In many services, the quality or value provided by the service increases with the time the service provider

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1 MANAGEMENT SCIENCE Vol. 57, No. 1, January 2011, pp issn eissn informs doi /mns INFORMS Quality Speed Conundrum: Trade-offs in Customer-Intensive Servies Krishnan S. Anand David Eles Shool of Business, University of Utah, Salt Lake City, Utah 84112, M. Fazıl Paç, Senthil Veeraraghavan The Wharton Shool, University of Pennsylvania, Philadelphia, Pennsylvania In many servies, the quality or value provided by the servie inreases with the time the servie provider spends with the ustomer. However, longer servie times also result in longer waits for ustomers. We term suh servies, in whih the interation between quality and speed is ritial, as ustomer-intensive servies. Ina queueing framework, we parameterize the degree of ustomer intensity of the servie. The servie speed hosen by the servie provider affets the quality of the servie through its ustomer intensity. Customers queue for the servie based on servie quality, delay osts, and prie. We study how a servie provider faing suh ustomers makes the optimal quality speed trade-off. Our results demonstrate that the ustomer intensity of the servie is a ritial driver of equilibrium prie, servie speed, demand, ongestion in queues, and servie provider revenues. Customer intensity leads to outomes very different from those of traditional models of servie rate ompetition. For instane, as the number of ompeting servers inreases, the prie inreases, and the servers beome slower. Key words: strategi ustomers; queueing games; servie operations; ost disease History: Reeived September 22, 2008; aepted August 17, 2010, by Sampath Rajagopalan, operations and supply hain management. Published online in Artiles in Advane Deember 10, Introdution Festina lente [Make haste slowly]. Motto of Aldus Manutius ( ) In a wide variety of servie industries, providing good ustomer servie requires a high level of diligene and attention. We refer to suh servies as ustomer-intensive servies. Examples of suh servies are health are, legal and finanial onsulting, and personal are (suh as spas, hairdressing, beauty are, and osmetis). Eonomists have noted that some industries in the servie setor, inluding health servies and eduation, have lagged signifiantly in their produtivity growth, despite rapid produtivity improvements overall, in the last few deades (Triplett and Bosworth 2004, Varian 2004). For example, in the last deade, the health-are industry displayed a negative annual growth of 0 4% (Triplett and Bosworth 2004, pp ). We note that low-produtivity industries 1 are predominantly ustomer intensive. 2 A major diffiulty in improving produtivity in suh ustomer-intensive servies is the sensitivity of the servie quality provided to the speed of servie: as the servie speed inreases, the quality of servie inevitably delines. Often, the only way to inrease produtivity without sarifiing quality is to inrease apaity investments, whih inreases osts. This phenomenon has been termed Baumol s ost disease (Baumol 1993). Primary health-are pratie in 1 It is diffiult to ompare produtivity per se between different industries; what an be ompared is their produtivity growths over time. The literature (f. Triplett and Bosworth 2004) desribes industries with low-produtivity growth as low-produtivity industries. Of ourse, a sustained period of low produtivity growth in an industry would lead to low produtivity relative to other industries. We thank the departmental editor for suggesting this distintion. 2 Customer-intensive servies are generally haraterized by high labor ontent, but high labor ontent need not imply high ustomerintensity (e.g., onstrution servies). 40

2 Management Siene 57(1), pp , 2011 INFORMS 41 the United States epitomizes this problem. Beause of high levels of demand, dotors need to rush between patients, 3 spending most of their time treating aute illnesses a proess that is also dissatisfying to patients (Yarnall et al. 2003). As Surowieki (2003, p. 27) notes: Cost disease isn t anyone s fault. (That s why it s alled a disease.) you an ontrol drug osts and limit expensive new proedures, but, when it omes to, say, hospital are and dotor visits, the only way to improve produtivity is to shrink the size of the staff and have dotors spend less time with patients (or treat several patients at one). Thus the Hobson s hoie: to lower pries you have to lower quality. Thus, primary health-are servies provide a lear ontext for the quality degradation assoiated with a servie system strethed to work at a fast pae while trying to serve a large number of patients. The above examples suggest that fousing exlusively on improving produtivity by inreasing the speed of servie leads to a redution in the value of the servie provided. On the other hand, inreasing the servie value by inreasing the time spent serving eah ustomer has its pitfalls. First, it inreases ustomers waiting times beause of ongestion effets from the slower servie times. Seond, it inreases the ost of the servie as the produtivity (number of ustomers served) falls. The first effet leads to lower ustomer value; the seond, to higher pries. In this paper, we study how a servie provider an make the optimal quality speed trade-off in the fae of strategi ustomers ustomers who join the servie only if the utility (the value of the servie net of ongestion osts) exeeds the prie harged by the servie provider. Congestion osts are an outome of the aggregate prourement deisions of all onsumers in the market every ustomer who joins the servie imposes a negative externality (in the form of additional expeted waiting time) on all other ustomers. In turn, the trade-off faed by the provider of a ustomer-intensive servie between quality (servie value) and servie speed forms the rux of our model. The extant aademi researh has not addressed the interation between servie value and servie speed, or its onsequenes. In general, the extant literature treats servie value and servie times as independent performane metris, despite the fat that their interation is ritial for ustomer-intensive servies. In our queueing model, ustomer intensity is indexed expliitly by the parameter. The greater the ustomer intensity of the servie, the higher the value 3 I would still be on the treadmill, seeing 30 patients a day and always rushing. Patients were dissatisfied I was dissatisfied (Kaminetsky 2004, p. 15). of. (The speial ase of = 0 orresponds to the traditional queueing model, in whih the servie value is independent of the servie speed.) We find that modeling ustomer intensity leads to outomes very different from those of traditional queueing models. To give a flavor of these differenes, we mention two suh insights: (i) We find that the servie provider slows down (i.e., inreases its servie time) as the ustomer intensity of the servie inreases. Thus, the equilibrium value of the servie provided to ustomers is always inreasing in ustomer intensity. As a onsequene, suh servies are likely to have partial market overage. (ii) We find that ompetition in servie rates does not dampen pries in fat, the prie harged by the servie provider inreases as the number of ompeting servers inreases. Furthermore, the equilibrium waiting osts are invariant with respet to the number of ompeting servers (even as the prie inreases) Related Literature The existing researh in servie operations treats quality and speed as independent performane metris. To our knowledge, there is no preedent in the queueing literature that models the ustomer intensity of a servie or studies the interations between servie quality and servie speed arising from ustomer intensity. A number of papers address the deision making of ustomers who hoose whether or not to join a queue based on rational self-interest, as in our model. Our paper differs from all of the extant literature in that we expliitly model the dependene of servie quality on servie duration and explore the resulting equilibrium behavior of ustomers as well as the servie provider s servie rate and priing deisions. Admission fees have long been onsidered an important tool to ontrol ongestion in servie queues, dating bak to the seminal paper by Naor (1969). Edelson and Hildebrand (1975) extend Naor s (1969) model by analyzing unobservable servie queues. Following Mendelson and Whang (1990), papers that explore equilibrium queue joining, priing and/or servie rate deisions inlude Afehe (2006), Armony and Haviv (2000), Cahon and Harker (2002), Chen and Frank (2004), Chen and Wan (2003), Gilbert and Weng (1998), Kalai et al. (1992), Lederer and Li (1997), Li (1992), and Li and Lee (1994). We refer the reader to Hassin and Haviv s (2003) exellent review of this literature. Other notable papers that explore the interation between servie quality and ongestion inlude Allon and Federgruen (2007), Chase and Tansik (1983), Gans (2002), Hopp et al. (2007), Lovejoy and Sethuraman (2000), Oliva and Sterman (2001), Png and Reitman (1994), Ren and Wang (2008), Veeraraghavan and Debo (2009), and Wang et al. (2010). Researh artiles that aknowledge the existene of interations between servie duration and quality in

3 42 Management Siene 57(1), pp , 2011 INFORMS different domains inlude Kostami and Rajagopalan (2009) (dynami deisions), de Veriourt and Zhou (2005) (routing unresolved all-baks), Lu et al. (2008) (manufaturing rework), Hasija et al. (2009) (an empirial study of all enters), de Veriourt and Sun (2009) (judgement auray), and Wang et al. (2010) (medial diagnosti servies). In these papers, the ustomer demand is assumed to be exogenous and/or priing deisions are absent. 2. A Model of Customer-Intensive Servie Provision We onsider a monopolist providing a ustomerintensive servie to a market of homogeneous, rational onsumers. We model the monopolist servie setting using an unobservable M/M/1 queueing regime. 4 We use the M/M/1 model in the interests of expositional simpliity; however, we an show that all our analytial results extend to general servie distributions. Customers arrive at the market aording to a Poisson proess at an exogenous mean rate. We shall refer to as the potential demand for the servie. We assume that ustomers are homogeneous in their valuations of the servie, and inur a waiting ost of per unit of time spent in the system. Upon arrival, every ustomer deides whether to proure the servie (join the queue) or quit (balk from the servie) based on the value of the servie, the expeted waiting ost and the prie. The servie rate of the servie provider is assumed to be ommon knowledge. The effetive demand for the servie (i.e., the effetive arrival rate),, is the aggregate outome of all ustomers deisions (joining or balking). For any ustomer, the expeted waiting time in an M/M/1 system is as follows: 5 1 W = if 0 < otherwise. Before we formalize our model of ustomerintensive servies, we disuss the lassial queueing model, whih will serve as a useful benhmark The Classial Queueing Model The lassial queueing model (e.g., Naor 1969, Edelson and Hildebrand 1975) assumes that ustomers reeive 4 The M/M/1 queueing approximation has of ourse been applied to a large variety of settings too numerous to be listed here. See Green and Savin (2008) for an appliation to primary health are and Brahimi and Worthington (1991) on outpatient appointment systems. 5 For an M/G/1 system, the mean waiting times an be alulated by the Pollazek Khinhin formula (Ross 2006). (1) a servie value V b that is independent of the servie rate b (or, equivalently, of the servie time b =1/ b ). (This model will serve as a useful benhmark for our analysis of ustomer-intensive servie queues and is indexed throughout this paper by the subsript b.) In the lassial queueing model, inreasing the servie rate (i.e., reduing the servie time spent with eah ustomer) always results in higher revenues beause it allows the firm to serve more ustomers and/or lower their expeted waiting time. In this paper, we depart from the lassial assumption that the servie value remains unaffeted by hanges in the servie rate Modeling Value in Customer-Intensive Servies In ustomer-intensive servies, the quality of the servie provided to a ustomer (and hene, servie value) inreases with the time spent in serving the ustomer. In our model, servie quality is refleted in the servie value funtion V, whih inreases with the mean servie time. Furthermore, in most situations, the marginal value to ustomers from an inrease in servie time are diminishing. Therefore, we model ustomer-intensive servies by onstruting the servie value funtion V as a nondereasing and onave funtion of the mean servie time. 6 Speifially, we let V = V b + / b / + or, simply expressed in servie rates, V = V b + b + (2) where x + =max x 0. 7 The parameter 0 aptures the ustomer intensity of the servie provided. It determines the sensitivity of the servie value to the servie speed and is a desriptor of the nature of the servie. Clearly, higher values of suggest a stronger dependene of the servie value on the servie time (highly ustomer-intensive tasks). When is zero, the value of the servie provided equals V b ; this ase is equivalent to the lassial queueing model. Thus, as disussed previously, V b serves as a benhmark servie value. Seond, for all, when the servie rate is b =1/ b, the value of the servie provided is V b. Therefore, b b ould be onsidered a benhmark servie rate (time), providing a servie value V b to ustomers. 6 Customer intensity depends only on the relationship between the servie time and the servie value for a ustomer. Thus, a highly ustomer-intensive servie need not be a high-ontat servie (Lovelok 2001). 7 We an generalize V to be onvex and dereasing in. Similarly, we an generalize V to be an inreasing and onave funtion of. Although this leads to more analytial omplexity in the model, our onlusions remain idential.

4 Management Siene 57(1), pp , 2011 INFORMS Customers Queue Joining Deision Rational ustomers arrive to the system aording to a Poisson proess at rate and deide whether to join the (unobservable) servie queue. The potential demand (market size),, prie, p, servie rate,, waiting ost per unit time, and the resulting servie value, V, are ommon knowledge to all arriving ustomers. We model the queue-joining deisions of ustomers as in Hassin and Haviv (2003) and fous on symmetri equilibrium queue-joining strategies beause all ustomers are homogeneous. Let e p denote the equilibrium probability that a ustomer would join the queue at a server whose servie rate is and admission prie is p. 8 Thus, the equilibrium deision of ustomers e p is based on the value of the servie, the prie, and the expeted ost of waiting. Three market outomes full, zero, or partial market overage are possible, depending on the market size and other parameters. These outomes are as follows: (1) For full overage, if the net utility is nonnegative for a ustomer even when all the other potential ustomers join (i.e., V p+ W 0), then every ustomer will join the queue in equilibrium (i.e., e p =1). (2) For no overage, if the net utility is not positive for a ustomer joining the queue even when no other ustomer joins the queue, (i.e., V p+/ 0), then no one joins the queue (i.e., e p =0). (3) For partial overage, when p+/ <V <p+w, eah ustomer plays a mixed strategy in equilibrium, meaning that eah ustomer joins the queue with the same probability e p 0 1 and balks with probability 1 e p 0 1. Therefore, the equilibrium arrival rate is e p = e p and satisfies the ondition V p =W e p Charaterization of the Servie Rate Deision Spae Clearly, the interation between the servie speed and the servie value imposes a onstraint on the servie provider s feasible operating region (i.e., the range of servie rates and pries he an hoose from while still drawing ustomers). In this setion, we haraterize the feasible range of servie rates for the servie provider, whih is maximized at p =0. We do this for two reasons: (i) This haraterization will be useful when we formulate and solve the servie provider s revenue-maximization problem, with the prie determined endogenously, in the next setion. (ii) The haraterization of the feasible spae itself illustrates the impat of ustomer intensity on business and ustomer outomes. 8 We indiate the equilibrium values of the various model variables by the subsript e. Figure 1 v( ) Net Servie Value (V / ) and Operating Region F Shown for =0 (Dotted Curve), =1 (Dashed Curve), and =3 (Thik Curve) = 3 = 1 0 A 1 (3) A 1 (0) 5 A 2 (3) 10 A 2 (1) A 1 (1) = 0 Notes. However, for >0 the servie rates that provide nonnegative net value are bounded in the interval A 1 A 2. This implies that for a ustomer-intensive servie of type, the ustomer experienes a derease in net value if the servie time exeeds a threshold. A servie should be at least valuable enough that a ustomer does not mind waiting during the proess of servie provision. Therefore, the value V must exeed /, the expeted waiting osts during the servie; i.e., V / 0. This ondition ensures that a ustomer an expet nonnegative net value from the servie (at p =0), at least when no other ustomer preedes him in the queue. Note that a ustomer s servie prourement imposes negative externalities on others, as the expeted waiting ost, /, inreases with the effetive demand,. Rewriting V / 0, we have V b + b / or, equivalently, A 1 A 2, where A 1 A 2 are the solutions for to the quadrati V b + b =/. Thus, A 1 = V b + b V b + b 2 4 / 2. The servie has to be fast enough. No one will wait forever even if the servie value is high. (Note that A 1 0 =lim 0 A 1 =/V b.) A 2 = V b + b + V b + b 2 4 / 2. The servie annot be too fast. It is not possible to provide valuable servie at very high servie speeds. This additional onstraint is unique to ustomer-intensive servies. (Observe that A 2 0 =lim 0 A 2 =.) For a ustomer-intensive servie of type, we denote this operating servie-rate region by F = A 1 A 2. 9 Figure 1 shows the operating region and the assoiated net servie value for any servie rate in the operating region, for various. Figure 1 9 As long as V b / b, F is nonempty, 0.

5 44 Management Siene 57(1), pp , 2011 INFORMS shows that the servie provider an hoose from a larger range of servie rates when the servie is not very ustomer intensive (i.e., when is small). When =0, the net servie value is inreasing in the servie rate in the entire range F 0 = A 1 0. When >0, the net servie value is unimodal in the region F, and thus our results are appliable to servies in whih ustomers net value dereases after a servie time threshold. 3. Servie Provider s Revenue Maximization The servie provider s objetive is to maximize his revenues with respet to the servie rate,, and the prie, p. The servie provider s revenue funtion is given by R p =p e p where e p is the equilibrium demand indued at the setting p Therefore, the objetive funtion of the servie provider is given by { } max R p max max p e p (3) p 0 F F 0 p V Thus, we solve the servie provider s revenue maximization problem in two steps. First, we find the optimal prie p for a given servie rate,. Then, using p, we find the revenue-maximizing servie rate in the operating region F. Reall that for any F, the net servie value derived by a ustomer is negative, and hene no ustomer will join the servie. Conversely, for eah F, there exists a nonnegative prie at whih the servie provider an attrat ustomers. Hene, we fous on F. Also reall that W =1/ (by Equation (1)); hene, the value derived by any ustomer at the arrival rate is V p / Customers will join the servie until this value is driven to zero. The equilibrium demand, e p, as a funtion of the prie is given as follows: if 0 p V e p = V p if V <p V 0 if V <p Using (4), it is easy to verify that for a given the equilibrium demand, e p, is a noninreasing funtion of the prie. The following proposition derives the servie provider s optimal priing poliy for a given servie rate. (The proofs of all results are provided in the appendix.) (4) Proposition 1. Consider a ustomer-intensive servie of type. For any servie rate F, the optimal prie equals V p = V V / if 0 ˆ if ˆ < where ˆ = /V. The resulting equilibrium arrival rate is equal to e p = ˆ if 0 ˆ if ˆ < The orresponding equilibrium revenues are R p = p e p. Proposition 1 derives the optimal prie and the equilibrium demand (arrival rate) for any arbitrary servie rate. We find a threshold ˆ that defines the maximum number of ustomers the servie provider would serve at a given servie speed. When ˆ, the servie provider lears the market. However, when the potential demand is higher (i.e., for all >ˆ ), the servie provider serves exatly ˆ ustomers and denies the rest by making adjustments to the admission prie p. In eah ase, the servie provider extrats all the onsumer surplus. This result is driven by the negative externality that eah ustomer imposes on all other ustomers in the form of an inrease in their waiting osts. Thus, to aommodate an additional ustomer, the servie provider has to ompensate all of its urrent ustomers for the additional waiting osts they inur by dereasing the prie. As the arrivals to the system inrease, serving every additional ustomer requires an additional redution in prie, whih eventually leads to the senario (at = ˆ ) in whih the inrease in demand does not make up for the revenues lost due to the orresponding prie redution. Hene, for large values of, the servie provider limits the number of ustomers admitted to the system to ˆ by harging a suitable admission prie. Therefore, as long as remains higher than the threshold ˆ, small flutuations in potential demand do not affet the optimal prie and, hene, revenues. Proposition 1 showed that for any F there exists a prie p that maximizes the servie provider s revenues. Having derived the optimal prie for eah servie rate, we now analyze the servie provider s optimal servie rate deision. In the next setion, we analyze the ase of partial market overage ( >ˆ ). We analyze the ase of full market overage ( ˆ ) in 3.2.

6 Management Siene 57(1), pp , 2011 INFORMS Partial Market Coverage In this setion, we assume that the potential demand is high enough that the servie provider s optimal prie and servie rate deisions are not onstrained by the availability of potential ustomers. (We an show that this ondition translates, mathematially, to > V b + b 2 / 2.) To derive the optimal servie rate under partial overage, we first establish that the equilibrium demand and prie urves (as a funtion of the servie rate) are unimodal (details are given in the appendix). The unimodality properties of both demand and prie are outomes of the tension between servie value and waiting osts as follows: Fousing exlusively on delivering a high-value servie requires setting a slow servie rate. This leads to high ustomer waiting osts and low demand. High waiting osts also translate to a low prie, beause the maximum prie the servie provider an harge is the value of the servie net of waiting osts. On the other hand, inreasing the servie rate to minimize waiting osts leads to a low servie value (and hene, low demand and a low prie). Thus, both demand and prie are maximized at some intermediate servie rates in F. Furthermore, beause the servie provider s revenues are a produt of the equilibrium demand and the prie, the revenue-maximizing servie rate is an interior point in F Thus, we see that even in markets where the potential demand is very large (i.e., ), inreasing the servie speed does not lead to an inrease in effetive demand for ustomer-intensive servies beause of the drop in servie quality. Thus, partial market overage is a byprodut of the ustomer intensity of servies. Building on these observations, Proposition 2 provides the equilibrium outomes from the maximization of (3), the servie provider s objetive funtion. Proposition 2. For a ustomer-intensive servie of type >0, and when >, (i) the optimal servie rate is equal to = V b + b / 2 ; (ii) the orresponding optimal prie is equal to p = V b + b 2 /2; (iii) the demand at the optimal prie and servie rate equals e p = V b + b 2 / 2 =. Therefore, the optimal revenue for the servie is equal to R p = V b + b 2 2 / 4. Proposition 2 shows that there exists a unique, interior servie rate in F that maximizes revenues. Proposition 2(i) shows that the optimal servie rate,, is dereasing in : as the servie beomes more ustomer intensive, the servie provider has a greater inentive to slow down and spend more time on eah ustomer. We also see this in the expression for the equilibrium servie value. From Equation (2), the servie value provided to ustomers in equilibrium is V = V b + b /2, whih is inreasing in From Proposition 2(ii), we note that the optimal prie, p, is unimodal in dereasing for < / 2 b and inreasing for >/ 2 b. We saw that as the servie beomes more ustomer intensive, the optimal servie time inreases. However, this does not imply that the net value of the servie provided also inreases with ustomer intensity. This is demonstrated by Proposition 2(ii), beause the optimal prie traks the net value of the servie. For low (i.e., </ 2 b ), ongestion effets dominate the inrease in servie value as inreases. Hene, as the task beomes more ustomer intensive (i.e., inreases), the prie falls. However, for high values ( >/ 2 b ), the optimal prie is inreasing in : The inreased servie value from a longer servie time dominates any inrease in the equilibrium waiting ost. The equilibrium demand e p is also determined by the trade-off between waiting osts and the servie value, and behaves similarly to the optimal prie. At low values of, waiting osts are more sensitive to small inreases in than is the servie value. Hene, ongestion onsiderations dominate in this range. For higher values of, the reverse is true the servie value is more sensitive to inreases in than the waiting ost. The net effet is that the equilibrium demand is unimodal dereasing in for <V 2 b / and inreasing in for >V 2 b / (Proposition 2(iii)). Finally, Proposition 2 aptures the effet of the delay parameter on servie outomes. Interestingly, the optimal servie rate,, is independent of the waiting ost, ; i.e., if ustomers are more impatient, the additional waiting ost does not result in a faster servie. As one might expet, higher waiting osts lead to both lower pries, p, and lower equilibrium demand, e p. Consequently, the optimal revenues, R p, derease with inreased waiting osts Analysis of Value Prie Demand Interations. We shed further light on the subtle interations among prie, demand, and servie value as the servie rate hanges in ustomer-intensive servies. The equilibrium prie, equilibrium demand, waiting osts, and the servie value to ustomers are outomes of these omplex interations. Lemma 1 studies the relationship between the equilibrium prie and the equilibrium demand at any servie rate. Lemma 1 (Property of -Symmetry). For a ustomer-intensive servie of type, p and e p have the following symmetri relationship around the optimal servie rate for any given F : p + = e p where =.

7 46 Management Siene 57(1), pp , 2011 INFORMS Figure 2 Illustration of the Symmetry of p (Thik Line) and e p (Dotted Curve) Around for Customer-Intensive Servies of Types =2 and =0 5 (for V b =10 and b =5) (a) ( = 2) (b) ( = 0.5) Λ Region 1 Region 2 Region 3 Λ Region 1 Region 2 Region 3 Demand Prie * Prie * Demand A 1 * A 2 A 1 * Notes. The optimal servie rate and the orresponding equilibrium demand are = e p. The maximum throughput, indued by the servie rate, is. A 2 Lemma 1 learly demonstrates that pries and effetive demand are two levers related to eah other by the ustomer-intensity parameter. To better understand the impliations of -symmetry between prie and demand, we divide the operating region into three subregions as shown in Figure 2. Region 1 orresponds to low servie rates, Region 2 orresponds to intermediate servie rates, and Region 3 orresponds to high servie rates. When the servie rate is low (Region 1), there is an overinvestment in time of servie, from both the ustomers and the servie provider s perspetives. Although the servie provided is of high value, the ost of waiting is also high. In other words, inreasing the servie rate would improve eah ustomer s servie value (net of waiting osts), as well as the servie provider s total revenues. In Region 1, inreasing the servie rate will lead to some loss of servie value; however, the gains from the waiting ost redution dominate the loss in servie value. (At low values of, waiting osts drop preipitously as inreases.) Hene, the net servie value provided to a ustomer is inreasing in the servie rate. This allows the servie provider to harge ustomers a higher prie. Furthermore, this servie rate inrease leads to higher throughput. By inreasing the servie rate, the servie provider therefore has the opportunity to simultaneously inrease the prie and the number of ustomers served, thus inreasing his revenues. For intermediate servie rates (Region 2), inreasing the servie rate no longer inreases the net value of the servie for the ustomer, beause the redution in the servie value is greater than the redution in waiting osts. Therefore, at any given prie in this region, inreasing the servie rate leads to lower equilibrium demand (and onsequently, lower revenues). However, by simultaneously inreasing the servie rate and lowering the prie, the servie provider an inrease the demand. As the servie rate is inreased in Region 2, the net effet of lower prie and higher demand is to inrease revenues up to the point (see Figure 2). Beyond this point, revenues start falling. When the servie rate provided is in Region 3, dereasing the servie rate is desirable, beause it leads to a higher prie and higher equilibrium demand. In this region, the gain in servie value from dereasing the servie rate is greater than the losses arued from the inrease in ustomer waiting osts. Thus, as the servie rate is redued in Region 3, the equilibrium demand inreases in spite of the inrease in the equilibrium prie. To summarize, servie rates in both Region 1 and Region 3 are untenable in equilibrium. The optimal servie rate must lie in the intermediate servie rate region (i.e., Region 2). Figure 2 also illustrates the potential for servie systems with very different servie value propositions to earn idential revenues. A servie provider may hoose to provide high-quality servie at a high prie to a limited number of ustomers, or it may provide lower-quality servie at a lower prie to a large number of ustomers. Comparable revenues may be attained through either of these servie strategies. Modeling ustomer intensity through allows us to apture the presene of suh options in servie provision Full Market Coverage In 3.1, we analyzed the equilibrium in markets with partial overage. Proposition 3 below derives the

8 Management Siene 57(1), pp , 2011 INFORMS 47 equilibrium in markets in whih full overage is possible. We find that ustomer intensity plays a similar, important role in these markets thus, the insights for partial overage derived in 3.1 ontinue to hold under full overage. Proposition 3. For a ustomer-intensive servie of type >0, and when, (i) the optimal servie rate is equal to = + / ; (ii) the orresponding optimal prie is equal to p = V b + b 2 ; (iii) the equilibrium demand e p at the optimal prie and servie rate is. Proposition 3(i) shows that, just as in the ase of partial market overage, the optimal servie rate dereases in. The servie provider spends more time on eah ustomer as the servie beomes more ustomer intensive. As one would expet for full overage, e p = i.e., the servie provider serves all ustomers in equilibrium (Proposition 3(iii)). Thus, as the ustomer intensity inreases, the optimal servie rate falls, whereas the equilibrium arrival rate remains unhanged at. This leads to inreased waiting osts as inreases. In fat, the expeted waiting ost is. In equilibrium, ustomers wait longer (i.e., the ongestion inreases) as the servie beomes more ustomer intensive. From Proposition 3(ii), we see that the optimal prie is onvex in. We first fous on the ase of < b In this range, when </ b 2, the optimal prie is dereasing in. Yet we saw that when inreases, the servie provider inreases the servie time with every ustomer, whih would inrease the servie value provided (Proposition 3(i)). As inreases in this range of parameter values, the higher waiting ost (due to the inreased servie time) dominates the inrease in servie value, leading to a degradation in the net value of the servie provided to ustomers. To aommodate this loss, the servie provider has to ut the prie as inreases. Thus, if we ompare two servies of low ustomer intensity (with </ b 2 ), the more ustomer-intensive servie will be more ongested but less expensive than the other. However, when is high (>/ b 2 ), the gains in servie value are signifiant enough to offset the inrease in waiting osts as inreases. Therefore, both the optimal prie and the servie time (or, servie value) inrease in. Comparing two servies that are both highly ustomer intensive ( >/ b 2 ), the servie with higher is both more expensive and more ongested than the other. Finally, the ase when b is similar to the first ase above: The prie is dereasing in while the servie value is inreasing in for the entire range of In this ase, the greater the ustomer intensity, the more ongested but heaper the servie will be. 4. Model with Servie Rate Competition In this setion, we onsider the effet of multiple ompeting servers owned by a single servie provider (firm) that provides a servie of ustomer intensity. Although the servie provider sets an admission prie to maximize total revenues, the individual servers have the flexibility to set their own servie speed/quality (for example, onsider primary are physiians who belong to the same health network or the same hospital that, in turn, determines the admission prie for patient visits). For ease of exposition, we initially restrit our attention to a firm with two servers and then show how our results extend to multiple servers. The firm sets the admission prie p to maximize its total revenues. Eah server individually deides its servie rate to maximize its own revenues under the admission prie p set by the firm. Arriving ustomers deide whether to join the system, and if they join, whih server to go to, based on the servie value offered by the servers, waiting osts at the servers, and the prie. We fous on the Nash equilibrium of the system omprised of the firm, the servers, and ustomers making all these deisions. We model eah server as an M/M/1 queueing regime. The queue joining deision of a ustomer is given by j 1 2 p, for j =0 1 2, where 0 denotes the probability of balking, and 1 and 2 denote the probability of joining queue 1 and queue 2, respetively. Under pure strategies, i.e., i =1 for some i, either one server gains all the ustomers ( ), or none of the servers serves any ustomers. We prove that none of these outomes are possible in equilibrium. We thus fous on mixed strategies. Again, as in 3, we divide our analysis into two ases based on market overage. When the market is suffiiently large, i.e., 2 = V b + b 2 /, we show that, in equilibrium, both servers hoose their servie rates as if they were monopolies, and the firm hooses the single-server monopoly prie. When the market is small, i.e., when < V b + b 2 /, we show that the firm hooses a prie suh that all of the onsumer surplus is extrated, and the market is fully overed by the firm. Proposition 4. When the market is suffiiently large, given by the ondition 2 = V b + b 2 /, the servers at as monopolists. The optimal servie rate set by server i is given by i = V b + b / 2 for i =1 2. The firm s optimal prie is p = V b + b 2 /2. Proposition 4 simply states that in a large enough market, the prie harged by the servie provider remains unaffeted by ompetition within his network. All our insights on ustomer-intensive servies, derived for the single-server monopoly in 3, ontinue to hold.

9 48 Management Siene 57(1), pp , 2011 INFORMS When the market is smaller, i.e., when < V b + b 2 /, ompetition affets the servers and the firm s strategies. The servers ompete by adjusting their servie rates, whereas the firm adjusts its admission prie for the servie. We find that the net values (V i W i i, for i =1 2) provided by the servers are equal and positive in equilibrium. Server i s equilibrium demand is i = /2; thus, the entire market is overed by the two servers (i.e., = ). The servie provider extrats the entire onsumer surplus by harging an appropriate prie p. Proposition 5. When the potential demand for the servie is low, i.e., <2 = V b + b 2 /, the two servers share the market demand equally in equilibrium by setting their servie rates to e i = /2+ / for i =1 2 Proposition 5 shows that the equilibrium servie rate e i is less than the optimal servie rate under monopoly (reall Proposition 3(i)). Thus, under servie-rate ompetition, the firm provides a higher servie value at a slower rate through its servers than it would if there were only one server. Moreover, we find that ustomers expeted waiting osts in the multiserver ase (W e i /2 = ) areidential to those under monopoly (W = ). Therefore, the servie value net of waiting osts inreases under server ompetition for market share. Our strutural results ontinue to hold when there are n >2 servers ompeting on servie rates. We find that (i) in small markets ( <n =n V b + b 2 / 2 )), eah additional server indues every server to slow down further; (ii) otherwise, the market is large enough that eah server ats as a loal monopolist. Proposition 6. When there are n servers, full market overage is assured for <n =nv b + b 2 2 The servie provider harges an admission prie pn =V b + b /n 2, whih is stritly inreasing and onave in the number of servers, n. Proposition 6 shows that under full market overage, the firm s prie (and therefore, the total revenues) inreases with the number of ompeting servers. As a speial ase, we see that the equilibrium prie under ompeting servers is greater than the monopoly prie. The results of Proposition 6 are driven by the impat of ustomer intensity on servie rates. For ustomer intensive servies, the greater the ompetition, the slower the equilibrium servie rates hosen by the servers. Although the servers ompete among themselves for ustomers, they hoose to provide higher servie value over faster servie rates, whih in turn allows the firm to harge higher admission pries. In pratie, there may be investment osts to hire and maintain servers. In suh ases, the servie provider needs to alulate the optimal number of servers as a trade-off between the additional revenues earned by adding servers and the inremental investment osts. Suppose the ost of additional servers is inreasing and onvex. Beause the prie pn is inreasing and onave in the number of ompeting servers (as established in Proposition 6), the servie provider will inrease the number of servers until the marginal revenue from adding one more server is exeeded by the marginal ost. Beause pn is inreasing in, the optimal number of servers is, eteris paribus, inreasing in ustomer intensity. 5. Summary, Insights, and Future Diretions We have argued that the results from traditional queueing models are not appliable to ustomerintensive servies, wherein the servie quality is sensitive to the time spent with the ustomer. The trade-off between quality and speed is at the rux of the servie provider s problem, and his hoie of an intermediate servie rate in the fae of rational ustomers reflets this trade-off. Thus, our model provides fundamentally new insights into the nature of ustomer-intensive servies. In the disussion below, we fous on a ouple of these insights and examine the related empirial evidene in the ontext of primary are servies. However, the onlusions from our model are appliable aross a wide variety of industries Servie Speed and Market Coverage An impliation of servie degradation with speed is that servie-level (quality) targets are met only at slow servie times, neessitating a larger investment in apaity/servers. This inreases the osts of providing the servie. Thus, Baumol s ost disease (disussed in 1) is a onsequene of the ustomer intensity of the servie. What ould exaerbate this disease is our analytial result that as the servie gets more ustomer intensive, the servie provider slows down and inreases the time spent with eah ustomer. For a highly ustomer-intensive servie suh as primary are, the servie provider gains by fousing on servie quality and spending adequate time with eah patient, rather than by inreasing throughput by speeding up the servie. Paradoxially, this approah leads to greater revenues and servie value. Reent empirial researh findings in primary are servies onfirm our onlusions. Chen et al. (2009) and Mehani et al. (2001) examine primary are visit data in the United States between 1989 and 2005, and show that primary are visit durations have inreased

10 Management Siene 57(1), pp , 2011 INFORMS 49 (i.e., the average servie rate is slower) with an aompanied inrease in servie value. It is optimal for firms providing primary are servies to invest in highquality, slower servie, and therefore, partial market overage is likely to be observed. It has been doumented that an inreasing fration of the U.S. population resorts to emergeny room visits beause of the lak of adequate aess to primary are (Pitts et al. 2008). Slowdown and longer servies have also reated a new primary are model, termed onierge mediine. Conierge primary are praties announe and limit the number of patients they aept and offer them highly ustomized primary are, spending as muh time as needed with eah patient, with minimal delays. In return, onierge physiians harge higher fees that also have the effet of limiting the demand for the servie, thus reduing ongestion. For example, MDVIP, 10 founded in 2000, is a national network of 250+ physiians who provide preventive and personalized health are. Conierge dotors affiliated with MDVIP are for a maximum of 600 patients eah. MDVIP, MD 2 International, 11 Current Health, 12 and Qliane Primary Care 13 are some leading onierge primary are firms in the market. The longer duration of patient are in onierge pratie leads to the servie provider providing more valuable servie to a limited number of ustomers. Partial market overage is likely to be observed in other highly ustomer-intensive servies, suh as legal/finanial onsulting, eduational servies, and other health-are-related servies Priing Under Servie Rate Competition We find that as the number of ompeting servers inreases, every server slows down further. As a onsequene, server ompetition enhanes the servie value delivered in equilibrium, while holding the equilibrium ongestion (waiting) osts onstant, and exerts an upward pressure on the prie harged by the provider of the ustomer-intensive servie. These results, whih are in sharp ontrast to previous queueing researh, are driven by ustomer intensity. For ustomer-intensive servies suh as primary are, adding servie agents improves quality but may not redue ongestion. Servie rate ompetition among the agents leads to the desirable outome of higher quality but also to a higher admission prie. In primary are settings, there is strong empirial evidene of higher pries when the number of primary are physiians in a market inreases. This empirial finding in the seminal paper by Pauly and Satterthwaite (1981) has subsequently found support in several studies that onfirm prie inreases due to ompetition in primary are servie provision (see Gaynor and Haas-Wilson 1999 and referenes therein). The theoretial explanations offered for suh observations of prie inreases have been based on tait ollusion and informational ineffiienies. In ontrast to these explanations, our paper posits that suh inreases in prie an emerge naturally under servie ompetition, when the value of the servie inreases with the time spent in serving the ustomer Future Diretions Several diretions seem promising for future researh. One extension ould be to model and study the effets of different kinds of market heterogeneity. Competing servers ould vary in their ustomer intensities (e.g., based on their hoies of patient-are models or investments in training agents). Whether ustomer-intensity differentiation is a viable ompetitive strategy or not is an interesting researh question. A seond extension would be to model multiple servie providers that independently set their pries and servie rates, whih would require a model of full-prie ompetition. Simplifying suh a model in other ways (suh as eliminating ustomer hoie by assuming an exogenously speified joining rate) and/or employing other methodologies, suh as omputational approahes, might be required. Another interesting extension of this researh would be to model information asymmetry espeially in ustomer intensity. Presumably, there are oasions when ustomers do not know the exat ontent of the servie offered. Debo et al. (2008) model inentive effets in the ontext of suh redene servies; similar issues are pertinent to ustomer-intensive servies. Aknowledgments A previous version of this paper was a Finalist in the 2009 INFORMS Junior Faulty paper ompetition. The authors thank the anonymous reviewers, an assoiate editor, and the departmental editor for their valuable suggestions. Speial thanks go to Philipp Afèhe, Mor Armony, Barış Ata, John Birge, Gérard Cahon, Franis de Vériourt, Jak Hershey, Ananth Iyer, Ashish Jha, Hsiao-hui Lee, Raj Rajagopalan, Alan Sheller-Wolf, Robert Shumsky, Anita Tuker, Ludo van der Heyden, Eitan Zemel, and Yong-Pin Zhou for their thoughts and disussions during various stages of this paper. The authors also thank seminar partiipants at Carnegie Mellon University, Cornell University, Northwestern University, University of Chiago, University of Maryland, University of Pennsylvania, the Utah Winter Operations Conferene 2010, and University of Rohester s Workshop on Information Intensive Servies; the reviewers and partiipants of the 2009 Servie SIG Conferene at MIT; and the judges of the 2009 INFORMS JFIG Competition.

11 50 Management Siene 57(1), pp , 2011 INFORMS They aknowledge finanial support from the Fishman- Davidson Center at the Wharton Shool, University of Pennsylvania. Appendix Proof of Proposition 1. We begin by showing the optimal prie, p for >A 2. In this ase, the servie provider annot serve all potential ustomers even when the prie is equal to zero. The equilibrium arrival rate, e p, is determined by the following equation in this ase: V p =W e p (5) The revenue of the servie provider, R p, is given by ( ) R p =p (6) V p Reall that the servie value is the upper bound for the prie, i.e., V p. Therefore, the revenue funtion is onave in the prie, p, for the set of admissible pries (for 0 p V ), beause the seond-order ondition is negative: 0> 2 R p 2 = p 2 V p 2p 2 V p 3 The optimal prie, maximizing the servie provider s revenues for a given servie rate, is found using the firstorder ondition: 0= R p = p V p p V p 2 The unique solution of the first-order ondition within the set of admissible pries, p 0 V, isp =V V /. Plugging p into Equation (5) we find the resulting equilibrium arrival rate: e p = V The equilibrium arrival rate, e p, is independent of the potential demand,. This shows that the optimal prie given servie rate is equal to V V / for all /V. So far, we have derived the optimal prie p for all /V. To omplete the proof, we need to derive the optimal prie for < /V. Note that the servie provider an serve all potential ustomers at a nonnegative prie for /V. For a given servie rate, the equilibrium demand, e p, is dereasing in prie. Therefore, the maximum number of ustomers that an be served (maximum throughput) at rate,, is found by setting the prie equal to zero. Using the following equation we find : V = = V If is greater than the potential demand, then the servie provider an serve all potential ustomers, harging a prie greater than zero. For < /V, we have = /V /V >, beause V / for all F. For < /V, all arriving ustomers join the queue if the net utility of joining when all others join the queue at prie p is nonnegative, i.e., V p W 0. The net utility dereases in prie, and it is nonnegative for p V /. Thus, the servie provider s revenue as a funtion of prie an be written as p if 0 p V W ( ) p V p R p = if V W <p V 0 if p>v Differentiating the revenue funtion with respet to prie, we get if 0 p V R p V p p V p 2 = p if V <p V (8) 0 if p>v The revenue, R p is learly inreasing in prie for p V /. Inreasing the prie further at p =V / will derease the demand (throughput) but it may still inrease the revenues. Note that the revenue funtion for p>v / is equivalent to the revenue funtion given by Equation (6), whih is maximized at p =V V /. The revenues derease in prie at p =V / beause V / >V V / for < /V : V = /V > As a result, the optimal prie at servie rate, for < /V, isp =V /. The resulting equilibrium arrival rate is equal to e p =. Therefore, V W p = V V / Thus, we have derived p for all. if 0 ˆ if ˆ < Preparatory Results for Lemmas 1, 2, and Proposition 3. Before we prove the lemmas, we prove two main preparatory results. Result 1. Servie provider s revenue funtion R p is nondereasing in demand,. (7) (9)

12 Management Siene 57(1), pp , 2011 INFORMS 51 Proof. For a given servie rate and prie p, the servie provider s revenue as a funtion of the potential demand,, is given as follows: p if W V p ( ) p R p = V p (10) if W 0 <V p <W 0 if V p W 0 To show that the revenue funtion R p is ontinuous in, we only need to show that it is ontinuous at the transition point satisfying W =V p. Note that W is dereasing in for. Rewriting the transition point, W =V p, and solving for, we get / =V p = / V p, whih shows that R p is ontinuous in for 0. Clearly, R p is inreasing in for W V p and onstant in for W >V p. This proves that R p is nondereasing in for 0. Result 2. For =max F e p, the optimal prie, p, and the resulting equilibrium arrival rate, e p, have the following symmetri relationship around = V b + b / 2 for any F : p + = e p where = V b + b / 2. The desired result is obtained by plugging in = + and = into p and e p (derived in Proposition 1), respetively. Remark 1. As long as the typial servie value V b is greater than the expeted waiting ost during a typial servie, / b, i.e., V b >/ b, we have a nonempty operating region, F, for a ustomer-intensive servie of type. Having proven Results 1 and 2, we are now ready to prove the lemmas. Proof of Proposition 2. (i) For >A 2, the objetive funtion in Equation (3) is unimodal in the servie rate,. The revenue funtion R p = V b + b 2 V b + b + is ontinuous in for A 1 A 2. The revenue funtion is differentiable in for A 1 A 2 : The first derivative, dr p =V d b + b 2 V b + b 2 Vb + b exists and is ontinuous for A 1 A 2. The first derivative, dr p /d =0, rosses 0 at three points: A 1, = V b + b / 2, and A 2. Note that A 1 A 2 for >0. As a result, is either the unique maximizer or the unique minimizer of R p for A 1 A 2. We show that maximizes the revenue funtion: R A 1 p A 1 and R A 2 p A 2 are equal to zero beause both the optimal prie, p A i, and the resulting equilibrium arrival rate, e A i p A i, are learly equal to zero, beause the servie value, V, is equal to the waiting ost during the servie, /, at these points. Now, we show that R p is greater than zero. The optimal revenue funtion R p = V b + b 2 2 / 4 >0 for all >0, beause V b + b 2 >0 from Remark 1. This shows that R p is inreasing in for A 1 and dereasing in for A 2, whih proves that the optimal servie rate is = V b + b / 2. (ii) From Proposition 1, p = V b + b 2 /2. (iii) From Proposition 1, e p = V b + b 2 / 2 This proves the optimality of the above servie setting for >A 2. We now show that the above operating setting is optimal, even for all e p = V b + b 2 2. The revenue funtion R p is nondereasing in the potential demand,, for all p 0 from Result 1. Therefore, the optimal revenue R p is nondereasing in for all 0. The servie provider an ahieve R V b + b / 2 V b + b 2 /2 by serving V b + b 2 / 2 ustomers at rate = V b + b / 2, harging prie p = V b + b 2 /2 for all V b + b 2 / 2. In other words, the optimal revenue for >A 2 an be ahieved by the idential operating setting (prie and servie rate) when the potential demand is lower than A 2 ( V b + b 2 / 2 A 2 ). The optimal revenues are nondereasing in the potential demand,, and therefore the above setting is optimal for all = V b + b 2 / 2 beause it is optimal for >A 2 and A 2 > V b + b 2 / 2. Proof of Lemma 1 ( -Symmetry). Proposition 2 indiates that defined in Result 2 is equal to the optimal servie rate,, for >. The result of the Lemma immediately follows from Result 2, plugging in for. Lemma 2. For any >0, the optimal prie for a given servie rate, p, and the resulting equilibrium demand, e p, are unimodal in the servie rate,. Proof of Lemma 2. For >, p and e p are unimodal (inreasing and then dereasing) in the servie rate,. We will prove that p is unimodal in. Unimodality of e p follows from the -symmetry property. For >, the optimal prie for servie rate F is p =V V /. The optimal prie p is equal to zero for =A i, where i =1 2. We pik an interior point = V b + b / 2 in the operating region, F. Clearly, A 1 A 2 for >0. The optimal prie for, p is nonnegative as long as the ondition in Remark 1 holds. For 0, p = V b + b 2 >0 V 2 b + b 2 4 >0 If V b >/ b, then V b + b 2 4 > V b b 2 >0, beause <V b b from Remark 1. We will prove the unimodality of p by showing that the first derivative dp /d rosses 0 only one in A 1 A 2, and this point is the maximizer of the prie,

13 52 Management Siene 57(1), pp , 2011 INFORMS p, with respet to for A 1 A 2. Hene, the first-order ondition is satisfied at a unique, interior point: dp d = + V b + b 2 2 V / =0 The first derivative is ontinuous for A 1 A 2. Reorganizing the terms, we an write the first-order ondition as 2 2 V / = V b + b For notational onveniene, let V b + b =K. Therefore, V =K. Plugging in K and squaring both sides of the equation, we get K / = 2 K 2, whene K = 3 K (11) Note that the left-hand side (LHS) of Equation (11) is onstant with respet to. We show that the right-hand side (RHS) rosses this onstant only one for A 1 A 2. The right-hand side, 3 K, is unimodal in the servie rate d 3 K /d = 2 3K 4 Clearly, the right-hand side is inreasing in for < 3K/ 4 and dereasing for >3K/ 4. We now show that the RHS term in Equation (11) is less than K 2 / 4 2 when =A 1 and greater than K 2 / 4 2 when =A 2, whih proves that the right-hand side rosses the left-hand side only one in the operating region, F, beause the right-hand side is unimodal in. If we plug in K for the value of A i, weget A i = K K 2 4 for i =1 2 2 Let RHS = 3 K. Let LHS=K 2 / 4 2. Then we have RHS A 1 = K K LHS and RHS A 2 = K+ K LHS 4 2 whih yield the desired result. Note that the weak inequalities in the above equations are strit if >0 and they are equalities if = 0 beause lim 0 RHS A 1 = lim 0 RHS A 2 =LHS. The point satisfying the first-order ondition in the operating region is a maximizer beause p A 1 =p A 2 =0. Furthermore, the prie is positive in the operating region, F, beause the optimal prie is positive at an interior point, i.e., p >0, and the first derivative of p rosses zero only one. Thus, we have shown that p is unimodal (inreasing and then dereasing) in the servie rate. Result 3. For a ustomer-intensive servie of type >0, the revenue-maximizing, equilibrium demand e p is stritly lower than =max F e p. The servie rate leading to is greater than the optimal servie rate, i.e., =argmax F e p >. The equilibrium demand, e p, is still inreasing in the servie rate, and the optimal prie, p, is dereasing in the servie rate, at the optimal servie rate,. Proof. The revenue-maximizing equilibrium demand, e p, is smaller than the maximum equilibrium demand, =max F e p. By definition we have > e p. We know that e p is unimodal in from Lemma 2. We first show that e p is inreasing in the servie rate at =. Differentiating the equilibrium demand e p with respet to, weget d e p K =1 d K K Evaluating the first derivative at,weget d e p d =1 2 /K = To prove the result, we need to show that 1 2 / K 0. Reorganizing the equation, we get K 2 4, whih holds for all K 0, whih follows from Remark 1: Remark 1 suggests that V b >/ b. Reall that K =V b + b. Therefore, the ondition in Remark 1 is equivalent to K b K V b >. Both b and V b are nonnegative. Therefore, we an rewrite the ondition of Remark 1 as follows: K V b V b > for V b 0 K (12) Inequality (12) holds for all V b 0 K, and therefore it holds for the value of V b that maximizes K V b V b in 0 K. The expression, K V b V b, is maximized when V b =K/2. Then, max K V b V b =K 2 /4 K2 0 V b K 4 > Therefore, Remark 1 implies the result K 2 4, whih in turn implies that d e p d 0 = Thus, we have shown that e p is inreasing in the servie rate at =. Therefore, the throughputmaximizing servie rate is. We begin haraterizing the optimal prie that lears the market demand, with the following lemma, whih in turn helps us build onditions for the proof of Proposition 3. Lemma 3. For any ustomer-intensive servie of type, when the potential demand is suh that <, there exist 1 and 2 in F suh that the optimal prie p lears the market for all 1 2. Proof of Lemma 3. When the potential demand is less than, there exist 1 and 2 in F suh that e p for 1 2. The result of the Lemma follows from the fat that e A i p A i =0 for i =1 2 and from the unimodality of e p (Lemma 2). Lemma 3 shows that for low values of potential demand, there exists a losed interval of servie rates 1 2 F, in whih it is always optimal to lear the market. A orresponding prie p an be hosen so that the market

14 Management Siene 57(1), pp , 2011 INFORMS 53 demand is leared for any servie rate in this interval. Note that = e 1 p 1 = e 2 p 2. If the potential demand,, is higher than, then the servie provider annot lear the market at any servie rate in the operating region F. Using the results of Lemma 3 and Proposition 1, we an write the resulting equilibrium arrival rate as e p ˆ = = V ˆ = V if A 1 < 1 if 1 2 if 2 A 2 (13) A representative example of the equilibrium demand leading to full market overage an be seen in the right panel of Figure 2. Note that the servie provider overs the entire demand in the interval 1 2. Again applying Proposition 1, we an rewrite the optimal prie, p, for a given servie rate as follows: V V / if A 1 < 1 p = V W if 1 2 (14) V V / if 2 A 2 Therefore, the equilibrium revenue equals R p = p e p. We an now derive the servie provider s optimal servie rate and prie using the above revenue funtion. Proof of Proposition 3. We will show that when the demand is low ( < V b + b 2 / 2 ), the servie provider s optimal servie rate is = + /, the prie is p =V b + b 2, and the optimal equilibrium demand is equal to (i.e., the market overage is full). (i) For the small market senario, the servie provider s objetive funtion is given by K 2 K + if A 1 < 1 ( V ) R p = if 1 2 K 2 K + if 2 <A 2 (15) The objetive funtion is ontinuous in as = e p and V W =p at 1 and 2, whih implies that the revenues are equal at the transition points between regions. Reall that Lemma 3 shows that there exists 1 and 2 suh that all potential ustomers join the queue at the optimal prie, p, for all 1 2 when <. < V b + b 2 / 2 and > V b + b 2 / 2 from Result 3. Therefore, A 1 < 1 < 2 <A 2 in the small market senario. Let Region A be A 1 < 1, let Region B be 1 < 2, and let Region C be 2 < <A 2. We will show that the optimal servie rate is in Region B for < V b + b 2 / 2. Note that in Region A and Region C the objetive funtion is equivalent to that of the large market senario (R p = K 2 K +), whih is maximized at =K/ 2 (Proposition 2). The servie rate providing full overage 2 = max e p = is greater than K/ 2 beause e p is unimodal by Lemma 2, and = argmax F e p >K/ 2 by Result 3. Therefore, the objetive funtion in Equation (15) is dereasing in in Region C. We show that the objetive funtion is inreasing in in Region A by showing that 1 <K/ 2. By definition, e 1 p 1 =. The equilibrium demand e p is unimodal in from Lemma 2, and argmax F e p >K/ 2 from Result 3. These fats imply that for < V b + b 2 ( ( )) K K = 2 e 2 p 2 1 < K 2 As a result, the servie rate that maximizes the objetive funtion is in Region B. Differentiating the objetive funtion, R p, with respet to the servie rate, we get [ ] K 2 1 K if A 1 < < 1 ( ) dr p = 2 d if 1 2 [ ] K 2 1 K if 2 <A 2 The first-order ondition is given by ( ) 0= dr p = d 2 (16) Reall that = e p = / K at = i for i =1 2. Therefore, we an write 1 = + 1 / K 1. Plugging this value into dr p /d, weget dr p d [ K 2 1 = = 1 1 The value of dr p /d is positive for 1 K/ 2. The servie rate 1 is monotonially inreasing in for 0 beause e p is unimodal (inreasing and then dereasing) in (Lemma 2). Therefore, the objetive funtion is inreasing in at = 1, beause 1 K/ 2. This proves that the optimal servie rate, ]

15 54 Management Siene 57(1), pp , 2011 INFORMS, is greater than 1 for low potential demand ( < K / 2 ). Similarly, we show that R p is dereasing in at = 2 by plugging in + 2 / K 2 for 2 : dr p d [ K 2 2 = = 2 2 ] <0 The above value is negative beause 2 >K/ 2. Therefore, an interior point of 1 2 satisfies the first-order ondition for <K/ 2 /. The unique solution of the first-order ondition for 1 2 is = + /, whih proves the result of Proposition 3. (ii) The optimal prie, p, is equal to V b + b 2, from Proposition 1. (iii) The resulting equilibrium demand, e p, is equal to the potential demand by Proposition 1. Proof of Proposition 4. Proposition 4 shows that for V b + 2 /, ompeting servers an ahieve monopoly revenues by using the optimal monopoly operating setting. Note that the potential demand must be at least two times the optimal monopoly equilibrium demand, e p. We begin by showing that for any potential demand 0, a server serving at rate and harging prie p is better off in the single-server setting than in the multiserver setting. Reall that the revenue of the server in the single-server setting is given by R p. Let R p be the revenue of server 1, providing servie with rate 1, when server 2 is providing servie with rate 2 at prie p in the two-server ompetition ase. The revenue of a server serving at rate in the single-server setting is greater than R 1 2 p, the revenue it arues under the two-server ompetition with the same servie rate for 2 F and p>0. p p 1e 2 p R 1 2 p = if W >V p / if V p W max 0 V 2 p W if V p W 0 max 0 V 2 p W (17) where 1e 1 2 p is the equilibrium arrival rate for server 1. Let us examine the first line in Equation (17). In the multiserver ompetition setting, server 1 an serve all potential ustomers if the net value of an arriving ustomer from joining server 1 when all other ustomers join server 1 is nonnegative and greater than the net value of joining server 2 when no other ustomer joins server 2, i.e., V p W max 0 V 2 p W 2 0. Reall that in the single-server setting, nonnegativity of the net value (V p W 0) is suffiient to serve all potential ustomers. Clearly, server 1 is better off in a single-server setting than in a multi-server setting when V p W 0. Let us examine the ase in whih W >V p / (line 2 in Equation (17)). In the single-server setting, ustomers join the queue until the net utility from joining equals zero. However, in a multiserver setting, the net utility of joining server 2 may be positive when e p ustomers join server 1 and e p ustomers join server 2. Therefore, ustomers will deviate to server 2 until an equilibrium is reahed, i.e., until the net utility from joining server 1 equals the net utility from joining server 2. Hene, R p R 1 2 p for all 2 F, and p 0, whih implies max R p max R 1 2 p (18) F p 0 F p 0 for all 0. Therefore, in the multiserver setting when W > V p /, the equilibrium arrival rate for server 1, 1e 2 p, is less than or equal to the monopoly equilibrium arrival rate, e p = / V p. For V b + b 2 /, the optimal equilibrium demand in the single-server setting is given by e p = V b + b 2 / 2. In the two-server setting, the servie provider an serve 2 e p ustomers, harging p when both servers serve at rate i =, hene doubling the optimal monopoly revenues. In this ase, the equilibrium demand at eah server is equal to e p. Clearly, the net value of an arriving ustomer from joining either server is equal to zero; therefore, an arriving ustomer is indifferent among the options to join server 1, join server 2, and to balk from the servie. Hene this setting is an equilibrium for ustomers. When the servie provider harges prie p, is a Nash equilibrium for the servers, beause they ahieve the maximum revenue (shown in the LHS of the Equation (12)) by hoosing (i.e., is the best response of a server to any servie rate adopted by the other server). Proof of Proposition 5. Proposition 5 indiates that for < V b + 2 / there exists a symmetri Nash equilibrium e e suh that all potential ustomers proure the servie and agents equally share the potential demand. Furthermore, we have e = /2+ /. We prove the proposition by showing that none of the players (the servers and ustomers) have inentive to deviate from e. An arriving ustomer s net value from joining server i when half of the ustomers join server i is given by NV i i p /2 =V i p / i /2 Let e maximize NV i i p /2 : Differentiating NV i i p /2 with respet to i,weget NV i i p /2 = + i i /2 2 The seond-order ondition indiates that NV i i p /2 is onave in i for i /2: 2 NV i i p /2 2 = 2 i i /2 <0 3 The first-order ondition is satisfied at e = /2+ / ; hene, e maximizes NV i p /2. The resulting net value is given by NV e p /2 =V b + b /2 2 p

16 Management Siene 57(1), pp , 2011 INFORMS 55 Clearly the net value for an arriving ustomer from joining server i is nonnegative for p V b + b /2 2, when agents serve at rate e and ustomers mix equally between servers 1 and 2. It an be shown that, server i, serving /2 ustomers, has no inentive to deviate from e, beause doing so dereases the net value for an arriving ustomer joining server i, whih will result in lower equilibrium demand for server i. Given this, it follows that e e is a Nash equilibrium for servie agents for all p V b + b /2 2. The maximum prie that an be harged by the servie provider is p2 =V b + b /2 2. The servie provider optimizes the revenues by harging p2 and fully extrating the onsumer surplus. This result extends to n servers and holds for all <n V b + 2 / 2. Proof of Proposition 6. To prove the proposition, we first derive the equilibrium servie rate for n agents and the resulting optimal prie, when <n V b + b 2 / 2. 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