Performance Estimation of Contact Centre with Variable Skilled Agents

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1 Performance Estimation of ontact entre with Variable Skilled Agents Tibor Mišuth, Erik hromý, and Ivan Baroňák Abstract Our paper deals with ontact centre quality level estimation using Erlang mathematical model. We focus on influence of agent skill level on contact centre metrics and compare the result against original model. We present method for parameters bounds estimation using Erlang s formula. Moreover we emphasize the influence of staff skill level requirements on the overall quality of service the ontact entre provides. Index Terms ontact entre, Erlang model, Quality of Service I. ITRODUTIO OTAT centre is one of many other ways how the institution is able to cover all its communication requirements towards clients and partners. It is a structured communication system consisting of both human and technological resources, which improves the communication between organization and customers. It is more or less a software and hardware upgrade of a private branch exchange which is the primary medium for a call access to the organization. It combines telephone processes and data processing to achieve the most effective business transactions. The ontact entre consists of many elements of which the most important is Automatic all Distribution (AD) module. It is responsible for routing of incoming queries to appropriate service group and agent. umber of agents in ontact entre has great impact on quality of service it provides. Thus staff requirements estimation based on mathematical model and verified by simulation are extremely handy during ontact entre design process. We analyze the influence of variability in agents skills level on the contact centre performance and propose a modification Tibor Mišuth is with the Slovak University of Technology, Faculty of Electrical Engineering and Information Technology, Dept. of Telecommunications, Ilkovičova 3, Bratislava, Slovakia ( tibor.misuth@ stuba.sk). Erik hormý is with the Slovak University of Technology, Faculty of Electrical Engineering and Information Technology, Dept. of Telecommunications, Ilkovičova 3, Bratislava, Slovakia ( erik.chromy@ stuba.sk). Ivan Baroňák is with the Slovak University of Technology, Faculty of Electrical Engineering and Information Technology, Dept. of Telecommunications, Ilkovičova 3, Bratislava, Slovakia ( ivan.baronak@ stuba.sk). to original model that covers this phenomenon. The next parts of this paper are structured as follows. First the Erlang queuing model is presented. In section 3 a possible model modification is presented. Finally, the are validated by comparing to simulation measurements. II. ERLAG MODEL OF OTAT ETRE Each component of AD (or contact centre) system can be more or less precisely converted to a mathematical model. Since the contact centres usually handle large amount of requests per time unit, the vast majority of these models are based on queuing theory. Accuracy of depends on a correct model selection. Also the precise description of input parameters and variable dependencies can significantly affect them. Danish mathematician A. K. Erlang is very closely tied to the origin of queuing theory [1]. He published the paper concerning application of statistics in telephone service in 199. This document begins further research of queuing theory. Together with Markov chains theory his ideas helped to define and describe more complicated queuing models. Even now, a century later since the Erlang models have been published, are these one of the basic approaches used in parameter estimation of telecommunication systems. Since the contact centre is considered to be a telecommunication system the models are valid for it as well. Both Erlang and Poisson process based Markovian models use the same assumptions and following requirements must be met [2]: number of sources (requests population) is much greater than the number of servers, requests are generated randomly and independently of each other, average number of requests per time unit from all sources is constant, request handling time is a random variable with exponential distribution, queuing is based on the First in First out (FIFO) algorithm. In this paper we focus on the second Erlang s model, so called Erlang model. Presence of simple FIFO queue eliminates overload situations. If a temporary overload situation occurs, the request is put into the queue until any of the agents become available, i.e. the handling of any 391

2 previous requests is finished. The FIFO strategy ensures the caller waiting the longest time to be served the first. Erlang model [2] is originally defined as function of two variables: the number of agents and the traffic load A (expressed in Erl). Based on these parameters it calculates the probability P (1) that the arriving request is not assigned to the agent immediately and it has to wait in the queue. A! ( A) P = (1) (, A) 1 i A A + i= i!! ( A) If we know the rate of calls per time unit λ and the average number of served requests per the same time unit (so the average handling time is 1/) then the traffic load can be easily evaluated as [3] λ A =. (2) Furthermore we define the variable ρ, that represents the average utilization of each agent as [1], [4], [5] λ =. (3) For system stability reason (i.e. the number of requests present in the queue does not extend to infinity) the value of is required to be less than 1 (stability criterion). From the conjunction of (1), (2) and (3) we can obtain following formula P (, ) ( )! ( 1 ) i ( ) ( ) + i!! ( ) = (4) 1 i= 1 If we start from Markov M/M/m/ queuing model we can derive identical equation that will define probability that m or more requests are present in the queuing system so the new incoming request will be inserted into the queue [1], [4], and [5]. This means that Markov M/M/m/ and Erlang models are identical and this relationship can be easily proved analytically. Since the Erlang model contains a queue, there are some more parameters and variables that can be measured and more or less influenced by model inputs. From the callers point of view the most important value is the waiting time or the time the request spends in the queue before it is served by an agent. This value is a random variable described by the probability density function [5] 1 P τ = F W ( τ ) = ( A) τ (5) 1 P e τ > Based on this formula we can calculate the average waiting time (or average speed of answer ASA) W P W = (6) ( A) and using Little s theorem [5] and (6) we can obtain the average number of requests in the queue Q λ A Q = λ W = P = P (7) ( A) ( A) The general definition of probability density function of any statistical distribution and its properties [6] gives us an opportunity to derive Grade of Service () parameter value from equation (5) once the Acceptable Waiting Time (AWT) value is known [1], [5] ( A)AWT = 1 P e (8) defines the percentage of incoming calls that are answered no later than defined AWT (usually sec.). Following equation is valid for the average number of requests K in queuing system [5] A K = + P = A + P = A + Q (9) 1 ( A) Again, the Little s theorem allows us to obtain the average time T the request will spend in the system K A + Q 1 1 P T = = = + W = + (1) λ λ A ( ) III. VARIABLY SKILLED AGETS As we mentioned earlier the Erlang model is one of the most frequently used contact centre dimensioning tools. Unfortunately its basic form is limited to only one service group (one queue and several agents) and considers all agents as equal in terms of their skills and call handling capability. In reality each agent has his / her own skill level therefore the average handling time among agents is different. Based on the agent s skill level the average number of calls the agent is capable of handling per time unit can be denoted as i. It is straightforward that average handling time is the reciprocal value. Then the total number of calls that can be received by all agents in service group (or contact centre) is = (11) TOT i= 1 i and average call handling time is 1 1 T H = = TOT i= 1 i (12) At this point the stability criterion (3) can be easily checked by considering TOT as. Furthermore the system with variably skilled agents can be now compared to original Erlang model. IV. RESULTS For comparison and verification purposes the simulation using MATLAB was established. Simulation input parameters are following [8, 9]: λ 1 = 667 calls per hour first run λ 2 = 333 calls per hour second run 392

3 = agents T H1 = 1 sec => 24 calls per hour first run T H2 = 3 sec => 12 calls per hour second run In each simulation run, agents are divided between two skill levels expressed by certain value of agent s average handling time. Both groups are equal in terms of agent count. The average call handling times for each group of agents and simulation run is stated in Table I. These values are chosen in order to maintain the total average number of calls handled per time unit the same as in case of original Erlang model with agents. Then the are relevant for comparison. TABLE I AVERAGE AGET ALL HADLIG TIME 1. run 2. run higher skilled sec = 28 calls / 2 sec. = 15 calls / agents hour. hour lower skilled agents Two round simulation is used in order to analyze influence of call handling time variance on contact centre significant parameters. Variance of the average call handling time for both groups of agents for the first simulation run is 1321 whereas 1 for the second run. A. alculation Table II. and Table III. show the calculation [7], [1] for all relevant parameters using the original Erlang model and equations (1) to (1) with set to 24 (first run) and 12 (second run) respectively. TABLE II FIRST ROUD - ALULATIO RESULTS P K T Q W 28 95, ,6 127,2 686,6 7,2 99,3 29,3 45,1 243,5 17,3 93,5 35,9 95,8 3 58,7 35,2 189,9 7,4 39,9 56,3 92, ,1 31,7 171,1 3,9 21,1,6 89, , ,1 2,2 12,1,6 86, ,3 29,1 157,3 1,4 7,3 87,3 84, ,5 28,6 154,5,8 4,5 91,9 81, ,3 28,3 152,8,5 2,8 94,9 79,4 36 9,4 28,1 151,7,3 1,7 96,8 77,2 37 6, ,1,2 1,1 98,1,1 P 1 sec. = calls / hour sec. = 9 calls / hour TABLE III SEOD ROUD ALULATIO RESULTS K Q T W 28 94,5 132,6 1433,5 14,8 1133,5 7,1 99, ,6 44,3 478,9 16,6 178,9 31,4 95, ,9 377,4 7,2 77,4 92, ,5 31,6 341,1 3,8 41,1 64,2 89, ,6 29,9 323,7 2,2 23,7 74,7 86, ,1 314,3 1,3 14,3 82,4 84, ,3 28,6 38,8,8 8, , ,1 28,3 35,4,5 5,4 91,9 79,3 36 9,3 28,1 33,4,3 3,4 94,7 77,1 37 6,4 27,9 32,1,2 2,1 96,5 B. Table IV. and Table V. show the simulation for all relevant parameters for both first and second round. In both cases agents are divided into two skill level groups with respective average call handling time and count per hour. P TABLE IV FIRST ROUD SIMULATIO RESULTS K Q T W 28 94, ,7 89,2 484,9 8,2 99, ,1 43,7 235,6 15,9 85,5 37,8 95,9 3 35,4 19,7 7,4 55,1 93, ,4 31,9 172,6 4 21,7 71,5 89, ,2 3,4 163,5 2,4 13,1 79,4 87, ,5 29,7 159,9 1,6 8,7 85,8 85, ,4 28,7 155,1,8 4,3 92,1 82, ,6 28,6 154,3,6 3,1 94,5 36 1,3 28,4 153,2,4 2 96, ,2 28,5 153,5,2 1,3 97,7 76,3 P TABLE V SEOD ROUD SIMULATIO RESULTS K Q T W 28 9,4, ,7 464,3 12,2 98, ,5 43,9 473,7 16,1 173,8 31,4 95,7 3 59,4 35,5 384,3 7,6 82,3 49,1 93, ,8 31,9 345,7 4,1 44,2 63,5 89, ,5 29,9 324,9 2,2 23,6,5 86, ,4 317,5 1,4 14,9 81,7 84, ,1 313,5 1 11,2 86,6 82, ,7 38,7,6 6,4 9,4,3 36 1,4 28,5 39,2,4 4,4 93,7 78,1 37 7,1 28,4 36,2,3 2,8 95,9 76 Probability of queuing (%) Fig alculated Simulation umber of agents () Probability of queuing first run Straightforward comparison of corresponding tables shows some differences. From the callers point of view the most 393

4 Avg. waiting time (s) Agent utilization (%) Fig. 2. umber of agents () Average waiting time first run umber of agents () Fig. 4. Agent utilization first run Important characteristics are probability of call queuing (P ), average waiting time (how much time does the caller spend waiting for an agent) W and Grade of Service () level (the probability of call reception in seconds). Fig. 1, Fig. 2 and Fig. 3 display the relation of these variables to the number of agents present in contact centre () and compares the simulation and calculation for these parameters for the first run. It can be seen all metrics suffered some degradation. The probability of queuing P rose about.5% that led to approx. 5.9% increment of average waiting time. Finally the parameter worsens of.5%. From the callers point of view it implicate nothing more than increased waiting time and globally less satisfaction with contact centre services. From managers point of view the agent utilization is an important parameter. If the utilization is low, the agents are Grade of Service (%) 87,5 62,5 37,5 25 umber of agents () Fig. 3. Grade of Service first run Probability of queuing (%) Fig. 5. Avg. waiting time (s) Fig alculated Simulation umber of agents () Probability of queuing second run umber of agents () Average waiting time second run 394

5 Grade of Service (%) 87,5 62,5 37,5 25 umber of agents () Fig. 7. Grade of Service second run Agent utilization (%) umber of agents () Fig. 8. Agent utilization second run not working effectively, however if it is too high, they work under pressure. Fig. 4 compares the agent utilization obtained from calculation using original Erlang model and simulation with variably skilled agents. It is clearly visible the average value is higher in case of unbalanced staff. In this case, the are up to 1.2% worse comparing to calculation. The second simulation run further proved the degradation of metrics. The probability of queuing P (Fig. 5) increased about.%. The average waiting time W increased more significantly of about 14.2 % (Fig. 6) together with approx 2.5% decrease of (Fig. 7). The agent utilization (Fig. 8) rose again around 1.%. The direct comparison between two simulation runs showed higher degree of parameters degradation for the second round. Since the variance of agents skills was significantly higher for this simulation execution. That leads to conclusion the variance of skill level affects the contact centre performance. The more differently skilled agents, the less performance of contact centre can be expected. From the other point of view the simulation confirmed the original Erlang model that considers agents at equal skill level can be efficiently used. The output then can be considered as lower / upper bound for particular parameter and real a few percent worse should be expected. V. OLUSIO The aim of this paper is to show some of the approaches towards ontact centre modelling and simulation that leads to parameter estimation and verification. The Erlang model is probably one of the most widely used for this purpose. The greatest advantage of Erlang models is their simplicity and ability to describe the most important operation situations of ontact centres with acceptable level of accuracy. In real operations, agents in contact centre are at different skill levels. This is not compliant with original Erlang model that considers all agents as equal. However our simulation showed that upon using slightly modified input values, the provided by model are quite accurate. AKOWLEDGMET This work is a part of research activities conducted at Slovak University of Technology Bratislava, Faculty of Electrical Engineering and Information Technology, Department of Telecommunications, within the scope of the projects VEGA o. 1/565/9 Modelling of traffic parameters in G telecommunication networks and services and the partial result of the Research & Development Operational Programme for the project Support of enter of Excellence for Smart Technologies, Systems and Services, ITMS 26215, co-funded by the ERDF. REFEREES [1] L. Unčovský, Stochastic models of operational analysis, ALFA, Bratislava, 19, 416 pages, ISB [2] Diagnostic Strategies, Traffic Modeling and Resource Allocation in all enters, eedham (Mass., USA), Diagnostic Strategies, 3, accessed on , Available: [3] G. Koole, all enter Mathematics : A scientific method for understanding and improving contact centers,. Vrije Universiteit, Amsterdam, , 68 pages, accessed on , Available: [4] J. Polec and T. Karlubíková, Stochastic models in telecommunications, FABER, Bratislava, 1999, 128 pages, ISB [5] G. Bolch, S. Greiner, H. de Meer, and K. S. Trivedi, Queueing etworks and Markov hains, 2nd ed, John Wiley, Hoboken (ew Jersey, USA), c6, 878 pages, ISB [6] Z. Riečanová, J. Horváth, V. Olejček, B. Riečan, and P. Volauf, umerical methods and mathematic statistics, ALFA, Bratislava, August 1987, 496 pages, ISB [7] T. Misuth, E. hromý, I. Baroňák, Method for Fast Estimation of ontact entre Parameters Using Erlang Model, In: The Third International onference on ommunication Theory, Reliability, and 395

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