Quality-Speed Conundrum: Tradeoffs in Customer-Intensive Services

Transcription

1 Quality-Speed Conundrum: Tradeoffs in Customer-Intensive Servies Krishnan S Anand M Fazıl Paç Senthil K Veeraraghavan Working paper, OPIM Department, Wharton Shool, Philadelphia PA April 2010 Abstrat In many servies, the quality or value provided by the servie inreases with the time spent 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 In a queueing framework, we parameterize the degree of ustomer-intensity of the servie The servie speed hosen by the servieprovider affets the quality of the servie through its ustomer-intensity Customers queue for the servie based on the quality of the servie, delay osts and prie We study how a servie provider makes the optimal quality-speed tradeoff in the fae of suh rational ustomers 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 servers inreases, the prie inreases and the servers beome slower Keywords: Customer-Intensity, Servie Operations, Strategi Customers, Queues, Cost Disease 1 Introdution Festina Lente [Make haste slowly] 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 David Eles Shool of Business, University of Utah, Salt Lake City, UT kanand@utahedu The Wharton Shool, University of Pennsylvania, Philadelphia, PA mpa@whartonupennedu Corresponding author The Wharton Shool, University of Pennsylvania, Philadelphia, PA senthilv@whartonupennedu 1

2 of suh servies are health are, legal and finanial onsulting, and personal are (suh as spas, hair-dressing, 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 in the last few deades (Triplett and Bosworth 2004, Varian 2004) For example, in the last deade, the health are industry displayed a negative growth, at 04% (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) Surowieki (2003) illustrates this point: Cost disease isn t anyone s fault (That s why it s alled a disease) [ ] 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 In sum, primary health are pratie in the United States epitomizes the problem of Baumol s ost disease Due to 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 for patients (Yarnall et al 2003) The primary health are servie, whih provides the fous of this paper, learly demonstrates 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 aforementioned examples posit that fousing predominantly 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 due to ongestion effets 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 DE for suggesting this point 2 Customer-intensive servies are generally haraterized by high labor ontent, but high labor ontent (eg, low-teh manufaturing, onstrution) need not imply high ustomer-intensity 3 I was seeing 30 people a day and always rushing Patients were dissatisfied I was dissatisfied Bernard Kaminetsky, MD, FACP, (formerly NYU, urrently with MDVIP) in his testimony to the Joint Eonomi Committee of the United States Congress, April 28,

3 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 tradeoff in the fae of rational ustomers To summarize, ustomers in our model are strategi in that they 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, sine every ustomer who joins the servie imposes a negative externality (in the form of additional waiting time) on all other ustomers In turn, the tradeoff between the quality (servie value) and the servie speed faed by the provider of a ustomer-intensive servie 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 by a parameter α The greater the ustomer-intensity of the servie, the higher the value of α We fous on two key insights from our analytial model 1 We find that the servie provider slows down (ie, inreases its servie-time) as the ustomer-intensity of the servie inreases (ie as α 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 2 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 servers inreases Furthermore, the equilibrium waiting osts are invariant with respet to the number of servers Related Literature: The extant aademi 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 the extant literature by expliitly modeling the dependene of servie quality on servie duration, and exploring the resulting equilibrium behavior of ustomers, and the servie rate and priing deisions of the servie provider 3

4 Admission fees have long been onsidered an important tool in ontrolling 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 Papers that explore equilibrium queue joining, priing, and/or servie rate deisions when ustomers have to wait for servies follow the paper by Mendelson and Whang (1990) Reent papers 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 (2003) s 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), Lu et al (2008), Oliva and Sterman (2001), Png and Reitman (1994), Ren and Wang (2008), Veeraraghavan and Debo (2009) and Wang et al (2008) Researh artiles that aknowledge the existene of interation between servie duration and quality in different areas inlude Lu et al (2008) (reworks in manufaturing), Hasija et al (2009) (empirial data on all enters), de Veriourt and Sun (2009) (judgement auray), and Wang et al (2008) (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 ustomer-intensive servie to a market of homogenous, rational onsumers We model the monopolist servie setting using an unobservable M/M/1 queueing regime 4 We use the M/M/1 model for the simpliity of exposition However, we an show that all of our analytial results extend for general servie distributions Customers are rational, and arrive to the market aording to a Poisson proess at an exogenous mean rate Λ We shall refer to Λ as the potential demand for the servie We assume that ustomers are homogenous in the 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, expeted ost of waiting and the prie 4 Suh queueing approximations have been applied for primary health are settings See Green and Savin (2008) for primary are settings, and Brahimi and Worthington (1991) for outpatient appointment systems 4

5 The servie rate µ of the servie provider is assumed to be ommon knowledge The effetive demand (effetive arrival rate), λ, 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 W (µ, λ) = { } 1 (if 0 λ < µ), (otherwise) µ λ Before we set up our model of ustomer-intensive servies, we examine the value provided in lassial queueing models (1) 21 The Classial Queueing Model The extant queueing models (eg Naor 1969, Edelson and Hildebrand 1975) assume that ustomers reeive a servie value V b, that is independent of the servie rate µ b (equivalently, the servie time τ b = 1/µ b ) We will refer to this setting as the lassial queueing model (This will serve as the benhmark for our analysis of ustomer-intensive servie queues, and from this point forward, will be indiated by the subsript b) In the lassial queueing model, inreasing the servie rate, whih redues the servie time spent with eah ustomer, always results in higher revenues, as it allows the firm to serve more ustomers and/or lower their expeted waiting time Critially, the servie value remains unaffeted by hanges in the servie rate In this paper, we depart from the assumption that the servie value does not vary with the servie rate 22 Modeling Value in Customer-Intensive Servies In many ustomer-intensive servies, the value of the servie provided to a ustomer inreases with the servie time We model the quality of the servie through its assoiated servie value V (τ) whih inreases with the mean servie time τ Furthermore, in most situations, the marginal return to inreased servie time is diminishing We model ustomer-intensive servies by onstruting the servie value funtion V (τ) as a non-dereasing and onave funtion of servie time τ 6 Speifially, we let V (τ) = (V b +α/τ b α/τ) + or simply expressed 5 For an M/G/1 queue, the mean waiting times an be alulated by the Pollazek-Khinhin formula (Ross 2006) 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) 5

6 in servie rates as, 7 V (µ) = (V b + αµ b αµ) + (2) where x + = max(x, 0) The parameter α 0 determines the sensitivity of the servie value to servie speed, and is an assoiated desriptor of the nature of the servie We denote α as the ustomer-intensity of the servie provided Clearly, higher values of α suggest a stronger dependene of the servie value on the servie time (highly ustomer-intensive tasks) When α is zero, the value of the servie provided equals V b This ase is equivalent to the lassial queueing model Thus, as argued above, V b ould be thought of as a benhmark servie value Seondly, for any α, when the servie rate is µ b = 1/τ b, the value of the servie provided is V b Therefore, µ b (τ b ) ould be onsidered a benhmark servie rate (time), providing a servie value V b to ustomers 23 Customers Queue Joining Deision Rational ustomers arrive to the system aording to a Poisson proess at rate Λ, and deide whether to join the unobservable servie queue Potential demand, Λ, prie, p, servie rate, µ, and resulting servie value, V (µ), are ommon knowledge to all arriving ustomers We model the queue joining deisions of ustomers as in Hassin and Haviv (2003), and onsider symmetri equilibrium queue joining strategies sine all ustomers are homogenous γ e (µ, p) denote the probability that any ustomer would join a queue at a server whose servie rate is µ and admission prie is p Let Thus, the equilibrium deision of ustomers γ e (µ, p) is based on the value of the servie, the prie and the expeted ost of waiting 8 Three market outomes full, zero or partial market overage are possible, depending on the market size Λ and other parameters These outomes are: 1 Full overage: if the net utility is non-negative for a ustomer even when all of the other ustomers join (ie, V (µ) (p + W (µ, Λ)) 0), then every ustomer will join the queue in equilibrium (ie γ e (µ, p) = 1) 2 No overage: if the net utility is negative for a ustomer joining the queue even when no other ustomer joins the queue, (ie, V (µ) (p + /µ) < 0), then no one joins the queue (ie, γ e (µ, p) = 0) 3 Partial overage: when p+/µ < V (µ) < p+w (µ, Λ), eah ustomer plays a mixed strategy in equilibrium, meaning that eah ustomer joins the queue with the same probability γ e (µ, p) (0, 1) and balks with probability 1 γ e (µ, p) (0, 1) 7 We an generalize V (µ) to be onvex and dereasing in µ While this leads to more analytial omplexity in the model, the onlusions remain idential We defer the tehnial details to Appendix C 8 The subsript e denotes the equilibrium value 6

7 Therefore, the equilibrium arrival rate is λ e (µ, p) = γ e (µ, p)λ and satisfies the ondition V (µ) p = W (µ, λ e (µ, p)) 24 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 operating region (ie the spae of servie rates he an hoose from) Even if there were no queues, for a ustomer to expet non-negative net value (servie value minus waiting ost during the servie), the servie value, V (µ), must exeed the expeted ost of waiting A servie should be at least valuable enough that a ustomer should not mind waiting during the proess of servie provision Therefore, the value V (µ) must exeed the ost of waiting during the servie, ie, V (µ) /µ 0 This ondition ensures that a µ ustomer an expet non-negative net value from the servie when no other ustomer joins the queue Note that a ustomer s servie prourement imposes negative externalities on others, as the expeted waiting ost,, inreases with the effetive demand, λ µ λ Rewriting V (µ) /µ 0, we have V b + αµ b αµ /µ, whih in turn implies that A 1 (α) µ A 2 (α), where A 1 (α), A 2 (α) are solutions to V b + αµ b αµ = /µ, whih is a quadrati in µ Thus, µ A 1 (α) = V b+αµ b (V b +αµ b ) 2 4α 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 α 0 A 1 (α) = V b ) µ A 2 (α) = V b+αµ b + The servie annot be too fast It is not possible to (V b +αµ b ) 2 4α provide valuable servie at very high speeds of servie (Note that A 2 (0) = lim α 0 A 2 (α) = ) For a ustomer-intensive servie of type α, we denote this operating servie-rate region by F(α) = [A 1 (α), A 2 (α)] 9 The operating region and assoiated net servie value for any given servie rate in the operating region are shown in Figure 1 Figure 1 shows that the servie provider an hoose from a larger range of servie rates when the servie is not very ustomerintensive When α = 0, the net servie value inreases in servie rate, µ [A 1 (0), ) 9 As long as V b > /µ b, F(α) is non-empty 7

8 Figure 1: The net servie value (V (µ) /µ) and the operating region F(α) shown for α = 0 (dotted urve), α = 1 (dashed urve) and α = 3 (thik urve) However, for α > 0 the servie rates that provide non-negative net value are bounded in the interval [A 1 (α), A 2 (α)] 3 Servie Provider s Revenue Maximization The servie provider s objetive is to maximize the revenues with respet to the servie rate, µ and the prie, p Therefore, the objetive funtion of the servie provider is given by: max {p 0,µ F(α)} {R(µ, p) = pλ e (µ, p)} max {µ F(α)} { max{0 p V (µ)} {R(µ, p)} } (3) Thus, we solve the servie provider s revenue maximization problem in two steps First, we find the optimal prie p(µ) for a given servie rate, µ Then, using p(µ), we find the revenue maximizing servie rate in the operating region F(α) Reall that for any µ F(α), the net servie value derived by a ustomer is negative, and hene no ustomer will join the servie Conversely, for eah µ F(α), there exists a non-negative prie at whih the servie provider an attrat ustomers Hene we fous on µ F(α) The equilibrium demand, λ e (µ, p), as a funtion of the prie is given as follows: λ e (µ, p) = Λ if 0 p V (µ) W (µ, Λ) µ V (µ) p if V (µ) W (µ, Λ) < p V (µ) W (µ, 0) 0 if V (µ) W (µ, 0) < p Using the expression in Equation (1), it is easy to verify that the equilibrium demand, λ e (µ, p), is a non-inreasing funtion of the prie The following proposition presents the servie provider s optimal priing poliy for a given servie rate µ (The proofs of all results (4) 8

9 appear in Appendix A) Proposition 1 Consider a ustomer-intensive servie of type α For any servie rate µ F(α), the optimal prie equals: where ˆλ(µ) = µ p (µ) = { V (µ) W (µ, Λ) if 0 Λ ˆλ(µ) V (µ) V (µ)/µ if ˆλ(µ) < Λ µ The resulting equilibrium arrival rate is equal to: V (µ) (5) λ e (µ, p (µ)) = { Λ if 0 Λ ˆλ(µ) ˆλ(µ) if ˆλ(µ) < Λ (6) and the orresponding equilibrium revenue equals R(µ, p (µ)) = p (µ)λ e (µ, p (µ)) Proposition 1 shows the optimal prie and equilibrium demand (arrival rate) for any arbitrary servie rate µ We note that the threshold ˆλ(µ) defines the maximum number of ustomers the servie provider would serve at a given servie speed µ When Λ < ˆλ(µ), the servie provider lears the market and also extrats all onsumer surplus at prie p(µ) However, when the potential demand is higher (ie, when Λ ˆλ(µ)), the servie provider serves ˆλ(µ) ustomers To aommodate more ustomers, the servie provider has to ompensate ustomers for the additional waiting osts they inurred, by dereasing the prie As the arrivals to the system inrease, serving every additional ustomer requires a larger redution in prie, whih eventually leads to the senario (at λ = ˆλ(µ)) in whih the inrease in demand does not make up for the revenue lost due to the orresponding prie redution Hene, for large Λ, the servie provider limits the number of ustomers admitted to the system to ˆλ(µ), by harging a suitable admission prie Therefore, as long as Λ remains higher than the threshold ˆλ(µ), small flutuations in potential demand would not affet the optimal prie, and hene, revenues Proposition 1 shows that for any µ F(α), there exists a prie p (µ) that maximizes the servie provider s revenue Having derived the optimal prie for eah servie rate µ, we now fous on the servie provider s optimal servie rate deision In the following setion, we examine a market in whih the servie provider offers partial market overage 9

10 31 Partial Market Coverage In this setion, we assume that the potential demand Λ is high enough that the servie provider s optimal prie and servie rate deisions are not onstrained by the volume of ustomer demand (Mathematially, Λ > λ α V b+αµ b 2 α ) Before we establish the optimal servie rate under partial overage, we demonstrate that the equilibrium demand and prie urves (as a funtion of the servie rate) are unimodal (details in Appendix A) The unimodality property implies that fousing exlusively on delivering a high value servie (ie, a slow servie rate) leads to very high ustomer waiting osts On the other hand, inreasing the servie rate to minimize waiting osts leads to a low servie value At an intermediate servie value (in F(α)), the value of the servie net of waiting osts is maximized The firm harges a prie based on this intermediate servie value Even in markets where the potential demand is very large (ie, Λ ), 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 by-produt of ustomer-intensive servies Building on these observations, Proposition 2 provides the equilibrium outomes from the maximization in (3) Proposition 2 For a ustomer-intensive servie of type α > 0, 1 The optimal servie rate is equal to µ = V b+αµ b 2 The orresponding optimal prie is equal to p (µ ) = V b+αµ b 2 α 2 3 The demand at the optimal prie and servie rate equals λ e (µ, p (µ )) = V b+αµ b 2 α = λ α Therefore, the optimal revenue for the servie is equal to R(µ, p (µ )) = (V b+αµ b 2 α) 2 4α Proposition 2 shows that there exists a unique, interior servie rate µ in F(α) that maximizes revenues Proposition 21 shows that the optimal servie rate, µ, is dereasing in α: as the servie beomes more ustomer-intensive, the servie provider has a higher inentive to spend more time on eah ustomer We an see this learly by omparing the equilibrium servie value to the benhmark Using equation (2), the servie value provided to ustomers in equilibrium is V (µ ) = (V b + αµ b )/2 Thus, V (µ ) > V b α > V b /µ b 10

11 From Proposition 22, we note that the optimal prie, p (µ ), is unimodal in α dereasing for α < /µ 2 b and inreasing for α > /µ2 b We saw that as the servie beomes more ustomerintensive, 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 22, sine the optimal prie traks the net value of the servie If α is low (ie, α < /µ 2 b ), the ongestion effets dominate the inrease in servie value as α inreases Hene, as the task beomes more ustomer-intensive (ie, α inreases), the prie falls However, for high α values (α > /µ 2 b ), the inreased servie value from a longer servie time dominates any inrease in the equilibrium waiting ost Therefore, the optimal prie is inreasing in α The equilibrium demand λ e (µ, p (µ )) is also determined by the tradeoff between waiting osts and the value of servie, and behaves similarly to the optimal prie At low values of α, waiting osts are more sensitive to small inreases in α than the servie value Hene, ongestion effets dominate in this range For higher values of α, the reverse is true the servie value is more sensitive to inreases in α than the waiting ost The net effet is that the equilibrium demand is unimodal dereasing in α for α < Vb 2 /, and inreasing in α for α > Vb 2 / (Proposition 23) Finally, Proposition 2 aptures the effet of the delay parameter on the servie outomes The optimal servie rate, µ, is independent of the waiting ost, Interestingly, 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 the lowering of pries, p (µ ), but they also lead to lower equilibrium demand λ e (µ, p (µ )) Consequently, the optimal revenues, R(µ, p (µ )), derease with inreased waiting osts 311 Analysis of Value-Prie-Demand Interations We shed further light on the subtle interations among prie, demand, and servie value as a funtion of servie rates in ustomer-intensive servies The equilibrium prie, equilibrium demand, waiting osts and the net value of the servie to ustomers are outomes of these omplex interations Lemma 1 studies the relationship between prie and demand Lemma 1 [Property of α-symmetry:] For a ustomer-intensive servie of type α, p (µ) and λ e (µ, p (µ)) have the following symmetri relationship around the optimal servie rate µ for any given µ F(α): p (µ + ɛ) = αλ e (µ ɛ, p (µ ɛ)), where ɛ = (µ µ ) 11

12 Lemma 1 learly demonstrates that pries and effetive demand are two levers related to eah other by the ustomer-intensity parameter α To better illustrate the property we derived in Lemma 1, we divide the operating region F(α) into 3 sub-regions as shown in Figure 2 Figure 2: The left panel illustrates partial market overage, and the right panel illustrates full market overage, for the same Λ In the figures, V b = 10 and µ b = 5 The symmetry of p (µ) (denoted by the thik line) and λ e (µ, p (µ)) (denoted by the dotted urve) around µ for ustomer-intensive servies of types α = 2, 05 The optimal servie rate µ and the orresponding equilibrium demand is λ α = λ e (µ, p (µ )) The maximum throughput is λ α, indued by the servie rate µ Details of the α-symmetry property between prie and demand in a ustomer-intensive servie an be found in Appendix B 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 the two servie strategies Modeling ustomer-intensity through α allows us to apture the presene of suh options in servie provision 32 Full Market Coverage Having derived the harateristis of the equilibrium in markets where only partial overage ours, we now analyze markets in whih full overage is possible Proposition 3 indiates that the onlusions derived in Setion 31 ontinue to hold for full-overage 12

13 Proposition 3 Under low potential demand Λ < λ α for any ustomer-intensive servie of type α: 1 The optimal servie rate is equal to µ = Λ + /α 2 The orresponding optimal prie is equal to p (µ ) = V b + αµ b αλ 2 α 3 The equilibrium demand at the optimal prie and servie rate is Λ (ie, λ e (µ, p (µ )) = Λ) Proposition 3(1), shows that the optimal servie rate µ dereases in α (just as in the ase of partial overage) The servie provider spends more time on eah ustomer as the servie beomes ustomer-intensive Before examining part (2), let us examine the effets of this result (3) on the equilibrium arrival rates The equilibrium arrival rates are onstant sine the servie provider serves all ustomers in equilibrium Therefore, note that as ustomerintensity α inreases, the optimal servie rate falls, while the equilibrium arrival rate remains unhanged This leads to inreased waiting osts, as α inreases In fat, the expeted waiting ost is α In equilibrium, if the market demand is low, ustomers wait longer as the servie beomes more ustomer-intensive Part (2) of Proposition 3 notes that the optimal prie is onvex in α When Λ < µ b, for α < (µ b, the optimal prie is dereasing in α, and for α > Λ) 2 (µ b it is inreasing in α Λ) 2 From Proposition 3(1), we noted that when α inreases, the servie provider inreases the servie time with every ustomer (whih inreases servie value) When α < (µ b, the Λ) 2 higher waiting ost (due to inreased servie time) dominates any inrease in servie value, leading to a degradation in the net value of the servie for the ustomers To aommodate this loss, the servie provider has to ut pries as α inreases Hene, the optimal prie is dereasing in α for this region If we ompare two servies of low ustomer-intensity (with α < (µ b ), the more ustomer-intensive servie will be both less expensive and less Λ) 2 ongested than the other However, when α is high (> (µ b ), the gains in servie value are signifiant enough to Λ) 2 dominate equilibrium waiting osts as α inreases Therefore, the optimal prie is inreasing in α when ustomer-intensity is high (ie α is greater than the desribed threshold) In ontrast to the above omparison in whih α was low, when we ompare two servies that are both highly ustomer-intensive (α > (µ b ), the servie with higher α is both more Λ) 2 expensive and more ongested than the other 13

14 4 Model with Servie Rate Competition In this setion, we onsider the effet of multiple ompeting servers owned by a single servie provider (firm) who provides a servie of ustomer-intensity α Although the servie provider sets an admission prie to maximize total revenues, the individual servers have the flexibility to set their own servie speed (and hene, quality) We fous on an example of primary are physiians who belong to the same health network or the same hospital whih, in turn, determines the admission prie for patient visits We initially restrit our attention to a firm with two servers for modeling ease, 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 All of these deisions are made simultaneously by the firm, the servers and the ustomers We model eah server as an M/M/1 queueing regime The queue joining deision of a ustomer is given by γ j (µ 1, µ 2, p, Λ), for j = 0, 1, 2, where γ 0 ( ) denotes the probability of balking, and γ 1 ( ) and γ 2 ( ) denote the probability of joining queue 1 and 2, respetively Under pure strategies, ie γ i = 1 for some i, either one of the servers serves all of the ustomers (Λ), or none of the servers serves any ustomers We find that none of these outomes are observed in equilibrium We thus fous on mixed onsumer strategies Again, as in Setion 3, we divide our analysis into two ases based on market overage When the market is suffiiently large, ie Λ 2 V b+αµ b 2 α = V b+αµ b 2 α, we α 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, ie when Λ < V b+αµ b 2 α, we show that the firm hooses a prie suh that all of the onsumer surplus α is extrated, and the market is fully overed by the firm Proposition 4 When the market is suffiiently large, given by the ondition Λ V b+αµ b 2 α α, the servers at as monopolists The optimal servie rate set by server i is given by: µ i = V b +αµ b for i = 1, 2 The firm s optimal prie is p = V b+αµ b 2 α 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 of our insights on ustomer-intensive servies, derived for the single-server monopoly in Setion 3, ontinue to hold 14

15 When the market is smaller, ie when Λ < V b+αµ b 2 α, ompetition affets the servers α and the firm s strategies The servers ompete by adjusting their servie rates, while the firm harges an admission prie for the servie We find that the net values (V (µ i ) W (µ i, λ i ) p, for i = 1, 2) provided by the servers are equal and positive in equilibrium Server i s equilibrium demand is λ i (ie, Λ = λ 1 + λ 2 ) orresponding prie p = Λ/2; thus, the entire market is overed by the two servers The servie provider extrats all onsumer surplus by a harging a Proposition 5 When the potential demand for the servie is low, Λ < V b+αµ b 2 α, the two α servers share the market demand, by setting their servie at µ e i = Λ + /α i 2 Proposition 5 shows that in equilibrium, the firm provides higher servie value at a slower rate through its servers than it would if there were only one server (see Proposition 31) However, note that ustomers expeted waiting ost in the multi-server ase (W (µ e i, Λ/2) = α) is idential to that in the orresponding single server ase (W (µ, Λ) = α) Therefore, the servie value net of the waiting ost inreases with the number of servers ompeting for market share Our strutural results ontinue to hold when there are n (> 2) servers ompeting on servie rates (i) In small markets (Λ < n V b+αµ b 2 α ), 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 when Λ < n V b+αµ b 2 α The servie provider harges an admission prie p n = V b +αµ b αλ/n 2 α p n is stritly inreasing and onave in 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 15

16 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 Sine the prie p n 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 Sine p n is inreasing in α, 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 ustomer-intensive servies, wherein the servie quality is sensitive to the time spent with the ustomer The tradeoff 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 tradeoff Thus our model provides fundamentally new insights into the nature of ustomer-intensive servies We fous on two key insights and examine empirial evidene in the ontext of primary are servies 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/servie rates This raises the osts of providing the servie Thus, Baumol s ost disease (disussed in Setion 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 the quality of servie by spending adequate time with eah patient, rather than on 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 over the period of , and show that primary are visit durations have inreased (ie, the average servie rate is slower) with an aompanied inrease in servie value It is optimal for firms providing primary are servies 16

17 to invest in a high-quality slower-servie, and therefore, partial market overage is likely to be observed Slowdown and longer servies have reated a new primary are model, termed onierge mediine Conierge dotors limit the number of patients they aept, and offer them a 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 ( founded in 2000, is a national network of 250+ physiians who provide preventive and personalized health are Conierge dotors affiliated to MDVIP are for a maximum of 600 patients eah MDVIP ( MD 2 International ( Current Health ( and Qliane Primary Care ( are some leading onierge primary are firms in the market Priing under Servie Rate Competition: We find that as the number of ompeting servers inreases, the servers slow down further As a onsequene, server-ompetition inreases the prie harged by the servie-provider of the ustomer-intensive servie, and enhanes the servie value in equilibrium, while holding the equilibrium ongestion (waiting) osts onstant These results, whih are in sharp ontrast to previous queuing researh, are driven by ustomer-intensity For ustomer-intensive servies suh as primary are servies, 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 a higher admission prie In primary are settings, there is a strong empirial evidene of inreased prie observations when the number of primary are physiians in a market inreases This empirial finding in the seminal paper by Pauly and Satterthwaite (1981) have been subsequently observed in several studies whih onfirm prie inreases due to the ompetition in primary are servie provision proess (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 servie inreases with servie time Future Diretions: Several diretions seem promising for future researh One option is to study different kinds of market heterogeneity Competing servers ould vary in their 17

18 ustomer-intensities (for eg based on the 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 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 offered servie Debo et al (2008) model inentive effets in the ontext of suh redene servies; similar issues are pertinent to ustomer-intensive servies Aknowledgements: 2009 INFORMS JFIG Finalist The authors would like to thank the anonymous reviewers, an assoiate editor and the departmental editor for their valuable suggestions Our speial thanks to Philipp Afehe, Baris Ata, 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 and Eitan Zemel We also thank seminar partiipants at Carnegie Mellon University, Cornell University, Northwestern University, University of Maryland, University of Pennsylvania, Utah Winter Conferene 2010, University of Rohester s Workshop on Information Intensive Servies and the reviewers of the Servie SIG Conferene and 2009 INFORMS JFIG Competition We aknowledge finanial support from the Fishman-Davidson Center at the Wharton Shool, University of Pennsylvania Referenes Afehe, Philipp, 2006 Inentive-ompatible Revenue Management in Queueing systems: Optimal Strategi delay and other Delay Tatis University of Toronto working paper Allon, G, A Federgruen 2007 Competition in Servie Industries Operations Researh, 55(1), Armony M, M Haviv 2000 Prie and Delay Competition between Two Servie Providers European Journal of Operational Researh, 147(1), Baumol, W J 1993 Health Care, Eduation, and the Cost Disease: A Looming Crisis for Publi Choie Publi Choie, 77(1), Brahimi, M, D J Worthington 1991 Queueing models for out-patient appointment systems A ase study Journal of Operational Researh Soiety, 42(9), Cahon, G, P Harker 2002 Competition and Outsouring with Sale Eonomies Management Siene, 48(10), Chase, RB, DA Tansik 1983 The Customer Contat Model for Organization Design Management Siene, 29(9), Chen, L M, W R Farwell, A K Jha 2009 Primary Care Visit Duration and Quality: 18

19 Does Good Care Take Longer? Arhives of Internal Mediine, 56(6), Chen, H, M Frank 2004 Monopoly Priing When Customers Queue, IIE Transations, 36(6), Chen, H, Y Wan 2003 Prie ompetition of make-to-order firms, IIE Transations, 35(9), de Veriourt, F, P Sun 2009 Judgement Auray Under Congestion In Servie Systems Duke University working paper Debo, LG, LB Toktay, L N Van Wassenhove 2008 Queueing for Expert Servies Management Siene, 54(8), Edelson, NM, DK Hildebrand 1975 Congestion Tolls for Poisson Queuing Proess Eonometria, 43(1), Gans, N 2002 Customer Loyalty and Supplier Quality Competition Management Siene, 48(2), Gaynor, M, D Haas-Wilson 1999 Change, Consolidation, and Competition in Health Care Markets The Journal of Eonomi Perspetives, Vol 13, No 1, pp Gilbert, S M, Z K Weng 1998 Inentive Effets Favor Non-Consolidating Queues in a Servie System: The Prinipal Agent Perspetive Management Siene, 44(12), Green, L, S Savin 2008 Reduing Delays for Medial Appointments: A Queueing Approah Operations Researh, 56(6), Hasija, S, E Pinker, RA Shumsky 2009 Work Expands to Fill the Time Available: Capaity Estimation and Staffing under Parkinson s Law M& SOM, 12(1), 1 18 Hassin, R, M Haviv 2003 To Queue or not to Queue: Equilibrium behavior in queuing systems Kluwer Aademi Publishers, Norwell, MA Hopp, WJ, S M R Iravani, G Y Yuen 2007 Operations Systems with Disretionary Task Completion Management Siene, 53(1), Kalai, E, M Kamien, M Rubinovith 1992 Optimal Servie Speeds in a Competitive Environment Management Siene, 38(8), Kaminetsky, B 2004 Testimony to the Joint Eonomi Committee of the United States Congress Consumer-Direted Dotoring: The Dotor is in, Even if Insurane is Out Wednesday, April 28, 2004 Lederer, P J, L Li 1997 Priing, prodution, sheduling, and delivery-time ompetition Operations Researh, 45(3), Li, L 1992 The role of inventory in delivery time-ompetition Management Siene, 38(2) Li, L, Y S Lee 1994 Priing and Delivery-Time Performane in a Competitive Environment Management Siene, 40(5), Lovejoy, W, K Sethuraman 2000 Congestion and Complexity Costs in a Plant with Fixed Resoures that Strives to Make Shedule M & SOM 2(3),

20 Lovelok, C 2001 Servie Marketing, People, Tehnology and Strategy Prentie Hall, Upper Saddle River, NJ Lu, L, J Van Mieghem, C Savaskan 2008 Inentives for Quality Through Endogenous Routing M & SOM, 11(2), Mehani, D, D MAlpine, M A Rosenthal 2001 Are patients visits with physiians getting shorter? New England Journal of Mediine, 344(3), Mendelson, H, and S Whang 1990 Optimal Inentive-Compatible Priority Priing for the M/M/1 Queue Operations Researh, 38(5), Naor, P 1969 The Regulation of Queue Sizes by Levying Tolls Eonometria, 37(1), Oliva, R, R J Sterman 2001 Cutting Corners and Working Overtime: Quality Erosion in the Servie Industry Management Siene, 47(7), Png, I, D Reitman 1994 Servie Time Competition RAND Journal of Eonomis, 25(4), Pauly, M V, M A Satterthwaite 1981 The Priing of Primary Care Physiians Servies: A Test of the Role of Consumer Information The Bell Journal of Eonomis, 12(2), Ren, Z J, X Wang 2008 Should Patients be Steered to High Volume Hospitals? An Empirial Investigation of Hospital Volume and Operations Servie Quality Boston University working paper Ross, SM 2006 Introdution to Probability Models Aademi Press Ninth Edition Surowieki, J 2003 What Ails us? The New Yorker, July 7, 2003 Triplett, J E, B P Bosworth 2004 Produtivity in the US Servies Setor, New Soures of Eonomi Growth Brookings Institution Press, Washington, DC Varian, H 2004 Eonomi Sene; Information Tehnology May Have Been What Cured Low Servie-Setor Produtivity NY Times, Published: February 12, 2004 Veeraraghavan, S, L G Debo 2009 Joining Longer Queues: Information Externalities in Queue Choie Manufaturing and Servie Operations Management, 11(4), Wang, X, L G Debo, A Sheller-Wolf 2008 Design and Analysis of Diagnosti Servie Centers Carnegie Mellon University working paper Yarnall, K S H, K I Pollak, T Ostbye, K M Krause and J L Mihener, 2003 Primary Care: Is There Enough Time for Prevention? Amerian Journal of Publi Health, 93(4),

21 Appendix A: Proofs Proof of Proposition 1: We begin by showing the optimal prie, p (µ) for Λ > A 2 (α) In this ase, the servie provider annot serve all potential ustomers even when the prie is equal to zero The equilibrium arrival rate, λ e (µ, p), is determined by the following equation in this ase: The revenue of the servie provider, R(µ, p), is given by: V (µ) p = W (µ, λ e (µ, p)) (1) ( ) R(µ, p) = p µ (2) V (µ) p Reall that the servie value is upper-bound for the prie, ie V (µ) p Therefore, the revenue funtion is onave in the prie, p, for the set of admissible pries (for 0 p V (µ)), as the seond order ondition is negative: 0 > δ2 R(µ, p) δp 2 2 = (V (µ) p) 2p 2 (V (µ) p) 3 The optimal prie, maximizing the servie provider s revenues for a given servie rate µ, is p (µ) = V (µ) V (µ)/µ We find the optimal prie using the first order ondition: 0 = δr(µ, p) δp = µ V p p (V p) 2 The first order ondition is satisfied at p admissible pries, p [0, V (µ)] equilibrium arrival rate: = V (µ) V (µ)/µ, whih is in the set of Plugging p (µ) into equation (1) we find the resulting λ e (µ, p (µ)) = µ µ V (µ) The equilibrium arrival rate, λ e (µ, p (µ)), is independent of the potential demand, Λ This shows that the optimal prie at servie rate µ is equal to V (µ) V (µ)/µ for all Λ µ µ So far, we have derived the optimal prie V (µ) p for all Λ µ µ V (µ) To omplete the proof we need to derive the optimal prie for Λ < µ µ Note V (µ) that the servie provider an serve all potential ustomers at a non-negative prie for Λ µ µ For a given servie rate µ, the equilibrium demand, λ V (µ) e(µ, p), is dereasing in prie Therefore, the maximum number of ustomers that an be served (maximum throughput) at rate µ, Λ(µ), is found by setting the prie equal to zero Using the following equation we 1

22 find Λ(µ) V (µ) = µ Λ(µ) Λ(µ) = µ V (µ) If Λ(µ) is greater than the potential demand Λ, then the servie provider an serve all potential ustomers, harging a prie greater than zero For Λ < µ µ, Λ(µ) > Λ: V (µ) Λ(µ) = µ µ µ > Λ, sine V (µ) for all µ F(α) V (µ) V (µ) µ For Λ < µ µ V (µ) all arriving ustomers join the queue, if the net utility of joining when all others join the queue at prie p is non-negative, ie V (µ) p W (µ, Λ) 0 The net utility dereases in prie, and it is non-negative for p V (µ) Inreasing the µ Λ prie further redues the equilibrium arrival rate Thus the servie provider s revenue as a funtion of prie an be written as: R(µ, p) = (3) pλ ( ) if 0 p V (µ) W (µ, Λ) p µ V (µ) p if V (µ) W (µ, Λ) < p V (µ) µ 0 if p > V (µ), µ Differentiating the revenue funtion with respet to prie we get: δr(µ, p) δp = Λ if 0 p V (µ) µ Λ µ p if V (µ) < p V (µ) (4) V (µ) p (V (µ) p) 2 µ Λ µ 0 if p > V (µ), µ The revenue, R(µ, p) is learly inreasing in the prie for p V (µ) Inreasing µ Λ the prie further at p = V (µ) will derease the demand (throughput) but it may still µ Λ inrease the revenues Note that the revenue funtion for p > V (µ) is equivalent to µ Λ V (µ) the revenue funtion given by equation (2), whih is maximized at p = V (µ) The µ revenues derease in prie at p = V (µ) beause V (µ) > V (µ) V (µ) for µ Λ µ Λ µ Λ < µ µ V (µ) : V (µ) µ = µ (µ µ V (µ) ) > As a result, the optimal prie at servie rate µ, for Λ < µ µ Λ µ V (µ) is p (µ) = V (µ) µ Λ 2

23 The resulting equilibrium arrival rate is equal to λ e (µ, p (µ)) = Λ { p V (µ) W (µ, Λ) if 0 Λ (µ) = ˆλ(µ) V (µ) V (µ)/µ if ˆλ(µ) < Λ (5) Thus we have derived p for all Λ Preparatory Results for Lemma 2 through Proposition 3: Before we prove the lemmas, we prove two main preparatory results Result 1: Servie provider s revenue funtion R(µ, p) is non-dereasing in the potential demand, Λ Proof: For a given servie rate µ and prie p, the servie provider s revenue as a funtion of the potential demand, Λ, is given as follows: R(µ, p) = R(µ, p) is ontinuous in Λ pλ ( ) if W (µ, Λ) V (µ) p p µ V (µ) p if W (µ, 0) < V (µ) p < W (µ, Λ) 0 if V (µ) p < W (µ, 0) To show this, we only need to show that the funtion is ontinuous at the transition point W (µ, Λ) = V (µ) p Note that the last region, V (µ) p < W (µ, 0), is independent of the potential demand, Λ W (µ, Λ) = is inreasing in Λ for Λ µ Rewriting the transition point, W (µ, Λ) = µ Λ V (µ) p, and solving for Λ we get: µ Λ = V (µ) p Λ = µ V (µ) p Therefore R(µ, p) is ontinuous in Λ for Λ 0 Clearly, R(µ, p) is inreasing in Λ for W (µ, Λ) V (µ) p and onstant in Λ for W (µ, Λ) > V (µ) p This proves that R(µ, p) is non-dereasing in Λ for Λ 0 Result 2: For Λ λ α = max {µ F(α)} {λ e (µ, p (µ))}, the optimal prie, p (µ), and the resulting equilibrium arrival rate, λ e (µ, p (µ)), have the following symmetri relationship around β = ( V b +αµ b ) for any µ F(α) p (β + ɛ) = αλ e (β ɛ, p (β ɛ)), (6) where ɛ = µ V b+αµ b Proof: For Λ > λ α = max {µ F(α)} {λ e (µ, p (µ))}, the optimal prie for a given servie rate is 3