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Pricing and Capacity Sizing for Systemswith Shared Resources: ApproximateSolutions and Scaling Relations

Constantinos Maglaras Assaf ZeeviGraduate School of Business, Columbia U niversity, 3022 Broadwai/, New York, New York 10027

[email protected] [email protected]

T his paper considers pricing and capacity sizing decisions, in a single-class Markovianmodel n:Totivated by communication and information services. The service provider isassumed to operate a finite set of processing resources that can be shared among users; how-ever, this shared mode of operation results in a service-rate degradation. Users, in turn, aresensitive to the delay implied by the potential degradation in service rate, and to the usagefee charged for accessing the system. We study the equilibrium behavior of such systemsin the specific context of pricing and capacity sizing under revenue and social optimizationobjectives. Exact solutions to these pro blems can only be obtained via exhaustive simulations.In contrast , we pursue approximate solutions that exploi t large-capacity asymptotics. Economicconsiderations and natural scaling relations demonstrate that the optimal operational modefor the system is close to "heavy traffic." This, in turn, supports the derivation of simpleapproxim ate solutions to economic optimization problem s, via asymptotic m ethods that com-pletely alleviate the need for simulation. These approximations seem to be extremely accu-rate. The main insights that are gleaned in the analysis follow: congestion costs are "small,"the optimal price admits a tw o-part decom position, and the joint capacity sizing and pricingproblem decouples and admits simple analytical solutions that are asymptotically optimal.All of the above phenomena are intimately related to statistical economies of scale that arean intrinsic part of these systems.{Shared Resources; Heavy Traffic; Equilibrium; Pricing; Many-Serve r Limits)

h m1. Introduction and OverviewIn recent years, there has been explosive growtthe usage of the Internet and, in particular, in thescope of services made available under the auspices ofthe so-called "World Wide Web" (WWW). Examplesinclude Internet telephony, streaming audio, e-mail,and information retrieval, to name a few. In addi-tion, many Internet visionaries foresee the futureof personal computing and, in particular, of hand-held appliances, as mainly providing an entry portto the WWW, with most common software beingaccessed on remote servers rather than locally on the

2003 INFORMS

desktop or hand-held device. The rapid developmenof this technology is a constant challenge for servicproviders, as economic objectives, viz, revenue management, are blended in with the structural propertieof these systems.This paper focuses on a class of systems that delivecomm unication and information services with the folowing four features: (1) large capacity, (2) an uppbound on the service rate that can be allotted teach user, (3) resources that can be shared amonthe users, and (4) elastic demand. They bear masimilarities with traditional OR/OM (operationresearch/operations m anagement) models: dem and i

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MAGLARAS AND ZEEVIPricing an d Capacity Sizing for Systems with Shared Resources

stochastic, customers are sensitive toboth price andcongestion effects, and there is a finite amount ofresources (processing and storage capacity) that con-strains the performance of the system. In contrastwith traditional models, however, typically there is nophysical inventory of goods and no queueing-relatedphenomena in these systems; congestion effects arisefrom the sharing of resources by the users. Thus,delays are not the result of jobs or users waiting toreceive service, rather, they arise from a rate of servicethat is lower than nominal. In particular, anynum-ber of users can simultaneously access the service, butwhen the combined demand exceeds a nominal load,the system's processing capacity gets divided amongthem, and the rate of service experienced by the usersdegra des. On the other hand, wh en the deman d is lessthan the nominal load, resources cannot be "pooled"to provide better-than-nominal service to theusers.Internet telephony, or bandwidth provision via cablemodems are examples of physical systems that sharethese features.^

Our formulation assumes that the pricing mecha-nism used by the service provider is a fixed "pay-per-access" charge, i.e., users pay a static usage feefor accessing the system. This is easy to implement,transparent tothe user, and provides a good startingpoint for the more complex problem of dynamic pric-ing. Taking into account the users' sensitivity to priceand service level, and capacity-related costs (initialinvestment, operational costs, and equipment devalu-ation), the service provider is faced with the problemof jointly selecting the system's capacity and a fixedpricing rule to optimize system profits {monopoly pric-ing) or total utility (social pricing).

This paper strives to contribute to the analysis andunderstanding of the above problems along threedimensions. The first dimension concerns model for-mulation. We propose a simple Markovian modelfor the dynamics of systems with finite capacity andresource sharing capabilities (queueing and inven-tory effects areabsent in our formulation). This is' In many applications, there is a binding constraint on the rateat which the user can access these services (typically due to othercapacity constraints, e.g., the maximum speed supporting the users'connection to the network), as well as intrinsic limitations on theextent to which resources can be pooled on the server end.

essentially an M/M/oo model where the total avail-able processing capacity shared by all busy serversis constrained and the capacity that can be ded-icated to each individual user, therefore, is upperbounded. User utility depends on the fixed usagefee as well as the anticipated service degradation;these effects, in turn, are captured via a probabilis-tic choice model that determines user behavior andaggregate demand. An intrinsic "feedback" betweencongestion and demand is present (an increase in th eformer reduces the latter and vice versa), due to theredynamics, so we invoke an equilibrium frameworkthat determines the system's nominal operating point.Despite the M arkovian nature of the model, the intrin-sic equilibrium structure renders the exact analysisof the optimization objectives essentially intractable,thus, one has to resort to approximate procedures.

The second dimension highlights the relation andconnections between economic analysis, and the opti-mal operation of the system. We essentially show thatif the demand for services is elastic, then economicoptimization of the class of systems described inthe previous paragraph is achieved in an operationalregime where high resource utilization and positiveprobability of congestion are both present. One infor-mal interpretation of this result is that "heavy trafficis economically optimal," with the understanding thatthis statement holds under structural and economicassumptions on the model and demand process. Thisobservation allows us, in turn, to port analysis toolsthat support straightforward approximate solutions tothe original economic optimization problems. More-over, the optimal operating regime provides a simple(but, perhaps, not immediately obvious) classificationof the various scaling relations that prevail in suchlarge-capacity systems that are optimized subject toeconomic criteria. Most notably, users experience onlyminor service degradation effects inspite of the highresource utilization, which is the key driver behindthe economic optimality of the heavy-traffic regime.These effects are directly related to the nomina l capac-ity in a ma nner tha t r evea l s statistical economies of scale,and are similar to what has been observed in otherlarge stochastic service systems such as call centersand communication networks.

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Finally, the third dimension concerns the mode ofanalysis a n d solution methodology. As previously men-t ioned, economic arguments lead to the optimali ty ofheavy traffic, which, in turn, allows us to harnesssome simple analytical insights from that framework.In part icular , we develop a tractable approximationto the original economic optimization problems thatyield analytical characterizations of the equilibriumbehavior, as well as the optimal pricing and capacitydecis ions. Simple "rules of thumb" for both pric ingand capacity sizing follow from basic scaling rela-t ions . This mode of analysis culminates in a "recipe"that involves s imple numerical calculat ions, thus,completely c ircumventing the need for tedious opti-mization rooted on exact analysis. The latter approachis used, however, to illustrate the accuracy of theresults obtained via these approximations.

The outline of this paper is as follows. The introduc-tion concludes with a short literature review. Section 2presents the model and formulates the optimal pric-ing problem. Section 3 reviews some machinery, andanalyzes the behavior of large-capacity systems withshared resource s. Section 4 stud ies the pricing pro b-lem under a revenue maximization object ive , and 5extends the analysis to the problem of jointly opti-mi zing ove r system cap acity and p rice. Section 6 dis-cusses the social pricing counterpart. Finally, 7 offerssome concluding remarks. The proofs are re legated tothe append ix .

The work we present here is most closely relatedto , and inspired by, two quite disparate s trands ofresearch. The first is the framew ork advo cated inMendelson (1985) and, subsequently , extended inMende lson and Whang {1990}; a closely related paperis Stidh am (1985). O ur basic setu p in wh ich the ser-vice system is embedded in a n:iicroecononiic frame-work, and the optimal pric ing problem is s tudied asan equil ibr ium model , fo l low these papers . Althoughqueueing per se is not exphcit ly present in our prob-lem, the second stream of research that inspires ouranalysis is rooted in queueing theory. In their semi-nal paper, Halfin and Whitt (1981) present a power-ful framework for analyzing queueing systems with"many" servers . Whit t (1992) discusses how scalingrelations that emerge in the so-called Halfin-Whittregime give r ise to s imple "rules of thumb" for the

design and dimensioning of large-scale service sytems (e .g . , custom er contact centers) . Ou r appro acb lends the economic se tup in Mende lson and Whan(1990), in which the problem is phrased, with the analytical tools in Halfin and Whitt (1981) that are ultmately used to solve it.

The l i tera ture s tudying the use of pric ing to manage the impact of externalities in congestion-sensitivsystems is extensive and appears to date back tNa or (1969); it has since been exten ded and general ized in various direct ions by numerous authorsIn terms of blending an equil ibr ium analysis witinsights from heavy-traffic queueing theory, the recenpaper by Van Mieghem (2000) is probably the mosakin to ours, though the setup, objective, and use othe asymptotic analysis are different . Revenue management in the context of Internet service provisiowas recently studied by Paschalidis and Tsitsikli(2000) in a Markov decis ion-process framew ork wh ercustomers are only assum ed to be price sensi t iveAlthough they focus on dynamic pric ing, an important conclusion of their study is that static pricinrules can achieve near optimal performance. A similar insight was derived in Gallego and van Ryzi(1994) in the "classical" context of selling a set ogoods within a f in i te t ime horizon. A recent reviewof the revenue management l i tera ture can be found iMcGill and van Ryzin (1999).

The study of large-capacity service systems in thspirit of Halfin and Whitt (1981) is currently an activarea of research. Garnett et al. (2002) study approxiniate design rules for large call centers, while Borset al. (2000) consider dimensioning of call centersthe latter is related to our capacity sizing problemin 5. The flavor of the an alysis in th is pap er is closto that in the work of Armony and Maglaras (2004and Whit t (2004). Both papers deal with mult iservequeue ing sys tems wi th conges t ion-sens i t ive demanbut without economic considerat ions, and s tudy thsys tem equ i l ib r ium behav io r us ing asympto t ic methods . The rela t ion between economics and heavy traffic has also been highlighted in a recent paper byHarrison (2003), thou gh the framew ork and a sym ptotic regime are both different than ours.

The model that we posi t with capacity constra intand shared resources is closely related to the recen

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MAGLARAS AND ZEEVIPricing and Capacity Sizing for Systems toith Shared Resources

work of Das and Srikant (2000), who analyze the per-formance of a congested link in a packet-switchednetwork, also appealing to the limit theory of Halfinand Whitt (1981). Finally, we refer to Courcoubetiset al. (2000) and Gibbens and Kelly (1999) and thereferences therein for an overview of Internet-relatedpricing literature that is tangentially related to thispaper.2. Problem Formulat ion

The System Model. External arrivals to the sys-tem, in the form of connection requests, are modelled asa hom ogeno us Poisson process with rate A. Each userhas an i.i.d. service requirement, which is assumed tobe exponentially distributed with known mean l/fi.Our stylized system model attempts to capture threeimportant features of the physical system: (1) finitecapacity, (2) no resource pooling when the systemis underutilized, and (3) the capability to share pro-cessing resources among users. We model this as anM/M/oo system with capacity constrained to C units.Specifically, let N{t) denote the number of users con-nected at hme t > 0; then, the processing rate experi-enced by the user isN{t) C.Without loss of generality, we take C to be an inte-ger corresponding to the number of nominal processingresources, assuming that such a notion is applica-ble. The capacity limitations play a crucial role inour modelling framework. Each nominal processingresource can be dedicated to handle a separate userrequest, providing service at unit rate (measured, e.g.,in Kbits/second), and resources cannot be pooledwhen N{t) < C. However, when the number of con-nection requests exceeds the nominal capacity C, pro-cessing resources are shared in an egalitarian wayamong users; the system is then said to be operatingin a congested state. The system dynamics are equiva-lent to those of an M/M/C queue, though queueingeffects are absent in our conceptual formulation andquality of service is succinctly summarized in the ser-vice rate or throughput rate that is allocated to theuser.

Economic Structure, User Choice Model, andSystem Equilibrium. We assume that the serviceprovider charges a fixed cormection fee $p > 0 whenusers access the system. Service degradation, due tolost throughput when the system is congested, is man-ifested in the form of delay experienced by the users.In communication and information services, userstypically observe the rate of transmission, which indi-cates the degradation in service. In particular, it isreasonable (and, as we argue, not restrictive for ourpurposes) to assume that the rate of service serves asa proxy to assess delay in the following manner. Therate delivered to the user at time t > 0 can be writtenas min (C /N( f), 1), and the user conceives delay as theinverse of the latter. Associating the nominal servicelevel with a unit delay (i.e., the situation where a ded-icated resource is allocated to the user), we define theu s e r ' s conceived excess delay^ as

0

c< C,

- 1 N ( O > C . (1)We assume that a cost of $i > 0 is associated w ith eachunit of excess delay that the user experiences.^ Usershave independent identically distributed (i.i.d.) reser-vation prices for the service, which are independentof the arrival process and service requirements. Theyjoin the system if their reservation price is greater orequal to the steady-state expected value of the "fullprice." Specifically, user valuations are denoted byu > 0 and are distributed according to a p robabilitydistribution P. We will assume that the cumulativedistribu tion function f is continu ously differentiable,the density / is everywhere positive on its supp ort,and the first moment is finite.

As previously indicated, we will focus our atten-tion on the equilibrium steady-state behavior of the sytem. An equilibrium roughly corresponds to a deman d^ In fact, in large-capacity systems D(t)/fi, due to a pathwise ver-sion of Little's law, is asymptotically a correct approximation to theactual excess delay. In our asymptotic analysis, both formulationslead to the same analysis and results.^ Perhaps, it is more realistic to assume that delay costs are con-vex increasing, as in Van Mieghem (2000), however, as we showin 3, systems with large capacity are characterized by "small"excess delay. Hence, convex delay costs reduce to the linear struc-ture assumed above via a first-order Taylor series approximation.

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rate A* and a resulting congestion cost qW such thatboth are time-independent and together satisfy thedemand relationship'*

A*(p) := AP{v > p-\-qED*), (2)where A is the maximal potential arrival rate into thesystem. (The maximal potential arrival rate may alsobe interpreted as the effective "market size" for theapplications offered by the service provider,) In par-ticular, given the reservation price and the full costof accessing the service, the risk-neutral user decidesto join the system with probability P{v > p + qED').In the sequel, a superscript (') is used when variousquantities are considered relative to the equilibriumdistribution.

We note that users' knowledge of their averageexcess delay may be due either to repeated visits tothe system in which the excess delay is observed,or alternatively, we may assume that the serviceprovider makes this information available to the usersto facilitate their decision whether to join the sys-tem or not. The reader should also note that theexpected excess delay (ED*) depends upon the price pthough this dependence will be suppressed for nota-tional clarity. The following result establishes the intu-itive fact that the system always admits a uniqueequilibrium .**

PROPOS IT ION 1. For each capacity C > 0 and pricep>0, there exists a unique equilibrium.

*To be precise, we say that for some price p, the system admits aunique equilibrium if there exists a unique steady-state probabilitydistribution for the process (N(f) : f > 0), such that the expectedexcess delay per user when taken w.r.t. to this distribution, ED",induces a time-homogenous external arrival rate A* through (2)and A', in turn, is consistent with the aforementioned steady-statedistribution.''It is more realistic to assume that state-dependent information isprovided orobserved by the users, however, this information stmc-ture entails a transient analysis that is more involved and will notbe pursued.''The intuition behind this result is straightforward: for a fixedprice, A is strictly decreasing in the delay, while the delay ED isstrictly increasing in p and, thus, in A. These observations indicatethat a unique fixed point should exist. This point is the so-soughtequilibrium.

The user choice model determines the nature othe demand function (the two notions are essentiaequivalent). That is, the choice model defines thexogenous arrival rate into the system according(2). The following examples illustrate how diffeent admissible user choice models induce differedemand functions. With slight abuse of notation s\{x) = AP{v > x).

EXAMPLES. (I) Linear demand: if v is uniformlytributed U[0, A/a], then \{x) = A - ax; (2) Exponendemand: if v is exponentially distributed with meI/a, then \{x) = Ae'"^'; and (3) Iso-elastic demand:is distributed Pareto with shape parameter a > 0, the\(x) = Ax^".Now, theelasticity of a demand function at a prX is given by

e{x) := - dX{x) \{x)A demand function is said to be elastic over an intval [fl, b\ if e{x) > 1 for all x e [a , b\. The key economassumption that we impose is the following.

ASSUMPTION 1. The demand function X{x) is elastover the set \x:Q< X{x) < Cfx\.

For example, in the first two of the three exampleof above demand functions, demand iselastic for cetain values of the price x and inelastic for others; ithe third model, elasticity isconstant and equal to thshape parameter a. Intuitively, a demand functionelastic if a decrease in price (and increase in demandresult in an increase in the revenue rate x\{x). Iterms of the probabilistic primitives, it is equivalento saying that xf{x)/f{x) > 1, where f () = 1 - F(Empirical studies of the demand for bandwidth seemto support the assumption of elastic demand, foexample, Lanning et al. (2000) estimated the elasticitto be around 1.4-1.5/^ While such detailed studies are not available for many other morecently introduced information services, we expect this assumption to hold true, especially in this early stage of their adoption. Fexample, informal inspection of the cable modem industry seemto indicate that, at least in its initial phase, prices have been dropping while demand hasbeen increasing in a manner that overarevenues have also been increasing.

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The Service Provider's Objective. We assume thatthe service provider has at her disposal the 4-tuple{q,\,P,lx) summarizing the congestion cost, thechoice model and its parameters, and the servicerequirement. Suppose that capacity is fixed and givenat C. With the price mechanism fixed, the serviceprovider's objective is to choose the connection priceip) so as to optimize an economic objective. Specif-ically, we consider the goal of revenue maximiza-tion under the unique system equilibrium. (Section6 discusses extensions to the problem of social wel-fare maximization.) This formulation assumes the ser-vice provider is operating in a monopolistic context,Under any pricing decision p, the system evolves asa continuous-time Markov chain. Because every userwho connects to the system pays $p, and the proba-bility of this happening in the next St time units isroughly \8t, we have that the revenue rate in equilib-rium is

(3)We will consider the following two formula-

tions:(1} Optimal pricing: Capacity of the system is fixed

a priori. A price p is sought that optimizes systemrevenues.

(2) joint capacity sizing and pricing: Capaci ty Ctogether with a price p are sought that jointly opti-mize system revenues.

Both problems are considered also w.r.t. a socialwelfare objective in 6.

Mode of Analysis. As we have noted in theintroduction, solving the aforementioned optimiza-tion problems is not straightforward. The reason isthat the equilibrium framew^ork requires us to solvethe steady-state distribution of an M/M/C model,and this cannot be done in closed-form. {The distribu-tion can only be expressed via a finite sum, where thenumber of terms as well as their magnitude dependson C, and the dependence on A is quite compli-cated.) Perhaps more importantly, we believe that thismode of analysis only reveals some of the potentiallyinteresting insights pertaining to the operation andeconomic value of this class of ser\'ice systems. (Forfurther discussion, see 3.3.)

The other viable option is simulation-based opti-mization searching for the pair (A*, ED') that satisfies(2) for the given price p. This is quite cumbersome,as it involves generating a large number of samplepaths to accurately estimate the steady-state quan-tities of interest. Furthermore, the equilibrium pairs(A", ED*) must be determined for a wide range ofprices to search for the optimal usage fee p. The classof systems we are attempting to modei here typicallyhas large nominal capacity, thus, the simulation-basedapproach tends to be computationally intensive. Justlike the direct numerical approach, this method isnot very revealing. Our approach will emphasize nat-ural scaling relations resulting in "rules of thumb"that generate some insight at the expense of settlingfor approximate solutions. It will also illuminate aninteresting connection between economic optimiza-tion and heavy-traffic limit theory, which is of interestin its own right.

3. Analysis of Large-CapacitySystemsThis section motivates and develops an approximat-ing model for the original system when its capacityis "large"; these approximations highlight a series offundamental scaling relations.

3.1. Background and Underlying AssumptionsTo make the forthcoming results more transparent,we first consider the "physical" modes in which thesystem can operate. Recall that the steady-state prob-ability of congestion is given by P(congestion) ;=P(D* > 0) = P(N* > C), where N* denotes the num-ber of steady-state connections in a system withcapacity C in equilibrium. Then, following Garnettet al. (2002), we consider the following three modes ofoperation:

Cost-driven regime: The system is undercapaci-tated, users almost always experience service degra-dation. In this regime, P(congestion) ^ 1,

Rationalized regitiie: The system has balancedcapacity, users experience degradation some ofthe times they use the system. In this regime,P(congestion) ^ f G (0,1).

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Quality-driven regime: The system is overcapaci-ta ted , users hardly ever exper ience serv ice degrada-t ion . In th is regime, P(congest ion) ^ 0.From a purely opera t ional s tandpoint , the' cost-driven regime will tend to underemphas ize conges t ioneffects, while the quality-driven regime will tend tostress quality of service, at the expense of overcapaci-ta t ing the sys tem. The rationalized reg ime may po ten -tially offer a good compromise in t e r m s of t r ad ing offcongestion costs, capacity costs, and utilization.

T he use of the probability of congest ion as aproxy for qual i ty of service is c o m m o n . As it wass h o w n in Halfin and Whitt (1981) and la ter arguedin Whitt (1992), this is a na tu ra l and canonical mea-sure of the system opera t ional character is t ics . In fact,in their paper, Halfin and Whitt (1981) showed thatin an M/M/C system, P(congestion) -* f e (0,1) as Cgrows large if and only if the load in the system is ofthe form p = l - y/'JC) note that this is a s ta temen t forthe ra t ional ized regime. The paramete r s v and y arere la ted through a one-to-one expression given later,and the system behavior bas ica l ly depends on thispa ramete r . In add i t ion , the "natura l" sca les emergingwhen capaci ty is large are of orde r ^capac i ty . Specif-ically, queue lengths tend to be of tha t order and, as aresul t , the wait ing t imes wil l be of o r d e r l / ^ c a p a c i t y(this is the t ime required to clear the backlogs) .

These observat ions have important des ign andopera t ion ramif ica t ions for service networks . Forexample , sys tems that s t r ive to offer a certain level ofqual i ty of service as m e a s u r e d by the probabil i ty ofcongest ion v, can do so whi le ope ra t ing at increas ingutilization rates as given by 1 - y/VC as their capac-it y C grows larger . This demonstra tes the statisticaleconomies of scale enjoyed by la rge serv ice sys tem s andexpla ins , for example , the high utilization rates expe-r ienced in modern-day cal l centers tha t go h a n d inhand with good qual i ty of service (see, e.g., G a r n e t te t al. 2002) . Another important manifes ta t ion of thisfact is cap tu red by the familiar square-root staffing rulethat dictates that to hand le a load of A reques t s perunit t ime, the system should have about A -f- yy/Xservers (assuming /x = 1).

The rema inde r of this section will characterize theasympto t ic pe r fo rmance of our mode l ope ra t ing in

the rationalized regime, while its capacity and potetial demand grow proportionally large. In particulawe will derive here an analogue of Proposition 2.1 oHalfin and Whitt (1981) that characterizes congestioin an M/M/C queue. Our analysis extends the onin Halfin and Whitt (1981) along the following diretions: congestion effects are manifested through shaing rather than queueing; the arrival rate into thsystem is governed by rational user behavior ancaptured via an equilibrium analysis; and the systemanager has the added capability of selecting a statprice to affect demand. The key questions that wbe addressed are the following: (1) How large is thcongestion term given that a nontrivial fraction of aarriving users will suffer some level of congestion(2) What is its effect on the equilibrium demand fthe service? (3) What are (if any) the implications the pricing rules that one needs to consider? Armewith these insights, 4 will show that, if one considethe objective of revenue maximization, the rationaized regime results in the optimal mode from a pueconomics standpoint.

The validity and applicability of the theory ware about to expose hinges on one simple assumption: Capacity and potential demand (market size) arlarge. In particular, we will let the potential demanA and the corresponding capacity C grow proportionally large according to Cju = KA for some appropria/ 0, while the user valuation distribution F is untered.* That is, the market potential A increases, thuser characteristics (valuations) stay the same, and thsystem size grows proportionally to A to serve thlarger market. Finally, in the absence of congestioeffects p defined by

p : F~^{Cfx/A) = F~^{K) (is the pr ice tha t ma tches demand to capacity, i.eAP{v >p) Cfx."The assumption that the valuation distribution is unaltered aschanges may require that some other parameters of the distributioare scaled appro priately as functions of A. This depends on the spcific form of that distribution. For example, in the linear demanmodel V -~ U[0, \/a] and, thus, to keep the distribution unalteas A changes, one needs to scale theprice sensitivity a proportioally to .'\. For the other twoexamples, the valuation distributioonly depend on the parameter a and, thus, require no change.

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3.2. Main Resu ltsWe now turn to one of the main theoretical buildingblocks that are the key in deriving approximate solu-tions to the pricing and capacity sizing problems. Inthe sequel, we will specify results in terms of "C c" highlighting that C is a system quantity. Implicitin this statement is that A is also growing such thatC/i = K A . (T O recover an approximation for the behav-ior of a given system of interest, one should just"plug in" the appropriate values of C and K into thederived expressions.) In the sequel, we make explicitthe dependence of the price on the system capacityusing the notation p{c).

T H E O R E M 1. Assume that potential demand A and thecapacity C grow proportionally large such that Cfi K Afor some K > 0. Then, for any usage fee p{C) > 0 such that

IP ( c o n g e s t i o n ) -* ve{O,l) as C-^ oo,

it follows that( i) user demand scales as X' = Cfx- y's/Cfx^

and system utilization scales as

(5)

zvhere y* is uniquely defined in terms of v.(ii) Theusage fee p{C) must be of the form

where p was defined in (4), and TT is a function of y*.(iii) System congestion scales as

E[delay per user] = ED' = V Cwhere d is an explicit function of y*.

Hereafter, we use the notation f{x) = o{g{x)) ifand only if f(x)/g{x) * 0 and f(x) % g{x) to meanoc, depending on the context and where no confusionarises. The theorem demonstrates that in the rational-ized regime, the system operates "close" to full uti-lization according to1 p'(C) ^ y/VC and congestion

is low, roughly d/y/C. The latter highlights the statis-tical economies of scale that such systems enjoy, and tfact that theassociated congestion becomes a second-order phe nom enon. T his specifies the appropriateregime for asymptotic analysis. In addition, the pricecontrol must admit the decomposition p ^p-\- TT/SJC,where p is explicitly identified in terms of exogenousproblem parameters (A , ju.) and the capacity (C). Notethat due to the assumption that potential demandand capacity scale proportionally, p is, indeed, a con-stant independent of the system size. T his implies thatthe pricing problems reduce to selecting the optimalvalue of 77" that will optimally balance the lost rev-enues with the congestion costs.

Discussion. We turn to several comments on thecontent of the theorem, and the various quantitiesappearing there.

(1) Converse Implications. The theorem is in fact aif and only if result. T hat is, the assumption that theprobability of congestion has a nondegenerate limitand each of the conditions (i)-(iii) are all equivalent.We do not spell this out, however, the proof of thetheorem should suggest why this is true.

(2) Parameter Relations. The second-order parameters {TT, y, d) are all uniquely determined by j ' , thelimiting congestion probability. For our purposes, itwill be more useful to consider TT as the free variable.In fact, because our study focuses on pricing objec-tives andprice is the intrinsic design variable, second-order "congestion-related effects" {p,y,d) shouldreally be considered subordinate to TT. We will pur-su e the analysis assuming that this shift in focus hasbeen made. To explain the nature of the infinitesimalparameters, first note that c is the limit of P(N' > C)as C ^ CO, and is given explicitly in (2.3) of Halfinand Whitt (1981):

(6)where 4>{x) and {x) denote the density and cumlative distribution function of a standard normalrandom variable. Note that v (0,1) implies that0 < 7 < oo. Given the one-to-one relationship in (6),henceforth, we will omit further reference to f and

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focus attention on y. The following gives relationsbetween y, d, and TT (see the Appendix for details):

whered{y) = (7)

^r--^ =

The expression for d{y) can be easily derivedby noting that d := lim^^^^ED*. Plugging in theexpression for the expected excess delay given in(9), namely, TCED* =pP (N * > C)/(v/C(l - p )) , notingthat N / C ( 1 - p ) ^ -y , P(iV* > C) -* j ^ , and p ^ 1 as C ^oc, we get the stated relation using the expression forr given in (6).

It remains to establish the uniqueness of y (itsunique value was denoted by y* in the statement ofthe theorem). This is resolved in the next p roposition,where (8) follows from (7).

P R O P O S I T I O N 2. Assume (5) in Theorem 1 holds and fixany value of TT ^U. Then. y'iTr) > 0 is uniquely definedas the solution of

(8)(3) System Equilibrium Calculations. Note that thesystem equilibrium is now characterized via the

second-order arrival rate y*, which, in turn, canbe obtained through a simple numerical computa-tion. As previously discussed, y' is the key quan-tity that is used to derive approximations for theequilibrium demand rate X* ^ C/i- y*sfCpL, and theequilibrium congestion cost

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user service requirements. The underlying Markovchain in this case is identical to that of an M/M/1system, where the server works at rate C instead ofunit rate. Our interest, as before, is focused on large-capacity systems with high traffic flows. Therefore,we assume that the demand rate scales as A(C) = AC,while the average service requirement is, as before,held fixed at \/^.

Let W denote the total time a user spends in thesystem in steady-state. Then, for an M/M/1 system,Little's law yields that EW = l /(C/i - CA). Now, if wefix the utilization p = A/ju,, then the expected time inthe system decays like 1/C as C grows large. In theso-called rationalized regime, i.e., where the arrivalrate scales like A{C) = Cji- yVC/.i as C grows large,the expected time in the system will scale like l/'/C.These scaling relations are consistent with the onespresent in an M/M/C system: if p < 1, then the con-gestion experienced due to resource sharing decaysas 1/C as C grows large; if the system is assumedto operate in the rationalized regime, the congestionscales like l/\/C, Because the congestion delay seemsto drive the user-choice dynamics, it would appearthat the simpler M/M/1 model should give rise tothe same insights derived on the basis of the M/M/Cmodel (that does not support complete resource pool-ing). A closer inspection, however, reveals that thetwo models are actually quite different.

In the sequel, we focus on the rationalized regime(the following section will establish its economic opti-mality). The key step is to parse the expected time inthe system EW into two components: the actual ser-vice time and the excess delay. Then, for the M/M/1mod el, the congestion cost is of order l/'/C, while theservice tinie is of order 1/C/x, and the total numberof jobs in the system is of order \/C. In sharp con-trast, in the M/M/C system, the congestion term isof order l/>/C, while the time in service is of order1//A, and the total number of jobs in the system isroughly C, plus or minus fluctuations of order Vc.Thus, the models give rise to fundamentally differ-ent dynamics along the following two components:(1) the number of users in the system, and (2) the rel-ative time scales in the system. In particular, in theM /M /1 system, congestion (^ l/ V C ); actual t ime in

service (^l/CjU,). In contrast, in the M/M/C system,congestion (^1/V^) ; actual time in service (^I/JLI),The above differences have an immediate bearingon economic analysis. To this end, note that the prob-ability of a user joining the system is P{v > p + q x(congestion)). While this expression does not includethe time in service, this is implicitly assumed to beaccounted for in the valuation v. In other words,one could rewrite the choice probability as P{v' >Pc + *?//* +

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asser ted in Theorem 1 and, consequently , tbe ra t io-nal ized regim e is economical ly opt im al . The m ainassumption under ly ing our analys is is tha t capaci tyis "scarce ," and th is is spel led out by assuming thatde m an d is elastic (i.e., > 1) in the feasible ra ng eA e [0, C M ) .

Informally , the e las t ic i ty assumption implies tha tthe service provider is always "better off" (as mea-sured hy her monopolis t ic object ive) inducing ahigher equilibrium arrival rate. That is , in the absenceof congestion costs, she should set a price that inducesthe maximum arr ival ra te sus ta inable by the sys tem,given by Cfx. This, of course, puts the system in heavytraffic and results in significant congestion costs, thus,reducing the equi l ibr ium arr ival ra te . The re levantquest ion is : By how muc h?

Extending the revenue rate definition in (3), weset, for each C > 0, R{p, C) := pAP{v > p + qtD'), anddenote the opt imal pr ice p''"'{C) e argmax^R{p, C).The associated equilibrium traffic intensity is denotedhy p""(C), making the dependence of these quant i-ties on C explicit (in the sequel, this may be omittedwh en n o confusion ar ises). The fo l lowing propo si-tion asserts that under demand elasticity, profit max-imizat ion implies tha t la rge-capaci ty sys tems opera tein heavy traffic.

P R O P O S I T I O N 3 . Let Assumption 1 hold and assume,in ad dition, that capacity scales proportionally to potentialdemand. Then, p""(C) -> 1 ns C oo.

The intuition behind this result is fairly straight-forward . First , Theorem 1 has a lready es tabl ishedthat as the system approaches heavy traffic along the"ra t iona l ized" regime (p(C) 1 - y/-/C), congest ioncos t s behave l ike O( l / \ /C) and become neg l ig ib le .While th is does not imply that the sys tem managerwould choose to opera te the sys tem in th is regime,it does simultaneously establish the feasibility of sus-ta in ing h igh u t i l iza t ion and good qual i ty of serv ice .The demand elasticity implies that if congestion isnegligible, the service provider will price to induce ahigh ra te of demand, because th is resul ts in increasedrevenues. (In a recent paper, Harrison (2003) consid-ers an example of a queueing system that is dr ivenby i ts economic and physica l s t ructure to opera te in

heavy traffic; the setup and analysis there are qudifferent from the one we pursue here).

Proposi t ion 3 suggests tha t the opt imal pr ice ougto be close to p, which is the s ta t ic pr ice tha t "p lacethe system in heavy traffic. It says nothing, howevabout the ac tual s t ructure of the opt imal pr ice , tassociated level of congestion, how close the systeis to the heavy-traffic regime, and what the revenureal ized by the opt imized system are . Theo rem addresses a l l of these quest ions .

T H E O R E M 2. Let Assumption 1 hold and assume, addition, that cap acity scales proportionally to potentdeman d. Then, for large C, the optimal price is given by

TT 'p ( c ) p h hwhere p is the price that places the system in heavy trafic given in (4), an d TT"" is the defined via the followioptimization problem:

77 = (1where y'{TT) is defined fo r each TT eU as the unique sotion of (8).

Given (10) the first-order condition that characteizes the revenue maximizing pr ice is

P ~F{p) dd{y)

1 (1This equat ion is used to def ine the opt imal seconorder social price TT'"', taking as given the definitioof p ( the heavy-traff ic- inducing pr ice) , and the n ot ioof the equi l ibr ium y'iir).

Discu ss ion and Ramif ica t ions . One key in s igthat fo l lows from P roposi t ion 3 and Theo rem 2 is theconomical ly opt imized systems with large capai ty , shared resources , and e las t ic dem and operaunder h igh nominal resource u t i l iza t ion ra tes , i .ethey give rise to heavy traffic as the nominal opeat ing poin t . I t is important to note tha t a l though ths ta tement of Theorem 2 resembles Theorem 1 , i t not der ived from i t . However , the converse implictions of Th eorem 1 (see Rem ark 1) establish tha t thstructure of the optimal price in itself implies th

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the system operates in the rationalized regime. Con-sequently, the equilibrium traffic intensity is given byp""{C) = 1 - y''"'/'/C + o{l/VC), where y" " := y*(-7r"")satisfies (8), or equivalently, the equilibrium demandis given by C/x - y'''"VC/x -i- o{VC). These expressionsdescribe the precise way by which the system shouldapproach heavy traffic to optimally balance revenueswith congestion costs. Moreover, the mode of opera-tion for the optimized system is such that the proba-bility of congestion is moderate, i.e., congestion is notcompletely avoided, nor is it the standard fare.

Theorem 2 specifies the form of the optimal price,and a simple numerical optimization routine com-putes it to within high accuracy. Specifically, thistheorem asserts that we can restrict attention to therationaUzed regime, and to pricing rules of the formp = P + 7T/\/C. Let us first give some intuition pertain-ing to the derivation of the optimal price. Using thescaling relations of Theorem 1, the revenue rate canbe expressed as

ITVc + 0Cfxp - - TT )

where the first equality is a restatement of Theorem 1,Parts (i) and (ii). Hence, as capacity grows large, theproblem of selecting the optimal static price to max-imize revenues reduces to the problem of choosing77 e IR to minimize the second-order term of lost rev-enues above, which is exactly (10) in the statementof the proposition. The trade-off in (10) is evident: as77 increases, we have that A*{p) decreases and, thus,7'{77) increases (as a consequence of Theo rem 1). Rev-enue maximization essentially reduces to a second-order analysis that answers the following question:How far from heavy traffic should the system be tooptimally balance lost revenues and congestion costs?Algorithniically, we can compute the optimal priceas follows. Taking as input the endogenous variable,i.e., fixed capacity (C), and the exogenous parametersof the problem, viz, potential demand (A), user choicemodel (f), and user mean service requirement

we proceed as follows. First, set the parameter p thatappe ars in the limit problem in terms of original prob-lem data as in (4), viz, p = F~^{K), where K = Cfx/A.Second, compute the equilibrium y*(7r) using (8) foreach 77 in a fine grid. Finally, evaluate py*{TT) - TT foeach value of 77 in the grid, and seek the m inim umvalue. The system revenues can be computed usingthe expansions for R(p, C) given above.The closed-form expressions derived thus far canbe used to analytically or numerically check the sensi-tivity of various performance quantities with respectto the congestion parameter q (as well as other modelparameters). The key insights derived from this sen-sitivity analysis are (1) first-order effects, in gen-eral, and the heavy-traffic price p, in particular, areindependent of q; (2) for fixed price, the equilib-rium arrival rate and the associated revenue rate aredecreasing in q; (3) the revenue maximizing price isincreasing in q; and (4) the equilibrium dem and at therevenue maximizing price, the associated congestioncost, and the resulting revenues are decreasing in Cj.

Num erical Results. We now turn to some numeri-cal results that illustrate the accuracy of the proposedapproximations. We consider a system with capacityC, potential d em and A = 2.5 - C (connection requestsper minute), /A = 1, and a linear demand model witha 1 C. User sensitivity to congestion is describedvia the cost g $1 per unit of lost throughput (peruser). Figure 1 depicts three graphs: revenue, price,and congestion cost as a function of capacity. Eachgraph has two plots: one describing results for thesimulation-based optimization and the other depict-ing the results obtained via the proposed approxima-tions. Two things, essentially one of the same, standout on close inspection.First, we observe that the approximation is quiteaccurate across a range of system sizes. In particular,the optimal price p, which was derived via exhaus-tive simulation-based optimization has the predictedbehavior ofp^p + Tr/ -/C, where p = 1.5 for this prob-lem data. To reiterate this point, the simulation-basedoptimization did not assume this structure, rather, itemerges as a consequence of the profit maximizationobjective. The exhaustive search for the optimum istedious given the necessity to seek equilibrium pairs

that satisfy (2). The accuracy of the

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Figure 1 Accuracy of the Approximations: (a) Opt imal Revenue Rate; (b) Ihe Opt imal Price; and (c) Expected C ongest ion Cost per User , qlD'

300

1.550 250 300X 10

g'2oo

1 1 ^ ^

\

- ^ ^ ^ - ^

1 ~ 1

1

~-^ +^ ,i 1

1 asymptct ic4 s imulated

1_

50 100 200 250 30050System CapacityNote. T h e" + " plots correspond to the simulation-based o ptimization results, while the solid iine plots correspond to the values derived from the approxim(second-order) analysis.simple approximation derived from Theorem 2 com-pletely alleviates this computational effort. The sec-ond interesting point is not explicitly depicted inthe graphs, though it can be inferred in an obviousmanner: the "optin:\ized" system operates in "heavytraffic." In fact, it is easy to see that the utilizationbehaves l ike {1-p'(C)) ^ O{1/VC).

5. Joint CapacitySizing and PricingSo far, capacity (C) wa s taken to be exog enou sly fixed,or the outcotne of some a priori optimiz ation. We nowconsider the problem of jo int ly choosing the systemcapacity and the price to maximize profits. Inputs tothis design problem are the model parameters sum-marized b y the 4-tuple (g ,A ,/x ,P ) and the cost ofcapacity (appropria te ly amortized over t ime), whichis assumed to be l inear and given by $w(i per unitof capacity. That is, we are assuming that there are

no economies of scale in the cost structure (in pratice, this is typically attrib uted to increased systecomplexity) .

The monopolis t ic service provider is faced with tfol lowing profi t maximization problem:

m a x R ( p , C) wCti.C,p>0 '^ ^ (1

Let us denote by C"" an d p"" the corresponding maimizers (not necessarily unique). To ensure the sevice provider can always extract some positive proby operating such a set of resources, we assume thP{v > w) = F(zv) > 0. Once again, exact solutions adifficult to derive in the absence of a simple charaterization of the equilibrium behavior. Our approatargets approximate solutions, and b uilds on the foudations developed in 3 and 4. The key result wobtained in Theorem 2, where it was shown that felastic demand (e > 1) and any choice of capacity the revenue maximizing price places the system the rationalized regime, where congestion costs p

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user are of order i/s/C, equilibrium demand is A* =Cfi - y'VCfi, and the optimal price is of the form p =P + TT/V/C. The same result holds true when C is adecision variable, provided that the dem and elasticityis still valid. Specifically, for any value of capacity C(including the optimal choice C""), the revenue max-imizing price places the system in the rationalizedregime, where

Rip, C) piC) +- \/Cii{p{C)y'{7T) ~ T T ) .

Substituting the above into the profit maximizationproblem (12), we arrive at the following approxim ateformulation:

max [Cfj.p{C)-wC/i]-\/C/i{p(C)y''{TT)-TT).C> 0 , JreRv ' - , Capacity Sizing Pricing

For large systems, the two terms are essentiallydecoupled suggesting the foUou'ing heuristic.

(i) Capacity Sizing: Choose the capacity C"" thatmaximizes the nominal profit rate assuming that thesystem will operate in heavy traffic and neglectingany stochastic effects. This fixes the first-order priceterm p :p{C"') that places the system in heavytraffic.

(ii) Pricing: Given the optimal choice for capacityC"", choose the second-order price component TT' '" tominimize the performance degradation due to con-gestion.

A solution C"' to the capacity sizing problem willserve to approximate C"", the optimal capacity deci-sion, while p"" p + TT""/v C'" will approximate theoptimal price p"",

A more explicit characterization of the approximatesolution (given by the decoupling heuristic) can beobtained if we are willing to assume a first-order opti-mality condition for the optimization problem (i). Forexample, problem (i) is concave in the design vari-able C for all three demand models mentioned in 2.The unique solution satisfies C'" in (0, A/jU.), whichimplies that the increasing costs of capacity force theoptimized system to operate in a region where the

demand is elastic. The first-order condition for anoptimizer is

Recall that p{C) = T'HK), where K = C/A/A, thus, (13)can be rewritten as fjLF~^(K)-K/f{F~^{K)) = zv. Underthe aforementioned concavity assumption, this equa-tion has a unique solution denoted by k, whichdetermines the optimal capacity as a fraction of thepotential dem and, i.e., C''"'jx = kA. Because P{v > iv)>0, it is easy to deduce that k (0,1). Given k, prob-lem (ii) can be used to select the second-order pricecorrection term TT'''" =argmin{F"'(K)y(77) 7r: TT eIR}.This is also uniquely determined by k. In summary,the proposed solution is to set

C" = kand

TT 'O

(14)where k is determined by (13). The solutions C'^"',p'^"'scale with A according to (14). In the s equel, it will beconvenient to recognize this dependence by writingC"'(A),p''"(A). The associated profit rate is '^(A) ^R{p""{A),C''"'{A))-wC""iA)fi. As the market sizerealized via the potential dema nd A grows large, theapproximate analysis becomes exact and the perfor-man ce of this h euristic becom es optimal. '

T H E O R E M 3. Let Assumption 1 hold. Then, the revenuerate under the capacity and price pair C""(A) ,p"" (A) i sasymp totically optima l in the sense that

1 fls A{A)where :^A) = max\R{p{A), C(A)) - wC{A): C(A)/i. >0, p(A)>0} .'" Previous asymptotic results were phrased in terms of C growinglarge. Given tha t C is now a decision variable, the natural proxy forsystem size is the potential demand A, which reflects the market

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Given that the profit rate is of the form

the asymptotic optimality result asserts that the pro-posed heuristic correctly matches the first-order profitterm of the optimally designed system. This impliesthat asymptotically the optimal level of capacity is"close" to kA and that the optimal price is close topTIK 11

Finally, the following numerical example depictedin Figure 2 illustrates the accuracy of this approximatesolution. The results are derived for a linear demandmodel with A = 200 connection requests/min, a =50, i = 1, jLt :=: 1 per min, and w = $1. For the lineardemand model.

The capacity sizing problem becomesmax I Cii wCu, \.

The corresponding optimizer is given byC ' V = - - ^ = 75.

Also,2a 2Optimizing over the second-order term rr in the wayoutlined above gives p''" = $2,604. In contrast, theoptimal capacity level and price obtained via exhaus-tive simulation was C"' = 79 and f" = $2,529. Thedifferences are, indeed, small (and asymptoticallynegligible).

The use of a second-order "correction" to con-trol the congestion costs also appears in Borst et al.(2000), where this is achieved by adjusting the sys-tem's capacity. In contrast to our work, they do notmodel the choice behavior (demand is fixed) andthere is no pricing or equilibrium analysis. To recapit-ulate, the key insight that emerges is that the capacity'' The same approach may still be applicable under more generalcost struc tures, e.g.. Theorem 3 can be extended to the case of lineardemand and quadratic cost of capacity. The key requirement is thatcapacity costs do not dominate revenues as capacity grows large.

sizing and pricing problems decouple and both caneasily solved. Capacity is selected to maximize proits to first order, while pricing is selected to optimabalance revenues w ith congestion costs; the optimasized and priced system operates in heavy traffic.

6. Social W elfare O ptim izatio nWe now turn to a brief discussion of iarge-capacsystems that operate under a social welfare objecBecause the main results, as well as their derivatioare essentially mirroring the approach taken in tprevious section, we will focus here only on a sketof the main ideas.Social Welfare Maximization (Social Pricing). Thformulation assumes that the system is to be operatwith the objective of maximizing the total utility.equilibrium, the total expected cost per connectiongiven by p-\-q\ED'. Define

vf{v) dv (1to be the total value generated per unit time for all sscribers that select to join the system. The systemwvalue created per unit time is

U{p) =user5

= V{p)-q\'ED% (1and the service provider's objective is to choose thprice p to maximize the systemwide value.We proceed using standard arguments to charaterize the socially optimal price (see Mendelson anWhang 1990). It is convenient to consider l^() anU () as functions of the rate variable A, rather thathe price p (with slight abuse of notation). The opmal pricing problem will then be phrased in termof choosing the optimal equilibrium demand rate Awhich, in turn, will uniquely define the socially opmal price p. Differentiating U with respect to A, aevaluating the derivative at the point A' (let us denothis as 1/(A*) for simplicity) gives the following firsorder condition:

^ED'(A)= 0.

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F i g u r e 2 J o i n t C a p a c i ty S i z i n g a n d P r i c i n g : A = 2 0 0 R e q u e s t s / M J n , a = 5 0 , Q = 1 ; f i = 1 p e r M i n , iv = $ 1104

55 60 65 70 95

N o t e . D e c o u p l e d c a l c u l a t i o n :75 80 85

Capacity (C )^ 7 5 a n d p " ^ = $ 2 , 6 0 4 . O p t i m a l v a l u e s o b t a i n e d v i a s i m u l a t i o n : C " " ^ 7 9 a n d p = $ 2 , 5 2 9

100 105

From A* = A F { p -F qED'), it follows that the utilityof the marginal user who joins the system when theequilibrium demand rate is A* is V"(A*) =p + qD* f"^(A'/A). Substituting this in the above expressiongives the socially optimal price, p^"^(17)

That is, it is socially optimal to charge each user theexternality (or congestion) cost that he imposes on thesystem. Exact evaluation of the socially optimal pricei s , again, complicated due to the equilibrium formu-lation and, thus, relies on exhaustive simulation ornumerical approximation of the steady-state equilib-rium probabilities. As we show next, the correspond-ing asymptotic characterization is much simpler towork with, and the task of social optimization is com-putationally trivial.An argument similar to that of 4 shows that max-imizing social welfare will also "drive" the systemto operate in the rationalized regime. Counterparts to

Proposition 3 and Theorem 2 can be derived underthe same modelling assumptions with one change;here, we do not require demand elasticity.^^ Using thescaling relations derived in Theorem 1, we can derivethe appropriate social optimization objective for theasymptotic system, and proceed with the second-order analysis to approximate the optimal price. Inthe following derivation, we will use the fact thatED * = d7 V C + o( l /VC), where d' := d(y*) (see 3).Starting from (16), we can write

'^ In the absence of congestion costs, the social value function U(-)becomes (J(p) = A / ' ' vf{v) d v , which is decreasing in p. That is, aswe lower price and A(p) increases, tbe systemwide value increases.This characteristic of the social value function is essentially theequivalent of the elasticity assumption and will drive the econom-ically optimized system to heavy traffic.

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(18)Because y = {f{pJ)/F{p)){TT -i- qd"), asymptotically thesocial optimization problem reduces to choosingthe second-order price term TT to m.m^^^{py*{TT) -^qd'iy*{TT))\, making the dependence on y' an d TTexplicit. Recall also that d{y) = i'{y)/y. This is read-ily solvable given the characterization of the equi-librium y*(7T) in (8). Taking derivatives with respectto y, the first-order optimality condition is given byd/dy'(py^ +qd^) = 0, which implies that

p=-q ddiy)dy (19)Note the similarity between (19) and (11) derivedunder the revenue maximization objective.

Discussion. Expression (19) is closely related to(17), and has a natural interpretation. Recall that theoptimal pricing rule is of the form p p - \ - TT/VC,thus, for large-capacity systems, each subscriber paysp . The right-hand side of (19) is the externality cost,where the negative sign accounts for the fact that asy" increases, the arrival rate into tbe system decreases.Hence, the socially optimal equilibrium correspondsto the operating point y'*'", where the externality costq{dd*/dy') equals the first-order price p paid by theusers. Parenthetically, it follows from (11) that underrevenue maximization, the service provider chargesa fixed premium over the externality cost. Moreover,using (19) and the explicit expression for d" = d(y*)given in (7), we can evaluate the term dd'/dy" asa function of y*, and solve for the socially opti-mal equilibrium denoted by y'"^ This characteriza-tion, together with the definition of the equilibriumin (8), specify the socially optimal price, viz, p'^'^ =p -h TT^^VVC- w h e re TT"" = (F{p)/f{p))y"-'' -

Joint Capacity Sizing and Social Pricing. Thisproblem is treated in an identical marmer to the rev-enue maximization counterpart. The objective is nowto solve tbe problemmax A / " vf{v)dv~\'qEDn-

which leads to the following asymptotic formulatior ,^ 1

max A / vf(v) dv - ivCixC>O.Tre_tt L JpiO JCapacity Sizing

PricingFor large systems, this problem, therefore, decouplinto two parts; (i) choose the capacity to maximizthe social surplus, and (ii) choose the price n to miimize performance degradation due to congestioProblem (i) is a simple maximization of a concavfunction. (This follows from the general assumptionon the choice mod el, and the form of the heavy-traffprice p = f"'(K).) Exploiting the decoupling of thcapacity sizing and pricing decisions, we can compare the optimal capacity levels and prices for thapproximating revenue and social optimization problems. Let C, e^S and p'"' = p " " -K TT""/S/C, p ^ "psoc _ ^soc^y^ ^^ ^j^p optimal capacities and pricfor the two approximating problems, essentially optmizing the respective objectives up to and includinsecond-order terms. Then, simple analysis shows th

'" ' > C"' and f " (2Moreover, one can show that asymptotically as thmarket size grows large, this implies that the actuoptimal capacity and price decisions will also bordered in the same way, i.e., C'^'"^ > C"" and p''"'

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become more "efficient" as their capacity increases.A tractable asymptotic analysis leads to several struc-tural insights.(i) Operating Regime: The system should operateclose to "heavy traffic," where nominally all resourcesare fully utilized. This is optimal in the context ofrevenue maximization (under the assumption thatdemand is elastic), and under a social optimizationiibjective.

(ii) Pricing: The optimal price admits a simpledecomposition; the service provider comp utes a priceto bring the system to "heavy traffic" and, subse-quently, applies a second-order correction term thatdepends on the size of the system to optimally bal-ance congestion effects.

(iii) }oint Capacity and Pricing Decisions: Given thenatural scaling relationships intrinsic to such systems,the capacity sizing and pricing problems decouple.Capacity is chosen to maximize profits (or total sys-tem value) assuming that the system is fully utilizedand neglecting stochastic variability. The price is thenadjusted tooptimally balance congestion costs.

Several interesting directions of future researcharise. One natural extension concerns a system thatsupports differentiated services. In the information andcommunication service context, in particular, for sys-tems that share resources, the natural first step is toconsider a menu with two service grades: "guaran-teed" and "best effort." The former refers to users thatare guaranteed a constant rate of service irrespectiveof the state of the system, while the latter refers tousers that share the rem aining capacity (and, thu s, areprone to service degradation). To this end, the Halfin-Whitt (1981) many-server asymptotic regime supportscertain diffusion approximations that can be usedto facilitate the study differentiated services (somepreliminary results along these lines are derived inMaglaras and Zeevi 2002).

While this paper has focused on steady-state analy-sis, the diffusion approximations that were pioneeredby Halfin and Whitt (1981) could be used to approx-imate transient behavior in such systems, addinganother important layer to the current static analy-sis. Finally, an even more challenging problem con-cerns the dynamic pricing mechanism that extends

the static fixed-price setting considered herein. In par-ticular, the results in the current paper strongly sug-gest tbat this would revolve around second-orderanalysis aswell.AcknowledgmentsThe authors and the manuscript both benefitted from the construc-tive comments of two anonymous referees and the associate editor.Appendix. ProofsFor notational clarity, we will denote the capacity of the system asC II. Because most of our results concern the asymptotic regimwhere C (thus, n) grows arbitrarily large, various variables andquantities will be appropriately indexed by a subscript ii.

PROOF OF PROPOSITION 1. Fix a capacity C,, = n, and price p>0,such that P{v > p) > 0 (otherwise, there is nothing to prove). Fixf > 0 and, with some abuse of notation, let the expected excessdelay for an arrival rate \(p + q) bedenoted by ED(O. The equi-librium demand rate and expected excess delay are now definedvia the solution ^* of the set of equationsand

if such a solution exists. To this end, let h(^) = i-E[D(f)]. Now, Ais a decreasing function when considered w,r,t. the variable ^,an d^ (the expected excess delay per user) is, in turn, increasing in A.This suggests that there exists a fixed point ^' that solves the aboveequations. To make this rigorous, note that

because ED(f) isdecreasing in , Differentiability of ED(^) followsfrom two observations. First, ED', considered here as a function ofA', is continuously differentiable in [0, n/i.). This follows from (9)and the expressions for the steady-state probability of congestion,P (N > Ji), for an M/M/n queue (see, e.g., Halfin and Whitt 1981).Second, A(-) is continuously differentiable due to the smoothnessof the choice model. Thus, by the chain rule, ED(^) is continuousand differentiable. Because /;(0) < 0 and /?(oo) > 0, it follows thath(^) 0 has a unique solution ^, and the associated variables A' :A(p + q^') andED(^') - f' characterize the unique equilibrium ofthe finite capacity system. Finally, the traffic intensity is then givenby p" : A'/("M)- which isstrictly less than 1. D

PROOF OF THEOREM 1, The convergence of the sequence of equi-librium traffic intensities follows from Proposition 1 in Halfin andWhitt (1981). First, note that the Markov chain associated with thesystem under investigation in this paper is identical tothat associ-ated with anM/M/n system. Haifin and Whitt (1983) considered asequence of MjM/n queues and showed that lim,, .^IP(N,, > ) =( (0, 1) if and only if lim,,^^ y/n{\ - pJ ^ y > 0, where fiy) =(l>(y)/{y^iy) + rf'(7))-Applying their result to our sequence of sys-tems (with capacities C,,) operating in equilibrium, we get thatP(congestion) - J- (0,1) implies that ^i(\ - p'^) - y' > 0. By

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Proposition 1 in Halfin and Whitt (1981), it also follows thatflie woconditions are equivalent. The next step is to establish the limit forthe expected congestion cost (part (iii)); later on, this will be com-bined with the structure of the customer choice model to derive theasymptotic decomposition of the pricing rule (part (ii)).

The starting point is the relation for the expected excess deiaygiven in(9), which we repeat here for completeness ED* =plP(N; >)/("( l - p;)). From here, using Halfin and Whitt (1981), it followsthat if v ^ ( l - p ; ) - * y', then -/JiED " -* d(y") := f (y ') /y".In the sequel, we will also use the notation d{y) when we wantto make explicit itsdependence on y. The right-hand side in theabove expression is d{y-) appearing in part (iii) of the statementof the main result. Finally, we will use (i) and (iii) toestablish thedesired price decomposition. For n sufficiently large, (iii) impliesthat ED* =d/s/n-\-o{\/JTi). Using this expansion, Taylor's theorem,and the smoothness of the choice distribution in thedefinition of

A', we get thatA: = A,,P(i;>

where the last equality follows from (i), i.e., v^(l -p") -* y". Now,by assumption A^ ^ np.K'\ thus, the last equality implies thatAnP(i' > pn) M/x + 8y/np. + o{s/n) for some appropriate choice of5 R. This, in turn, implies that the price p,, must be of the formp ,, =p + Tr/,/n + o{l/^), where p is selected such that P{V>P) K;that is, p = f"' ( K ) . This establishes part (ii) of the theorem. Finally,using this structural form of p,,, Taylor's theorem, and the smooth-ness of the choice model distribution, we can express A; in theform

(21)Using An = nfiK~\ we h a v e in the limit as K ^ oc that y' = (77- +W(r))/(p)/F(p).

P R O O F O F P R O P O S I T I O N 2. C o n s i d e r any c o n v e r g i n g s u b s e q u e n c e|n,) such that ^ ( 1 -p- ,) -* 7, > 0. As n o t e d in the proof ofT h e o r e m 1, we h a v e t h a t ^ ^ D ^ , -* (y ^) ^ f ( y , ) / y , . N e x t , we takea Taylor expansion of the expression for A ,:

Rewrite A,, as n^p./{p) to get that

which implies that y^ =w e get that

.

^)). Solving for

(22)

The expression for d{y-) and (22) imply that the e q u i l i b r i u mmust satisfy (8). Now, observe that (8) can be rewrit ten as h{y)' ? " ' f ( p ) / / ( p ) - y - y " ' 0 ( y ) ( y < I ' ( y ) + p;,"') < (! -5 /2 ) , say, foinfinitely many n. Consider now theprice sequence p,, = p, whep = f"' ( K ) . i-e., the price that places the system inheavy traffic. Thassumptions on the choice model together with the above implthat for some d' > 0, we have p >p^(l-l-6') for infinitely many nNow, due to demand elasticit)', wehave x{x) t asa: i and, thuusing the assumed smoothness of the choice model, we have thafor some 5" > 0,

l imsupin contradiction to the claimed optimality of \p',,"'\. Thus, itmustthat liminf^_^p >\-8, and because 5 was arbitrary, this impliethat for the profit maximizing price sequence liminf,, .^p > which together with (23) establishes that p;/" -^ 1 under the profmaximization objective. This concludes the proof, a

P R O O F OF T HEOREM 2. The main idea is to establish that trationalized regime leads to profit maximization, and then appeato Theorem 1 in3 toconclude that the price structure must be othe form asserted in the current theorem. The only added task ito establish that the second-order price correction n is, indeed, tlisolution to the given optimization problem, stated in the theoremThe proof isdivided into three steps.Step 1. Proposition 3 asserts that p"" -* 1 asH - 00. We will no

determine the rate of this convergence, which, in turn, will impl

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that -* p. To this end, consider the price sequence {p"" \ that is rev-enue maximizing, i.e., p ' , " ' e argmaXj,iR(p, H )I for each n > 1, Here,R(p, n) extends the definition in (3) denoting the revenue rate for asystem with capacity C,, ^ n, and nnaking this dependence explicit.In what follows, all system quantities are considered in equilibrium,and we omit the superscript (*) for notational clarity. Let us denotethe resulting expected excess delay by d,, : ED,,, Recall from (9),we have that d,, = p,,P (N,, > n)/{n{\-p,,)). Suppose that p -^1so that liminf,, ^ tt(l - p ) < M for some finite positive constantM > q/i/p. Then, itmust be that JTi{\ - p) = o(i), where a,, = o{\)if -* 0 as JI -* 00. By Proposition 1 of Halfin and Whitt (1981),we then have that P (N,, > n) -> 1 and , thus, limsup,,_^i/,, > M"' .Now, _

thus, it follows that p ' , , " 'often, it follows that lim inf,,hm mf

p. But because d >M"' infinitelyp < p - q/M and, consequently.

R(p. n) Mpwhere the right-hand side is strictly less than 1 by the choiceof M. Therefore, pcannot be the revenue maximizing price,in contradiction. We conclude that for any positive constant M,l i m i n f , , _ ^ n ( l - p 7 ) > M and, therefore, because M is arbitrary,we must have (1 -p""} -* oc. Thus, d,, = o(l) and this yields thatP',,"' -^ P-

Step 2. Here, we establish that p'J" ^ p incombination with thedemand elasticity assumption imply that ^/n(1 - p"") -* y. First,because A,, = nfi{\ - y,,), we have

ntj.A,, = nfi =F({7r,, + qd,,) + 0(n(TT,,+qd,,]-)- (24)

where -y,, :- 1 -p and IT,, =p'J" -p. (The second equality abovefollows from the equilibrium condition that "ties together" d,,, IT,,,an d y ,, (see (22)), Thus, we can express the reven ue rate as follows:R(p,7", n) = \,,p',"' = y, , -TT,,)+O(mr^y,,),

using (9), it follows that iimsup,, _^demand elasticity assumption implies

In addition, the

dp \F(p)thus, it follows that f(p) < pf(p). Consequently,Now, if ^/ny,, -> 7 > 0, then 4'(y,,)/^/n - c > 0, which minimizesthe rate of growth of the lost revenues due to second-order effects.Thus, 1/v^ is the economically optimal rate of convergence toheavy traffic.

Step 3. To conclude the proof, we again appeal toProposition 1of Halfin and Whitt (1981), which asserts that if v ^ ( l - p',;") -* 1,then P {N,, > H) -* c e (0, 1), Consequently, the rationalized regimeis the economically optimal regime, viz, supporting the best-possible rate of growth of revenue (up to second-order effects).Moreover, byTheorem 1, it follows that the price sequence corre-sponding to this regime is of the form

P " =1

Thus, 7J-,, ^ n/.Jn. Finally, the optimal choice of ir is obtained byminimizing the effects of the second-order term I/'(T,,)- Spellingthis out, using (25), we have TT"" = zx^mm^^^\y'{TT)p- TT], wherehere, we have made explicit the fact that y depends on -n(through (8)), D

PROOF OF THEOREM 3. Consider the profit maximization problemmax |K(p(A), C(A}) - wC{A)p.: C(A), p(A) > 0| and define K(A) :=C(A)^/A, We will prove that K(A) ^- K e (0,1) as A ^ x, wherek isdefined viaC = k\/fi, the unique root of (13). It then followsthat the decoupled heuristic of (14) isasymptotically optimal.

To shorten notation, the superscript "rm" will be dropped in thesequel. First, recall that k e (0,1) and that C(A)iU,/A = k for all A.Also, note that p(C(A)) = F,"'(J

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policy. This is dearly true because our proposed heuristic achievesthat rate, and the optimal policy can only do better. It follows thatK = a. Using the asymptotic expansions for theprofit rates underthe optimal policy and under the heuristic of (14), we imm ediatelyestablish the asymptotic optimality result, D '

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