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Routing and Staffing to Incentivize Servers i n Many Server Systems

Routing and Staffing to Incentivize Servers i n Many Server Systems. Amy Ward (USC) Raga Gopalakrishnan (Caltech/CU-Boulder/USC) Adam Wierman (Caltech) Sherwin Doroudi (CMU). S ervice systems are staffed by humans. m. strategic servers. system performance.

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Routing and Staffing to Incentivize Servers i n Many Server Systems

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  1. Routing and Staffing to Incentivize Servers in Many Server Systems Amy Ward (USC) Raga Gopalakrishnan (Caltech/CU-Boulder/USC) Adam Wierman (Caltech) Sherwin Doroudi (CMU)

  2. Service systems are staffed by humans. m strategic servers system performance

  3. Service systems are staffed by humans. m Routing and Staffing to IncentivizeServers strategic servers system performance Queueing games: • Strategic arrivals • Service/price competition Classic Queueing: Assumes fixed (arrival and) service rates. [Hassin and Haviv 2003] This talk: Impact of strategic server on system design • Blue for strategic service rates • Yellow for routing/staffing policy parameters • Pink is to highlight.

  4. Outline • The M/M/1 Queue – a simple example • Model for a strategic server • The M/M/N Queue • Classic policies in non-strategic setting • Impact of strategic servers Routing Staffing which idle server gets the next job? how many servers to hire?

  5. M/M/1/FCFS ? λ m=1/μ m strategic server What is the service rate? Values idleness Cost of effort utility function

  6. Outline • The M/M/1 queue – a simple example • Model for a strategic server • The strategic M/M/N queue • Classic policies in non-strategic setting • Impact of strategic servers Scheduling Staffing

  7. M/M/N/FCFS m1 m2 mN scheduling strategic servers Nash equilibrium symmetric Why symmetric? This is fair. (Server payment is fixed.) existence? performance?

  8. Outline • The M/M/1 queue – a simple example • Model for a strategic server • The strategic M/M/N queue • Classic policies in non-strategic setting • Impact of strategic servers Scheduling Staffing

  9. M/M/N/FCFS m1 m2 scheduling mN When servers are not strategic… • Fastest-Server-First (FSF) is asymptotically optimal for . • Longest-Idle-Server-First (LISF) is asymptotically optimal subject to fairness (idleness distribution). [Lin and Kumar1984] [Armony 2005] [Atar 2008] [Armonyand Ward 2010]

  10. M/M/N/FCFS m1 m2 mN scheduling Q: Which policy does better – FSF or its counterpart, SSF? Theorem:No symmetric equilibrium exists under either FSF or SSF. Q: How about Longest-Idle-Server-First (LISF)? Theorem:All idle-time-order-based policies result in the same symmetric equilibrium as Random. Also, (Haji and Ross, 2013). Q: Can we do better than Random? Answer:Yes, but …

  11. M/M/N/FCFS m1 m2 mN Random Theorem: For every λand N, under mild conditions on c, there exists a unique symmetric equilibrium service rate μ* under Random. Furthermore, U(μ*)>0. What is the symmetric equilibrium service rate? First order condition:

  12. Proposition: Under Random routing, Gumbel (1960) for the fully heterogeneous case. Problem: This is a mess!!! There is no hope to use this to decide on a staffing level.

  13. Outline • The M/M/1 queue – a simple example • Model for a strategic server • The strategic M/M/N queue • Classic policies in non-strategic setting • Impact of strategic servers Scheduling Staffing

  14. M/M/N/FCFS m m staffing Random m When servers are not strategic… Q: How many servers to staff? Objective: Minimize total system cost Answer: Square root staffing is asymptotically optimal. Halfin and Whitt (1981) and Borst, Mandelbaum and Reiman (2004)

  15. M/M/N/FCFS staffing m m m Random When servers are strategic… Q: How many servers to staff? Objective: Minimize total system cost Problem: Explicit expression is unknown. Fortunately, there is hope if we let λbecome large.

  16. M/M/N/FCFS staffing m m m Random When servers are strategic… 1. Rate-independent staffing 2. Rate-dependent staffing

  17. M/M/N/FCFS staffing m m m Random Such a solution is not desirable. In order that there exists μ*,λ with The cost function blows up at rate λ. Eliminates square-root staffing. Must staff order λmore. we must staff

  18. M/M/N/FCFS staffing m m m Random What is a? Fluid scale cost. Set Since servers are strategic. Theorem: The staffingNλ is asymptotically optimal in the sense that

  19. M/M/N/FCFS staffing m m m Random Example: Suppose Then Convexity helps. Efficiency is decreased.

  20. Concluding remarks • We need to rethink optimal system design to account for how servers respond to incentives (i.e., when servers are strategic)! $$$$ M/M/N/FCFS m FSF,SSF LISF m ? = Random m We solved for an asymptotically optimal staffing There is a loss of efficiency.

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