Simulation Study of an Inbound Call Centersjayaswa/projects/Simulation... · Simulation Study of an Inbound Call Center Submitted by ... the inter-arrival times since the modeling

DISCRETE EVENT SIMULATION MSCI 632 SPRING ‘05

COURSE INSTRUCTOR DR. J. BOOKBINDER

PROJECT REPORT

Simulation Study of an Inbound Call Center

Submitted by

SACHIN JAYASWAL

&

GAURAV CHHABRA

Table of Contents Abstract ........................................................................................................................................... 3 1 Introduction............................................................................................................................. 4 2 Problem Definition.................................................................................................................. 5 3 Call Centre Data...................................................................................................................... 6

3.1 Data Collection ............................................................................................................... 6 3.2 Inter-Arrival Times Distribution..................................................................................... 7 3.3 Service Times Distribution ........................................................................................... 11 3.4 Balking and Reneging................................................................................................... 15

4 Simulation Model.................................................................................................................. 16 5 Verification, Validation and Testing..................................................................................... 19 6 Experimentation and Results ................................................................................................ 21 7 Conclusions and Future Research Directions ....................................................................... 29 References..................................................................................................................................... 31 List of Tables Table 3-1Goodness of Fit Test output for Lognormal Inter-Arrival Times.................................. 11 Table 3-2 Goodness of Fit Test output for Lognormal Inter-Arrival Times................................. 14 Table 6-1 Service Levels: Regular Customers (5 minutes) Priority Customers (1 minute) ......... 23 Table 6-2 Service Levels: Regular Customers (3 minutes) Priority Customers (1 minute) ......... 27 Table 6-3 Service Levels: Regular Customers (2 minutes) Priority Customers (1 minute) ......... 27 Table 6-4 Service Levels: Regular Customers (2 minutes) Priority Customers (30 seconds)...... 29 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....................................................... 10 Figure 3-4 Difference Graph for Log Normal Distribution of Inter-Arrival Times ..................... 10 Figure 3-5 Log Logistic Distribution for Service Times .............................................................. 12 Figure 3-6 Pearson5 Distribution for Service Times .................................................................... 13 Figure 3-7 Log Normal Distribution for Service Times ............................................................... 13 Figure 3-8 Difference Graph for Log Normal Distribution of Service Times.............................. 14 Figure 3-9 Renege Probability Distribution.................................................................................. 16 Figure 4-1 Simulation Model in Arena Environment ................................................................... 18 Figure 6-1 Service Level for Priority Customers versus Number of Agents................................ 24 Figure 6-2 Abandonment Rate for Priority Customers versus Number of Agents....................... 25 Figure 6-3 Abandonment Rate of Regular Customers versus Number of Agetns........................ 25 Figure 6-4 Average Waiting Time for Priority Customers versus Number of Agents................. 26 Figure 6-5 Average Waiting Time for Regular Customers versus Number of Agents ................ 26 Figure 6-6 Sensitivity Analysis on Service Level thresholds for Regular Customers.................. 28

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.

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 90% of callers wait for less than 30 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

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 90 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 90 percent of regular calls should be attended within 5 minutes (SL_RC1).

iv. Abandonment rate for regular calls should not exceed 10 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

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 sub-

section 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 sub-

section below.

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 10% to 50% 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 90% to answer the calls within one minute for

the priority customers and a service level of 90% to answer the calls within five minutes

for the regular customers. Also, the maximum permissible abandonment rate is set as 5%

and 10% 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 90% 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

each 15 minute interval. As an example, if there were 20 incoming calls in a time step of 15

minutes between 1330 hours to 1345 hours, then the inter-arrival time was assumed to be 15/20

(=0.75) 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(5.8906, 1.2856) Shift=+0.25189

ii. LogLogistic(0.31440, 0.16754, 2.9835)

iii. Lognorm(0.21240, 0.12779) Shift=+0.30195

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.

PEARSON5(5.8906, 1.2856) SHIFT=+0.25189

0

1

2

3

4

5

6

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

< >5.0% 5.0%90.0% 0.3759 0.7578

Figure 3-1 Pearson5 Distribution for Inter-Arrival Times

LOGLOGISTIC(0.31440, 0.16754, 2.9835)

0

1

2

3

4

5

6

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

>5.0% 5.0%90.0% 0.3768 0.7639

Figure 3-2 Log Logistic Distribution for Inter-Arrival Times

LOGNORM(0.21240, 0.12779) SHIFT=+0.30195

0

1

2

3

4

5

6

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

>5.0% 5.0%90.0% 0.3749 0.7560

Figure 3-3 Log Normal Distribution for Inter-Arrival Times

LOGNORM(0.21240, 0.12779) SHIFT=+0.30195

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Figure 3-4 Difference Graph for Log Normal Distribution of Inter-Arrival Times

Chi-Sq A-D K-S

Test Value 6.742 0.3697 0.117

P Value 0.2406 N/A N/A

Rank 4 3 3

C.Val @ 0.75 2.6746 N/A N/A

C.Val @ 0.5 4.3515 N/A N/A

C.Val @ 0.25 6.6257 N/A N/A

C.Val @ 0.15 8.1152 N/A N/A

C.Val @ 0.1 9.2364 N/A N/A

C.Val @ 0.05 11.0705 N/A N/A

C.Val @ 0.025 12.8325 N/A N/A

C.Val @ 0.01 15.0863 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, 3.9236, 2.5351)

ii. Pearson5(5.0463, 26.680) Shift=+0.47587

iii. Lognorm(5.2343, 3.5904) Shift=+1.8160

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

the service times for the same reasons cited in previous section. Further, we used the K-S test

statistic for the distribution ranking.

LOGLOGISTIC(2.1757, 3.9236, 2.5351)

0.00

0.05

0.10

0.15

0.20

0.25

2 4 6 8

10 12 14 16

>5.0% 5.0%90.0%3.40 14.71

Figure 3-5 Log Logistic Distribution for Service Times

PEARSON5(5.0463, 26.680) SHIFT=+0.47587

0.00

0.05

0.10

0.15

0.20

0.25

2 4 6 8

10 12 14 16

< >5.0% 5.0%90.0%3.37 13.82

Figure 3-6 Pearson5 Distribution for Service Times

LOGNORM(5.2343, 3.5904) SHIFT=+1.816

0.00

0.05

0.10

0.15

0.20

0.25

2 4 6 8

10 12 14 16

< >5.0% 5.0%90.0%3.37 13.80

Figure 3-7 Log Normal Distribution for Service Times

LOGNORM(5.2343, 3.5904) SHIFT=+1.8160

Values

-6

-4

-2

0

2

4

6

8

2 4 6 8

10 12 14 16

Figure 3-8 Difference Graph for Log Normal Distribution of Service Times

Chi-Sq A-D K-S

Test Value 4.4 0.2277 0.08944

P Value 0.6227 N/A N/A

Rank 4 3 3

C.Val @ 0.75 3.4546 N/A N/A

C.Val @ 0.5 5.3481 N/A N/A

C.Val @ 0.25 7.8408 N/A N/A

C.Val @ 0.15 9.4461 N/A N/A

C.Val @ 0.1 10.6446 N/A N/A

C.Val @ 0.05 12.5916 N/A N/A

C.Val @ 0.025 14.4494 N/A N/A

C.Val @ 0.01 16.8119 N/A N/A

Table 3-2 Goodness of Fit Test output for Lognormal Inter-Arrival Times

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 already-

spent 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.

Renege probability distribution that we modeled based on our literature review was a piece-

wise linear function of wait time. The renege probability distribution is assumed to be CONT

[(0.0, 1), (0.25, 2), (0.40, 3), (0.50, 4), (0.70, 5), (1.0, 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

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7

Waiting Time (Minutes)

Ren

ege

Prob

abili

ty

Figure 3-9 Renege Probability Distribution

4 Simulation Model

Figure 4-1 shows the simulation model, built in Arena 7.01, 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

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).

Call arrival

Sp customerAssign priority to

Special Customer?Tr ue

False

Regular customerAssign priority to

Call Service

Call Served

Assign Search #

Dispose

Duplicate CallsO r iginal

Duplicat e

Found

Not Found

queueSearc h Serv er

O r iginal

Rem oved Ent it y

queueRem ov e from

tim eDelay for renege

Server Idle?Tr ue

False

Balk ?Tr ue

False

customer BalksRecord Sp.

Sp Customer?Tr ue

False

customer BalksRecord Regular

Spc customer?Tr ue

False

RenegesRecord Sp

RenegesRecord Reg

CustoomersRecord Sp

CustomersRecord Regular

Spcl cust?Tr ue

False

delaysp cust cummul

compltd srvcRecord # sp cust

cust compltd srvcRecord # Reg

DelayAssign sp cust

DelayAssign Reg cust

delayReg cust cummul

sp within t ime limit?Tr ue

False

within limitRecord sp #

limit?Reg cust within t ime

Tr ue

False

within limitRecord Reg #

Sp cust avg delay

delayReg cust avg

hangupsPercent Sp cust

hangupsPercent Reg cust

srv within limitPercent Sp cust

srv within limitPercent Reg cust

Assign Attributes

0

0

0

S i m u l a t i

00:00:00

0

0

0

0

0

0

0

0

0

p c ust Avg W a it T im

0. 0 60. 00.0

1.0

Percen tag e Sp cu sto mers Han g u p s

0. 0 60. 00.0

1.0

0

0

0

0

Percen tag e Reg cu sto mers Han g u p s

0. 0 60. 00.0

1.0

0 0

0 0

0

e g c ust Avg W a i t T im

0. 0 60. 00.0

1.0

S p cu st av g d

0 . 0 0

R eg cu st av g d

0 . 0 0

p e r c e n t s p c u s t with in time limit

0 . 0 0

0

p e r c e n t Re g c u s t with in time limit

percent sp cust w ithin tim e lim it

0 . 0 0

percent sp cust w ithin tim e lim it0 . 0 0

Pe rc e n t s p c u s t s e rv e d w i

0 . 0 6 0 . 0

0.01.0

P ercent Reg cust served within limit

0 . 0 6 0 . 0

0.1.0

00 . 0 0

Figure 4-1 Simulation Model in Arena Environment

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 2001). 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.

Static VV&T techniques are concerned with assessing the accuracy of a model based

upon characteristics of the static simulation model design (Siwsher 2001). 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.

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 30%. The results of the 3 scenarios thus evaluated are presented in Table 6-2 to

Table 6-4.

We performed 20 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 90% 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 90% by using the t-statistic value at 90% and using the variance calculated above with

the same number of replications. Table 6-1 to Table 6-4 reports these confidence intervals at

90%.

% of priority Calls

Number of Agents Performance Measures

12 13 14

Service Level – Priority [80.61,83.63] [86.35,89.37] [92.74,94.30]

Service Level – Regular [98.30,98.70] [98.78,98.98] [98.82,99.06]

Abandonment Rate – Priority [5.08,6.18] [3.98,4.82] [2.36,3.00]

Abandonment Rate – Regular [18.64,21.40] [10.74,13.34] [4.74,5.98]

Avg. wait time – Rapid [0.46,0.52] [0.33,0.39] [0.18,0.22]

Avg. wait time – Regular [0.91,1.03] [0.56,0.68] [0.30,0.38]

50%

Agent Utilization [0.980.0.980] [0.962,0.962] [0.931,0.931]







40%

Agent Utilization [0.982,0.982] [0.965,0.965] [0.932,0.932]







30%




Abandonment Rate – Priority

[4.61,6.57] [2.97,4.41] [2.27,3.07]

20% Abandonment Rate – Regular

[14.69,16.31] [7.68,9.28] [3.78,4.84]







10%


Table 6-1 Service Levels: Regular Customers (5 minutes) Priority Customers (1 minute)

For (P = 10%, S = 12) we were able to achieve SL_RC1 (At least 90 percent of regular

calls should be attended within 5 minutes), however we were unable to achieve SL_PC1 (At

least 90 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 10 percent) were not met. We therefore

need to increase agents to meet the targets. For (P = 10%, 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 90% confidence SL_PC1

cannot be ascertained; the lower limit of the confidence interval being 89.36% which is less than

the desired level of 90%. This can be further ascertained by increasing the number of replications

in the simulation. We observe that for (P=10%, S = 14) we are able to achieve all the service

levels with 90% 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 = 10%) the




average waiting time for the regular customers varies between 0.29 to 1 minute and for the

priority customers it varies between 0.17 to 0.42 minutes. The average agent utilization should

drop with the increase in the number of agents intuitively, and this can be observed for (P =

10%) 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 = 10%, S= 13) and (P = 10%, 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

75

80

85

90

95

100

12 13 14

Number of Agents

Perc

enta

ge o

f Pri

ority

C

usto

mer

s w

aitin

g le

ss

than

1 m

inut

e P = 50%P = 40%P = 30%P = 20%P = 10%

Figure 6-1 Service Level for Priority Customers versus Number of Agents

Abandonment Rate of Priority Customers

0123

4567

12 13 14

Number of Agents

Aba

ndon

men

t Rat

e

P = 50%P = 40%P = 30%P = 20%P = 10%

Figure 6-2 Abandonment Rate for Priority Customers versus Number of Agents

Abandonment Rate of Regular Customers

0

5

10

15

20

25

12 13 14

Number of Agents

Aba

ndon

men

t Rat

e

P = 50%P = 40%P = 30%P = 20%P = 10%

Figure 6-3 Abandonment Rate of Regular Customers versus Number of Agetns

Average Waiting Time for Priority Customers

0

0.1

0.2

0.3

0.4

0.5

0.6

12 13 14

Number of Agents

Ave

rage

Wai

ting

Tim

e (M

inut

es)

P = 50%P = 40%P = 30%P = 20%P = 10%

Figure 6-4 Average Waiting Time for Priority Customers versus Number of Agents

Average Waiting Time for Regular Customers

0

0.2

0.4

0.6

0.8

1

1.2

12 13 14

Number of Agents

Ave

rage

Wai

ting

Tim

e (M

inut

es)

P = 50%P = 40%P = 30%P = 20%P = 10%

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 30%. We observed a decrease in the service

level (SL_RC1) by varying the maximum allowable service time from 5 minutes to 3 minutes

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

12 13 14

Service Level - Priority [82.73,85.65] [90.09,92.23] [93.52,94.86]

Service Level - Regular [92.03,93.95] [96.82,97.52] [98.24,98.68]

Abandonment Rate - Priority [5.01,6.51] [3.42,4.70] [2.18,2.98]

Abandonment Rate - Regular [15.79,17.59]



[8.54,10.22] [4.30,5.08]


Avg. wait time - Regular [0.95,1.07] [0.57,0.65] [0.31,0.37]

30%


% of priority Calls


Number of Agents

12 13 14







30%


Sensitivity Analysis - Service Level for Regular Customers

0204060

80100120

12 13 14

Number of Agents

Perc

enta

ge o

f Reg

ular

C

usto

mer

s

Service Level RegularCustomers - 3minutesService Level RegularCustomers - 2minutes

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 90% 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 90% of the calls answered within 30

seconds for priority customers and 90% 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.

% of priority Calls


Number of Agents

14 15 16




Abandonment Rate – Regular [4.30,5.08]

Table 6-4 Service Levels: Regular Customers (2 minutes) Priority Customers (30 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

[1.93,2.77] [0.76,1.16]



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 non-

stationary 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.

References

[1] Andrews, B.H. and Parsons, H.L., “Establishing telephone-agent staffing levels through

economic optimization,” Interfaces, Vol. 23(2), 1993, pp. 15-20.

[2] Balci, O., “Verification, validation and testing,” In: Banks J, editor. The handbook of

simulation. New York: Wiley, 1997, pp. 335-393.

[3] Dawson, K., “The call Center Handbook,” Flatiron Publishing, New York, 1996.

[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. 504-523.

[5] Kelton, W. D., Sadowski, R. P., and Sadowski, D. A., “Simulation with Arena,”

McGraw-Hill, New York, 1998.

[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. 28-29.

[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. 1-21.

[8] Parkman, C., “Simulation of a Fast-Food Operation Where Dissatisfied Customers

Renege,” The Journal of Operational Research Society, Vol. 38(2), 1987, pp. 137-148.

[9] Saltzman, R.M. and Mehrotra, V., “A call center uses simulation to drive strategic

change,” Interfaces, Vol. 31(3), 2001, pp. 87-101.

[10] Swisher, J.R., Jacobson, S.H., Jun, J.B., and Balci, O., “Modeling and analyzing a

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Simulation Study of an Inbound Call Centersjayaswa/projects/Simulation... · Simulation Study of an Inbound Call Center Submitted by ... the inter-arrival times since the modeling

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