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Dispatching to Incentivize Fast Service in Multi-Server Queues Raga Gopalakrishnan and Adam Wierman California Institute of Technology Sherwin Doroudi Carnegie Mellon University 6/8/2011MAMA 2011

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

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

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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 2 m

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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 2 m

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

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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 2 m

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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 2 Note: We assume a fixed payment model.

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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 2 ( 1 2 ) is a Nash equilibrium if, for each server, U i ( 1 2 ; ) = max i ½ U i ( i 3-i ; )

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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 2

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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 2

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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 2

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

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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 2

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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 2

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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 2 What about rate-based policies in general? 0 – SSF FSF RANDOM Policy parameter (r)

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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 2 What about rate-based policies in general? 0 – SSF FSF RANDOM Policy parameter (r)

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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 2 What about rate-based policies in general?

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Simulation 1 2 3 –20 204060 –1 Policy parameter (r) Log [Mean response time] –10

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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 2 Design a dispatch policy that: leads to a symmetric Nash equilibrium in the service rates: ( ) minimizes the mean response time, E [T], at ( )

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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 2 Note: We assume a fixed payment model.

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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 2 Note: We assume a fixed payment model. More than 2 servers More general queueing models Other payment models Other utility functions

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Dispatching to Incentivize Fast Service in Multi-Server Queues Raga Gopalakrishnan and Adam Wierman California Institute of Technology Sherwin Doroudi Carnegie Mellon University 6/8/2011MAMA 2011

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[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

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