedet o the cotuous The we shall secfy a otmal average value of rate ˆ by dervatve of the crtera fucto C ( ˆ wth resect ˆ We shall comute a ecessary codto of the exstece of the mmum value settg the frst dervatve ( 7 equal to zero Thus we have ( ˆ d [ c ˆ c d ( ˆ ] dc S ( 73 dˆ dˆ Emloymet of geeral form of dervatve ( 73 leads to equatos of hgh orders whose soluto s ossble oly wth comutatoal aroach ad t does ot let us aalyse the soluto cosderato of the costs Let? deote a otmal average rate of servce from ( 73 We have aother roblem how to otmse a otmal rate? for every servce ot Deote h as a dfferece of the o- tmal average rate? ad the average rate? If t s ˆ ˆ > the h h ad f t s ˆ ˆ < the h h Next deoteϕ,,, The otmal average servce rate we ca exress as ˆ ˆ h After reform we have doe? h h ( ϕ h ( ϕ h h So, to obta a otmal average servce rate? for the th servce ot t s eeded to revse

7 Some queue models wth dfferet servce rates 5 tal rates accordg to h ϕ h ϕ ( ˆ ˆ 8 Coclusos: From the revous results t follows that basc models of the queue wth equal servce rates ad the queues wth the dfferet rates have a more smle relatosh tha we would exect Establshg the dfferet rates of the servce eables a geeralsato of the soluto also for the queues wth the urelable servce ots whch frame a temoral mo-del betwee the basc model ad the model wth the dfferet servce rates The above descr-bed method otmsg a average rate of servce also allows us to use t for otmsg all of the revous models of the queue wth a resect to ther a costs So we have a sold commo techque for solvg ad otmsg the whole class of the queue models Acowledgemet The solved roblem s a art of the research roect suorted by the Scetfc grat Agecy of Mstry of Educato of the Slova Reublc ad the Slova Academy of Sceces uder grat No /7/ Refereces [] Gross, D, Harrs, C, M: Fudametals of Queueg Theory, Wley, New Yor 985, 998 [] Saaty, T, L: Elemets Of Queueg Theory ad Its Alcatos, Wley, New Yor, 963 [3] Vecetel, E, S: Issledovae oerac, Sovetsoe rado, Mosva 97 RNDr REBO JÚLIUS, ŽILINSKÁ UNIVERZITA V ŽILINE, DP FAKULTY RIADENIA A INFORMATIKY, BAKALÁRSKA, 97 PRIEVIDZA e mal : rebo@utcds