Designing a Telephone Call Center with Impatient Customers

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1 Designing a Telephone Call Center with Impatient Customers with Ofer Garnett Marty Reiman Sergey Zeltyn Appendix: Strong Approximations of M/M/ + Source: ErlangA QEDandSTROG FIAL.tex

2 M/M/ + M System Poisson arrivals-rate λ Service times exp(µ statistically identical agents attending to single queue Service policy FCFS Customers patience: exp(θ Q = {Q(t, t } - number of customers in the system Birth & Death: transition diagram Everything calculable via stationary distribution (λ/µ k π k k! π k = k ( λ (λ/µ π k > µ + (j θ! j=+ where π = (λ/µ k k ( λ (λ/µ + k! µ + (j θ! k= k=+ j=+ 2

3 M/M/ + M Characteristics otation: P {Ab} - abandonment probability (fraction P {W ait > } - waiting probability (fraction P {Block} - blocking probability in an M/M// system. Stationary distribution always exists (Sandwiched between infinite-server models 2. P {Ab} = θ E[W ait] Proof: λp {Ab} = θ E[umber in queue], now use Little. 3. P {Ab} increases monotonically in θ, λ P {Ab} decreases monotonically in, µ (Bhattacharya and Ephremides (99 4. P {Ab} P {Block} (Boxma and de Waal ( lim P {Ab} = ( lim ρ 3

4 Exact Calculations V - virtual waiting time = waiting time of a customer with infinite patience (test customer. X - customer s patience (X exp(θ, independent of V. Wait V X - actual waiting time. Performance measures of the form E[f(V, X]: Calculable, in numerically stable procedures (4CallCenters. f(v, x E[f(V, X] {v>x} P {V > X} = P {Ab} (t, (v x P {W ait > t} (t, (v x {v>x} P {W ait > t; Ab} (v x {v>x} E[W ait; Ab] (v x (t, (v x {v>x} E[W ait; W ait > t; Ab] g(v x E[g(W ait] From these obtain more natural measures, for example P {Ab W ait > t} = P {W ait > t; Ab} P {W ait > t} 4

5 QED M/M/ + M (θ θ Following Halfin-Whitt (98 and Flemming-Simon-Stolyar (996 Theorem: Let α lim P {W ait > } β lim ( ρ λ µ + β λ µ γ lim P {Ab} P {Ab} γ Then > α > iff > β > iff > γ > in which case α = w( β, µ/θ [ ] γ = θ/µ h(β µ/θ β α Here w(x, y = h(x = [ + h( xy ], yh(x φ(x, hazard rate of std. normal Φ(x 5

6 Designing a QED Call Center (Zeltyn (Approximate Performance Measures P {W ait > } [ ] W ait E E[S] W ait > [ + h(rβ ] µ, r = rh( β θ r [h(rβ rβ] P {Ab} P {Ab W ait > } h(rβ rβ r h(rβ rβ r [ + h(rβ rh( β ] P { W ait E[S] > P { Ab t } W ait > W ait E[S] > t } [ ] W ait E E[S] Ab Φ (rβ + t/r Φ(rβ h(rβ + t/r rβ r [ ] 2 h(rβ rβ rβ r Here Φ(x = Φ(x, h(x = φ(x/ Φ(x, hazard rate of (,. 6

7 ew Operational Regimes (M/M/ + ρ P {Wait > } P {Ab} Efficiency driven Quality driven R ɛr + ɛ ɛ R + ɛr ɛ QED R + β R β α(β < β < < α < (β Compare with Previous (M/M/, Halfin-Whitt E R + ɛ ɛ Q R + ɛr + ɛ QED R + β R β α(β < β < < α < 7

8 Sequence of M/M/ + M, =, 2,... Motivation: Focus: Insight, umerical stability Large Call Centers λ, large. Framework: Sequence of M/M/ + M systems, indexed by Q = {Q (t, t }: total number in system; V = {V (t, t }: virtual wait of an infinite-patience customer Parameters λ, µ, θ θ θ θ θ impatient patient rational Offered load R = λ µ (ρ = R / Approximations of Process and Stationary Distribution ˆQ (t = [Q (t ], t ˆV (t = V (t 8

9 Approximating Queueing and Waiting Q = {Q (t, t } : Q (t = number in system at t. ˆQ = { ˆQ (t, t } : stochastic process obtained by centering and rescaling: ˆQ = Q ˆQ ( : stationary distribution of ˆQ ˆQ = { ˆQ(t, t } : process defined by: ˆQ (t d ˆQ(t. ˆQ (t t ˆQ ( ˆQ(t t Q( Approximating (Virtual Waiting Time ˆV = V ˆV = [ ] + µ ˆQ (Puhalskii, 994 9

10 Weak Convergence of Stationary Distribution Assume lim ( ρ = β, < β <. Allow θ to vary with. ˆQ ( converges iff at least one of the following prevails:. θ θ or θ 2. β > in which case ˆQ ( d ˆQ(, where Density function f(x of ˆQ( : θ, β > f(x = θ θ f(x = { A φ(β + x x A 2 exp( βx x > B φ(β + x x ( B 2 φ β µ/θ + x θ/µ x > θ f(x = { Cφ(β + x x x >

11 Weak Convergence of Processes ρ β ; λ µ + β λ µ Theorem: ˆQ d ˆQ if ˆQ ( d ˆQ(. Theorem: ˆV d ˆV d = µ ˆQ +. θ Patient d ˆQ(t = f( ˆQdt + 2µ db(t { µ(β + x x f(x = µβ x > ( OU ˆQ BM ˆQ > M/M/ Erlang C θ Impatient d ˆQ(t = µ(β + ˆQ(tdt + 2µ db(t dy (t Y ( =, Y, ˆQ dy = (ROU ˆQ M/M// Erlang B θ fixed Rational d ˆQ(t = f( ˆQdt + 2µ db(t { µ(β + x x f(x = (µβ + θx x > ( OU ˆQ OU ˆQ > (B - standard Brownian Motion (β - as before

12 Designing a Call Center - Selecting a Model Very impatient customers - M/M// model Very patient customers - M/M/ model Balanced abandoning - M/M/ + M, θ fixed What if General Patience with distribution function G? Steady-state formulae prevail with θ G (. 2

13 M/M/+G System Customers patience: general distribution G, with g = G (. Steady-State Results Baccelli & Hebuterne (98 - virtual wait, abandon probability. Brandt & Brandt (999, 22 - stationary-queue distribution. QED Regime Main Case: G( =, g > (no balking. µ Use M/M/ + M formulae: θ g, r =. g Hence, P{Ab} g E[W ]. Special Cases: Balking (G( >, g > ; g = : k-th derivative of G at zero positive, k 2 (Erlang, Phase-type; g = : Delayed patience distributions (const, c+exp(θ. Asymptotic analysis possible for all special cases, which yields varying convergence rates. 3

14 Appendix M/M/ + M : Strong Approximations Parameters: λ, µ, θ, Building Blocks: A i, independent Poisson ( processes Model: Q = {Q(t, t } total number in system Q(t = Q( +A (λt arrivals ( A 2 µ [Q(u ]du services ( A 3 θ [Q(u ] + du abandons Strong Approximations: on the same probability space with A i s, there are SBM B i s such that A i (t = t + B i (t + O(log t, t. Approximate Q by substituting above A i (t t + B i (t. 4

15 Approximation: Q(t = Q(t + o (, u.o.c., via A i (t t + B i (t Q(t = Q( + λt + B (λt ( µ [ Q(u ]du B 2 µ [ Q(u ]du ( θ [ Q(u ] + du B 3 θ [ Q(u ] + du Insight? Limit theorems (FSLL, FCLT as, for Q, hence for Q : For example, Q (t = Q (t + o(. 5

16 QED M/M/ + : Approximations, Limits For simplicity Q ( : all servers busy, no queue. Recall λ = µ µb, µ, θ Theorem: Strong Approximation. Q (t = + ˆQ(t + o ( u.o.c., as or [Q(t ] d ˆQ(t t, where d ˆQ(t = [ µβ + µ ˆQ (t θ ˆQ + (t ] dt + 2µ db(t ; namely ˆQ is a diffusion with µ(x = { µβ θx x µβ µx x, σ2 (x 2µ. FSLL Q (t u.o.c., a.s. FCLT [ Q ] = [Q ] d ˆQ 6

17 QED M/M/ + : FSLL Q (t = + λ t + B (λ t ( µ [ Q (u ]du B 2 µ [ Q (u ]du FSLL:... µ θ [ Q (u ] + du B 3 ( Q (t = + λ t + ( B λ t [ ] Q (u du ( [ ] B 2 µ Q (u du Observe and λ t µt B i(t since λ = µ µβ FSLL for SBM. If indeed Q (t Q(t, then Q(t = + µt µ [Q(u ]du θ which has a unique solution Q(t, t. [Q(u ] + du Theorem (FSL: As, Q (t u.o.c. a.s. Proof: Gronwall s inequality 7

18 QED M/M/: FCLT Q (t = + λ t + B (λ t ( µ [ Q (u]du B 2 µ [ Q (u]du θ [ Q (u ] + du B 3 ( FCLT ˆQ (t = [ Q (t ] = consists of the following ingredients:. µβt + ( B [µ β ] t 2. +µ t µ = µ 3. θ ˆQ (udu B 2 (µ [ ] Q (t [ ] Q (u du B 2 ( ˆQ + (udu B 3 ( θ [ Q (u Q + (udu ] du 8

19 Basic Properties of Brownian Motion Let B d = Standard Brownian Motion. Then, Self-similarity B(t d = B(t Additivity ( B i (C i t = d B C i t i i d = i C i B(t, if B i independent SBM Time-Change B( τ (t d B(τ(t, if τ d τ and, say, τ is continuous deterministic. 9

20 Recall. µβt + B [µ 2. +µ t µ = µ 3. θ ( β ] t ˆQ (udu B 2 (µ [ ] Q (u du B 2 ( ˆQ + (udu B 3 ( θ [ Q (u Q + (udu ] du If indeed ˆQ d ˆQ, then. d µβt + B (µt 2. d µ ˆQ (udu B 2 (µt 3. d θ ˆQ + (udu B 3 ( Theorem (FCLT As, ˆQ d ˆQ, where d ˆQ(t = [ µβ + µ ˆQ (t θ ˆQ + (t]dt + 2µ db(t 2

21 Customer-Focused Queueing Theory 2 abandonment in Direct-Banking ot scientific Reason to Abandon Actual Abandon Perceived Abandon Perception Time (sec Time (sec Ratio Fed up waiting (77% ot urgent (% Forced to (4% Something came up (6% Expected call-back (3% Rational Abandonment from Invisible Queues (with Shimkin. 2

Markovian N-Server Queues (Birth & Death Models)

Markovian N-Server Queues (Birth & Death Models) Markovian -Server Queues (Birth & Death Moels) - Busy Perio Arrivals Poisson (λ) ; Services exp(µ) (E(S) = /µ) Servers statistically ientical, serving FCFS. Offere loa R = λ E(S) = λ/µ Erlangs Q(t) = number