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Proceedings of the 2010 Winter Simulation Conference

B. Johansson, S. Jain, J. Montoya-Torres, J. Hugan, and E. Yücesan, eds.

DOES THE ERLANG C MODEL FIT IN REAL CALL CENTERS?

Thomas R. Robbins D. J. Medeiros

East Carolina University The Pennsylvania State University

Department of Marketing and Supply Chain Industrial and Manufacturing Engineering

3212 Bate Building 310 Leonhard Building

Greenville, NC 27858, USA University Park, PA 16802, USA

Terry P. Harrison

The Pennsylvania State University

Smeal College of Business

459 Business Building

University Park, PA 16802, USA

ABSTRACT

We consider the Erlang C model, a queuing model commonly used to analyze call center performance.

Erlang C is a simple model that ignores caller abandonment and is the model most commonly used by

practitioners and researchers. We compare the theoretical performance predictions of the Erlang C model

to a call center simulation model where many of the Erlang C assumptions are relaxed. Our findings in-

dicate that the Erlang C model is subject to significant error in predicting system performance, but that

these errors are heavily biased and most likely to be pessimistic, i.e. the system tends to perform better

than predicted. It may be the case that the model’s tendency to provide pessimistic (i.e. conservative) es-

timates helps explain its continued popularity. Prediction error is strongly correlated with the abandon-

ment rate so the model works best in call centers with large numbers of agents and relatively low utiliza-

tion rates.

1 INTRODUCTION

A call center is a facility designed to support the delivery of some interactive service via telephone

communications; typically an office space with multiple workstations manned by agents who place and

receive calls (Gans, Koole et al. 2003). Call centers are a large and growing component of the U.S. and

world economy and are estimated to employ approximately 2.1 million call center agents (Aksin, Armo-

ny et al. 2007). Large scale call centers are technically and managerially sophisticated operations and

have been the subject of substantial academic research. The literature focused on call centers is quite

large, with thorough and comprehensive reviews provided in (Gans, Koole et al. 2003) and (Aksin, Ar-

mony et al. 2007). Empirical analysis of call center data is given in (Brown, Gans et al. 2005).

Call centers are examples of queuing systems; calls arrive, wait in a virtual line, and are then serviced

by an agent. Call centers are often modeled as M/M/N queuing systems, or in industry standard terminol-

ogy - the Erlang C model. The Erlang C model makes many assumptions which are questionable in the

context of a call center environment. Specifically the Erlang C model assumes that calls arrive at a

known average rate, and that they are serviced by a defined number of statistically identical agents with

service times that follows an exponential distribution. Most significantly, Erlang C assumes all callers

wait as long as necessary for service without hanging up. The model is used widely by both practitioners

and academics.

2853978-1-4244-9864-2/10/$26.00 ©2010 IEEE

Robbins, Medeiros and Harrison

Recognizing the deficiencies of the Erlang C model, many recent papers have advocated using alter-

native queuing models and staffing heuristics which account for conditions ignored in the Erlang C mod-

el. The most popular alternative is the Erlang A model, a simple extension of the Erlang C model that al-

lows for caller abandonment. For example, in a widely cited review of the call center literature (Gans,

Koole et al. 2003) , the authors state “For this reason, we recommend the use of Erlang A as the standard

to replace the prevalent Erlang C model.” Another widely cited paper examines empirical data collected

from a call center (Brown, Gans et al. 2005) and these authors make a similar statement; “using Erlang-

A for capacity-planning purposes could and should improve operational performance. Indeed, the model

is already beyond typical current practice (which is Erlang-C dominated), and one aim of this article is to

help change this state of affairs.”

The purpose of this study is to systematically analyze the fit of the Erlang C model in realistic call

center situations. We seek to understand the nature and magnitude of the error associated with the model,

and develop a better understanding of what factors influence prediction error.

The remainder of this paper is organized as follows. In Section 2 we review the Erlang C model and

highlight the relevant literature. In section 3 we present a general model of a steady state call center envi-

ronment and review the simulation model we developed to evaluate it. In section 4 we evaluate the per-

formance of the Erlang C model. We conclude in Section 5 with summary observations and identify fu-

ture research questions.

2 QUEUING MODELS AND THE ASSOCIATED LITERATURE

Queuing models are used to estimate system performance of call centers so that the appropriate staff-

ing level can be determined to achieve a desired performance metric such as the Average Speed to An-

swer, or the Abandonment percentage. The most common queuing model used for inbound call centers is

the Erlang C model (Gans, Koole et al. 2003; Brown, Gans et al. 2005). A Google search on “Erlang C

Calculator” generates about 700,000 items including a large number of downloadable applications to cal-

culate staffing requirements based on the Erlang C model.

The Erlang C model (M/M/N queue) is a very simple multi-server queuing system. Calls arrive ac-

cording to a Poisson process at an average rate of

λ

. By nature of the Poisson process interarrival times

are independent and identically distributed exponential random variables with mean

1

λ

−

. Calls enter an

infinite length queue and are serviced on a First Come – First Served (FCFS) basis. All calls that enter

the queue are serviced by a pool of

n

homogeneous (statistically identical) agents at an average rate

of

n

µ

. Service times follow an exponential distribution with a mean service time of

1

µ

−

.

The steady state behavior of the Erlang C queuing model is easily characterized, see for example

(Gans, Koole et al. 2003). The offered load, a unit-less quantity often referred to as the number of Er-

langs, is defined as

R

λ µ

. The traffic intensity (aka utilization or occupancy) is defined as

(

)

N R N

ρ λ µ

= .

Given the assumption that all calls are serviced, the traffic intensity must be strictly less than one or

the system becomes unstable, i.e. the queue grows without bound. This system can be analyzed by solv-

ing a set of balance equations and the resulting steady state probability that all N agents are busy is

{ }

1 1

0 0

1

0 1

! ! ! 1

m m m

N N

m m

R R R

P Wait

m m N R N

− −

= =

> = − +

−

∑ ∑

(1)

Equation (1) calculates the proportion of callers that must wait prior to service, an important measure of

system performance. Another relevant performance measure for call centers managers is the Average

Speed to Answer (ASA).

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Robbins, Medeiros and Harrison

[ ]

{ }

[ ]

{ }

0 | 0

1 1 1

0

1

i i

ASA E Wait P Wait E Wait Wait

P Wait

N

µ ρ

= > ⋅ >

= > ⋅ ⋅ ⋅

−

(2)

A third important performance metric for call center managers is the Telephone Service Factor (TSF),

also called the “service level.” The TSF is the fraction of calls presented which are eventually serviced

and for which the delay is below a specified level. For example, a call center may report the TSF as the

percent of callers on hold less than 30 seconds. The TSF metric can then be expressed as

{

}

{

}

{

}

( )

1

1 0 | 0

1 ( , )

i i

N T

i

TSF P Wait T P Wait P Wait T Wait

C N R e

µ ρ

− −

≤ = − > ⋅ > >

= − ⋅

(3)

A fourth performance metric monitored by call center managers is the Abandonment Rate; the propor-

tion of all calls that leave the queue (hang up) prior to service. Abandonment rates cannot be estimated

directly using the Erlang C model because the model assumes no abandonment occurs.

A substantial amount of research analyzes the behavior of Erlang C model, much of it seeks to estab-

lish simple staffing heuristics based on asymptotic frameworks applied to large call centers. (Halfin and

Whitt 1981) develop a formal version of the square root staffing principle for M/M/N queues in what has

become known as the Quality and Efficiency Driven (QED) regime. (Borst, Mandelbaum et al. 2004) de-

velop a framework for asymptotic optimization of a large call center with no abandonment.

As is the case with any analytical model, the Erlang C model makes many assumptions, several of

which are not wholly accurate. In the case of the Erlang C model several assumptions are questionable,

but clearly the most problematic is the no abandonment assumption, as even low levels of abandonment

can dramatically impact system performance (Gans, Koole et al. 2003). Many call center research papers

however analyze call center characteristics under the assumption of no abandonment (Jennings and Man-

delbaum 1996; Green, Kolesar et al. 2001; Green, Kolesar et al. 2003; Borst, Mandelbaum et al. 2004;

Wallace and Whitt 2005; Gans and Zhou 2007).

The Erlang C model assumes also that calls arrive according to a Poisson process. The interarrival

time is a random variable drawn from an exponential distribution with a known arrival rate. Several au-

thors assert that the assumption of a known arrival rate is problematic. Both major call center reviews

(Gans, Koole et al. 2003; Aksin, Armony et al. 2007) have sections devoted to arrival rate uncertainty.

(Brown, Gans et al. 2005) perform a detailed empirical analysis of call center data. While they find that a

time-inhomogeneous Poisson process fits their data, they also find that arrival rate is difficult to predict

and suggest that the arrival rate should be modeled as a stochastic process. Many authors argue that call

center arrivals follow a doubly stochastic process, a Poisson process where the arrival rate is itself a ran-

dom variable (Chen and Henderson 2001; Whitt 2006; Aksin, Armony et al. 2007). Arrival rate uncer-

tainty may exist for multiple reasons. Arrivals may exhibit randomness greater than that predicted by the

Poisson process due to unobserved variables such as the weather or advertising. Call center managers at-

tempt to account for these factors when they develop forecasts, yet forecasts may be subject to significant

error. (Robbins 2007) compares four months of week-day forecasts to actual call volume for 11 call cen-

ter projects. He finds that the average forecast error exceeds 10% for 8 of 11 projects, and 25% for 4 of

11 projects. The standard deviation of the daily forecast to actual ratio exceeds 10% for all 11 projects.

(Steckley, Henderson et al. 2009) compare forecasted and actual volumes for nine weeks of data taken

from four call centers. They show that the forecasting errors are large and modeling arrivals as a Poisson

process with the forecasted call volume as the arrival rate can introduce significant error. (Robbins, Me-

deiros et al. 2006) use simulation analysis to evaluate the impact of forecast error on performance meas-

ures demonstrating the significant impact forecast error can have on system performance.

Some recent papers address staffing requirements when arrival rates are uncertain. (Bassamboo, Har-

rison et al. 2005) develop a model that attempts to minimize the cost of staffing plus an imputed cost for

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Robbins, Medeiros and Harrison

customer abandonment for a call center with multiple customer and server types when arrival rates are va-

riable and uncertain. (Harrison and Zeevi 2005) use a fluid approximation to solve the sizing problem for

call centers with multiple call types, multiple agent types, and uncertain arrivals. (Whitt 2006) allows for

arrival rate uncertainty as well as uncertain staffing, i.e. absenteeism, when calculating staffing require-

ments. (Steckley, Henderson et al. 2004) examine the type of performance measures to use when staffing

under arrival rate uncertainty. (Robbins and Harrison 2010)develop a scheduling algorithm using a sto-

chastic programming model that is based on uncertain arrival rate forecasts.

The Erlang C model also assumes that the service time follows an exponential distribution. The me-

moryless property of the exponential distribution greatly simplifies the calculations required to character-

ize the system’s performance, and makes possible the relatively simple equations (1)-(3). If the assump-

tion of exponentially distributed talk time is relaxed, the resulting queuing model is the

/ /

M G N

queue,

which is analytically intractable (Gans, Koole et al. 2003) and approximations are required. However,

empirical analysis suggests that the exponential distribution is a relatively poor fit for service times. Most

detailed analysis of service time distributions find that the lognormal distribution is a better fit (Mandel-

baum, Sakov et al. 2001; Gans, Koole et al. 2003; Brown, Gans et al. 2005).

Finally, the Erlang C model assumes that agents are homogeneous. More precisely, it is assumed that

the service times follow the same statistical distribution independent of the specific agent handling the

call. Empirical evidence supports the notion that some agents are more efficient than others and the dis-

tribution of call time is dependent on the agent to whom the call is routed. In particular more experienced

agents typically handle calls faster than newly trained agents (Armony and Ward 2008). (Robbins 2007)

demonstrated a statistically significant learning curve effect in an IT help desk environment.

3 CALL CENTER SIMULATION

3.1 The Modified Model

In this section we present a revised model of a call center, relaxing key assumptions discussed previously.

In our model calls arrive at the call center according to a Poisson process. Calls are forecasted to arrive at

an average rate of

ˆ

λ

. The realized arrival rate is

λ

, where

λ

is a normally distributed random variable

with mean

ˆ

λ

, standard deviation

λ

σ

and coefficient of variation

ˆ

c

λ λ

σ λ

= . The choice of the normal dis-

tribution gives us a symmetric distribution centered on the forecasted value. A disadvantage of the nor-

mal distribution is the possibility of generating negative values. However, in our experiments the mean

value is sufficiently positive, a minimum of 5 standard deviations, that this is not a concern. The time re-

quired to process a call by an average agent is a lognormally distributed random variable with

mean

1

µ

−

and standard deviation

µ

σ

. Arriving calls are routed to the agent who has been idle for the long-

est time if one is available. If all agents are busy the call is place in a FCFS queue. When placed in

queue a proportion of callers will balk; i.e. immediately hang up. Callers who join the queue have a pa-

tience time that follows a Weibull distribution. If wait time exceeds their patience time the caller will ab-

andon. Calls are serviced by agents who have variable relative productivity

i

r

. Agent productivity is as-

sumed to be a normally distributed random variable with a mean of 1 and a standard deviation of

r

σ

. An

agent with a relative productivity level of 1, for example, serves calls at the average rate. An agent with

a relative productivity level of 1.5 serves calls at 1.5 times the average rate, an agent with a productivity

level of .75 serves calls at .75 times the average rate. Given the mean productivity level of 1, on average

calls are served at the rate

λ

.

3.2 Experimental Design

In order to evaluate the performance of the Erlang C against the simulation model we conduct a series of

designed experiments. Based on the assumptions for our call center discussed previously, we define the

following set of nine experimental factors.

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Robbins, Medeiros and Harrison

Table 1: Experimental Factors

Factor Low High

1 Number of Agents 10 100

2

Offered Utilization (

ˆ

ρ

)

65% 95%

3 Talk Time (mins) 2 20

4

Patience

β

60 600

5

Forecast Error CV (

c

λ

)

0 .2

6

Patience

α

.75 1.25

7 Talk time CV .75 1.25

8 Probability of Balking 0 .25

9 Agent Productivity Standard Deviation 0 .15

The forecasted arrival rate in the simulation is a quantity derived from other experimental factors by

ˆ

ˆ

N

λ ρ µ

= (4)

Given the relatively large number of experimental factors, a well designed experimental approach is re-

quired to efficiently evaluate the experimental region. A standard approach to designing computer simu-

lation experiments is to employ either a full or fractional factorial design (Law 2007). However, the fac-

torial model only evaluates corner points of the experimental region and implicitly assumes that responses

are linear in the design space. Given the anticipated non-linear relationship of errors we chose to imple-

ment a Space Filling Design based on Latin Hypercube Sampling (LHS) as discussed in (Santner, Wil-

liams et al. 2003). Given a desired sample of n points, the experimental region is divided into n

d

cells. A

sample of n cells is selected in such a way that the centers of these cells are uniformly spread when pro-

jected onto each axis of the design space. While the LHS design is not perfectly orthogonal like a factori-

al design, the design does provide for a low correlation between input factors greatly reducing the risk of

multicollinearity. We chose our design point as the center of each selected cell.

3.3 Simulation Model

The model is evaluated using a straightforward discrete event simulation model. The purpose of the

model is to predict the long term, steady state behavior of the queuing system. The model generates ran-

dom numbers using the a combined multiple recursive generator (CMRG) based on the Mrg32k3a genera-

tor described in (L'Ecuyer 1999). Common random numbers are used across design points to reduce out-

put variance. To reduce any start up bias we use a warm up period of 5,000 calls, after which all statistics

are reset. The model is then run for an evaluation period of 25,000 calls and summary statistics are col-

lected. For each design point we repeat this process for 500 replications and report the average value

across replications.

The specific process for each replication is as follows. The input factors are chosen based on the ex-

perimental design. The average arrival rate is calculated based on the specified talk time, number of

agents, and offered utilization rate according to equation (4). A random number is drawn and the rea-

lized arrival rate is set based on the probability distribution of the forecast error. That arrival rate is then

used to generate Poisson arrivals for the replication. Agent productivities are generated using a normal

distribution with mean one and standard deviation

p

σ

. Each new call generated includes an exponential-

ly distributed interarrival time, a lognormally distributed average talk time, a Weibull distributed time be-

fore abandonment, and a Bernoulli distributed balking indicator. When the call arrives it is assigned to

the longest idle agent, or placed in the queue if all agents are busy. If sent to the queue the simulation

model checks the balking indicator. If the call has been identified as a balker it is immediately aban-

doned, if not an abandonment event is scheduled based on the realized time to abandon. Once the call has

been assigned to an agent, the realized talk time is calculated by multiplying the average talk time and the

agent’s productivity. The agent is committed for the realized talk time. When the call completes the

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Robbins, Medeiros and Harrison

agent processes the next call from the queue, or if no calls are queued becomes idle. If a call is processed

prior to its time to abandon, the abandonment event is cancelled. If not, the call is abandoned and re-

moved from the queue when the patience time expires.

After all replications of the design point have been executed the results are compared to the theoreti-

cal predictions of the Erlang C model. We calculate the error as the difference between the theoretical

value and the simulated value. We make a relative error calculation so that the sign of the error indicates

the bias in the calculation. In our experiment we evaluated an LHS sample of 1,000 points.

4 EXPERIMENTAL ANALYSIS

4.1 Summary Observations

Based on our analysis we can make the following summary observations:

• The Erlang C model is, on average, subject to a reasonably large error over this range of parameter

values.

• Measurement errors are highly positively correlated across performance measures.

• The Erlang C model is on average pessimistically biased (the real system performs better than pre-

dicted) but may become optimistically biased when utilization is high and arrival rates are uncertain.

• Measurement error is high when the real system exhibits higher levels of abandonment. The error is

strongly positively correlated with realized abandonment rate and predicted ASA.

• The Erlang C model is most accurate when the number of agents is large and utilization is low.

• Errors decrease as caller patience increases.

We will now review our experimental results in more detail.

4.2 Correlation and Magnitude of Errors

The magnitude of errors generated by using the Erlang C model across our test space is high on average,

and very high in some cases. The errors across the key metrics are highly correlated with each other, and

highly correlated with the realized abandonment rate. Table 2 shows a correlation matrix of the errors

generated from the Erlang C model.

Table 2: Error Correlation Matrix

Simulated

Abandonment

Rate

Prob Wait

Error

ASA

Error

TSF

Error

Utilization

Error

Simulated Abandonment Rate 1.000

Prob Wait Error .867 1.000

ASA Error .766 .722 1.000

TSF Error -.880 -.987 -.759 1.000

Utilization Error .970 .861 .745 -.873 1.000

Correlations between measure errors are strong. The measured errors all move, on average, in an

optimistic or pessimistic direction together. ProbWait and ASA are positively correlated; it is desirable

for both these measures to be low. ProbWait is negatively correlated with TSF; a measure for which a

high value is desirable. Measurement error is also highly correlated with abandonment rate. Given the

high correlation between measures we will utilize ProbWait as a proxy for the overall error of the Erlang

C model.

Average error rates are reasonably high under the Erlang C model, with errors being pessimistically

skewed. Figure 1 shows a histogram of the ProbWait error.

2858

Robbins, Medeiros and Harrison

0

5

10

15

20

25

30

Percent

Error (Theoretical - Simulation)

Prob Wait Error

Figure 1: Histogram of Erlang C Prob Wait Errors

The average error is 7.96%, and the data has a strong positive skew; 72% of the errors being positive.

The largest error is 49.4%, the smallest is -8.0 %.

4.3 Drivers of Erlang C Error

Having established that error rates are high under the Erlang C model, we now turn out attention to cha-

racterizing the drivers of that error. As discussed in the previous section, Erlang C errors are highly cor-

related with the realized abandonment rate. The notion that abandonment is a major driver of errors in the

Erlang C model is further illustrated in Figure 2. This graph shows the error in the ProbWait measure on

the vertical axis and the abandonment rate from the simulation analysis on the horizontal axis.

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Error in Probability of Wait Estimate

(Theoretical - Simulation)

Abandonment Rate from Simulation

Error in Probability of Wait Calculations vs. Abandonment

Figure 2: Scatter Plot of Erlang C Errors and Abandonment Rate

2859

Robbins, Medeiros and Harrison

The graph clearly shows that as abandonment increases, the error in the ProbWait measure increases as

well. The graph also reveals that optimistic errors, i.e. errors in which the system performed worse than

predicted, only occur with relatively low abandonment rates. The average abandonment rate for optimis-

tic predictions was .74%. The graph also reveals that significant error can be associated with even low to

moderate abandonment rates. For example, for all test points with abandonment rates of less than 5%, the

average error for ProbWait is 4.8%. For test points in which abandonment ranged between 2% and 5%

the average ProbWait error is 12.2%.

To assess how each of the nine experimental factors impacts the error, we perform a regression analy-

sis. The dependent variable is the ProbWait error. For the independent variable we use the nine experi-

mental factors normalized to a [-1,1] scale. This normalization allows us to better assess the relative im-

pact of each factor. The LHS sampling method provides an experimental design where the correlation

between experimental factors is low, greatly reducing risks of multicollinearity. The results of the regres-

sion analysis are shown in Table 3.

Table 3: Regression Analysis of ProbWait Error

Regression Analysis

R² 0.746

Adjusted R² 0.744 n 1000

R 0.864 k 9

Std. Error 0.058 Dep. Var.

Prob Wait Error

ANOVA table

Source SS df MS F p-value

Regression 9.6689 9 1.0743 323.87 8.38E-288

Residual 3.2839 990 0.0033

Total 12.9529 999

Regression output confidence interval

variables coefficients std. error t (df=990) p-value 95% lower 95% upper

Intercept 0.0797 0.0018 43.745 1.52E-233 0.0761 0.0832

Num Agents -0.0721 0.0032 -22.778 1.11E-92 -0.0783 -0.0658

Utilization Target 0.1500 0.0032 47.365 1.09E-256 0.1438 0.1562

Talk Time 0.0184 0.0032 5.829 7.53E-09 0.0122 0.0246

Patience -0.0134 0.0032 -4.233 2.52E-05 -0.0196 -0.0072

AR CV -0.0260 0.0032 -8.206 7.05E-16 -0.0322 -0.0198

Talk Time CV -0.0035 0.0032 -1.096 .2734 -0.0097 0.0027

Patience Shape -0.0027 0.0032 -0.858 .3912 -0.0089 0.0035

Probability of Balking 0.0228 0.0032 7.172 1.44E-12 0.0165 0.0290

Agent Heterogeneity 0.0050 0.0032 1.585 .1133 -0.0012 0.0112

Given the normalization of the experimental factors, the magnitude of the regression coefficients pro-

vides a direct assessment of the impact that a factor has on the measurement error. The factor that most

strongly influences the error is the offered utilization, the magnitude of its coefficient being more than

twice the value of the next measure and more than five times the magnitude of all other factors. The size

of the call center, measured as the number of agents, has a major impact on errors. Factors related to wil-

lingness to wait, i.e. Patience, Patience Shape, and Probability of Balking, all have low to moderate im-

pacts, but with exception of Patience Shape are statistically significant. Talk time is also a statistically

significant factor with a moderate impact. The variability of talk time and agent heterogeneity both have

low impacts that are not statistically significant.

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Robbins, Medeiros and Harrison

The most important drivers of Erlang C errors are the size and utilization of the call center. This is

further illustrated in Figure 3. This graph shows the results of an experiment where the number of agents

and utilization factors are varied in a controlled fashion. All other experimental factors are held at their

mid-point.

-10%

0%

10%

20%

30%

40%

50%

10 15 20 2 5 30 35 40 45 50 55 60 65 70 75 8 0 85 9 0 95 100

Error in Erlang C Probability of Wait Calculation

(Theoretical-Simulated)

Number of Agents

Prob of Wait Error by Number of Agents and Offered Utilization

0.65

0.75

0.85

0.95

Figure 3: Erlang C ProbWait Errors by Call Center Size and Utilization

This graph demonstrates that the Erlang C model tends to provide relatively poor predictions for

small call centers. This error tends to decrease as the size of the call center increases. However, the

graph also illustrates that for busy centers the error remains high. For a very busy call center, running at

95% offered utilization, the error rate remains at 30%, even with a pool of 100 agents. The errors tend to

track with abandonment; abandonment rates increase with utilization and decrease with the agent pool.

The conclusion that abandonment behavior drives the Erlang C error is further illustrated in Figure 4.

In this experiment we systematically vary two Willingness to Wait parameters. Specifically, we vary the

balking probability and the β factor of patience distribution.

-0.02

-0.01

0

0.01

0.02

0.03

0.04

60 90 120 150 180 210 2 40 270 300 3 30 36 0 390 420 450 480 510 540 570 600

Error in Erlang C Probability of Wait Calculation

(Theoretical-Simulated)

Patience (β

ββ

β)

Prob of Wait Error by Patience β

ββ

β and Balking Rate

0

0.125

0.25

Figure 4: Erlang C ProbWait Errors by Willingness to Wait

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Robbins, Medeiros and Harrison

This analysis verifies that the more likely callers are to balk, the higher the error rate. The analysis al-

so shows that when callers are more patient, the error rates decrease. The more likely callers are to aban-

don, either immediately or soon after being queued, the higher the abandonment rate and the less accurate

the Erlang C measures become.

An additional factor of interest is the uncertainty associated with the arrival rate. While it’s overall

effect is not large, about 1.8%, it has effects that are dissimilar to other experimental factors as illustrated

in Figure 5. This graph shows the results of an experiment that varies the coefficient of variation of the

arrival rate error and the number of agents while holding all other factors at their mid-points.

-10%

-5%

0%

5%

10%

15%

20%

10 15 20 ' 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Error in Erlang C Probability of Wait Calculation

(Theoretical-Simulated)

Number of Agents

Prob of Wait Error by Number of Agents and Arrival Rate Uncertainty

0

0.1

0.2

Figure 5: Erlang C ProbWait Errors by Call Center Size and Forecast Error

This experiment shows that for small call centers arrival rate uncertainty has a small effect, but that

effect becomes more pronounced for larger call centers. It is also worth noting that arrival rate uncertain-

ty has an optimistic effect, and for high levels of uncertainty the model exhibits an optimistic bias. Arriv-

al rate uncertainty is a major factor leading to an optimistic estimate from the Erlang C model; of the

21.9% of test points with an optimistic bias the average arrival rate uncertainty cv was 14.0%. Since ar-

rival rate uncertainty tends to bias the prediction in the opposite direction of most other factors, it also has

the effect of reducing error in many situations. For example, high utilization tends to bias the estimate

pessimistically, a bias reduced when arrival rate uncertainty is present.

5 SUMMARY AND CONCLUSIONS

The Erlang C model is commonly applied to predict queuing system behavior in call center applica-

tions. Our analysis shows that when we test the Erlang C model over a range of reasonable conditions

predicted performance measures are subject to large errors. The Erlang C model works reasonably well

for large call centers with low to moderate utilization rates, but factors that tend to generate caller aban-

donment; such as high utilization, small agent pools, and impatient callers, cause the model error to be-

come quite large. While the model tends to provide a pessimistic estimate, arrival rate uncertainty will

either reduce that bias or lead to a optimistic bias. It may be the case that the model’s tendency to provide

pessimistic (i.e. conservative) estimates helps explain its continued popularity. It is clear that great care

must be taken before using the Erlang C model to make any calculations that require a high level of preci-

sion.

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Our future research is focused on analyzing the increasingly popular Erlang A model and comparing

it’s performance to the Erlang C model to test the growing consensus that Erlang A is a superior model

for call center analysis.

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AUTHOR BIOGRAPHIES

THOMAS R. ROBBINS is an Assistant Professor in the department of Marketing and Supply Chain at

East Carolina University. He holds a PhD in Business Administration and Operations Research from

Penn State University, an MBA from Case Western Reserve and a BSEE from Penn State. Prior to be-

ginning his academic career he worked in professional services for approximately 18 years. His email

address is <robbinst@ecu.edu>.

D. J. MEDEIROS is Associate Professor of Industrial Engineering at Penn State University. She holds a

Ph.D. and M.S.I.E from Purdue University and a B.S.I.E. from the University of Massachusetts at Am-

herst. She has served as track coordinator, Proceedings Editor, and Program Chair for WSC. Her re-

search interests include manufacturing systems control and CAD/CAM. She is a member of IIE and

SME. Her email address is <djm3@psu.edu>.

TERRY P. HARRISON is the Earl P. Strong Executive Education Professor of Business and Professor

of Supply Chain and Information Systems at Penn State University. He holds a Ph.D. and M.S. degree in

Management Science from the University of Tennessee and a B.S in Forest Science from Penn State.

He was formerly the Editor-in-Chief of Interfaces and is currently Vice President of Publications for

INFORMS. His mail address is <tharrison@psu.edu>.

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