Staffing and Routing in Large-Scale Service Systems with Heterogeneous-

Servers

Mor ArmonyStern School of Business, NYUINFORMS 2009

Joint work with Avi Mandelbaum

Motivation: Call Centers

The Inverted-V Model

NK

K

K 21

• Calls arrive at rate (Poisson process).

• K server pools.

• Service times in pool k are exponential with rate k and are non-preemptive

• Customers abandon from the queuewith rate

N1

1

¹ 2 > ¹ 1

¹ > ¹

Experienced employees on averageprocess requests faster than new hires.Gans, Mandelbaum and Shen (2007)

…

Our Focus

Routing: When an incoming call arrives to an empty queue, which agent pool should take the call?

Staffing: How many servers should be working in each pool?

¹ 2 > ¹ 1

¹ > ¹

x = y

NK

K

K 21

N1

1

…

Background: Human Effects in Large-Scale Service

Systems

M/M/N

M/M/N+M+M/M/N+

M/M/N+M

M/M/N++

Halfin & Whitt ’81

Borst et al ’04

Garnett et al ’02

Mandelbaum & Zeltyn ’08

Why Consider Abadonment?

Even little abandonment can have a significant effect on performance:

– An unstable M/M/N system (>1) becomes stable with abandonment.

– Example (Mandelbaum & Zeltyn ‘08): Consider =2000/hr, =20/hr. Service level target: “80% of customers should be served within 30 seconds”:

• 106 agents (=0)• 95 agents (=20 (average patience of 3 minutes), P(ab)=6.9%)• 84 agents (=60 (average patience of 1 minute), P(ab)=16.8%)

Problem Formulation

policy routing somefor ,)(

,EW

,)(s.t.

)(...)(min 11

abP

W

TWP

NCNC KK

Challenges:

•Asymptotic regimes: QED, ED, ED+QED are all relevant

•Asymptotic optimality: No natural lower bound on staffing

•Assumptions: For delay related constraints, FCFS is sub-optimal. Work conservation assumption required when >

our focus

Asymptotic Regimes(Mandelbaum & Zeltyn 07)

¹ 2 > ¹ 1

¹ > ¹

x = y

,)()1(:iff

)(lim :ED

1

oN

abP

K

k kk

Baron & Milner

07

,:iff

10,)0(lim :QED

1oN

WP

K

k kk

)exp(1 ,

,)1( iff

)exp(0,)(lim :QEDED

1

T

oN

TTWP

K

k kk

Solution approach

¹ 2 > ¹ 1

¹ > ¹

x = y

Original Joint Staffing and Routing problem:

policy routing somefor ,)(s.t.

)(...)(min 11

TWP

NCNC KK

Our approach: 1. Given a “sensible” staffing vector,solve the routing problem:

),(

)(min

N

TWP

2. Show that the proposed staffing vector is is asymptotically feasible.

3.Minimize staffing cost over the asymptotically feasible region.

The Routing Problem

¹ 2 > ¹ 1

¹ > ¹

x = y

Proposition: The preemptive Faster Server First (FSF) policy is optimal within FCFS policies if either of these assumptions holds:1≤ min{1,…,K}, or2.Only work-conserving policies are allowed.

),(

)(min

N

TWP

For a given staffing vector:

Asymptotically Optimal Routing

in the QED Regime (T=0)Proposition:

The non-preemptive routing policy FSF is asymptotically optimal in the QED regime

Proof: State-space collapse: in the limit faster servers are always busy.

The preemptive and non-preemptive policies are asymptotically the same

The ED+QED Asymptotic Regime

¹ 2 > ¹ 1

¹ > ¹

x = y

)exp(1,

,)()1( 1

T

oNK

k kk

NK

K

K 21

N1

1

…

Routing solution: All work conserving policiesare asymptotically optimal

Proof: All these policies are asymptoticallyequivalent to the preemptive FSF.

)(limsupmin :problem Routing TWP

Asymptotically Feasible Region

N1

N2

1N1+2N2 ≥ (1-)+√

policy routing somefor ,)(s.t.

)(...)(min 11

TWP

NCNC KK

))(/(1 1Δ dist. exp Tg-G(T))(α-G(T),G

N1

1N1+2N2 ≥ (1-)+√

N2

Asymptotically Optimal Staffing

)1(...s.t.

)(...)(min

11

11

KK

KK

NN

NCNC

Asymptotic Optimality Definition

M/M/N+G (M&Z): |N-N*|=o(√)

model w/o abandonment (QED): Natural lower bound

KK

KK

NN

NCNC

...s.t.

)(...)(min

11

11Centering factor:Stability bound

model w/abandonment: No natural lower bound.

)1(...s.t.

)(...)(min

11

11

KK

KK

NN

NCNCCentering factor: Fluid level solution

Asymptotically Optimal Staffing

Focus: C(N)=c1N1p+…+cKNK

p

• Let C=inf {C(N) | ¹1N1+…¹KNk=(1-)¸}

• Definition (Asymptotic Optimality)1. N* Asymptotically Feasible and2. (C(N*)-C)/(C(N)- C) = 1 (in the limit)

If =0, replace 2. by C(N*)-C(N)=o(p-1/2)

Summary: M/M/N+ in ED+QED

• Simple Routing: All work-conserving policies

• Staffing: Square-root “safety” capacity (ED+QED regime as an outcome)

• Challenges:– FCFS assumption– Robust definition of asymptotic optimality

• Opportunities:– General Skill-based routing in ED+QED