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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

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

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

• Demand (call arrival) uncertainty

• Aim to satisfy quality of service (QoS) targets• Most calls served with short waiting time• Most calls served before abandoned

• 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

• Capacity expansion is very costly• Capital investment costs are large

• Significant costs to hiring, training, and scheduling

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)

Co-sourcing in call centers• Q: What is co-sourcing?

• Why co-source?

• Reduce internal staffing costs• Reduce wait time and abandonment• Mitigate capacity constraints• Maintain service at peak demand

• Why co-source?• Reduce internal staffing costs

• Reduce wait time and abandonment• Mitigate capacity constraints• Maintain service at peak demand

• Why co-source?• Reduce internal staffing costs• Reduce wait time and abandonment

• Mitigate capacity constraints• Maintain service at peak demand

• 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)

• Examples of external contractors:• Convergys Corporation (150 locations)

• Wipro (4 million calls/mo. by 15k employees)

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)

The basic co-sourcing model

Poisson arrivals (λ)

x portion

Call Center’s Agents

1− x portion

Contractor’s Agents

x portion

1− x portion

x portion

1− x portion

...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

Non-stationary demandDaily call volumes at a major Pittsburgh healthcare provider

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?

Our research questionsWe explore a model involving a single call centerand an external contractor in a setting wheredemand changes over time.

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

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

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)

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

• Contractor has ∞ servers each w/ rate µc

• Calls abandon the queue w/ rate θ

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)

The Contract

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 ]

• 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 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 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 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 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,

• 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.

• where the contractor is serving N(t) calls at time t,

• and ψ(N(t)) = αN(t) + βN(t)2; α, β > 0.

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

The fluid model

• Let n→∞

• Stochastic variation vanishes; deterministicnon-stationarity variation dominates

The fluid model

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

The fluid model with abandonments• Consider Mt/M/1 +M w/ finite horizon [0, T ]

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

Q(t) B(t)x(t)λ(t)

capacity s

Q(t) B(t)x(t)λ(t)

drains at rate θQ(t)

drains at rate µB(t) = B(t)

capacity s

Q(t) B(t)x(t)λ(t)

capacity s

Q(t) B(t)x(t)λ(t)

capacity s

Q(t) B(t)x(t)λ(t)

capacity s

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)

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)

(P (t)λe(t)− E[αN(t) + βN(t)2]

• is solved at λe(t) = (1− x(t))λ(t).

The contractor’s problem• Contractor modeled w/ stochastic Mt/M/∞

• In reality a contractor can accept an arrivalrate λe(t),

maxλe(t)

The contractor’s problem• Contractor modeled w/ stochastic Mt/M/∞• In reality a contractor can accept an arrival

rate λe(t),

maxλe(t)

rate λe(t),

maxλe(t)

rate λe(t),

maxλe(t)

rate λe(t),

maxλe(t)

rate λe(t),

maxλe(t)

rate λe(t),

maxλe(t)

Solving the contractor’s problem

• maxλe(t)

• 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

(P (T )−

α+ β

)cosh (µc(T − t))

µP ′(T ) sinh(µc(T − t))−

tλe(r) sinh((r − t)µc)dr

• P (t) is market-clearing when λe(t) = (1− x(t))λ(t).

Solving the contractor’s problem• max

λe(t)

E[αN(t) + βN(t)2] = (α + β)h(t) + βh(t)2

• maxλe(t)

]• λe(t) = 1

2βmax{0, µ2

P (t) =α+ β

(P (T )−

α+ β

λe(t)

E[αN(t) + βN(t)2] = (α + β)h(t) + βh(t)2

• maxλe(t)

]• λe(t) = 1

2βmax{0, µ2

P (t) =α+ β

(P (T )−

α+ β

λe(t)

E[αN(t) + βN(t)2] = (α + β)h(t) + βh(t)2

• maxλe(t)

]• λe(t) = 1

2βmax{0, µ2

P (t) =α+ β

(P (T )−

α+ β

λe(t)

E[αN(t) + βN(t)2] = (α + β)h(t) + βh(t)2

• maxλe(t)

• 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

(P (T )−

α+ β

λe(t)

E[αN(t) + βN(t)2] = (α + β)h(t) + βh(t)2

• maxλe(t)

]• 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

(P (T )−

α+ β

λe(t)

E[αN(t) + βN(t)2] = (α + β)h(t) + βh(t)2

• maxλe(t)

]• λe(t) = 1

2βmax{0, µ2

P (t) =α+ β

(P (T )−

α+ β

λe(t)

E[αN(t) + βN(t)2] = (α + β)h(t) + βh(t)2

• maxλe(t)

]• λe(t) = 1

2βmax{0, µ2

P (t) =α+ β

(P (T )−

α+ β

λe(t)

E[αN(t) + βN(t)2] = (α + β)h(t) + βh(t)2

• maxλe(t)

]• λe(t) = 1

2βmax{0, µ2

P (t) =α+ β

(P (T )−

α+ β

The call center’s problem

• Given prices P (t), the call center must choosethe allocation x(t) that maximizes its profits.

maxx(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)

[P (t)x(t)λ(t)− CQ(t)

maxx(t)

revenue

staffing costs co-sourcing costs wait costs aband. costs

maxx(t)

revenue staffing costs

co-sourcing costs wait costs aband. costs

maxx(t)

revenue staffing costs co-sourcing costs

wait costs aband. costs

maxx(t)

revenue staffing costs co-sourcing costs wait costs

aband. costs

maxx(t)

Call center fluid model illustration revisited

Q(t) B(t)x(t)λ(t)

capacity s

Solving the call center’s problem

• maxx(t)

• 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

Solving the call center’s problem• max

and zero.

time intervals

(Ui, Oi+1)

0Q(t)dt

and zero.

time intervals

(Ui, Oi+1)

0Q(t)dt

and zero.

time intervals

(Ui, Oi+1)

0Q(t)dt

and zero.

time intervals

(Ui, Oi+1)

0Q(t)dt

and zero.

time intervals

(Ui, Oi+1)

0Q(t)dt

and zero.

time intervals

(Ui, Oi+1)

0Q(t)dt

and zero.

time intervals

(Ui, Oi+1)

0Q(t)dt

and zero.

time intervals

(Ui, Oi+1)

0Q(t)dt

Integrating over intervals

• maxx(t)

• 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!

• maxx(t)

∑Ki=1

(∫ Oi

Ui−1Q(t)dt+

∫ Ui

OiQ(t)dt

)• Q: What is

∫ Oi

OiQ(t)dt

i.e., Q(Oi) = Q(Ui) = 0

• maxx(t)

∑Ki=1

(∫ Oi

Ui−1Q(t)dt+

∫ Ui

OiQ(t)dt

)• Q: What is

∫ Oi

OiQ(t)dt

i.e., Q(Oi) = Q(Ui) = 0

• maxx(t)

∑Ki=1

(∫ Oi

Ui−1Q(t)dt+

∫ Ui

OiQ(t)dt

• Q: What is∫ Oi

OiQ(t)dt

i.e., Q(Oi) = Q(Ui) = 0

• maxx(t)

∑Ki=1

(∫ Oi

Ui−1Q(t)dt+

∫ Ui

OiQ(t)dt

)• Q: What is

∫ Oi

OiQ(t)dt

i.e., Q(Oi) = Q(Ui) = 0

• maxx(t)

∑Ki=1

(∫ Oi

Ui−1Q(t)dt+

∫ Ui

OiQ(t)dt

)• Q: What is

∫ Oi

OiQ(t)dt

i.e., Q(Oi) = Q(Ui) = 0

• maxx(t)

∑Ki=1

(∫ Oi

Ui−1Q(t)dt+

∫ Ui

OiQ(t)dt

)• Q: What is

∫ Oi

OiQ(t)dt

i.e., Q(Oi) = Q(Ui) = 0

• maxx(t)

∑Ki=1

(∫ Oi

Ui−1Q(t)dt+

∫ Ui

OiQ(t)dt

)• Q: What is

∫ Oi

OiQ(t)dt

i.e., Q(Oi) = Q(Ui) = 0

• maxx(t)

∑Ki=1

(∫ Oi

Ui−1Q(t)dt+

∫ Ui

OiQ(t)dt

)• Q: What is

∫ Oi

OiQ(t)dt

i.e., Q(Oi) = Q(Ui) = 0

Integrating Q(t) over (Oi, Ui)

• maxx(t)

• 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

∫ Ui

Oi(x(t)λ(t)− s)dt

maxx(t)

(P (t)x(t)λ(t)− C

K∑i=1

∫ Ui

(x(t)λ(t)− s)dt

• maxx(t)

Oix(t)λ(t)dt

• A:∫ Ui

Oi(θQ(t) + s) dt =

∫ Ui

Oix(t)λ(t)dt

• Solving:∫ Ui

OiQ(t)dt = 1

∫ Ui

maxx(t)

K∑i=1

∫ Ui

(x(t)λ(t)− s)dt

• maxx(t)

Oix(t)λ(t)dt

• A:∫ Ui

Oi(θQ(t) + s) dt =

∫ Ui

Oix(t)λ(t)dt

• Solving:∫ Ui

OiQ(t)dt = 1

∫ Ui

maxx(t)

K∑i=1

∫ Ui

(x(t)λ(t)− s)dt

• maxx(t)

Oix(t)λ(t)dt

• A:∫ Ui

Oi(θQ(t) + s) dt =

∫ Ui

Oix(t)λ(t)dt

• Solving:∫ Ui

OiQ(t)dt = 1

∫ Ui

maxx(t)

K∑i=1

∫ Ui

(x(t)λ(t)− s)dt

• maxx(t)

Oix(t)λ(t)dt

• A:∫ Ui

Oi(θQ(t) + s) dt =

∫ Ui

Oix(t)λ(t)dt

• Solving:∫ Ui

OiQ(t)dt = 1

∫ Ui

maxx(t)

K∑i=1

∫ Ui

(x(t)λ(t)− s)dt

• maxx(t)

Oix(t)λ(t)dt

• A:∫ Ui

Oi(θQ(t) + s) dt =

∫ Ui

Oix(t)λ(t)dt

• Solving:∫ Ui

OiQ(t)dt = 1

∫ Ui

maxx(t)

K∑i=1

∫ Ui

(x(t)λ(t)− s)dt

• maxx(t)

Oix(t)λ(t)dt

• A:∫ Ui

Oi(θQ(t) + s) dt =

∫ Ui

Oix(t)λ(t)dt

• Solving:∫ Ui

OiQ(t)dt = 1

∫ Ui

maxx(t)

K∑i=1

∫ Ui

(x(t)λ(t)− s)dt

• maxx(t)

Oix(t)λ(t)dt

• A:∫ Ui

Oi(θQ(t) + s) dt =

∫ Ui

Oix(t)λ(t)dt

• Solving:∫ Ui

OiQ(t)dt = 1

∫ Ui

maxx(t)

K∑i=1

∫ Ui

(x(t)λ(t)− s)dt

• maxx(t)

Oix(t)λ(t)dt

• A:∫ Ui

Oi(θQ(t) + s) dt =

∫ Ui

Oix(t)λ(t)dt

• Solving:∫ Ui

OiQ(t)dt = 1

∫ Ui

maxx(t)

K∑i=1

∫ Ui

(x(t)λ(t)− s)dt

Back to the call center’s optimization

• maxx(t)

K∑i=1

∫ Ui

(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 :

eθ(r−Oi)(λ(r)x(r)− s)dr < 0

}Oi+1 = inf

{t ≥ Ui :

e(r−Ui)(λ(r)x(r)− s)dr > 0

}0 ≤ x(t) ≤ 1

• maxx(t)

K∑i=1

∫ Ui

(x(t)λ(t)− s)dt

• We need to find, not only x(t), but the Ui’sand Oi’s

Ui = inf

{t ≥ Oi :

}Oi+1 = inf

{t ≥ Ui :

}0 ≤ x(t) ≤ 1

• maxx(t)

K∑i=1

∫ Ui

(x(t)λ(t)− s)dt

and Oi’s

Ui = inf

{t ≥ Oi :

}Oi+1 = inf

{t ≥ Ui :

}0 ≤ x(t) ≤ 1

• maxx(t)

K∑i=1

∫ Ui

(x(t)λ(t)− s)dt

and Oi’s

Ui = inf

{t ≥ Oi :

}Oi+1 = inf

{t ≥ Ui :

}0 ≤ x(t) ≤ 1

• maxx(t)

K∑i=1

∫ Ui

(x(t)λ(t)− s)dt

and Oi’s

Ui = inf

{t ≥ Oi :

}Oi+1 = inf

{t ≥ Ui :

}0 ≤ x(t) ≤ 1

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

x(t) = 0 if P (t)− CI − e−(Oi+1−t)ΓULi < 0

x(t) ∈ [0, 1] otherwise.

Solution to the optimization problem• We solve a continuous-time stochastic dynamic program tracking

x(t) = 1 if P (t)− (CI + Cθ

x(t) = 0 if P (t)− (CI + Cθ

x(t) = 1 if P (t)− (CI + Cθ

x(t) = 0 if P (t)− (CI + Cθ

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

x(t) = 0 if P (t)− (CI + Cθ

x(t) = 1 if P (t)− (CI + Cθ

x(t) = 0 if P (t)− (CI + Cθ

x(t) = 1 if P (t)− (CI + Cθ

x(t) = 0 if P (t)− (CI + Cθ

Solution insights

x(t) = 1 if P (t)− (CI + C

x(t) = 0 if P (t)− (CI + Cθ

xt ∈ [0, 1] otherwise.x(t) = 1 if P (t)− CI − e−(Oi+1−t)ΓUL

• 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.

Solution insightsx(t) = 1 if P (t)− (CI + C

x(t) = 0 if P (t)− (CI + Cθ

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

(P (T )−

α+ β

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

Existence of equilibrium• Finally, we address price-allocation equilibrium existence

P (t) =α+ β

(P (T )−

α+ β

• Currently limited by technical assumptions on state at time T

• Basic idea exploits form of price function:

P (t) =α+ β

(P (T )−

α+ β

P (t) =α+ β

(P (T )−

α+ β

P (t) =α+ β

(P (T )−

α+ β

P (t) =α+ β

(P (T )−

α+ β

P (t) =α+ β

(P (T )−

α+ β

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

P (t) =α+ β

(P (T )−

α+ β

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

P (t) =α+ β

(P (T )−

α+ β

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

P (t) =α+ β

(P (T )−

α+ β

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)

P (t) =α+ β

(P (T )−

α+ β

P (t) =α+ β

(P (T )−

α+ β

• 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)?

Any questions?

Thank you!

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