Staffing and Routing in Large-Scale Service Systems with Heterogeneous-Servers Mor Armony Stern School of Business, NYU INFORMS 2009 Joint work with Avi Mandelbaum TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA A A A A A AA A

Motivation: Call Centers

The Inverted-V Model NKNK KK 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 queue with rate  N1N1 11 Experienced employees on average process 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? NKNK KK N1N1 1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 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) Baron & Milner 07

Solution approach Original Joint Staffing and Routing problem: Our approach: 1. Given a “sensible” staffing vector, solve the routing problem: 2. Show that the proposed staffing vector is is asymptotically feasible. 3.Minimize staffing cost over the asymptotically feasible region.

The Routing Problem Proposition: The preemptive Faster Server First (FSF) policy is optimal within FCFS policies if either of these assumptions holds:  ≤ min{  1,…,  K }, or 2.Only work-conserving policies are allowed. 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 NKNK KK N1N1 11 …  Routing solution: All work conserving policies are asymptotically optimal Proof: All these policies are asymptotically equivalent to the preemptive FSF.

Asymptotically Feasible Region N1N1 N2N2  1 N 1 +  2 N 2 ≥ (1-  ) +  √

N1N1 N2N2 Asymptotically Optimal Staffing

Asymptotic Optimality Definition M/M/N+G (M&Z): |N-N*|=o(√ )  model w/o abandonment (QED): Natural lower bound Centering factor: Stability bound  model w/abandonment: No natural lower bound. Centering factor: Fluid level solution

Asymptotically Optimal Staffing Focus: C(N)=c 1 N 1 p +…+c K N K p Let C=inf {C(N) | ¹ 1 N 1 +… ¹ K N k =(1-  ) ¸ } Definition (Asymptotic Optimality) 1.N* Asymptotically Feasible and 2.(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