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Raga Gopalakrishnan Caltech CU-Boulder / USC Adam Wierman (Caltech) Amy R. Ward (USC)

Scheduling and staffing strategic servers. Raga Gopalakrishnan Caltech CU-Boulder / USC Adam Wierman (Caltech) Amy R. Ward (USC) Sherwin Doroudi (CMU). Journal reviews Call centers Crowdsourcing Cloud computing Enterprise data centers …. service systems. m. strategic servers.

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Raga Gopalakrishnan Caltech CU-Boulder / USC Adam Wierman (Caltech) Amy R. Ward (USC)

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  1. Scheduling and staffing strategic servers Raga GopalakrishnanCaltech CU-Boulder / USC Adam Wierman (Caltech) Amy R. Ward (USC) Sherwin Doroudi (CMU)

  2. Journal reviews • Call centers • Crowdsourcing • Cloud computing • Enterprise data centers • … service systems m strategic servers system performance

  3. Journal reviews • Call centers • Crowdsourcing • Cloud computing • Enterprise data centers • … service systems m Scheduling and staffingstrategic servers strategic servers system performance Classic Queueing: Assumes fixed (arrival and) service rates, fixed control/policies. This talk: Impact of strategic servers on optimal system design Queueing games: • Strategic arrivals • Service/price competition [Hassin and Haviv 2003]

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

  5. M/M/1/FCFS l m m cost idleness strategic server utility function LHS RHS 1 / l m* 0 l

  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 l m1 m2 mN scheduling strategic servers • Blue for strategic service rates • Yellow for control/policy parameters symmetricNash equilibrium Nash equilibrium 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 l m2 scheduling mN When servers are not strategic… • Fastest-Server-First (FSF) is asymptotically optimal for . • Longest-Idle-Server-First (LISF) is asymptotically fair (idleness distribution). • Random is naïve, and easily implementable. [Lin et al. 1984] [Véricourt et al. 2005] [Armony 2005] [Atar 2008] [Armony et al. 2010]

  10. M/M/N/FCFS l 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. Q: Can we do better than Random? (ask me later!) Answer: Yes!

  11. M/M/N/FCFS l m1 m2 mN Random Theorem:For every ,, under mild conditions on c, there exists a unique symmetric equilibrium under Random. Q: What does look like? First order condition:

  12. First order condition: Theorem:Under Random scheduling, suppose a tagged server works at rate , and the other servers work at rate . Then, where , and is the Erlang-C formula, Problem: This is a mess!!!

  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 l m Random staffing m When servers are not strategic… Q: What staffing policy should the system manager adopt? Objective: minimize total system cost: Answer: Square-root staffing: asymptotically optimal [Borst et al. 2004]

  15. M/M/N/FCFS l m m m Random staffing When servers are strategic… Q: What staffing policy should the system manager adopt? Objective: minimize total system cost: Problem: Explicit expression unknown! Hope: Perhaps feasible to solve when is large. approximate by taking the limit as

  16. M/M/N/FCFS l m m m Random staffing When servers are strategic… Rate-independent staffing for some function Need to staff in a very narrow way in order to ensure unique equilibrium Rate-dependent staffing for some function (ask me later!)

  17. for some function Feasibility: We are interested in solutions for which: Theorem: Feasibility is satisfied if and only if as. Furthermore, , where: • Eliminates square-root staffing () • Need to staff more servers!

  18. for some function Feasibility: We are interested in solutions for which: Theorem (asymptotic optimality): Suppose as , where . Then, as

  19. Concluding remarks • We need to rethink optimal system design when servers are strategic! $ M/M/N/FCFS m l $$$$ m ? Random ? $$ m loss of efficiency?

  20. Scheduling and staffing strategic servers RagavendranGopalakrishnanCaltech CU-Boulder / USC Adam Wierman (Caltech) Amy R. Ward (USC) Sherwin Doroudi (CMU)

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