Markov Models and Their Use for Calculations of Important Traffic Parameters of Contact Center

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1 Markov Models and Their Use for Calculations of Iportant Traffic Paraeters of Contact Center ERIK CHROMY, JAN DIEZKA, MATEJ KAVACKY Institute of Telecounications Slovak University of Technology Bratislava Ilkovicova 3, 82 9 Bratislava SLOVAKIA chroy@ut.fei.stuba.sk, jan.diezka@gail.co, atej.kavacky@stuba.sk MIROSLAV VOZNAK Departent of Telecounications VSB - Technical University of Ostrava 7. listopadu 5, Ostrava Poruba CZECH REPUBLIC iroslav.voznak@vsb.cz Abstract: - The appropriate odel and consecutive design is the basis for building of successful Contact center. Excessive coplexity of odel can be soeties contra-productive. The ain goal of this work is to analyze of Markov odel M/M// properties in Contact center environent. This odel offers wide range of iportant traffic paraeters calculations of the syste, while keeping transparent and siple structure the key factor in design of Contact center. The second part of the paper deals with the odelling of contact center equipped with Interactive Voice Response (IVR) syste. IVR syste is odeled by Markov odel M/M/ /. It results in the less nuber of needed agents in contact center. Key-Words: - M/M//, M/M/ /, Erlang C forula, Autoatic Call Distribution, Interactive Voice Response, Quality of Service Introduction Contact center is coplex counication syste which arises as upgrade of private branch Exchange and offers integrated syste for counication with custoers whether by telephone call, e-ail or even by text chat with contact center agent. Contact center [7] consist of: Private Branch exchange (PBX), Autoatic Call Distribution (ACD), Coputer Telephony Integration (CTI), Interactive Voice Response (IVR), Call Manageent Syste (CMS), Voice Recording (VR), Capaign anager (CM), agents and supervisors workplaces, eail and web server (see Fig. ). It is very difficult to describe such coplex syste by atheatical odels []. Therefore in the next we will see under ter Contact center only the syste for processing of telephone calls [9]. Fig.. Contact center architecture. 2 Contact Center as Queuing Syste The syste in which soe nuber of service stations serves large volue of requests is tered as queuing syste [5]. The Contact center is such queuing syste. In this case the requests are represented by custoers calling to Contact center ISSN: Issue, Volue, Noveber 2

2 and service stations are represented by Contact center agents. The task of these agents is to handle custoer requests. The general exaple of queuing syste is depicted in the Fig. 2. Zzzzzzzz Waiting queue... 2 Served custoers Service stations Fig. 2. Illustration of Queuing syste. Let us assue that syste in the Fig. 2 is Contact center with service stations or agents. Requests are generated by custoers randoly and independently each other calling into Contact center. Agents are serving callers one by one and in the case if all agents are busy, another custoers have to wait in the waiting queue until soe agent will be free. Call handling tie is rando and is not dependent on handling tie of other calls. In this case we can this Contact center nae as stochastic queuing syste. Statistical observation as shown that incoing telephone calls into Contact center siulate Poisson distribution of rando variable, handling tie is represented by exponential distribution []. Behaviour of such syste can be described by Markov processes. 2. Autoatic Call Distribution Autoatic Call Distribution (ACD) belongs to the basic software features, that is needed to the contact center realization. Agents assigned to the individual service groups (Fig. 3) process the sae type of incoing calls. ACD classifies and routes incoing calls according to prograed rules e.g. according to the called nuber, arrival tie of call, length of the waiting queue, etc. Then the calls are routed to the agent service groups (sales departent, technical departent, custoer service, etc.). Through the ACD the calls can be routed to the next free, or to the longest free agent of the particular service group. The result is the unifor distribution of working load on all agents of the service group or processing the ajor of phone calls by one of the best agents. Fig. 3. Service group with special agent. The task of the ACD is to connect the calling custoer with appropriate agent - it is a coplex operation with these iportant issues: custoer request for the service, agent profil, agents state (free or busy), state of the waiting queues, syste load in real tie, alternative resource (in the case of load). Contact centers are built in order to effective resources utilization (whether technical or huan). Thereat, we have also to think about the situation that there is no free agent with appropriate skills for the custoer. Therefore, there ust be waiting queues in contact center. 2.2 Waiting Queues in Contact Center On the basis of requested service the call is routed to the waiting queue assigned for the identified service. The behaviour of the call in the waiting queue is according to the FIFO (First In First Out) rule thus, the call at the beginning of the waiting queue will be routed to the agent as the first. The special event occurs when the caller is specified as VIP (Very Iportant Person). Then the call is not placed to the end of the waiting queue, but to the beginning. The calls rest in waiting queue till: release of soe agent of called service group, expiration of tie interval assigned for the custoers waiting, session cancelation fro the caller side. For callers assigned into the waiting queue it is possible according to the predefined scenario to play up the different announceents (the position of the ISSN: Issue, Volue, Noveber 2

3 caller in the waiting queue or forecast of the waiting tie), usic or ringing tone. The other possibility for the caller is to offer also inforation about new products without change of the position in the waiting queue. If the call is not processed till defined tie, this call can be routed to other service group (i.e. call overflow), or it can be played another announceent. In service group, the individual agent can have special skills (e.g. when the agent knows soe languages) and for this agent, in case of free, is assigned call that needs his special skills with the axiu priority. 2.3 Poisson and Exponential Distribution Nuber of incoing calls per tie unit represents the discrete rando variable. For description of rando variable, we use its probability function defined in discrete for for rando variable X, which takes values x as follows: f ( x) = P( X = x) () It is obvious, that f ( x) =, where H is the x H set of all values which the rando variable X can obtain. For description of incoing requests into the Queuing Syste the Poisson distribution [3] is the ost coonly used, for which is valid: λ ( k) = P( X = k) = λ, (2) p k = f e k! where E (X ) = λ, D (X ) = λ, λ represents the paraeter of distribution and specifies ean nuber of requests per tie unit. The tie which elapses between incoing of individual requests is also rando variable. Based on the independence of the individual requests, it is possible for Poisson input flow to declare, that this continuous rando variable has exponential distribution with ean value / λ [4]. Density of this rando variable is defined as follows (t represents the inter-arrival tie) [3]: t f ( t) = λe λ (3) The rando variable based on Poisson distribution is the ost frequently used atheatical odel for description of incoing calls into the contact center. 3 Markov Processes Markov processes offer flexible and very efficient tool for description and analysis of dynaical properties of stochastic queuing systes [2], [3]. Let T = {t, t,, t n } is set of various ties, while = t < t < < t n and let S = {s, s 2,, s } is set of all states in which the syste can be in various ties. Then such syste can be described by Markov processes, if the following condition holds (4): ( X s s ) P = (4) n t + n + X tn n i.e. probability that syste is in tie t n+ in state s n+, depends only on assuption that in tie t n the syste was in state s n and does not depends on in which states and which ties the syste was prior to tie t n [2]. 4 Markov Model M/M// Fro the large nuber of Markov odels of queuing systes, it is possible to choose the ost appropriate odel for each syste according to various criterions (coplexity, coputation costingness, etc.) [9]. Following part of our work tends to utilization of Markov odel M/M// in the Contact center environent. 4. Basic Assuptions of Model Let λ represents ean value of arrival rate of requests into syste and let µ represents ean rate of request handling, while ean handling tie of one requests is T serv = /µ. Graphical representation of M/M// syste is depicted in the Fig. 4. Fig. 4. Markov odel M/M//. In the Fig. 4 we can see the behaviour of the syste. The requests randoly arrive with Poisson distribution with ean value λ, these requests are handled, while handling tie is rando and can be described by exponential distribution with scale µ. Therefore syste is fulfilled with rate λ (transition rate fro state k to state k+, where k is nuber of requests in the syste). On the other hand, syste is eptying with rate kµ (fro state k+ to state k). This behaviour is valid until state k, in case of ore requests k in the syste as nuber of ISSN: Issue, Volue, Noveber 2

4 agents, agents can handle requests only with rate µ and every another incoing request have to wait in waiting queue for free agent, this waiting queue has unliited capacity (length). 4.2 Calculation Capacity of Model Through the basic equations of Markov odel M/M// and their odifications it is possible to calculate large nuber of iportant traffic paraeters of Contact center [2] - [5]. Syste stability ρ is as follows (5): λ ρ = (5) µ where represents nuber of agents. The condition ρ < ust be valid in order to syste stability. Paraeter P Q represents the probability that calling custoer will have to wait in the waiting queue for free agent (6): PQ = ( ρ ) P!( ρ ) (6) where P represents probability that syste is epty (7): k ( ρ ) ( ρ ) P = + =!! ρ (7) k k Mean nuber of requests in the whole syste is N (8): ρ N = ρ + P (8) ( ρ) Q The request spend in the syste ean tie T (9): µ + P Q T = µ λ (9) Mean nuber of requests waiting in the queue Q is as follows (): Q = ρ P ( ρ ) Q () while request spend in the waiting queue ean tie W (): W = ρ PQ ( ρ ) λ () Fro definition of paraeters λ and µ we have (2): λ A = (2) µ where A represents traffic load of the syste in Erlangs. 4.3 Erlang C Forula and M/M// Iediate rejection of call in case of occupation of all agents (as expected in Erlang B forula) is fro point of provided services in contact center bad solution. This shortness is eliinated in second Erlang forula - Erlang C forula. In the case of the call can t be iediately served, the call is placed into the waiting queue with infinite length. If one of the agent is free, the call fro waiting queue is autoatically assigned to this agent. In the case of epty waiting queue, the agent is free and he waits for the next call. Erlang C forula is defined as the function of two variables: nuber of agents N and traffic load A. Based on these values it is possible to deterine the probability of P C, that the incoing call will be not served iediately, but it have to wait in waiting queue. By substitution of (2) into (5) and further substitution of (5) into (6) and (7) and by odification of equations we have (3): P C A, =! ( A) k A A + (3) ( A) ( A) k= k!! Equation (3) is known as Erlang C forula [], where P C represents probability that request will have to wait for handling in the waiting queue. It is obvious that equations (6) and (3) are the sae. Fro the analytical results we can see that Erlang C forula and Markov odel are identically []. By utilization of this knowledge and by extension of Erlang forula with paraeter GoS (Grade of Service), representing level of offered services [], [6], it is possible to extend the coputing capacity of M/M// odel with this paraeter (4): GoS = µ ( A) T P W Q e (4) where GoS represents ratio of incoing calls that will be assigned to agent until selected tie T W. ISSN: Issue, Volue, Noveber 2

5 On the basis of realized calculations [6] it is possible to use Erlang B forula for calculations of following paraeters of contact center: nuber of agents needed to handle defined traffic load (A) at certain probability of call blocking (P B ), axiu possible traffic load of contact center at constant nuber of agents, probability of call rejection at defined load and nuber of agents. And it is possible to use Erlang C forula for calculations of following paraeters of contact center: nuber of agents needed to handle A at defined axiu values for probability of call waiting in waiting queue (P C ) and P B, Grade of Service (GoS) at defined nuber of agents, defined A and average call processing tie, axiu possible A at defined nuber of agents (contact center stability), P B at defined A and nuber of agents, P C at defined A and nuber of agents. 5 Calculation of Iportant Traffic Paraeters of Contact Center The right design of Contact center requires the precise estiation of basic traffic paraeters of Contact center, e.g. ean rate of telephone call arrivals into syste and ean handling tie of one call. Unless otherwise stated, all calculations of traffic paraeters of Contact center are tied with basic values: λ = 6 incoing calls per hour, call handling tie is T serv = 5 inutes. Fro (2) results that average traffic load of such Contact center is 5 Erl. 5. Calculation of Required Nuber of Agents The nuber of agents is the basic traffic paraeter needed in design of Contact center. In the case of accurately estiated values λ and T serv the subsequent calculation of required nuber of agents according to (6) is relative siple. Fro the Fig. 5 we can see the relation between probability of waiting in the queue and raising nuber of agents. Let axial probability P Q in odeled Contact center does not exceed 5 %. Fro the Fig. 5 and Tab. it is obvious that the first suitable nuber of agents is =. P Q [%] ,6 % Nuber of agents Fig. 5. Relation between P Q [%] and nuber of agents. Also fro the Fig. 5 we can see that if we have agents, the probability P Q = 3,6 %. It eans that fro calling custoers into Contact center up to 4 custoers have to wait in the waiting queue. Other values of traffic paraeters of Contact center are stated in the Tab.. Table. Traffic paraeters if A = 5 Erl. P Q T W GoS N Q [%] [in] [in] [%] ρ 8 6,73 5,28 5,28,28,28 87,6,63 9 8,5 5, 5,,, 94,6,56 3,6 5,4 5,4,4,4 97,8,5,5 5, 5,,, 99,7,45 2,59 5, 5,,, 99,7,42 Note : In the Tab. we can see that traffic paraeters N, T and Q, W reach up the sae values. It is necessary to reind that λ = 6. In Markov odel M/M//, and siilarly in others queuing systes following equation is valid (5): N = λ T (5) In the original calculation all ties are stated in hours. For better view, the ties T and W in the Tab. are stated in inutes, thus ultiplied by 6, what is by coincidence also value of λ. Equation (5) is called Little law, which says that average nuber of requests in the syste equals ean tie which the request spend in the syste ultiplied by ean rate of request arrivals [2], [5], [7]. ISSN: Issue, Volue, Noveber 2

6 5.2 Traffic Paraeters and Traffic Load Variation After calculation of required nuber of agents ( = ) it is necessary to analyze the properties of designed Contact center at different traffic loads. Firstly, it is necessary to know where the traffic liits of designed Contact center are. The basic representation is given in the Fig. 6. P Q [%] = A [Erl] = Fig. 6. Relation between P Q [%] and A [Erl]. = 2 Fro the relation between P Q [%] and A [Erl] we can see that with raising traffic load the probability that custoer will have to wait for handling in the waiting queue is rising sharply. For coparison, also dependencies between P Q [%] and A [Erl] for the systes with 8 and 2 agents are depicted in the Fig. 6. The better view of odeled Contact center behaviour give the graphs figure out in the Fig. 7 and Fig. 8. N, Q N Q = A [Erl] = 8 = 2 Fig. 7. Relation between nuber of requests in syste/queue and A [Erl]. increasing is very sharply. The sae breaking point can be seen in Fig. 8 between dependencies of the ean tie, that request will spend in the syste, and traffic load. T, W [in] T [in] W [in] A [Erl] = 8 = = 2 Fig. 8. Relation between the ean tie that request spend in the syste/queue [in] and A [Erl]. Breaking point position gives the axiu traffic load which can be handled by Contact center without change in the nuber of agents or in the call handling tie. Fro the Fig. 7 and Fig. 8 it is obvious that breaking point for Contact center with nuber of agents = and ean call handling tie T serv = 5 inutes represents the value A = 9 Erl. Precise values are stated in the Tab. 2. Traffic load value 9 Erl eans in other words 8 incoing calls per hour at ean call handling tie 5 inutes. Table 2. Traffic paraeters if =. A P λ Q T W N Q [Erl] [%] [in] [in] req 4 48,88 4, 5,,, ,6 5,4 5,4,4,4 6 72,3 6,5 5,3,5, ,7 7,52 5,37,52, ,92 9,64 6,2,64, ,87 5,2 8,34 6,2 3,34 5 The Tab. 2 also contains paraeter req. This paraeter says about required nuber of agents at given traffic load in order to preserve axial probability P Q under level of 5%. Note 2: Little law (5) described in Note is also valid here. Fro the trace of dependencies between average nuber of requests in the syste/queue and traffic load (see Fig. 7) we can see that initially it is increasing oderately and alost linearly until it reaches breaking point value of traffic load, then ISSN: Issue, Volue, Noveber 2

7 6 Contact Center as Multi-phase Queuing Syste The previous part of the paper was dealt with Contact center as syste in which incoing calls are served directly by Contact center agents, respectively these calls have been waiting in the queue for free agent. In fact, there are often Contact centers in which the first contact with custoer is ade by IVR (Interactive Voice Response) unit [7]. The ain tasks of IVR unit are identification of calling custoer, to recognize the atter of his proble and after then the call is transferred to the ost skilled agent that will solve given proble. This way can effectively shorten the call handling tie by tie needed for custoer identification and find out the atter of his proble. Modeling of Contact center with IVR unit is very coplicated, the wide range of statistical calculations and detailed knowledge of Markov processes are needed [8]. The ore siple way is utilization of ultiphase queuing syste (QS) theory (Fig. 9). QS QS 2 Fig. 9. Multi-phase queuing syste. Let QS 2 is Contact center odeled by Markov odel M/M// as till now, with nuber of agents =. Let QS represents the IVR unit odeled by Markov odel M/M/ /. 6. Interactive Voice Response Interactive Voice response (IVR) represents the abbreviation for the electronically synthetized voice output syste, that will greet the custoer and provide hi the possibility of the session with service or agent. The syste can be used for counication in ore languages. Therefore the IVR application can at the beggining of the session request the caller to choice the counication language. Through the IVR function the selection of needed service and identification of the caller can be autoatically recognized. After connection of the caller with the agent, the agent autoatically see on the desktop of his PC the ost iportant data about the custoer. Thus, the agent of contact center is copetent to serve the caller. The success of the IVR syste is based on the caller s experience. The IVR application ust give the caller always the possibility of choice. The IVR syste enables: iniizing of interruption in realized session, providing of interactive custoer oriented services, which are available not only within working hours, providing of inforation on deand by voice, fax and e-ail, providing of IVR services for autoatic transactions, autoatic voice recognition, serving the calls also when the agents are busy, providing the inforation about estiated waiting tie, interactive waiting queues, that offer services - e.g. ordering within waiting (the caller save his order in waiting queue). 6.2 Markov Model M/M/ / Contrary to M/M// odel, this odel assues unliited nuber of service stations. It eans that every request incoing to syste is iediately handled. In this syste no waiting queue exists and infinite nuber of requests can be siultaneously in the syste [3]. It is obvious that such syste is always stable, i.e. the syste will never be overloaded by requests that it can not handle. λ N = (6) µ Mean tie which the request will spend in the syste is (7): T = = µ T obs. (7) 6.3 Multi-phase Queuing syste M/M/ / + M/M// Let traffic paraeters of queuing syste QS represented by M/M/ / odel has index and traffic paraeters of queuing syste QS 2 M/M// has index 2. The one of the assuptions of both systes is unliited requests population. Therefore such ultiphase queuing syste is called open. In the open syste the following equation is valid [3]: λ = λ2 = λ. (8) ISSN: Issue, Volue, Noveber 2

8 Total nuber of requests in the syste is (9): N = N + N 2. (9) Total tie which request will spend in the syste is (8): T = T + T 2. (2) 6.4 Contact Center odel with IVR Let IVR in Contact center is represented by M/M/ / odel and Contact center itself is represented by M/M// odel described in the previous part. Let the ean handling tie of one call T serv = 5 in. is divided as follows: the request spend in IVR unit average tie T serv = in. Average tie of call handling by agent is shortened by IVR to T serv2 = 4 in. Average nuber of requests incoing to Contact center is λ = 6 calls per hour. Model of this syste is depicted in the Fig..... Fig.. Contact center odel with IVR. 6.5 Ipact of IVR on Modeled Contact Center Ipact of IVR syste on odeled Contact center is figure out in the Fig.. Nuber of requireents Nuber of requireents in syste after handling in IVR Nuber of requireents in IVR Total nuber of requireents in syste with IVR Total nuber of requireents in syste without IVR A [Erl] Fig.. Ipact of IVR on odeled Contact Center. Fro the course of dependencies between average nuber of requests in the syste and traffic load A [Erl] it is obvious that breaking point has oved till behind value A = Erl, contrary to 9 Erl in the syste without IVR unit. The Contact center with IVR is hence stable with this volue of traffic load, when the syste without IVR can not be already used. Through the IVR unit the Contact center is able to handle traffic load up to Erl, what represents 32 incoing calls per hour at ean call handling tie 5 inutes. It is obvious that if the odeled Contact center was designed for handling traffic load up to 9 Erl, while required nuber of agents was, after adding the IVR unit into the syste, the lower nuber of agents will be needed under the sae conditions, what eans great savings of operational costs. Precise values are stated in the Tab. 3. Table 3. Coparison of systes with/without IVR. A P N [Erl] N 2 N Q Savings [%] [%] without IVR 5 5,4 3,6 with IVR 5, 4,6 5,6 8 5,9 2 with IVR 5, 4,2 5,2 9 2,38 Fro the Tab. 3 we can see that if A = 5 Erl approxiately the sae nuber of requests on average is in the syste with or without IVR unit. The difference lies in the different nuber of required agents. The syste with IVR unit is alost the sae as the syste without IVR when 8 agents are eployed, but such syste fails in condition P Q < 5 %. The syste with IVR with 9 agents actually overreaches the syste without IVR with agents in value P Q. One agent in such syste represents % of total nuber of agents. The use of IVR unit brings relative high savings in needed nuber of agents. The savings have ainly financial character, because the costs for contact centers agent are every onth periodically repeated. This fact is stated in the Tab. 3 as paraeter Savings. 7 Conclusion The basic assuptions for success achieving are above quality of service [8]. In case of services provided by contact center are agents very iportant and therefore it is necessary to deterine their required nuber. When there are a few agents, it causes reduction the nuber of custoers (the custoers are not disposed to wait in waiting queue ISSN: Issue, Volue, Noveber 2

9 too long). On the other hand, when there are lot of agents, so they are unused and they also increases the financial costs of contact center traffic. It is therefore necessary to give attention to optial nuber of agents in contact center. Markov odel M/M// offers interesting tool for Contact Center analysis. The ain advantage of this odel is the wide spectru of iportant traffic paraeters of Contact Center calculations: P Q - probability that calling custoer will have to wait in the waiting queue for free agent, P - probability of epty syste, N - ean nuber of requests in the whole syste, T - ean tie that the request spend in the syste, W - ean tie that the request spend in the waiting queue, Q - ean nuber of requests waiting in the queue. Other possibilities bring interconnection of Markov odel with Erlang C odel, especially the estiation of Grade of Service, which can be one of the ain paraeters in QoS evaluation of odeled Contact Center [2], [3]. Aong the disadvantages of this odel there are soe restrictive basic assuptions of the odel, ainly assuptions of unliited requests population and infinite capacity of waiting queue. In spite of these liitations, we can iniize the, for exaple by chosen axial value of traffic load A or through T W paraeter in GoS estiation. Interesting results offer adding of IVR unit into odeled Contact Center. Using ulti-phase queuing syste theory and interconnection of M/M// odel with M/M/ / odel creates fro Markov M/M// odel ultiately coplex tool for Contact Center analysis. Acknowledgeent This work is a part of research activities conducted at Slovak University of Technology Bratislava, Faculty of Electrical Engineering and Inforation Technology, Departent of Telecounications, within the scope of the projects VEGA No. /565/9 Modelling of traffic paraeters in NGN telecounication networks and services and ITMS Support for Building of Centre of Excellence for SMART technologies, systes and services II. This work has been also supported by the Ministry of Education of the Czech Republic within the project LM25. References: [] DIAGNOSTIC STRATEGIES: Traffic Modeling and Resource Allocation in Call Centers, Available on Internet: < c_modeling.pdf> [2] G. Bolch, S. Greiner, H. de Meer, S. Trivedi, Queuing Networks and Markov Chains, John Wiley & Sons, Inc., Hoboken, New Jersey, 26. [3] J. Polec and T. Karlubíková, Stochastic odels in telecounications, Published by Jozef Murgaš fund for telecounications in FABER, 999,. edition, ISBN [4] Washington University in St. Louis, The Departent of Coputer Science & Engineering: Queuing theory, Available on Internet: < quick-qt.pdf> [5] M. Veeraraghavan, M/M/ and M/M/ Queuing Systes, 2. March 24, < res/qt.pdf> [6] Ch. Hischinua, M. Kanakubo and T. Goto, An Agent Scheduling Optiization for Call Center, In: The 2 nd IEEE Asia-Pacific Services Coputing Conference, 27. [7] I. Baroňák and E. Chroý, Contact center part of odern counication infrastructure, In: Telekounikace, no., Noveber, 24, pp [8] J. Wang and R. Srinivasan: Staffing a Call Center with Interactive Voice Response Units and Ipatient Calls, In: IEEE/SOLI 28, IEEE International Conference, 28. [9] T. Mišuth and I. Baroňák, Perforance Forecast of Contact Centre with Differently Experienced Agents, In: Elektrorevue. ISSN Vol. 5, 6..2 (2), art. no. 97. [] E. Chroy, T. Misuth and M. Kavacky, Erlang C Forula and Its Use In the Call Centers, In: Journal AEEE - Inforation and Counication Technologies and Services, Volue 9, Nuber, pp. 7-3, March 2, ISSN [] M. Mrajca, J. Serafin and Z. Brabec, Optiization of the NGOSS Change Manageent Process Siulation Model, In: The th International Conference KTTO 2, June 22-24, Szczyrk, Poland, 2, pp. 4-8, ISBN ISSN: Issue, Volue, Noveber 2

10 [2] B. Kyrbashov, I. Baroňák, M. Kováčik and V. Janata, Evaluation and Investigation of the Delay in VoIP Networks, In: Radioengineering, ISSN 2-252, Vol. 2, No. 2 (2), pp [3] J. Mičuch and I. Baroňák, Coparison of Two Probability Theories Providing QoS, In: Elektrorevue, ISSN , Vol. 5, 6..2, art. no 2. [4] K. Vastola, Interarrival ties of a Poisson process, Troy (New York, USA): Rensselaer Polytechnical Institue, 5.March 996. Available on Internet: < erf/node32.htl>. [5] I. Baroňák and P. Kvačkaj, A New CAC Method Using Queuing Theory, In: Radioengineering. ISSN Vol. 7, No. 4 (28), Part. II, pp [6] E. Chroy, J. Diezka and M. Kovacik, Traffic Analysis in Contact Centers, In: The th International Conference KTTO 2, June 22-24, Szczyrk, Poland, 2, pp. 9-24, ISBN [7] M. Fernandes, C. Lia and J. Schiiguel, Strategic Manageent Using VoIP Technology: a Case Study in a Call Center Copany, In: WSEAS TRANSACTIONS on COMMUNICATIONS, Issue, Volue, January 2, pp , ISSN: [8] M. Voznak and J. Rozhon, Methodology for SIP Infrastructure Perforance Testing, In: WSEAS TRANSACTIONS on COMPUTERS, Issue, Volue 9, Septeber 2, pp. 2-2, ISSN [9] D. Tran and T. Pha, A Cobined Markov and Noise Clustering Modeling Method For Cell Phase Classification, In: WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE, Issue 3, Volue 3, March 26, pp. 6-66, ISSN About Authors Erik Chroy was born in Velky Krtis, Slovakia, in 98. He received the Master degree in telecounications in 25 fro Faculty of Electrical Engineering and Inforation Technology of Slovak University of Technology (FEI STU) Bratislava. In 27 he subitted PhD work fro the field of Observation of statistical properties of input flow of traffic sources on virtual paths diensioning and his scientific research is focused on optiizing of processes in convergent networks. Nowadays he works as assistant professor at the Institute of Telecounications of FEI STU Bratislava. Jan Diezka was born in Dolny Kubin, Slovakia in 988. He is a student at the Institute of Telecounications, Faculty of Electrical Engineering and Inforation Technology of Slovak University of Technology (FEI STU) Bratislava. He focuses on application of Erlangs' forulas in Contact Centers. Matej Kavacky was born in Nitra, Slovakia, in 979. He received the Master degree in telecounications in 24 fro Faculty of Electrical Engineering and Inforation Technology of Slovak University of Technology (FEI STU) Bratislava. In 26 he subitted PhD work Quality of Service in Broadband Networks. Nowadays he works as assistant professor at the Institute of Telecounications of FEI STU Bratislava and his scientific research is focused on the field of quality of service and private telecounication networks. Miroslav Voznak holds position as an associate professor with Departent of telecounications, Technical University of Ostrava. He received his Master and PhD degree in telecounications, dissertation thesis Voice traffic optiization with regard to speech quality in network with VoIP technology fro the Technical University of Ostrava, in 995 and 22, respectively. Topics of his research interests are Next Generation Networks, IP telephony, speech quality and network security. ISSN: Issue, Volue, Noveber 2