Abandonment and Customers Patience in Tele-Queues. The Palm/Erlang-A Model.

Transcription

1 Sergey Zeltyn February 25 STAT 991. Service Engineering. The Wharton School. University of Pennsylvania. Abandonment and Customers Patience in Tele-Queues. The Palm/Erlang-A Model. Based on: Mandelbaum A. and Zeltyn S. The Palm/Erlang-A Queue, with Applications to Call Centers. Lecture note to Service Engineering course. A Dec4.pdf Mandelbaum A. Service Engineering course, Technion. No abandonment in models of the previous lecture. However, abandonment takes place and can be very significant and very important. 1

2 Example 1. Catastrophic situation. Call center of telephone company. Average wait 72 sec, 81% calls answered (Saturday, 6/11/99) Average wait 217 sec, 53% calls answered (Thursday, 25/11/99) Average wait 376 sec, 24% calls answered (Sunday, 21/11/99) 2

3 Example 2. Moderate abandonment. (Moderate, but still very important.) Health Insurance. Charlotte Center. ACD report. Time Calls Answered Abandoned% ASA AHT Occ% # of agents Total 2,577 19,86 3.5% % 8: % % : % % : % % :3 1,152 1, % % : 1,33 1, % % :3 1,364 1, % % : 1,38 1,28 7.2% % :3 1,272 1,247 2.% % : 1,179 1,177.2% % :3 1,174 1,16 1.2% % : 1, % % :3 1, % % : 1,173 1,82 7.8% % :3 1,212 1, % % : 1,137 1, % % :3 1,169 1, % % : 1,17 1,59 4.3% % : % % : % % : % % : % % 5.8 3

4 Abandonment Important Practically One of two customer-subjective performance measures (2 nd =Redials); Lost business (now); Poor service level (future losses); 1-8 costs (out-of-pocket vs. alternative); Self-selection: the fittest survive and wait less; Must account for (carefully) in models and measures. Otherwise, wrong picture of reality: misleading performance measures, hence staffing. Unstable models (vs. robustness). Abandonment also Interesting Theoretically Queueing Science (Paradigm: experiment, measure, model, validate); Research: OR + Psychology + Marketing (Modelling: steady-state, transient, equilibrium); Wide Scope of Applications: in addition to Phone, VRU/IVR: opt-out-rates; Internet: business-drivers (6% and more). 4

5 The Erlang-A (Palm, M/M/n+M) Model Simplest model with abandonment, used by well-run call centers. Building blocks: λ Poisson arrival rate. µ Exponential service rate. n number of service agents. θ individual abandonment rate. agents arrivals λ queue 1 2 abandonment θ n µ Patience time τ exp(θ): time a customer is willing to wait for service; Offered wait V : waiting time of a customer with infinite patience; If τ V, customer abandons; otherwise, gets service; Actual wait W q = min(τ, V ). 5

6 Erlang-A vs. Erlang-C 48 calls per min, 1 min average service time, 2 min average patience probability of wait vs. number of agents average wait vs. number of agents probability of wait Erlang A Erlang C average waiting time, sec Erlang A Erlang C number of agents number of agents If 5 agents: M/M/n M/M/n+M M/M/n, λ 3.1% Fraction abandoning 3.1% - Average waiting time 2.8 sec 3.7 sec 8.8 sec Waiting time s 9-th percentile 58.1 sec 12.5 sec 28.2 sec Average queue length Agents utilization 96% 93% 93% The fittest survive and wait less - much less. Abandonment reduces workload when needed at high-congestion periods. 6

7 Erlang-A: Birth-and-Death Process L(t) number-in-system at time t (served plus queued); L = {L(t), t } Markov birth-and-death process. Transition-rate diagram Steady-state equations: λπ j = (j + 1) µπ j+1, j n 1 λπ j = (nµ + (j + 1 n)θ) π j+1, j n. Steady-state distribution: where π = π j = n j= (λ/µ) j π, j! j λ nµ + (k n)θ k=n+1 (λ/µ) j j! + j=n+1 j k=n+1 j n (λ/µ) n π, j n + 1, n! λ nµ + (k n)θ (λ/µ) n n! 1. Numerical drawback: infinite sums. 7

8 Stability Erlang-A is always stable! d j death-rate in state j, < j < : j min(µ, θ) d j j max(µ, θ). Bounds are death rates of M/M/ queues with service rates min(µ, θ) and max(µ, θ). Proof of stability: π 1 = n (λ/µ) j j= j! j= + j=n+1 j k=n+1 (λ/ min(µ, θ)) j (Use that nµ + (k n)θ k min(µ, θ).) j! λ nµ + (k n)θ = e λ/ min(µ,θ). (λ/µ) n n! 8

9 Steady-state distribution via special functions (Palm): Gamma function: Γ(x) = Incomplete Gamma function: t x 1 e t dt, x >. γ(x, y) = y tx 1 e t dt, x >, y. A(x, y) = xey y x γ(x, y) = 1+ j=1 y j j k=1(x + k), x >, y. Recall E 1,n blocking probability in M/M/n/n (Erlang-B): E 1,n = (λ/µ) n n! n j= (λ/µ) j j! = (λ/µ)n e λ/µ Can be calculated also via recursion. Then one can show: where π j = π n = π n π n 1 Γ(n + 1) γ(n + 1, λ/µ). n! j! (λ ) n j, j n, µ ( ) λ j n θ ( j n nµ k=1 θ +, j n + 1, k) E 1,n 1 + [ A ( nµ θ, λ θ ) 1 ] E1,n. 9

10 Operational Performance Measures The most popular performance measure is P {W q T ; Sr} (or even worse P {W q T Sr}). We recommend: P {W q T ; Sr} - fraction of well-served; P {Ab} - fraction of poorly-served. or a four-dimensional refinement: P {W q T ; Sr} - fraction of well-served; P {W q > T ; Sr} - fraction of served, with potential for improvement (say, a higher priority on next visit); P {W q > ɛ; Ab} - fraction of poorly-served; P {W q ɛ; Ab} - fraction of those whose service-level is undetermined. Properties of P{Ab}: P{Ab} increases monotonically in θ, λ; P{Ab} decreases monotonically in n, µ (Bhattacharya and Ephremides (1991)) M/M/n+G: if E[τ] is fixed, deterministic patience minimizes P{Ab} (Mandelbaum and Zeltyn (24)) 1

11 Additional important performance measures: Delay probability P{W q > }; Average wait E[W q ]; ASA (Average Speed of Answer) used extensively in call centers; usually defined as E[W q Sr]; Agents occupancy ρ = Average queue-length E[L q ]. λ (1 P{Ab}). nµ Operational Performance Measures: calculation via 4CallCenters Performance measures of the form E[f(V, τ)]. Calculable, in numerically stable procedures. For example, f(v, τ) E[f(V, τ)] 1 {v>τ} P{V > τ} = P{Ab} 1 (t, ) (v τ) P{W q > t} 1 (t, ) (v τ)1 {v>τ} P{W q > t; Ab} (v τ)1 {v>τ} E{W q ; Ab} g(v τ) E[g(W q )] 11

12 Operational Performance Measures: calculation via 4CallCenters Erlang-A parameters: λ = 3 calls per hour, 1/µ = 2 min, n = 1, 1/θ = 2 min. Target times T = 3 sec, ɛ = 1 sec. P{W q T ; Sr} = 71.1%; P{W q > T ; Sr} = 87.5% 71.1% = 16.4%; P{W q > ɛ; Ab} = 12.5% 3.9% = 8.6%; P{W q ɛ; Ab} = 3.9%. Delay probability P{W q > } = 1% 45.8% = 54.2%. 12

13 Additional performance measures Average Time in Queue = E[W q ] = 15 sec; ASA = E[W q Sr] = 13.8 sec; Agents Occupancy ρ = 87.5%; Average Queue Length E[L q ] =

14 Operational Performance Measures: calculation via special functions For example, P{W q > } = P[Ab W q > ] = j=n E[W q W q > ] = 1 θ π j = 1 ρa ( nµ θ, λ θ 1 ρa ( nµ θ, λ θ A ( nµ 1 + ( A ( nµ θ, λ θ ) ρ, 1 ) + 1 ρ θ, λ ) θ E1,n ) ), 1 E1,n. Operational Performance Measures: calculation via M/M/n+G formulae M/M/n+G generalization of Erlang-A, patience times distributed with cdf G( ). See Mandelbaum A. and Zeltyn S. (24) M/M/n+G queue. Summary of performance measures; Zeltyn S. (24) Call centers with impatient customers: exact analysis and many-server asymptotics of the M/M/n+G queue, Ph.D. Thesis. Explained how to adapt M/M/n+G to Erlang-A: G(x) = 1 e θx, θ >. 14

15 The relation P{Ab}/E[W q ] Theoretical: In Erlang-A (and other queueing models with exp(θ) patience): P{Ab} = θ E[W q ]. Proof. Balance equation: Little s formula: Substitute (2) into (1). θ E[L q ] = λ P{Ab}. (1) E[L q ] = λ E[W q ]. (2) Empirical relations Israeli bank: yearly data.8 hourly data aggregated Probability to abandon Average waiting time, sec Probability to abandon Average waiting time, sec The graphs are based on 4158 hour intervals. 15

16 .9 Retail U.S. bank.14 Telesales Probability to abandon (aggregated) Average wait, sec (aggregated) Probability to abandon (aggregated) Average wait, sec (aggregated) Retail significant abandonment during first seconds of wait. Linear patterns with non-zero intercepts Probability to abandon Israeli data: new customers Average waiting time, sec Probability to abandon VRU-time included in wait Average wait (VRU + queue), sec Left-hand plot exp patience with balking: with probability p, exp(θ) with probability (1 p). Right-hand plot delayed patience: c + exp(θ), c >. 16

17 Erlang-A: parameter estimation and prediction Estimation: inference from historical data (e.g. exp, normal) were parameters assumed fixed over time. Prediction: forecast behavior of sample outside of original data set. Arrivals (λ) Typically Poisson, time-varying rates, constant at 15/3/6 min scale; Significant uncertainty concerning future rates prediction; Predict separately daily volumes and fraction of arrivals per time interval. Services (µ) Typically stable from day to day estimation; Can change depending on time-of-day; Typically, service time talk time. First approach: service time = talk time + wrap-up time (after-call work) +... ; Second approach: service time = Total Working Time Total Idle Time Number of Served Customers. 17

18 Number of agents (n) Output of WFM software given λ, µ, θ, performance goals. One gets number of FTE s (Full Time Equivalent positions). Agents on schedule = FTE s RSF (Rostered Staff Factor) (RSF > 1). Reasons: absenteeism, unscheduled breaks,... Obtaining historical data on n can be hard. Patience (θ) Observations are censored! (heavily) Customer abandoned patience τ known; Customer served offered wait V known τ > V. Avoiding direct uncensoring : use P{Ab} = θ E[W q ]..8 hourly data aggregated Probability to abandon Average waiting time, sec Probability to abandon Average waiting time, sec Regression average patience (1/θ) sec. 18

19 Estimating patience distribution Are patience times really exponential? To uncensor data use Kaplan-Meier (product-limit) estimator. Output: estimates of survival function and hazard rate. Empirical hazard rates of patience times U.S. bank Israeli bank hazard rate time, sec hazard rate x time, sec Israeli bank: regular vs. priority customers 14 Time, sec 28 19

20 Time Israeli bank: service types Figure 16: Survival curves for time willing to wait (Nov. Dec.) Survival IN NE NW PS Time IN Internet Assistance; NE Stock Transactions; NW New Customers; PS Regular anomalies. In spite of the fact that one expects the true distributions to be skewed to the right, the estimated distributions are severely truncated. This is especially true for types PS and NE because Conclusions: they are more heavily censored. See Figure 16. A result of this is that the estimated means for PS and NE calls are much smaller than the estimated medians, while the opposite relation holds for Patience times are, in general, non-exponential; Most tele-customers are very patient; 26 Kaplan-Meier is very informative concerning patience qualitative patterns (abandonment peaks, comparisons,... ); Kaplan-Meier can be problematic concerning estimation of quantitative characteristics (mean, variance, median). E[τ] = S(x)dx, where S(x) - survival function of patience. However, S(x) not reliable for large x. Question: can we apply Erlang-A with non-exponential patience? 2

21 Fitting a simple model to a complex reality Erlang-A Formulae vs. Data Averages (Israeli Bank) P{Ab} E[W q ] 25 Probability to abandon (data) Waiting time (data), sec Probability to abandon (Erlang A) Waiting time (Erlang A), sec 1 P{W q > }.9.8 Probability of wait (data) Probability of wait (Erlang A) 21

22 Conclusions: Points: hourly data vs. Erlang-A output; Formulae with continuous n used; Patience estimated via P{Ab}/E[W q ] relation; Erlang-A estimates close upper bounds. 22

23 Fitting a simple model to a complex reality: Patience index How to define (im)patience? Theoretical Patience Index = Calculation can be difficult. = time willing to wait time required to wait average patience average offered wait. (3) Empirical Patience Index = % served % abandoned. (4) Easily calculable from ACD reports. If τ and V exponentially distributed, (4) is MLE of (3). Patience index empirical vs. theoretical Theoretical Index Empirical Index 23

24 PATIENCE INDEX How to Define? Measure? Manage? Statistics Time Till Interpretation 36K served (8%) 2 min.? must = expect 9K abandon (2%) 1 min.? willing to wait Time willing to wait of served is censored by their wait. Uncensoring (simplified) 36K Willing to wait = = 9 min. 9K Expect to wait K 1 = = 2.25 min. 36K 4 Patience Index = time willing time expect definition = 4 = # served/wait > # abandon/wait > measure Supported by ongoing research (Wharton)

25 Customer-Focused Queueing Theory Waiting experience can be summarized by: 1. Time that a customer expects to wait; 2. Time that a customer is willing to wait (τ, patience or need); 3. Time that a customer must wait (V, offered wait); 4. Time that a customer actually waits (W q = min(τ, V )); 5. Time that a customer perceives waiting. Customer-Focused Queueing Theory Experienced customers 1=3; Rational 2 abandonment customers in4=5; Direct-Banking Then left with (τ, V, W q ), as introduced before. Not scientific 2 abandonment in Direct-Banking: perceived vs. actual waiting. Reason to Abandon Actual Abandon Perceived Abandon Perception Time (sec) Time (sec) Ratio Fed up waiting (77%) Not urgent (1%) Forced to (4%) Something came up (6%) Expected call-back (3%) Rational Abandonment25from Invisible Queues (with

26 Adaptive behavior of impatient customers Question: Do customers adapt their patience to system performance (offered wait)? Israeli bank: Internet-support customers Rational abandonment from invisible queues: Mandelbaum, Shimkin, Zohar. 26

27 Advanced features of 4CallCenters Advanced profiling Vary input parameters of Erlang-A and display output (performance measures) in a table or graphically. Example: 1/µ = 2 minutes, 1/θ = 3 minutes; λ varies from 4 to 23 calls per hour, in steps of 1; n varies from 2 to 12. Probability to abandon Average wait 8.% 14 7.% 12 6.% 1 %Abandon 5.% 4.% 3.% Average Time in Queue (secs) % 4 1.% 2.% Calls per Interval EOS curve Calls per Interval EOS curve Red curve: EOS (Economies-Of-Scale). Why the two graphs are similar? 27

28 Multiple performance goals. Advanced staffing queries Example: 1/µ = 4 minutes, 1/θ = 5 minutes; λ varies from 1 to 12, in steps of 5. Performance targets: P{Ab} 3%; P{W q < 2 sec; Sr}.8. 4CallCenters output 28

29 4CallCenters. Advanced staffing queries. Dynamics of staffing level and performance. Recommended staffing level Target performance measures 9 3.5% 92% 8 3.% 9% 7 2.5% 88% 6 Number of Agents 5 4 %Abandon 2.% 1.5% 86% 84% %Served within 2 sec 3 1.% 82% 2.5% 8% Calls per Interval.% 78% Calls per Interval %Abandon %Abandon Target %Served within 2 sec %Served within 2 sec Target EOS: 1 agents needed for 1 calls per hour but only 83 for 12 calls per hour. 29