The Impact of Server Incentives on Scheduling Raga Gopalakrishnan and Adam Wierman California Institute of Technology Sherwin Doroudi Carnegie Mellon University 7/7/2011INFORMS APS 2011

Scheduling in Multi-Server Queues How should the dispatcher be designed? FCFS dispatcher 11 22 mm

Commonly Studied Dispatch Policies Fastest Server First (FSF) [Lin et al. 1984] [Véricourt et al. 2005] [Armony 2005] RANDOM Dispatch Policy (  ) FCFS dispatcher 11 22 mm 

What if servers are people? 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] Example: Call Centers FCFS dispatcher 11 22 mm 

What if servers are people? Longest Idle Server First (LISF) [Atar 2008] [Armony et al. 2010] LISF has good “fairness” properties. [Atar 2008] Example: Call Centers FCFS dispatcher 11 22 mm 

What if people can react? This Talk: How should the dispatcher be designed if servers are strategic? FCFS dispatcher 11 22 mm 

M/M/m/FCFS Model servers choose  i є [1/m  ∞) to maximize: U i (  1,  2,…,  m ;  ) = I i (  1,  2,…,  m ;  ) – c(  i ) utilityidle timecost Note: We assume a fixed payment model. (increasing, convex) dispatcher 11 22 mm  

M/M/2/FCFS Model servers choose  i є [1/2  ∞) to maximize: U i (  1,  2 ;  ) = I i (  1,  2 ;  ) – c(  i ) utilityidle timecost (increasing, convex) dispatcher 11 22   Note: We assume a fixed payment model.

Goal U i (  1  2 ;  ) = I i (  1  2 ;  ) – c(  i ) Design a dispatch policy that: leads to a symmetric Nash equilibrium in the service rates: (     ) minimizes the mean response time, E [T], at (     ) Design a dispatch policy that: leads to a symmetric Nash equilibrium in the service rates: (     ) minimizes the mean response time, E [T], at (     ) Design a dispatch policy that: leads to a symmetric Nash equilibrium in the service rates: (     ) minimizes the mean response time, E [T], at (     ) M/M/2/FCFS dispatcher 11 22   (  1  2 ) is a Nash equilibrium if, for each server, U i (  1  2 ;  ) = max  ’ i ≥ ½  U i (  ’ i  3-i ;  )

What about well-known policies? Fastest Server First (FSF) Wrong incentive No symmetric equilibrium U i (  1  2 ;  ) = I i (  1  2 ;  ) – c(  i ) M/M/2/FCFS dispatcher 11 22  

What about well-known policies? Slowest Server First (SSF) Right incentive No symmetric equilibrium U i (  1  2 ;  ) = I i (  1  2 ;  ) – c(  i ) M/M/2/FCFS dispatcher 11 22  

What about well-known policies? RANDOM Unique symmetric equilibrium under mild assumptions that guarantee voluntary participation: c’(½) 0. U i (  1  2 ;  ) = I i (  1  2 ;  ) – c(  i ) M/M/2/FCFS dispatcher 11 22  

U i (  1  2 ;  ) = I i (  1  2 ;  ) – c(  i ) M/M/2/FCFS dispatcher 11 22   Can we do better than RANDOM? Longest Idle Server First (LISF) Equivalent to RANDOM.

Can we do better than RANDOM? 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. What about idle-time-based policies in general? All idle-time-based policies are equivalent and result in the same unique symmetric equilibrium as RANDOM. U i (  1  2 ;  ) = I i (  1  2 ;  ) – c(  i ) M/M/2/FCFS dispatcher 11 22  

Can we do better than RANDOM? The probability that an idle server i gets the next job is proportional to  i r, where r e R is a policy parameter. What about rate-based policies in general? ∞ 0 ∞ – SSF FSF RANDOM Policy parameter (r) U i (  1  2 ;  ) = I i (  1  2 ;  ) – c(  i ) M/M/2/FCFS dispatcher 11 22  

Can we do better than RANDOM? Any rate-based policy with r є {-2,-1,0,1} admits a unique symmetric Nash equilibrium. U i (  1  2 ;  ) = I i (  1  2 ;  ) – c(  i ) M/M/2/FCFS dispatcher 11 22   What about rate-based policies in general? ∞ 0 ∞ – SSF FSF RANDOM Policy parameter (r)

Can we do better than RANDOM? There exists a bounded interval for r outside of which, no rate-based policy admits a symmetric Nash equilibrium. U i (  1  2 ;  ) = I i (  1  2 ;  ) – c(  i ) M/M/2/FCFS dispatcher 11 22   What about rate-based policies in general? ∞ 0 ∞ – SSF FSF RANDOM Policy parameter (r)

Can we do better than RANDOM? 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. U i (  1  2 ;  ) = I i (  1  2 ;  ) – c(  i ) M/M/2/FCFS dispatcher 11 22   What about rate-based policies in general?

Simulation – –1 Policy parameter (r) Log [Mean response time] –10

Summary ∞ 0 ∞ – SSFFSF Random, Idle-time- based Random Policy parameter (r) Mean response time ∞ 0 ∞ – ∞ U i (  1  2 ;  ) = I i (  1  2 ;  ) – c(  i ) M/M/2/FCFS dispatcher 11 22   Design a dispatch policy that: leads to a symmetric Nash equilibrium in the service rates: (     ) minimizes the mean response time, E [T], at (     )

M/M/2/FCFS Model servers choose  i є [1/2  ∞) to maximize: U i (  1,  2 ;  ) = I i (  1,  2 ;  ) – c(  i ) utilityidle timecost (increasing, convex) dispatcher 11 22   Note: We assume a fixed payment model.

M/M/2/FCFS Future Work servers choose  i є [1/2  ∞) to maximize: U i (  1,  2 ;  ) = I i (  1,  2 ;  ) – c(  i ) utilityidle timecost (increasing, convex) dispatcher 11 22   Note: We assume a fixed payment model. More than 2 servers More general queueing models Other payment models Other utility functions

The Impact of Server Incentives on Scheduling Raga Gopalakrishnan and Adam Wierman California Institute of Technology Sherwin Doroudi Carnegie Mellon University 7/7/2011INFORMS APS 2011

[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. References