Empirical Analysis of a Telephone Call Center

Anat Sakov

Joint work with Avishai Mandelbaum, Sergey Zeltyn, Larry Brown, Linda Zhao and Heipeng Shen

Statistics Seminar, Tel Aviv University, 29.5.01

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Introduction

A call center is a service network in which agents provide telephone-based services.Consists of: callers (customers), servers (agents) and queues.Growing rapidly.Sources of data: ACD, CTI, surveys.Availability of data.

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Israeli Bank Call Center

Types of service Regular services on checking/saving

accounts. Regular services in English. Internet technical support. Information for prospective customers. Stock trading. Outgoing calls.

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Israeli Bank Call Center (cont’)

Agents 8 regular agents. 5 Internet-support agents. 1 shift supervisor.

Working hours Weekdays: 7a.m.-24a.m. Friday: 7a.m.-14p.m. Saturday: 8p.m.-24p.m.

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Research GoalsStandard assumptions: Arrival rate is a Poisson process. Service time is Exponential. Little empirical evidence to support assumptions. Checking assumptions, and developing (with OR

researchers) realistic models.Estimation of customer’s patience while waiting, has not been studied. When better understood, can be used by call center

managers (e.g. for better staffing). Requires data on the individual call basis.

As a first of its kind, can serve as a prototype for future work (e.g. larger centers).

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Specific QuestionsChecking standard queueing theory assumptions.Interrelation between components of queueing model.What can be said on behavior of customers. Are they patience ? Do customers with different service type, behave

differently ? Analyzing individual customers (e.g. to set

priorities).

Analysis of an individual agent (e.g. “learning curve” of new agents; skill-based routing).

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Statistical Challenges

Massive amounts of data (‘our’ center is very small compared to centers in US, and even Israel). Issues in Survival Analysis High % of censoring (we have 80%, can reach 95%

or even higher). Smooth estimation of hazard function and

confidence bands for it. Dependence between censoring and failure times. More …

Interdisciplinary research.

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Incoming Call – Event History

Abandon~15%

Abandon~5%

End of Service ~80%

End of Service

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Description of Data

About 450,000 calls.All calls during 1999.Information on calls Customer ID. Priority (high/low). Type of service. Date. Time call enters VRU.

Time spent in VRU. Time in queue (could be 0). Service time. Outcome: HANG / AGENT. Server name.

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Sample Datavru+line call_id customer_id pri type date vru_entry vru_exit vru_tAA0101 47530 17878737 2 PE 991101 7:14:55 7:14:59 4AA0101 47533 4345955 2 PS 991101 8:17:40 8:17:46 6AA0101 47536 69438695 1 PS 991101 8:50:34 8:50:40 6AA0101 47540 55279970 2 NE 991101 9:18:40 9:18:46 6AA0101 47546 57379505 2 PS 991101 9:56:01 9:56:07 6

q_start q_exit q_time outcome ser_start ser_exit ser_time server7:14:59 7:15:17 18 AGENT 7:15:16 7:23:16 480 YITZ8:17:46 8:18:01 15 AGENT 8:18:00 8:18:35 35 KAZAV8:50:40 8:51:59 79 HANG 0:00:00 0:00:00 0 NO_SERVER9:18:46 9:18:58 12 AGENT 9:18:57 9:25:12 375 VICKY9:56:07 9:56:52 45 HANG 0:00:00 0:00:00 0 NO_SERVER

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Calls per Month

0

15,000

30,000

45,000

Jan

Feb

Mar

Apr

May

June

July

Aug

Sep

Oct

Nov

Dec

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Types of Service

68.1%

15.2%

8.9%4.7% 2.8% 0.4%

0

10

20

30

40

50

60

70

Regular New Cust. Stock Internet Outgoing English

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% of Internet Calls per Month

0.9% 1.4%3.1%

8.5%10% 9.4%

0

4

8

12

Jan

Feb

Mar

Apr

May

June

July

Aug

Sep

Oct

Nov

Dec

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Queueing Process

Three components Arrival to the system. Queue while waiting to an agent. Service. Interrelations between the three

components.

Waiting time is a function of arrival rate and of service time.

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Queueing Time

60% of calls waited in queue (positive wait) Average wait – 98 seconds. SD – 105. Median – 62. Exponential looking.

Can we say that the customers are patience ? How can we define patience ?

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Queueing Time (cont’)

Average waiting time and SD for Customers reaching an agent: 105 (111). Customers abandoning: 79 (104).

Numbers vary by type and priorities.Can 105 and 79 be estimates for the mean waiting time until reaching an agent and until abandoning ? No.

To estimate mean time until abandoning, customers

reaching an agent are censored by abandoning customers, and vice versa.

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Queueing Time (Cont’)

Denote time willing to wait, by R.Denote time needed to wait, by V.Observe W=min(R,V ) and δ=1(W=V ).To estimate the distribution of R, about 75%-80% censoring. If assumes R and V are both Exponential and independent E(V ) = 131 (compared to 105). E(R ) = 393 (compared to 78).

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Queueing Time (Cont’)

To avoid parametric assumptions, use Kaplan-Meier, to estimate survival function.

E(V ) = 141 (compared to 105).E(R ) = 741 (compared to 78).

Depending on which observation is last, either E(R ) or E(V ) is downward biased.

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Waiting Times Survival Curves

Time

Surv

ival

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Stochastic Ordering

The stochastic ordering says that customers are willing to wait, more than what they need to wait.

This suggests that customers are patience.

We obtained the same picture for different types of service and different months.

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Time Willing to Wait

Time

Surv

ival

Survival function

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Hazard Estimation

Shows local behavior.

Raw hazard are building blocks for Kaplan-Meier.

Noisy and unstable at tails.

Would like to estimate hazard function smoothly (later, construct confidence bands).

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HEFT /HARELet

Model (Kooperberg, Stone and Truong (1995 JASA)),

The are splines basis functions.

Plug into joint likelihood, and estimate coefficients using maximum-likelihood.

;)|(1

)|()|(

xtFxtf

xt

Rxtxt )|(log)|(

p

jjj xtBxt

1)|()|(

jB

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HEFT/HARE (cont’)

HARE – HAzard REgression Use linear splines in time and covariates, and their

interactions. Cox proportional model is a special case.

Additivity in time and covariates indicates that proportionality assumption holds.

HEFT – Hazard Estimation with Flexible Tails. No covariates. Cubic splines in time. Include additional two log terms. Fit Weibull and

Pareto very well.Can use bootstrap to construct confidence bands.

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HEFT/HARE (software)

Implementation in Splus.

Pick model in an adaptive manner. Using stepwise addition/deletion. Add/drop terms to maintain hierarchy. Use BIC criteria.

Fits the tails well.

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Time Willing To Wait

Hazard rate

Time

Haza

rd

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Validity of Analysis

A basic assumption in Survival Analysis is independence of time to failure and censoring time.A message which informs customers about their location in queue, might affect their patience.Nevertheless, the picture is informative.We ignore this and other types of dependence, as well.

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Other Approaches

Apply nonparametric regression to obtain smooth estimates of hazard (regress raw hazard on time). Super-smoother; Kernel; LOWESS.

Not as good at the tails. Local polynomial (LOCFIT). Has a module

to estimate hazard.

Gave qualitatively same picture.

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Service Time

Overall Regular service

New customers

Internet Stock

Mean 188

181 111 381 269

SD 240 207 154 485 320

Med 114 117 64 196 169

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Short Service Times

Jan-Oct Nov-Dec

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Service Time (cont’)Survival curve, by types

Time

Surv

ival

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Service Time (cont’)Hazard rate

Time

Surv

ival

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Service Time (cont’)

Standard assumption is that service time distribution is Exponential (for mathematical convenience).Density, survival function and hazard, do not support this assumption.We found that log-normal is a very good fit to service time. Holds for different types of service. Holds for different times of days. We are in the process of examining how service

time vary by time of day.Can use regression.

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Arrival process

Four levels of presentation. Differ by their time scale.

Top three levels are required to support staffing. Yearly – supports strategic decisions; how many

agents are needed (affects hiring and training). Monthly – supports tactical decisions; given total

number of agents needed, how many permanent. Daily – supports operational decisions; staffing is

made to fit “rush hours”, weekdays, weekends. All the above exhibit predictable variability.

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Arrival Process (cont’)Yearly Monthly

Daily Hourly

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Arrival process (cont’)

Hourly picture – depict stochastic pattern.Arrivals are typically random. Usual assumptions: Many potential, statistically identical callers. Very small probability for each to call, at any given

minute. Decisions to call are independent of each other. Under the above assumptions, Poisson process.

Further decomposition by types, shows that behavior vary by type of service.

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Checking Poisson Assumption

Arrival rate being a Poisson process is a standard assumption in queueing processes.The daily picture suggests that the rate is not constant over the day, hence, inhomogeneous Poisson.We are in the process of checking this. Consider the difference in times between

successive calls. Expect to behave like an exponential sample. Our checks indicates that indeed

inhomogeneous Poisson.

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Analyzing Individual Customer

Analysis of an individual customer can be used for example to update his priority.Most customers calls more than once during the year.

Average

SD Median

Overall 16 64 5

Regular service

14 51 4

Stock trading 83 188 5

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Obsessive Callers

Calls/year

Calls/day

(ave)

Service

Average

Service

Stock Tradin

g

1471 5 95% 9.5 min.

Regular

Service

1996 8 92% 2 min.