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

Servers

Mor Armony

Based on joint papers with Avi Mandelbaum and

Amy Ward

¸ < ¹

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

N1

1

¹ 2 > ¹ 1

¹ > ¹

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

…

The Problem

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

Talk Outline

• M/M/N+ (Armony ‘05)• M/M/N++M (Armony & Mandelbaum

’08)• M/M/N++☺ (Armony & Ward ’08)

The Problem: M/M/N+

¹ 2 > ¹ 1

¹ > ¹

x = y

NK

K

K 21

N1

1

…

Minimize C1(N1) + ::: + CK (NK )Subject to P (wait > 0) · ®;

Assumption: FCFS

For some routing policy

The Routing Problem

¹ 2 > ¹ 1

¹ > ¹

x = y

Minimize C1(N1) + ::: + CK (NK )Subject to P (wait > 0) · ®;

For some routing policy

• For N1=N2=1 optimal routing is of a threshold form (the slow server problem)

• For general N, structure of optimal routing is an open problem (de Vericourt & Zhou)

• The optimal preemptive policy is FSFP (Proof: Sample-path argument)

The Asymptotic RegimeHalfin-Whitt (QED)

¹ 2 > ¹ 1

¹ > ¹

x = y

oN

K

k kk

1

, As

X̂ ¸ = X ¸ ¡ N ¸p

N ¸hX̂ ¸

i += scaled queue length

hX̂ ¸

i ¡= scaled # of idle servers

NK

K

K 21

N1

1

…

Asymptotically Optimal Routing

Proposition: The non-preemptive routing policy FSF is asymptotically optimal

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

The preemptive and non-preemptive policies are asymptotically the same

Note: Thresholds are not-needed: The Halfin-Whitt regime is different from the conventional

heavy- traffic regime (Teh & Ward ’02).

Asymptotically Feasible Region

Asymptotic Feasibility

• Proposition: Under FSF

if and only if

where

provided that

lim¸ ! 1

P (wait > 0) = ® 0· ®· 1;

¹ 1N1 + ¹ 2N2 + ::: + ¹ K NK ¼¸ + ±p

¸; 0 · ±· 1

®=·1+

(±=p

¹ 1)©(±=p

¹ 1)Á(±=

p¹ 1)

¸¡ 1

;

liminf¸ ! 1 N1=N > 0.

Asymptotically Optimal Staffing

• All solutions of the form

have approximately the same cost

• Let C=inf {C(N) | ¹1N1+…+¹KNk=¸}

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

¹ 1N1 + ¹ 2N2 + ::: + ¹ K NK = ¸ + ±p

¸; 0< ±< 1

Asymptotically Optimal Staffing

Staffing Example:Homogeneous Cost Function

• Problem:

• Solve:

• To obtain:

• Note:

Minimize C1Np1 + C2N

p2 + ::: + CK N p

K ; p> 1Subject to P (wait > 0) · ®; for some routing policy

Minimize C1N p1 + C2N

p2 + ::: + CK N p

K ; p> 1Subject to ¹ 1N1 + ¹ 2N2 + ::: + ¹ K NK ¸ ¸ + ±

p¸

Nk

Nj=

µ¹ k=Ck

¹ j =Cj

¶1=(p¡ 1)

(1)

¹ 1N1 + ¹ 2N2 + ::: + ¹ K NK = ¸ + ±p

¸ (2)

N1=N > 0

Summary: M/M/N+• Routing: FSF• Staffing: Square-root safety capacity (QED

regime as an outcome)• Under FCFS non-idling is asymptotically

optimal• For non-idling policies: min P(W>0) min EW• Outperforming M/M/N• Faster servers are never idle• All idleness is experienced by the slowest

servers

Adding Fairness

Fairness in Call Center

¹ 2 > ¹ 1

¹ > ¹

x = y

Call centers care aboutEmployee burnout and turnover.

Some call centers address fairness byrouting to the server that has idled the longest (LISF).

How does LISF perform?

Do any other fair policies perform better?

NK

K

K 21

N1

1

…

The Fairness Problem

¹ 2 > ¹ 1

¹ > ¹

x = y

Minimize C1(N1)+…+CK(NK)

Subject to:

E(Waiting time)· W

E[# of idle servers of pool k]=

fk

E[Total # of idle servers]

* f1 + f2 + … + fK = 1

Assumption: Non-idling

NK

K

K 21

N1

1

…

The Fairness Problem: Routing

Minimize E[Waiting Time]Subject to:

E[# of idle servers of pool k]= fk

E[Total # of idle servers]

Analysis: Sample-path arguments are not straightforward even if preemption is allowed.

¹ 2 > ¹ 1

¹ > ¹

x = y

MDP Approach: Routing(Assumption: non-idling)

Q=1 Q=2 Q=31,1

1,00,0

0,1

= 1+ 2 N1 = N2 = 1

2

1

1

2Pslow

Pfast

Infinite state space

Numeric Example

MDP as an LP

• Complexity: Polynomial in N, Exponential in K• Solution: Switching curve (Difficult to

characterize explicitly).• How does solution perform vs. LISF?• Staffing search: Too long!!!• Instead, we propose an asymptotic approach.

Threshold Routing Control

)0 and(with

levelsat control dA threshol

0

11

K

K

LN L

N L...L

x = y2

NL1 L3 L2

FSF

FSFw/opool 3 FSF w/o pool 2

0

FSFw/opool 4

Outline of Asymptotic Analysis

• Formulation of a Diffusion Control Problem (DCP)

• Solution of DCP: Multi-Threshold Control

• Note: Resulting Diffusion has Discontinuous Drift

• Policy Translation: Multi-Threshold Policy

• Policy Adjustment: -Threshold Policy

• Establishing Asymptotic Optimality

²-Threshold Policy

X

Death rate

slope ¹2

slope ¹1

L N

Asymptotic Performance (Simulation)

1 = 1, 2 = 2, = 1, = 1.5, 2 = 2 = 3, N1=300, N2=200, ¸=674

A Simulation Comparison of the Threshold and LWISF Policy

0.0000002.000000

4.0000006.000000

8.00000010.000000

12.000000

0 0.5 1

Slow Server Idleness Proportion

E[N

um

be

r o

f W

ait

ing

C

us

tom

ers

]

Threshold

LWISF

Literature Review

• MDP approach to constrained optimization– Gans and Zhou (2003), Bhulai and Koole (2003)

• The Limit Regime– Halfin and Whitt (1981)

• The Inverted V (and more general) Models– Tezcan (2006), Atar (2007), Atar & Shwartz

(2008), Atar, Shaki & Shwartz (2009), Tseytlin (2008)

- Gurvich and Whitt (2007)

• Customer / Flow Fairness literature– Harchol-Balter and Wierman (2003, 2007)– Jahn et al (2005) & Schulz and Stier-Moses (2006)

• Fairness literature in HRM

Summary

• Server Heterogeneity: Effect on Staffing and Routing

• Incorporation of customer abandonment

• Incorporation of server fairness

• Simple routing schemes (priorities and threshold) • Simple staffing schemes (square-root safety

staffing)

Further Research

• Multi-skill environment (ongoing with Kocaga)

• LWISF policy (ongoing with Gurvich)• Non-idling assumption• Incorporate abandonment

(M/M/N++M+☺)• Other fairness criteria• Server compensation schemes

Acknowledgement: Rami Atar, Ashish Goel, Itay Gurvich, Tolga Tezcan & Assaf Zeevi