Re k Gull u Bo gazi˘ci University Industrial Engineering ... · Pricing in Service Systems with...

Pricing in Service Systems with Strategic Customers

Refik Gullu

Bogazici UniversityIndustrial Engineering Department

Istanbul, Turkey

Refik Gullu (Bogazici University) YEQT VIII Eindhoven 1 / 86

Service Systems with Strategic Customers

Before the talk

My professor’s advice on queueing theory versus game theory

mathematical difficulty versus conceptual maturity

How to learn new stuff?

teaching, writing a code, thesis supervision



Before the talk

A review from a personal perspective

A great place to start reading:Rafael Hassin and Moshe Haviv, To queue or not the queue:equilibrium behavior in queueing systems, Springer, 2003.http://www.math.tau.ac.il/ hassin/book.html

A follow up survey book, “Rational Queueing” to appear soon



Outline

A simple example of an “unobservable” queue

Parameter uncertainty

Effect of delay information

Observable queues: residual service time

Multiple customer types: identical price

Multiple customer types: differentiation

An inventory model



A Framework Example

A student cafeteria in a university

University administration regulates the price

possibly in the form of a subsidy

Students arrive according to a Poisson process

Λ is the rate of potential students

A single server with exponential rate µ > Λ.



A Framework Example

Cafeteria is sort of far away from the main building

Students decide eating there or not before observing the congestion

There are other dining facilities on campus

Once a decision is made, it can not be changed

Students are identical

with respect to their valuation of the service, value of time, and theirbehaviour towards riskall are rational decision makers



A Framework Example

R : the value of service (as judged by students)

c : the unit cost for waiting

p : fee for dining at the cafeteria

The expected utility of a student from the service

R− p− cE[sojourn time]

the system is at steady state



A Framework Example: equilibrium behaviour

As students are “identical”, their equilibrium behaviour is expected tobe the same

Each student choose to enter the cafeteria with probability q

Let U(qtagged, qothers) be the utility of a tagged student when all theothers behave with qothers

Best response against qothers: U(q′, qothers) ≥ U(q, qothers) for all q.

Symmetric Nash equilibrium: best response against itself

U(qe, qe) ≥ U(q, qe) for every q



A Framework Example: equilibrium behaviour

0 ≤ qe(p) ≤ 1 is the equilibrium probability of joining the cafeteriawhen the fee is p

λe(p) = Λqe(p) < µ

For effective arrival rate λ < µ

w(λ) = 1/(µ− λ)

3 cases need to be examined



Case 1

Nobody else is joining and

p+ cw(0) > R =⇒ R < p+ c1

µ

µ <c

R− p

qe(p) = 0λe(p) = 0

w(λe(p)) = 1/µ



Case 2

If everybody else is joining and

p+ cw(Λ) ≤ R =⇒ R ≥ p+ c1

µ− Λ

µ ≥ Λ +c

R− p

qe(p) = 1λe(p) = Λ

w(λe(p)) = 1/(µ− Λ)



Case 3

p+ cw(0) ≤ R < p+ cw(Λ)

R = p+ cw(λe(p))

qe(p) = λe(p)/Λw(λe(p)) = 1/(µ− λe(p))

λe(p) = µ− c

R− p



Administration’s Problem: social optimization

The administration cares about the overall performance

Solves the following problem

max0≤λ≤Λ

{λ(R− c 1

µ− λ)}

strictly concave

maximum occurs at

λ∗ = µ−√cµ

R

λ∗ ≥ 0 (by R ≥ c/µ)




By considering the constraint λ ≤ Λ

λ∗ = min{Λ, µ−√cµ

R}

if Λ ≥ µ−√

cµR

optimal objective function value

(√Rµ−

√c)2

w(λ∗) =√

Rcµ




if Λ ≤ µ−√

cµR

optimal objective function value

Λ(R− c

µ− Λ)

w(λ∗) = 1µ−Λ



By assuming R ≥ c/µ

λe(0) = µ− c

R≥ µ−

√cµ

R= λ∗

Individual optimization leads to longer queues than imposed by socialoptimization

Admission fee can regulate this



Revenue maximization

Let pm be the admission fee charged for dining

pm = R− cw(λ)max

0≤λ≤Λpmλ

Same as the social optimization objective

The socially optimal arrival rate can be induced by the fee

pm = p∗ = R− cw(λ∗) = R−

√cR

µ

λe(p∗) = λ∗, profit = (

√Rµ−

√c)2



Rafael Hassin and Moshe Haviv, To queue or not the queue: equilibriumbehavior in queueing systems, Springer, 2003. (Chapter 3)

Bell, C. E., Stidham, Jr., 1983, Individual versus Social Optimization inthe Allocation of Customers to Alternate Servers, Management Science,29, 831-839.



Parameter Uncertainty

The preceding model considered an “unobservable” system

The queue length or the waiting times upon arrival are unobservable

Need to be careful

Many things are known and/or intelligently computable: service rate,expected waiting time, service value, etc.

These parameters are known with certainty



Uncertainty in the Service Rate

Suppose µ, the service rate, can take two values

µ =

{µ1 with probability αµ2 with probability 1− α

µ1 > µ2

Do students know the realised service rate?

No, they are “uninformed”Yes, they are “informed”, and the server charges either a different feeor the same fee



Service Rate Uncertainty

For the “uninformed” case

v = (R− p)/cv = α

µ1−λ + 1−αµ2−λ in equilibrium

v is a solution of a nonlinear equation

Πun = λ(R− cv)




“informed” case with two prices

λi = µi − cR−pi , pi = R−

√cRµi

Πin2 =

α(√Rµ1 −

√c)2 + (1− α)(

√Rµ2 −

√c)2 if R ≥ c

µ2

α(√Rµ1 −

√c)2 if c

µ1< R < c

µ2

0 if R ≤ cµ1




“informed” case with a single price

λi = µi − cR−p (rate of arrivals given the service rate)

If p is small enough to attract customers for both values of µ

average arrival rate for the single price p

λ = α(µ1 −c

R− p) + (1− α)(µ2 −

c

R− p)

= µ− c

R− p




Maximizing pλ = pµ− pcR−p

(R− p)2 = Rc/µ

=⇒ p = R−

√Rc

µ

Resulting profit(√Rµ−

√c)2




But a higher price may be chosen: so that customers opt out whenµ = µ2.

p = R−√

Rcµ1

Resulting profit: α(√Rµ1 −

√c)2

Two profit terms are equal for

R

c= η =

(1−√α√

µ−√µ1α

)2




Πin1 =

(√Rµ−

√c)2 if R

c ≥ ηα(√Rµ1 −

√c)2 if 1

µ1≤ R

c ≤ η0 if R

c ≤1µ1



Parameters Uncertainty

The service provider benefits from revealing the service rate, and frompricing accordingly

Πin2 ≥ Πin

1 ≥ Πun

As the variability in service rate increases, Πin2 increases

The server provider may lose (1− α) fraction of the customers butextracts higher revenue from the remaining



Parameters Uncertainty

Hassin, R., 2007, Information and Uncertainty in a Queuing System,Probability in the Engineering and Informational Sciences, 21, 361-380.

waiting cost uncertainty

service valuation uncertainty



Delay Information

So far: “unobservable” systems

What if the customers are revealed information on the possible delaybefore they decide to join or not

”observable” systems

Three levels of information

1. No information (same as before)

2. Partial information: how many customers are in front ofme?

3. Full information: what is my exact waiting time?



Delay Information

M/M/1 type service system

W : waiting time (in the queue)

θ: customer type parameter, a random variable, θ ∈ [0, 1] with cdf H,pdf h

c(t): cost of waiting t time units

U = R− θE[c(W )]

Previously: θ ≡ 1, c(t) = ct



Delay Information

Scale R and c(t) so that R = 1

Assume that c(0) > 1

Customers with θ > 1/c(0) balk

Scale Λ (ignore them) to λ, and assume (new) c(0) = 1

Customers with θ ≈ 1 are also attracted to join when W = 0.

U(no waiting) = 1− θ



Delay Information: who stays in the system?

Information I, a random variable

Want: U |I = 1− θEW [c(W )|I] ≥ 0

Given information I = i, an arriving customer stays if

θ ≤ θi =1

EW [c(W )|I = i]

Pr{stays|I = i} = H(θi)

Fraction of customers who stay: EI [H(θI)]

ThroughputλEI [H(θI)]



Average utility

Define

J(θ) =1

θ

∫ θ

0H(x)dx

Average utility

u = E[U+] = Eθ,I [(1− θEW [c(W )|I])+]

= EI

[∫ θI

0(1− xEW [c(W )|I])h(x)dx

]= EI

[H(θI)− (1/θI)

∫ θI

0xh(x)dx

]= EI [J(θI)]



Case 1: No information

The equilibrium arrival rate:

λNI = λH

(1

E[c(WNI)]

)ρNI = λNI/µ

c(s) =∫∞

0 e−stc(t)dt LST of c(t).

Pr{WNI > t} = ρNIe−µ(1−ρNI)t

E[c(WNI)] = (1− ρNI) + ρNIµ(1− ρNI)c(µ(1− ρNI))



Case 1: No information

An example: Uniform customers with Linear Cost

c(t) = 1 + t

c(s) = 1s + 1

s2

λNI =λ

1 + ρNI/(µ(1− ρNI))

=⇒ (1− µ)(ρNI)2 + (µ+ λ)ρNI − λ = 0

=⇒ ρNI =−(µ+ λ) +

√(λ+ µ)2 + 4λ(1− µ)

2(1− µ)

πNIn = (1− ρNI)(ρNI)n



Case 2: Partial information

The service provider tells the arriving customer N(t)

The customer computes cn = E[c(W )|N(t) = n]

Stays if θ ≤ θn = 1/cn

Birth-death process with state dependent arrival rate λn = λH(θn)

Steady-state probabilities

Θn =

n−1∏m=0

H(θm), Θ =

∞∑n=1

Θn(λ/µ)n

πPI0 = 1/(1 + Θ)

πPIn = Θn(λ/µ)nπPI0



Case 2: Partial information

An example: Uniform customers with Linear Cost

c(t) = 1 + t

θm = 11+m

µ, Θn =

∏n−1m=0

11+m

µ

Θn(λ/µ)n = Γ(µ)Γ(µ+n)λn

πPI0 =1

1 + γ(µ, λ)λ1−µeλ, πPIn = πPI0

Γ(µ)

Γ(µ+ n)λn

Γ(x) =

∫ ∞0

tµ−1e−tdt and γ(µ, λ) =

∫ λ

0tµ−1e−tdt



Case 3: Full information

The service provider tells the arriving customer workload V (t) = v

Critical point: θv = 1/c(v)

Effective arrival rate λ(v) = λH(θv)

f(v): the pdf of the stationary workload V

under linear cost and uniform customers

πFI0 =1

1 + λeµµ−(λ+1)γ(λ+ 1, µ)

fFI(v) = λπFI0 (1 + v)λe−µv, v > 0



Benefit of Information

The service provider would like to have large throughput λEI [H(θI)]

Customers would like to have a large average utility EI [J(θI)]

EI

[1

θI

∫ θI

0H(x)dx

]What is the impact of information on these measures?

Clearly the answer depends on H(x).




There are cases where the service provider and the customers arealigned

H(x) = xα, α > 0

J(θ) = 1θ

∫ θ0 x

αdx = 1α+1θ

α = 1α+1H(θ)

Average utility ∼ Throughput

More information is better for all parties




Consider constant θ

No information: throughput is λ for sufficiently small λ

Partial information: there is a threshold n∗ beyond which customersdo not join

throughput < λ

The service provider may hide information

Essentially whether information beneficial to one party or the otherdepends on the shape of H(θ)




If H(1/x) is convex in x ≥ 1, then more information benefits theservice provider by increasing throughput

If J(H−1(y)) is convex on [0, 1], then more information benefits thecustomers by increasing the average utility.

H(x) =γe−γx

1− e−γx ∈ [0, 1]

=⇒ γ ≤ 2 (h(x) does not decrease too rapidly)




If H(1/x) is convex, then πPI0 ≤ πNI0

πNI0 = 1− λ

µH

(1

E[cNNI ]

)πPI0 = 1− λ

µE

[H

(1

cNPI

)]πPI0 > πNI0 =⇒ NPI ≤st NNI =⇒ E[cNPI ] ≤ E[cNNI ]

=⇒ H

(1

E[cNNI ]

)≤ E

[H

(1

cNPI

)]by convexity of H(1/x) and Jensen’s inequality

a contradiction



Guo, P., Zipkin, P., 2007, Analysis and Comparison of Queues withDifferent Levels of Delay Information, Management Science, 53, 962-970.

Guo, P., Zipkin, P., 2009, The Effects of the Availability of Waiting-timeInformation on a Balking Queue, EJOR, 198, 199-209.



Observable Queues: Residual Service Time

Consider an observable M/G/1 queue

The arriving customer observes the queue length before joining

If the service time is exponential

Customer joins if the number in the system is less than⌊R

cE[Service Time]

⌋With non-exponential service times

Residual service time matters




A customer who observes n customers upon arrival joins withprobability qn

(q1, q2, . . .)

The behaviour of others has an effect on the assessment of residualservice time

E[RSTn] = E[residual ST when the arriving customer finds n in the system]

E[RSTn] = fn(q1, q2, . . . , qn) (a recursive expression)

E[RST1] =E[ST ]

1− G(Λq1)− 1

Λq1

Suppose deterministic service time and q1 is highinformation about the current service state




Once q1, q2, . . . , qn−1 are known

if nE[ST ] + fn(q1, . . . , qn−1, 1) ≤ R/c =⇒ qn = 1if nE[ST ] + fn(q1, . . . , qn−1, 0) ≥ R/c =⇒ qn = 0otherwise nE[ST ] + fn(q1, . . . , qn−1, q) = R/c =⇒ qn = q

Intuition: q1 ≥ q2 ≥ q3 ≥ · · ·Which turns out to be wrong!




ST = 1 with probability ε (small), ST = 0 with probability 1− εE[RST1] = 1

1−e−Λq1− 1

Λq1

Solve ε+ E[RST1] = R/c to find Λq1.

Λ ≤ Λ1 =⇒ q1 = q2 = 1Λ1 < Λ ≤ Λ2 =⇒ 0 < q1 < 1, q2 = 1Λ > Λ2 =⇒ 0 < q1 < q2 < 1

The fact that there are two customers (one in service) means that thewe are probably nearing the end of the current service time, and theservice time of the next customer is very likely to be zero anyway




If the service time distribution is of type “decreasing mean residuallife” (DMRL)

that is, E[ST − t|ST > t] is monotone decreasing in t

There is ne as the smallest integer satisfying

nE[S] + fn(1, 1, 1, . . . , 1) ≥ R/c

qe ∈ [0, 1) satisfying

neE[S] + fn(1, 1, 1, . . . , qe) = R/c

qn =

1 n < neqe n = ne0 n > ne




Intuition: waiting time difference between states n and n+ 1

(n+ 1)E[ST ] +E[RSTn+1]−nE[ST ]−E[RSTn] > E[ST ]−E[RSTn] ≥ 0

Waiting times are increasing in nThere exist at most one n with mixed strategy[RSTn] is increasing in q for (q1, . . . , qn−1, q)Hence qe is unique.

“Avoiding the crowd” versus “following the crowd”



Naor, P., 1969, The Regulation of Queue Size by Levying Tolls,Econometrica, 37, 15-24.

Kerner, Y., 2011, Equilibrium Joining Probabilities for an M/G/1 queue,Games and Economic Behavior, 71, 521-526.

Haviv, M., Kerner, Y., 2007, On Balking from an Empty Queue, QueueingSystems, 55, 239-249.

Kerner, Y., 2008, The Conditional Distribution of the Residual ServiceTime in the Mn/G/1 Queue, Stochastic Models, 24, 364-375.

Manou, A., Economou, A., Karaesmen, F., 2014, Strategic Customers in aTransportation Station: When Is It Optimal to Wait?, OperationsResearch, 62, 910-925.



Multiple Customer Classes-identical price

Things get complicated with multiple customer classes

Consider two classes of customers with M/G/1 type service facility

R1, R2, c1, c2, Λ1, Λ2

Suppose that the customers are either

indistinguishable to the service provideror price discrimination is not possible

Service times are identically distributed

Customers are treated in a FCFS manner

They can not observe the queue length




λ = λ1 + λ2, the equilibrium arrival rate

zi = Ri − (ci/µ), i = 1, 2

ui(λ, p) = Ri − p− ci(wQ(λ) + (1/µ)) = zi − p− ciwQ(λ)

z1 ≥ z2, and p ≤ zi (otherwise does not join)

ui(λ, p) is strictly decreasing and concave in λ

since wQ(λ) is convex increasing




For any price p and 0 ≤ λ < µ,

c1 ≤ c2 =⇒ u1(λ, p) ≥ u2(λ, p)If c1 > c2, there is a critical value λ so that

λ > λ =⇒ u1(λ, p) < u2(λ, p)

As system gets more congested, the one with higher sensitivity todelay hurts more




By solving ui(λ, p) = 0 for λ and p

λi(p) =2µ2(zi − p)

2µ(zi − p) + ci(1 + cv2)

pi(λ) = zi −λci(1 + cv2)

2µ(µ− λ)

maximum arrival rate for i for a given price p

maximum price i is willing to pay for total arrival rate λ




How do customers behave for a given price p?

Depends on c1 and c2

If c1 ≤ c2, then Λ1 and λ2(p) are compared

λ1(p) ≥ λ2(p)

Λ1 ≥ λ2(p) =⇒ (q1e , q

2e) = (min{1, λ1(p)/Λ1}, 0)

Λ1 < λ2(p) =⇒ (q1e , q

2e) = (1,min{1, (λ2(p)− Λ1)/Λ2})




Suppose Λ1 ≥ λ1(p) ( =⇒ ≥ λ2(p))

First: Class-1 enters with rate λ1(p)

u2(λ1(p), p) ≤ u1(λ1(p), p) = 0

=⇒ q2e = 0

Next: Class-2 does not enter the system

u1(λ, p) > 0 ⇐⇒ λ < λ1(p)

u1(λ, p) is decreasing in λ

=⇒ q1e = λ1(p)/Λ1




How the server provider sets the price?

maxp λ(p)p

If c1 ≤ c2, then depends on the market size of Type-1

Λ1“large” =⇒ pe = max{p∗1, p1(Λ1)}

and (q1e , q

2e) = (min{1, λ1(p∗1)/Λ1}, 0)

p∗1 = arg max pλ1(p)

p∗1 = z1 −√c1(1 + cv2)(c1(1 + cv2) + 2µz1)− c1(1 + cv2)

2µ



Zhou, W., Chao, X., Gong, X., 2014, Optimal Uniform Pricing Strategy ofa Service Firm when Facing Two Classes of Customers, POM, 23, 676-688.



Multiple Customer Types-differentiation

M customer types

Ri, ci, ΛiSome questions:

what will be the “control policy”what will be the price/delay menu?what is the information structure: who knows what?

Incentive compatibility

pi + ciwi ≤ pj + ciwj j 6= i

individual rationalityRi ≥ pi + ciwi



Multiple Customer Types-Revenue Maximization Model

maxp,W,u

∑Mi=1 piλi

λi = Λi Pr{Ri ≥ pi + ciwi} ∀ ipi + ciwi ≤ pj + ciwj j 6= i

M∑i=1

λi < µ



Multiple Customer Types-General Results

If the service provider observes the types of the customers

Set priorities with a work conserving discipline (cµ rule)

If the service provider does not observe the types

Set priorities with strategic delaysWork conserving discipline may not be optimalDelay cost minimization is not the dominant criterionStrategic delay (for low priority items) deters high priority customerspurchasing a low priority menuThis accomplishes incentive compatibility



Multiple Customer Types-An example (Allon, 2010)

M/M/1 queue with µ = 1

λ1 = 0.2, R1 = 100, c1 = 20

λ2 = 0.3, R2 = 30, c2 = 4

Under cµ rule, w1 = 1/(µ− λ1) = 1.25,w2 = 1/(µ(1− ρ1)(1− ρ1 − ρ2)) = 2.5

p1 = R1 − c1w1 = 100− 20(1.25) = 75p2 = R2 − c2w2 = 30− 4(2.5) = 20Revenue: 0.2(75) + 0.3(20) = 21

Overall delay cost=2.5(4) + 1.25(20) = 35

Not IC: 75 + 20(1.25) > 20 + 20(2.5) = 70



Multiple Customer Types-An example

Highest IC price under cµ:

p1 = p2 + c1w2 − c1w1 = 20 + 20(2.5)− 20(1.25) = 45

with revenue=45(0.2) + 20(0.3) = 15

If one can have w2 ← 2.5 + 1 = 3.5

Price for Type-2 becomes: p2 = 30− (3.5)4 = 16

IC Type 1 price

p1 = 16 + 20(3.5)− 20(1.25) = 61

Revenue=61(0.2) + 16(0.3) = 17.

Overall delay cost=3.5(4) + 1.25(20) = 39



A manufacturing and service system

A common part is used for service activity of heterogenous customers

AudiA4 and VW Passat use the same engine, transmission and someother featuresDesign for after-sales-service

Customers have different sensitivity for waiting and service valuation

Service provider keeps a common spare parts inventory

Operates with a base stock policy (base stock level y)

Parts are replenished through a finite capacity system

M/M/1 but can be generalized



A manufacturing and service system

Whenever there is on-hand stock, customer demand is satisfiedirrespective of the type

If on-hand stock is zero, customers have to wait

Non-preemptive priorities

A customer is tagged with an incoming part (irrespective of the type)

y = net inventory + outstanding parts (N)



An Illustration

Manufacturer

N=0



An Illustration

Manufacturer

N=1

W=0



An Illustration

Manufacturer

N=1



An Illustration

Manufacturer

N=2

W=0



An Illustration

Manufacturer

N=2



An Illustration

Manufacturer

N=3

W=0



An Illustration

Manufacturer

C1

N=4



An Illustration

Manufacturer

N=5

C1C2



An Illustration

Manufacturer

N=6

C1C2C3



An Illustration

Manufacturer

N=5

C1C2

W>0



The service provider determines

base stock level yprice menu p = (p1, . . . , pM )the priority scheme

Customers react by arriving with λ = (λ1, . . . , λM )

Ri = pi + ciE[waiting time]

i = 1, 2, . . . ,M



The distribution of the outstanding parts

M/M/1: Pr{N = k} = ρk(1− ρ)

ρ =∑Mi=1 λi/µ

By PASTA property

E[waiting with y ≥ 0] = Pr{N ≥ y}E[waiting in a standard queue]

= ρyE[waiting in a standard queue]



The service provider’s problem

maxy,λ,u

{M∑i=1

λiRi − ρyM∑i=1

ciλiwi − hE[(y −N)+]

}

u = (u(1), u(2), . . . , u(M)) the priority order



Main Results

Restricted to work-conserving disciplines:

cµ rule is optimal: c1 ≥ c2 ≥ · · · cMGiven λ, optimal base stock level:

y∗(λ) = min

{y ≥ 0 : ρy+1 ≤ h

h+ vH(λ)

}

v = (1− ρ)/ρ, H(λ) =

M∑i=1

ciλiE[waiting(λ)]

Prices given by the first order conditions are incentive compatible



A reduction:

If Ri ≥ Rj and ci ≤ cj , then Type-i dominates Type-j

λ∗j = 0 (p∗j = Rj)

ci = c, Rk = max{Ri} or Ri = R, ck = min{ci}

maxy≥0,ρ∈[0,1)

{Rµρ− cρy ρ

1− ρ− h(y − ρ

1− ρ(1− ρy))

}y(ρ) = min{y ≥ 0 : ρy+1 ≤ h/(c+ h)}



A continuous approximation: Pr{N ≥ y} ≈ e−vy

Variable Value ∂/∂h ∂/∂c

λ µ−√µK/R ↓ ↓

y(√RµK −K

)/h ↓ ↑, ↓

E[W ]

√R/

(µK) ↓ ↓

p R− c√R/

(µK) ↑ ↓

K h log(1 + c/h)



K = h log(1 + c/h)→ c as h→∞Compare p above with

p = R−

√cR

µ= R− c

√R

µc

Optimal profit: (√Rµ−

√K)2

K < c =⇒ (profit with y > 0) ≥ (profit with y = 0)

Attracts more demand (with smaller price) and achieves a higher totalprofit.



Mendelson, H., Wang, S., 1990, Optimal Incentive-Compatible PriorityPricing for the M/M/1 Queue, Operations Research, 38, 870-883.

Afeche, P., 2013, Incentive-Compatible Revenue Management in QueueingSystems: Optimal Strategic Delay, MSOM, 15, 423-443.

Allon, G., 2010, Pricing and Scheduling Decisions, Wiley Encyclopedia ofOperations Research and Management Science edited by James J.Cochran, Wiley.

Maglaras, C., Yao, J., Zeevi, A., 2013, Optimal Price and DelayDifferentiation in Queueing Systems, Working Paper.

Guler, G., Bilgic, T., Gullu, R., 2014, Joint Inventory and Pricing Decisionswhen Customers are Delay Sensitive, to appear in IJPE.



Some less explored topics

Alternative cost and reward structures

“willingness to wait”: E[(W −WtoW )+]

Correlation of R and c

Competition and cooperation among service providers

Distribution free bounds

maxλ

minfR

λE[(R− cw(λ))+]

Estimation errors in parameters

Bounded rationality (Huang, Allon and Bassamboo, 2014)


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