Modeling of Contact Center Traffic E. Chromy, S. Petran and M. Kavacky * Abstract Paper deals with operation of the contact center and its modeling. Erlang B and Erlang C formulas are used for our calculations. Modeling is enriched with call redial situations and with parameter Grade of Service. Through the particular calculations it is possible to estimate the required number of agents for model solutions while considering various traffic parameters: probability of blocking, probability of enqueue, waiting time in queue, etc. Keywords Contact Center, Erlang formulas, Quality of Service, Traffic modeling. I. INTRODUCTION N order to provide services, or help with the problem Ifor large number of customers from one place with comparatively less costs a contact centers have built their place in every major company. They represent a large number of sophisticated interconnection of software and hardware resources that allows the customers to access the contact center by various communication channels. Parameters of contact centers can be well estimated and then calculated by applying an appropriate model. There are many models of contact center based on mathematical and statistical knowledge. However, with a large number of models brings a big difference in computational complexity and in their accuracy. Our work addresses two models: Erlang B and Erlang C formula. It examines the limits of call rejection and the corresponding number of contact center agents. Erlang B formula is in Section 3 enhanced by repeated calls (in the case that the customer is rejected). II. CONTACT CENTER Contact center as a connection of customer and the company is due to the trend of new technologies equivalent to personal contact in service offering and support for customers. In laic terms, this connection is just an ordinary tele- E. Chromy is with the Institure of Telecommunications, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology Bratislava, Ilkovicova 3, 119 Bratislava, SK (phone: +1 7951; fax: +1 7951; e-mail: chromy@ut.fei.stuba.sk). S. Petran is with the Institure of Telecommunications, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology Bratislava, Ilkovicova 3, 119 Bratislava, SK (e-mail: petran.stefan@gmail.com). M. Kavacky is with the Institure of Telecommunications, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology Bratislava, Ilkovicova 3, 119 Bratislava, SK (e-mail: matej.kavacky@stuba.sk). phone call or virtual contact with company employees - agents. In fact, the connection is based on a large set of technical solutions in the background. Customers have different ways to communicate with the contact center: phone call, SMS, chat, e-mail, video, VoIP (Voice over Internet Protocol) and others. Integration between the computer and the telephone device provides technology CTI (Computer Telephony Integration), through the different standards. One of the most important components of the contact center is the ACD (Automatic Call Distribution), which contains of complex routing algorithm of interactions. The challenge is to optimize the time the request spent in system. ACD is therefore responsible for routing calls to the IVR (Interactive Voice Recording), agent or waiting queue. The supervisor is responsible for agents and for help with problems of agents. Contact center is a queuing system based on the theories of queues, which can represent different types of communication [1,, 3]. More information about Quality of Service in [ - 13] III. ERLANG B FORMULA The main feature of one of the simplest mathematical models, Erlang B formula, is the lack of queues. Request is upon arrival routed directly to the agent. However, if none of the agents is available, the request is blocked, or rejected. In this case the customer is leaving and the request is lost. Due to this, the model is also referred as a system with loss [1], [15]. Erlang B formula uses three basic parameters: A traffic load [erl], N number of agents (number of concurrent served requests), P B probability of blocking/rejection of request. If the values of parameters A and N are known, it is possible to calculate the blocking probability [15] by the Erlang B formula (1):,!! However, if you do not know the value of the traffic load A, but the value of incoming requests λ is known and the (1) ISBN 97--7--5 11
average number of requests μ for the same unit of time, A can be expressed as: () Thanks to the iterative numerical methods we can backward state the original load A (7), in which the contact center can handle the requests [15]. 1,. (7) Then by substituting equation () into equation (1) we have equation (3):,, 1! 1! Through this substitution the same equation is derived as in the Markov model M/M/n/n [15]. A) Erlang B formula and call repetition Erlang B formula can be enhanced with assumption of generation traffic with repetitive calls in the case if the previous call was not successful, or it was rejected. (3) IV. ERLANG C FORMULA Rejection of the caller request is in the quality of contact center major disadvantage. This lack addresses the Erlang C formula, which incorporates a waiting queue. If the request can not be served immediately, it is placed in the queue with unlimited length where it is waiting to be served by the first available agent. If the waiting queue is empty, the agent waiting for the next request [17]. Erlang C formula uses three basic parameters: A traffic load [erl], N - number of agents (number of concurrent served requests), P C probability of request enqueue.,!!! () Figure 1. Principle of enhanced Erlang B formula. In this case the Erlang B formula includes a new variable, r factor (recall factor), which reflects the probability that the customer is repeatedly trying to call a contact center agent [1]..,. () In this enhanced case the traffic is composed of two parts (). The factor A, traffic load generated by the first call attempts and r factor, which represents the traffic load of repeated calls. P B is the blocking probability from the original Erlang B formula. In the Fig. 1 under the variable C are expressed served requests and under the variable O rejected requests. Served requests can be expressed by equation (5).., (5) In the same way we can express the rejected requests O ()..,. 1 () If the value of the traffic load A is not known, but we know the rate of incoming requests λ and the average number of requests μ for the same unit of time, A can be expressed as follows: (9) The variable η represents the load of a single agent (1): μ (1) By substitution of (9), (1) into equation () we can express equation (11):,!!!! 1!! 1 (11) ISBN 97--7--5 111
Equation (11) is identical to the equation for the Markov model M/M/n/ [1]. By adding the waiting queue to system the modeling of contact centers is enriched by several traffic parameters. An important factor for the customers is a waiting time in the queue, which is described by a distribution function: 1, (1) By use of (1) it is possible to calculate the average length of time spent in the queue, from the enqueue until the request is taken by free agent: (13) It is possible to express the average number of requests in the queue: (1) By use of general definition of the distribution function (11) it is possible to derive an equation for calculating the parameter GoS (Grade of Service). GoS parameter expresses the percentage of calls that are answered until predefined time AWT (Acceptable Waiting): 1 (15) V. MODELING OF TRAFFIC PARAMETERS OF CONTACT CENTER We have used Matlab software for calculating of the traffic parameters of contact center. Matlab offers a variety of useful and simple tools suitable for modeling of contact center traffic. B) Modeling with Erlang B formula For the calculation of traffic parameters or blocking probability P B the rate of incoming requests and request handling time are defined. By using a gradual increase of the number of agents it is possible to show the progress of rejection probability P B, or other useful courses. Traffic values of the model contact center are following: λ = 3 (number of incoming requests per hour) μ = /hour (average number of answered requests) P B = 5% (probability of call rejection). Blocking probability P B for certain number of agents can be expressed by equation (3). From a practical point of view it is unacceptable if the contact center is refusing the requests. On the contrary, it is desired to have the most answered requests. This is the reason why among the defined variables is also P B, which gives the maximum threshold of the probability of rejected requests. 1 Average number of requests K in the system is: 1 And average time that request spent in the system is: 1 1 (1) (17) Blocking probability - Pb [%] 1 1 1 Pb = 5% 1 1 3 Another important parameter is an average utilization of agents η: 1!!! (1) Figure. Probability of request rejection. In the Fig. we can see as a request rejection probability P B depends on the number of agents N. The course is decreasing. From Fig. it is also obvious that to fulfill the requirement of P B (red line) for the model with a traffic load of A = 15 erl we need N = agents. When operating with agents the probability of request rejection is P B =.5%. In order ISBN 97--7--5 11
to fulfill the condition of 5%, the higher number of agents is not required. Fig. 3 shows the progress of the request rejection probability P B for increasing traffic load of 1% and %. Blocking probability - Pb [%] Figure 3. Increasing of traffic load. When the traffic load is increased to 1%, the required number of agents is raised to and with an increase to % we need agents. Therefore, if the operator of the call center is taking into account an increase of traffic while maintain the maximum probability of call blocking P B at 5%, it must take into consideration the possibility of employment of more agents. When considering the possibility of repeated calls the Erlang B model contains R factor. Blocking probability - Pb [%] 1 1 1 1 1 17 1 19 1 3 5 7 9 3 7 5 3 1 Traffic without R factor Traffic with R factor (r = 1,) Traffic with R factor (r =,5) Figure. Application of R factor. Traffic load A = 15 erl (1%) Traffic load A = 1,5 erl (11%) Traffic load A = 1 erl (1%) Pb = 5% Pb = 5% 1 19 1 If the initial traffic load of the contact center is A = 15. erl and we consider parameter r =.5 (which means that every second refused request will return), then the total system load is A = 15.3 erl () when the number of agents is N =. The value of the blocking probability is P B = 5.17%, and it is depicted in the Fig. Therefore there is a situation where the operation with redials (for r =.5) with the number of agents N = exceeds the defined threshold blocking probability of 5%. Therefore an operator of contact center must increase the number of call center agents by one. The same consideration applies when each rejected call will return (r = 1.). This case is also shown in the Fig. C) Modeling with Erlang C formula Since the Erlang B formula is due to lack of waiting queue not suitable for contact center modeling, we will model it by use of Erlang C formula. The model contains unlimited waiting queue and provides a greater set of traffic parameters. Thanks to this, the call center operator can focus on specific problems and their better solutions. The input variables are λ, μ from which by use of equation (9) traffic load can be calculated. At certain number of agents N we have from equation () the probability that call will be not immediately answered and it will be enqueued. The values of input variables are: λ = 3 (number of incoming requests per hour), μ = /hour (average number of answered requests), AWT = 3 seconds (acceptable waiting time), P C = 5% (probability of enqueue). Traffic load A is the same as in the previous model (i.e. 15. erl). From the notation of the basic equation () for the Erlang C formula implies the condition that the number of agents can not fall below the traffic load of the contact center (i.e. N> A). Unlike the previous model, Erlang C model never refuses requests, but it places them to the queue. Value P C = 5% represents the fact that each twentieth request entering the contact center will be placed to the queue with unlimited duration. Probability of enqueue - Pc [%] 1 1 1 1 1 1 3 Figure 5. Probability of request enqueue. Pc = 5% ISBN 97--7--5 113
From the Fig. 5 we can see that the probability of request enqueue decreases while the number of agents is increasing. If the contact center model with defined quantities has to meet the maximum probability of enqueue P C, we need just N = 3 agents. The value of the probability of enqueue P C is 3.% for given number of agents and for traffic load A = 15 erl. As mentioned above, Erlang C formula allows us to estimate the broader set of traffic parameters. Of defined values we can calculate following parameters: traffic load A, average number of requests in waiting queue Q, average number of requests in whole system K, average time T, that request spent in the system, average utilization of agents η, average time W, that request spent in the waiting queue, Grade of service GoS. Grade of Service - GoS [AWT = 3 s] 1 9 7 5 3 1 17 1 19 1 3 5 Figure. Parameter GoS. Fig. shows the GoS parameter, which is very important from the perspective of the customer and his evaluation of the contact center quality. Progress of the parameter GoS has exponential nature of growth in respect of the number of agents. Substantial change of GoS parameter is caused by any change in the number of agents, but beyond a certain limit the change of GoS is only minimal. Due to the high marginal probability of the enqueue P C, GoS value is equal to 99.%. In this way sized contact center says that only one percent of the requests will be queued to wait over to serve more than 3 seconds. GoS value is very high and thus opens the space for possible reduction of the number of agents in the contact center. The reduction of agents will cause decrease of the GoS value, but it would have not a large impact on QoS of contact center. VI. CONCLUSION The paper presents results of traffic modeling in contact centers through Erlang B and Erlang C formulas. Each of our models gives required number of agents for satisfying defined QoS: if it is probability of request rejection (Erlang B) or probability of request enqueue (Erlang C). For Erlang B formula there is also presented the case of instant call repetition if the call was rejected. Due to this fact it is shown that for defined contact center parameters the condition of defined probability of request rejection is not fulfilled in this model, therefore the operator has to increase the number of contact center agents. The calculation of GoS for Erlang C model has shown that grade of service of contact center for required agents is on the very high level. The customer is served fast, what is the main purpose of contact centers. Similarly to Erlang formulas there are also Markov models, e.g. M/M/m/K model which allow us to calculate the following parameters: probability of call rejection in the case of fully occupied waiting queue, total number of requests in contact center, probability of empty contact center. ACKNOWLEDGEMENT This work is a part of research activities conducted at Slovak University of Technology Bratislava, Faculty of Electrical Engineering and Information Technology, Institute of Telecommunications, within the scope of the project VEGA No. 1/1/11 Analysis and proposal for advanced optical access networks in the NGN converged infrastructure utilizing fixed transmission media for supporting multimedia services and Support of Center of Excellence for SMART Technologies, Systems and Services II., ITMS 19, co-funded by the ERDF. REFERENCES [1] I. Baronak, I.: Kontaktné centrum. Slovenská Technická univerzita v Bratislave, Nakladateľstvo STU, Vydanie 1., 1, ISBN 97--7-31-1. [] L. Unčovský L.: Stochastické modely operačnej analýzy. Alfa, 19. [3] T. Misuth, and I. Baronak, Packet Loss Probability Estimation using Erlang B and M/G/1/K Models in Modern VoIP Networks, Journal of Electrical and Electronics Engineering, Istanbul University, Vol. 1, Iss. (1), ISSN 133-91, pp. 13-191. [] M. Halas, and S. Klucik, Modelling the Probability Density Function of IPTV Traffic Packet Delay Variation, Advances in Electrical and Electronic Engineering, Vol. 1, No. (1), ISSN 133-137, pp. 59-3. [5] S. Klucik, DiffServ Queueing Methods Comparison, Telecommunications and Signal Processing TSP-1: 33rd International Conference on Telecommunications and Signal Processing, Baden near Vienna, ISBN 97--7--5 11
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