Staffing and Routing in Large-Scale Service Systems with Heterogeneous-Servers Mor Armony Stern...
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Staffing and Routing in Large-Scale Service Systems with Heterogeneous-
Servers
Mor ArmonyStern School of Business, NYUINFORMS 2009
Joint work with Avi Mandelbaum
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
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)