DISCRETE EVENT SIMULATION COURSE INSTRUCTOR DR. J. BOOKBINDER PROJECT REPORT. Simulation Study of an Inbound Call Center SACHIN JAYASWAL

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1 DISCRETE EVENT SIMULATION MSCI 632 SPRING 5 COURSE INSTRUCTOR DR. J. BOOKBINDER PROJECT REPORT Simulation Study of an Inbound Call Center Submitted by SACHIN JAYASWAL & GAURAV CHHABRA

2 Table of Contents Abstract Introduction Problem Definition Call Centre Data Data Collection Inter-Arrival Times Distribution Service Times Distribution Balking and Reneging Simulation Model Verification, Validation and Testing Experimentation and Results Conclusions and Future Research Directions References List of Tables Table 3-1Goodness of Fit Test output for Lognormal Inter-Arrival Times Table 3-2 Goodness of Fit Test output for Lognormal Inter-Arrival Times Table 6-1 Service Levels: Regular Customers (5 minutes) Priority Customers (1 minute) Table 6-2 Service Levels: Regular Customers (3 minutes) Priority Customers (1 minute) Table 6-3 Service Levels: Regular Customers (2 minutes) Priority Customers (1 minute) Table 6-4 Service Levels: Regular Customers (2 minutes) Priority Customers (3 seconds) List of Figures Figure 3-1 Pearson5 Distribution for Inter-Arrival Times... 9 Figure 3-2 Log Logistic Distribution for Inter-Arrival Times... 9 Figure 3-3 Log Normal Distribution for Inter-Arrival Times... 1 Figure 3-4 Difference Graph for Log Normal Distribution of Inter-Arrival Times... 1 Figure 3-5 Log Logistic Distribution for Service Times Figure 3-6 Pearson5 Distribution for Service Times Figure 3-7 Log Normal Distribution for Service Times Figure 3-8 Difference Graph for Log Normal Distribution of Service Times Figure 3-9 Renege Probability Distribution Figure 4-1 Simulation Model in Arena Environment Figure 6-1 Service Level for Priority Customers versus Number of Agents Figure 6-2 Abandonment Rate for Priority Customers versus Number of Agents Figure 6-3 Abandonment Rate of Regular Customers versus Number of Agetns Figure 6-4 Average Waiting Time for Priority Customers versus Number of Agents Figure 6-5 Average Waiting Time for Regular Customers versus Number of Agents Figure 6-6 Sensitivity Analysis on Service Level thresholds for Regular Customers... 28

3 Abstract This paper examines the design and development of a simulation model of a call center environment. Two classes of call center customers are considered, priority customers and regular customers. The Call Center operation under study guarantees a higher service level to its priority customers. An animated simulation model is developed in ARENA to capture the impact on various system performance measures such as abandonment rate, average waiting time, agent utilization and service level, based on the call mix of the priority and regular customers. Distributions of the inter-arrival and service times are drawn from a health care call center data and are modeled into the simulation model. Balking and Reneging affects are considered in the modeling when the customer finds a server busy. We find the optimal number of agents required to serve the call center operations in order to meet the business objectives of minimal target service levels and abandonment rates set by the management. Verification, Validation and Testing categorized into informal, static and dynamic techniques are used throughout the design and development of the call center simulation model. A terminating simulation study is conducted and confidence intervals are constructed on the measures of performance. Sensitivity analysis is done by varying the target service levels and call-mix of the priority and regular customers.

4 1 Introduction The past decade has witnessed a rapid growth in Call Center industry as businesses have increasingly embraced the idea of using telephone as a means to provide to their customers services like telemarketing, technical support, etc. Call Centers are locations where calls are placed, or received, in high volume for the purpose of sales, marketing, customer service, telemarketing, technical support or other specialized business activity (Dawson 1996). Call Center operations are now part of many manufacturing and service industries. A finance company, for example, may have its call center operations that provide its customers with online information on its various kinds of financial products available; a software company may have its call center operations to provide technical support to its customers while a health clinic may provide online health services. All these are examples of Inbound Call Centers where calls are received in high volumes from customers seeking services. All Inbound call centers face the classical planning problems of forecasting and scheduling under uncertainty. Customer calls at these centers are attended by staffs called agents. Most call centers target a specific level of service to its customers. Service level may be defined as the percentage of callers who wait on hold for less than a particular period of time. For example, a particular call center may aim to have 9% of callers wait for less than 3 seconds. Other related measures of service level may be the average customers wait time on hold or abandonment rate, which is defined as the percentage of callers who hang up while on hold before talking to an agent. It is quite intuitive that customer abandonment rates and customer waiting times are highly correlated. High abandonment rates result in forming a negative impression towards the company and likely loss in business due to the economic value associated with customer dissatisfaction. In order to be able to guarantee a specific level of customer service, a call center

5 needs to carefully plan for the level of staffing (agents) to match its demands. Too low a staff level may render the attainment of the target service level unattainable while a staff level higher than actually needed to meet the actual demand serves to increase the cost thereby squeezing the profit margin. 2 Problem Definition The problem that we propose to study here is that of a typical Call Center. It has two classes of customers - Priority Customers and Regular Customers. The Call Center guarantees a service level to its priority customers, which is comparatively better than that for regular customers Priority customers could be paying a service charge in return for a prompt service or they could be some special customers contributing towards significant business value to the organization. This guaranteed service level can be achieved by keeping a sufficiently large number of agents to serve the calls. We used the following target service levels for our study: i. At least 9 percent of high priority calls should be attended within 1 minute (SL_PC1). ii. Abandonment rate for high priority calls should not exceed 5 percent (SL_PC2). iii. At least 9 percent of regular calls should be attended within 5 minutes (SL_RC1). iv. Abandonment rate for regular calls should not exceed 1 percent (SL_RC2). We developed an animated simulation model to study the problem. The reasons for using a simulation study was that the problem required the modeling of two classes of customers, and the ease in representation of one of the most of important dynamic features of the system call abandonment while gathering output on a variety of performance measures. The objective of the

6 simulation study is to find the optimum number of agents to achieve these service levels and at the same time maximizing their utilization. We used the following performance measures for the study, the first three defined for each class of customers: i. Average waiting time: Amount of time the customer waits in the queue before getting served. ii. Service Level: Percentage of customers served who spent less than the target time on hold. iii. Abandonment Rate: Percentage of customers who hung up without being served. iv. Agent Utilization: Percentage of time that the agent was busy talking to a customer. 3 Call Centre Data 3.1 Data Collection We collected a limited amount of data on call arrivals and service times from a call centre servicing Health industry. The system under study requires the following input data: i. Pattern of call arrivals: This is specified in terms of probability distribution of number of calls in a given time. This is further discussed in the Inter-Arrival Times Distribution subsection below. ii. Pattern of call service: This is specified in terms of probability distribution of amount of time to serve a call. This is further discussed in the Service Times Distribution subsection below.

7 iii. Call mix: This is defined in terms of the proportion of calls that are high priority. We did a sensitivity analysis by varying the call mix from 1% to 5% and observed the optimal number of agents required with other business constraints remaining unchanged. iv. Balking Percentage: This is specified in terms of percentage of callers who hang up as soon as they find that the server is busy before talking to an agent. This is further discussed in the sub-section, Balking and Reneging, below. v. Reneging Probability: This is defined as the probability that a customer abandons a call after waiting some time in the queue before getting served. This is further discussed in the Balking and Reneging sub-section below. vi. Target service levels: For each class of customers, this is specified in terms of a lower limit on percentage of customers whose calls should be attended to within a target amount of time and an upper limit on abandonment rates. As discussed in the previous section, we have fixed a service level of 9% to answer the calls within one minute for the priority customers and a service level of 9% to answer the calls within five minutes for the regular customers. Also, the maximum permissible abandonment rate is set as 5% and 1% for priority and regular customers respectively. We further conducted a sensitivity analysis by changing the threshold times in queue for both the regular and priority customer with the service levels fixed at 9% for both classes of customers. 3.2 Inter-Arrival Times Distribution Data on call arrivals was available in the form of number of incoming calls every 15 minutes during the hours of operation. The inter-arrival time was assumed to be constant within

8 each 15 minute interval. As an example, if there were 2 incoming calls in a time step of 15 minutes between 133 hours to 1345 hours, then the inter-arrival time was assumed to be 15/2 (=.75) minutes. We used BestFit to find the distribution of the inter-arrival times. The following were the top three best fits for the inter-arrival time data based on the Kolmogorov-Smirnov Goodness of Fit test. i. Pearson5(5.896, ) Shift= ii. LogLogistic(.3144,.16754, ) iii. Lognorm(.2124,.12779) Shift= Even though Pearson5 and Log Logistic distributions were ranked higher than the Log Normal distribution, we chose the Log Normal distribution for our simulation model to generate the inter-arrival times since the modeling software Arena, used for the simulation study, does not support the Pearson5 and Log Logistic distributions. The distribution of the K-S test statistic itself does not depend on the underlying cumulative distribution function being tested. This advantage coupled with the advantage that it is an exact test as compared to chi-squared goodness of fit test, which depends a lot on the adequate sample size for the approximations to be valid (since we had limited data on arrival), were the main reasons for using K-S test for ranking the distributions. Finally, the K-S test tends to be more sensitive near the centre of the distribution than at the tails, which is advantageous to us, since the objective of our simulation model is to come up with an upper limit on the staffing level for which the data for busy hours holds more importance.

9 PEARSON5(5.896, ) SHIFT= < 5.% 9.% 5.% > Figure 3-1 Pearson5 Distribution for Inter-Arrival Times LOGLOGISTIC(.3144,.16754, ) % 9.% 5.% > Figure 3-2 Log Logistic Distribution for Inter-Arrival Times

10 6 LOGNORM(.2124,.12779) SHIFT= % 9.% 5.% > Figure 3-3 Log Normal Distribution for Inter-Arrival Times LOGNORM(.2124,.12779) SHIFT= Figure 3-4 Difference Graph for Log Normal Distribution of Inter-Arrival Times

11 Chi-Sq A-D K-S Test Value P Value.246 N/A N/A Rank N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A Table 3-1Goodness of Fit Test output for Lognormal Inter-Arrival Times 3.3 Service Times Distribution The call service time data, the time that an agent serves a customer on a call, was given in seconds. The time unit was converted to minutes and BestFit was used to find the distribution. The following were the top three best fits for the service time data based on the Kolmogorov- Smirnov Goodness of Fit test. i. LogLogistic(2.1757, , ) ii. iii. Pearson5(5.463, 26.68) Shift= Lognorm(5.2343, 3.594) Shift= Even though the Log Logistic and Pearson5 distributions were ranked higher than the Log Normal distribution, we chose the Log Normal distribution for our simulation model to generate

12 the service times for the same reasons cited in previous section. Further, we used the K-S test statistic for the distribution ranking..25 LOGLOGISTIC(2.1757, , ) % 9.% 5.% > Figure 3-5 Log Logistic Distribution for Service Times

13 .25 PEARSON5(5.463, 26.68) SHIFT= < 5.% 9.% 5.% > Figure 3-6 Pearson5 Distribution for Service Times.25 LOGNORM(5.2343, 3.594) SHIFT= < 5.% 9.% 5.% > Figure 3-7 Log Normal Distribution for Service Times

14 LOGNORM(5.2343, 3.594) SHIFT= Values Figure 3-8 Difference Graph for Log Normal Distribution of Service Times Chi-Sq A-D K-S Test Value P Value.6227 N/A N/A Rank N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A Table 3-2 Goodness of Fit Test output for Lognormal Inter-Arrival Times

15 3.4 Balking and Reneging Some call centre customers may decide not to join the system upon arrival if the server is busy, i.e. there is no agent to serve their call. Such customers are said to have balked. Others may leave after spending some time in the queue. They are said to have reneged. Of course there are also those who stay on until service completion. The balking and reneging of customers in the service systems is a common occurrence in all real life queuing systems, and have a direct impact on the quality of service delivered despite an inherent complexity involved in its estimation. Harris, Hoffman, and Saunders (1987) argued that the human behavior is such that a very long wait often keeps the caller on the line because of an increased expectation and an already incurred overhead although the future delays in any M/M/c queue are independent of alreadyspent waiting time. In the simulation model that they developed for the IRS telephone taxpayer information system, they assigned different hold-on probabilities based on accumulated waiting time. Parkan (1987) developed a simulation model for the operation of a fast food kiosk where dissatisfied customers renege. He argued that the customers do not join a queue with the intention of reneging, and therefore, it is more appropriate not to presume that a person will balk at a queue or renege from it later with a certain probability. He further argues that one must look into the decision process that leads to balking or reneging of a person and try to understand the relationship between this process and the operational characteristics (such as speed and quality of service) of the system. His approach on reneging decisions of customers in a queue is based on Bayesian analysis where he assumes a gamma distribution of the initial expectation of the waiting time before getting serviced common to all customers of the estimated waiting time.

16 Renege probability distribution that we modeled based on our literature review was a piecewise linear function of wait time. The renege probability distribution is assumed to be CONT [(., 1), (.25, 2), (.4, 3), (.5, 4), (.7, 5), (1., 6)]. We argue that the probability of reneging will decrease with waiting time only until a certain amount of waiting time after which the customer tend to lose patience and the reneging probability increases thereafter. This can be observed in the Figure 3-9 below where the slope of the probability distribution function decreases until a certain time and increases thereafter. We further assume that customers who join the queue will wait at least for one minute before deciding to renege. Renege Probability Distribution Renege Probability Waiting Time (Minutes) 4 Simulation Model Figure 3-9 Renege Probability Distribution Figure 4-1 shows the simulation model, built in Arena 7.1, a module-based graphical simulation software package (Kelton, Sadowski, and Sadpwsk, 1998), for the system under study. The Calls are generated using the Call Arrival module with the Inter-Arrival time distribution. The call mix is decided using a 2-way by chance decision module. The calls on arrival are tagged with the following attributes: Service time, Renege Time, Customer Number

17 and the Arrival Time. Upon arrival, the entity is duplicated with the same value of the attributes. The original entity joins the queue if it does not balk. The duplicated entity is delayed using the Delay Module by its renege time obtained from the Renege Probability Distribution. The original entity balks with a probability of 5% using the Decide module. The duplicated entity after being delayed enters a Search module, which searches the queue before the server to see if the corresponding original entity is still present in the queue. If present, it is reneged using the remove module; otherwise the original entity has already been served. After the Process module, the customer waiting time, and the percentage of calls that waited for less than the upper limit on wait time are noted. Similarly, statistics are calculated for the percentage renege and percentage balk for each type of calls, the sum of which gives the abandonment rate. Finally, the served calls are disposed off the system using the Dispose Module, Call Served. We defined several essential run characteristics, such as the number of replications and the length of each replication, in the Simulation model. Model parameters that were held constant for a given scenario, such as the percentage of Priority class customers defining the call mix, the service level target answer times by call type, etc. were also defined. The distributions for drawing random variates for inter-arrival time, service time, and the amount of time a caller waits on hold before abandoning were defined in the simulation model. During execution, the model tracks a number of system performance measures for each class of calls and continually updates their values: the percentage of callers who hung up without getting served (Abandoned), the average number of minutes served callers spent on hold (Average Queue Time), and the percentage of served customers who spent less than the target time on hold (Service Level).

18 Cal ar r ival Assign Attributes Special Cust omer? False Si m ul at i :: Tr ue Assign priority to Regular customer Assign priority to Sp customer Server Idle? False Record Regular Cust omers Tr ue Balk? Record Sp Cust oomer s False Tr ue Duplicate Cals Cal Service O r igin al Duplicat e Percentage Sp customers Hang ups p cust Avg Wait Tim percentspcustw ithintim elim it Percentage Reg customers Hang upse g cust Avg W ait Tim percentspcustw ithintim elim it Sp Customer? False Sp cust avg d. Spcl cust? False Sp cust avg delay Record # sp cust compltd srvc Percent Reg cust hangups Tr ue Delay for renege time Reg cust avg d. Record Sp. customer Balks Record Regular customer Balks Searc h Serv er queue Found Spc customer? False Percent Sp cust Percent Sp cust Tr ue hangups srv within limit Assign sp cust sp cust cummul Delay delay sp within time limit? Percent Reg cust srv within limit Not Found Tr ue Assign Search # Rem o v e fro m queue O r iginal Record Sp Reneges Record Reg Reneges Reg cust avg delay Rem oved Ent it y Record # Reg cust compltd srvc Assign Reg cust Delay Reg cust within time limit? Recor d Reg # within limit p er centsp custwith in timelimit. per centre g c u s twith in timelimit. False Reg cust cummul delay Tr ue Dispose False Percent sp cust served wi 1.. Percent Reg cust served within limit Tr ue Recor d sp # wit hin limit Cal Ser ved Figure 4-1 Simulation Model in Arena Environment

19 5 Verification, Validation and Testing Simulation model verification, validation and testing (VV&T) plays an important role in any simulation study. VV&T is the structured process of increasing one s confidence in a model, thereby providing a basis for confidence in modeling study s results (Swisher 21). Model verification substantiates that the model has been properly transformed from one form to another (e.g. from a flowchart to an executable program). Model validation, on the other hand, substantiates that the model behaves with sufficient accuracy in light of the study s objectives. We were able to ascertain that the model we developed closely reflects the call center operation. However, we could not employ the validation techniques such as high face validity (a model which on surface seems reasonable to people who are knowledgeable about the system under study) or Turing Test (comparison of the model outputs to those observed in actual system) due to the lack of industrial contacts and the unavailability of the data on true system performance measures of the call center under study. Finally, model testing is the process of revealing errors in a model. Testing procedures may be designed to perform either model verification or model validation. The VV&T techniques used throughout the design and development of the call center simulation model can be categorized into informal, static or dynamic (Balci 1997). Balci (1997) states that well-structured informal VV&T techniques applied under formal guidelines can be very effective. Informal VV&T techniques employed in the call center simulation modeling effort were review and walk-through. Arena has a completely graphical user interface, many automated bookkeeping features that greatly reduce the likelihood of programming error, and debugging capabilities that allow the user to stop execution and examine the values of any variable or caller attribute.

20 Static VV&T techniques are concerned with assessing the accuracy of a model based upon characteristics of the static simulation model design (Siwsher 21). They do not require the computational execution of the model. Static VV&T technique employed in the modeling effort was fault/failure analysis. We examined under what conditions the model should logically fail. This helped us in identifying the logic problems in the definition of the call flow and made it easier to define the possible paths that the customer call may take in the model. Dynamic VV&T techniques require model execution and are intended to evaluate the model based on its execution behavior (Balci 1997). Examples of dynamic VV&T techniques applied to the simulation model include assertion checking, debugging, functional testing, and sensitivity analysis. The feasibility of critical state variables was monitored using the assertion checking technique. The simulation program was developed and debugged modularly to avoid any critical bug fixes in the final model. Further we ran the model with simplifying assumptions (adequate servers, no reneging, etc.) which is an essential part of debugging. Functional testing is used to assess the accuracy of a model based upon its outputs, given a specific set of inputs (Balci 1997). The model was tested with several arrival call rates. As an example with low call rates (achieved by changing the parameters of the Inter-Arrival time distribution) the agent utilization was low for a fixed number of agents. Finally, sensitivity analysis on the number of agents, call-mix, and the threshold values of time on hold for regular and priority customers was done. We discuss this aspect of the VV&T technique in the Experimentation and Results section of the paper.

21 6 Experimentation and Results The call center is a terminating system that begins each morning empty of calls and ends hours later when agents go home after serving their last calls. We took each replication of the model to be exactly 12 hours, so even though the calls in the system at the end of the day were not served to completion, we counted them as served. We evaluated our results for the following scenarios: i. Specific combinations of the number of agents (S) and the percentage of priority (P) class callers having the service levels defined in Section 2, Problem Definition of this paper. We used three values of S and five values of P leading to a total of 15 scenarios. The results for these 15 scenarios are presented in Table 6-1. ii. We did a sensitivity analysis by changing our service levels defined previously by fixing P at 3%. The results of the 3 scenarios thus evaluated are presented in Table 6-2 to Table 6-4. We performed 2 independent replications for each scenario and Arena s output analyzer calculated the summary statistics for the various performance measures discussed in Section 2, Problem Definition. Arena generates a default of 95% confidence intervals across the mean. We converted these confidence intervals at 9% by calculating the approximate measure of variance at 95% using the half-length and the number of replications. We then constructed the confidence interval at 9% by using the t-statistic value at 9% and using the variance calculated above with the same number of replications. Table 6-1 to Table 6-4 reports these confidence intervals at 9%.

22 % of priority Calls Performance Measures Number of Agents Service Level Priority [8.61,83.63] [86.35,89.37] [92.74,94.3] Service Level Regular [98.3,98.7] [98.78,98.98] [98.82,99.6] Abandonment Rate Priority [5.8,6.18] [3.98,4.82] [2.36,3.] 5% Abandonment Rate Regular [18.64,21.4] [1.74,13.34] [4.74,5.98] Avg. wait time Rapid [.46,.52] [.33,.39] [.18,.22] Avg. wait time Regular [.91,1.3] [.56,.68] [.3,.38] Agent Utilization [ ] [.962,.962] [.931,.931] Service Level Priority [81.4,84.14] [87.27,89.67] [93.12,94.84] Service Level Regular [98.48,98.82] [98.82,99.2] [98.86,99.8] Abandonment Rate Priority [5.4,6.96] [3.7,4.86] [2.12,3.1] 4% Abandonment Rate Regular [16.99,19.51] [9.59,11.71] [4.63,6.7] Avg. wait time Rapid [.45,.51] [.32,.36] [.17,.21] Avg. wait time Regular [.93,1.3] [.56,.68] [.29,.37] Agent Utilization [.982,.982] [.965,.965] [.932,.932] Service Level Priority [82.73,85.65] [9.9,92.23] [93.52,94.86] Service Level Regular [98.77,98.97] [98.87,99.5] [98.95,99.11] Abandonment Rate Priority [5.1,6.51] [3.42,4.7] [2.18,2.98] 3% Abandonment Rate Regular [15.79,17.59] [8.54,1.22] [4.3,5.8] Avg. wait time Rapid [.44,.48] [.29,.33] [.18,.2] Avg. wait time Regular [.95,1.7] [.57,.65] [.31,.37] Agent Utilization [.982,.982] [.964,.964] [.936,.936] Service Level Priority [83.89,86.85] [89.83,92.17] [93.43,95.47] Service Level Regular [98.83,99.5] [98.95,99.13] [98.99,99.15] Abandonment Rate Priority [4.61,6.57] [2.97,4.41] [2.27,3.7] 2% Abandonment Rate Regular [14.69,16.31] [7.68,9.28] [3.78,4.84]

23 Avg. wait time Rapid [.42,.46] [.27,.31] [.16,.2] Avg. wait time Regular [.94,1.6] [.53,.63] [.28,.34] Agent Utilization [.983,.983] [.963,.963] [.933,.933] Service Level Priority [84.6,87.46] [89.36,93.] [92.37,95.9] Service Level Regular [98.95,99.15] [99.4,99.18] [99.4,99.18] Abandonment Rate Priority [3.43,6.81] [3.9,4.97] [1.29,3.17] 1% Abandonment Rate Regular [13.24,14.46] [7.12,8.48] [3.83,4.61] Avg. wait time Rapid [.38,.42] [.25,.31] [.17,.21] Avg. wait time Regular [.9,1.] [.54,.64] [.29,.37] Agent Utilization [.983,.983] [.966,.966] [.937,.937] Table 6-1 Service Levels: Regular Customers (5 minutes) Priority Customers (1 minute) For (P = 1%, S = 12) we were able to achieve SL_RC1 (At least 9 percent of regular calls should be attended within 5 minutes), however we were unable to achieve SL_PC1 (At least 9 percent of high priority calls should be attended within 1 minute). Further SL_PC2 (Abandonment rate for high priority calls should not exceed 5 percent) and SL_RC2 (Abandonment rate for regular calls should not exceed 1 percent) were not met. We therefore need to increase agents to meet the targets. For (P = 1%, S = 13) we are able to achieve all the service levels (SL_PC1, SL_PC2, SL_RC1, SL_RC2) if we just consider the means for these performance measures. However, as can be seen in Table 6-1, at 9% confidence SL_PC1 cannot be ascertained; the lower limit of the confidence interval being 89.36% which is less than the desired level of 9%. This can be further ascertained by increasing the number of replications in the simulation. We observe that for (P=1%, S = 14) we are able to achieve all the service levels with 9% confidence. We further observe a drop in the average waiting times for both the regular and the priority customers as we increase the number of agents. For (P = 1%) the

24 average waiting time for the regular customers varies between.29 to 1 minute and for the priority customers it varies between.17 to.42 minutes. The average agent utilization should drop with the increase in the number of agents intuitively, and this can be observed for (P = 1%) case where it drops from 98.3% to 93.7% as we increase the number of agents from 12 to 14. Finally, if the confidence intervals for a performance measure overlap with an increase in the number of agents, then we cannot claim that there is a significant difference in the performance achieved. In the (P = 1%, S= 13) and (P = 1%, S = 14) the confidence intervals for the abandonment rate for the priority customers overlap and hence we cannot conclude that there is a significant improvement in performance even though there is a decrease in the mean abandonment rate by increasing the number of agents from 13 to 14. Figure 6-1 to Figure 6-5 shows the plots of various output performance measures against the number of agents at different percentages of priority class callers. Service Level for Priority Customers Percentage of Priority Customers waiting less than 1 minute Number of Agents P = 5% P = 4% P = 3% P = 2% P = 1% Figure 6-1 Service Level for Priority Customers versus Number of Agents

25 Abandonment Rate of Priority Customers Abandonment Rate Number of Agents P = 5% P = 4% P = 3% P = 2% P = 1% Figure 6-2 Abandonment Rate for Priority Customers versus Number of Agents Abandonment Rate of Regular Customers 25 Abandonment Rate Number of Agents P = 5% P = 4% P = 3% P = 2% P = 1% Figure 6-3 Abandonment Rate of Regular Customers versus Number of Agetns

26 Average Waiting Time for Priority Customers.6 Average Waiting Time (Minutes) P = 5% P = 4% P = 3% P = 2% P = 1% Number of Agents Figure 6-4 Average Waiting Time for Priority Customers versus Number of Agents Average Waiting Time for Regular Customers 1.2 Average Waiting Time (Minutes) P = 5% P = 4% P = 3% P = 2% P = 1% Number of Agents Figure 6-5 Average Waiting Time for Regular Customers versus Number of Agents An overall of 14 agents seemed to be optimal to guarantee the target service levels for the range of call mix. However, with 14 agents we observed that we were able to achieve almost 98% service level (SL_RC1) for regular calls because of a high threshold limit of 5 minutes. We investigated the sensitivity of the percentage of calls answered to the service level threshold in minutes by setting the percentage of priority calls at 3%. We observed a decrease in the service level (SL_RC1) by varying the maximum allowable service time from 5 minutes to 3 minutes

27 and to 2 minutes. The results are presented in Table 6-2 and Table 6-3 below and Figure 6-6 shows the results of this sensitivity analysis. % of priority Calls Performance Measures Number of Agents Service Level - Priority [82.73,85.65] [9.9,92.23] [93.52,94.86] Service Level - Regular [92.3,93.95] [96.82,97.52] [98.24,98.68] Abandonment Rate - Priority [5.1,6.51] [3.42,4.7] [2.18,2.98] 3% Abandonment Rate - Regular [15.79,17.59] [8.54,1.22] [4.3,5.8] Avg. wait time Rapid [.44,.48] [.29,.33] [.18,.2] Avg. wait time - Regular [.95,1.7] [.57,.65] [.31,.37] Agent Utilization [.982,.982] [.964,.964] [.936,.936] Table 6-2 Service Levels: Regular Customers (3 minutes) Priority Customers (1 minute) % of priority Calls Performance Measures Number of Agents Service Level Priority [82.73,85.65] [9.9,92.23] [93.52,94.86] Service Level Regular [79.63,83.69] [9.31,92.63] [95.24,96.76] Abandonment Rate Priority [5.1,6.51] [3.42,4.7] [2.18,2.98] 3% Abandonment Rate Regular [15.79,17.59] [8.54,1.22] [4.3,5.8] Avg. wait time Rapid [.44,.48] [.29,.33] [.18,.2] Avg. wait time Regular [.95,1.7] [.57,.65] [.31,.37] Agent Utilization [.982,.982] [.964,.964] [.936,.936] Table 6-3 Service Levels: Regular Customers (2 minutes) Priority Customers (1 minute)

28 Sensitivity Analysis - Service Level for Regular Customers Percentage of Regular Customers Number of Agents Service Level Regular Customers - 3 minutes Service Level Regular Customers - 2 minutes Figure 6-6 Sensitivity Analysis on Service Level thresholds for Regular Customers By performing the sensitivity analysis on the service levels to threshold service times for regular customers described above, we observed that with 14 agents 9% of the regular calls could be attended by an agent within two minutes. We therefore observed the change in system performance if the target service levels are changed to 9% of the calls answered within 3 seconds for priority customers and 9% within 2 minutes for regular customers. We found that to assure this service level we need to increase the number of agents from 14 to 16. The results are presented in Table 6-4 below.

29 % of priority Calls Performance Measures Number of Agents Service Level Priority [82.3,84.58] [89.58,92.] [94.89,96.21] Service Level Regular [95.24,96.76] [97.11,98.53] [98.54,98.86] Abandonment Rate Priority [2.18,2.98] [.73,1.47] [.54,1.6] 3% Abandonment Rate Regular [4.3,5.8] [1.93,2.77] [.76,1.16] Avg. wait time Rapid [.18,.2] [.9,.11] [.3,.5] Avg. wait time Regular [.31,.37] [.13,.19] [.5,.7] Agent Utilization [.935,.935] [.892,.892] [.849,.849] Table 6-4 Service Levels: Regular Customers (2 minutes) Priority Customers (3 seconds) 7 Conclusions and Future Research Directions We obtained several output performance measures from the simulation model of the call center operation that we developed under the assumed business constraints on service levels and abandonment rates. The sensitivity analysis on the performance measures that we conducted by varying our business constraints helped us in identifying the appropriate service levels that can be provided to both the regular and priority customers and assisted us to derive the optimal number of agents required to service the priority and regular class of customers. An important area of future research is to model a scalar performance measure, constructed as a weighted linear combination of several output performance measures that we discussed in the paper. This is because the optimal call center configuration, from the perspective of the management should simultaneously maximize the call center profits by employing lesser number of agents and hence lesser number of trunk setup and maintenance cost, customer satisfaction of both the regular and priority customers by providing good service levels, and staff satisfaction by

30 having lower agent utilization rates. However, determining an optimal staffing level is complicated by the conflicting nature of these objectives. For example a call center configuration that maximizes the profits may create lower service levels and higher agent utilization. Therefore, addressing this problem through multi-variate analysis would provide a simple, concise and intuitive measurement of the call center effectiveness. Hence, we propose to develop a univariate measure composed of multiple attributes as a means of providing information to decision-makers through a single measurement. Further, we need to conduct a fractional factorial experimental design to determine the significance of several input measures and to obtain good estimates of the main effects and some higher-order interactions of the altering input parameters. Further, we haven t considered the transient state of the system in this paper. Generally, there are fewer number of calls when the call center operation begins. We need to identify the warm-up period of the system before it reaches a steady state and truncate those observations for evaluating our performance measures. We propose to use the Welch s procedure of plotting the moving average by adjusting the window size to obtain the length of warm-up period. Further, we propose to design a reneging distribution that models the human behavior more closely given the problem at hand. We have used a piece-wise linear function of waiting time to model the reneging distribution in this paper with a lower probability of dropping the call till a certain threshold time and a higher probability of dropping the call thereafter, with no calls dropping in the first minute and all calls dropped within six minutes of waiting time. We need to examine the actual data on call drops to model our reneging distribution. Finally, we propose to use a nonstationary process of inter-arrival times owing to varying traffic at different times of the day. This would also help us in deriving different staffing levels for different times of the day.

31 References [1] Andrews, B.H. and Parsons, H.L., Establishing telephone-agent staffing levels through economic optimization, Interfaces, Vol. 23(2), 1993, pp [2] Balci, O., Verification, validation and testing, In: Banks J, editor. The handbook of simulation. New York: Wiley, 1997, pp [3] Dawson, K., The call Center Handbook, Flatiron Publishing, New York, [4] Harris, C.M., Hoffman, K.L., and Saunders, P.B., Modeling the IRS Telephone Taxpayer Information System, Operations Research, Vol. 35(4), 1987, pp [5] Kelton, W. D., Sadowski, R. P., and Sadowski, D. A., Simulation with Arena, McGraw-Hill, New York, [6] Mehrotra, V., Profozich, D. and Bapat, V., "Simulation: The best way to design your call center," Telemarketing and Call Center Solutions, Vol. 16(5), 1997, pp [7] Mehrotra, V., "The call center workforce management cycle," Proceedings of the 1999 Call Center Campus, Purdue University Center for Customer-Driven Quality, Vol. 27, 1999, pp [8] Parkman, C., Simulation of a Fast-Food Operation Where Dissatisfied Customers Renege, The Journal of Operational Research Society, Vol. 38(2), 1987, pp [9] Saltzman, R.M. and Mehrotra, V., A call center uses simulation to drive strategic change, Interfaces, Vol. 31(3), 21, pp [1] Swisher, J.R., Jacobson, S.H., Jun, J.B., and Balci, O., Modeling and analyzing a physician clinic environment using discrete-event (visual) simulation, Computers and Operations Research, Vol. 28, 21, pp