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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)

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strategic servers system performance Service systems are staffed by humans.

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

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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 RoutingStaffing which idle server gets the next job? how many servers to hire?

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

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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 SchedulingStaffing

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

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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 SchedulingStaffing

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M/M/N/FCFS scheduling 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 Kumar 1984] [Armony 2005] [Atar 2008] [Armony and Ward 2010]

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M/M/N/FCFS 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. Q: Can we do better than Random? Answer: Yes, but … Also, (Haji and Ross, 2013).

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M/M/N/FCFS Random First order condition: What is the symmetric equilibrium service rate? 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.

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

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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 SchedulingStaffing

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M/M/N/FCFS When servers are not strategic… Random 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) staffing

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M/M/N/FCFS When servers are strategic… Random staffing 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.

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

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

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

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

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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 FSF,SSF LISF There is a loss of efficiency. $$$$ ? We solved for an asymptotically optimal staffing = Random

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