ERLANG C FORMULA AND ITS USE IN THE CALL CENTERS

Transcription

1 IFORTIO D OUITIO TEHOLOGIES D SERVIES, VOL. 9, O., RH 2 7 ERLG FORUL D ITS USE I THE LL ETERS Er HROY., Tbor ISUTH., atj KVKY. Dpartmnt of Tlcommuncatons, Faculty of Elctrcal Engnrng and Informaton Tchnology, Slova Unvrsty of Tchnology, Ilovčova 3, 82 9 Bratslava, Slova Rpublc [email protected], [email protected], [email protected] bstract. Ths papr dals wth calculaton of mportant paramtrs of th all ntr usng th Erlang formula and thn rsults ar vrfd through smulatons. Erlang formula s dfnd as a functon of two varabls: th numbr of agnts and th load. On th bas of thr valus t s possbl to dtrmn th probablty, that th ncomng call wll not b srvd mmdatly, but t wll hav to wat n th watng quu. Smulatons satsfy th assumpton of arov modls. Kywords all ntr, Erlang Formula, arov odls, ualty of Srvc.. Introducton all ntr s dynamcal, tchncal systm (pacag of tchncal qupmnts hardwar, softwar and human sourcs) dsgnd for ffctv connctng popl wth th rqurmnts for srvc wth oprator or wth systms abl to satsfy thr rqurmnts. Th cor of th all ntr s utomatc all Dstrbuton (D). Each of th componnts of th D can b dscrbd wth som prcson by mans of mathmatcal tools and causalts. Snc th D systms procss a larg numbr of ncomng rqusts, th majorty of modls s basd on th prncpls of mathmatcal statstcs. Th rght choc of a statstcal modl s abl to nsur th suffcnt accuracy of th rsults. It s ssntal to dscrb th dpndncy of nput varabls and paramtrs that can gratly affct th accuracy of th rsults. Th modlng of all ntr paramtrs s possbl through arov modls, but also through Erlang formulas. Ths papr dals wth calculaton of mportant paramtrs of th all ntr (whch affct propr opraton of such quung systm) usng th Erlang formula and thn rsults ar vrfd through smulatons. Ths smulatons satsfy th assumpton of arov modls.. Erlang formula and //m/ modl Immdat rjcton of call by rason of occupaton of all agnts (as xpctd n Erlang B formula) s n trms of provdd srvcs by all ntr napproprat soluton. Ths shortnss s lmnatd n scond Erlang formula Erlang. In th cas that th call cannot b srvd mmdatly, th call s placd nto th watng quu wth unlmtd lngth. If th rlas of on of th agnts happns, t s automatcally assgnd to th followng call from th quu. If th watng quu s mpty, th agnt s fr and h wats for nxt call. Erlang formula [5] s n th orgnal form dfnd as a functon of two varabls: th numbr of agnts and th load. On th bas of thr valus t s possbl to dtrmn th probablty (), that th ncomng call wll not b srvd mmdatly, but t wll hav to wat n th watng quu. whr,, (). (2) ow, w us th rlatonshp btwn load (2), th avrag numbr of calls pr tm λ and th avrag numbr of rqusts procssd pr tm μ. xt, w dfn th varabl η, whch rprsnts th load of agnt as [2, 3, 7]:. (3) By substtutng (2) and (3) nto quaton () w hav: 2 DVES I ELETRIL D ELETROI EGIEERIG ISS 84-39

2 8 IFORTIO D OUITIO TEHOLOGIES D SERVIES, VOL. 9, O., RH 2 2 DVES I ELETRIL D ELETROI EGIEERIG ISS 84-39,, (4) what corrspondng wth th rlaton for th probablty that th rqust n quung systm //m/ wll b placd nto th quu,.. n th systm s mor than m rqusts [2, 3, 7]. ow t s analytcally drvd, that th Erlang modl () and th arov modl //m/ (4) ar dntcal. It s possbl by usng of th basc form for th Erlang formula () to calculat th valu of paramtr (maxmum load) at a nown numbr of agnts and th probablty of watng. Du to complxty of th analytcal xprsson of ths unnown paramtrs, numrcal mthod for solvng s usd. By addng th watng quu nto th systm, many othr paramtr varabls that can b montord and also affctd wll appar. n mportant factor n trms of callr s quu watng tm. Ths valu s random varabl dscrbd by dstrbuton functon [7]:, F. (5) Thn t s possbl to calculat th avrag quu watng tm (avrag call watng tm n th quu bfor assgnng a call to agnt):. (6) and by applyng of Lttl thorm [7] and formula (2) w gt th avrag numbr of rqusts n watng quu as follows:. (7) By usng of gnral dfnton of dstrbuton functon of any statstcal dstrbuton [4] and by applyng ts proprts on th dstrbuton functon (5), w can drv th formula for calculaton of th paramtr GoS (Grad of Srvc) (prcntag of calls, that ar answrd, or assgnd to th agnt bfor th dfnd thrshold T ccptabl atng Tm) by nown valu of T [2, 6]: T GoS. (8) Th avrag numbr of rqusts n th systm K (and also th avrag numbr of occupd lns) s [7]: K, (9) whr w gt from Lttl thorm a valu of th avrag tm T, that th rqurmnt spnd n th systm: K T. () Othr mportant paramtr s th avrag utlzaton of agnts η [2]: n. () Erlang formula s th bass for th analyss of paramtrs and smulaton of th call cntr. Its shortnss s th assumpton of th unlmtd watng quu. Ths s not a problm n trms of avalabl storag capacty, and thrfor th watng quu could b potntally unlmtd, but no ral callr wll wat too long. Thrfor, th lmtaton of watng prod rprsnts othr paramtr that s undr consdraton n th spcal arov modls..2 //m/ modl wth lmtd lngth of watng quu lmtd numbr of rqurmnts placd n th quu of quung systm can b dscrbd by arov modl //m/, whr th maxmum numbr of rqusts n th systm s gratr or qual to th numbr of srvrs m= [2]. Th probablty p [2], that n th systm occurs xactly rqurmnts (.. mpty) s dfnd as: p (2) and thn w can dfn th probablty of th call rjcton B [2] as: p B (3) and th probablty, that th call wll b assgnd nto th watng quu [2] as: p. (4) Furthrmor, for th valus [2] and K [2] can b calculatd as p ) )( ( 2, (5) p K. (6)

3 IFORTIO D OUITIO TEHOLOGIES D SERVIES, VOL. 9, O., RH 2 9 Th tm charactrstcs can b calculatd on th bass of Lttl thorm [7]. Ths rlatons ar rlatvly complcatd. It s thrfor possbl a consdraton, f th approxmaton usng standard Erlang formula s not suffcnt, or what ar th condtons, that such approxmaton s suffcntly xact. If th avrag numbr of rqusts n th systm K wll b far lss than th lmt, t s possbl to apply Erlang formula wthout thnng of th maxmum capacty. Th mor closly wll b th valu K to th lmt, th lss accurat rsults Erlang formula wll provd. 2. Th prncpl of ralzd smulatons Th basc of th smulatd algorthm conssts of thr blocs: st up of nputs, traffc smulatons, procssng of masurd valus and thr prsntaton. 2. odlng of nputs Th basc paramtrs ar: avrag numbr of ncomng calls nto th call cntr pr tm λ, avrag tm of call procssng by agnt /μ, numbr of agnts. Th group of tm paramtrs conssts of: th total lngth of smulatd traffc n sconds T SI, tm stp of smulaton T STE (by dfault scond), tm to stady stat T B. Vctor wth arrval tms of ach call basd on th avrag numbr of rqusts pr tm unt and lngth of smulaton s cratd by random gnrator. Thr s usd a proprty about xponntal dstrbuton of lngth of th ntrvals btwn arrvals. Gnratd random varabls ar thn tstd by th statstcal ch-quadrat [4]. Thr s tst of vctor consstncy wth xponntal dstrbuton on sgnfcanc lvl α =,5. In th cas of unsuccssful tst, th ntr vctor s randomly gnratd onc agan. umbr of gnratd calls s by 2 % hghr than th avrag numbr of calls that should ntr nto th call cntr through th smulatd prod. Furthrmor, vctor wth srvc tms of ndvdual calls that wll b allocatd durng th smulaton, s gnratd for ach agnt basd on hs avrag srvc tm. ll ths opratons ar prformd n advanc by rason of hgh rat and ffcncy of TLB by worng wth vctors and matrcs. Th gnraton of st of valus s thn fastr than a gradual gnraton of ndvdual valus durng th smulaton. 2.2 Traffc smulaton Th cor of smulaton s ralzd as a cycl, n whch th ach traton rprsnts on tm stp n smulaton. In th ach traton s th vctor of ncomng calls compard wth actual tm and f t s ncssary, th call ntrs nto th smulatd call cntr n propr tmr. oncurrntly th tm of ncomng call s rcordd for latr calculatons. all s placd nto th watng quu or drctly assgnd to a fr agnt. Th tm ndx s stord for latr calculatons (avrag watng tm) at th momnt of call assgnmnt. Furthrmor, n th ach traton th status and th occupancy of all agnts ar chcd. If any of th agnts s fr, t wll b assgnd th frst call from th watng quu. If mor agnts ar fr, th call s assgnd to agnt that ddn t wor for th longst tm. If thr s not a call n th watng quu, th agnt rmans as fr and xpcts th arrval of anothr rqust nto th systm. Durng th on traton thr s possblty of ntry nto th systm and also assgnmnt to th agnt of mor than on call smultanously. Th numbr of calls n th systm s stord n ach stp so thn t s possbl to dtrmn th numbr of calls n th watng quu. Th mathmatcal modl of quung systm s dfnd for th stady stat. It mans, that n th momnt of paramtrs montorng, th smulaton runs nfntly long tm. Th run of th smulaton s trmnatd aftr prdfnd smulaton tm. Only th calls that ar trmnatd (and ar thus srvd by agnt) bfor xpraton of th smulatd tm ar ncludd nto th statstcs. 2.3 rocssng of th smulaton rsults Ths phas of th algorthm nsurs th procssng of all masurd data durng th smulaton (tm of vnts gnraton, numbr of calls n th systm, agnt occupancy, ). Basd on ths th partcular paramtrs montord n th call cntr ar calculatd. Thus obtand smulaton rsults can b thn compard wth th xpctd valus obtand by calculatons through th mathmatcal modl. Th gnral varabls avalabl from th rsults of th smulatd modl ar: ral lngth of th smulaton (calculaton tm), numbr of smulatd calls, avrag srvc tm pr on call (/μ), avrag numbr of calls n th systm (K), avrag tm spnt by usr n th systm (T), 2 DVES I ELETRIL D ELETROI EGIEERIG ISS 84-39

4 IFORTIO D OUITIO TEHOLOGIES D SERVIES, VOL. 9, O., RH 2 numbr calls placd n th watng quu, probablty of blocng B, rspctvly probablty of watng, avrag utlzaton of agnts η. 3. alculatons usng th Erlang modl Erlang modl wors n th basc form wth 3 nput paramtrs (,, ). By us of rlaton (2) t s possbl to dvd th valu of load (gnratd by ncomng calls) nto 2 componnts: λ and /μ. alculator thus always wors wth 4 valus, whl 3 of thm act as nput paramtrs and th last paramtr s th output paramtr. By addng of th watng quu w can obtan a st of nw paramtrs that can b calculatd and thn compard wth th smulaton rsults: avrag numbr of rqusts n th watng quu (usng (7)), avrag numbr of rqusts n th call cntr K (usng (9)), avrag watng tm n th watng quu (usng (6)), avrag tm spnt n th call cntr T (usng ()), valu of GoS for T=2 s (usng (8)), avrag utlzaton of agnts η (usng ()). robablty of nsrton nto watng quu can b ntrd n thr dffrnt varants: drct ntry of valu, ntry of avrag watng tm, ntry of GoS. hn usng a as nput varabl, ths varabl s consdrd to th uppr lmt. 3. alculaton of th paramtr c Th calculaton of th unnown valu of probablty of watng can b asly ralzd by basc rlaton for Erlang modl (). Th dvdng of two larg numbr could lad to numrcal rrors and th obtand rsult mght not b accurat. Thrfor w drvd an altrnatv rlaton, that s dntcal to th orgnal () (n trms of rsult):. (7), Smlarly, w can us Hornr schm [8] also for mor ffcnt calculatons. 3.2 alculaton of th paramtr nalytcal calculaton accordng to () rspctvly (7) would b vry dffcult. Th soluton s thrfor through a numrcal mthod. Th asst tchnqu s to gradually ncras of th numbr of agnts and contnuous chcng th stop condtons. Ths dpnds on th form, n whch th nput valu s nsrtd (thus drct, or as, or as GoS). t th momnt, whr th currnt valu n th calculaton s lss or qual to th rqurd valu, th ncssary numbr of agnts s found. Th mplmntaton uss ths da, but applyng of th rlaton (7) n vry stp of th calculaton t s possbl to us th currnt rsult obtand from th prvous traton cycl for th lowr valu of. So, ths s vry qucly mthod to fnd th unnown valu. 3.3 alculaton of th paramtr λ or /μ Th calculaton of on of ths unnown quantts n trms of Erlang modl mans th fnd out of th load, that can th systm procss at th spcfd paramtrs and (rspctvly and GoS paramtrs). onsquntly, w can calculat th scond on by applyng th formula (2) and on of th valu λ and /μ. In trms of mplmntaton, th fndng of th unnown valu s th most dffcult of all thr combnatons. s th soluton w can us th fatur, that allows to sarch th valu of unnown x, for whch t holds f(x) =. In ths cas, th functon f(x) for nput s: ( x, ) _. (8) IUT If th nput valu s dfnd as th avrag watng tm, rspctvly GoS, w can us th followng substtutons by (6) and ():, (9) GoS T. (2) In both cass w hav th valu μ. Thrfor, f w nd th soluton for th unnown valu / μ, w must fnd th soluton for th valu μ and only thn to calculat th load by (2). Th functons f(x) for nput (by us of substtuton (9)) ar: x ( x, ), (2) rspctvly, f μ s unnown, thn:,. 36 x x 36 x (22) If th nput paramtr s GoS, th substtuton (2) s usd and th functon for paramtr λ s: GoS x, x T, (23) rspctvly, f μ s unnown, thn: 2 DVES I ELETRIL D ELETROI EGIEERIG ISS 84-39

5 IFORTIO D OUITIO TEHOLOGIES D SERVIES, VOL. 9, O., RH 2, 36 x 4. Smulatons GoS x T 36x. (24) Th whol procss of smulaton for Erlang modl s shown n th followng fg. : mathmatcal modl and rsults also obtand by smulatons. Tab.. Th rsults of calculatons c GoS η K T [s] [%] [s] [%] [%] 28 95, ,6 27,2 686,6 7,2 99, ,3 45, 243,5 7,3 93,5 35,9 95,8 3 58,7 35,2 89,9 7,4 39,9 56,3 92,6 3 45, 3,7 7, 3,9 2, 7,6 89, , 3 62, 2,2 2, 8,6 86, ,3 29, 57,3,4 7,3 87,3 84,2 34 8,5 28,6 54,5,8 4,5 9,9 8,7 35 3,3 28,3 52,8,5 2,8 94,9 79,4 36 9,4 28, 5,7,3,7 96,8 77,2 37 6,5 28 5,,2, 98, 75, Tab. 2. Th rsults of Erlang modl smulatons Fg.. rocss of smulaton for Erlang modl 4. Smulaton rsults Th paramtrs of smulatd modl ar: 667 ncomng calls pr hour, avrag call srvc tm s 5 sconds, smulaton tm of th call cntr wor s 3 hours wth scond stp (run to stady stat s hour). Th calls that cannot b procssd mmdatly ar placd nto th watng quu (accordng to Erlang formula). For th systm stablty t s ncssary to satsfy th condton <. Thrfor th calculatons and smulatons ar ralzd for th numbr of agnts n th rang of 28 to 37. In ths rang ar th most notabl changs obsrvd n output paramtrs. Th tabl and tabl 2 shows th rsults obtand by calculaton usng th c GoS η K T [s] [%] [s] [%] [%] 28 9,9 88, , ,3 98, , ,4 83, ,6 3 59,3 35, 89 7,2 38,8 55, ,2 32,4 74 4,5 24, 69, , ,2,9 8,3 86, , 29,2 58,4 7,3 87,5 84,4 34 9,2 28,8 55,9 4,7 9,5 82, ,3 53,5 2,5 95,5 79,4 36 8,7 28 5,3,5 97,4 77, 37 6,4 28, 5,2 98,2 75,3 Obtand smulaton rsults and calculatons ar vry smlar. Th dffrncs xst only n th cas of th mnmum numbr of agnts. Howvr, t s probably that any company wll not carry on th call cntr wth xtrmly poor qualty of srvc [9,,, 2] dlvrng (manly th vry long watng tm). From th callr pont of vw, th most mportant paramtrs ar th avrag watng tm n th watng quu and paramtr GoS (ths cas s valuatd for T=2 sconds). Th callr xpcts th lowst valu of and also th valu GoS, whch s clos to %. From th call cntr pont of vw, th most mportant paramtrs ar th numbr of agnts and thr utlzaton η, bcaus ths two varabls sgnfcantly affct th fnancal dmands of srvc. Th am of th oprator s to mnmz th numbr of agnts and to maxmz thr utlzaton. Th am of th analyss s thrfor to fnd such a mnmum numbr of agnts, whn th opraton paramtrs ar yt on th suffcnt lvl. s sutabl w can consdr th GoS paramtr on lvl 8% and th avrag watng tm about sconds. 2 DVES I ELETRIL D ELETROI EGIEERIG ISS 84-39

6 2 IFORTIO D OUITIO TEHOLOGIES D SERVIES, VOL. 9, O., RH 2 Fg. 2. Smulaton rsults Th fg. 2 shows th charactrstc curv of th mportant call cntr smulaton rsults accordng to Erlang modl assumptons n rlaton wth th numbr of agnts. Th charactrstc curv of th avrag watng tm has a vry strong xponntal charactr and thus a slght ncras of th numbr of agnt (about to 2 agnts) can brng a sgnfcant mprovmnt of ths paramtr. smlar, vn though lss aggrssv, s th charactrstc curv of th probablty. GoS paramtr also xponntally convrgs to th lvl % and w can s that a small chang n th numbr of agnts can brng sgnfcant mprovmnt. Th charactrstc curv of agnts load η s n th dsplayd rang almost lnar. ccordng to th abov mntond rqurmnts on th provdd qualty of srvc by th call cntr s n ths cas possbl to consdr 32 agnts as suffcnt. By addng two agnts t s possbl to shortn th avrag watng tm by half and to ncras th valu of GoS paramtr at % on vry dcnt lvl (9%). Th utlzaton of agnts dos not dcras blow 8% and thrfor t dos not crat unncssarly long pauss, whn th agnts wr rdundant. 5. oncluson Basd on calculatons and smulatons t can b statd, that n trm of smplcty and accuracy of obtand rsults Erlang formula s applcabl for call cntr smulatons. Howvr, ts shortnss s th possblty of calculatons only for on srvc group, and also th nd to dfn for all agnts th sam srvc tm. Dspt ths lmtatons, t s possbl to us th basc Erlang formula also for calculatons for call cntr wth svral srvc groups. In ths cas whn t s possbl to dtrmn th probablty that calls ar routd to th ndvdual srvc groups, thn t s possbl to us th basc Erlang formula for calculatons for ach srvc group ndvdually. Th numbr of ncomng calls pr tm unt s th alquot porton of th total numbr of ncomng calls nto th call cntr. Th Erlang formula calculaton can b usd n th cas of dffrnt prformanc of agnts. It s possbl to dtrmn th valu of th avrag call procssng tm or avrag numbr of th procssd calls pr tm unt by dvdng th total numbr of procssd calls pr tm unt of all agnts and th total numbr of agnts n srvc group. For th purpos of furthr study t would b ntrstng to xpand th smulatons by mor ndpndnt srvc groups at th sam tm and th random dstrbuton of call btwn thm accordng to dfnd probablts. nothr ntrstng possblty could b dffrnt avrag srvc tm for ndvdual agnts. Fnally thr s possblty to smulat th mpact of unqual prformanc of agnts on th rsults of th whol call cntr. cnowldgmnts Ths wor s a part of rsarch actvts conductd at Slova Unvrsty of Tchnology Bratslava, Faculty of Elctrcal Engnrng and Informaton Tchnology, Dpartmnt of Tlcommuncatons, wthn th scop of th projcts VEG o. /565/9 odlng of traffc paramtrs n G tlcommuncaton ntwors and srvcs and ITS Support for Buldng of ntr of Excllnc for SRT tchnologs, systms and srvcs II. Rfrncs [] BERGEVI, R., YTT,. ontact cntrs for dumms. [onln]. Indanapols (I) (US): ly ublshng, 25, 8 p. ISB valabl from: < [2] UČOVSKÝ, L. Stochastcé modly opračnj analýzy. st d. Bratslava: LF, 98, 46 p. ISB [3] OLE, J., KRLUBÍKOVÁ, T. Stochastcé modly v tlomunácách. st d. Bratslava: Fond Jozfa urgaša pr tlomunác n.f. vo vydavatľstv FBER, 999, 28 p. ISB [4] RIEČOVÁ, Z. t al. umrcal mthods and mathmatcal statstcs. st d. Bratslava: LF, august p. ISB [5] Dagnostc Stratgs. Traffc odlng and Rsourc llocaton n all ntrs. dham, US: Dagnostc Stratgs, 23. valabl from: < f>. [6] KOOLE, G. all ntr athmatcs: scntfc mthod for undrstandng and mprovng contact cntrs. mstrdam: Vrj Unvrstt, 68 p. [ctd ]. valabl from: < [7] BOLH G., t al. uung twors and arov hans. 2 nd d. Hobon, w Jrsy, US: John ly, 26, 878 p. ISB [8] FORSYTHE, G. E., LOL,.., OLER,.B. omputr thods for athmatcal omputatons. Uppr Saddl Rvr, w Jrsy, US: rntc Hall, 977, 259 p. ISB [9] OSOLDOVÁ,., BROŇÁK, I.: thods n T and I twors. In w Informaton and ultmda Tchnologs. IT 2 DVES I ELETRIL D ELETROI EGIEERIG ISS 84-39

7 IFORTIO D OUITIO TEHOLOGIES D SERVIES, VOL. 9, O., RH 2 3 2: Brno, zch Rpublc, , Brno Unvrsty of Tchnology, pp. 2-23, ISB [] BLOGH, T., LUKÁROVÁ, D., EDVEKÝ,.: rformanc of Round Robn-Basd uu Schdulng lgorthms. In: TR 2: Th Thrd Intrnatonal onfrnc on ommuncaton Thory, Rlablty and ualty of Srvc, thns, Grc, , IEEE omputr Socty, 2, pp. 56-6, ISB [] BROŇÁK, I., IČUH, J.: rvntv thods Supportng os n I. In: w Informaton and ultmda Tchnologs. IT 29, Brno, zch Rpublc, , Brno Unvrsty of Tchnology, pp. 5-23, ISB [2] VOZK,., ROZHO, J.: thodology for SI Infrastructur rformanc Tstng, SES TRSTIOS on OUTERS, Issu, Volum 9, pp. 2-2, Sptmbr 2, ISS bout uthors... Er HROÝ was born n Vľý Krtíš, Slovaa, n 98. H rcvd th astr dgr n tlcommuncatons n 25 from Faculty of Elctrcal Engnrng and Informaton Tchnology of Slova Unvrsty of Tchnology (FEI STU) Bratslava. In 27 h submttd hd wor from th fld of Obsrvaton of statstcal proprts of nput flow of traffc sourcs on vrtual paths dmnsonng and hs scntfc rsarch s focusd on optmzng of procsss n convrgnt ntwors. owadays h wors as assstant profssor at th Dpartmnt of Tlcommuncatons of FEI STU Bratslava. Tbor IŠUTH s a studnt of hd. study at Dpartmnt of Tlcommuncatons, Faculty of Elctrcal Engnrng and Informaton Tchnology of Slova Unvrsty of Tchnology Bratslava. H focuss on applcaton of Erlangs' quatons both n classc tlcommuncaton ntwors and modrn I ntwors. atj KVKÝ was born n tra, Slovaa, n 979. H rcvd th astr dgr n tlcommuncatons n 24 from Faculty of Elctrcal Engnrng and Informaton Tchnology of Slova Unvrsty of Tchnology (FEI STU) Bratslava. In 26 h submttd hd wor ualty of Srvc n Broadband twors. owadays h wors as assstant profssor at th Dpartmnt of Tlcommuncatons of FEI STU Bratslava and hs scntfc rsarch s focusd on th fld of qualty of srvc and prvat tlcommuncaton ntwors. 2 DVES I ELETRIL D ELETROI EGIEERIG ISS 84-39