Post on 18-Jul-2020
transcript
Dynamic Market Prices for CallCenters with Co-Sourcing Subject
to Non-stationary Demand
Sherwin Doroudi
University of Minnesota
Department of Industrial & Systems Engineering
Joint work with Mohammad Mousavi (Target Lab)
May 16, 2018
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 1 / 31
Call centers
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 2 / 31
Call centers — big business
• Call centers are crucial for providing directcustomer support
• Over 2 million employees across ≈ 6,800 callcenters in the U.S. [Site Selection Group 2015]
• Lower overseas wages often leads tooutsourcing
• Contemporary trends: chat and automation
• 67% of customers want live customer service[inContact 2017]
• Huge impact on customer satisfaction
• Huge impact on an organization’s bottom line
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 3 / 31
Call centers — big business
• Call centers are crucial for providing directcustomer support
• Over 2 million employees across ≈ 6,800 callcenters in the U.S. [Site Selection Group 2015]
• Lower overseas wages often leads tooutsourcing
• Contemporary trends: chat and automation
• 67% of customers want live customer service[inContact 2017]
• Huge impact on customer satisfaction
• Huge impact on an organization’s bottom line
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 3 / 31
Call centers — big business
• Call centers are crucial for providing directcustomer support
• Over 2 million employees across ≈ 6,800 callcenters in the U.S. [Site Selection Group 2015]
• Lower overseas wages often leads tooutsourcing
• Contemporary trends: chat and automation
• 67% of customers want live customer service[inContact 2017]
• Huge impact on customer satisfaction
• Huge impact on an organization’s bottom line
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 3 / 31
Call centers — big business
• Call centers are crucial for providing directcustomer support
• Over 2 million employees across ≈ 6,800 callcenters in the U.S. [Site Selection Group 2015]
• Lower overseas wages often leads tooutsourcing
• Contemporary trends: chat and automation
• 67% of customers want live customer service[inContact 2017]
• Huge impact on customer satisfaction
• Huge impact on an organization’s bottom line
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 3 / 31
Call centers — big business
• Call centers are crucial for providing directcustomer support
• Over 2 million employees across ≈ 6,800 callcenters in the U.S. [Site Selection Group 2015]
• Lower overseas wages often leads tooutsourcing
• Contemporary trends: chat and automation
• 67% of customers want live customer service[inContact 2017]
• Huge impact on customer satisfaction
• Huge impact on an organization’s bottom line
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 3 / 31
Call centers — big business
• Call centers are crucial for providing directcustomer support
• Over 2 million employees across ≈ 6,800 callcenters in the U.S. [Site Selection Group 2015]
• Lower overseas wages often leads tooutsourcing
• Contemporary trends: chat and automation
• 67% of customers want live customer service[inContact 2017]
• Huge impact on customer satisfaction
• Huge impact on an organization’s bottom line
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 3 / 31
Call centers — big business
• Call centers are crucial for providing directcustomer support
• Over 2 million employees across ≈ 6,800 callcenters in the U.S. [Site Selection Group 2015]
• Lower overseas wages often leads tooutsourcing
• Contemporary trends: chat and automation
• 67% of customers want live customer service[inContact 2017]
• Huge impact on customer satisfaction
• Huge impact on an organization’s bottom line
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 3 / 31
Call centers — big business
• Call centers are crucial for providing directcustomer support
• Over 2 million employees across ≈ 6,800 callcenters in the U.S. [Site Selection Group 2015]
• Lower overseas wages often leads tooutsourcing
• Contemporary trends: chat and automation
• 67% of customers want live customer service[inContact 2017]
• Huge impact on customer satisfaction
• Huge impact on an organization’s bottom line
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 3 / 31
Features of call centers
• Finitely may agents or “servers” (often 100+)
• Demand (call arrival) uncertainty• Aim to satisfy quality of service (QoS) targets
• Most calls served with short waiting time• Most calls served before abandoned
• Capacity expansion is very costly• Capital investment costs are large• Significant costs to hiring, training, and scheduling
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 4 / 31
Features of call centers
• Finitely may agents or “servers” (often 100+)
• Demand (call arrival) uncertainty• Aim to satisfy quality of service (QoS) targets
• Most calls served with short waiting time• Most calls served before abandoned
• Capacity expansion is very costly• Capital investment costs are large• Significant costs to hiring, training, and scheduling
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 4 / 31
Features of call centers
• Finitely may agents or “servers” (often 100+)
• Demand (call arrival) uncertainty
• Aim to satisfy quality of service (QoS) targets• Most calls served with short waiting time• Most calls served before abandoned
• Capacity expansion is very costly• Capital investment costs are large• Significant costs to hiring, training, and scheduling
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 4 / 31
Features of call centers
• Finitely may agents or “servers” (often 100+)
• Demand (call arrival) uncertainty• Aim to satisfy quality of service (QoS) targets
• Most calls served with short waiting time• Most calls served before abandoned
• Capacity expansion is very costly• Capital investment costs are large• Significant costs to hiring, training, and scheduling
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 4 / 31
Features of call centers
• Finitely may agents or “servers” (often 100+)
• Demand (call arrival) uncertainty• Aim to satisfy quality of service (QoS) targets
• Most calls served with short waiting time
• Most calls served before abandoned
• Capacity expansion is very costly• Capital investment costs are large• Significant costs to hiring, training, and scheduling
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 4 / 31
Features of call centers
• Finitely may agents or “servers” (often 100+)
• Demand (call arrival) uncertainty• Aim to satisfy quality of service (QoS) targets
• Most calls served with short waiting time• Most calls served before abandoned
• Capacity expansion is very costly• Capital investment costs are large• Significant costs to hiring, training, and scheduling
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 4 / 31
Features of call centers
• Finitely may agents or “servers” (often 100+)
• Demand (call arrival) uncertainty• Aim to satisfy quality of service (QoS) targets
• Most calls served with short waiting time• Most calls served before abandoned
• Capacity expansion is very costly
• Capital investment costs are large• Significant costs to hiring, training, and scheduling
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 4 / 31
Features of call centers
• Finitely may agents or “servers” (often 100+)
• Demand (call arrival) uncertainty• Aim to satisfy quality of service (QoS) targets
• Most calls served with short waiting time• Most calls served before abandoned
• Capacity expansion is very costly• Capital investment costs are large
• Significant costs to hiring, training, and scheduling
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 4 / 31
Features of call centers
• Finitely may agents or “servers” (often 100+)
• Demand (call arrival) uncertainty• Aim to satisfy quality of service (QoS) targets
• Most calls served with short waiting time• Most calls served before abandoned
• Capacity expansion is very costly• Capital investment costs are large• Significant costs to hiring, training, and scheduling
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 4 / 31
Co-sourcing in call centers
• Q: What is co-sourcing?
• A: Practice where some calls served byexternal contractor
• Only a portion of the service capacity isprovided “in-house”
• Why co-source?• Reduce internal staffing costs• Reduce wait time and abandonment• Mitigate capacity constraints• Maintain service at peak demand
• Examples of external contractors:• Convergys Corporation (150 locations)• Wipro (4 million calls/mo. by 15k employees)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 5 / 31
Co-sourcing in call centers• Q: What is co-sourcing?
• A: Practice where some calls served byexternal contractor
• Only a portion of the service capacity isprovided “in-house”
• Why co-source?• Reduce internal staffing costs• Reduce wait time and abandonment• Mitigate capacity constraints• Maintain service at peak demand
• Examples of external contractors:• Convergys Corporation (150 locations)• Wipro (4 million calls/mo. by 15k employees)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 5 / 31
Co-sourcing in call centers• Q: What is co-sourcing?
• A: Practice where some calls served byexternal contractor
• Only a portion of the service capacity isprovided “in-house”
• Why co-source?• Reduce internal staffing costs• Reduce wait time and abandonment• Mitigate capacity constraints• Maintain service at peak demand
• Examples of external contractors:• Convergys Corporation (150 locations)• Wipro (4 million calls/mo. by 15k employees)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 5 / 31
Co-sourcing in call centers• Q: What is co-sourcing?
• A: Practice where some calls served byexternal contractor
• Only a portion of the service capacity isprovided “in-house”
• Why co-source?• Reduce internal staffing costs• Reduce wait time and abandonment• Mitigate capacity constraints• Maintain service at peak demand
• Examples of external contractors:• Convergys Corporation (150 locations)• Wipro (4 million calls/mo. by 15k employees)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 5 / 31
Co-sourcing in call centers• Q: What is co-sourcing?
• A: Practice where some calls served byexternal contractor
• Only a portion of the service capacity isprovided “in-house”
• Why co-source?
• Reduce internal staffing costs• Reduce wait time and abandonment• Mitigate capacity constraints• Maintain service at peak demand
• Examples of external contractors:• Convergys Corporation (150 locations)• Wipro (4 million calls/mo. by 15k employees)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 5 / 31
Co-sourcing in call centers• Q: What is co-sourcing?
• A: Practice where some calls served byexternal contractor
• Only a portion of the service capacity isprovided “in-house”
• Why co-source?• Reduce internal staffing costs
• Reduce wait time and abandonment• Mitigate capacity constraints• Maintain service at peak demand
• Examples of external contractors:• Convergys Corporation (150 locations)• Wipro (4 million calls/mo. by 15k employees)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 5 / 31
Co-sourcing in call centers• Q: What is co-sourcing?
• A: Practice where some calls served byexternal contractor
• Only a portion of the service capacity isprovided “in-house”
• Why co-source?• Reduce internal staffing costs• Reduce wait time and abandonment
• Mitigate capacity constraints• Maintain service at peak demand
• Examples of external contractors:• Convergys Corporation (150 locations)• Wipro (4 million calls/mo. by 15k employees)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 5 / 31
Co-sourcing in call centers• Q: What is co-sourcing?
• A: Practice where some calls served byexternal contractor
• Only a portion of the service capacity isprovided “in-house”
• Why co-source?• Reduce internal staffing costs• Reduce wait time and abandonment• Mitigate capacity constraints
• Maintain service at peak demand
• Examples of external contractors:• Convergys Corporation (150 locations)• Wipro (4 million calls/mo. by 15k employees)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 5 / 31
Co-sourcing in call centers• Q: What is co-sourcing?
• A: Practice where some calls served byexternal contractor
• Only a portion of the service capacity isprovided “in-house”
• Why co-source?• Reduce internal staffing costs• Reduce wait time and abandonment• Mitigate capacity constraints• Maintain service at peak demand
• Examples of external contractors:• Convergys Corporation (150 locations)• Wipro (4 million calls/mo. by 15k employees)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 5 / 31
Co-sourcing in call centers• Q: What is co-sourcing?
• A: Practice where some calls served byexternal contractor
• Only a portion of the service capacity isprovided “in-house”
• Why co-source?• Reduce internal staffing costs• Reduce wait time and abandonment• Mitigate capacity constraints• Maintain service at peak demand
• Examples of external contractors:
• Convergys Corporation (150 locations)• Wipro (4 million calls/mo. by 15k employees)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 5 / 31
Co-sourcing in call centers• Q: What is co-sourcing?
• A: Practice where some calls served byexternal contractor
• Only a portion of the service capacity isprovided “in-house”
• Why co-source?• Reduce internal staffing costs• Reduce wait time and abandonment• Mitigate capacity constraints• Maintain service at peak demand
• Examples of external contractors:• Convergys Corporation (150 locations)
• Wipro (4 million calls/mo. by 15k employees)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 5 / 31
Co-sourcing in call centers• Q: What is co-sourcing?
• A: Practice where some calls served byexternal contractor
• Only a portion of the service capacity isprovided “in-house”
• Why co-source?• Reduce internal staffing costs• Reduce wait time and abandonment• Mitigate capacity constraints• Maintain service at peak demand
• Examples of external contractors:• Convergys Corporation (150 locations)• Wipro (4 million calls/mo. by 15k employees)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 5 / 31
Costs associated with co-sourcing
• Preparation costs
• Inferior service quality (real or perceived)
• Less specialized staff
• Less loyal staff
• Reduced operational control
• Must pay the contractor (we assume per call)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 6 / 31
Costs associated with co-sourcing
• Preparation costs
• Inferior service quality (real or perceived)
• Less specialized staff
• Less loyal staff
• Reduced operational control
• Must pay the contractor (we assume per call)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 6 / 31
Costs associated with co-sourcing
• Preparation costs
• Inferior service quality (real or perceived)
• Less specialized staff
• Less loyal staff
• Reduced operational control
• Must pay the contractor (we assume per call)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 6 / 31
Costs associated with co-sourcing
• Preparation costs
• Inferior service quality (real or perceived)
• Less specialized staff
• Less loyal staff
• Reduced operational control
• Must pay the contractor (we assume per call)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 6 / 31
Costs associated with co-sourcing
• Preparation costs
• Inferior service quality (real or perceived)
• Less specialized staff
• Less loyal staff
• Reduced operational control
• Must pay the contractor (we assume per call)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 6 / 31
Costs associated with co-sourcing
• Preparation costs
• Inferior service quality (real or perceived)
• Less specialized staff
• Less loyal staff
• Reduced operational control
• Must pay the contractor (we assume per call)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 6 / 31
Costs associated with co-sourcing
• Preparation costs
• Inferior service quality (real or perceived)
• Less specialized staff
• Less loyal staff
• Reduced operational control
• Must pay the contractor (we assume per call)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 6 / 31
The basic co-sourcing model
Poisson arrivals (λ)
x portion
Call Center’s Agents
...
1− x portion
Contractor’s Agents
...
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 7 / 31
The basic co-sourcing model
Poisson arrivals (λ)
x portion
Call Center’s Agents
...
1− x portion
Contractor’s Agents
...
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 7 / 31
The basic co-sourcing model
Poisson arrivals (λ)
x portion
Call Center’s Agents
...
1− x portion
Contractor’s Agents
...Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 7 / 31
Non-stationary demand
Daily call volumes at a major Pittsburgh healthcare provider
Mirzaeian, Mousavi, and Kharoufeh - POMS 2016
• Demand (λ) depends on time in a systematic fashion
• Co-sourcing decisions (x) should depend on time too
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 8 / 31
Non-stationary demandDaily call volumes at a major Pittsburgh healthcare provider
Mirzaeian, Mousavi, and Kharoufeh - POMS 2016
• Demand (λ) depends on time in a systematic fashion
• Co-sourcing decisions (x) should depend on time too
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 8 / 31
Non-stationary demandDaily call volumes at a major Pittsburgh healthcare provider
Mirzaeian, Mousavi, and Kharoufeh - POMS 2016
• Demand (λ) depends on time in a systematic fashion
• Co-sourcing decisions (x) should depend on time too
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 8 / 31
Non-stationary demandDaily call volumes at a major Pittsburgh healthcare provider
Mirzaeian, Mousavi, and Kharoufeh - POMS 2016
• Demand (λ) depends on time in a systematic fashion
• Co-sourcing decisions (x) should depend on time too
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 8 / 31
Our research questions
We explore a model involving a single call centerand an external contractor in a setting wheredemand changes over time.
• Question 1: How should the market price ofco-sourcing a call change over time alongsidedemand, so as to incentivize the contractor toaccept all calls sent to them?
• Question 2: How much and how often shouldthe call center co-source calls, and how doesthis change over time?
• Question 3: What does an equilibrium (ifany) look like with respect to pricing andco-sourcing decisions?
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 9 / 31
Our research questionsWe explore a model involving a single call centerand an external contractor in a setting wheredemand changes over time.
• Question 1: How should the market price ofco-sourcing a call change over time alongsidedemand, so as to incentivize the contractor toaccept all calls sent to them?
• Question 2: How much and how often shouldthe call center co-source calls, and how doesthis change over time?
• Question 3: What does an equilibrium (ifany) look like with respect to pricing andco-sourcing decisions?
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 9 / 31
Our research questionsWe explore a model involving a single call centerand an external contractor in a setting wheredemand changes over time.
• Question 1: How should the market price ofco-sourcing a call change over time alongsidedemand, so as to incentivize the contractor toaccept all calls sent to them?
• Question 2: How much and how often shouldthe call center co-source calls, and how doesthis change over time?
• Question 3: What does an equilibrium (ifany) look like with respect to pricing andco-sourcing decisions?
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 9 / 31
Our research questionsWe explore a model involving a single call centerand an external contractor in a setting wheredemand changes over time.
• Question 1: How should the market price ofco-sourcing a call change over time alongsidedemand, so as to incentivize the contractor toaccept all calls sent to them?
• Question 2: How much and how often shouldthe call center co-source calls, and how doesthis change over time?
• Question 3: What does an equilibrium (ifany) look like with respect to pricing andco-sourcing decisions?
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 9 / 31
Our research questionsWe explore a model involving a single call centerand an external contractor in a setting wheredemand changes over time.
• Question 1: How should the market price ofco-sourcing a call change over time alongsidedemand, so as to incentivize the contractor toaccept all calls sent to them?
• Question 2: How much and how often shouldthe call center co-source calls, and how doesthis change over time?
• Question 3: What does an equilibrium (ifany) look like with respect to pricing andco-sourcing decisions?
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 9 / 31
Literature & Positioning
• Co-Sourcing in Service Industry• Aksin et al. (2008) - nonstationary demand w/ static
contract design• Gurvich and Perry (2012) - capacity planning / co-sourcing
threshold (no pricing)• Kocaga et al. (2015) - staffing and co-sourcing under static
pricing
• Mirzaeian, Mousavi and Kharoufeh (2016) - closest to our
model but with stationary demand and static prices
• Our fluid framework follows Liu and Whitt (2012)
• Positioning: first look at co-sourcing w/dynamic pricing
• Why it matters: Non-stationarity is morerealistic, and temporary overload gives rise tofeatures absent in stationary settings
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 10 / 31
Literature & Positioning• Co-Sourcing in Service Industry
• Aksin et al. (2008) - nonstationary demand w/ staticcontract design
• Gurvich and Perry (2012) - capacity planning / co-sourcingthreshold (no pricing)
• Kocaga et al. (2015) - staffing and co-sourcing under staticpricing
• Mirzaeian, Mousavi and Kharoufeh (2016) - closest to our
model but with stationary demand and static prices
• Our fluid framework follows Liu and Whitt (2012)
• Positioning: first look at co-sourcing w/dynamic pricing
• Why it matters: Non-stationarity is morerealistic, and temporary overload gives rise tofeatures absent in stationary settings
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 10 / 31
Literature & Positioning• Co-Sourcing in Service Industry
• Aksin et al. (2008) - nonstationary demand w/ staticcontract design
• Gurvich and Perry (2012) - capacity planning / co-sourcingthreshold (no pricing)
• Kocaga et al. (2015) - staffing and co-sourcing under staticpricing
• Mirzaeian, Mousavi and Kharoufeh (2016) - closest to our
model but with stationary demand and static prices
• Our fluid framework follows Liu and Whitt (2012)
• Positioning: first look at co-sourcing w/dynamic pricing
• Why it matters: Non-stationarity is morerealistic, and temporary overload gives rise tofeatures absent in stationary settings
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 10 / 31
Literature & Positioning• Co-Sourcing in Service Industry
• Aksin et al. (2008) - nonstationary demand w/ staticcontract design
• Gurvich and Perry (2012) - capacity planning / co-sourcingthreshold (no pricing)
• Kocaga et al. (2015) - staffing and co-sourcing under staticpricing
• Mirzaeian, Mousavi and Kharoufeh (2016) - closest to our
model but with stationary demand and static prices
• Our fluid framework follows Liu and Whitt (2012)
• Positioning: first look at co-sourcing w/dynamic pricing
• Why it matters: Non-stationarity is morerealistic, and temporary overload gives rise tofeatures absent in stationary settings
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 10 / 31
Literature & Positioning• Co-Sourcing in Service Industry
• Aksin et al. (2008) - nonstationary demand w/ staticcontract design
• Gurvich and Perry (2012) - capacity planning / co-sourcingthreshold (no pricing)
• Kocaga et al. (2015) - staffing and co-sourcing under staticpricing
• Mirzaeian, Mousavi and Kharoufeh (2016) - closest to our
model but with stationary demand and static prices
• Our fluid framework follows Liu and Whitt (2012)
• Positioning: first look at co-sourcing w/dynamic pricing
• Why it matters: Non-stationarity is morerealistic, and temporary overload gives rise tofeatures absent in stationary settings
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 10 / 31
Literature & Positioning• Co-Sourcing in Service Industry
• Aksin et al. (2008) - nonstationary demand w/ staticcontract design
• Gurvich and Perry (2012) - capacity planning / co-sourcingthreshold (no pricing)
• Kocaga et al. (2015) - staffing and co-sourcing under staticpricing
• Mirzaeian, Mousavi and Kharoufeh (2016) - closest to our
model but with stationary demand and static prices
• Our fluid framework follows Liu and Whitt (2012)
• Positioning: first look at co-sourcing w/dynamic pricing
• Why it matters: Non-stationarity is morerealistic, and temporary overload gives rise tofeatures absent in stationary settings
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 10 / 31
Literature & Positioning• Co-Sourcing in Service Industry
• Aksin et al. (2008) - nonstationary demand w/ staticcontract design
• Gurvich and Perry (2012) - capacity planning / co-sourcingthreshold (no pricing)
• Kocaga et al. (2015) - staffing and co-sourcing under staticpricing
• Mirzaeian, Mousavi and Kharoufeh (2016) - closest to our
model but with stationary demand and static prices
• Our fluid framework follows Liu and Whitt (2012)
• Positioning: first look at co-sourcing w/dynamic pricing
• Why it matters: Non-stationarity is morerealistic, and temporary overload gives rise tofeatures absent in stationary settings
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 10 / 31
Literature & Positioning• Co-Sourcing in Service Industry
• Aksin et al. (2008) - nonstationary demand w/ staticcontract design
• Gurvich and Perry (2012) - capacity planning / co-sourcingthreshold (no pricing)
• Kocaga et al. (2015) - staffing and co-sourcing under staticpricing
• Mirzaeian, Mousavi and Kharoufeh (2016) - closest to our
model but with stationary demand and static prices
• Our fluid framework follows Liu and Whitt (2012)
• Positioning: first look at co-sourcing w/dynamic pricing
• Why it matters: Non-stationarity is morerealistic, and temporary overload gives rise tofeatures absent in stationary settings
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 10 / 31
Literature & Positioning• Co-Sourcing in Service Industry
• Aksin et al. (2008) - nonstationary demand w/ staticcontract design
• Gurvich and Perry (2012) - capacity planning / co-sourcingthreshold (no pricing)
• Kocaga et al. (2015) - staffing and co-sourcing under staticpricing
• Mirzaeian, Mousavi and Kharoufeh (2016) - closest to our
model but with stationary demand and static prices
• Our fluid framework follows Liu and Whitt (2012)
• Positioning: first look at co-sourcing w/dynamic pricing
• Why it matters: Non-stationarity is morerealistic, and temporary overload gives rise tofeatures absent in stationary settings
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 10 / 31
Outline / “Plan of attack”
• Present the “stochastic” queueing model
• Present “economic” aspects of our model
• Move from stochastic model to fluid model
• Find equilibrium pricing given allocation
• Find optimal co-sourcing allocation given prices
• Prove existence of a price-allocation equilibrium(work in progress)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 11 / 31
Outline / “Plan of attack”
• Present the “stochastic” queueing model
• Present “economic” aspects of our model
• Move from stochastic model to fluid model
• Find equilibrium pricing given allocation
• Find optimal co-sourcing allocation given prices
• Prove existence of a price-allocation equilibrium(work in progress)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 11 / 31
Outline / “Plan of attack”
• Present the “stochastic” queueing model
• Present “economic” aspects of our model
• Move from stochastic model to fluid model
• Find equilibrium pricing given allocation
• Find optimal co-sourcing allocation given prices
• Prove existence of a price-allocation equilibrium(work in progress)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 11 / 31
Outline / “Plan of attack”
• Present the “stochastic” queueing model
• Present “economic” aspects of our model
• Move from stochastic model to fluid model
• Find equilibrium pricing given allocation
• Find optimal co-sourcing allocation given prices
• Prove existence of a price-allocation equilibrium(work in progress)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 11 / 31
Outline / “Plan of attack”
• Present the “stochastic” queueing model
• Present “economic” aspects of our model
• Move from stochastic model to fluid model
• Find equilibrium pricing given allocation
• Find optimal co-sourcing allocation given prices
• Prove existence of a price-allocation equilibrium(work in progress)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 11 / 31
Outline / “Plan of attack”
• Present the “stochastic” queueing model
• Present “economic” aspects of our model
• Move from stochastic model to fluid model
• Find equilibrium pricing given allocation
• Find optimal co-sourcing allocation given prices
• Prove existence of a price-allocation equilibrium(work in progress)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 11 / 31
Outline / “Plan of attack”
• Present the “stochastic” queueing model
• Present “economic” aspects of our model
• Move from stochastic model to fluid model
• Find equilibrium pricing given allocation
• Find optimal co-sourcing allocation given prices
• Prove existence of a price-allocation equilibrium(work in progress)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 11 / 31
The Mt/M/s +M queueing model
• Non-stationary Poisson arrivals w/ rate λ(t)
• All other distributions are exponential
• Call center has s servers each w/ rate µ = 1
• Contractor has ∞ servers each w/ rate µc• Calls abandon the queue w/ rate θ
• Also known as the Erlang-A model
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 12 / 31
The Mt/M/s +M queueing model
• Non-stationary Poisson arrivals w/ rate λ(t)
• All other distributions are exponential
• Call center has s servers each w/ rate µ = 1
• Contractor has ∞ servers each w/ rate µc• Calls abandon the queue w/ rate θ
• Also known as the Erlang-A model
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 12 / 31
The Mt/M/s +M queueing model
• Non-stationary Poisson arrivals w/ rate λ(t)
• All other distributions are exponential
• Call center has s servers each w/ rate µ = 1
• Contractor has ∞ servers each w/ rate µc• Calls abandon the queue w/ rate θ
• Also known as the Erlang-A model
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 12 / 31
The Mt/M/s +M queueing model
• Non-stationary Poisson arrivals w/ rate λ(t)
• All other distributions are exponential
• Call center has s servers each w/ rate µ = 1
• Contractor has ∞ servers each w/ rate µc• Calls abandon the queue w/ rate θ
• Also known as the Erlang-A model
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 12 / 31
The Mt/M/s +M queueing model
• Non-stationary Poisson arrivals w/ rate λ(t)
• All other distributions are exponential
• Call center has s servers each w/ rate µ = 1
• Contractor has ∞ servers each w/ rate µc
• Calls abandon the queue w/ rate θ
• Also known as the Erlang-A model
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 12 / 31
The Mt/M/s +M queueing model
• Non-stationary Poisson arrivals w/ rate λ(t)
• All other distributions are exponential
• Call center has s servers each w/ rate µ = 1
• Contractor has ∞ servers each w/ rate µc• Calls abandon the queue w/ rate θ
• Also known as the Erlang-A model
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 12 / 31
The Mt/M/s +M queueing model
• Non-stationary Poisson arrivals w/ rate λ(t)
• All other distributions are exponential
• Call center has s servers each w/ rate µ = 1
• Contractor has ∞ servers each w/ rate µc• Calls abandon the queue w/ rate θ
• Also known as the Erlang-A model
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 12 / 31
The Contract
• Let x(t) be the proportion of calls the callcenter accepts at time t
• We call x(t) the allocation.
• All unaccepted calls are co-sourced
• At time t, call center faces arrival rate x(t)λ(t)
• Meanwhile, contractor faces (1− x(t))λ(t)
• Call center pays contractor for co-sourcing
• Market price of co-sourcing at time t: P (t)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 13 / 31
The Contract
• Let x(t) be the proportion of calls the callcenter accepts at time t
• We call x(t) the allocation.
• All unaccepted calls are co-sourced
• At time t, call center faces arrival rate x(t)λ(t)
• Meanwhile, contractor faces (1− x(t))λ(t)
• Call center pays contractor for co-sourcing
• Market price of co-sourcing at time t: P (t)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 13 / 31
The Contract
• Let x(t) be the proportion of calls the callcenter accepts at time t
• We call x(t) the allocation.
• All unaccepted calls are co-sourced
• At time t, call center faces arrival rate x(t)λ(t)
• Meanwhile, contractor faces (1− x(t))λ(t)
• Call center pays contractor for co-sourcing
• Market price of co-sourcing at time t: P (t)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 13 / 31
The Contract
• Let x(t) be the proportion of calls the callcenter accepts at time t
• We call x(t) the allocation.
• All unaccepted calls are co-sourced
• At time t, call center faces arrival rate x(t)λ(t)
• Meanwhile, contractor faces (1− x(t))λ(t)
• Call center pays contractor for co-sourcing
• Market price of co-sourcing at time t: P (t)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 13 / 31
The Contract
• Let x(t) be the proportion of calls the callcenter accepts at time t
• We call x(t) the allocation.
• All unaccepted calls are co-sourced
• At time t, call center faces arrival rate x(t)λ(t)
• Meanwhile, contractor faces (1− x(t))λ(t)
• Call center pays contractor for co-sourcing
• Market price of co-sourcing at time t: P (t)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 13 / 31
The Contract
• Let x(t) be the proportion of calls the callcenter accepts at time t
• We call x(t) the allocation.
• All unaccepted calls are co-sourced
• At time t, call center faces arrival rate x(t)λ(t)
• Meanwhile, contractor faces (1− x(t))λ(t)
• Call center pays contractor for co-sourcing
• Market price of co-sourcing at time t: P (t)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 13 / 31
The Contract
• Let x(t) be the proportion of calls the callcenter accepts at time t
• We call x(t) the allocation.
• All unaccepted calls are co-sourced
• At time t, call center faces arrival rate x(t)λ(t)
• Meanwhile, contractor faces (1− x(t))λ(t)
• Call center pays contractor for co-sourcing
• Market price of co-sourcing at time t: P (t)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 13 / 31
The Contract
• Let x(t) be the proportion of calls the callcenter accepts at time t
• We call x(t) the allocation.
• All unaccepted calls are co-sourced
• At time t, call center faces arrival rate x(t)λ(t)
• Meanwhile, contractor faces (1− x(t))λ(t)
• Call center pays contractor for co-sourcing
• Market price of co-sourcing at time t: P (t)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 13 / 31
Economic features of the model
• The call center• receives a reward R per call received (anywhere),• less a staffing expenses CI per call served in-house,• less waiting costs Cw per unit time per call queued,• less abandonment costs Cab per call abandoned,• and pays P (t) to co-source at time t.
• The contractor• receives P (t) per call accepted at time t,• but experiences a cost ψ(N(t)) per unit time,
• where the contractor is serving N(t) calls at time t,• and ψ(N(t)) = αN(t) + βN(t)2; α, β > 0.
• Both sides want to maximize their earningsover a finite time horizon: [0, T ]
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 14 / 31
Economic features of the model
• The call center
• receives a reward R per call received (anywhere),• less a staffing expenses CI per call served in-house,• less waiting costs Cw per unit time per call queued,• less abandonment costs Cab per call abandoned,• and pays P (t) to co-source at time t.
• The contractor• receives P (t) per call accepted at time t,• but experiences a cost ψ(N(t)) per unit time,
• where the contractor is serving N(t) calls at time t,• and ψ(N(t)) = αN(t) + βN(t)2; α, β > 0.
• Both sides want to maximize their earningsover a finite time horizon: [0, T ]
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 14 / 31
Economic features of the model
• The call center• receives a reward R per call received (anywhere),
• less a staffing expenses CI per call served in-house,• less waiting costs Cw per unit time per call queued,• less abandonment costs Cab per call abandoned,• and pays P (t) to co-source at time t.
• The contractor• receives P (t) per call accepted at time t,• but experiences a cost ψ(N(t)) per unit time,
• where the contractor is serving N(t) calls at time t,• and ψ(N(t)) = αN(t) + βN(t)2; α, β > 0.
• Both sides want to maximize their earningsover a finite time horizon: [0, T ]
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 14 / 31
Economic features of the model
• The call center• receives a reward R per call received (anywhere),• less a staffing expenses CI per call served in-house,
• less waiting costs Cw per unit time per call queued,• less abandonment costs Cab per call abandoned,• and pays P (t) to co-source at time t.
• The contractor• receives P (t) per call accepted at time t,• but experiences a cost ψ(N(t)) per unit time,
• where the contractor is serving N(t) calls at time t,• and ψ(N(t)) = αN(t) + βN(t)2; α, β > 0.
• Both sides want to maximize their earningsover a finite time horizon: [0, T ]
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 14 / 31
Economic features of the model
• The call center• receives a reward R per call received (anywhere),• less a staffing expenses CI per call served in-house,• less waiting costs Cw per unit time per call queued,
• less abandonment costs Cab per call abandoned,• and pays P (t) to co-source at time t.
• The contractor• receives P (t) per call accepted at time t,• but experiences a cost ψ(N(t)) per unit time,
• where the contractor is serving N(t) calls at time t,• and ψ(N(t)) = αN(t) + βN(t)2; α, β > 0.
• Both sides want to maximize their earningsover a finite time horizon: [0, T ]
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 14 / 31
Economic features of the model
• The call center• receives a reward R per call received (anywhere),• less a staffing expenses CI per call served in-house,• less waiting costs Cw per unit time per call queued,• less abandonment costs Cab per call abandoned,
• and pays P (t) to co-source at time t.
• The contractor• receives P (t) per call accepted at time t,• but experiences a cost ψ(N(t)) per unit time,
• where the contractor is serving N(t) calls at time t,• and ψ(N(t)) = αN(t) + βN(t)2; α, β > 0.
• Both sides want to maximize their earningsover a finite time horizon: [0, T ]
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 14 / 31
Economic features of the model
• The call center• receives a reward R per call received (anywhere),• less a staffing expenses CI per call served in-house,• less waiting costs Cw per unit time per call queued,• less abandonment costs Cab per call abandoned,• and pays P (t) to co-source at time t.
• The contractor• receives P (t) per call accepted at time t,• but experiences a cost ψ(N(t)) per unit time,
• where the contractor is serving N(t) calls at time t,• and ψ(N(t)) = αN(t) + βN(t)2; α, β > 0.
• Both sides want to maximize their earningsover a finite time horizon: [0, T ]
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 14 / 31
Economic features of the model
• The call center• receives a reward R per call received (anywhere),• less a staffing expenses CI per call served in-house,• less waiting costs Cw per unit time per call queued,• less abandonment costs Cab per call abandoned,• and pays P (t) to co-source at time t.
• The contractor
• receives P (t) per call accepted at time t,• but experiences a cost ψ(N(t)) per unit time,
• where the contractor is serving N(t) calls at time t,• and ψ(N(t)) = αN(t) + βN(t)2; α, β > 0.
• Both sides want to maximize their earningsover a finite time horizon: [0, T ]
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 14 / 31
Economic features of the model
• The call center• receives a reward R per call received (anywhere),• less a staffing expenses CI per call served in-house,• less waiting costs Cw per unit time per call queued,• less abandonment costs Cab per call abandoned,• and pays P (t) to co-source at time t.
• The contractor• receives P (t) per call accepted at time t,
• but experiences a cost ψ(N(t)) per unit time,• where the contractor is serving N(t) calls at time t,• and ψ(N(t)) = αN(t) + βN(t)2; α, β > 0.
• Both sides want to maximize their earningsover a finite time horizon: [0, T ]
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 14 / 31
Economic features of the model
• The call center• receives a reward R per call received (anywhere),• less a staffing expenses CI per call served in-house,• less waiting costs Cw per unit time per call queued,• less abandonment costs Cab per call abandoned,• and pays P (t) to co-source at time t.
• The contractor• receives P (t) per call accepted at time t,• but experiences a cost ψ(N(t)) per unit time,
• where the contractor is serving N(t) calls at time t,• and ψ(N(t)) = αN(t) + βN(t)2; α, β > 0.
• Both sides want to maximize their earningsover a finite time horizon: [0, T ]
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 14 / 31
Economic features of the model
• The call center• receives a reward R per call received (anywhere),• less a staffing expenses CI per call served in-house,• less waiting costs Cw per unit time per call queued,• less abandonment costs Cab per call abandoned,• and pays P (t) to co-source at time t.
• The contractor• receives P (t) per call accepted at time t,• but experiences a cost ψ(N(t)) per unit time,
• where the contractor is serving N(t) calls at time t,
• and ψ(N(t)) = αN(t) + βN(t)2; α, β > 0.
• Both sides want to maximize their earningsover a finite time horizon: [0, T ]
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 14 / 31
Economic features of the model
• The call center• receives a reward R per call received (anywhere),• less a staffing expenses CI per call served in-house,• less waiting costs Cw per unit time per call queued,• less abandonment costs Cab per call abandoned,• and pays P (t) to co-source at time t.
• The contractor• receives P (t) per call accepted at time t,• but experiences a cost ψ(N(t)) per unit time,
• where the contractor is serving N(t) calls at time t,• and ψ(N(t)) = αN(t) + βN(t)2; α, β > 0.
• Both sides want to maximize their earningsover a finite time horizon: [0, T ]
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 14 / 31
Economic features of the model
• The call center• receives a reward R per call received (anywhere),• less a staffing expenses CI per call served in-house,• less waiting costs Cw per unit time per call queued,• less abandonment costs Cab per call abandoned,• and pays P (t) to co-source at time t.
• The contractor• receives P (t) per call accepted at time t,• but experiences a cost ψ(N(t)) per unit time,
• where the contractor is serving N(t) calls at time t,• and ψ(N(t)) = αN(t) + βN(t)2; α, β > 0.
• Both sides want to maximize their earningsover a finite time horizon: [0, T ]
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 14 / 31
The fluid model
• Imagine the Mt/M/1 system
• Consider arrival rate nλ(t) with ns servers
• Let n→∞• Stochastic variation vanishes; deterministic
non-stationarity variation dominates
• Fluid builds w/ rate λ(t) and drains w/ rate s
• Limit approximation for the stochastic systemor model in its own right
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 15 / 31
The fluid model
• Imagine the Mt/M/1 system
• Consider arrival rate nλ(t) with ns servers
• Let n→∞• Stochastic variation vanishes; deterministic
non-stationarity variation dominates
• Fluid builds w/ rate λ(t) and drains w/ rate s
• Limit approximation for the stochastic systemor model in its own right
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 15 / 31
The fluid model
• Imagine the Mt/M/1 system
• Consider arrival rate nλ(t) with ns servers
• Let n→∞• Stochastic variation vanishes; deterministic
non-stationarity variation dominates
• Fluid builds w/ rate λ(t) and drains w/ rate s
• Limit approximation for the stochastic systemor model in its own right
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 15 / 31
The fluid model
• Imagine the Mt/M/1 system
• Consider arrival rate nλ(t) with ns servers
• Let n→∞
• Stochastic variation vanishes; deterministicnon-stationarity variation dominates
• Fluid builds w/ rate λ(t) and drains w/ rate s
• Limit approximation for the stochastic systemor model in its own right
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 15 / 31
The fluid model
• Imagine the Mt/M/1 system
• Consider arrival rate nλ(t) with ns servers
• Let n→∞• Stochastic variation vanishes; deterministic
non-stationarity variation dominates
• Fluid builds w/ rate λ(t) and drains w/ rate s
• Limit approximation for the stochastic systemor model in its own right
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 15 / 31
The fluid model
• Imagine the Mt/M/1 system
• Consider arrival rate nλ(t) with ns servers
• Let n→∞• Stochastic variation vanishes; deterministic
non-stationarity variation dominates
• Fluid builds w/ rate λ(t) and drains w/ rate s
• Limit approximation for the stochastic systemor model in its own right
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 15 / 31
The fluid model
• Imagine the Mt/M/1 system
• Consider arrival rate nλ(t) with ns servers
• Let n→∞• Stochastic variation vanishes; deterministic
non-stationarity variation dominates
• Fluid builds w/ rate λ(t) and drains w/ rate s
• Limit approximation for the stochastic systemor model in its own right
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 15 / 31
The fluid model with abandonments
• Consider Mt/M/1 +M w/ finite horizon [0, T ]
• Think of fluid as a continuous quantity
• Fluid arrives to server if possible, queues othw.
• Served only at server; abandons only in queue
• Stochastic distributions act as proportions
• B(t) ∈ [0, s] fluid in server at time t
• After v time, e−µvB(t) of this fluid remains
• Q(t) ∈ [0,∞) fluid in queue at time t
• Assuming no service (for illustration), after vtime e−θvQ(t) of this fluid remains
• Q(t) > 0 implies B(t) = s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 16 / 31
The fluid model with abandonments• Consider Mt/M/1 +M w/ finite horizon [0, T ]
• Think of fluid as a continuous quantity
• Fluid arrives to server if possible, queues othw.
• Served only at server; abandons only in queue
• Stochastic distributions act as proportions
• B(t) ∈ [0, s] fluid in server at time t
• After v time, e−µvB(t) of this fluid remains
• Q(t) ∈ [0,∞) fluid in queue at time t
• Assuming no service (for illustration), after vtime e−θvQ(t) of this fluid remains
• Q(t) > 0 implies B(t) = s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 16 / 31
The fluid model with abandonments• Consider Mt/M/1 +M w/ finite horizon [0, T ]
• Think of fluid as a continuous quantity
• Fluid arrives to server if possible, queues othw.
• Served only at server; abandons only in queue
• Stochastic distributions act as proportions
• B(t) ∈ [0, s] fluid in server at time t
• After v time, e−µvB(t) of this fluid remains
• Q(t) ∈ [0,∞) fluid in queue at time t
• Assuming no service (for illustration), after vtime e−θvQ(t) of this fluid remains
• Q(t) > 0 implies B(t) = s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 16 / 31
The fluid model with abandonments• Consider Mt/M/1 +M w/ finite horizon [0, T ]
• Think of fluid as a continuous quantity
• Fluid arrives to server if possible, queues othw.
• Served only at server; abandons only in queue
• Stochastic distributions act as proportions
• B(t) ∈ [0, s] fluid in server at time t
• After v time, e−µvB(t) of this fluid remains
• Q(t) ∈ [0,∞) fluid in queue at time t
• Assuming no service (for illustration), after vtime e−θvQ(t) of this fluid remains
• Q(t) > 0 implies B(t) = s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 16 / 31
The fluid model with abandonments• Consider Mt/M/1 +M w/ finite horizon [0, T ]
• Think of fluid as a continuous quantity
• Fluid arrives to server if possible, queues othw.
• Served only at server; abandons only in queue
• Stochastic distributions act as proportions
• B(t) ∈ [0, s] fluid in server at time t
• After v time, e−µvB(t) of this fluid remains
• Q(t) ∈ [0,∞) fluid in queue at time t
• Assuming no service (for illustration), after vtime e−θvQ(t) of this fluid remains
• Q(t) > 0 implies B(t) = s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 16 / 31
The fluid model with abandonments• Consider Mt/M/1 +M w/ finite horizon [0, T ]
• Think of fluid as a continuous quantity
• Fluid arrives to server if possible, queues othw.
• Served only at server; abandons only in queue
• Stochastic distributions act as proportions
• B(t) ∈ [0, s] fluid in server at time t
• After v time, e−µvB(t) of this fluid remains
• Q(t) ∈ [0,∞) fluid in queue at time t
• Assuming no service (for illustration), after vtime e−θvQ(t) of this fluid remains
• Q(t) > 0 implies B(t) = s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 16 / 31
The fluid model with abandonments• Consider Mt/M/1 +M w/ finite horizon [0, T ]
• Think of fluid as a continuous quantity
• Fluid arrives to server if possible, queues othw.
• Served only at server; abandons only in queue
• Stochastic distributions act as proportions
• B(t) ∈ [0, s] fluid in server at time t
• After v time, e−µvB(t) of this fluid remains
• Q(t) ∈ [0,∞) fluid in queue at time t
• Assuming no service (for illustration), after vtime e−θvQ(t) of this fluid remains
• Q(t) > 0 implies B(t) = s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 16 / 31
The fluid model with abandonments• Consider Mt/M/1 +M w/ finite horizon [0, T ]
• Think of fluid as a continuous quantity
• Fluid arrives to server if possible, queues othw.
• Served only at server; abandons only in queue
• Stochastic distributions act as proportions
• B(t) ∈ [0, s] fluid in server at time t
• After v time, e−µvB(t) of this fluid remains
• Q(t) ∈ [0,∞) fluid in queue at time t
• Assuming no service (for illustration), after vtime e−θvQ(t) of this fluid remains
• Q(t) > 0 implies B(t) = s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 16 / 31
The fluid model with abandonments• Consider Mt/M/1 +M w/ finite horizon [0, T ]
• Think of fluid as a continuous quantity
• Fluid arrives to server if possible, queues othw.
• Served only at server; abandons only in queue
• Stochastic distributions act as proportions
• B(t) ∈ [0, s] fluid in server at time t
• After v time, e−µvB(t) of this fluid remains
• Q(t) ∈ [0,∞) fluid in queue at time t
• Assuming no service (for illustration), after vtime e−θvQ(t) of this fluid remains
• Q(t) > 0 implies B(t) = s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 16 / 31
The fluid model with abandonments• Consider Mt/M/1 +M w/ finite horizon [0, T ]
• Think of fluid as a continuous quantity
• Fluid arrives to server if possible, queues othw.
• Served only at server; abandons only in queue
• Stochastic distributions act as proportions
• B(t) ∈ [0, s] fluid in server at time t
• After v time, e−µvB(t) of this fluid remains
• Q(t) ∈ [0,∞) fluid in queue at time t
• Assuming no service (for illustration), after vtime e−θvQ(t) of this fluid remains
• Q(t) > 0 implies B(t) = s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 16 / 31
The fluid model with abandonments• Consider Mt/M/1 +M w/ finite horizon [0, T ]
• Think of fluid as a continuous quantity
• Fluid arrives to server if possible, queues othw.
• Served only at server; abandons only in queue
• Stochastic distributions act as proportions
• B(t) ∈ [0, s] fluid in server at time t
• After v time, e−µvB(t) of this fluid remains
• Q(t) ∈ [0,∞) fluid in queue at time t
• Assuming no service (for illustration), after vtime e−θvQ(t) of this fluid remains
• Q(t) > 0 implies B(t) = s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 16 / 31
Call center fluid model illustration
Q(t) B(t)
x(t)λ(t)
drains at rate θQ(t) drains at rate µB(t) = B(t)
replaces drained fluid at
server when possible
capacity s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 17 / 31
Call center fluid model illustration
Q(t) B(t)x(t)λ(t)
drains at rate θQ(t) drains at rate µB(t) = B(t)
replaces drained fluid at
server when possible
capacity s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 17 / 31
Call center fluid model illustration
Q(t) B(t)x(t)λ(t)
drains at rate θQ(t)
drains at rate µB(t) = B(t)
replaces drained fluid at
server when possible
capacity s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 17 / 31
Call center fluid model illustration
Q(t) B(t)x(t)λ(t)
drains at rate θQ(t) drains at rate µB(t) = B(t)
replaces drained fluid at
server when possible
capacity s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 17 / 31
Call center fluid model illustration
Q(t) B(t)x(t)λ(t)
drains at rate θQ(t) drains at rate µB(t) = B(t)
replaces drained fluid at
server when possible
capacity s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 17 / 31
Call center fluid model illustration
Q(t) B(t)x(t)λ(t)
drains at rate θQ(t) drains at rate µB(t) = B(t)
replaces drained fluid at
server when possible
capacity s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 17 / 31
Fluid model assumptions
• Finite time horizon: [0, T ]
• Technical assumptions on λ(t) (continuity, etc.)
• Fluid is scheduled “first-come-first-served”
• System alternates between underloaded andoverloaded time intervals finitely many times(we will make this clear soon)
• Contractor model is still stochastic(because quadratic cost implies variancematters)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 18 / 31
Fluid model assumptions
• Finite time horizon: [0, T ]
• Technical assumptions on λ(t) (continuity, etc.)
• Fluid is scheduled “first-come-first-served”
• System alternates between underloaded andoverloaded time intervals finitely many times(we will make this clear soon)
• Contractor model is still stochastic(because quadratic cost implies variancematters)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 18 / 31
Fluid model assumptions
• Finite time horizon: [0, T ]
• Technical assumptions on λ(t) (continuity, etc.)
• Fluid is scheduled “first-come-first-served”
• System alternates between underloaded andoverloaded time intervals finitely many times(we will make this clear soon)
• Contractor model is still stochastic(because quadratic cost implies variancematters)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 18 / 31
Fluid model assumptions
• Finite time horizon: [0, T ]
• Technical assumptions on λ(t) (continuity, etc.)
• Fluid is scheduled “first-come-first-served”
• System alternates between underloaded andoverloaded time intervals finitely many times(we will make this clear soon)
• Contractor model is still stochastic(because quadratic cost implies variancematters)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 18 / 31
Fluid model assumptions
• Finite time horizon: [0, T ]
• Technical assumptions on λ(t) (continuity, etc.)
• Fluid is scheduled “first-come-first-served”
• System alternates between underloaded andoverloaded time intervals finitely many times(we will make this clear soon)
• Contractor model is still stochastic(because quadratic cost implies variancematters)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 18 / 31
Fluid model assumptions
• Finite time horizon: [0, T ]
• Technical assumptions on λ(t) (continuity, etc.)
• Fluid is scheduled “first-come-first-served”
• System alternates between underloaded andoverloaded time intervals finitely many times(we will make this clear soon)
• Contractor model is still stochastic(because quadratic cost implies variancematters)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 18 / 31
The contractor’s problem
• Contractor modeled w/ stochastic Mt/M/∞• In reality a contractor can accept an arrival
rate λe(t),
• where λe(t) ∈ [0, (1− x(t))λ(t))] for all t.
• Given an allocation function, x(t), we callprices P (t) market clearing, if
• it is in the contractor’s best interest to acceptall offered calls, that is if
maxλe(t)
1
T
∫ T
0
(P (t)λe(t)− E[αN(t) + βN(t)2]
)dt
• is solved at λe(t) = (1− x(t))λ(t).
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 19 / 31
The contractor’s problem• Contractor modeled w/ stochastic Mt/M/∞
• In reality a contractor can accept an arrivalrate λe(t),
• where λe(t) ∈ [0, (1− x(t))λ(t))] for all t.
• Given an allocation function, x(t), we callprices P (t) market clearing, if
• it is in the contractor’s best interest to acceptall offered calls, that is if
maxλe(t)
1
T
∫ T
0
(P (t)λe(t)− E[αN(t) + βN(t)2]
)dt
• is solved at λe(t) = (1− x(t))λ(t).
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 19 / 31
The contractor’s problem• Contractor modeled w/ stochastic Mt/M/∞• In reality a contractor can accept an arrival
rate λe(t),
• where λe(t) ∈ [0, (1− x(t))λ(t))] for all t.
• Given an allocation function, x(t), we callprices P (t) market clearing, if
• it is in the contractor’s best interest to acceptall offered calls, that is if
maxλe(t)
1
T
∫ T
0
(P (t)λe(t)− E[αN(t) + βN(t)2]
)dt
• is solved at λe(t) = (1− x(t))λ(t).
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 19 / 31
The contractor’s problem• Contractor modeled w/ stochastic Mt/M/∞• In reality a contractor can accept an arrival
rate λe(t),
• where λe(t) ∈ [0, (1− x(t))λ(t))] for all t.
• Given an allocation function, x(t), we callprices P (t) market clearing, if
• it is in the contractor’s best interest to acceptall offered calls, that is if
maxλe(t)
1
T
∫ T
0
(P (t)λe(t)− E[αN(t) + βN(t)2]
)dt
• is solved at λe(t) = (1− x(t))λ(t).
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 19 / 31
The contractor’s problem• Contractor modeled w/ stochastic Mt/M/∞• In reality a contractor can accept an arrival
rate λe(t),
• where λe(t) ∈ [0, (1− x(t))λ(t))] for all t.
• Given an allocation function, x(t), we callprices P (t) market clearing, if
• it is in the contractor’s best interest to acceptall offered calls, that is if
maxλe(t)
1
T
∫ T
0
(P (t)λe(t)− E[αN(t) + βN(t)2]
)dt
• is solved at λe(t) = (1− x(t))λ(t).
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 19 / 31
The contractor’s problem• Contractor modeled w/ stochastic Mt/M/∞• In reality a contractor can accept an arrival
rate λe(t),
• where λe(t) ∈ [0, (1− x(t))λ(t))] for all t.
• Given an allocation function, x(t), we callprices P (t) market clearing, if
• it is in the contractor’s best interest to acceptall offered calls, that is if
maxλe(t)
1
T
∫ T
0
(P (t)λe(t)− E[αN(t) + βN(t)2]
)dt
• is solved at λe(t) = (1− x(t))λ(t).
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 19 / 31
The contractor’s problem• Contractor modeled w/ stochastic Mt/M/∞• In reality a contractor can accept an arrival
rate λe(t),
• where λe(t) ∈ [0, (1− x(t))λ(t))] for all t.
• Given an allocation function, x(t), we callprices P (t) market clearing, if
• it is in the contractor’s best interest to acceptall offered calls, that is if
maxλe(t)
1
T
∫ T
0
(P (t)λe(t)− E[αN(t) + βN(t)2]
)dt
• is solved at λe(t) = (1− x(t))λ(t).
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 19 / 31
The contractor’s problem• Contractor modeled w/ stochastic Mt/M/∞• In reality a contractor can accept an arrival
rate λe(t),
• where λe(t) ∈ [0, (1− x(t))λ(t))] for all t.
• Given an allocation function, x(t), we callprices P (t) market clearing, if
• it is in the contractor’s best interest to acceptall offered calls, that is if
maxλe(t)
1
T
∫ T
0
(P (t)λe(t)− E[αN(t) + βN(t)2]
)dt
• is solved at λe(t) = (1− x(t))λ(t).
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 19 / 31
Solving the contractor’s problem
• maxλe(t)
1
T
∫ T
0
(P (t)λe(t)− E[αN(t) + βN(t)2]
)dt
• Idea: Let h(t) ≡ E[N(t)] =∫ t
0λe(t− v)e−µcv dv.
• Moreover, N(t) is Poisson distributed, so
E[αN(t) + βN(t)2] = (α + β)h(t) + βh(t)2
• maxλe(t)
[P (t)(h′(t) + µch(t))− (α + β)h(t)− βh(t)2
]• λe(t) = 1
2βmax{0, µ2
cP (t)− P ′′(t)− (α + β)µc}• Solving the differential equation for P (t), we have:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
tλe(r) sinh((r − t)µc)dr
• P (t) is market-clearing when λe(t) = (1− x(t))λ(t).
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 20 / 31
Solving the contractor’s problem• max
λe(t)
1
T
∫ T
0
(P (t)λe(t)− E[αN(t) + βN(t)2]
)dt
• Idea: Let h(t) ≡ E[N(t)] =∫ t
0λe(t− v)e−µcv dv.
• Moreover, N(t) is Poisson distributed, so
E[αN(t) + βN(t)2] = (α + β)h(t) + βh(t)2
• maxλe(t)
[P (t)(h′(t) + µch(t))− (α + β)h(t)− βh(t)2
]• λe(t) = 1
2βmax{0, µ2
cP (t)− P ′′(t)− (α + β)µc}• Solving the differential equation for P (t), we have:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
tλe(r) sinh((r − t)µc)dr
• P (t) is market-clearing when λe(t) = (1− x(t))λ(t).
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 20 / 31
Solving the contractor’s problem• max
λe(t)
1
T
∫ T
0
(P (t)λe(t)− E[αN(t) + βN(t)2]
)dt
• Idea: Let h(t) ≡ E[N(t)] =∫ t
0λe(t− v)e−µcv dv.
• Moreover, N(t) is Poisson distributed, so
E[αN(t) + βN(t)2] = (α + β)h(t) + βh(t)2
• maxλe(t)
[P (t)(h′(t) + µch(t))− (α + β)h(t)− βh(t)2
]• λe(t) = 1
2βmax{0, µ2
cP (t)− P ′′(t)− (α + β)µc}• Solving the differential equation for P (t), we have:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
tλe(r) sinh((r − t)µc)dr
• P (t) is market-clearing when λe(t) = (1− x(t))λ(t).
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 20 / 31
Solving the contractor’s problem• max
λe(t)
1
T
∫ T
0
(P (t)λe(t)− E[αN(t) + βN(t)2]
)dt
• Idea: Let h(t) ≡ E[N(t)] =∫ t
0λe(t− v)e−µcv dv.
• Moreover, N(t) is Poisson distributed, so
E[αN(t) + βN(t)2] = (α + β)h(t) + βh(t)2
• maxλe(t)
[P (t)(h′(t) + µch(t))− (α + β)h(t)− βh(t)2
]• λe(t) = 1
2βmax{0, µ2
cP (t)− P ′′(t)− (α + β)µc}• Solving the differential equation for P (t), we have:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
tλe(r) sinh((r − t)µc)dr
• P (t) is market-clearing when λe(t) = (1− x(t))λ(t).
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 20 / 31
Solving the contractor’s problem• max
λe(t)
1
T
∫ T
0
(P (t)λe(t)− E[αN(t) + βN(t)2]
)dt
• Idea: Let h(t) ≡ E[N(t)] =∫ t
0λe(t− v)e−µcv dv.
• Moreover, N(t) is Poisson distributed, so
E[αN(t) + βN(t)2] = (α + β)h(t) + βh(t)2
• maxλe(t)
[P (t)(h′(t) + µch(t))− (α + β)h(t)− βh(t)2
]
• λe(t) = 12β
max{0, µ2cP (t)− P ′′(t)− (α + β)µc}
• Solving the differential equation for P (t), we have:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
tλe(r) sinh((r − t)µc)dr
• P (t) is market-clearing when λe(t) = (1− x(t))λ(t).
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 20 / 31
Solving the contractor’s problem• max
λe(t)
1
T
∫ T
0
(P (t)λe(t)− E[αN(t) + βN(t)2]
)dt
• Idea: Let h(t) ≡ E[N(t)] =∫ t
0λe(t− v)e−µcv dv.
• Moreover, N(t) is Poisson distributed, so
E[αN(t) + βN(t)2] = (α + β)h(t) + βh(t)2
• maxλe(t)
[P (t)(h′(t) + µch(t))− (α + β)h(t)− βh(t)2
]• λe(t) = 1
2βmax{0, µ2
cP (t)− P ′′(t)− (α + β)µc}
• Solving the differential equation for P (t), we have:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
tλe(r) sinh((r − t)µc)dr
• P (t) is market-clearing when λe(t) = (1− x(t))λ(t).
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 20 / 31
Solving the contractor’s problem• max
λe(t)
1
T
∫ T
0
(P (t)λe(t)− E[αN(t) + βN(t)2]
)dt
• Idea: Let h(t) ≡ E[N(t)] =∫ t
0λe(t− v)e−µcv dv.
• Moreover, N(t) is Poisson distributed, so
E[αN(t) + βN(t)2] = (α + β)h(t) + βh(t)2
• maxλe(t)
[P (t)(h′(t) + µch(t))− (α + β)h(t)− βh(t)2
]• λe(t) = 1
2βmax{0, µ2
cP (t)− P ′′(t)− (α + β)µc}• Solving the differential equation for P (t), we have:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
tλe(r) sinh((r − t)µc)dr
• P (t) is market-clearing when λe(t) = (1− x(t))λ(t).
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 20 / 31
Solving the contractor’s problem• max
λe(t)
1
T
∫ T
0
(P (t)λe(t)− E[αN(t) + βN(t)2]
)dt
• Idea: Let h(t) ≡ E[N(t)] =∫ t
0λe(t− v)e−µcv dv.
• Moreover, N(t) is Poisson distributed, so
E[αN(t) + βN(t)2] = (α + β)h(t) + βh(t)2
• maxλe(t)
[P (t)(h′(t) + µch(t))− (α + β)h(t)− βh(t)2
]• λe(t) = 1
2βmax{0, µ2
cP (t)− P ′′(t)− (α + β)µc}• Solving the differential equation for P (t), we have:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
tλe(r) sinh((r − t)µc)dr
• P (t) is market-clearing when λe(t) = (1− x(t))λ(t).
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 20 / 31
Solving the contractor’s problem• max
λe(t)
1
T
∫ T
0
(P (t)λe(t)− E[αN(t) + βN(t)2]
)dt
• Idea: Let h(t) ≡ E[N(t)] =∫ t
0λe(t− v)e−µcv dv.
• Moreover, N(t) is Poisson distributed, so
E[αN(t) + βN(t)2] = (α + β)h(t) + βh(t)2
• maxλe(t)
[P (t)(h′(t) + µch(t))− (α + β)h(t)− βh(t)2
]• λe(t) = 1
2βmax{0, µ2
cP (t)− P ′′(t)− (α + β)µc}• Solving the differential equation for P (t), we have:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
tλe(r) sinh((r − t)µc)dr
• P (t) is market-clearing when λe(t) = (1− x(t))λ(t).
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 20 / 31
The call center’s problem
• Given prices P (t), the call center must choosethe allocation x(t) that maximizes its profits.
maxx(t)
1
T
∫ T
0[Rλ(t)− CIx(t)λ(t)− P (t)(1− x(t))λ(t)− CwQ(t)− CabθQ(t)] dt
revenue staffing costs co-sourcing costs wait costs aband. costs
• Letting P (t) = P (t)−CI and C = Cw +Cabθ:
maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 21 / 31
The call center’s problem
• Given prices P (t), the call center must choosethe allocation x(t) that maximizes its profits.
maxx(t)
1
T
∫ T
0[Rλ(t)− CIx(t)λ(t)− P (t)(1− x(t))λ(t)− CwQ(t)− CabθQ(t)] dt
revenue staffing costs co-sourcing costs wait costs aband. costs
• Letting P (t) = P (t)−CI and C = Cw +Cabθ:
maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 21 / 31
The call center’s problem
• Given prices P (t), the call center must choosethe allocation x(t) that maximizes its profits.
maxx(t)
1
T
∫ T
0[Rλ(t)− CIx(t)λ(t)− P (t)(1− x(t))λ(t)− CwQ(t)− CabθQ(t)] dt
revenue
staffing costs co-sourcing costs wait costs aband. costs
• Letting P (t) = P (t)−CI and C = Cw +Cabθ:
maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 21 / 31
The call center’s problem
• Given prices P (t), the call center must choosethe allocation x(t) that maximizes its profits.
maxx(t)
1
T
∫ T
0[Rλ(t)− CIx(t)λ(t)− P (t)(1− x(t))λ(t)− CwQ(t)− CabθQ(t)] dt
revenue staffing costs
co-sourcing costs wait costs aband. costs
• Letting P (t) = P (t)−CI and C = Cw +Cabθ:
maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 21 / 31
The call center’s problem
• Given prices P (t), the call center must choosethe allocation x(t) that maximizes its profits.
maxx(t)
1
T
∫ T
0[Rλ(t)− CIx(t)λ(t)− P (t)(1− x(t))λ(t)− CwQ(t)− CabθQ(t)] dt
revenue staffing costs co-sourcing costs
wait costs aband. costs
• Letting P (t) = P (t)−CI and C = Cw +Cabθ:
maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 21 / 31
The call center’s problem
• Given prices P (t), the call center must choosethe allocation x(t) that maximizes its profits.
maxx(t)
1
T
∫ T
0[Rλ(t)− CIx(t)λ(t)− P (t)(1− x(t))λ(t)− CwQ(t)− CabθQ(t)] dt
revenue staffing costs co-sourcing costs wait costs
aband. costs
• Letting P (t) = P (t)−CI and C = Cw +Cabθ:
maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 21 / 31
The call center’s problem
• Given prices P (t), the call center must choosethe allocation x(t) that maximizes its profits.
maxx(t)
1
T
∫ T
0[Rλ(t)− CIx(t)λ(t)− P (t)(1− x(t))λ(t)− CwQ(t)− CabθQ(t)] dt
revenue staffing costs co-sourcing costs wait costs aband. costs
• Letting P (t) = P (t)−CI and C = Cw +Cabθ:
maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 21 / 31
The call center’s problem
• Given prices P (t), the call center must choosethe allocation x(t) that maximizes its profits.
maxx(t)
1
T
∫ T
0[Rλ(t)− CIx(t)λ(t)− P (t)(1− x(t))λ(t)− CwQ(t)− CabθQ(t)] dt
revenue staffing costs co-sourcing costs wait costs aband. costs
• Letting P (t) = P (t)−CI and C = Cw +Cabθ:
maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 21 / 31
The call center’s problem
• Given prices P (t), the call center must choosethe allocation x(t) that maximizes its profits.
maxx(t)
1
T
∫ T
0[Rλ(t)− CIx(t)λ(t)− P (t)(1− x(t))λ(t)− CwQ(t)− CabθQ(t)] dt
revenue staffing costs co-sourcing costs wait costs aband. costs
• Letting P (t) = P (t)−CI and C = Cw +Cabθ:
maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 21 / 31
Call center fluid model illustration revisited
Q(t) B(t)x(t)λ(t)
drains at rate θQ(t) drains at rate µB(t) = B(t)
replaces drained fluid at
server when possible
capacity s
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 22 / 31
Solving the call center’s problem
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Idea: Express∫ T
0Q(t)dt in terms of x(t). But how?
• Observation: Q(t) alternates between being positive
and zero.
• Q(t) is positive (zero) on overloaded (underloaded)
time intervals
• A special case: a sub-interval of an underload interval
can be critically loaded: Q(t) = 0 and B(t) = s
• Overloaded intervals (Oi, Ui) and underloaded intervals
(Ui, Oi+1)
• Assumption: finitely many interval switchovers in [0, T ]
• Let’s use these intervals to calculate∫ T
0Q(t)dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 23 / 31
Solving the call center’s problem• max
x(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Idea: Express∫ T
0Q(t)dt in terms of x(t). But how?
• Observation: Q(t) alternates between being positive
and zero.
• Q(t) is positive (zero) on overloaded (underloaded)
time intervals
• A special case: a sub-interval of an underload interval
can be critically loaded: Q(t) = 0 and B(t) = s
• Overloaded intervals (Oi, Ui) and underloaded intervals
(Ui, Oi+1)
• Assumption: finitely many interval switchovers in [0, T ]
• Let’s use these intervals to calculate∫ T
0Q(t)dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 23 / 31
Solving the call center’s problem• max
x(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Idea: Express∫ T
0Q(t)dt in terms of x(t). But how?
• Observation: Q(t) alternates between being positive
and zero.
• Q(t) is positive (zero) on overloaded (underloaded)
time intervals
• A special case: a sub-interval of an underload interval
can be critically loaded: Q(t) = 0 and B(t) = s
• Overloaded intervals (Oi, Ui) and underloaded intervals
(Ui, Oi+1)
• Assumption: finitely many interval switchovers in [0, T ]
• Let’s use these intervals to calculate∫ T
0Q(t)dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 23 / 31
Solving the call center’s problem• max
x(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Idea: Express∫ T
0Q(t)dt in terms of x(t). But how?
• Observation: Q(t) alternates between being positive
and zero.
• Q(t) is positive (zero) on overloaded (underloaded)
time intervals
• A special case: a sub-interval of an underload interval
can be critically loaded: Q(t) = 0 and B(t) = s
• Overloaded intervals (Oi, Ui) and underloaded intervals
(Ui, Oi+1)
• Assumption: finitely many interval switchovers in [0, T ]
• Let’s use these intervals to calculate∫ T
0Q(t)dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 23 / 31
Solving the call center’s problem• max
x(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Idea: Express∫ T
0Q(t)dt in terms of x(t). But how?
• Observation: Q(t) alternates between being positive
and zero.
• Q(t) is positive (zero) on overloaded (underloaded)
time intervals
• A special case: a sub-interval of an underload interval
can be critically loaded: Q(t) = 0 and B(t) = s
• Overloaded intervals (Oi, Ui) and underloaded intervals
(Ui, Oi+1)
• Assumption: finitely many interval switchovers in [0, T ]
• Let’s use these intervals to calculate∫ T
0Q(t)dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 23 / 31
Solving the call center’s problem• max
x(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Idea: Express∫ T
0Q(t)dt in terms of x(t). But how?
• Observation: Q(t) alternates between being positive
and zero.
• Q(t) is positive (zero) on overloaded (underloaded)
time intervals
• A special case: a sub-interval of an underload interval
can be critically loaded: Q(t) = 0 and B(t) = s
• Overloaded intervals (Oi, Ui) and underloaded intervals
(Ui, Oi+1)
• Assumption: finitely many interval switchovers in [0, T ]
• Let’s use these intervals to calculate∫ T
0Q(t)dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 23 / 31
Solving the call center’s problem• max
x(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Idea: Express∫ T
0Q(t)dt in terms of x(t). But how?
• Observation: Q(t) alternates between being positive
and zero.
• Q(t) is positive (zero) on overloaded (underloaded)
time intervals
• A special case: a sub-interval of an underload interval
can be critically loaded: Q(t) = 0 and B(t) = s
• Overloaded intervals (Oi, Ui) and underloaded intervals
(Ui, Oi+1)
• Assumption: finitely many interval switchovers in [0, T ]
• Let’s use these intervals to calculate∫ T
0Q(t)dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 23 / 31
Solving the call center’s problem• max
x(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Idea: Express∫ T
0Q(t)dt in terms of x(t). But how?
• Observation: Q(t) alternates between being positive
and zero.
• Q(t) is positive (zero) on overloaded (underloaded)
time intervals
• A special case: a sub-interval of an underload interval
can be critically loaded: Q(t) = 0 and B(t) = s
• Overloaded intervals (Oi, Ui) and underloaded intervals
(Ui, Oi+1)
• Assumption: finitely many interval switchovers in [0, T ]
• Let’s use these intervals to calculate∫ T
0Q(t)dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 23 / 31
Solving the call center’s problem• max
x(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Idea: Express∫ T
0Q(t)dt in terms of x(t). But how?
• Observation: Q(t) alternates between being positive
and zero.
• Q(t) is positive (zero) on overloaded (underloaded)
time intervals
• A special case: a sub-interval of an underload interval
can be critically loaded: Q(t) = 0 and B(t) = s
• Overloaded intervals (Oi, Ui) and underloaded intervals
(Ui, Oi+1)
• Assumption: finitely many interval switchovers in [0, T ]
• Let’s use these intervals to calculate∫ T
0Q(t)dt
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 23 / 31
Integrating over intervals
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ T
0Q(t)dt in terms of x(t)
• Observation:∫ T0Q(t)dt =
∑Ki=1
(∫ Oi
Ui−1Q(t)dt+
∫ Ui
OiQ(t)dt
)• Q: What is
∫ Oi
Ui−1Q(t)dt? (hint: underloaded)
• A: Zero, so now we only need∫ Ui
OiQ(t)dt
• Observation: in overload, we have B(t) = s
• Observation: overload starts and ends with Q(t) = 0,
i.e., Q(Oi) = Q(Ui) = 0
• Hint: what goes into the queue must come out!
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 24 / 31
Integrating over intervals
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ T
0Q(t)dt in terms of x(t)
• Observation:∫ T0Q(t)dt =
∑Ki=1
(∫ Oi
Ui−1Q(t)dt+
∫ Ui
OiQ(t)dt
)• Q: What is
∫ Oi
Ui−1Q(t)dt? (hint: underloaded)
• A: Zero, so now we only need∫ Ui
OiQ(t)dt
• Observation: in overload, we have B(t) = s
• Observation: overload starts and ends with Q(t) = 0,
i.e., Q(Oi) = Q(Ui) = 0
• Hint: what goes into the queue must come out!
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 24 / 31
Integrating over intervals
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ T
0Q(t)dt in terms of x(t)
• Observation:∫ T0Q(t)dt =
∑Ki=1
(∫ Oi
Ui−1Q(t)dt+
∫ Ui
OiQ(t)dt
)• Q: What is
∫ Oi
Ui−1Q(t)dt? (hint: underloaded)
• A: Zero, so now we only need∫ Ui
OiQ(t)dt
• Observation: in overload, we have B(t) = s
• Observation: overload starts and ends with Q(t) = 0,
i.e., Q(Oi) = Q(Ui) = 0
• Hint: what goes into the queue must come out!
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 24 / 31
Integrating over intervals
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ T
0Q(t)dt in terms of x(t)
• Observation:∫ T0Q(t)dt =
∑Ki=1
(∫ Oi
Ui−1Q(t)dt+
∫ Ui
OiQ(t)dt
)
• Q: What is∫ Oi
Ui−1Q(t)dt? (hint: underloaded)
• A: Zero, so now we only need∫ Ui
OiQ(t)dt
• Observation: in overload, we have B(t) = s
• Observation: overload starts and ends with Q(t) = 0,
i.e., Q(Oi) = Q(Ui) = 0
• Hint: what goes into the queue must come out!
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 24 / 31
Integrating over intervals
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ T
0Q(t)dt in terms of x(t)
• Observation:∫ T0Q(t)dt =
∑Ki=1
(∫ Oi
Ui−1Q(t)dt+
∫ Ui
OiQ(t)dt
)• Q: What is
∫ Oi
Ui−1Q(t)dt? (hint: underloaded)
• A: Zero, so now we only need∫ Ui
OiQ(t)dt
• Observation: in overload, we have B(t) = s
• Observation: overload starts and ends with Q(t) = 0,
i.e., Q(Oi) = Q(Ui) = 0
• Hint: what goes into the queue must come out!
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 24 / 31
Integrating over intervals
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ T
0Q(t)dt in terms of x(t)
• Observation:∫ T0Q(t)dt =
∑Ki=1
(∫ Oi
Ui−1Q(t)dt+
∫ Ui
OiQ(t)dt
)• Q: What is
∫ Oi
Ui−1Q(t)dt? (hint: underloaded)
• A: Zero, so now we only need∫ Ui
OiQ(t)dt
• Observation: in overload, we have B(t) = s
• Observation: overload starts and ends with Q(t) = 0,
i.e., Q(Oi) = Q(Ui) = 0
• Hint: what goes into the queue must come out!
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 24 / 31
Integrating over intervals
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ T
0Q(t)dt in terms of x(t)
• Observation:∫ T0Q(t)dt =
∑Ki=1
(∫ Oi
Ui−1Q(t)dt+
∫ Ui
OiQ(t)dt
)• Q: What is
∫ Oi
Ui−1Q(t)dt? (hint: underloaded)
• A: Zero, so now we only need∫ Ui
OiQ(t)dt
• Observation: in overload, we have B(t) = s
• Observation: overload starts and ends with Q(t) = 0,
i.e., Q(Oi) = Q(Ui) = 0
• Hint: what goes into the queue must come out!
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 24 / 31
Integrating over intervals
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ T
0Q(t)dt in terms of x(t)
• Observation:∫ T0Q(t)dt =
∑Ki=1
(∫ Oi
Ui−1Q(t)dt+
∫ Ui
OiQ(t)dt
)• Q: What is
∫ Oi
Ui−1Q(t)dt? (hint: underloaded)
• A: Zero, so now we only need∫ Ui
OiQ(t)dt
• Observation: in overload, we have B(t) = s
• Observation: overload starts and ends with Q(t) = 0,
i.e., Q(Oi) = Q(Ui) = 0
• Hint: what goes into the queue must come out!
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 24 / 31
Integrating over intervals
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ T
0Q(t)dt in terms of x(t)
• Observation:∫ T0Q(t)dt =
∑Ki=1
(∫ Oi
Ui−1Q(t)dt+
∫ Ui
OiQ(t)dt
)• Q: What is
∫ Oi
Ui−1Q(t)dt? (hint: underloaded)
• A: Zero, so now we only need∫ Ui
OiQ(t)dt
• Observation: in overload, we have B(t) = s
• Observation: overload starts and ends with Q(t) = 0,
i.e., Q(Oi) = Q(Ui) = 0
• Hint: what goes into the queue must come out!
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 24 / 31
Integrating Q(t) over (Oi, Ui)
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ Ui
OiQ(t)dt in terms of x(t)
• Q: How much fluid has entered the queue?
• A: Integrating over the arrival rate:∫ Ui
Oix(t)λ(t)dt
• Q: How much fluid has left the queue? (hint: B(t) = s)
• A:∫ Ui
Oi(θQ(t) + s) dt =
∫ Ui
Oix(t)λ(t)dt
• Solving:∫ Ui
OiQ(t)dt = 1
θ
∫ Ui
Oi(x(t)λ(t)− s)dt
maxx(t)
∫ T
0
(P (t)x(t)λ(t)− C
θ
K∑i=1
∫ Ui
Oi
(x(t)λ(t)− s)dt
)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 25 / 31
Integrating Q(t) over (Oi, Ui)
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ Ui
OiQ(t)dt in terms of x(t)
• Q: How much fluid has entered the queue?
• A: Integrating over the arrival rate:∫ Ui
Oix(t)λ(t)dt
• Q: How much fluid has left the queue? (hint: B(t) = s)
• A:∫ Ui
Oi(θQ(t) + s) dt =
∫ Ui
Oix(t)λ(t)dt
• Solving:∫ Ui
OiQ(t)dt = 1
θ
∫ Ui
Oi(x(t)λ(t)− s)dt
maxx(t)
∫ T
0
(P (t)x(t)λ(t)− C
θ
K∑i=1
∫ Ui
Oi
(x(t)λ(t)− s)dt
)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 25 / 31
Integrating Q(t) over (Oi, Ui)
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ Ui
OiQ(t)dt in terms of x(t)
• Q: How much fluid has entered the queue?
• A: Integrating over the arrival rate:∫ Ui
Oix(t)λ(t)dt
• Q: How much fluid has left the queue? (hint: B(t) = s)
• A:∫ Ui
Oi(θQ(t) + s) dt =
∫ Ui
Oix(t)λ(t)dt
• Solving:∫ Ui
OiQ(t)dt = 1
θ
∫ Ui
Oi(x(t)λ(t)− s)dt
maxx(t)
∫ T
0
(P (t)x(t)λ(t)− C
θ
K∑i=1
∫ Ui
Oi
(x(t)λ(t)− s)dt
)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 25 / 31
Integrating Q(t) over (Oi, Ui)
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ Ui
OiQ(t)dt in terms of x(t)
• Q: How much fluid has entered the queue?
• A: Integrating over the arrival rate:∫ Ui
Oix(t)λ(t)dt
• Q: How much fluid has left the queue? (hint: B(t) = s)
• A:∫ Ui
Oi(θQ(t) + s) dt =
∫ Ui
Oix(t)λ(t)dt
• Solving:∫ Ui
OiQ(t)dt = 1
θ
∫ Ui
Oi(x(t)λ(t)− s)dt
maxx(t)
∫ T
0
(P (t)x(t)λ(t)− C
θ
K∑i=1
∫ Ui
Oi
(x(t)λ(t)− s)dt
)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 25 / 31
Integrating Q(t) over (Oi, Ui)
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ Ui
OiQ(t)dt in terms of x(t)
• Q: How much fluid has entered the queue?
• A: Integrating over the arrival rate:∫ Ui
Oix(t)λ(t)dt
• Q: How much fluid has left the queue? (hint: B(t) = s)
• A:∫ Ui
Oi(θQ(t) + s) dt =
∫ Ui
Oix(t)λ(t)dt
• Solving:∫ Ui
OiQ(t)dt = 1
θ
∫ Ui
Oi(x(t)λ(t)− s)dt
maxx(t)
∫ T
0
(P (t)x(t)λ(t)− C
θ
K∑i=1
∫ Ui
Oi
(x(t)λ(t)− s)dt
)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 25 / 31
Integrating Q(t) over (Oi, Ui)
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ Ui
OiQ(t)dt in terms of x(t)
• Q: How much fluid has entered the queue?
• A: Integrating over the arrival rate:∫ Ui
Oix(t)λ(t)dt
• Q: How much fluid has left the queue? (hint: B(t) = s)
• A:∫ Ui
Oi(θQ(t) + s) dt =
∫ Ui
Oix(t)λ(t)dt
• Solving:∫ Ui
OiQ(t)dt = 1
θ
∫ Ui
Oi(x(t)λ(t)− s)dt
maxx(t)
∫ T
0
(P (t)x(t)λ(t)− C
θ
K∑i=1
∫ Ui
Oi
(x(t)λ(t)− s)dt
)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 25 / 31
Integrating Q(t) over (Oi, Ui)
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ Ui
OiQ(t)dt in terms of x(t)
• Q: How much fluid has entered the queue?
• A: Integrating over the arrival rate:∫ Ui
Oix(t)λ(t)dt
• Q: How much fluid has left the queue? (hint: B(t) = s)
• A:∫ Ui
Oi(θQ(t) + s) dt =
∫ Ui
Oix(t)λ(t)dt
• Solving:∫ Ui
OiQ(t)dt = 1
θ
∫ Ui
Oi(x(t)λ(t)− s)dt
maxx(t)
∫ T
0
(P (t)x(t)λ(t)− C
θ
K∑i=1
∫ Ui
Oi
(x(t)λ(t)− s)dt
)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 25 / 31
Integrating Q(t) over (Oi, Ui)
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ Ui
OiQ(t)dt in terms of x(t)
• Q: How much fluid has entered the queue?
• A: Integrating over the arrival rate:∫ Ui
Oix(t)λ(t)dt
• Q: How much fluid has left the queue? (hint: B(t) = s)
• A:∫ Ui
Oi(θQ(t) + s) dt =
∫ Ui
Oix(t)λ(t)dt
• Solving:∫ Ui
OiQ(t)dt = 1
θ
∫ Ui
Oi(x(t)λ(t)− s)dt
maxx(t)
∫ T
0
(P (t)x(t)λ(t)− C
θ
K∑i=1
∫ Ui
Oi
(x(t)λ(t)− s)dt
)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 25 / 31
Integrating Q(t) over (Oi, Ui)
• maxx(t)
∫ T
0
[P (t)x(t)λ(t)− CQ(t)
]dt
• Goal: Express∫ Ui
OiQ(t)dt in terms of x(t)
• Q: How much fluid has entered the queue?
• A: Integrating over the arrival rate:∫ Ui
Oix(t)λ(t)dt
• Q: How much fluid has left the queue? (hint: B(t) = s)
• A:∫ Ui
Oi(θQ(t) + s) dt =
∫ Ui
Oix(t)λ(t)dt
• Solving:∫ Ui
OiQ(t)dt = 1
θ
∫ Ui
Oi(x(t)λ(t)− s)dt
maxx(t)
∫ T
0
(P (t)x(t)λ(t)− C
θ
K∑i=1
∫ Ui
Oi
(x(t)λ(t)− s)dt
)
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 25 / 31
Back to the call center’s optimization
• maxx(t)
∫ T
0
(P (t)x(t)λ(t)− C
θ
K∑i=1
∫ Ui
Oi
(x(t)λ(t)− s)dt
)• We need to find, not only x(t), but the Ui’s
and Oi’s
• We can write an optimization problem whereUi’s and Oi’s are defined by constraints:
Ui = inf
{t ≥ Oi :
∫ t
Oi
eθ(r−Oi)(λ(r)x(r)− s)dr < 0
}Oi+1 = inf
{t ≥ Ui :
∫ t
Ui
e(r−Ui)(λ(r)x(r)− s)dr > 0
}0 ≤ x(t) ≤ 1
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 26 / 31
Back to the call center’s optimization
• maxx(t)
∫ T
0
(P (t)x(t)λ(t)− C
θ
K∑i=1
∫ Ui
Oi
(x(t)λ(t)− s)dt
)
• We need to find, not only x(t), but the Ui’sand Oi’s
• We can write an optimization problem whereUi’s and Oi’s are defined by constraints:
Ui = inf
{t ≥ Oi :
∫ t
Oi
eθ(r−Oi)(λ(r)x(r)− s)dr < 0
}Oi+1 = inf
{t ≥ Ui :
∫ t
Ui
e(r−Ui)(λ(r)x(r)− s)dr > 0
}0 ≤ x(t) ≤ 1
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 26 / 31
Back to the call center’s optimization
• maxx(t)
∫ T
0
(P (t)x(t)λ(t)− C
θ
K∑i=1
∫ Ui
Oi
(x(t)λ(t)− s)dt
)• We need to find, not only x(t), but the Ui’s
and Oi’s
• We can write an optimization problem whereUi’s and Oi’s are defined by constraints:
Ui = inf
{t ≥ Oi :
∫ t
Oi
eθ(r−Oi)(λ(r)x(r)− s)dr < 0
}Oi+1 = inf
{t ≥ Ui :
∫ t
Ui
e(r−Ui)(λ(r)x(r)− s)dr > 0
}0 ≤ x(t) ≤ 1
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 26 / 31
Back to the call center’s optimization
• maxx(t)
∫ T
0
(P (t)x(t)λ(t)− C
θ
K∑i=1
∫ Ui
Oi
(x(t)λ(t)− s)dt
)• We need to find, not only x(t), but the Ui’s
and Oi’s
• We can write an optimization problem whereUi’s and Oi’s are defined by constraints:
Ui = inf
{t ≥ Oi :
∫ t
Oi
eθ(r−Oi)(λ(r)x(r)− s)dr < 0
}Oi+1 = inf
{t ≥ Ui :
∫ t
Ui
e(r−Ui)(λ(r)x(r)− s)dr > 0
}0 ≤ x(t) ≤ 1
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 26 / 31
Back to the call center’s optimization
• maxx(t)
∫ T
0
(P (t)x(t)λ(t)− C
θ
K∑i=1
∫ Ui
Oi
(x(t)λ(t)− s)dt
)• We need to find, not only x(t), but the Ui’s
and Oi’s
• We can write an optimization problem whereUi’s and Oi’s are defined by constraints:
Ui = inf
{t ≥ Oi :
∫ t
Oi
eθ(r−Oi)(λ(r)x(r)− s)dr < 0
}Oi+1 = inf
{t ≥ Ui :
∫ t
Ui
e(r−Ui)(λ(r)x(r)− s)dr > 0
}0 ≤ x(t) ≤ 1
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 26 / 31
Solution to the optimization problem
• We solve a continuous-time stochastic dynamic program tracking
Q(t) (in overload) and s−B(t) (in underload).
• There exist values ΓOLi ,ΓUL
i constant in t (that we can calculate)
such that the optimal x(t) is as follows:
1. In critically loaded subintervals, the x(t) = s/λ(t).2. In the i-th overloaded interval
x(t) = 1 if P (t)− (CI + Cθ
)− e−θ(Ui−t)ΓOLi > 0
x(t) = 0 if P (t)− (CI + Cθ
)− e−θ(Ui−t)ΓOLi < 0
xt ∈ [0, 1] otherwise.
3. In the i-th (not critically) underloaded interval,x(t) = 1 if P (t)− CI − e−(Oi+1−t)ΓUL
i > 0
x(t) = 0 if P (t)− CI − e−(Oi+1−t)ΓULi < 0
x(t) ∈ [0, 1] otherwise.
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 27 / 31
Solution to the optimization problem• We solve a continuous-time stochastic dynamic program tracking
Q(t) (in overload) and s−B(t) (in underload).
• There exist values ΓOLi ,ΓUL
i constant in t (that we can calculate)
such that the optimal x(t) is as follows:
1. In critically loaded subintervals, the x(t) = s/λ(t).2. In the i-th overloaded interval
x(t) = 1 if P (t)− (CI + Cθ
)− e−θ(Ui−t)ΓOLi > 0
x(t) = 0 if P (t)− (CI + Cθ
)− e−θ(Ui−t)ΓOLi < 0
xt ∈ [0, 1] otherwise.
3. In the i-th (not critically) underloaded interval,x(t) = 1 if P (t)− CI − e−(Oi+1−t)ΓUL
i > 0
x(t) = 0 if P (t)− CI − e−(Oi+1−t)ΓULi < 0
x(t) ∈ [0, 1] otherwise.
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 27 / 31
Solution to the optimization problem• We solve a continuous-time stochastic dynamic program tracking
Q(t) (in overload) and s−B(t) (in underload).
• There exist values ΓOLi ,ΓUL
i constant in t (that we can calculate)
such that the optimal x(t) is as follows:
1. In critically loaded subintervals, the x(t) = s/λ(t).2. In the i-th overloaded interval
x(t) = 1 if P (t)− (CI + Cθ
)− e−θ(Ui−t)ΓOLi > 0
x(t) = 0 if P (t)− (CI + Cθ
)− e−θ(Ui−t)ΓOLi < 0
xt ∈ [0, 1] otherwise.
3. In the i-th (not critically) underloaded interval,x(t) = 1 if P (t)− CI − e−(Oi+1−t)ΓUL
i > 0
x(t) = 0 if P (t)− CI − e−(Oi+1−t)ΓULi < 0
x(t) ∈ [0, 1] otherwise.
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 27 / 31
Solution to the optimization problem• We solve a continuous-time stochastic dynamic program tracking
Q(t) (in overload) and s−B(t) (in underload).
• There exist values ΓOLi ,ΓUL
i constant in t (that we can calculate)
such that the optimal x(t) is as follows:
1. In critically loaded subintervals, the x(t) = s/λ(t).
2. In the i-th overloaded intervalx(t) = 1 if P (t)− (CI + C
θ)− e−θ(Ui−t)ΓOL
i > 0
x(t) = 0 if P (t)− (CI + Cθ
)− e−θ(Ui−t)ΓOLi < 0
xt ∈ [0, 1] otherwise.
3. In the i-th (not critically) underloaded interval,x(t) = 1 if P (t)− CI − e−(Oi+1−t)ΓUL
i > 0
x(t) = 0 if P (t)− CI − e−(Oi+1−t)ΓULi < 0
x(t) ∈ [0, 1] otherwise.
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 27 / 31
Solution to the optimization problem• We solve a continuous-time stochastic dynamic program tracking
Q(t) (in overload) and s−B(t) (in underload).
• There exist values ΓOLi ,ΓUL
i constant in t (that we can calculate)
such that the optimal x(t) is as follows:
1. In critically loaded subintervals, the x(t) = s/λ(t).2. In the i-th overloaded interval
x(t) = 1 if P (t)− (CI + Cθ
)− e−θ(Ui−t)ΓOLi > 0
x(t) = 0 if P (t)− (CI + Cθ
)− e−θ(Ui−t)ΓOLi < 0
xt ∈ [0, 1] otherwise.
3. In the i-th (not critically) underloaded interval,x(t) = 1 if P (t)− CI − e−(Oi+1−t)ΓUL
i > 0
x(t) = 0 if P (t)− CI − e−(Oi+1−t)ΓULi < 0
x(t) ∈ [0, 1] otherwise.
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 27 / 31
Solution to the optimization problem• We solve a continuous-time stochastic dynamic program tracking
Q(t) (in overload) and s−B(t) (in underload).
• There exist values ΓOLi ,ΓUL
i constant in t (that we can calculate)
such that the optimal x(t) is as follows:
1. In critically loaded subintervals, the x(t) = s/λ(t).2. In the i-th overloaded interval
x(t) = 1 if P (t)− (CI + Cθ
)− e−θ(Ui−t)ΓOLi > 0
x(t) = 0 if P (t)− (CI + Cθ
)− e−θ(Ui−t)ΓOLi < 0
xt ∈ [0, 1] otherwise.
3. In the i-th (not critically) underloaded interval,x(t) = 1 if P (t)− CI − e−(Oi+1−t)ΓUL
i > 0
x(t) = 0 if P (t)− CI − e−(Oi+1−t)ΓULi < 0
x(t) ∈ [0, 1] otherwise.
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 27 / 31
Solution insights
x(t) = 1 if P (t)− (CI + C
θ)− e−θ(Ui−t)ΓOL
i > 0
x(t) = 0 if P (t)− (CI + Cθ
)− e−θ(Ui−t)ΓOLi < 0
xt ∈ [0, 1] otherwise.x(t) = 1 if P (t)− CI − e−(Oi+1−t)ΓUL
i > 0
x(t) = 0 if P (t)− CI − e−(Oi+1−t)ΓULi < 0
x(t) ∈ [0, 1] otherwise.
• Outside of special “indifference points” and critically loaded
intervals, we co-source “all or nothing,” which does not only
depend on the type of interval.
• As we persist in an interval, the price threshold on total
co-sourcing rises or falls depending on the sign of the Γi value.
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 28 / 31
Solution insightsx(t) = 1 if P (t)− (CI + C
θ)− e−θ(Ui−t)ΓOL
i > 0
x(t) = 0 if P (t)− (CI + Cθ
)− e−θ(Ui−t)ΓOLi < 0
xt ∈ [0, 1] otherwise.x(t) = 1 if P (t)− CI − e−(Oi+1−t)ΓUL
i > 0
x(t) = 0 if P (t)− CI − e−(Oi+1−t)ΓULi < 0
x(t) ∈ [0, 1] otherwise.
• Outside of special “indifference points” and critically loaded
intervals, we co-source “all or nothing,” which does not only
depend on the type of interval.
• As we persist in an interval, the price threshold on total
co-sourcing rises or falls depending on the sign of the Γi value.
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 28 / 31
Solution insightsx(t) = 1 if P (t)− (CI + C
θ)− e−θ(Ui−t)ΓOL
i > 0
x(t) = 0 if P (t)− (CI + Cθ
)− e−θ(Ui−t)ΓOLi < 0
xt ∈ [0, 1] otherwise.x(t) = 1 if P (t)− CI − e−(Oi+1−t)ΓUL
i > 0
x(t) = 0 if P (t)− CI − e−(Oi+1−t)ΓULi < 0
x(t) ∈ [0, 1] otherwise.
• Outside of special “indifference points” and critically loaded
intervals, we co-source “all or nothing,” which does not only
depend on the type of interval.
• As we persist in an interval, the price threshold on total
co-sourcing rises or falls depending on the sign of the Γi value.
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 28 / 31
Solution insightsx(t) = 1 if P (t)− (CI + C
θ)− e−θ(Ui−t)ΓOL
i > 0
x(t) = 0 if P (t)− (CI + Cθ
)− e−θ(Ui−t)ΓOLi < 0
xt ∈ [0, 1] otherwise.x(t) = 1 if P (t)− CI − e−(Oi+1−t)ΓUL
i > 0
x(t) = 0 if P (t)− CI − e−(Oi+1−t)ΓULi < 0
x(t) ∈ [0, 1] otherwise.
• Outside of special “indifference points” and critically loaded
intervals, we co-source “all or nothing,” which does not only
depend on the type of interval.
• As we persist in an interval, the price threshold on total
co-sourcing rises or falls depending on the sign of the Γi value.
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 28 / 31
Existence of equilibrium
• Finally, we address price-allocation equilibrium existence
• Currently limited by technical assumptions on state at time T• Basic idea exploits form of price function:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
t(1− x(r))λ(r) sinh((r − t)µc)dr
• Prices need only consider future allocation
• Given terminal price information, P (T ), P (T ′), we construct ΓULi ,
ΓOLi , x(t), P (t), Ui, and Oi values jointly by “crawling
backward” in time• Based on interval type at t, compute Γi threshold• Based on P (t) and Γi threshold, determine if x(t− ε) is 0 or 1 (or
critical) for arbitrarily small ε > 0• Now we can compute P (t− ε)• Meanwhile, check “inverted” definitions of Ui and Oi to see if we
have crossed into a new interval type (or become critical)
• Hope that construction method proves existence
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 29 / 31
Existence of equilibrium• Finally, we address price-allocation equilibrium existence
• Currently limited by technical assumptions on state at time T• Basic idea exploits form of price function:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
t(1− x(r))λ(r) sinh((r − t)µc)dr
• Prices need only consider future allocation
• Given terminal price information, P (T ), P (T ′), we construct ΓULi ,
ΓOLi , x(t), P (t), Ui, and Oi values jointly by “crawling
backward” in time• Based on interval type at t, compute Γi threshold• Based on P (t) and Γi threshold, determine if x(t− ε) is 0 or 1 (or
critical) for arbitrarily small ε > 0• Now we can compute P (t− ε)• Meanwhile, check “inverted” definitions of Ui and Oi to see if we
have crossed into a new interval type (or become critical)
• Hope that construction method proves existence
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 29 / 31
Existence of equilibrium• Finally, we address price-allocation equilibrium existence
• Currently limited by technical assumptions on state at time T
• Basic idea exploits form of price function:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
t(1− x(r))λ(r) sinh((r − t)µc)dr
• Prices need only consider future allocation
• Given terminal price information, P (T ), P (T ′), we construct ΓULi ,
ΓOLi , x(t), P (t), Ui, and Oi values jointly by “crawling
backward” in time• Based on interval type at t, compute Γi threshold• Based on P (t) and Γi threshold, determine if x(t− ε) is 0 or 1 (or
critical) for arbitrarily small ε > 0• Now we can compute P (t− ε)• Meanwhile, check “inverted” definitions of Ui and Oi to see if we
have crossed into a new interval type (or become critical)
• Hope that construction method proves existence
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 29 / 31
Existence of equilibrium• Finally, we address price-allocation equilibrium existence
• Currently limited by technical assumptions on state at time T• Basic idea exploits form of price function:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
t(1− x(r))λ(r) sinh((r − t)µc)dr
• Prices need only consider future allocation
• Given terminal price information, P (T ), P (T ′), we construct ΓULi ,
ΓOLi , x(t), P (t), Ui, and Oi values jointly by “crawling
backward” in time• Based on interval type at t, compute Γi threshold• Based on P (t) and Γi threshold, determine if x(t− ε) is 0 or 1 (or
critical) for arbitrarily small ε > 0• Now we can compute P (t− ε)• Meanwhile, check “inverted” definitions of Ui and Oi to see if we
have crossed into a new interval type (or become critical)
• Hope that construction method proves existence
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 29 / 31
Existence of equilibrium• Finally, we address price-allocation equilibrium existence
• Currently limited by technical assumptions on state at time T• Basic idea exploits form of price function:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
t(1− x(r))λ(r) sinh((r − t)µc)dr
• Prices need only consider future allocation
• Given terminal price information, P (T ), P (T ′), we construct ΓULi ,
ΓOLi , x(t), P (t), Ui, and Oi values jointly by “crawling
backward” in time• Based on interval type at t, compute Γi threshold• Based on P (t) and Γi threshold, determine if x(t− ε) is 0 or 1 (or
critical) for arbitrarily small ε > 0• Now we can compute P (t− ε)• Meanwhile, check “inverted” definitions of Ui and Oi to see if we
have crossed into a new interval type (or become critical)
• Hope that construction method proves existence
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 29 / 31
Existence of equilibrium• Finally, we address price-allocation equilibrium existence
• Currently limited by technical assumptions on state at time T• Basic idea exploits form of price function:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
t(1− x(r))λ(r) sinh((r − t)µc)dr
• Prices need only consider future allocation
• Given terminal price information, P (T ), P (T ′), we construct ΓULi ,
ΓOLi , x(t), P (t), Ui, and Oi values jointly by “crawling
backward” in time• Based on interval type at t, compute Γi threshold• Based on P (t) and Γi threshold, determine if x(t− ε) is 0 or 1 (or
critical) for arbitrarily small ε > 0• Now we can compute P (t− ε)• Meanwhile, check “inverted” definitions of Ui and Oi to see if we
have crossed into a new interval type (or become critical)
• Hope that construction method proves existence
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 29 / 31
Existence of equilibrium• Finally, we address price-allocation equilibrium existence
• Currently limited by technical assumptions on state at time T• Basic idea exploits form of price function:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
t(1− x(r))λ(r) sinh((r − t)µc)dr
• Prices need only consider future allocation
• Given terminal price information, P (T ), P (T ′), we construct ΓULi ,
ΓOLi , x(t), P (t), Ui, and Oi values jointly by “crawling
backward” in time
• Based on interval type at t, compute Γi threshold• Based on P (t) and Γi threshold, determine if x(t− ε) is 0 or 1 (or
critical) for arbitrarily small ε > 0• Now we can compute P (t− ε)• Meanwhile, check “inverted” definitions of Ui and Oi to see if we
have crossed into a new interval type (or become critical)
• Hope that construction method proves existence
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 29 / 31
Existence of equilibrium• Finally, we address price-allocation equilibrium existence
• Currently limited by technical assumptions on state at time T• Basic idea exploits form of price function:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
t(1− x(r))λ(r) sinh((r − t)µc)dr
• Prices need only consider future allocation
• Given terminal price information, P (T ), P (T ′), we construct ΓULi ,
ΓOLi , x(t), P (t), Ui, and Oi values jointly by “crawling
backward” in time• Based on interval type at t, compute Γi threshold
• Based on P (t) and Γi threshold, determine if x(t− ε) is 0 or 1 (orcritical) for arbitrarily small ε > 0
• Now we can compute P (t− ε)• Meanwhile, check “inverted” definitions of Ui and Oi to see if we
have crossed into a new interval type (or become critical)
• Hope that construction method proves existence
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 29 / 31
Existence of equilibrium• Finally, we address price-allocation equilibrium existence
• Currently limited by technical assumptions on state at time T• Basic idea exploits form of price function:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
t(1− x(r))λ(r) sinh((r − t)µc)dr
• Prices need only consider future allocation
• Given terminal price information, P (T ), P (T ′), we construct ΓULi ,
ΓOLi , x(t), P (t), Ui, and Oi values jointly by “crawling
backward” in time• Based on interval type at t, compute Γi threshold• Based on P (t) and Γi threshold, determine if x(t− ε) is 0 or 1 (or
critical) for arbitrarily small ε > 0
• Now we can compute P (t− ε)• Meanwhile, check “inverted” definitions of Ui and Oi to see if we
have crossed into a new interval type (or become critical)
• Hope that construction method proves existence
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 29 / 31
Existence of equilibrium• Finally, we address price-allocation equilibrium existence
• Currently limited by technical assumptions on state at time T• Basic idea exploits form of price function:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
t(1− x(r))λ(r) sinh((r − t)µc)dr
• Prices need only consider future allocation
• Given terminal price information, P (T ), P (T ′), we construct ΓULi ,
ΓOLi , x(t), P (t), Ui, and Oi values jointly by “crawling
backward” in time• Based on interval type at t, compute Γi threshold• Based on P (t) and Γi threshold, determine if x(t− ε) is 0 or 1 (or
critical) for arbitrarily small ε > 0• Now we can compute P (t− ε)
• Meanwhile, check “inverted” definitions of Ui and Oi to see if wehave crossed into a new interval type (or become critical)
• Hope that construction method proves existence
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 29 / 31
Existence of equilibrium• Finally, we address price-allocation equilibrium existence
• Currently limited by technical assumptions on state at time T• Basic idea exploits form of price function:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
t(1− x(r))λ(r) sinh((r − t)µc)dr
• Prices need only consider future allocation
• Given terminal price information, P (T ), P (T ′), we construct ΓULi ,
ΓOLi , x(t), P (t), Ui, and Oi values jointly by “crawling
backward” in time• Based on interval type at t, compute Γi threshold• Based on P (t) and Γi threshold, determine if x(t− ε) is 0 or 1 (or
critical) for arbitrarily small ε > 0• Now we can compute P (t− ε)• Meanwhile, check “inverted” definitions of Ui and Oi to see if we
have crossed into a new interval type (or become critical)
• Hope that construction method proves existence
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 29 / 31
Existence of equilibrium• Finally, we address price-allocation equilibrium existence
• Currently limited by technical assumptions on state at time T• Basic idea exploits form of price function:
P (t) =α+ β
µc+
(P (T )−
α+ β
µc
)cosh (µc(T − t))
−1
µP ′(T ) sinh(µc(T − t))−
2β
µc
∫ T
t(1− x(r))λ(r) sinh((r − t)µc)dr
• Prices need only consider future allocation
• Given terminal price information, P (T ), P (T ′), we construct ΓULi ,
ΓOLi , x(t), P (t), Ui, and Oi values jointly by “crawling
backward” in time• Based on interval type at t, compute Γi threshold• Based on P (t) and Γi threshold, determine if x(t− ε) is 0 or 1 (or
critical) for arbitrarily small ε > 0• Now we can compute P (t− ε)• Meanwhile, check “inverted” definitions of Ui and Oi to see if we
have crossed into a new interval type (or become critical)
• Hope that construction method proves existenceSherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 29 / 31
Future research questions
• Can we generalize our equilibrium result anduse our analysis to explore simple examplesnumerically?
• Can we gain insight into what allows criticallyloaded subintervals to persist? How frequentlyare we indifferent to co-sourcing? Does thishappen at only finitely many isolated points?
• Do our insights carry over into the stochasticsetting with a moderate number of servers?
• What if the number of servers is allowed tochange as a function of time (i.e., we are givens(t) rather than s)?
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 30 / 31
Future research questions
• Can we generalize our equilibrium result anduse our analysis to explore simple examplesnumerically?
• Can we gain insight into what allows criticallyloaded subintervals to persist? How frequentlyare we indifferent to co-sourcing? Does thishappen at only finitely many isolated points?
• Do our insights carry over into the stochasticsetting with a moderate number of servers?
• What if the number of servers is allowed tochange as a function of time (i.e., we are givens(t) rather than s)?
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 30 / 31
Future research questions
• Can we generalize our equilibrium result anduse our analysis to explore simple examplesnumerically?
• Can we gain insight into what allows criticallyloaded subintervals to persist? How frequentlyare we indifferent to co-sourcing? Does thishappen at only finitely many isolated points?
• Do our insights carry over into the stochasticsetting with a moderate number of servers?
• What if the number of servers is allowed tochange as a function of time (i.e., we are givens(t) rather than s)?
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 30 / 31
Future research questions
• Can we generalize our equilibrium result anduse our analysis to explore simple examplesnumerically?
• Can we gain insight into what allows criticallyloaded subintervals to persist? How frequentlyare we indifferent to co-sourcing? Does thishappen at only finitely many isolated points?
• Do our insights carry over into the stochasticsetting with a moderate number of servers?
• What if the number of servers is allowed tochange as a function of time (i.e., we are givens(t) rather than s)?
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 30 / 31
Future research questions
• Can we generalize our equilibrium result anduse our analysis to explore simple examplesnumerically?
• Can we gain insight into what allows criticallyloaded subintervals to persist? How frequentlyare we indifferent to co-sourcing? Does thishappen at only finitely many isolated points?
• Do our insights carry over into the stochasticsetting with a moderate number of servers?
• What if the number of servers is allowed tochange as a function of time (i.e., we are givens(t) rather than s)?
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 30 / 31
Any questions?
Thank you!
Sherwin Doroudi (UMN - ISyE) Call Centers with Co-Sourcing May 16, 2018 31 / 31