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The Impact of Server Incentives o n Scheduling

The Impact of Server Incentives o n Scheduling. Raga Gopalakrishnan and Adam Wierman California Institute of Technology Sherwin Doroudi Carnegie Mellon University. Scheduling in Multi-Server Queues. m 1. FCFS. l. m 2. dispatcher. m m. How should the dispatcher be designed?.

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The Impact of Server Incentives o n Scheduling

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  1. The Impact of Server Incentives on Scheduling Raga Gopalakrishnan and Adam Wierman California Institute of Technology Sherwin Doroudi Carnegie Mellon University INFORMS APS 2011

  2. Scheduling in Multi-Server Queues m1 FCFS l m2 dispatcher mm How should the dispatcher be designed?

  3. Commonly Studied Dispatch Policies m1 FCFS l m2 dispatcher P mm Dispatch Policy (P) • Fastest Server First (FSF) • [Lin et al. 1984] [Véricourt et al. 2005] [Armony 2005] • RANDOM

  4. What if servers are people? m1 FCFS l m2 dispatcher P mm Example: Call Centers • Fair distribution of idle time is an important measure of employee satisfaction. [Cohen-Charash et al. 2001] [Colquitt et al. 2001] [Whitt 2006] • FSF is not a “fair” policy. [Armony 2005]

  5. What if servers are people? m1 FCFS l m2 dispatcher P mm Example: Call Centers • Longest Idle Server First (LISF) • [Atar 2008] [Armonyet al. 2010] • LISF has good “fairness” properties. [Atar 2008]

  6. What if people can react? m1 FCFS l m2 dispatcher P mm This Talk: How should the dispatcher be designed if servers are strategic?

  7. Model m1 M/M/m/FCFS l = 1 m2 dispatcher P mm servers choosemiє[1/m,∞) to maximize: Ui(m1,m2,…,mm; P) = Ii(m1,m2,…,mm; P) – c(mi) Note: We assume a fixed payment model. utility idle time cost (increasing, convex)

  8. Model m1 M/M/2/FCFS l = 1 dispatcher P m2 servers choosemiє[1/2,∞) to maximize: Ui(m1,m2; P) = Ii(m1,m2; P) – c(mi) Note: We assume a fixed payment model. utility idle time cost (increasing, convex)

  9. Goal m1 M/M/2/FCFS Ui(m1,m2;P) = Ii(m1,m2;P) – c(mi) (m1,m2) is a Nash equilibrium if, for each server, Ui(m1,m2 ; P) = maxm’i≥ ½Ui(m’i,m3-i ; P) l = 1 dispatcher P m2 • Design a dispatch policy that: • leads to a symmetric Nash equilibrium in the service rates: (m*,m*) • minimizes the mean response time, E[T], at (m*,m*) • Design a dispatch policy that: • leads to a symmetric Nash equilibrium in the service rates: (m*,m*) • minimizes the mean response time, E[T], at (m*,m*) • Design a dispatch policy that: • leads to a symmetric Nash equilibrium in the service rates: (m*,m*) • minimizes the mean response time, E[T], at (m*,m*)

  10. What about well-known policies? m1 M/M/2/FCFS Ui(m1,m2;P) = Ii(m1,m2;P) – c(mi) l = 1 dispatcher P m2 • Fastest Server First (FSF) • Wrong incentive • No symmetric equilibrium

  11. What about well-known policies? m1 M/M/2/FCFS Ui(m1,m2;P) = Ii(m1,m2;P) – c(mi) l = 1 dispatcher P m2 • Slowest Server First (SSF) • Right incentive • No symmetric equilibrium

  12. What about well-known policies? m1 M/M/2/FCFS Ui(m1,m2;P) = Ii(m1,m2;P) – c(mi) l = 1 dispatcher P m2 • RANDOM • Unique symmetric equilibrium under mild assumptions that guarantee voluntary participation: c’(½) < 5/6, c”’(m) > 0.

  13. Can we do better than RANDOM? m1 M/M/2/FCFS Ui(m1,m2;P) = Ii(m1,m2;P) – c(mi) l = 1 dispatcher P m2 • Longest Idle Server First (LISF) • Equivalent to RANDOM.

  14. Can we do better than RANDOM? m1 M/M/2/FCFS Ui(m1,m2;P) = Ii(m1,m2;P) – c(mi) l = 1 dispatcher P What about idle-time-based policies in general? m2 • Suppose there are |I(t)| idle servers in the system (1 ≤ |I(t)| ≤ 2). • These servers are ranked in the order in which they last became idle. • The next job in the queue is then routed according to a probability distribution on this ranking. All idle-time-based policies are equivalent and result in the same unique symmetric equilibrium as RANDOM.

  15. Can we do better than RANDOM? m1 M/M/2/FCFS Ui(m1,m2;P) = Ii(m1,m2;P) – c(mi) l = 1 dispatcher P What about rate-based policies in general? m2 • The probability that an idle server i gets the next job is proportional to mir, where r eR is a policy parameter. FSF SSF RANDOM ∞ ∞ 0 – Policy parameter (r)

  16. Can we do better than RANDOM? m1 M/M/2/FCFS Ui(m1,m2;P) = Ii(m1,m2;P) – c(mi) l = 1 dispatcher P What about rate-based policies in general? m2 Any rate-based policy with r є{-2,-1,0,1} admits a unique symmetric Nash equilibrium. FSF SSF RANDOM ∞ ∞ 0 – Policy parameter (r)

  17. Can we do better than RANDOM? m1 M/M/2/FCFS Ui(m1,m2;P) = Ii(m1,m2;P) – c(mi) l = 1 dispatcher P What about rate-based policies in general? m2 There exists a bounded interval for r outside of which, no rate-based policy admits a symmetric Nash equilibrium. FSF SSF RANDOM ∞ ∞ 0 – Policy parameter (r)

  18. Can we do better than RANDOM? m1 M/M/2/FCFS Ui(m1,m2;P) = Ii(m1,m2;P) – c(mi) l = 1 dispatcher P What about rate-based policies in general? m2 Any rate-based policy that admits a symmetric Nash equilibrium, admits a unique symmetric Nash equilibrium. Further, among all such policies, E[T] at symmetric equilibrium is increasing in r.

  19. Simulation Log [Mean response time] 3 2 1 –10 –20 20 40 60 Policy parameter (r) –1

  20. Summary m1 M/M/2/FCFS Ui(m1,m2;P) = Ii(m1,m2;P) – c(mi) l = 1 dispatcher P m2 ∞ Random, Idle-time-based • Design a dispatch policy that: • leads to a symmetric Nash equilibrium in the service rates: (m*,m*) • minimizes the mean response time, E[T], at (m*,m*) SSF Random FSF Mean response time ∞ ∞ 0 – Policy parameter (r) 0 ∞ ∞ – Policy parameter (r)

  21. Model m1 M/M/2/FCFS l = 1 dispatcher P m2 servers choosemiє[1/2,∞) to maximize: Ui(m1,m2; P) = Ii(m1,m2; P) – c(mi) Note: We assume a fixed payment model. utility idle time cost (increasing, convex)

  22. Future Work • More than 2 servers • More general queueing models m1 M/M/2/FCFS l = 1 dispatcher P m2 • Other utility functions servers choosemiє[1/2,∞) to maximize: • Other payment models Ui(m1,m2; P) = Ii(m1,m2; P) – c(mi) Note: We assume a fixed payment model. utility idle time cost (increasing, convex)

  23. The Impact of Server Incentives on Scheduling Raga Gopalakrishnan and Adam Wierman California Institute of Technology Sherwin Doroudi Carnegie Mellon University INFORMS APS 2011

  24. References • [Lin et al. 1984] Optimal control of a queueing system with two heterogeneous servers. • [Cohen-Charash et al. 2001] The role of justice in organizations: A meta-analysis. • [Colquitt et al. 2001] Justice at the millennium: A meta-analytic review of 25 years of organizational justice research. • [Véricourt et al. 2005] Managing response time in a call-routing problem with service failure. • [Armony 2005] Dynamic routing in large-scale service systems with heterogeneous servers. • [Whitt 2006] The impact of increased employee retention on performance in a customer contact center. • [Atar 2008] Central limit theorem for a many-server queue with random service rates. • [Armony et al. 2010] Fair dynamic routing in large-scale heterogeneous-server systems. • [Armony et al. 2010] Blind fair routing in large-scale service systems.

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