Service Engineering - Buisness Management - Lecture Notes, Study notes of Business Administration

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Service Engineering

QED Q's

Q uality & E fficiency D riven

Telephone Call/Contact Centers

Contents

1. Background: Service Engineering, Call Centers, WFM
2. Operational Regime: Quality-Driven, Efficiency-Driven

QED Q's = Quality & Efficiency Driven

in an M/M/N (Erlang-C) world

1. Intuition; Dimensioning

Research-Partners

QED/QD/ED Q's:

Garnett, Reiman: M/M/N+M-Patience(Erlang-A )

Zeltyn: M/M/N+G- patience

Jelelnkovic, Momcilovic:G/G/N w/G=D, finite-support

Kaspi, Ramanan: G/G/N+ G

Massey, Reiman,Rider, Stolyar: Service Networks

Dimensioning:

Borst, Reiman; Zeltyn: Erlang-C and A

Armony,Gurvich: V- and Reversed- V

Massey, Whitt; Jennings, Feldman, Rozenshmidt:

Stablizing Time-Varying Q's

SBR:

Atar, Reiman; Stolyar: Control

Atar, Shaikhet: Null-Controllability

History

"The Life of Work ofA.A.Markov,"by Basharin et al,

Linear Algebra and its Applications, 2004.

Erlang Models (B/ C) :

Erlang, A.K. "Solutions of Some Problems in the Theory of

Probabilities of Significance in Automatic Telephone

Exchanges", Elektroteknikeren, 1917; English translation in

the "Life and Work of A.K. Erlang" 1948, by Brokemeyer

et al.

Square-Root Staffing / Dimensioning:

Erlang, A.K.: "On the Rational Determination of the

Number of Circuits", 1924; first published in the "Life…,"

1948; proofs on page 120-6.

Impatient Customers(Erlang-A/ Irritation) :

Palm, C."Etude des Delais D'Attente", Ericsson Technics,

Palm, C.: "Methods of Judging the Annoyance Caused by

Congestion", Tele, 1953.

Erlang-C = M/M/N

arrivals (^) queueACD

agents

Rough Performance Analysis

Offered load R =! " E(S)

= 400 " 3:45 = 1500 min./30 min. = 50 Erlangs

Occupancy # = R/N

= 50/100 = 50%

Quality-driven : 100 agents, 50% utilization

$ Can increase offered load - by how much?

Erlang-C N=100 E(S) = 3:45 min.

! /hr^ #^ E(Wq ) = ASA^ % Wait = 0

Quality-driven : 100 agents, 50% utilization

$ Can increase offered load - by how much?

Erlang-C N=100 E(S) = 3:45 min.

! /hr^ #^ E(Wq ) = ASA^ % Wait = 0

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%

Changing N ( Staffing ) in Erlang-C

E(S) = 3: !/hr N^ OCC^ ASA^ % Wait = 0 1585 100 99.1% 3:34 12%

Changing N ( Staffing ) in Erlang-C

E(S) = 3: !/hr N^ OCC^ ASA^ % Wait = 0 1585 100 99.1% 3:34 12% 1599 100 99.9% 59:33 0%

Changing N ( Staffing ) in Erlang-C

E(S) = 3: !/hr N^ OCC^ ASA^ % Wait = 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 Rationalized Operation

Efficiently driven, in the sense that OCC > 95%; Quality-Driven, 50% answered immediately

QED Regime = Quality- and Efficiency-Driven Regime

Economies of Scale in a Frictionless Environment

Above: R = 100, N = R + 5, 50% delayed.

, Safety-Staffing N = &R + 0 R ' , 0 > 0.

QED Theorem ( Halfin-Whitt , 1981)

Consider a sequence of M/M/N models, N=1,2,3,…

Then the following 3 points of view are equivalent:

1 Customer (^) N N lim^ P

• 2

{Wait > 0} = 3 , 0 < 3 < 1;

1 Server N lim+ 2 N ( 1 - # (^) N ). 0 , 0 < 0 < 2 ;

1 Manager N % R ) 0 R , R. !" E(S) large;

Here

1 (^1) ( )

(^45)

3 0 h ,

where h (,) is the hazard-rate of standard-normal.

Extremes: Everyone waits : 3. 1 8 0. 0 Efficiency-driven

No one waits : 3. 0 8 0. 2 Quality-driven

1

19

The Halfin-Whitt Delay Function P( E )

Beta

Alpha

M/M/N (Erlang-C) with Many Servers: N ↑ ∞

Q Q+

0 1 2 N-1 (^) N N+

Q(0) = N : all servers busy, no queue.

Recall E 2 ,N =

[ 1 + TTNN,N^ −^1 −,N 1

]− 1

[ 1 + (^) ρN^1 E^ − 1 ,NρN − 1

]− 1 .

Here TN − 1 ,N = (^) λN E^11 ,N − 1 ∼ (^) N μ × h(^1 −β)/√N ∼ (^) h(−^1 β/μ)√N which applies as √N (1 − ρN ) → β, −∞ < β < ∞. Also TN,N − 1 = (^) N μ(1^1 − ρN ) ∼ (^) β^1 √/μN which applies as above, but for 0 < β < ∞. Hence, E 2 ,N ∼

[ 1 + (^) h(−ββ)

]− 1 , assuming β > 0.

QED: N ∼ R + β√R for some β, 0 < β < ∞ ⇔ λN ∼ μN − βμ√N ⇔ ρN ∼ 1 − √βN , namely (^) Nlim →∞√N (1 − ρN ) = β.

Theorem (Halfin-Whitt, 1981) QED ⇔ (^) Nlim →∞E 2 ,N =

[ 1 + (^) h(−ββ)

]− 1 .

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