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For Review OnlyPricing and Capacity Sizing of a Service Facility: Customer

Abandonment Effects

Journal: Production and Operations Management

Manuscript ID POM-Oct-16-OA-0700.R1

Wiley - Manuscript type: Original Article

Keywords: Customer Abandonment, Pricing and Capacity Sizing, Profit Maximization, Heavy Traffic

Abstract:

This paper studies the effect of customer abandonment in the economic optimization of a service facility. Specifically, we consider how to jointly set the price and service capacity in order to maximize the steady-state expected profit, when the system is subject to customer abandonment and the price is paid only by customers that eventually receive service. We do this under the assumption that there is a high rate of prospective customer arrivals. Our analysis reveals that the economically optimal operating regime is consistent with the standard heavy traffic regime. Furthermore, we derive the following economical insight: when the capacity cost is sufficiently high, it can be advantageous for the system manager to "underinvest" in capacity and "take advantage of the abandonments" to trim congestion. Lastly, we show the loss in profit when customer abandonments are ignored is on the order of the square-root of the demand rate.

Production and Operations Management

For Review Only

Authors’ response to the reviewers’ reports on the manuscript“Pricing and Capacity Sizing of a Service Facility: Customer Abandon-ment Effects.”

We thank the Associate Editor and two reviewers for their valuable comments and sugges-tions that have enhanced the quality of the paper. Our responses to the issues raised by thereview team are detailed below. References to pages, equation numbers, and bibliographieshere are consistent with those in the revised manuscript.

In addition to carefully addressing the raised comments, when we were doing the revi-sion, we came across a technical problem in the paper Katsuda (2016) we were originallyreferencing in the proof of our main result. Specifically, we need Lemma 2 on the conver-gence of steady-state distributions and their moments. This allows us to establish our mainresults in Theorems 2 and 3 regarding asymptotic optimality of the proposed policy (22).The manuscript Katsuda (2016) (which was under revision) turned out as never published,and catching and figuring out the technical problem therein was the main reason why therevision took up long time. In the revision, we made sure of a solid methodological groundbased on the recent work of Huang and Gurvich (2018) published in Operations Research.This resulted in doing away with some technical assumptions on the model primitivesconsidered in the original submission.

Moreover, we were able to show that the two associated diffusion control problems(DCPs) indeed have a unique maximizer. The proofs required nontrivial analysis, such asLaplace method of asymptotic expansion for integrals. We provide more detailed expla-nation in the response to Reviewer 2’s comment 1. As a consequence, the heavy trafficregime is an economically optimal operating regime, whose optimal behavior (such as op-timal capacity imbalance) is characterized by the unique solution of the associated DCP.

Associate Editor

Comment 1: Writing of the paper. Both the reviewers comment on the writing of

the paper. The paper in its current structure is not well suited for the general audienceof the POMS journal. Further, as pointed by Reviewer 1, adding more explanation forthe non-expert readers would be helpful. In addition, the authors need to provide suffi-cient background and proper definition when introducing technical details (see Reviewer 1comment on this issue).

Response: In the revision, we made a major effort to make the paper more relevantfor the general audience of the POMS journal who might be less familiar with the languageof asymptotic analysis. Our goal was that anyone who understands a single server queuewith abandonment should be able to get the idea of the Introduction and appreciate thevalue of our contribution. Specifically, in the Introduction:

(i) We set up a pre-limit example (no limit “n → ∞”) in the context of make-to-orderproduction firm. See the Example. In that pre-limit model, we have a simulation

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result that compares between profit losses when there are abandonments and arenot. See Figure 2, which is developed without the “n” and shows a clear profitimprovement under a proposed policy with a suitable correction term accounting forstochasticity. This motivates the reader to start going through our asymptotic regimeframework and the solution to the associated optimization problem (diffusion controlproblem) in the subsequent sections.

(ii) We explain how to treat n in a given system (i.e., we define n as the maximum marketsize) and how the scaling regime we study relates to a practical proposed policy. Seethe paragraphs Our Approach and The Proposed Policy. These together withthe Example will also address to Reviewer 1’s comments (i) and (ii) below.

(iii) To enhance the coherence of a presentation and a readability, we revisit the Examplein Section 4 in order to highlight the key features of the proposed policy, and againin Section 6 in order to connect with the asymptotic performance results developedin Sections 3–5. See Figures 3 and 6.

We also made an effort to introduce the needed technical details in order to keep rigor.See also the response to Reviewer 1’s Overall Comment below. Lastly, we streamlinedthe needed assumptions on the primitives λ(·) and F (·) that are necessary for the SPPand the two DCP’s, as assumptions (A1), (A2), (A3), respectively. We have made explicitreferences to them in all statements.

Comment 2: Comparison with Markovian system and also with system withoutabandonment. The paper shows that the gap when one ignores the abandonment is of theorder of square root of n. Mathematically this statement is precise but it will be importantto explain the implications for a finite system (both reviewers comment on this). Also,this is linked to the comment 1. Lastly, given that only the behavior near zero of theabandonment distribution plays a role. A natural question to ask is whether modelinggeneral abandonment distribution plays a significant role. This question can perhaps beanswered via a numerical study.

Response: The issue regarding the implications for a finite system is addressed as initems (i) and (ii) in the response detailed above. See also item 2 in Our Contributionsin the Introduction, where we put the finding “the gap when one ignores the abandonmentis of the order of square root of n” into the context of the Example and Figure 2. Thesewill also address to Reviewer 2’s comment 2. Lastly, we illustrate the benefit of modelinggeneral abandonment distribution in Figure 6 and also in Proposition 4 where we concludethere could be a misestimation of the maximum profit. See also the response to Reviewer 1’scomment (iii) below, where it is mentioned, via a numerical study, that using the purelyMarkovian counterpart model can yield a large misspecification error of the maximumprofit.

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Reviewer 1

Overall Comment: I list many specific comments below, but would like to state that,in general, the writing is sometimes overly simplistic. I understand the need to write to ageneral readership, but I think that introducing technical details without sufficient expla-nations (such as Assumption A2), or writing things that have no real meaning (Equation(8), for example) only confuse both the layperson and the queueing expert.

Response: In the revision, we introduce technical details to make sure clearer under-standings, e.g., regarding the notion of heavy traffic in (5) and the convergence of stochasticprocesses in a rigorous sense in (14), (16). The earlier assumption (A2) is removed becausewe rely on the results of Huang and Gurvich (2018).

Main Recommendations:

Comment (i): It would help the non-expert to have a discussion on how to translate theasymptotic results to a finite system; in particular, to explain how to treat n in a givensystem with fixed λ(p) and µ.

Response: We provided a discussion on the meaning of the scaling parameter n andhow our proposed policy is based on the asymptotic results; see the paragraphs withtitles Our Approach and The Proposed Policy in the Introduction and the associatedExample and Figure 2.

Comment (ii): Figure 1 establishes that abandonment can result in significant reductionin profit, but also that the optimal arrival rate and service rate are not much differentthan in the M/M/1 case. Doesn’t this mean that the manager can essentially ignore theabandonment when making the price and service rate decisions? There is a discussion inSection 7 about the order of the loss, but the problem goes back to my first point, giventhe ambiguity of what n means for any given system.

Response: In the paragraph which contains (2), directly preceding the paragraph OurApproach, we added a discussion about “Can we use the solution from a system with noabandonment to approximate the solution to a system with abandonment?” Also, in theparagraph Our Approach on page 3, we explain the meaning of n for any given systemas the maximum market size; see (3) and (4).

Comment (iii): Since the M/M/1 + M can be solved without appealing to asymptoticanalysis, what is the benefit of considering the“real” distributions, namely, the G/G/1+Gqueue, instead of the Markovian counterpart? A numerical example can demonstrate thatthe asymptotic approximation may provide a better policy than the Markovian approxi-mation for a non-Markovian model.

Response: We carried out a numerical example and provide a remark in the “Note”right before Our Contributions in the Introduction, exemplifying that the Markoviancounterpart can yield a large misspecification error of the maximum profit.

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Specific Comments:

Comment 1: p.2 L.15 - isn’t the M/M/1 the simplest model? Also, Mendelson (1985)and Dewan and Mendelson (1990) don’t limit attention to the G/G/1 queue.

Response: We rephrased the sentence accordingly.

Comment 2: Writing of p.3 needs work: the writing of the first paragraph is overlysimplistic. The last two lines need to be rephrased.

Response: We provided more details about the direct numerical approach in the thirdparagraph in the Introduction. Those last two lines are rephrased with a paragraph startingOur Approach; please see below (2).

Comment 3: Writing in the first paragraph in p.4 is too obscured: the use of the termexpanding market; the service rate increases relative to what? At this point, it is not clearwhat you mean by scaling the abandonment distribution.

Response: We revised the paragraph into a new paragraph starting Our Approachin the Introduction. In particular, we describe our approach in the context of a lineardemand function, whose maximum market size grows with the parameter n.

Comment 4: p.4 second paragraph - there is subscript n which was not introduced yet.Moreover, n appears as a superscript later in the paper. – The tautology “may or maynot” keeps appearing. (Isn’t it always true that something may or may not hold?) Thiscan be OK in the writing as a figure of speech if it appears once or twice (although “notnecessarily”, or even a more elaborate explanation, seem like better choices to me), but itshould not appear in the semi-formal statement in lines 30-32, or in the abstract.

Response: The explicit meaning of the index n is now spelled out as: “The maximummarket size n is exactly the demand when the service is offered for free ...”, see the newparagraph starting Our Approach in the Introduction. Also, we provided a concreteexample on page 4 in order to put our approach into context. The phrase “may or maynot” is removed and we have also revised Abstract accordingly.

Comment 5: Item 4 in p.5 - Reading the paper for the first time, I didn’t understand whatyou mean by “heavy traffic”. My understanding of this term in a model with abandonmentis that it could also mean that ρ increases to 1 at a rate larger than

√n, or even to a constant

larger than 1.

Response: We have specified the term “standard heavy traffic regime” by providingan explicit expression in (5).

Comment 6: In p.7, the remark in the parenthesis L.11-12 needs to be rephrased, as wellas the sentence following it.

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Response: We have rephrased both of them.

Comment 7: The choice of what is assumed in formal assumptions is not clear to me.For example, why is the assumption that F is strictly increasing not included in (A0)?Why does the scaling of the arrival rate deserve a formal assumption, but the scalings ofthe interarrival and service times do not?

Response: We collect assumptions on the primitives λ(·) and F (·) that are necessaryfor the SPP and the two DCP’s, as (A1), (A2), (A3), respectively, and have made explicitreference to them in all statements. We assume the scaling of the arrival rate in (8) as astanding assumption.

Comment 8: Isn’t the last display in p.9 a triviality given that the star denotes theoptimal parameters? Do you mean (p? − c)µ? = π?

Response: Yes, that display is trivial and now removed.

Comment 9: p.10 L.30, is “superscript” a verb? In L.32, the subscript v appears, eventhough the notation is introduced only later.

Response: We rephrased it as “by inserting a superscript n into the appropriatesymbol.” The notation with the subscript (now w) is formally defined with more details(see the third paragraph in Section 3).

Comment 10: At the top of p.11 there is another assumption regarding F , making itunclear if F is still assumed to be strictly increasing. All the properties of F should begathered in one place.

Response: The assumption on continuity and strict monotonicity of F is mentionedin (A1). All other assumed properties of F are now gathered in Lemma 2 (and we refer toit subsequently).

Comment 11: p.12 L.10 - not clear what you mean by π is an upper bound on the limit,given that it is the limit in the display.

Response: The corresponding phrase is removed.

Comment 12: µn in (5) has no o(n) term, but does have it in (6).

Response: We have fixed it and also checked other places.

Comment 13: p.14 L.18 - time to empty the queue, not the system.

Response: The term ‘system’ has been replaced by ‘queue.’

Comment 14: Both sides of (8) are stochastic processes, so the the equation is wrong.This could work if the formal meaning of (8) is explained. Do you mean that the LHSconverges in distribution to V u.o.c? Also, I think the reference is Ward and Glynn (2005).

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Response: The accurate meaning of (8) (now, (14) in the revision) is explained as aweak convergence on D[0,∞) space by referencing to Ward and Glynn (2005) and Billings-ley (1999).

Comment 15: Not clear if the conjecture in p.15 L.14 is made for the paper. Do theresults rely on this conjecture?

Response: We admit ‘conjecture’ was a misused term. We formally present it asLemma 2 and provide a proof, which relies on Huang and Gurvich (2018).

Comment 16: The last paragraph in p.15 is obscured (and the first line needs rephrasing).Why don’t you write the exact reason why the technical condition is assumed?

Response: Resorting to the results of Huang and Gurvich (2018), we can now do awaywith that technical condition, and hence it has been removed.

Comment 17: p.17 L.23 - evidently, the monotonicity of the expected value cannotdirectly be seen, or otherwise there would be no need for Lemma 2. In that case, what’sthe purpose of writing the expression in the middle display?

Response: We agree and removed the phrase regarding the monotonicity of the ex-pected value and also the expected value expression in the middle display.

Comment 18: First sentence in p.18 needs to be rephrased.

Response: We rephrased and moved it to the Appendix, see (A21).

Comment 19: The statement of Proposition 3 has no conclusion for the solution to theDCP when (16) does not hold.

Response: The related statement is now removed.

Comment 20: p.19 L.30 - doesn’t Ward and Glynn (2003a) treat the purely Markoviancase?

Response: That is correct; Ward and Glynn (2003a) treats the purely Markovian case.We found the corresponding subsection is immaterial and redundant, so we removed it inthe revision.

Comment 21: I don’t understand the last sentence in Remark 1. What is “xc” thanheavy traffic?

Response: It was a typo, however, the associated sentence is now removed.

Comment 22: p. 21 L. 44 - is P ? the optimal profit?

Response: That is correct. We have corrected it.

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Comment 23: In Corollary 1 it is stated that the same conditions in Theorem 2 hold.Then why repeat the conclusion in Theorem 2 that (19) holds?

Response: In Corollary 1, we intend to emphasize that the asymptotically optimalpolicy v(θ?) has the same behavior as an optimal policy. We now modified it as “Assume(A1) and (A2)” and also replaced “(19) holds” by “the diffusion scaled profit Pn

opt → P?

as n→∞.”

Comment 24: p.22 L. 15 - sentence starting with “This is” needs rephrasing. Moreover,the reason for why it is unsatisfactory is not convincing; more explanations are needed.In particular, given the goal is to identify an asymptotically optimal policy, does anythingchange by considering the entire distribution?

Response: We rewrote the entire paragraph in order to explain the motivation of analternative hazard-rate scaling. See the first paragraph of Section 5. Under the hazard-ratescaling (where the abandonment distributions change with n), the resulting diffusion-scaledoptimal profit P?

2 is not the same as that of the unscaled case P?1.

Comment 25: The last paragraph of p.22 is badly written.

Response: That paragraph is now rewritten and also we made the hazard rate scalingassumption explicit as in (A2).

Comment 26: p.23 L. 7 - customer.

Response: It is fixed.

Comment 27: The statement of Theorem 3 needs to be precise. What does it mean“results parallel to. . .”?

Response: The statement of Theorem 3 has been revised with more clarifying details.

Comment 28: p.24 L.22 - sentence needs rephrasing. Abandoning customers are notpresent in the system.

Response: We have revised the entire section and this sentence was removed.

Comment 29: p.25 L.10 - “there” should be “this”.

Response: This is fixed now.

Comment 30: Figure 2 - Please remark on the profit P ? being negative.

Response: All figures are revised to refer to the finite systems (not the limit), so P ?

does not appear.

Comment 31: p.26 L.23 - will the plot have colors in publication?

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Response: All figures can be published without colors because we used distinct sym-bols.

Comment 32: p.30 L. 49 - “there is not a limit result” needs rephrasing.

Response: The associated sentence has been removed.

Comment 33: The proof of Proposition 1 builds on Proposition 2 which has extra as-sumptions not mentioned in Proposition 1.

Response: We have modified the proof of Proposition 1 (especially (A2)) withoutappealing to Proposition 1.

Comment 34: EC.2 L.14 - the rate of served customers must equal their arrival rate forany system, stable or not. Moreover, the system is stable regardless of the policy.

Response: The rate of served customers equal their arrival rate in the limit (such asin the fluid model), not in the prelimit. In the prelimit, the system is stable (regardless ofthe policy) due to abandonments, so the rate of the served customers cannot exceed theirarrival rate.

Comment 35: Proof of Proposition 2 has missing details. Where do you use Gamarnikand Zeevi (2006) (last sentence of the proof?) and a Lyapunov-function argument? Isthere a reference for the convergence of stationary distributions, or is it conjectured?

Response: We now resort to the results in Huang and Gurvich (2018) for the conver-gence of stationary distributions.

Comment 36: Proof of Theorem 3 - reference to the fact that V has a unique steadystate with the given density?

Response: We have cited Proposition 6.1(i) of Reed and Ward (2008).

Reviewer 2

Comment 1: My first concern about the paper is the lack of analysis for the case wherethe optimized system does not operate according to the heavy traffic regime. ...

Response: During the revision process, we were able to prove that both diffusioncontrol problems (DCPs) (21) and (26) indeed have a unique maximizer. See Lemmas 3and 4 and their proofs. It required nontrivial analysis, especially for the second DCP (26),and there is the following subtlety at the heart of the proofs: The range of capacity costc ∈ [0, p?], which is implied by the SPP solution in Lemma 1, is in fact contained in therange of c that is identified as the necessary and sufficient condition for the existence of thefinite DCP solution θ? (see the ranges in (A22) and (A28), corresponding to the DCP (21)and (26), respectively). Lemmas 4 and 5 imply Theorem 2, Corollary 1, Theorem 3, which

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in turn imply that the optimized system operates according to the heavy traffic regime insuch a way that the scaled capacity imbalance

√nθnopt → θ? ∈ IR.

Comment 2: As another contribution, the paper aims at exploring the impact of customerabandonments on the firm’s profit. To this end, the paper compares the firm’s optimalprofits in GI/GI/1 and GI/GI/1 +GI systems in Section 7. Although, the impact of cus-tomer abandonments is mentioned as a significant contribution, Section 7 provides minimalinsight and analysis (most of which is limited to numerical work). I recommend authors toexplore the impact of customer abandonments in a more rigorous way. For example, howdo model parameters (such as c, h0, σ, F

′(0)) affect the impact of customer abandonments?Can the paper prove new results regarding the effects of these parameters? Furthermore,what does it mean in real-life to have a percentage loss in profit is on the order of 1/

√n?

The paper argues that “when n = 100, the percentage profit loss that results from ignoringabandonments is around 10%” but what does n = 100 mean? Related to my comment in1, how does the customer abandonments affect the firm’s profits when heavy traffic is notthe optimal operating regime? I think the paper needs to study this question in order tofully understand the impacts of customer abandonments

Response: In the revision, we find that understanding the impact of ignoring aban-donments boils down to studying the scaling limit of percentage loss in profit

L(c, h0, F′(0), σ) ≡ P0 −P?

π;

see (30) and the related motivation/derivation. The more negative L(c, h0, F′(0), σ), the

more harmful is using a model that ignores abandonments. Below is a verbatim copy fromthe revised Section 6.

“An exact analysis of L(c, h0, F′(0), σ), such as taking partial derivatives, was in-

tractable to us because it involves cumbersome derivatives of the normal hazard rate func-tion hz(x) = φ(x)/(1− Φ(x)) and also its evaluation at the solution θ?. Alternatively, wecould use the known double inequality (

√x2 + 2+x)/2 < h(x) < (

√x2 + 4+x)/2 (cf. Yang

and Chu (2015)) and analytically check simple qualitative behaviors of L(c, h0, F′(0), σ)

in its parameters. In particular, it can be seen that the loss in profit L(c, h0, F′(0), σ)

becomes large as each of the input parameters becomes large, and thus the more harmfulis using a model that ignores abandonments; see Figures 4 and 5.”

Regarding the second issue, see item 2 in Our Contributions in the Introduction,where we put the finding “the gap when one ignores the abandonment is of the order ofsquare root of n” into the context of the Example and Figure 2. Regarding the implicationfor a finite system and the meaning of n, see items (i) and (ii) in the response to theAssociate Editor’s comment 1.

Comment 3: Last but not least, the paper is written almost like a technical note. As Ifar as I know from the papers I read and reviewed at POM journal, the journal is a general

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audience journal. Hence, the published papers should be written for a reader with a generalOM interest and knowledge. I do not have any specific recommendations to improve thepaper’s exposition but authors should make a major effort to make the paper more relevantfor people that are less accustomed to the language of asymptotic analysis.

Response: We made a major effort to make the paper more relevant for the generalaudience with OM interest and basic knowledge. The description towards that effort isdetailed in the response to the Associate Editor’s comment 1.

References

Huang, J., I. Gurvich. 2018. Beyond heavy-traffic regimes: Universal bounds and controls for thesingle-server queue. Operations Research 66(4) 1168–1188.

Katsuda, T. 2016. Heavy-traffic limit theorem and its application to stationary distribution con-vergence in a multiclass FIFO single-server queue with abandonment. Under Revision.

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For Review OnlyPricing and Capacity Sizing of a Service Facility:

Customer Abandonment Effects

October 16, 2018

Abstract

This paper studies the effect of customer abandonment in the economic optimization

of a service facility. Specifically, we consider how to jointly set the price and service

capacity in order to maximize the steady-state expected profit, when the system is

subject to customer abandonment and the price is paid only by customers that even-

tually receive service. We do this under the assumption that there is a high rate of

prospective customer arrivals. Our analysis reveals that the economically optimal op-

erating regime is consistent with the standard heavy traffic regime. Furthermore, we

derive the following economical insight: when the capacity cost is sufficiently high, it

can be advantageous for the system manager to “underinvest” in capacity and “take

advantage of the abandonments” to trim congestion. Lastly, we show the loss in profit

when customer abandonments are ignored is on the order of the square-root of the

demand rate.

Keywords: Customer Abandonment; Pricing and Capacity Sizing; Profit Maximization,

Heavy Traffic

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1 Introduction

There is a long history of using price and/or capacity size planning to control congestion in

a service facility. Many academic papers (such as Mendelson (1985), Dewan and Mendelson

(1990), Stidham (1992), and references in Stidham (2009)) study the service facility under

the assumption that all customers will wait for their service as long as necessary, which is

unrealistic. The objective of this paper is to understand the effect of customer abandonment

in the economic optimization of service facilities that can be modeled as queues.

Specifically, this paper studies how to jointly set the price and capacity size in order to

maximize the steady-state expected profit in a M/GI/1+GI queue (“+GI” refers to i.i.d.

abandonment (patience) times with a general distribution) with FCFS service when (i) the

demand for service is d(p) when the service is priced at $p > 0, (ii) the cost of capacity

that results in average service or processing rate µ is $cµ for some c > 0, and (iii) there

is a linear holding cost $h0 ≥ 0. (The holding cost may be zero because the loss of profit

due to customer abandonment still results in a sensible problem formulation.) Then, the

objective is to

maxp,µP(p, µ), where P(p, µ) ≡ pd(p)(1− Pa)− cµ− h0IE[Q(∞)], (1)

where Pa represents the steady-state fraction of customers that abandon, and Q(∞) is the

random variable that represents the steady-state queue-length.

The first question that arises is: How much does the presence of customer abandonments

affect the maximum profit in (1)? If the answer is “not much”, then there is no reason

to explicitly model the customer abandonments, and we can continue to use the solution

in the aforementioned papers that model the service facility without abandonment. To

gain intuition, we first restrict our analysis to the purely exponential setting. Solving (1)

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for a M/M/1 model (that assumes customers will wait as long as necessary for service) is

straightforward, because then Pa = 0 and there is a convenient closed form expression for

IE[Q(∞)]. The analysis for a M/M/1 +M model is more complicated; however, the birth-

death structure implies that the steady-state probabilities can be written explicitly, and

then can be used to provide expressions for Pa and IE[Q(∞)] (although those expressions

are not simple). Then (1) can be solved numerically.

Figure 1 below displays the solution to (1) in the purely exponential setting, as the

abandonment rate varies between 0 and 1, which is equivalent to the steady-state customer

abandonment probability varying between 0 and 0.30, for a linear demand function d(p) =

10 × max(1 − p/5, 0) for p > 0, and c = h0 = $1. Figure 1(a) establishes that customer

abandonments can result in a significant loss in profit, while Figure 1(b) suggests that the

optimal price and capacity size for the M/M/1 +M and M/M/1 systems are not too far

apart.

Then, the second question arises: Can we use the solution from a system with no aban-

donment to approximate the solution to a system with abandonment? Despite the seeming

promise of Figure 1(b), there are several reasons to worry about such an approach, which

motivates developing theory. First, such an approach ignores the relationship between

system utilization and the probability a customer abandons, which suggests that the first

term in (1) may not be approximated well. Second, the heavy traffic approximation for the

steady-state queue-length in a GI/GI/1 system has an exponential distribution (cf. Whitt

(2002)) whereas that for a GI/GI/1 +GI has a truncated normal distribution (cf. Ward

and Glynn (2005)), which suggests that the third term in (1) may not be approximated

well. Finally, this approach ignores stochasticity when h0 = 0. That is because when

h0 = 0, if abandonments are ignored (so that Pa = 0) and we require stability, then (1)

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P@abandonD

1

2

3

4

5Maximum Profit

M�M�1+M

M�M�1

(a) The maximum profit in a M/M/1 +M queueas compared to that in a M/M/1 queue.

0.00 0.05 0.10 0.15 0.20 0.25 0.30P@abandonD

1

2

3

4

5

6

M�M�1 Optimal Arrival Rate

M�M�1+M Optimal Arrival Rate

M�M�1 Optimal Service Rate

M�M�1+M Optimal Service Rate

(b) The solution to (1) in the purely exponentialsetting.

Figure 1: Comparison of two Markovian queueing systems.

becomes the deterministic optimization problem

maxp,µ{pd(p)− cµ} subject to d(p) ≤ µ. (2)

In this paper, we directly study (1), which provides the following structural insight: There

is one scaling (fluid) under which the solution to (1) is consistent with the solution to

the system with no abandonment and one finer scaling (diffusion) under which the two

solutions differ.

Our Approach. The problem (1) is analytically intractable when the inter-arrival, service,

and abandonment distributions are not all exponential. However, the problem becomes

tractable under a large market size assumption. That assumption is most easily interpreted

in the context of a linear demand function that has maximum market size n > 0 and price

responsiveness coefficient b > 0; i.e.,

d(p) = n×max(1− bp, 0) for p ≥ 0. (3)

The large market size assumption implies that n is large compared to b. In general, we

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assume demand becomes large in a way that preserves the structure of the demand function.

That is, for a given price p ≥ 0, we assume

d(p) = n× λ(p), with λ(0) = 1, (4)

and we let n grow large while all other parameters remain fixed. The maximum market

size n is exactly the demand when the service is offered for free since d(0) = n. The linear

demand function (3) clearly satisfies (4).

The Proposed Policy. Our proposed policy for setting the price and capacity size is

based on the solution to two optimization problems. The first is a static planning problem

that ignores stochasticity (that is, a deterministic optimization problem); see Section 2.

The second is an optimization problem that uses the steady-state of an underlying diffusion

approximation to provide a “stochasticity correction factor”; see Sections 3-5.

• Example: Consider a service facility having demand function (3) with b = 1 and n =

5000, deterministic service times, and an abandonment distribution that is Uniform(0, d) for

d = 1/365, 2/365, 7/365. Such a stochastic flow system, modeled as M/D/1+Uniform(0, d)

queue, is reasonable when the service facility is a make-to-order production firm having

yearly demand d(p) (thousands of orders per year), measures delivery times in days, and

puts together orders from component parts in a standardized fashion so that production

times have very little variance. We assume the customer orders are subject to deadlines as

d = 1, 2, 7 days. We focus on maximizing the revenue from selling products, and assume

h0 = 0 for simplicity. Then, the solution from a system with no abandonment is found by

solving (2), which has solution (p?, nµ?) for p? = (1 + c)/2 and µ? = λ(p?) = (1− c)/2 > 0.

We assume c = 0.1. As a concrete example, this could correspond to a customized bicycle

shop in which the cost to produce one bike is $0.1K, or $100, the associated selling price

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p? = $0.55K or $550, and the resulting yearly profit is approaching one million (in the

$700-$950K range). Figure 2 shows that the proposed policy outperforms the solution that

ignores abandonments. The simulation results1 in Figure 2 show that the proposed policy

can increase yearly profit by several percent (on the order of tens of thousands of dollars).

We will revisit this example in Section 4 to highlight the key feature of the proposed policy,

and again in Section 6 to connect with the asymptotic performance results developed in

Sections 3–5.

Note: Without appealing to asymptotic analysis, one might use the purely Marko-

vian model M/M/1 +M in the face of the general non-Markovian model, e.g., M/D/1 +

Uniform(0, d) as considered above. Numeric calculation demonstrates that the Marko-

vian counterpart (matching the mean parameters) can yield a large misspecification error

of the maximum profit (measured in percentage error) as 23.7%, 27.7%, 60.9%, under

d = 1/365, 2/365, 7/365, respectively. Moreover, a direct numerical approach to the prob-

lem (1) can not provide any useful insights on the optimal policy’s structure.

Our Contributions.

1. We establish that the solution to (1) is consistent with the standard heavy traffic regime.

More precisely, for a given demand function having maximum market size n, if (pn, µn)

solves (1), then for some θ ∈ IR,

d(pn)− µn√n

=nλ(pn)− µn√

n→ θ as n→∞. (5)

See Theorem 1.

As a consequence, the standard heavy traffic regime is an economically optimal oper-

1The profit shown is averaged over 5 simulation runs, each having around 1.125M arrivals, which impliedthe percentage of customers abandoning differed by less than 0.05% between runs. The simulation platformused was ExtendSim.

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1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

$650.00

$700.00

$750.00

$800.00

$850.00

$900.00

$950.00

1 2 7

Thousands

Proposed AbandonmentsIgnored %Decrease

Figure 2: Simulation results of the profit comparison in aM/D/1+Uniform(0, d) having c =0.1, yearly demand function given in (3) with n = 5000, b = 1, and d = 1/365, 2/365, 7/365.

ating regime.

2. We develop two approximating diffusion control problems (DCPs) that follow from (5).

One is simpler to solve and implement but is more fragile to abandonment distribution

characteristics (for example, is useless if the abandonment distribution density function is

0 or∞ at 0), whereas the other is not. Both give rise to an asymptotically optimal control

policy. See Theorems 2 and 3.

This refined analysis reveals that ignoring abandonments (as was done in Figure 2)

leads to a loss in profit on the order of the square root of the maximum market size

n (a small percentage in the context of Figure 2 since 1/√n = 1/

√5000 = 1.4%).

3. We find that when the cost of capacity is high enough, an asymptotically optimal policy

sets p and µ such that demand exceeds capacity; see Figure 3 in Section 4. This statement

can only be made for a model with abandonments; without abandonments, such a policy

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would give rise to a system without a stationary distribution, rendering the optimization

(1) meaningless.

When the cost of capacity is high enough, the system manager will want to take

advantage of abandonments to trim system congestion.

The remainder of this paper is organized as follows. We first review the most related

literature. Then, in Section 2 we set up and solve the deterministic relaxation to (1), also

known as the static planning problem (SPP). Section 3 establishes that an optimal policy

induces the system to operate in the standard heavy traffic regime. That motivates us in

Section 4 to formulate a diffusion control problem (DCP) based on the limiting behavior

of the diffusion-scaled profit. Our proposed policy follows from the solution to that DCP.

The aforementioned approximating DCP only involves the behavior of the abandonment

distribution at 0, and so in Section 5 we provide an alternative approximating DCP that

involves the full abandonment distribution and is valid for a broader family of abandonment

distributions. Section 6 examines the impact of ignoring customer abandonment on the

percentage loss in profit and provides some supporting numerics. The proofs of all results

in the main body are found in the Appendix (electronic companion).

Literature Review

The literature related to our work is concerned with determining the economically optimal

operating regime for a given service system. The majority of this work establishes that

the standard heavy traffic (or, Halfin-Whitt regime for the many server setting) is induced

by the optimal solution (rather than the heavy traffic condition being given) from, e.g.,

selecting optimal price and capacity size. We refer the reader to Armony and Maglaras

(2004a,b), Harrison (2003), Maglaras and Zeevi (2003), Plambeck and Ward (2006), and

Nair et al. (2016). All of these studies have assumed infinitely patient customers (i.e., no

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abandonment from the system).

Plambeck (2004) and Plambeck and Ward (2008) show that in manufacturing systems

with a high volume of prospective customers, the optimal price and capacity together

with optimal leadtime quotation result in heavy traffic. One main conclusion from the

paper Armony et al. (2009), in a purely exponential setting, is that it is very important

to understand customer impatience when setting capacity. In relation to that work, we

establish that such an insight continues to hold in more generality.

The capacity setting papers most closely related to ours are Bassamboo and Randhawa

(2010) and Randhawa (2013). Bassamboo and Randhawa (2010) assumes the demand

rate (no pricing control) and solves for the optimum capacity in both M/M/N + G and

M/M/1 + G queueing systems. In particular, in the case of the M/M/1 + G system, the

objective of the authors is to solve a cost minimization problem that is equivalent to (1), and

has capacity control but not pricing control (so only one decision variable). The authors

characterize conditions under which optimal capacity sizing induces the overloaded regime,

in which demand exceeds capacity, and the critically loaded regime, in which demand

equals capacity. The authors show that fluid approximations are extremely accurate in

the overloaded regime. That insight is also true in Randhawa (2013), which considers a

capacity sizing problem when customers balk but do not renege (that is, arriving customers

may decide not to join the system, but any customer that joins waits as long as is necessary

to obtain service). Very interestingly, and consistent with Remark 3 in Randhawa (2013),

we find that when both price and capacity are included as decision variables, the critically

loaded regime is the economically optimal operating regime. We then show the stronger

result that the economically optimal operating regime is the standard heavy traffic regime.

(In the critically loaded regime, the system load approaches one, but the rate is unspecified.

The standard heavy traffic regime specifies the rate.)

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Our asymptotic analysis relies heavily on past work that has developed heavy-traffic

approximations for the queueing systems with abandonment. The heavy traffic limit of

the queue-length process and the abandonment probability follow from the heavy traffic

limit for the offered waiting time process. The offered waiting time process, introduced in

Baccelli et al. (1984), tracks the amount of time an infinitely patient customer must wait

for service. Its heavy traffic limit when the abandonment distribution is left unscaled is a

reflected Ornstein-Uhlenbeck process (see Ward and Glynn (2005)), and its heavy traffic

limit when the abandonment distribution is scaled through its hazard rate is a reflected

nonlinear diffusion (see Reed and Ward (2008)).

2 The Static Planning Problem

We introduce the static planning problem (SPP) that ignores stochasticity. We then show

that that leads to also ignoring abandonments, as in (2). To do this, we adapt the reasoning

in Section 3 in Whitt (2006) from a many server setting to a single server setting. We

assume unit maximum market size (n = 1) for explanatory purposes, and then generalize.

Suppose the abandonment distribution is F . Note that when λ(p) is the demand rate

and µ is the capacity size (service rate), then λ(p) − µ is the rate at which incoming

customers abandon. In this deterministic model, if all customers wait w time units, then

λ(p)F (w) also equals the rate at which incoming customers abandon. Equating these two

rates shows that all customers wait for w = w(p, µ) time units, defined as the solution to

λ(p)F (w) = [λ(p)− µ]+,

whose solution is unique when F is continuous and strictly increasing. Next, since (1−F (x))

is the probability a customer that has waited x time units is still present in the system, it

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follows that∫ w0 λ(p)(1 − F (x))dx approximates the number of customers waiting. This

reasoning results in the following SPP

π ≡ maxp,µ

{pλ(p)

(1− [λ(p)− µ]+

λ(p)

)− cµ− h0

∫ w(p,µ)

0λ(p)(1− F (x))dx

}. (6)

For n ≥ 1, the same logic leads to

maxp,µ

{pnλ(p)

(1− [nλ(p)− µ]+

nλ(p)

)− cµ− h0

∫ w(p,µ)

0nλ(p)(1− F (x))dx

},

which has the maximized objective function value nπ, and a unique optimal solution

(p?, nµ?) if and only if (6) has unique optimal solution (p?, µ?), which is guaranteed by

the following sufficient condition.

(A1) The abandonment distribution function F is continuous and strictly increasing. The

function λ(p) has λ(0) > 0, and is strictly decreasing, convex, and continuous. Fur-

thermore, pλ(p) is strictly concave.

Lemma 1. Assume (A1). The unique solution to the SPP (6) is exactly the solution to

the following SPP

maxp,µ{pλ(p)− cµ} (7)

subject to λ(p) ≤ µ.

Furthermore, that solution (p?, µ?) has λ(p?) = µ?, c < p? and 0 < µ?.

We denote

π ≡ p?λ(p?)− cµ?.

The implication of Lemma 1 is that the consequence of ignoring stochasticity is ignoring

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abandonments. In particular, the solution to the deterministic optimization problem (2)

in the Introduction is exactly the solution to (7), scaled by n. That is, (p?, nµ?) solves (2),

and the resulting maximum profit is nπ.

3 The Behavior of an Optimal Policy

We now investigate how an optimal policy behaves as the market size n grows. Henceforth,

we use λn(·) to denote the demand function of the system with a maximum market size n,

namely, let

λn(p) = nλ(p). (8)

When n is regarded as fixed, this is exactly the demand function d(p) specified in (4) in the

Introduction. We refer to any process or quantity associated with the M/GI/1+GI model

having demand function λn(p) by inserting a superscript n into the appropriate symbol.

The model is built from three independent sequences of non-negative i.i.d. random

variables (ui, i ≥ 1), (vi, i ≥ 1), (di, i ≥ 1). We assume u1 is an exponential random

variable with unit mean, v1 has a unit mean and finite variance, and let the inter-arrival

and service time sequences be specified as

uni ≡ui

λn(p)and vni ≡

viµn. (9)

If the i-th customer, who arrives at time tni ≡∑i

j=1 uni and has service time vni , must wait

longer than di time units to reach the server, then s/he departs at time tni + di without

receiving service. (There can be customers initially present; however, since we are interested

in steady-state behavior, we do not specify those.) We assume a mild technical assumption

P (v1 < u1) > 0, which ensures the steady-state abandonment probability and steady-

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state offered waiting times are well-defined; see Lemma 2 in Baccelli et al. (1984). Also,

Theorem 3.2 in Kang and Ramanan (2012) ensures the existence of steady-state queue-

length distribution (under a very mild assumption such that the inter-arrival distribution

has a density, which is satisfied by an exponential distribution).

A policy w = {(pn, µn) : n ≥ 1} is a sequence of prices and capacity sizes. We let

Pnw denote the steady-state expected profit under an admissible policy w with maximum

market size n (equivalently, demand function λn); that is, Pnw ≡ P(pn, µn).

The following result establishes the connection between the problem we would like

to solve (1) and the deterministic optimization problem (7) that results from ignoring

stochasticity (and, therefore, also ignoring abandonments). Here and throughout (pnopt, µnopt)

solves (1) when the demand function is λn and

opt ≡ {(pnopt, µnopt) : n ≥ 1}

denotes an optimal policy (which may or may not be unique).

Proposition 1. (Optimality of Critical Load) Assume (A1). Under an optimal

policy opt,Pn

opt

n→ π, pnopt → p? and

µnopt

n→ µ? as n→∞.

Proposition 1 implies that the critically loaded regime is the economically optimal operating

regime, because the system load λn(pnopt)/µnopt approaches 1 as n tends to infinity. To see

that, note that λ(p?) = µ? from Lemma 1.

Remark 1. Because c < p? by Lemma 1, the condition in Bassamboo and Randhawa

(2010) that guarantees the critically loaded regime is the economically optimal operating

regime is satisfied.

Proposition 1 provides a necessary condition for us to use to develop a proposed policy

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v. Specifically, v should be such that

Pnvn→ π as n→∞. (10)

However, the condition (10) does not differentiate between a system that explicitly models

abandonments and a system that ignores them. This motivates us to use a stricter criterion,

that takes into account convergence rate. To do that, for any policy w, we define the

centered and scaled profit

Pnw ≡√n

(Pnwn− π

). (11)

Definition 1. We say that an admissible policy w is asymptotically optimal if lim infn→∞

Pnw ≥

lim supn→∞

Pnw′ , for any other admissible policy w′, and also lim infn→∞

Pnw > −∞.

Taking the other policy w′ to be opt, we see that an asymptotically optimal policy w must

have scaled profit Pnw/n that converges to π as fast as an optimal policy, assuming the

limits exist.

The question is: How do we translate the asymptotic optimality definition into a stricter

necessary condition to use to develop a proposed policy? For this, we first define the

capacity imbalance under any policy w,

θnw ≡λn(pn)− µn

n.

The capacity imbalance of an optimal policy approaches 0 as n → ∞ by Proposition 1.

The following result gives the convergence rate.

Theorem 1. (Optimality of Heavy Traffic Regime) Assume (A1). Under any

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asymptotically optimal policy w

lim supn→∞

√n|θnw| <∞.

Since an optimal policy is asymptotically optimal, Theorem 1 implies that the standard

heavy traffic regime is an economically optimal operating regime; i.e., (5) holds under

opt (at least on a subsequence). This provides a stricter necessary condition to use to

develop a proposed policy, because (5) implies λn(pn)/µn → 1 as n → ∞ but not vice

versa. Specifically, Theorem 1 motivates proposing a policy v that satisfies the heavy

traffic condition (5), repeated here for the reader’s convenience.

√nθnv =

nλ(pn)− µn√n

→ θ ∈ IR as n→∞.

4 The Proposed Policy

The proposed policy when the maximum market size is n has the form

pn = p? and µn = nµ? −√nθ + o(

√n) for some θ ∈ (−∞,∞). (12)

Under such a policy, the heavy traffic condition (5) holds, because

√nθnv =

nλ(p?)− nµ? +√nθ + o(

√n)√

n= θ + o(1),

where the first equality follows by definition and the second follows because λ(p?) = µ?.

Recalling that p? and µ? are the solution to the static planning problem (7), that was de-

rived by ignoring stochasticity, we interpret the parameter θ as a “stochasticity correction”

factor.

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Our purpose in this section is to understand the performance of a policy v defined

by (12) as a function of θ, in order to find a θ? that results in an asymptotically optimal

policy. The first step is to study the behavior of the centered and scaled profit under policy

v = v(θ) as n becomes large. Substitution, algebra, and using π = p?λ(p?) − cµ?, show

that

Pnv(θ) =√n

(pnλn(pn)(1− Pna )− cµn − h0IE[Qn(∞)]

n− π

)=√n

(p?λ(p?)(1− Pna )− c

(µ? − θ√

n+o(√n)

n

)− h0

IE[Qn(∞; θ)]

n− π

)= cθ + o(1)− p?λ(p?)

√nPna − h0IE

[Qn(∞; θ)√

n

]. (13)

We would like to take the limit as n→∞ in the above expression (13), in order to arrive

at an optimization problem to determine θ. This requires understanding the asymptotic

behavior of√nPna and IE [Qn(∞)/

√n], which relies on understanding the offered waiting

time process.

The offered waiting time process {V n(t), t ≥ 0} in the system having maximum market

size n tracks the amount of time a hypothetical customer arriving at time t ≥ 0 would

have to wait for service, which depends only on the service times of the customers waiting

at time t who eventually receive service (do not abandon). Equivalently, V n(t) represents

the workload; that is, the time required for the server to process all the jobs in the queue

at time t if no more jobs arrive. We use tni − to represent the workload at the instant the

i-th customer arrives, not including the i-th customer’s service requirement vni . Under a

given policy,

V n(t) ≡ V n(0) +

An(t)∑i=1

vni 1{V n(tni −) < di} −Bn(t) ≥ 0,

where An(t) ≡ max{i ≥ 0 :∑i

j=1 unj ≤ λn(pn)t} counts the number of customers that have

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arrived to the system by time t ≥ 0, Bn(t) ≡∫ t0 1{V n(s) > 0}ds is the cumulative server

busy time, and V n(0) is the possibly random initial state. The i-th customer abandons with

a probability P (V n(tni −) ≥ di|V n(0)) that depends on the initial conditions. The Markov

chain {V n (tni ) , i ≥ 1} having state space [0,∞) has a unique steady-state distribution by

Lemma 2 in Baccelli et al. (1984), and we let V n(∞) be a random variable having that

steady-state distribution. When the initial state is distributed according to the steady-

state distribution, the probability the i-th customer abandons is a random variable having

distribution

P (V n(tni −) ≥ di) = F (V n(∞)) for all i.

Then, the expected steady-state probability a customer abandons is

Pna ≡ IE[F (V n(∞))].

We impose the following assumption in this section.

(A2) The abandonment distribution F is differentiable with bounded derivative, in partic-

ular, 0 < F ′(0) <∞.

Theorem 1 in Ward and Glynn (2005), adapted to our setting, establishes that for V

defined in (15) below, when V n(0) = 0 and under policy v(θ) (which implies Assumption

1 in that same paper holds because the heavy traffic condition (5) holds),

√nV n(·)⇒ V (·), as n→∞, (14)

in the topology of weak convergence on D[0,∞); see, for example, Billingsley (1999) for

a discussion of this convergence concept. In the following display, (V,L) is the unique

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solution to the reflected stochastic differential equation

V (t) = σW (t) + θλ t− F

′(0)∫ t0 V (s)ds+ L(t) ≥ 0

subject to: L is non-decreasing, has L(0) = 0 and∫∞0 V (s)dL(s) = 0,

(15)

where {W (t) : t ≥ 0} denotes a one-dimensional standard Brownian motion, λ ≡ λ(p?)

and

σ2 ≡ λ−1(var(u1) + var(v1)). (16)

From Sections 18.3 and 18.4.1 in Browne and Whitt (1995), the steady-state distribution

V (∞) of V follows a truncated normal distribution on [0,∞) (see (19) for the mean of

V (∞)).

In contrast to the offered waiting time process, the queue-length process includes both

customers that will eventually receive service and those that will abandon. Still, given

a policy under which the probability a customer abandons is small when n is large, we

conjecture that Little’s law almost holds; i.e., that IE[Qn(∞)] ≈ nλIE[V n(∞)] (equiva-

lently, IE[Qn(∞)/√n] ≈ µ?IE[

√nV n(∞)]. That conjecture is consistent with the following

asymptotic relationship between the scaled queue-length and offered waiting time process

Qn(·)/√n− λ

√nV n(·)⇒ 0, as n→∞, (17)

in D[0,∞), which follows from Theorem 3 in Ward and Glynn (2005), adapted to our

setting. The process-level convergences in (14) and (17) suggest the weak convergence of

the associated steady-state distributions.

Lemma 2. Assume (A1) and (A2). Under the policy in (12),

IE[√nV n(∞)]→ IE[V (∞)] and IE

[Qn(∞)√

n

]→ µ?IE[V (∞)] as n→∞.

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The following result allows us to determine the limit of the centered and scaled profit.

Proposition 2. (Limiting Performance Measure Expressions) Assume (A1) and

(A2). As n→∞, under the policy in (12),

√nPna → F ′(0)IE[V (∞)] and IE

[Qn(∞)√

n

]→ µ?IE[V (∞)]. (18)

The limit of Pnv(θ) in (13) as n becomes large is immediate from Proposition 2, and

letting

g(θ) ≡ IE[V (∞; θ)] =θ

λF ′(0)+

σ√2F ′(0)

hz

(−θλσ

√2

F ′(0)

), (19)

it is seen that

limn→∞

Pnv(θ) = cθ − p?λ(p?)F ′(0)g(θ)− h0µ?g(θ)

= cθ − (p?F ′(0) + h0)µ?g(θ), (20)

where the second equality follows since λ(p?) = µ?. This leads to a sensible trade-off:

Increasing θ in (20) reduces the capacity in the proposed policy (12), which has marginal

savings c, but also incurs the associated congestion cost (p?F ′(0) + h0)µ?g(θ) (because

g(θ) is increasing in θ; see Lemma 3 below). The convergence in (20) leads to the following

optimization problem

maxθ∈IR{cθ − (p?F ′(0) + h0)µ

?g(θ)}. (21)

Lemma 3. We have that g(θ) ≡ IE[V (∞; θ)] is increasing in θ and the optimization

problem (21) has a unique maximizer θ? ∈ IR.

Our proposed policy is v(θ?), which substitutes θ? for θ in (12).

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Theorem 2. (Asymptotic Optimality) Assume (A1) and (A2). Any policy

v(θ?) ≡ {(p?, nµ? −√nθ? + o(

√n)), n ≥ 1} (22)

is asymptotically optimal and, furthermore,

Pnv(θ?) → P?, as n→∞, (23)

where P? ≡ cθ? − (p?F ′(0) + h0)µ?g(θ?) is the maximum objective value in (21).

The following corollary to Theorem 2 emphasizes that the asymptotically optimal policy

v(θ?) has the same behavior as an optimal policy (because an optimal policy must be

asymptotically optimal).

Corollary 1. Assume (A1) and (A2). Under an optimal policy opt, the scaled capacity

imbalance√nθnopt → θ? and the diffusion scaled profit Pnopt → P? as n→∞.

Our proposed policy results in a demand rate λn(p?) = nλ(p?) that exceeds the capacity

size nµ? −√nθ? whenever θ? > 0, which occurs when the cost of capacity c is large. By

Corollary 1, this is also true for an optimal policy (i.e., in the solution to (2)). This is

in contrast to any policy that could potentially be developed from a system that ignores

abandonments, because such a system must have demand strictly less than the capacity

size for stability (more specifically, the system must possess a steady-state distribution

so that the objective (1) with Pa = 0 is well-defined). The intuition is that it can be

advantageous for the system manager to “underinvest in capacity” and “take advantage of

the abandonments” to trim congestion (letting the abandonments stabilize the system).

• Example: Figure 3 illustrates the aforementioned phenomenon in a M/D/1 +

Uniform(0, d) system with h0 = 0 and maximum market size n = 5000 (the same set-

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ting as in Figure 2 in Section 1). Figure 3 further shows that as the customers are more

patient (as the “deadline” d increases) the optimal capacity size decreases at a slower rate

in c and the optimal policy gets closer to that of the system without abandonment.

0.15 0.2 0.25 0.3c, Cost per capacity

1400

1800

2200

2600

Cap

acity

siz

e in

the

prop

osed

pol

icy

d=1/365d=2/365d=7/365no abandonmentdemand rate

Figure 3: The capacity size in the proposed policy is nµ? −√nθ?, as compared to the

optimal policy nµ? when abandonments are ignored (that is, the solution to (2)), in thesetting in Figure 2 in the Introduction. Both polices have demand rate nλ(p?). Whenthe cost of capacity c is large, the proposed policy (lines with symbols) has capacity sizestrictly less than the demand rate (solid line), which cannot happen in a system that ignoresabandonments (dash-dot line).

5 Incorporating the Entire Abandonment Distribution

The optimization problem (21) that determines the stochasticity correction factor θ? de-

pends on the abandonment distribution only through the value F ′(0). However, F ′(0) is not

a very robust statistic (in contrast to mean and variance, for example). Furthermore, (21)

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is well-defined only when 0 < F ′(0) <∞, which is a restrictive assumption. For example,

a gamma distribution with shape parameter α > 1 has F ′(0) = 0, and one with 0 < α < 1

has F ′(0) = ∞. So, unless the shape parameter α = 1 (in which case the distribution

is exponential) the optimization problem (21) cannot be considered. These observations

motivate developing an optimization problem that incorporates more of the structure of

the abandonment distribution, and this is our purpose in this Section. The end result is

an alternative proposed policy, that can apply even when F ′(0) = 0 or F ′(0) = ∞. That

alternative proposed policy has the same structure as in (12), but uses an optimization

problem that is different than (21) to determine the stochasticity correction factor θ?.

More specifically, we apply the hazard rate scaling in Reed and Ward (2008), and to

do this, we must allow the abandonment distribution to change with n. Specifically, in

the queueing model described in the first paragraph of Section 3, replace the sequence

{di, i ≥ 1} with {dni , i ≥ 1}, which changes with n. Then, the maximum amount of

time the i-th arriving customer will wait for service is dni (instead of di). The cumulative

distribution function of dn1 is Fn, and its associated hazard rate function is hn.

(A3) Suppose

Fn(x) = 1− exp

(−∫ x

0hn(u)du

), x ≥ 0.

holds and hn(x) = h(√nx) for some hazard rate function h(·). Furthermore, assume

h(u) ≤ K(1 + u`) for some ` > 0 and K > 0.

Theorem 5.1 in Reed and Ward (2008), adapted to our setting, establishes

√nV n(·)⇒ V (·) as n→∞

in D[0,∞), where (V,L) is the unique solution to the reflected stochastic differential equa-

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tion

V (t) = σW (t) + θλ t−

∫ t0

[∫ V (s)0 h(u)du

]ds+ L(t) ≥ 0

subject to: L is non-decreasing, has L(0) = 0 and∫∞0 V (s)dL(s) = 0.

(24)

In a slight abuse of notation, in this Section, V (∞) is a random variable that has the steady-

state distribution of (24), which is given in Proposition 6.1(i) in Reed and Ward (2008).

Observe that (24) and (15) coincide when the abandonment distribution is exponential

with rate γ, because then F ′(0) = γ and h(u) = γ for all u ≥ 0. In general, the connection

between (24) and (15) can be seen by performing a Taylor series expansion of h about 0

(which follows the intuition provided in Reed and Tezcan (2012) for the GI/M/N + GI

model) as follows

h(u) = h(0) +∞∑i=1

ui

i!h(i)(0). (25)

Then, since h(0) = F ′(0), it follows that the linear portion of the infinitesimal drift in (15)

arises from the lowest order term of the above Taylor series expansion.

The optimization problem that replaces (21) to determine the stochasticity correction

factor θ? can be seen as follows. As in the previous subsection, for large n, we expect

IE

[Qn(∞)√

n

]≈ µ?IE [V (∞)] .

In contrast to the previous subsection, the intuition for the approximation for the aban-

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donment probability follows from (with explanation provided below)

√nPna = IE

[√nFn(V n(∞))

]= IE

[√n

(1− exp

(−∫ V n(∞)

0hn(u)du

))]

≈ IE

[√n

∫ V n(∞)

0hn(u)du

]

= IE

[∫ √nV n(∞)

0h(u)du

].

The first equality is by definition and the second by the assumption (A3). The approxima-

tion on the third line is from a Taylor series expansion of the exponential. The last equality

is by a change of variable. In the remainder of this section, the random variable V (∞) is

now in reference to the diffusion in (24). Then, we have the large n approximation for the

abandonment probability and steady-state queue-length as shown below.

Proposition 3. (Limiting Performance Measure Expressions) Assume (A1) and

(A3). Under the policy in (12),

√nPna → IE

[∫ V (∞)

0h(v)dv

]and IE

[Qn(∞)√

n

]→ µ?IE[V (∞)] as n→∞.

Lemma 4. We have that g(θ) ≡ IE[V (∞; θ)] is increasing in θ and the optimization

problem (26) below has a unique maximizer θ? ∈ IR.

Theorem 3. (An Alternative Asymptotically Optimal Policy) Assume (A1) and

(A3). Define θ?2 as the unique solution to the optimization problem

maxθ∈IR

{cθ − p?λ(p?)IE

[∫ V (∞;θ)

0h(u)du

]− h0µ?IE[V (∞; θ)]

}, (26)

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where IE[V (∞; θ)] is the steady-state mean of the diffusion given in (24), and we write

V (∞) = V (∞; θ) to emphasize the dependence of the steady-state distribution on θ. Then,

Theorem 2 and Corollary 1 hold for the policy

v(θ?2) ≡ {(p?, nµ? −√nθ?2 + o(

√n)), n ≥ 1}, (27)

except with P? replaced by P?2, defined as the maximum objective value in (26) as follows

P?2 ≡ cθ?2 − p?λ(p?)IE

[∫ V (∞;θ?2)

0h(u)du

]− h0µ?IE[V (∞; θ?2)].

Remark 2. Combining Proposition 1, Theorems 2 and 3, we obtain the following second

order approximation to the optimal profit in the n-th system:

P(pnopt, µnopt) = nπ +

√nP?

i + o(√n), (28)

for i = 1, 2 corresponding to when there is no hazard rate scaling and when there is such a

scaling, respectively.

Although using the optimization problem (21) to define the stochasticity correction

factor θ? is simpler than using the above optimization problem (26), its scope is limited

to a narrower family of abandonment distribution. Furthermore, the two limiting regimes

produce different approximations of the optimal profit in the n-th system. We expect

the regime with hazard rate scaling to result in an improved approximation, because that

regime captures the entire abandonment distribution. The following Proposition orders

the approximations produced by the two limiting regimes.

Proposition 4. Let P?1 be as in Theorem 2 and P?

2 as in Theorem 3. If h(·) is strictly

increasing, then P?1 > P?

2, and if h(·) is strictly decreasing, P?1 < P?

2.

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Under the scaling assumption (A3), an optimal policy satisfies

Pnopt → P?2 as n→∞.

Supposing h(·) is strictly decreasing, Proposition 4 implies that when the entire abandon-

ment distribution information is used, the approximated optimal profit in the n-th system

is larger than that based on the information only near the origin.

6 The Consequences of Ignoring Abandonments

In this section, we study the impact of ignoring abandonments on the loss in profit and

connect back to the concrete example in the Introduction (recall Figure 2) that shows the

proposed policy outperforms the solution that ignores abandonments. Suppose the modeler

chooses to ignore abandonments. Then, the policy

pn = p? and µn = nµ? −√nθ0 (29)

for

θ0 ≡ −σ√h0/2c

is known to be asymptotically optimal for the GI/GI/1 system; see (4.5) and Theorem 6.1

in Lee and Ward (2014).

Next, define θ?1 to be the unique solution to (21) and θ?2 to be the unique solution to

(26). Under either policy, the percentage loss in profit from an optimal policy is o(1/√n)

because for i ∈ {1, 2}

√nP(p?, nµ? −

√nθ?i )− P(pnopt, µ

nopt)

P(pnopt, µnopt)

=(Pnv(θ?) − P

nopt

)× n

Pnopt→ (P?

i −P?i )×

1

π= 0,

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as n → ∞ in the relevant asymptotic regime. The equality in the above display follows

from algebra, and the convergence follows from Proposition 1, Theorems 2 and 3 and

Corollary 1.

In contrast, under the policy (29), the percentage profit loss no longer is o(1/√n)

because

√nP(p?, nµ? −

√nθ0)− P(pnopt, µ

nopt)

P(pnopt, µnopt)

=(Pnv(θ0) − P

nopt

)× n

Pnopt→ (P0−P?

i )×1

π< 0, (30)

for i ∈ {1, 2} as n→∞, where (when there is no hazard rate scaling and recalling g(·) in

(19))

P0 ≡ cθ0 −(p?F ′(0) + h0

)µ?g(θ0). (31)

(The reason why P0 < P?i is because θ?i (i = 1, 2) is the unique maximizer of (21) and

(26), respectively.) Thus, to understand the impact of ignoring abandonments, we want

to analyze the scaling limit of percentage loss in profit. We do this in the simpler limiting

regime in which the hazard rate is not scaled, and study

L(c, h0, F′(0), σ) ≡ P0 −P?

1

π. (32)

The more negative L(c, h0, F′(0), σ), the more harmful is using a model that ignores aban-

donments.

Connection to the Example in the Introduction. Figure 2 in the Introduction

shows the simulation results of the profit comparison in a M/D/1 + Uniform(0, d) model

under the proposed policy based on the optimization problem (21) and the policy ignoring

abandonments. In the Example, the cost to produce one product is c = $0.1K and

the assumed linear demand (3) has maximum yearly market size n = 5000 and price

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responsiveness coefficient b = 1. With this, the SPP (7) yields the solution p? = $0.55K

and λ(p?) = µ? = 0.45, and this in turn provides σ = 1.49 according to the relation

(16). Assuming the holding cost h0 = 0 and noticing the abandonment rate parameter

F ′(0) = 1/d under the Uniform(0, d) distribution, one can easily solve the optimization

problem (21) numerically, under different deadline requirements d = 1/365, 2/365, 7/365.

Notice that the percentage profit loss in (30) is defined with respect to the optimal

profit P(pnopt, µnopt), however, its exact evaluation is a daunting task. Owing to Theorem 2

and Corollary 1, one can instead use the profit under asymptotically optimal policy (22).

Under the Example setting, we find that the percentage profit losses are

P(p?, nµ? −√nθ0)− P(p?, nµ? −

√nθ?1)

P(p?, nµ? −√nθ?1)

≈ −4.26%,−2.75%,−1.34%,

for d = 1/365, 2/365, 7/365, respectively. We also note that the simulation results in Figure

2 (i.e., the graph “% Decrease”) are −5.04%,−3.28%,−1.55% for d = 1/365, 2/365, 7/365,

respectively. Although there is some degree of discrepancy, the above theoretical predicted

values reasonably match with the simulation results.

Lastly, we want to examine the impact of ignoring abandonments by studying the be-

havior of L(c, h0, F′(0), σ) as any one of the four input parameters changes. An exact anal-

ysis of L(c, h0, F′(0), σ), such as taking partial derivatives, was intractable to us because it

involves cumbersome derivatives of the normal hazard rate function hz(x) = φ(x)/(1−Φ(x))

in (19) and also its evaluation at the solution θ?. Alternatively, we could use the known

double inequality (√x2 + 2 + x)/2 < h(x) < (

√x2 + 4 + x)/2 (cf. Yang and Chu (2015))

and analytically check simple qualitative behaviors of L(c, h0, F′(0), σ) in its parameters.

In particular, it can be checked that the loss in profit L(c, h0, F′(0), σ) becomes large as

each of the input parameters becomes large, and thus the more harmful is using a model

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For Review Only0.25 0.3 0.35 0.4 0.45

c

-12

-10

-8

-6

-4

-2L(

c)

(a) Impact of c on L(c, h0, F′(0), σ)

1 2 3 4 5 6 7 8h

0

-16

-14

-12

-10

-8

-6

L(h 0)

(b) Impact of h0 on L(c, h0, F′(0), σ)

Figure 4

that ignores abandonments; see Figures 4 and 5.

Impact of the Variability of Abandonment Distribution. Lastly, we examine

the impact of variability of abandonment distribution on the optimal profit. In doing so,

we consider the same parameter setting as in Figure 2 in the Introduction, except the

abandonment distribution now follows a gamma distribution G(α) having both the scale

and shape parameters as given α > 0 (hence mean 1). Then, the simpler optimization

problem (21) is no longer applicable unless α = 1, because F ′(0) = 0 if α > 1 and

F ′(0) = ∞ if 0 < α < 1. Hence, we need to scale the hazard rate function and consider

alternative optimization problem in (26). Figure 6 illustrates that variability lessens profits

under the proposed policy in (27).

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origin. Adv. in Appl. Probab. 19(3) 560–598.

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Electronic Companion to “Pricing and Capacity Sizing of a

Service Facility: Customer Abandonment Effects”

In this electronic companion, we provide the arguments to establish the results in the main

body. Section A provides the proofs for propositions and theorems, and Section B provides

the proofs for lemmas.

A Proofs of Propositions and Theorems

Proof of Proposition 1. Fix 4 > 0 and consider the policy w defined by

pn = p? and µn = nµ? +√n4. (A1)

In comparison to an optimal policy, Pnopt ≥ Pnw for each n and so

lim infn→∞

Pnoptn≥ lim sup

n→∞

Pnwn.

From substitution of (A1) into (1), the definition of λn in (8), and algebra,

Pnwn

=1

n

(p?nλ(p?) (1− Pna )− c

(nµ? +

√n4)− h0IE [Qn(∞)]

)= p?λ(p?)− c

(µ?+

4√n

)− p?λ(p?)Pna − h0

1

nIE [Qn(∞)] .

Since π = p?λ(p?)− cµ?, taking the limit in the above display shows that

lim infn→∞

Pnoptn≥ π, (A2)

if we can establish

Pna → 0 and1

nIE [Qn(∞)]→ 0 as n→∞, (A3)

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which we defer to the last paragraph of this proof. We also from (1) have the upper bound

Pnoptn≤ pnλ(pn)− cµ

n

n− pnλ(pn)Pna .

Since the rate that non-abandoning customers arrive cannot exceed the rate at which they

are served, λn(pn)(1− Pna ) ≤ µn, from which it follows that

Pna ≥[λ(pn)− µn/n]+

λ(pn).

We conclude that

Pnoptn≤ pnλ(pn)

(1− [λ(pn)− µn/n]+

λ(pn)

)− cµ

n

n,

and so optimizing over (pn, µn/n) in the right-hand side of the above expression shows

Pnopt/n ≤ π. Then,

lim supn→∞

Pnoptn− π ≤ 0. (A4)

The inequalities (A2) and (A4) imply that

limn→∞

Pnoptn

= π,

and this is true if and only if

pnopt → p? andµnoptn→ µ?,

because the solution to the SPP is unique by Lemma 1.

Finally, to complete the proof, we use the fact that the GI/GI/1 +GI queue is upper

bounded by a conventional GI/GI/1 queue without abandonment to prove (A3). Suppose

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WnC(∞) and QnC(∞) represent the respective steady-state wait time and queue-length in

a conventional GI/GI/1 queue having inter-arrival and service time sequences specified

as in (9), under policy w in (A1) (which are well-defined because the GI/GI/1 queue is

positive recurrent for each n when 4 > 0). In other words, the demand rate is nλ(p?),

the capacity size is nµ? +√n4, the inter-arrival time variance is var(u1)/(nλ(p?))2, and

the service time variance is var(v1)/(nµ? +√n4)2. From (12) in Marshall (1968) (see also

Kingman (1962))

IE [WnC(∞)] ≤

nλ(p?)(

var(u1)(nλ(p?))2

+ var(v1)(nµ?+

√n4)2

)2(

1− nλ(p?)nµ?+

√n4

) .

Algebra and the fact that λ(p?) = µ? imply that the right-hand side of the above expression

equals

1√n× 1

2×

var(u1)

λ(p?)+

var(v1)

µ? + 2 4√n

+ 4nλ(p?)

× (µ? +4/√n

4

),

which converges to 0 as n → ∞. Hence IE [WnC(∞)] → 0 as n → ∞. Since IE[V n(∞)] ≤

IE [WnC(∞)] for each n, we have IE[V n(∞)] → 0 as n → ∞, which implies V n(∞) ⇒ 0

as n → ∞, because V n(∞) is a non-negative random variable. The continuous mapping

theorem then shows F (V n(∞))⇒ 0 as n→∞, and, since F is bounded,

Pna ≡ IE [F (V n(∞))]→ 0 as n→∞.

Little’s law implies

IE [QnC(∞)]

nλ(p?)= IE [Wn

C(∞)]→ 0 as n→∞.

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Since IE[Qn(∞)] ≤ IE [QnC(∞)] for each n, we conclude

1

nIE[Qn(∞)]→ 0 as n→∞.

This completes the proof.

Proof of Theorem 1. The proof of this result requires the use of the following problem

that perturbs the constraint in (7)

π(ν) ≡ maxp,µ{pλ(p)− cµ} (A5)

subject to λ(p) ≤ µ+ ν.

It follows from Lemma 5.2 in Lee and Ward (2014) that the unique solution to (A5) is, for

ν ≤ µ?,

(p?(ν), µ?(ν)) = (p?, µ? − ν),

so that if π solves (7), then

π(ν)− π = cν.

The next step is to obtain an upper bound on Pnu under any admissible policy u =

{(pn, µn) : n ≥ 1}. First note that from algebraic manipulation

Pnu =pnnλ(pn)(1− Pna )− cµn − h0IE[Qn(∞)]− πn√

n

=√n

(pnλ(pn)− cµ

n

n− π

)− pnλ(pn)

√nPna − h0IE

[Qn(∞)√

n

].

Under any asymptotically optimal policy

pn → p? andµn

n→ µ? as n→∞,

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by Proposition 1, so that θn → 0 as n→∞, where we have dropped the subscript u on θn

for convenience in presentation. Then, from the definitions of π(θn) and θn,

pnλ(pn)− cµn

n− π ≤ π(θn)− π = cθn.

Hence for large enough n,

Pnu ≤ c√nθn − pnλ(pn)

√nPna − h0IE

[Qn(∞)√

n

]. (A6)

Now, suppose

lim supn→∞

√n|θn| =∞,

and consider any subsequence {ni} on which√ni|θni | → ∞. In the case that

√niθ

ni →

−∞, it follows immediately that Pnu → −∞. Hence consider the case that√niθ

ni → ∞.

Since the rate at which customers that eventually enter service must be less than or equal

to the capacity size (dropping the subscript i from ni), we get

λn(pn)(1− Pna ) ≤ µn,

or, equivalently

−Pna ≤µn − λn(pn)

λn(pn)= − θn

λ(pn). (A7)

Hence,

c√nθn − pnλ(pn)

√nPna − h0IE

[Qn(∞)√

n

]≤ c

√nθn − pnλ(pn)

√nPna

≤ c√nθn − pn

√nθn

= (c− pn)√nθn, (A8)

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where the second inequality follows from (A7). Since from Proposition 1 under any asymp-

totically optimal policy pn → p? as n→∞, and p? > c, it follows that for all large enough

n, c − pn < 0. Hence, (c − pn)√nθn → −∞ and so Pnu → −∞ from the bounds in (A6)

and (A8). We conclude that such a policy cannot be asymptotically optimal.

Proof of Proposition 2. From Lemma 2 and the fact that λ ≡ λ(p?) = µ?,

IE

[Qn(∞)√

n

]→ µ?IE[V (∞)] as n→∞.

Define V n(∞) ≡√nV n(∞), so that

V n(∞)⇒ V (∞) as n→∞ (A9)

from the proof of Lemma 2, and note that

√nPna = IE

[√nF

(V n(∞)√

n

)].

Also note that if {xn} is a sequence in [0,∞) for which xn → x as n→∞, then

∣∣∣∣√nF ( xn√n

)− F ′(0)x

∣∣∣∣→ 0 as n→∞ (A10)

because

∣∣∣∣√nF ( xn√n

)− F ′(0)x

∣∣∣∣ =

∣∣∣∣F (xn/√n)− F (0)

xn/√n

xn − F ′(0)x

∣∣∣∣=

∣∣F ′(ξn)xn − F ′(0)x∣∣ , for ξn ∈ (0, xn/

√n),

where the first equality follows from F (0) = 0 and the second is by the mean value theorem

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(recalling that F is assumed differentiable). Then,

∣∣F ′(ξn)xn − F ′(0)x∣∣ ≤ |F ′(ξn)− F ′(0)|xn + F ′(0)|xn − x| → 0 as n→∞

establishes (A10). The generalized continuous mapping theorem (Theorem 3.4.4 in Whitt

(2002)), (A9), and (A10) imply

√nF

(V n(∞)√

n

)⇒ F ′(0)V (∞) as n→∞. (A11)

Next, since for any v ≥ 0,

√nF

(v√n

)=F (v/

√n)− F (0)

v/√n

v = F ′(ξ)v for ξ ∈ (0, v/√n)

by the mean value theorem and the fact that F is differentiable, it follows that

√nF

(V n(∞)√

n

)≤ V n(∞) sup

x∈[0,∞)F ′(x).

Since F ′ is bounded by assumption, the family {√nF (V n(∞)/

√n)} is dominated by the

family {V n(∞) supx∈[0,∞) F′(x)}, which is uniformly integrable from the proof of Lemma

2. Hence {√nF (V n(∞)/

√n)} is uniformly integrable, which together with (A11), implies

IE

[√nF

(V n(∞)√

n

)]→ F ′(0)IE[V (∞)] as n→∞.

This completes the proof.

Proof of Theorem 2. The convergence in (23) follows from Proposition 2, as shown in

(20), and the representation for the diffusion-scaled profit in (13). Consider any admissible

policy w = {(pn, µn) : n ≥ 1}. It follows from Theorem 1 that it is sufficient to evaluate

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the limiting behavior of this policy on any subsequence {ni} having√niθ

ni → θ ∈ IR as

ni →∞. Recall from (A6) that

Pnw ≤ c√niθ

ni − pniλ(pni)√niP

nia − h0IE

[Qni(∞)√ni

].

Since it follows from Proposition 2 that on any subsequence {ni} having√niθ

ni → θ ∈ IR,

recalling that g(θ) ≡ IE[V (∞; θ)]

c√niθ


nia − h0IE

[Qni(∞)√ni

]→ cθ − p?λ(p?)F ′(0)g(θ)− h0µ?g(θ),

as ni → ∞, and also since θ? is a unique maximizer of the right-hand side of the above

display, as guaranteed from Lemma 3, we find

lim supn→∞

Pnw ≤ P? =cθ? − p?λ(p?)F ′(0)IE [V (∞; θ?)]− h0µ?IE[V (∞; θ?)]. (A12)

The proof is complete because (i) the policy v(θ?) attains the upper bound in (A12), (ii)

any policy w under which∣∣√niθni

∣∣ → ∞ as ni → ∞ has Pnw → −∞ as n → ∞ by

Theorem 1, and (iii) any policy under which√niθ

ni → θ 6= θ? as ni →∞ does not attain

the upper bound in (A12).

Proof of Corollary 1. We show the scaled capacity imbalance√nθnopt → θ? as n → ∞.

From Theorem 1,

lim supn→∞

√n|θnopt| <∞.

Consider any convergent subsequence {ni} on which

√niθ

niopt → θ ∈ IR as ni →∞,

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so that the standard heavy traffic assumption is satisfied. On this subsequence, Proposition

2 shows, recalling that g(θ) ≡ IE[V (∞; θ)],

c√niθ


nia − h0IE

[Qni(∞)√ni

]→ cθ − p?λ(p?)F ′(0)g(θ)− h0µ?g(θ),

as ni →∞. The definition of θ? as the unique solution to the DCP (21) implies that

cθ − p?λ(p?)F ′(0)g(θ)− h0µ?g(θ) ≤ cθ? − p?λ(p?)F ′(0)g(θ?)− h0µ?g(θ?),

with equality if and only if θ = θ?. Recall that opt must be asymptotically optimal. Since

the proposed policy v(θ?) achieves the upper bound in the above expression by Theorem 2,

the definition of asymptotic optimality is contradicted unless θ = θ?. Hence,√nθnopt → θ?

as n→∞. Next, we show limn→∞ Pnopt = P?. The proof has exactly the same structure as

the proof of Corollary 6.2 in Lee and Ward (2014). The same arguments leading to (A12)

show that,

lim supn→∞

Pnopt ≤ P?. (A13)

Also, from (23) in Theorem 2,

limn→∞

Pnv(θ?) = P?.

The policy opt must be asymptotically optimal, and so

lim infn→∞

Pnopt ≥ P?. (A14)

It follows from (A13) and (A14) that limn→∞ Pnopt = P?. This completes the proof.

Proof of Proposition 3. The proof structure closely follows those of Lemma 2 and Propo-

sition 2. From Lemma EC.4 of Huang and Gurvich (2018), the M/GI/1+GI systems under

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hazard-rate scaling (indexed by n) belong to a single queue family Q(H) for all sufficiently

large n. The tightness of the family {√nV n(∞) : n ≥ 1} readily follows from the first

part of Lemma 1(i) of Huang and Gurvich (2018). Hence, the process-level convergence

of Reed and Ward (2008), together with the aforementioned tightness, implies the steady-

state convergence of offered waiting time sequence√nV n(∞) ⇒ V (∞) as n → ∞. Then,

the convergence IE[Qn(∞)√

n

]→ µ?IE[V (∞)] as n → ∞ follows exactly as in the proofs of

Lemma 2 and Proposition 2.

Next, recall

√nPna = IE

[√nFn(V n(∞))

]= IE

[√n

(1− exp

(−∫ V n(∞)

0hn(u)du

))]

= IE

[gn

(∫ √nV n(∞)

0h(v)dv

)]

where gn(x) ≡√n(

1− exp(−x√n

)), x ≥ 0. Notice that gn(x) → x, as n → ∞, uni-

formly on compact sets, which in turn implies that gn(xn) → x whenever xn → x.

From a generalized continuous mapping theorem (cf. Theorem 3.4.4 in Whitt (2002)),

√nV n(∞)⇒ V (∞), and the uniform integrability of the family {[

√nV n(∞)]k : n ≥ 1} for

any k ≥ 1 (following from Lemma 1(i) of Huang and Gurvich (2018)), we have

√nPna = IE

[gn

(∫ √nV n(∞)

0h(v)dv

)]→ IE

[∫ V (∞)

0h(v)dv

]

as n→∞. This completes the proof.

Proof of Theorem 3. The proof follows exactly the same arguments as the proof of

Theorem 2.

Proof of Proposition 4. The proposition can be easily verified by noting the fact h(0) =

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F ′(0) and comparing the objective functions in (21) and (26).

B Lemma Proofs

Proof of Lemma 1. Lemma 2.1 in Lee and Ward (2014) shows there exists a unique

solution to (7) with the desired properties. Then, from the definition of a unique maximizer,

it suffices to check that, for (p, µ) 6= (p?, µ?) and p ≥ 0, µ ≥ 0,

pλ(p)

(1− [λ(p)− µ]+

λ(p)

)− cµ− h0λ(p)

∫ w(p,µ)

0(1− F (x))dx

< p?λ(p?)

(1− [λ(p?)− µ?]+

λ(p?)

)− cµ? − h0λ(p?)

∫ w(p?,µ?)

0(1− F (x))dx.

From Lemma 2.1 in Lee and Ward (2014), one has λ(p?) = µ? and therefore the right-hand

side of the above inequality becomes p?λ(p?)− cµ? = π. Since also h0 ≥ 0, it is enough to

show

pλ(p)

(1− [λ(p)− µ]+

λ(p)

)− cµ < π. (A15)

To show the inequality (A15) holds, first note that in the region {(p, µ) : λ(p) ≤ µ},

the left-hand side of (A15) equals to pλ(p)− cµ, which is less than π because the solution

to the SPP (7) is unique. Next, observe that in the region {(p, µ) : λ(p) > µ}, if p ≤ c,

then

pλ(p)

(1− [λ(p)− µ]+

λ(p)

)− cµ = (p− c)µ ≤ 0.

Since 0 < π by Lemma 2.1 in Lee and Ward (2014), it is enough to verify (A15) in the

region

{(p, µ) : λ(p) > µ, p > c}.

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To do this, it is helpful to define

ν ≡ λ(p)− µ,

and use the following equality

pλ(p)

(1− [λ(p)− µ]+

λ(p)

)− cµ = pλ(p)− cµ− pν.

Then, to complete the proof, we must show

pλ(p)− cµ− pν < π, when λ(p) > µ, p > c. (A16)

We show (A16) by further dividing the parameter regime into the following four sub-

cases:

(i) ν ≤ µ?;

(ii) ν > µ?, p > p?;

(iii) ν > µ?, p < p?, µ ≤ µ?;

(iv) ν > µ?, p < p?, µ > µ?.

Also, it is helpful to recall that since the revenue function R(p) ≡ pλ(p) is concave by

assumption (A1),

R(p) ≤ R(p?) +R′(p?)(p− p?). (A17)

Furthermore, recall that the KKT conditions in the proof of Lemma 2.1 in Lee and Ward

(2014) guarantee

R′(p?) = λ(p?) + p?λ′(p?) = cλ′(p?). (A18)

Case (i): Recall the optimization problem (A5) that perturbs the constraint in the SPP

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(7), which has solution π(ν) and satisfies π(ν) = π + νc. Then, (A16) follows from

pλ(p)− cµ− pν < π(ν)− pν = π − (p− c)ν.

Case (ii): It follows from (A17) that

pλ(p)− cµ− pν ≤ R(p?) +R′(p?)(p− p?)− cµ− pν.

Since ν > µ? and p > c, pν > cµ?. Then, also noting that R(p?)− cµ? = π, we find

R(p?) +R′(p?)(p− p?)− cµ− pν < π +R′(p?)(p− p?)− cµ.

The term R′(p?)(p− p?) is negative because R is concave and p > p?. Hence (A16) follows

from the previous two displays.

Case (iii): Simple algebra shows that

pλ(p)− cµ− pν = (p− c)µ < (p? − c)µ? = π.

(The last equality follows because λ(p?) = µ?.)

Case (iv): It follows from (A17) and ν > µ? that

pλ(p)− cµ− pν ≤ R(p?) +R′(p?)(p− p?)− cµ− pµ?.

Since R(p?) = π + cµ?,

R(p?) +R′(p?)(p− p?)− cµ− pµ? = π + c(µ? − µ) +R′(p?)(p− p?)− pµ?.

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Since µ? < µ, to show (A16), it is sufficient to show

R′(p?)(p− p?)− pµ? < 0.

From (A18), the fact that λ(p?) = µ?, and algebra,

R′(p?)(p− p?)− pµ? =(λ(p?) + p?λ′(p?)

)(p− p?)− pµ?

= −p?(λ(p?) + p?λ′(p?)

)+ pp?λ′(p?)

= −p?cλ′(p?) + pp?λ′(p?)

= p?λ′(p?)(p− c)

< 0,

because p > c and λ′(p?) < 0 from assumption (A1).

Proof of Lemma 2. Under the assumed conditions, we can invoke Lemma EC.3 of Huang

and Gurvich (2018), which concludes our M/GI/1 + GI systems (indexed by n) belong

to a single queue family Q(H) for all sufficiently large n. The tightness, as well as the

uniform integrability, of {√nV n(∞) : n ≥ 1} readily follows from the first part of Lemma

1(i) of Huang and Gurvich (2018). (Under critically loaded regime, as considered in our

setting, it can be seen ρ = 1, wp = 0 and the scaling parameter cp = 1/√λn = 1/

√nλ;

see Section EC.3.1 of Huang and Gurvich (2018).) Hence, the process-level convergence of

Ward and Glynn (2005), together with the aforementioned tightness, implies the steady-

state convergence of offered waiting time sequence√nV n(∞) ⇒ V (∞) as n → ∞. Then,

the uniform integrability implies IE[√nV n(∞)]→ IE[V (∞)] as n→∞.

Combining the preceding convergence with Corollary 1 of Huang and Gurvich (2018),

we conclude the second result IE[Qn(∞)√

n

]→ λIE[V (∞)] as n→∞. Indeed, this is because

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the term λ of Huang and Gurvich (2018) translates into λn(= nλ) in our setup and their

Corollary 1 implies IE[Qn(∞)] = λnIE[Wn(∞)], where Wn(∞) ≡ v∧V n(∞) is the steady-

state waiting time and v is the patience threshold drawn from the patience distribution F .

Therefore,

IE

[Qn(∞)√

n

]= λ

√nIE[Wn]

= λIE[√nv ∧

√nV n(∞)]→ λIE[V (∞)]

as n → ∞, from the fact that IE[√nV n(∞)] → IE[V (∞)] and

√nv → ∞ as n → ∞.

Recalling that λ = λ(p?) = µ? completes the proof.

Proof of Lemma 3. We show g(θ) ≡ IE[V (∞; θ)] is strictly increasing in θ. Recalling

IE[V (∞; θ)] =θ

λF ′(0)+

σ√2F ′(0)

hz

(−θλσ

√2

F ′(0)

),

a straightforward calculation shows that

g′(θ) =1

λF ′(0)

(1− h′z

(−θλσ

√2

F ′(0)

))> 0. (A19)

This follows from a property h′z(x) < 1 for any x ∈ R; see, e.g., (3) in Sampford (1953).

Next, let

fobj(θ) ≡ cθ −(p?F ′(0) + h0

)µ?g(θ)

be the objective function in (21), recalling that

g(θ) =θ

λF ′(0)+

σ√2F ′(0)

hz

(−θλσ

√2

F ′(0)

),

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for λ = λ(p?). Then

f′′

obj(θ) = −(p?F ′(0) + h0

)µ?g′′(θ) < 0,

because

g′′(θ) =1

λ2σF ′(0)

√2

F ′(0)h

′′z

(−θλσ

√2

F ′(0)

)> 0 (A20)

from the fact that the hazard rate function associated with a standard normal distribution

is convex (see, for example, (4) in Sampford (1953)). We conclude that fobj is strictly

concave. Therefore, to complete the proof, we must show a critical point exists; that is,

there is a unique solution to f′

obj(θ) = 0.

A critical point of fobj satisfies

c =(p?F ′(0) + h0

)µ?g′(θ). (A21)

A unique solution to (A21) exists if and only if

(p?F ′(0) + h0

)µ? lim

θ→−∞g′(θ) < c <

(p?F ′(0) + h0

)µ? lim

θ→∞g′(θ), (A22)

since g′(θ) is increasing in θ from (A20) and continuous. Straightforward calculus shows

h′z(x)→ 1 as x→∞ and h′z(x)→ 0 as x→ −∞. Hence from (A19)

limθ→−∞

g′(θ) = 0 and limθ→∞

g′(θ) =1

λ(p?)F ′(0),

and so

(p?F ′(0) + h0

)µ? lim

θ→−∞g′(θ) = 0 (A23)(

p?F ′(0) + h0)µ? lim

θ→∞g′(θ) =

(p?F ′(0) + h0

) 1

F ′(0)≥ p? > c, (A24)

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where we use λ(p?) = µ? and p? > c from Lemma 1 to derive (A24). Together (A22),

(A23), and (A24) guarantee a unique solution to (A21) exists.

Proof of Lemma 4. The proof structure is close to that of Lemma 3. While the first

assertion can be checked from a direct calculation of ddθIE[V (∞; θ)], our argument here is

based on a simple stochastic comparison result. Suppose θ1 ≤ θ2. It is straightforward to

see that the proof in Proposition 2 in Ward and Glynn (2003) can be extended to show

that on each sample path

V (t; θ1) ≤ V (t; θ2), for each t ≥ 0, (A25)

where V (t; θi) is as in (24) with θ replaced by θi, i = 1, 2. So, IE[V (t; θ1)] ≤ IE[V (t; θ2)]

for each t ≥ 0. It is known from ergodicity of {V (t)}t≥0 that (e.g., Corollary 5.11(2) in

Budhiraja and Lee (2007)) IE[V (t; θ)] → IE[V (∞; θ)] as t → ∞ and hence we conclude

that IE[V (∞; θ1)] ≤ IE[V (∞; θ2)].

Now, we establish the second assertion. Taking the derivative in the objective function

of (26), we determine a condition under which there exists a unique solution θ? to

c = p?λ(p?)d

dθIE

[∫ V (∞;θ)

0h(u)du

]− h0µ?

d

dθIE[V (∞; θ)]. (A26)

Recall the process V in (24) has a unique steady-state V (∞; θ) whose density, as given in

Proposition 6.1(i) in Reed and Ward (2008), is

p(x; θ) = M(θ) exp

(2

σ2

(θ

λx−

∫ x

0H(s)ds

)), x ≥ 0, (A27)

where H(s) ≡∫ s0 h(u)du is a cumulative hazard function and M(θ) ∈ (0,∞) is such that∫∞

0 p(x)dx = 1. Such an explicit steady-state density formula along with a basic calculus

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yields that the objective function of (26) is strictly concave in θ.

Let U(θ) denote the right hand side of (A26). A unique solution to (A26) exists if and

only if

limθ→−∞

U(θ) < c < limθ→∞

U(θ). (A28)

Recall the cost per capacity c must be such that 0 < c < p? from Lemma 1. Now, we verify

(A28) always holds. We first show limθ→−∞

U(θ) = 0. Our argument is based on a simple

stochastic comparison result. Suppose θ < 0. (The case θ ≥ 0 is immaterial since we are

concerning only about θ → −∞.) Let R = {R(t) : t ≥ 0} satisfy the stochastic equation

R(t) = σW (t)+ θλ t+L(t), where W is the same Brownian motion as the one used to define

V in (24). It is straightforward to see that the proof in Proposition 2 in Ward and Glynn

(2003) can be extended to show that on each sample path

V (t) ≤ R(t), for each t ≥ 0, (A29)

and so IE[V (t)] ≤ IE[R(t)] for each t ≥ 0. It is well-known that (cf. Corollary 1.1.1 and

Corollary 2.3.1 in Abate and Whitt (1987)) IE[R(t)]→ IE[R(∞; θ)] = λσ2/(2|θ|) as t→∞.

Hence {R(t) : t ≥ 0} is a uniformly integrable (UI) family, and therefore, from (A29),

{V (t) : t ≥ 0} is a UI family. Thus, IE[V (t; θ)] → IE[V (∞; θ)] as t → ∞ and we conclude

that IE[V (∞; θ)] ≤ IE[R(∞; θ)]. With this bound and a fact IE[R(∞; θ)] = λσ2/(2|θ|)→ 0

as θ → −∞, one has IE[V (∞; θ)] → 0 as θ → −∞. Moreover, Corollary 2.3.1 in Abate

and Whitt (1987) implies IE[[R(∞; θ)]2] = λ2σ4/(2θ2), which tends to zero as θ → −∞.

Thus, V (∞; θ) → 0 a.s. as θ → −∞, which implies IE[∫ V (∞;θ)

0 h(u)du]→ 0 as θ → −∞.

This, together with the fact that IE[V (∞; θ)] is monotonely decreasing as θ decreases and

limθ→−∞ IE[V (∞; θ)]→ 0, the mean value theorem concludes limθ→−∞ U(θ) = 0 in (A28).

It remains to show the upper limit of (A28) satisfies limθ→∞ U(θ) ≥ p?. (Recall c

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For Review Only

must be such that 0 < c < p? from Lemma 1.) In fact, we will show that if h0 = 0 then

limθ→∞ U(θ) = p?. (Henceforth, assume h0 = 0.) To do this, we use a variant of Laplace

method of asymptotic expansion for integrals, see, e.g., Chapter 3 of Miller (2006). Recall

H(x) ≡∫ x0 h(u)du and from (A27),

IE

[∫ V (∞;θ)

0h(u)du

]= IE[H(V (∞; θ))] =

∫∞0 H(x) exp

(2σ2

(θλx−

∫ x0 H(s)ds

))dx∫∞

0 exp(

2σ2

(θλx−

∫ x0 H(s)ds

))dx

.

(A30)

Now, we examine an asymptotic behavior of IE[H(V (∞; θ))] as θ →∞. For a real valued

function f(x), consider an integral I(f(·); θ) defined by

I(f(·); θ) ≡∫ ∞0

f(x) exp

(2

σ2

(θ

λx−

∫ x

0H(s)ds

))dx =

∫ ∞0

f(x) exp (θG(x)) dx,

where

G(x) ≡ 2

σ2λx− 2

σ2θ

∫ x

0H(s)ds.

Then, the integral in (A30) is given as I(H(·); θ)/I(1(·); θ), where 1(·) stands for a

constant function 1(x) = 1. Notice that G′(x) = 2σ2λ− 2σ2θ

H(x) and G′′(x) = − 2σ2θ

h(x) ≤ 0

for x ≥ 0 (because the hazard rate h(x) ≥ 0). So, G(x) has a unique maximizer x0 such

that G′(x0) = 0, which implies H(x0) = θλ (recall H(x) is a cumulative hazard rate and

hence increasing in x). Intuitively, for large values of θ, the integrand of I(f(·); θ) has one

narrow sharp peak around x0 and the integral value I(f(·); θ) is completely dominated by

that peak. From Section 3.4 of Miller (2006), the ratio of two integrals has an asymptote

given as

IE[H(V (∞; θ))] =I(H(·); θ)I(1(·); θ)

∼ H(x0) =θ

λas θ →∞,

where f(x) ∼ g(x) as x → ∞ means limx→∞ f(x)/g(x) = 1. Now, a direct calculation

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For Review Only

(using the ratio expression in (A30)) shows that ddθIE[H(V (∞; θ))] is continuous and non-

decreasing in θ, which implies, from, e.g., Exercise 4.58 in Francinou et al. (2014), that

d

dθIE[H(V (∞; θ))] ∼ 1

λas θ →∞.

Hence, we conclude that if h0 = 0 then limθ→∞ U(θ) = p?, that is, in (A26), one has

p?λ(p?) ddθIE[∫ V (∞;θ)

0 h(u)du]

converges to p? as θ → ∞. To sum up, limθ→∞ U(θ) ≥ p?

whenever h0 ≥ 0. This completes the proof.

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