Method for Fast Estimation of Contact Centre Parameters Using Erlang C Model

Transcription

1 200 Thrd Internatonal onference on ommuncaton Theory, Relablty, and Qualty of Servce Method for Fast Estmaton of ontact entre Parameters Usng Erlang Model Tbor Mšuth, Erk hromý, Ivan Baroňák Department of Telecommuncatons Faculty of Electrcal Engneerng and Informaton Technology, Slovak Unversty of Technology Ilkovčova 3, 82 9 Bratslava, Slovak Republc tbor.msuth@stuba.sk, erk.chromy@stuba.sk, van.baronak@stuba.sk bstract The paper deals wth ontact centre parameters estmaton. The deas are based on Erlang mathematcal model that s useful durng ontact centre desgn and mplementaton. We concentrate on modfcaton of the basc Erlangs formulae to provde resources non-demandng teratve method of ontact centre parameters calculaton. Especally we focus on the staffng requrements to meet certan requests handlng level wth mnmum queung. The accuracy of the computaton based on newly derved formulae s proofed aganst smulaton. Keywords-ontact entre; QoS; Erlang model I. ITRODUTIO The publc contact today s a must for almost every enterprse or organzaton. valablty of smpler, faster and more comfortable contact wth the clents, customers or trade partners can be a crucal advantage n busness competton. ontact centre s one of many other ways how the nsttuton s able to cover all ts communcaton requrements towards clents and partners. It s a structured communcaton system consstng of both human and technologcal resources, whch mproves the communcaton between organzaton and customers. It s more or less a software and hardware upgrade of a prvate branch exchange, whch s the prmary medum for a call access to the organzaton. It combnes telephone processes and data processng to acheve the most effectve busness transactons. The heart of the ontact centre conssts of utomatc all Dstrbuton (D) module and several other optonal modules D s responsble for correct and effcent routng of all types of requests (voce calls, e-mals, etc.) through the ontact centre to an adequate and sklled agent or system capable of handlng the request. For a company or ontact centre operator t s crucal to provde suffcent level of call handng qualty (expressed mostly n terms of watng tme) wth mnmum possble resources. We present an alternatve approach to staffng requrements calculaton. The next parts of ths paper are structured as follows. Frst we present some assumpton regardng telecommuncaton traffc and contact centre mathematcal models. ext the Erlang queung model s presented. In secton 3 a possble less resources demandng formulae are derved and presented. Fnally, the are valdated by comparng to smulaton measurements. II. MTHEMTIL MODEL OF OTT ETRE Each component of D (or contact centre) system can be more or less precsely converted to a mathematcal model. Snce the contact centres usually handle large amount of requests per tme unt, the vast majorty of these models are based on queung theory. ccuracy of depends on a correct model selecton. lso the precse descrpton of nput parameters and varable dependences can sgnfcantly affect them.. Input traffc model The rate of arrvng calls (or other requests) per tme unt s consdered as random varable. ontact centre s goal s to provde ts servces to a wde range of customers therefore we can assume the requests populaton to be unlmted. Once the populaton s unlmted the contact centre system can be denoted as queung system wth unlmted request populaton [], [2]. In such a wde populaton the request creators (.e. the callers) choose the start moment of ther call ndvdually and ndependently of each other. ny events (calls) are consdered ndependent from the mathematcal pont of vew f for any arbtrary set of requests, 2,... n and arrval probabltes P( ), P( 2 ),..., P( n ) durng randomly chosen tme nterval the followng formula s vald [3]: n n P(I ) = P( ). () í= = So the arrval probablty of n requests durng the same tme nterval equals to conjuncton of ndvdual requests arrval probabltes for gven nterval. It s obvous that the arrval rate of calls can acqure only countable set of values, so t s a dscrete varable. Incomng requests nto a queung system are mostly modelled by Posson dstrbuton [2], [3]. Followng formulae defne ts probablty densty functon, mean and varance: λ λ p k = f ( k) = P( X = k) = e k!, (2) E (X ) = λ, (3) D (X ) =λ, (4) where λ represents the average event arrval rate (.e. avg. number of requests per tme unt) /0 $ IEEE DOI 0.09/TRQ

2 The nter-arrval tme s a very mportant varable as well. Snce the arrval rate s a random varable the nter-arrval tme must be random varable as well. Based on the ndependence of each request the nter-arrval tme for Posson traffc flow s exponentally dstrbuted contnuous random varable wth the mean equals to /λ [4]. The probablty densty of ths random varable s defned by the followng formula [2], [3]: t f ( t) =λe λ (5) where t represents the nter-arrval tme (random varable). B. Servce group model based on Erlang The servce group conssts of a queue and at least one human agent. The agent s (from the queung theory pont of vew) a server where the requests are processed. In case of there are more requests at the same tme, the later comng requests has to wat n the queue. Dansh mathematcan. K. Erlang s very closely ted to the orgn of queung theory []. He publshed the paper concernng applcaton of statstcs n telephone servce n 909. Ths document begns further research of queung theory. Together wth Markov chans theory hs deas helped to defne and descrbe more complcated queung models. Even now, a century later snce the Erlang models have been publshed, are these one of the basc approaches used n parameter estmaton of telecommuncaton systems. Snce the contact centre s consdered to be a telecommuncaton system the models are vald for t as well. Both Erlang and Posson process based Markovan models use the same assumptons and followng requrements must be met [5]: number of sources (requests populaton) s much greater than the number of servers, requests are generated randomly and ndependently of each other, average number of requests per tme unt from all sources s constant, request handlng tme s a random varable wth exponental dstrbuton, queung s based on the Frst n Frst out (FIFO) algorthm. s we mentoned earler the frst two requrements are easly met by havng large group of potental callers (request creators). Usage of Posson dstrbuton for arrval process ensures, the average number of request per tme nterval s stable and equal to λ. The remanng two requrements were mplemented nto the smulaton applcaton. Erlang presented two basc models that dffer n presence of a queue. The smpler Erlang B model does not contan a queue and requests over the current capacty of the system (no dle agent avalable) are permanently lost. Ths model s more sutable for dmensonng of telephone trunks or data throughput capacty (n case of VoIP traffc). In ths paper we focus on the second Erlang s model, so called Erlang model. Presence of smple FIFO queue elmnates the man dsadvantage of the prevous model. If a temporary overload stuaton occurs, the request s put nto the queue untl any of the agents become avalable,.e. the handlng of any prevous requests s fnshed. The FIFO strategy ensures the caller watng the longest tme to be served the frst. Erlang model [5] s orgnally defned as functon of two varables: the number of agents and the traffc load (expressed n Erl). Based on these parameters t calculates the probablty P (6) that the arrvng request s not assgned to the agent mmedately and t has to wat n the queue.! ( ) P = (6) (, ) + = 0!! ( ) If we know the rate of calls per tme unt λ and the average number of served requests per the same tme unt µ (so the average handlng tme s /µ) then the traffc load can be easly evaluated as [6]: λ =. (7) µ Furthermore we defne the varable ρ, that represents the average utlzaton of each agent as [], [2], [7]: λ η =. (8) µ For system stablty reason (.e. the number of requests present n the queue does not extend to nfnty) the value of η s requred to be less than (stablty crteron). From the conjuncton of (6), (7) and (8) we can obtan followng formula: P (, ) ( η)! ( η) ( η) ( η) +!! ( η) η =. (9) = 0 If we start from Markov M/M/m/ queung model we can derve dentcal equaton that wll defne probablty that m or more requests are present n the queung system so the new ncomng request wll be nserted nto the queue [], [2], [7]. Ths means that Markov M/M/m/ and Erlang models are dentcal and ths relatonshp can be easly proved analytcally. Snce the Erlang model contans a queue, there are some more parameters and varables, can be measured and more or less nfluenced by model nputs. From the callers pont of vew the most mportant value s the watng tme or the tme the request spends n the queue before t s served by an agent. Ths value s a random varable descrbed by the probablty densty functon [7]: P τ = 0 ( τ) = µ ( ) F W τ. (0) e τ > 0 Based on ths formula we can calculate the average watng tme (or average speed of answer S) W: W =. () µ ( ) and usng Lttle s theorem [8] and (0) we can obtan the average number of requests n the queue Q: 82

3 λ Q= λ W = =. (2) µ ( ) ( ) The general defnton of probablty densty functon of any statstcal dstrbuton and ts propertes [3] gves us an opportunty to derve Grade of Servce (GoS) parameter value from equaton (0) once the cceptable Watng Tme (WT) value s known [], [7]: µ ( )WT GoS = P e. (3) GoS defnes the percentage of ncomng calls that are answered no later than defned WT (usually 20 sec.). Followng equaton s vald for the average number of requests K n queung system [7]: η K = η+ = + = + Q. (4) η ( ) gan, the Lttle s theorem allows us to obtan the average tme T the request wll spend n the system: K + Q T = = = + W = +. (5) λ λ µ µ µ III. ( ) EVLUTIO OF ERLG MODEL ll calculatons usng Erlang model are based on three varables, and P c, of whch any two can be the nput varables and the remanng one s computed by the formulae. Once all three values are known, more queung system parameters can be evaluated usng set of equatons n prevous secton. The computaton of a queung probablty P c once the traffc load and the number of agents are defned s the most straghtforward problem. To avod numercal errors (dvson of two relatvely large values), the basc formula (6) can be easly modfed nto (, ) = =.(6) k + + ( s) = k= 0 usng the same prncple as for Erlang B model n [8]. Furthermore the Horner scheme can be easly appled to reduce number of requred operatons to compute the sum part of the denomnator s as s = ( + s - ) * (/) (7) for =... Then s = s. (8) The remanng two stuatons ( or values to be calculated) are complcated to solve analytcally, but the numercal soluton provdes us the expected wth adequate accuracy. Probably the most mportant calculaton for a ontact centre provder s determnaton of number of requred agents to handle gven traffc load wth queung probablty not more than P max. The computaton s based on cyclc calculatons of P for ncrementng values of untl the result s below expected threshold. However usng (6) the stop condton can be easly transformed to MX s (9) MX and the for each step we can use + = +, (20) + s+ = ( + s). (2) Ths extremely shortens the solvng tme snce each teraton requres only a few smple operatons. Obvously to fulfll the stablty crteron the number of agents must be hgher than expected traffc load so the frst check of stop condton could be postponed untl >. The last combnaton of varables (.e. unknown traffc load the agents are capable handlng at queung probablty less or equal to P ) s the most complcated case. It requres fndng the soluton of f ( x) P ( x, ) _ = 0. (22) = P IPUT where and P _IPUT are defned and P (x,) s the queung probablty for current value of x. Snce the P (x,) s monotonous functon, only one real soluton exsts and ts nterval can be lmted to (0;> to fulfll the stablty crteron. ny avalable numercal soluton method for such an equaton can be appled. IV. MODEL VERIFITIO We decded to verfy the model s relablty and accuracy (ncludng the modfed calculatons) by means of smulaton [9]. It was mplemented n MTLB takng all Erlang model s requrements nto account (see above). The smulaton algorthm conssts of three basc sectons: nput parameters verfcaton, request arrval tmes determnaton and handlng tmes determnaton (based on exponental dstrbuton), smulaton wth dscrete tme steps ( sec), calculaton and presentaton. Both the arrval traffc flow and requests handlng tmes values are tested aganst ts statstcal dstrbuton usng χ- square test wth α = Snce the Erlang models descrbe the steady state of queung system, the very begnnng part of smulaton s dscarded untl the model reaches the steady state. The smulaton length was set to 30 hours of smulated traffc wth avg. arrval rate set to λ = 667 requests per hour and avg. handlng tme of 50 seconds. The number of agents was varable from 28 (mnmum requred to fulfl the stablty crteron snce = 27.8 Erl) up to 37. TBLE I. Pc K ERLG MODEL LULTIO RESULTS T Q W GoS ρ

4 Probablty of queung (%) TBLE II Pc K ERLG MODEL SIMULTIO RESULTS T Q W GoS ρ Drect comparson of the tables above shows that the smulaton are very smlar to Erlang model calculatons. The bggest dfference can be seen for low number of agents or the stuaton when the stablty crteron s matched by the mnmum number of agents. However the more overstaffed the contact centre s, the better precson of model s ganed. From the callers pont of vew the most mportant characterstcs are probablty of call queung (P ), average watng tme (how much tme does the caller spend watng for an agent) W and Grade of Servce (GoS) level (the probablty of call recepton n 20 seconds). Fg., Fg. 2 and Fg. 3 dsplay the relaton of these varables to the number of agents present n the contact centre (). alculated Smulaton umber of agents () vg. watng tme (s) Grade of Servce (%) alculated Smulaton umber of agents () Fgure 3. verage watng tme dependency on number of agents. The agent utlzaton decreases (Fg. 4) almost lnearly wth ncreasng number of agents. Ths nto fact, that small ncrement n number of agents does not cause steep agent utlzaton decrease and the ncreased costs of added staff s compensated by sgnfcant mprovements of relevant call handlng metrcs that leads to hgher satsfacton of all potental callers. Ths phenomenon s vsble the more the contact centre s operated near ts stablty bound n relaton to number of agents and ncomng traffc load. The most mportant decson of the contact centre manager s to defne equlbrum pont n number of agents. In our model case, accordng to smulaton and computaton 00 87, , ,5 25 alculated Smulaton umber of agents () Fgure 2. Grade of Servce level dependency on number of agents. Fgure. all queung probablty dependency on number of agents. 84

5 gent utlzaton (%) alculated umber of agents () Fgure 4. gent utlzaton dependency on number of agents agents should be suffcent to provde smooth operaton of the centre wth up to approx. 95% of all ncomng requests assgned to an agent n 20 seconds and average watng tme not more than 7 seconds. From the other pont of vew agents spend at least % of ther workng tme by call handlng and the remanng tme forms the dle perod that can be utlzed to e.g., related paper work, agent educaton etc. REFEREES [] L. Unčovský, Stochastc models of operatonal analyss, LF, Bratslava, 9, 46 pages, ISB [2] J. Polec and T. Karlubíková, Stochastc models n telecommuncatons, FBER, Bratslava, 999, 28 pages, ISB [3] Z. Rečanová, J. Horváth, V. Olejček, B. Rečan, and P. Volauf, umercal methods and mathematc statstcs, LF, Bratslava, ugust 987, 496 pages, ISB [4] K. Vastola, Interarrval tmes of a Posson process, [Onlne], Rensselaer Polytechncal Insttue, Troy (ew York, US), , accesed on , valable: [5] Dagnostc Strateges, Traffc Modelng and Resource llocaton n all enters, eedham (Mass., US), Dagnostc Strateges, 2003, accesed on , valable: [6] G. Koole, all enter Mathematcs : scentfc method for understandng and mprovng contact centers,. Vrje Unverstet, msterdam, , 68 pages, accesed on , valable: [7] G. Bolch, S. Grener, H. de Meer, and K. S. Trved, Queueng etworks and Markov hans, 2nd ed, John Wley, Hoboken (ew Jersey, US), c2006, 878 pages, ISB [8] S. Qao, Robust and Effcent lgorthm for Evaluatng Erlang B Formula, McMaster Unversty, Hamlton, Ontaro anada, , accesed on , valable: [9] M. Kavacký, Performance Smulatons of TM Swtch Model, In: 6^th Internatonal onference on Emergng elearnng Technologes and pplcatons, IET 2008, Stará Lesná, Slovak Republc, September - 3, 2008, ISB V. OLUSIO The am of ths paper s to show some of the approaches towards ontact center modellng and smulaton that leads to parameter estmaton and verfcaton. The Erlang model s probably one of the most wdely used for ths purpose. The greatest advantage of Erlang models s ther smplcty and ablty to descrbe the most mportant operaton stuatons of ontact centers wth acceptable level of accuracy. Smulaton show almost no dfference to calculaton s steady state of operaton. The basc form of Erlang model and ts formulae can be adjusted to provde hgher level of computaton accuracy and more mportantly less resource demandng algorthms. These modfed formulae enable calculaton of any model parameter usng fast teratve process. KOWLEDGEMET Ths work s a part of research actvtes conducted at Slovak Unversty of Technology Bratslava, Faculty of Electrcal Engneerng and Informaton Technology, Department of Telecommuncatons, wthn the scope of the projects VEG o. /0565/09 Modellng of traffc parameters n G telecommuncaton networks and servces and ITMS Support for Buldng of entre of Excellence for SMRT technologes, systems and servces. 85