MARKOV MODEL M/M/M/K IN CONTACT CENTER Erik CHROMY 1, Ja DIEZKA 1, Matej KAVACKY 1 1 Istitute of Telecommuicatios, Faculty of Electrical Egieerig ad Iformatio Techology, Slovak Uiversity of Techology Bratislava, Ilkovicova 3, 812 19 Bratislava, Slovak Republic chromy@ut.fei.stuba.sk, ja.diezka@gmail.com, kavacky@ut.fei.stuba.sk Abstract. Our paper deals with a cotact ceter dimesioig. The mai task of a cotact ceter is to offer services to customers. The most simple ad geuie way of commuicatio with the cotact ceter aget is by phoe. From the customer satisfactio poit of view, the shortest servig time is the most importat factor. It is kow that the highest operatig costs of the cotact ceter are costs for agets. Therefore, we eed to choose the right umber of agets ad effectively utilize them. For cotact ceter dimesioig we ca use various models. I this paper, we itroduce the Markov M/M/m/K model which offers the broad rage of parameters suitable for the cotact ceter sizig, e.g. required umber of agets based o the probability of call refusal ad probability of call equeue. The availability 24 hours ad 7 days per week eed t mea permaet occupacy of cotact ceter by agets. Durig light traffic (e.g. i the ight or small hours), it ca be satisfactory to operate oly self-service parts of cotact ceter. The paper is divided as follows. The chapter 2 describes basic priciples of the cotact ceter operatio ad summarizes the importat traffic parameters with impact o cotact ceter desig. The chapter 3 itroduces the cotact ceter as a queuig system. Tha the modelig of the cotact ceter through the Markov model M/M/m/K is preseted. Calculatios ad relatios of importat traffic parameters of the cotact ceter are give. The fial part of the paper deals with coclusios o Markov M/M/m/K model. Keywords Cotact ceter, M/M/m/K, queuig system. 1. Itroductio The cotact ceter belogs to coverget techologies ad allows compaies to provide services o four basic platforms: voice, data, video ad web. The processig of customer s phoe queries by agets of the cotact ceter is the basic cotact ceter service. Utilizatio of the self-service Iteractive Voice Respose (IVR) system is tightly coupled with processig of a telephoe call. For the correct operatio of the cotact ceter it is ecessary to esure: easy access, oe umber, access through other media (Iteret, GSM, WAP), specialized umbers (for VIP cliets or importat products), 24/7 availability. 2. Traffic Parameters i Cotact Ceter The operatio of cotact ceter is based o the followig priciple. Customers make phoe calls to the cotact ceter with various queries. The first cotact with callig customer is through IVR system. This system idetifies the customer ad it is capable of automatically respodig to some queries from customers. If the help of a aget is eeded, the IVR ca reroute the call to the Automatic Call Distributio module (ACD) which coects the calls with the most appropriate agets through special routig algorithms. I the case of more callig customers tha available agets the waitig queue will occur. The maagemet of waitig queues is also the task of ACD module. At the ed, the cotact ceter aget hadles the particular query. It is obvious that customer calls are radom, some of them will stay o the lie through the whole process (cotact with IVR, reroute by ACD, waitig i the queue), some will leave the lie after cotact with IVR, or durig waitig i the queue. Detailed iformatio about cotact ceter operatio ca be obtaied by moitorig of various traffic parameters. These traffic parameters ca be divided ito two groups accordig to their characteristics. They ca 264 212 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING
describe the cotact ceter or grade of Quality of Service (QoS) [1], [2], [3], [4], [5], [6], therefore, the quality of services offered by give cotact ceter. The essetial traffic parameters of cotact ceter are: traffic load of the cotact ceter A [erl], umber of agets m, average rate of icomig calls per hour λ [calls/h], average rate of served calls per hour μ [calls/h], average legth of call hadlig T c [s] ad holds that T c = 1/µ. The parameters listed above will form hereiafter the basic iput values for traffic modelig i the cotact ceter. 3. The Cotact Ceter as a Queuig System A queuig system is such system i which give umber of service statios hadle a large umber of queries. Also, cotact ceter is such system, the queries are geerated by callig customers ad service statios are represeted by cotact ceter agets. I geeral, oe aget ca hadle oe query at the same time. There ca be also situatio, whe the agets ca t hadle all icomig call immediately. I this case, the customers may leave a system or wait for a available aget i the waitig queue. Of course, istat hag-ups are uwated, therefore, the proper maagemet of the waitig queue is oe of the mai task i the cotact ceter. Accordig to the queuig theory, the properties ad behavior of the queuig system is described by five parameters: the way of arrival of iputs ito the system, the way of processig of queries, the umber of service statios, total capacity of the system, largeess of populatio of queries for this system. Large umber of queuig system exists ad the most appropriate queuig system should be used for solvig of a particular problem. Accordig to Kedall classifyig of queuig systems the system ca be marked as A/B/c/N/K, while [7]: A probability distributio of queries arrivals (iputs) ito system, B probability distributio of hadlig time required for query processig, c the umber of service statios (agets), N capacity of the system (umber of customers, queries, etc.), K restrictio of waitig the queue legth. For the deotatio of probability distributios of queries arrivals ito the queuig system ad hadlig time of queries (A ad B i Kedall classifyig) the followig symbols are used [7]: M expoetial distributio, E Erlag distributio, D determiistic distributio (time betwee arrivals of queries or hadlig time is costat), G geeral distributio, G1 Geeral distributio with idepedet iterarrival times. The way of waitig queue maagemet is the substatial problem i quality of service provisio i the queuig system. The admiistrator of queuig system has to esure o query will stay i waitig queue too log time. From this, we ca derive the umber of service statios (agets, servers, etc.) ad capacity of waitig queue. I high-speed etworks, there is cosiderable iterest i traffic arrival processes where successive arrivals are correlated. Such o-g1 arrival processes iclude the Markov modulated Poisso process (MMPP) [7]. 3.1. Basic Probability Distributios Call arrival ito the cotact ceter is i the most cases modeled by Poisso probability distributio. Stochastic variable X has Poisso distributio with the mea λ, if set of their possible values is H(X) = {, 1, 2,,, } ad the probability desity fuctio [8] is give by: k λ λ f(k) e, for k, 1, 2,,,. (1) k! Aother probability distributio ofte used i the cotact ceter modelig is a Expoetial distributio. Variable X has a expoetial distributio if the probability desity fuctio [8] is give by: x f ( x) e, for x. (2) Phase-type distributio - a method of Markovizig a o-markovia model ad as such has a wide applicability [7]. 4. Modelig of Cotact Ceter Number of agets has a direct impact o QoS of the cotact ceter. O the other had, the high umber of 212 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 265
agets has a egative impact o operatioal costs of cotact ceter. From the cotact ceter operatioal costs aalysis results that the sigificat part of costs cotais costs for huma sources, e.g. wages of agets, maagers ad supervisors of cotact ceter. These costs are mothly repeated therefore, the proper desig of the required umber of agets is the oe of the key tasks i the cotact ceter desig. Whe determiig the required umber of agets the followig three basic assumptios valid i the majority of cotact ceters should be take ito cosideratio [9]: aget has to hadle call immediately (delayed respose decreases QoS), call arrivals are ot periodical ad their umber always vary (accordig to daytime, day of the week, etc.), there is o liear relatio betwee umber of icomig calls ad the required umber of agets i order to esure QoS. There are various models of cotact ceter applicable to the determiatio of the required umber of agets or calculatios of importat traffic parameters, from basic Erlag models B ad C [1], through Markov queuig system models to complex o-markov models ad simulatios of the cotact ceter. 4.1. Markov M/M/m/K Model The biggest differece of M/M/m/K model [11], [12] agaist M/M/m/ model is a limited legth of waitig queue. With such limitatio, this model is sigificatly approachig the real situatio i the cotact ceter which is of limited capacity. The priciple of Markov M/M/m/K model is depicted i Fig. 1. λ μ λ λ λ λ λ 1 2... m-2 m-1 m m+1... K-1 K 2μ (m-1)μ mμ mμ mμ Fig. 1: Markov M/M/m/K model. Queries arrivig ito the system with rate, servers hadles particular queries with rate mµ, ad maximum K queries ca be i the system at the same time. All other icomig queries are refused. The customers call the cotact ceter radomly, with the Poisso distributio with mea λ. Therefore, there are λ calls per hour arrivig ito the cotact ceter. There are m agets i the cotact ceter ad each of them serves the callig customers with rate µ customers per hour. Hadlig time has a expoetial distributio with rate µ. There ca be maximum of K customers i the cotact ceter, while the maximum of m customers ca be served simultaeously. All other customers have to wait i the waitig queue util there is ay free aget. The parameter K therefore sigificatly limits the capacity L of waitig queue ad the capacity of waitig queue L is give by equatio (3): L K m. (3) I the case of full occupacy of the cotact ceter (K customers are i the cotact ceter) the each other icomig call is blocked. Probability of call blockig is P B. Probability of successful call ito the cotact ceter is 1 - P B. The customers are served by agets with rate mµ, while the rate i which the agets are occupied by customers is give by the value : 1 P B. (4) 4.2. Calculatio Capacity of Markov M/M/m/K Model The cotact ceter modeled by Markov M/M/m/K model is stable i every traffic load, because i the case of full occupacy each other call is blocked. The parameter ρ (5) is therefore used for determiatio of traffic load per server, or per aget [11]:. (5) m Probability that the system is empty [11] (there is ot ay query) is give by: ad m P i 1 1 Km1 i m m i m m 1, i! m! 1 for 1, (6) 1 m 1 i m m m 1!! P K m, i i m for 1. (7) Probability that i the system there are just queries [11] is give by: ad P P 1 m,!! P P for,1,, m, (8) m m m m! Pm P for m, m 1,, K, (9) where P m is the probability that just m queries are i the system. I the case that just K queries are i the system each other icomig call is blocked. Probability P(K), 266 212 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING
P B [%] P B [%] INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES specifies the probability of call blockig P B, ad from equatio (7) we have: P B m K P. (1) m! From the defiitio of the Markov M/M/m/K system results that i the case of m calls i the cotact ceter, each other icomig call is rerouted to the waitig queue util there are K queries i the cotact ceter, the each other icomig call is blocked. Probability of call equeue P Q ca be determied by use of theorem of full probability (11): referece value of traffic load of cotact ceter 5 erl the call arrivals rate is λ = 6 calls/h. However, agets of the cotact ceter are occupied with rate 58,74 calls/h. Other calls are waitig i the queue, or are blocked. Other calls are waitig i the queue or are blocked. 1 9 8 7 6 5 P p i 1. (11) i By use of theorem (11) it is possible to costruct a equatio for P Q (12): m 1 P Q 1 P P. (12) 1 5. Calculatio of Importat Traffic Parameters of Cotact Ceter Calculatio of particular parameters i the aalysis of the cotact ceter by Markov M/M/m/K model is based o the followig referece values: λ = 6 icomig calls per hour, service time is T serv = 5 miutes, the maximal umber of queries i the system K = 15. Thus, the average traffic load of such cotact ceter is 5 erl. 5.1. Calculatio of the Required Number of Agets The probability of call blockig P B ca be used as QoS parameter for determiatio of the required umber of agets i the cotact ceter modeled by the Markov M/M/m/K model. I Fig. 2 we ca see that this probability is sheer decreasig with each ew aget. This decreasig stops at the umber of agets m = 6, whe the P B is about 2 %, what is a acceptable value. Exact values are stated i Tab. 1. It must be oted that with each additioal aget also the capacity L of the waitig queue vary accordig to equatio (3). This property comes from the defiitio of M/M/m/K model ad i all calculatios of traffic parameters it is ecessary to take ito accout the variatio of the waitig queue capacity K. I Tab. 1 there is also parameter. From Tab. 1 we ca see that for example i the case of 6 agets ad 4 3 2 1 2 4 6 8 1 12 14 umber of agets Fig. 2: Relatio betwee P B [%] ad umber of agets m whe A = 5 erl ad K = 15. Tab.1: Traffic parameters if A = 5 erl. m P B [%] P Q [%] L ρ λ 5 7,4 81,42 1 1, 55,56 6 2,1 54,43 9,83 58,74 7,64 31,34 8,71 59,62 8,23 16,4 7,63 59,86 9,11 7,93 6,56 59,94 5.2. Traffic Parameters ad Traffic Load Variatio I Fig. 3 we ca see the relatio betwee probability of call blockig P B [%] ad traffic load A [erl] i the case of 6 agets. Whe the value of traffic load of 7 erl is exceeded, the probability of call blockig is above 5 %. Traffic load of 7 erl at the average call hadlig time T obs = 5 mi represets average of 84 icomig calls per hour. 5 45 4 35 3 25 2 15 1 5 2 4 6 8 1 12 14 16 18 2 A [Erl] Fig. 3: Relatio betwee P B [%] ad A [erl]. 5 % m = 6 m = 8 m = 1 212 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 267
P Q [%] INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES Figure 3 shows the cases whe the traffic load is icreasig ad the umber of agets is 8 or 1. The impact of icreasig cotact ceter traffic load A o probability of call equeue P Q ca be see i Fig. 4. Also, cases with icreasig traffic load ad 8 or 1 agets are depicted i Fig. 4. 1 9 8 7 6 5 4 3 2 1 m = 6 m = 8 m = 1 2 4 6 8 1 12 14 16 18 A [erl] Fig. 4: Relatio betwee P Q [%] ad A [erl]. Tab.2: Traffic parameters if m = 6. A [erl] P B [%] P Q [%] ρ λ 3,1 9,91,5 36, 4,25 28,12,67 47,88 5 2,1 54,43,83 58,74 6 7,83 78,28 1, 66,36 7 16,65 91,61 1,17 7,1 8 25,69 96,98 1,33 71,34 9 33,55 98,89 1,5 71,77 Table 2 icludes all iterestig traffic parameters for 6 agets. The model M/M/m/K does ot take ito accout the situatio whe the customer will hag up after he is equeued, so the call is lost. The model of the cotact ceter ca be exteded with parameters describig the waitig queue such as: capacity (legth) of the waitig queue L, umber of all calls i the cotact ceter (served ad waited) N, average umber of customers i the waitig queue Q, average time that the query will spet i the system T [s] (waitig time i the queue ad hadlig time by aget), average time that the query will spet i the waitig queue W [s]. So the iterestig results ca be obtaied by addig of calculatios of above parameters which ca better moitor the QoS level i the cotact ceter. Based o this it is possible to respod to other situatios that will occur i the cotact ceter. Ackowledgemets This work is a part of research activities coducted at Slovak Uiversity of Techology Bratislava, Faculty of Electrical Egieerig ad Iformatio Techology, Istitute of Telecommuicatios, withi the scope of the projects Grat programme to support youg researchers of STU - Modelig of Traffic i NGN Networks ad Support of Ceter of Excellece for SMART Techologies, Systems ad Services II., ITMS 26241229, co - fuded by the ERDF. 6. Coclusio The cotributio of this paper is the applicatio of M/M/m/K model o desiged cotact ceter model. This model is possible to use for dimesioig of cotact ceters ad predictio of their traffic parameters. By restrictio of waitig queue capacity this model is approachig the situatio i the cotact ceter. The calculatio capacity of the model allows the calculatios of the most of the importat traffic parameters of the cotact ceter. By use of Markov M/M/m/K model the followig QoS parameters of cotact ceter have bee determied: m the umber of agets i the cotact ceter, P B the probability that customer will be rejected, P Q the probability that customer will have to wait i the waitig queue. Refereces [1] VOZNAK, M., A. KOVAC ad M. HALAS. Effective Packet Loss Estimatio o VoIP Jitter Buffer. I: Networkig 212 Workshops. Prague: Spriger, 212, pp. 157-162. ISBN 978-3- 642-338-7. DOI: 1.17/978-3-642-339-4_21. [2] KYRBASHOV, B., I. BARONAK, M. KOVACIK ad V. JANATA. Evaluatio ad Ivestigatio of the Delay i VoIP Networks. Radioegieerig. 211, vol. 2, o. 2, pp. 54-547. ISSN 121-2512. [3] MICUCH, J. ad I. BARONAK. Implemetatio Admissio Cotrol Methods for VoIP Applicatios. I: Telecommuicatios ad Sigal Processig TSP-21: 33rd Iteratioal Coferece o Telecommuicatios ad Sigal Processig. Viea: Budapest: Asszisztecia Szervezo Kft., 21, pp. 391 395. ISBN 978-963-88981--4. [4] POLACEK, P. ad I. BARONAK. Ehaced Equivalet Capacity Method. I: Proceedigs of Iformatics 29: IADIS Multi Coferece o Computer Sciece ad Iformatio Systems. Algarve: IADIS Press, 29, pp. 192-196. ISBN 978-972-8924-86-7. 268 212 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING
[5] VOZNAK, M., M. HALAS, B. BOROWIK ad Z. KOCUR. Delay Model of RTP Flows i Accordace with M/D/1 ad M/D/2 Kedall's Notatio. Iteratioal Joural of Mathematics ad Computers i Simulatio. 211, vol. 3, iss. 3, pp. 242-249. ISSN 1998-159. [6] VOZNAK, M., F. REZAC ad M. HALAS. Speech Quality Evaluatio i IPsec Eviromet. I: Proceedigs of the 12th Iter. Coferece o Networkig, VLSI ad Sigal Processig: ICNVS 21. Cambridge: Uiversity of Cambridge, 21, pp. 49-53. ISBN 978-96-474-162-5. ISSN 179-5117. [7] BOLCH, G., S. GREINER, H. MEER ad K. S. TRIVEDI. Queuig Networks ad Markov Chais. 2d ed. New York: Joh Wiley & Sos, 26. ISBN 978--471-56525-3. [8] VOLAUF, Peter. Numerical ad statistical calculatios i MATLAB. Bratislava, 25. Diploma thesis. Slovak Techical Uiversity-STU. [9] HISHINUMA, Ch., M. KANAKUBO ad T. GOTO. A Aget Schedulig Optimizatio for Call Ceter. I: APSCC 7 Proceedigs of the 2d IEEE Asia-Pacific Services Computig Coferece. Tsukuba Sciece City: IEEE, 27, pp. 423 43. ISBN -7695-351-6. DOI: 1.119/APSCC.27.27. [1] DIAGNOSTIC STRATEGIES. Traffic Modelig ad Resource Allocatio i Call Ceters. Needham, MA, 23. Avaible at: www.fer.hr/_dowload/repository/a4_1traffic_modelig.pdf. [11] BLANC, J. P. C. TILBURG UNIVERSITY. Queueig Models: Aalytical ad Numerical Methods. Jauar 211. [12] STOLLETZ, Raik. Performace Aalysis ad Optimizatio of Iboud Call Ceters. Berli: Spriger-Verlag, 23. ISSN 75-845. ISBN 3-54-812-8. About Authors Erik CHROMY was bor i Velky Krtis, Slovakia, i 1981. He received the Master degree i telecommuicatios i 25 from Faculty of Electrical Egieerig ad Iformatio Techology of Slovak Uiversity of Techology Bratislava. I 27 he submitted Ph.D. work from the field of Observatio of statistical properties of iput flow of traffic sources o virtual paths dimesioig ad his scietific research is focused o optimizig of processes i coverget etworks. Nowadays he works as assistat professor at the Departmet of Telecommuicatios of Faculty of Electrical Egieerig ad Iformatio Techology of Slovak Uiversity of Techology Bratislava Bratislava. Ja DIEZKA was bor i Doly Kubi, Slovakia i 1988. He is a studet at the Istitute of Telecommuicatios, Faculty of Electrical Egieerig ad Iformatio Techology of Slovak Uiversity of Techology Bratislava. He focuses o applicatio of Erlags' formulas i Cotact Ceters. Matej KAVACKY was bor i Nitra, Slovakia, i 1979. He received the Master degree i telecommuicatios i 24 from Faculty of Electrical Egieerig ad Iformatio Techology of Slovak Uiversity of Techology Bratislava. I 26 he submitted Ph.D. work Quality of Service i Broadbad Networks. Nowadays he works as assistat professor at the Istitute of Telecommuicatios of Faculty of Electrical Egieerig ad Iformatio Techology of Slovak Uiversity of Techology Bratislava ad his scietific research is focused o the field of quality of service ad private telecommuicatio etworks. 212 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 269