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

1. 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.: AAAAAAAAAAAAA

2. Motivation: Call Centers

3. The Inverted-V Model  • 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 q q … NK N1 1 K Experienced employees on average process requests faster than new hires. Gans, Mandelbaum and Shen (2007)

4. 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?  q … NK N1 1 K

5. Background: Human Effects in Large-Scale Service Systems M/M/N+M M/M/N Halfin & Whitt ’81 Borst et al ’04 M/M/N+M+ M/M/N+ Garnett et al ’02 Mandelbaum & Zeltyn ’08 M/M/N++

6. Why Consider Abadonment? Even little abandonment can have a significant effect on performance: • An unstable M/M/N system (r>1) becomes stable with abandonment. • Example (Mandelbaum & Zeltyn ‘08): Consider l=2000/hr, m=20/hr. Service level target: “80% of customers should be served within 30 seconds”: • 106 agents (q=0) • 95 agents (q=20 (average patience of 3 minutes), P(ab)=6.9%) • 84 agents (q=60 (average patience of 1 minute), P(ab)=16.8%)

7. Problem Formulation our focus • 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 q>m.

8. Asymptotic Regimes(Mandelbaum & Zeltyn 07) Baron & Milner 07

9. 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. • Minimize staffing cost over the asymptotically feasible • region.

10. The Routing Problem For a given staffing vector: • Proposition: The preemptive Faster Server First (FSF) policy is optimal within FCFS policies if either of these assumptions holds: • q ≤ min{m1,…,mK}, or • Only work-conserving policies are allowed.

11. Asymptotically Optimal Routingin 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

12. The ED+QED Asymptotic Regime  q … NK N1 Routing solution: All work conserving policies are asymptotically optimal Proof: All these policies are asymptotically equivalent to the preemptive FSF. 1 K

13. Asymptotically Feasible Region N2 N1 m1N1+m2N2 ≥ (1-D)l+d√l

14. Asymptotically Optimal Staffing N2 N1 m1N1+m2N2 ≥ (1-D)l+d√l

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

16. Asymptotically Optimal StaffingFocus: C(N)=c1N1p+…+cKNKp • Let C=inf {C(N) | ¹1N1+…¹KNk=(1-D)¸} • Definition (Asymptotic Optimality) • N* Asymptotically Feasible and • (C(N*)-C)/(C(N)- C) = 1 (in the limit) If d=0, replace 2. by C(N*)-C(N)=o(lp-1/2)

17. 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