Staffing and Routing in Large-Scale Service Systems with Heterogeneous-Servers Mor Armony Based on joint papers with Avi Mandelbaum and Amy Ward 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 N1N1 11 Experienced employees on average process 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? 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

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+  NKNK KK N1N1 11 … Assumption: FCFS For some routing policy

The Routing Problem For some routing policy For N 1 =N 2 =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 FSF P (Proof: Sample-path argument)

The Asymptotic Regime Halfin-Whitt (QED) NKNK KK N1N1 1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

Asymptotically Optimal Staffing All solutions of the form have approximately the same cost Let C=inf {C(N) | ¹ 1 N 1 +…+ ¹ K N k = ¸ } Definition (Asymptotic Optimality) 1.N* Asymptotically Feasible and 2.(C(N*)-C)/(C(N)- C) · 1 (in the limit)

Asymptotically Optimal Staffing

Staffing Example: Homogeneous Cost Function Problem: Solve: To obtain: Note:

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 Call centers care about Employee burnout and turnover. Some call centers address fairness by routing to the server that has idled the longest (LISF). How does LISF perform? Do any other fair policies perform better? NKNK KK N1N1 11 …

The Fairness Problem Minimize C 1 (N 1 )+…+C K (N K ) Subject to: E(Waiting time) · W E[# of idle servers of pool k] = f k E[Total # of idle servers] * f 1 + f 2 + … + f K = 1 Assumption: Non-idling NKNK KK N1N1 11 …

The Fairness Problem: Routing Minimize E[Waiting Time] Subject to: E[# of idle servers of pool k] = f k E[Total # of idle servers] Analysis: Sample-path arguments are not straightforward even if preemption is allowed.

MDP Approach: Routing (Assumption: non-idling) Q=1Q=2Q=31,1 1,0 0,0 0,1   =  1 +  2 N 1 = N 2 = 1  22 11 11 22 Pslow 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 N L1L1 L 3 L2L2 FSF w/o pool 3 FSF w/o pool 2 0 FSF w/o pool 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 ¹ 2 slope ¹ 1 L N

Asymptotic Performance (Simulation)  1 = 1,  2 = 2,  = 1,  = 1.5,  2 = 2  = 3, N 1 =300, N 2 =200, ¸ =674

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