Service Engineering QED Q's Qualiy & Efficiency Driven Telephone Call/Conac Ceners
Service Engineering QED Q's Qualiy & Efficiency Driven Telephone Call/Conac Ceners INFORMS San-Francisco November 15, 2005 e.mail : avim@x.echnion.ac.il Websie: hp://ie.echnion.ac.il/serveng
Conens 1. Background: Service Engineering, Call Ceners, WFM 2. Operaional Regime: Qualiy-Driven, Efficiency-Driven QED Q's = Qualiy & Efficiency Driven in an M/M/N (Erlang-C) world 3. Inuiion; Dimensioning
Conens 1. Background: Service Engineering, Call Ceners, WFM 2. Operaional Regime: Qualiy-Driven, Efficiency-Driven QED Q's = Qualiy & Efficiency Driven in an M/M/N (Erlang-C) world 3. Inuiion; Dimensioning Leading o models, inference and ools: 4. Impaien (Abandoning) Cusomers (Erlang-A) 5. Predicably (Time) Varying Queues
Conens 1. Background: Service Engineering, Call Ceners, WFM 2. Operaional Regime: Qualiy-Driven, Efficiency-Driven QED Q's = Qualiy & Efficiency Driven in an M/M/N (Erlang-C) world 3. Inuiion; Dimensioning Leading o models, inference and ools: 4. Impaien (Abandoning) Cusomers (Erlang-A) 5. Predicably (Time) Varying Queues 6. General Service Times (M/D/N, M/LN/N, ) 7. Human/Smar Cusomers and Smar Sysems 8. Heerogeneous Cusomer Types and Parially Overlapping Server Skills (SBR)
Supporing Maerial (Downloadable) Gans, Koole, M: Telephone Call Ceners: Tuorial, Review and Research Prospecs. MSOM, 2003. Brown, Gans, M., Sakov, Shen, Zelyn, Zhao: "Saisical Analysis of a Telephone Call Cener: A Queueing-Science Perspecive." JASA, 2005. Feigin, M., Trofimov.: "Daa MOCCA: Models for Call Cener Analysis." Ongoing (available for use: 7GB or 20GB memory).
Research-Parners QED/QD/ED Q's: Garne, Reiman: M/M/N+M-Paience (Erlang-A) Zelyn: M/M/N+G-paience Jelelnkovic, Momcilovic: G/G/N w/ G=D, finie-suppor Kaspi, Ramanan: G/G/N+G Massey, Reiman, Rider, Solyar: Service Neworks Dimensioning: Bors, Reiman; Zelyn: Erlang-C and A Armony, Gurvich: V- and Reversed-V Massey, Whi; Jennings, Feldman, Rozenshmid: Sablizing Time-Varying Q's SBR: Aar, Reiman; Solyar: Conrol Aar, Shaikhe: Null-Conrollabiliy
Hisory "The Life of Work of A. A. Markov," by Basharin e al, Linear Algebra and is Applicaions, 2004. Erlang Models (B / C): Erlang, A.K. "Soluions of Some Problems in he Theory of Probabiliies of Significance in Auomaic Telephone Exchanges", Elekroeknikeren, 1917; English ranslaion in he "Life and Work of A.K. Erlang" 1948, by Brokemeyer e al. Square-Roo Saffing / Dimensioning: Erlang, A.K.: "On he Raional Deerminaion of he Number of Circuis", 1924; firs published in he "Life," 1948; proofs on page 120-6. Impaien Cusomers (Erlang-A / Irriaion): Palm, C. "Eude des Delais D'Aene", Ericsson Technics, 1937. Palm, C.: "Mehods of Judging he Annoyance Caused by Congesion", Tele, 1953.
Tele-Nes: Call/Conac Ceners Scope Examples Informaion (uni, bi-dir) #411, Tele-pay, Help Desks Business Emergency Police #911 Mixed Info + Emerg. Info + Bus. Tele-Banks, #800-Reail Uiliy, Ciy Halls Airlines, Insurance, Cellular Scale 10s o 1000s of agens in a single Call Cener X% of work force in call ceners (up o several millions) 70% of oal business ransacions in call ceners 20% growh rae of he call cener indusry Leading-edge echnology, bu 70% coss for people Trends: THE inerface for/wih cusomers Conac Ceners (E-Commerce/Mulimedia), ousourcing, Reails oules of 21-Cenury bu also he Swea-shops of he 21-Cenury
Erlang-C = M/M/N agens arrivals ACD queue
Rough Performance Analysis Peak 10:00 10:30 a.m., wih 100 agens 400 calls 3:45 minues average service ime 2 seconds ASA (Average Speed of Answer)
Rough Performance Analysis Peak 10:00 10:30 a.m., wih 100 agens 400 calls 3:45 minues average service ime 2 seconds ASA Offered load R =! " E(S) = 400 " 3:45 = 1500 min./30 min. = 50 Erlangs Occupancy # = R/N = 50/100 = 50%
Rough Performance Analysis Peak 10:00 10:30 a.m., wih 100 agens 400 calls 3:45 minues average service ime 2 seconds ASA Offered load R =! " E(S) = 400 " 3:45 = 1500 min./30 min. = 50 Erlangs Occupancy # = R/N = 50/100 = 50% $ Qualiy-Driven Operaion (Ligh-Traffic) $ Classical Queueing Theory (M/G/N approximaions) Above: R = 50, N = R + 50, % all served immediaely. Rule of Thumb: N = & R ) ( R, * 0 ' ( service-grade.
Qualiy-driven: 100 agens, 50% uilizaion $ Can increase offered load - by how much? Erlang-C N=100 E(S) = 3:45 min.!/hr # E(W q ) = ASA % Wai = 0 800 50% 0 100%
Qualiy-driven: 100 agens, 50% uilizaion $ Can increase offered load - by how much? Erlang-C N=100 E(S) = 3:45 min.!/hr # E(W q ) = ASA % Wai = 0 800 50% 0 100% 1400 87.5% 0:02 min. 88% 1550 96.9% 0:48 min. 35% 1580 98.8% 2:34 min. 15% 1585 99.1% 3:34 min. 12%
Qualiy-driven: 100 agens, 50% uilizaion $ Can increase offered load - by how much? Erlang-C N=100 E(S) = 3:45 min.!/hr # E(W q ) = ASA % Wai = 0 800 50% 0 100% 1400 87.5% 0:02 min. 88% 1550 96.9% 0:48 min. 35% 1580 98.8% 2:34 min. 15% 1585 99.1% 3:34 min. 12% $ Efficiency-driven Operaion (Heavy Traffic) 1 # W W W 0 N q % q q *.,, E( S) + E( S) N 1- # N = 3:45! N ( 1 # ). 1, # + 1 - N N Above: R = 99, N = R + 1, % all delayed. Rule of Thumb: N = & R ) / ', / * 0 service grade.
Changing N (Saffing) in Erlang-C E(S) = 3:45!/hr N OCC ASA % Wai = 0 1585 100 99.1% 3:34 12%
Changing N (Saffing) in Erlang-C E(S) = 3:45!/hr N OCC ASA % Wai = 0 1585 100 99.1% 3:34 12% 1599 100 99.9% 59:33 0%
Changing N (Saffing) in Erlang-C E(S) = 3:45!/hr N OCC ASA % Wai = 0 1585 100 99.1% 3:34 12% 1599 100 99.9% 59:33 0% 1599 100+1 98.9% 3:06 13% 1599 102 98.0% 1:24 24% 1599 105 95.2% 0:23 50%
Changing N (Saffing) in Erlang-C E(S) = 3:45!/hr N OCC ASA % Wai = 0 1585 100 99.1% 3:34 12% 1599 100 99.9% 59:33 0% 1599 100+1 98.9% 3:06 13% 1599 102 98.0% 1:24 24% 1599 105 95.2% 0:23 50% $ New Raionalized Operaion Efficienly driven, in he sense ha OCC > 95%; Qualiy-Driven, 50% answered immediaely QED Regime = Qualiy- and Efficiency-Driven Regime Economies of Scale in a Fricionless Environmen Above: R = 100, N = R + 5, 50% delayed., Safey-Saffing N = &R + 0 R ', 0 > 0.
QED Theorem (Halfin-Whi, 1981) Consider a sequence of M/M/N models, N=1,2,3, Then he following 3 poins of view are equivalen: 1 Cusomer lim N N +2 1 Server lim (1 - # ). 0 N +2 P {Wai > 0} = 3, 0 < 3 < 1; N, 0 < 0 < 2; 1 Manager N % R ) 0 R,.! " N R E(S) large; Here 3. & 61 ) 7 0 ' h( -0 ) 4 5-1, where h(,) is he hazard-rae of sandard-normal. Exremes: Everyone wais: 3. 1 8 0. 0 Efficiency-driven No one wais: 3. 0 8 0. 2 Qualiy-driven 1
, Safey-Saffing: Performance R =! " E(S) Offered load (Erlangs) N = R + # 0 "! R 0 = service-grade > 0 = R + 9, safey-saffing Expeced Performance: -1 % ) ' % Delayed P( ) 6 & 0 0. 1 ), * 0 (- ) 4 0 7 h 0 5 Erlang-C & Congesion index = E6 7 Wai E(S) Wai * ' 04 5. 1 9 ASA %? < > Wai * T Wai * 0;. e -T9 TSF = E(S) : Servers Uilizaion = R % 1-0 Occupancy N N 2
The Halfin-Whi Delay Funcion P( ) Alpha Bea 19
M/M/N (Erlang-C) wih Many Servers: N 0 1 2 N-1 N N+1 Q Q + Q(0) = N: all servers busy, no queue. Recall [ E 2,N = 1+ T ] 1 N 1,N = T N,N 1 [ 1+ 1 ρ ] 1 N. ρ N E 1,N 1 Here T N 1,N = 1 λ N E 1,N 1 1 Nµ h( β)/ N 1/µ h( β) N which applies as N (1 ρ N ) β, < β <. Also T N,N 1 = 1 Nµ(1 ρ N ) 1/µ β N Hence, which applies as above, bu for 0 < β <. E 2,N [ 1+ β ] 1, assuming β> 0. h( β) QED: N R + β R for some β, 0 < β < λ N µn βµ N ρ N 1 β N, namely lim N N (1 ρn )=β. Theorem (Halfin-Whi, 1981) QED lim E 2,N = N [ 1+ β h( β)] 1. 6
Approximaing Queueing and Waiing Q N = {Q N (), 0} : Q N () =number in sysem a 0. ˆQ N = { ˆQ N (), 0} : sochasic process obained by cenering and rescaling: ˆQ N = Q N N N ˆQ N ( ) : saionary disribuion of ˆQ N ˆQ = { ˆQ(), 0} : process defined by: ˆQ N () d ˆQ(). ˆQ N () ˆQ N ( ) N N ˆQ() Q( ) Approximaing (Virual) Waiing Time ˆV N = NV N ˆV = [ ] + 1 µ ˆQ (Puhalskii, 1994) 9
Economics: Qualiy vs. Efficiency Dimensioning: wih Bors and Reiman Qualiy D() delay cos ( = delay ime) Efficiency C(N) saffing cos (N = # agens) Opimizaion: N* minimizes Toal Coss 1 C >> D : Efficiency-driven 1 C << D : Qualiy-driven 1 C % D : Raionalized - QED Framework: Asympoic heory, M/M/N, N @ 2
Economics:, Safey-Saffing Coss: d = delay/waiing c = saffing Min Toal Coss Asympoically Opimal N * % R + y * F d D E c A B C R Here y * (r) % F r D E1) rg I / 2 C A -1 B H 1/ 2, 0 < r < 10 % F D2 ln E r C A 2I B 1/ 2, r large.
Square-Roo Safey Saffing: r = cos of delay / cos of saffing * N. R ) y ( r) R 22
, Safey-Saffing: Overview d D E c Simple Rule-of-humb: N * % R + y * F C A B Robus: covers also efficiency- and qualiy-driven Accurae: o wihin 1 agen (from few o many 100 s) ypically Relevan: Medium o Large CC do perform as above. Insrucive: In large call ceners, high resource uilizaion and service levels could coexis, which is enabled by economies of scale ha dominae sochasic variabiliy. R Example: 100 calls per minue, a 4 min. per call $ R = 400, leas number of agens 9 % R y * ( r ) R. * y 20, wih y * : 0.5 1.5 ; Safey saffing: 2.5% 7.5% of R=Min! $ Real Problem? Performance: N * % wai > 20 sec. Uilizaion 400 + 11 20% 97% 400 + 29 1% 93% 23
Arrivals: Inhomogeneous Poisson Figure 1: Arrivals (o queue or service) Regular Calls 10 30
Figure 12: Mean Service Time (Regular) vs. Time-of-day (95% CI) (n = 42613) Mean Service Time 100 120 140 160 180 200 220 240 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time of Day 30 29
Service Time Survival curve, by Types Means (In Seconds) NW (New) = 111 PS (Regular) = 181 Survival NE (Socks) = 269 IN (Inerne) = 381 Time 32 31
14 33
Beyond Daa Averages Shor Service Times Jan Oc: 7.2 %? 200 249 AVG: STD: AVG: 185 STD: 238 Nov Dec: AVG: 200 STD: 249 Log-Normal 45
Conens 1. Background: Service Engineering, Call Ceners, WFM 2. Operaional Regime: Qualiy-Driven, Efficiency-Driven QED Q's = Qualiy & Efficiency Driven in an M/M/N (Erlang-C) world 3. Inuiion, Moivaion Leading o models, inference and ools: 4. Impaien (Abandoning) Cusomers (Erlang-A) 5. Predicably (Time) Varying Queues 6. General Service Times (M/D/N, M/LN/N, ) 7. Human/Smar Cusomers and Smar Sysems 8. Heerogeneous Cusomer Types and Parially Overlapping Server Skills (SBR) 9. Daa MOCCA: MOdels for Call Cener Analysis 10. 4CallCeners: (Personal) Tool for WFM
Erlang-A (wih G-Paience): M/M/N+G los calls FRONT arrivals busy ACD queue abandonmen los calls
QED Theorem (Garne, M. and Reiman '02; Zelyn '03) Consider a sequence of M/M/N+G models, N=1,2,3, Then he following poins of view are equivalen: 1 QED %{Wai > 0} % 3, 0 < 3 < 1 ; 1 Cusomers %{Abandon} % / N, 0 < / ; 1 Agens OCC 0 ) / % 1 - -2 < 0 < 2 ; N 1 Managers N R ) 0 R %, R.! " E(S) no small; QED performance (ASA,...) is easily compuable, all in erms of 0 (he square-roo safey saffing level) see laer. Covers also he Exremes: 3 = 1 : N = R - / R Efficiency-driven 3 = 0 : N = R + / R Qualiy-driven
QED Approximaions (Zelyn) λ arrival rae, µ service rae, N number of servers, G paience disribuion, g 0 paience densiy a origin (g 0 = θ, if exp(θ)). N = λ µ + β λµ + o( λ), < β <. P{Ab} { P W> T } N { P Ab W> T } N Here ˆβ = µ β g 0 Φ(x) = 1 Φ(x), 1 [h(ˆβ) ˆβ ] [ µ + h(ˆβ) ] 1, N g 0 h( β) [ ] 1 g0 1+ µ h(ˆβ) Φ (ˆβ + g 0 µ T ) h( β) Φ( ˆβ) 1 g0 N µ [h (ˆβ + g 0 µ T ) ˆβ ]., h(x) = φ(x)/ Φ(x), hazard rae of N(0, 1). Generalizing Garne, M., Reiman (2002) (Palm 1943 53) No Process Limis
HW/GMR Delay Funcions! vs. " 1 0.9 0.8 Halfin-Whi Delay Probabiliy 0.7 0.6 0.5 0.4 0.3 0.2 0.1 QED Erlang-A 0-3 -2.5-2 -1.5-1 -0.5 0 0.5 1 1.5 2 2.5 3 Bea Halfin-Whi Garne(0.1) Garne(0.5) Garne(1) Garne(2) Garne(5) Garne(10) Garne(20) Garne(50) Garne(100)!/µ
Example: "Real" Call Cener (The "Righ Answer" for he "Wrong Reasons") Time-Varying (wo-hump) arrival funcions common (Adaped from Green L., Kolesar P., Soares J. for benchmarking.) 2500 2000 Calls per Hour 1500 1000 500 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Hour of Day Assume: Service and abandonmen imes are boh Exponenial, wih mean 0.1 (6 min.)
Time-Varying Arrivals Model M / M / N + M! Parameers "() µ?!? N = R + 0 R
Time-Varying Arrivals Model M / M / N + M! Parameers "() µ?!? N = R + 0 R Offered Load: R. E! ( - S), E( S). E J!( u) du -S Average # in M / M / 2
Time-Varying Arrivals Model M / M / N + M! Parameers "() µ?!? N = R + 0 R Offered Load: R. E! ( - S), E( S). E J!( u) du -S Average # in M / M / 2 Gives rise o TIME-STABLE PEFORMANCE (Why? Think M / M / N + M wih µ =!; And if µ #!, or generally: use he Ieraive Simulaion-Based Saffing Algorihm in Feldman, M., Massey and Whi, 2005.)
HW/GMR Delay Funcions! vs. " 1 0.9 0.8 Halfin-Whi Delay Probabiliy 0.7 0.6 0.5 0.4 0.3 0.2 0.1 QED Erlang-A 0-3 -2.5-2 -1.5-1 -0.5 0 0.5 1 1.5 2 2.5 3 Bea Halfin-Whi Garne(0.1) Garne(0.5) Garne(1) Garne(2) Garne(5) Garne(10) Garne(20) Garne(50) Garne(100)!/µ
Delay Probabiliy! Delay Probabiliy 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Targe Alpha=0.1 Targe Alpha=0.2 Targe Alpha=0.3 Targe Alpha=0.4 Targe Alpha=0.5 Targe Alpha=0.6 Targe Alpha=0.7 Targe Alpha=0.8 Targe Alpha=0.9
Abandon Probabiliy Abandon Probabiliy 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Tage Alpha=0.1 Tage Alpha=0.2 Tage Alpha=0.3 Tage Alpha=0.4 Tage Alpha=0.5 Tage Alpha=0.6 Tage Alpha=0.7 Tage Alpha=0.8 Tage Alpha=0.9 Abandon Probabiliy 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Tage Alpha=0.1 Tage Alpha=0.2 Tage Alpha=0.3 Tage Alpha=0.4 Tage Alpha=0.5 Tage Alpha=0.6 Tage Alpha=0.7 Tage Alpha=0.8 Tage Alpha=0.9
Real Call Cener: Empirical waiing ime, given posiive wai (1) $=0.1 (QD) (2) $=0.5 (QED) (3) $=0.9 (ED)
The "Righ Answer" (for he "Wrong Reasons") Prevalen Pracice N.!( ), E( S) & ' (PSA) "Righ Answer" N % R ) 0, R (MOL) R. E! ( - S), E( S)
The "Righ Answer" (for he "Wrong Reasons") Prevalen Pracice N. &!( ), E( S) ' (PSA) "Righ Answer" N % R ) 0, R (MOL) R. E! ( - S), E( S) Pracice % "Righ" 0 % 0 (QED) and!() % sable over service-duraions Pracice Improved N. &![ - E( S)], E( S) '
2500 2000 Calls per Hour 1500 1000 500 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Hour of Day QED Saffing ("=0 iff $=0.5) 2500 250 2000 200 1500 150 1000 100 500 50 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Arrived Saffing Offered Load 0
The "Righ Answer" (for he "Wrong Reasons") Prevalen Pracice N. &!( ), E( S) ' (PSA) "Righ Answer" N % R ) 0, R (MOL) R. E! ( - S), E( S) Pracice % "Righ" 0 % 0 (QED) and!() % sable over service-duraions Pracice Improved N. &![ - E( S)], E( S) ' When Opimal? for moderaely-paien cusomers: 1. Saisfizaion 8 A leas 50% o be serve immediaely 2. Opimizaion 8 Cusomer-Time = 2 x Agen-Salary
Time-Varying Arrivals:, Safey-Saffing Model M / M / N + M! Parameers "() µ?!? N = R + 0 R
Time-Varying Arrivals:, Safey-Saffing Model M / M / N + M! Parameers "() µ?!? N = R + 0 R µ =! : L ḍ Poisson( R ) % d N(R, R ), since / M / 2 M R. E! ( - S), E( S). E J!( u) du offered load -S
Time-Varying Arrivals:, Safey-Saffing Model M / M / N + M! Parameers "() µ?!? N = R + 0 R µ =! : L ḍ Poisson( R ) % d N(R, R ), since / M / 2 M R. E! ( - S), E( S). E J!( u) du offered load -S Given L % R + Z R, d Z. N(0,1) choose N = R + 0 R $ $ = P(W > 0) % P(L % N ) = P(Z % &) = 1 K(0) PASTA $ 0 = K 1 (1 $) ime-sable $ # P(W > 0)?
Time-Varying Arrivals:, Safey-Saffing Model M / M / N + M! Parameers "() µ?!? N = R + 0 R µ =! : L ḍ Poisson( R ) % d N(R, R ), since / M / 2 M R. E! ( - S), E( S). E J!( u) du offered load -S Given L % R + Z R, d Z. N(0,1) choose N = R + 0 R $ $ = P(W > 0) % P(L % N ) = P(Z % &) = 1 K(0) PASTA $ 0 = K 1 (1 $) ime-sable $ # P(W > 0)? Indeed, bu in fac TIME-STABLE PERFORMANCE
Time-Varying Arrivals:, Safey-Saffing Model M / M / N + M! Parameers "() µ?!? N = R + 0 R µ =! : L ḍ Poisson( R ) % d N(R, R ), since / M / 2 M R. E! ( - S), E( S). E J!( u) du offered load -S Given L % R + Z R, d Z. N(0,1) choose N = R + 0 R $ $ = P(W > 0) % P(L % N ) = P(Z % &) = 1 K(0) PASTA $ 0 = K 1 (1 $) ime-sable $ # P(W > 0)? Indeed, bu in fac TIME-STABLE PERFORMANCE (µ #!, or generally : Ieraive Simulaion-Based Algorihm)