1 MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol., No. 3, Suer 28, pp issn eissn infors doi.287/so INFORMS INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at Usage Restriction and Subscription Services: Operational Benefits with Rational Users Raandeep S. Randhawa McCobs School of Business, University of Texas at Austin, Austin, Texas 7872 Sunil Kuar Graduate School of Business, Stanford University, Stanford, California 9435 This paper studies a rental fir that offers reusable products to price- and quality-of-service sensitive custoers Netflix or Blockbuster can be thought of as the canonical exaple. Custoers perception of quality is deterined by their likelihood of obtaining the product or service iediately upon request. We study the alternatives of offering either a subscription option that liits the nuber of concurrent rentals in return for a flat fee per-unit tie, or a pay-per-use option with no such restriction. Custoers are assued to desire a noinal usage rate of the product, which they eet by adjusting their request rate in either option. Thus, they have a higher request rate in the subscription option. We propose a Markov chain odel for custoer behavior under the subscription option equivalent to the standard Poisson odel under the pay-per-use option. In a large arket setting, assuing exponential deand, we show that using the subscription option is ore profitable for the fir. Further, via a nuerical study, we show that this assuption is not essential for the result to hold. However, we show that the subscription option does not necessarily doinate the pay-per-use option in quality of service. The fir anages the trade-off between price and quality of service better in the subscription option. Moreover, we show that social welfare and the consuer surplus can also be higher in the subscription option, indicating that both the fir and the consuers can benefit fro the subscription option. Key words: subscription services; pay-per-use services; operational benefits; pricing; capacity sizing; finite custoer population; loss syste; on-off odel; diffusion approxiation History: Received: Deceber 9, 25; accepted: March 23, 27. Published online in Articles in Advance January 4, 28.. Introduction Many firs provide reusable products or services to custoers who request these products or services in soe rando fashion. Moreover, the duration of product use by the custoer is also rando in such systes. A canonical exaple of such a syste is a rental fir. The fir cannot always accoodate all custoer requests because the nuber of products stocked or the nuber of available servers is liited. A easure of the fir s quality of service as seen by a custoer is the likelihood of obtaining the product iediately upon request. To anage the variability in the presence of capacity constraints, the fir needs to set the capacity level, i.e., the nuber of products stocked, and the price so as to optially extract profit. Access to such rental firs is typically of two kinds: subscription or pay per use, with either option having an equal ease of ipleentation in any contexts. For exaple, in the case of DVD rentals, Netflix has eerged as a ajor player and it provides only a subscription option. On the other hand, Blockbuster ostly provides a pay-per-use option. With the introduction of Total Access, Blockbuster has also ade a foray into the subscription world. Of course, one would expect a fir to do better in the face of copetition by locking in custoers via subscription. In this paper, we study subscription fro a different angle. We study whether the fir is able to handle the inherent variability in the syste better if it offers a subscription option rather than a pay-per-use option. That is, we study whether the fir receives any operational benefit of offering a subscription option. We consider a rental fir serving price- and qualitysensitive custoers. The fir has the option of offering a subscription or a pay-per-use option. We assue that subscribers pay a subscription fee per-unit tie, 429
2 Randhawa and Kuar: Usage Restriction and Subscription Services 43 Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at independent of the usage, while the pay-per-use custoers pay a price each tie they use the product. In addition, we assue that subscribers are restricted to renting only one product at a tie, while pay-per-use custoers are not subject to such a restriction. This assuption is quite natural in a DVD rental setting, where custoers could be renting ultiple DVDs at a given tie. The nuber of concurrent rentals is either contractually restricted as in the case of Netflix or not restricted as in the pay-per-use odel of Blockbuster. Of course, rational custoers will consider the concurrent usage restriction and react accordingly. In our paper, in both the subscription and pay-per-use settings, the fir statically sets price and capacity levels and observes deand in response. The fir s objective is to axiize the difference between the revenues it obtains fro the custoers and the cost of aintaining the capacity. We assue that the custoer deand that is, the nuber of subscribers joining the syste or the rate at which pay-per-use custoers show up at the fir depends on both the prices the fir sets as well as the equilibriu likelihood of obtaining the product iediately upon request. The reader can think of this equilibriu as having been reached via repeated interactions with the custoers, and, hence, this likelihood is coon knowledge in the arket. We solve the fir s optiization proble for both the subscription option as well as the pay-per-use option and then copare the two using an asyptotic approach. The natural asyptotic regie considered is one where the arket for potential custoers is large. In coparing the two options, the first question that one ight ask is: Does restricting usage in the subscription option lead to a reduction in the inherent syste variability, resulting in higher quality of service at any choice of capacity or price? We show that this is not necessarily true. In fact, neither option need doinate the other in ters of quality of service. However, if profits are copared in a large arket with rational price- and quality-of-service sensitive custoers, the subscription option does doinate the equivalent pay-per-use option. For the case of exponential deand, we prove that the subscription option is indeed better for the fir. Through a nuerical study, we show that this result is robust with respect to the choice of deand function. Although the subscription option does not necessarily doinate the pay-per-use option on quality of service, the fir is able to better anage the trade-off between price and quality of service in the subscription option. Moreover, we show that the social welfare and the consuer surplus can also be higher in the subscription option, indicating that both the fir and the consuers can benefit fro the subscription option. However, this need not always be the case, with the fir soeties profiting fro the subscription option at the custoers expense. Our approach for solving the fir s optiization proble for either option is as follows. Denoting the size of the arket that is, the nuber of potential custoers by n, we first solve a noinal optiization proble at the On level. As usual, On denotes any quantity such that On/n is bounded as n. In such a syste, the natural scale at which stochastic variability anifests itself is O n. Any refineent to the noinal proble that considers the stochastic variability in the syste ust be at the O n level. The following exaple ay help illustrate this scale difference. Consider a newsvendor odel with deand arising fro n independent, identically distributed sources. In this case, the optial stock level coprises of the ean of the total deand, which is On, and a safety stock against variability, which is soe nuber of standard deviations of the total deand and is O n. The safety stock in the newsvendor odel is exactly analogous to the refineent to the noinal solution that we propose. By applying a law of large nubers arguent, Proposition shows that on the On scale, the syste operates in a deterinistic regie with essentially all requests being satisfied iediately. Furtherore, using a functional central liit theore proved in earlier work, we perfor a refined analysis on the O n scale to obtain prices and capacity levels optial on this scale. That is, these choices result in profits within o n of the optial values see Proposition 4. As usual, on denotes any quantity such that on/n as n. Furtherore, Proposition 3 shows that in this regie, pricing and capacity sizing are equivalent levers for the fir. That is, any refineent in prices can be iitated by an equivalent refineent in capacity levels for both options. This asyptotic approach allows us to copare the profits obtained under each option up to a resolution of o n, and we prove
3 Randhawa and Kuar: Usage Restriction and Subscription Services Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS 43 INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at the priary result of the paper; that is, the profit is higher for the subscription option by a quantity that is O n. Cobining this with the size of the possible error in estiating the profit allows us to conclude that in a large enough arket, the subscription option doinates. This paper is organized in the following anner. In 2, we propose a generic odel of custoer behavior when restricted to only one concurrent rental. We develop the analytical ethods required to calculate the asyptotically optial price and capacity levels that axiize the fir s profit. We solve both the noinal proble on the On scale and its variability refineent and show that pricing and capacity sizing are equivalent levers for the fir on the O n scale. In 3, we build the fraework necessary to have a unified odel of unrestricted custoer behavior so that we can copare the subscription and pay-per-use options. We show that this unified odel reduces the odel developed in 2 under usage restriction, and is equivalent to a standard Poisson odel in a payper-use setting without such restrictions. We solve the fir s optiization proble in pay-per-use settings using the achinery developed in 2. Section 4 copares the subscription option with an equivalent pay-per-use option and proves the ain result of the paper; that is, the fir s profits are higher when offering a subscription option. Section 5 discusses the conclusions and scope for future work. Finally, the appendix contains a suary of the asyptotic results in Randhawa 26 used in this paper and the proofs of all the results... Literature Review We use the approach in Mendelson and Whang 99 to build a icro-econoic fraework around our service syste and study optial pricing and capacity sizing at the syste equilibriu. We characterize the syste equilibriu as in Arony and Maglaras 24 and Whitt 23. These papers deal with ultiserver queueing systes with congestion-sensitive deand but without econoic considerations, and study the syste equilibriu behavior using asyptotic ethods. Perhaps the ost related to our work is a recent paper Maglaras and Zeevi 23 that studies pricing and capacity selection in a ultiserver queueing syste with Poisson arrivals. This paper uses the Halfin-Whitt see Halfin and Whitt 98 asyptotic results to develop estiates for congestion levels to copute optial price levels. Another application of optial pricing and capacity sizing in an asyptotic regie is in Plabeck and Ward 25, where the authors study static pricing, capacity selection, and dynaic scheduling using the traditional heavy traffic assuptions in an asseble-to-order setting. Optial capacity sizing under fixed, exogenous deand is studied in Borst et al. 24, where the authors use asyptotic ethods to copute the optial staffing level in a call center with the trade-off being between the agents cost and the quality of service provided. Although this paper deals with static pricing, work in Paschalidis and Tsitsiklis 2 and Gallego and van Ryzin 994 suggests that this is not a ajor liitation. Paschalidis and Tsitsiklis 2 study revenue anageent in the context of Internet service provision using a Markov decision process fraework where custoers are only assued to be price sensitive. Although they focus on dynaic pricing, an iportant conclusion of their study is that static pricing rules can achieve near-optial perforance. Gallego and van Ryzin 994 derive a siilar insight in the classical context of selling a set of goods within a finite-tie horizon. The theory of club goods in econoic literature also copares subscription and pay-per-use settings see Buchanan 965, Cornes and Sandler 986, Scotcher 985, Barro and Roer 987. The treatent there is acroscopic in the sense that the operational aspects due to the dynaics of custoer requests and rentals are not considered. The ost siilar paper in this literature to our work is Barro and Roer 987, which deonstrates that in a ski-lift pricing setting, a fir can achieve the sae profit by pricing per ride or charging a flat daily fee. We establish a siilar result in our setting where we observe that as long as a change is not induced in custoer behavior via a usage restriction, for exaple, usage-based pricing and pricing independent of usage generate identical profits for the fir see 4.5 for ore details. 2. Subscription 2.. A Model of Subscriber Behavior Under Usage Restriction We begin by describing a natural Markovian odel for custoer behavior in a subscription setting where
4 Randhawa and Kuar: Usage Restriction and Subscription Services 432 Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at the custoer is restricted to renting only one product at a tie. The odel is paraetrized by three paraeters,,, and. We associate each subscriber with a Markov chain having three states: on, off, and hold. The transition rate atrix is given by on off hold M = product available product not available product available product available Subscribers spend an exponentially distributed aount of tie with ean / in the off state, after which they request a product. If no products are available they transit to the hold state, otherwise, a product is assigned to the and they transit to the on state. In the hold state, they retry to obtain a product after every exponentially distributed aount of tie with ean / until a product is available. Once a product is assigned to the, they transit to the on state. In the on state, they use the product for an exponentially distributed aount of tie with ean /, after which they return the product and transit to the off state. We assue that all the ties spent by a subscriber in each state are independent, identically distributed, and independent of ties in other states. We also assue that a product cannot be assigned to ore than one subscriber at any tie. This subscriber odel is related to the classical Engset odel see Kleinrock 975 used in telecounications. In fact, for the special case in which the hold and off states are indistinguishable i.e., when the rates at which a subscriber tries to obtain a product fro the hold and off states are equal, the subscribers can be characterized by a two-state Markov chain; this is identical to the Engset odel. For the ost part, we will assue that =, which renders the hold and off states indistinguishable. When the assuption that = is not satisfied, we use the asyptotic liits derived in Randhawa and Kuar 27 to obtain insights based on coparative statics on the retrial rate using nuerical experients. These are discussed in 5. The paraeter choice is in fact intrinsic to the subscriber in the sense that each subscriber chooses to eet a noinal usage level. We elaborate on this intrinsic odel in 3, where we shall see the iportance of the underlying odel in coparing different contracts the fir offers its subscribers. We now build a deand function for the subscribers. We consider a arket with a total of n potential subscribers. In accordance with Naor s original idea see Naor 969, we assue that each potential subscriber has a value S for each service copleted. This value is assued to be the sae for every service copleted for each potential subscriber but ay differ aong potential subscribers. The valuations are distributed according to a distribution F and with density f. We shall assue that the density is strictly positive on and there is a value p e such that dx F x/dx < for x p e, where F denotes the tail of the cuulative density function. This assuption is equivalent to the deand function corresponding to F being elastic beyond the price p e and ensures a nontrivial solution to the pricing and capacity sizing proble. We assue that subscribers are individually rational, i.e., they join the syste only if joining ensures the a positive benefit. Subscribers sensitivity to the quality of service is captured through the denial probability; that is, the steady-state probability that upon request a subscriber does not receive the product. We define the denial probability by No. of denied attepts by tie t = li t No. of attepts by tie t The higher the denial probability the longer it takes a subscriber to obtain the product. In particular, denote the ean tie between successful attepts to obtain a product in steady-state for a subscriber by. We can then characterize as a function of the denial probability as follows. Lea. The ean tie between successful attepts to obtain a product in steady-state for a subscriber is = / + / + /. All the proofs, including that of this result, can be found in A.2. We assue that a potential subscriber joins the syste only if the value obtained per service, S, exceeds average cost per service, as follows. When the syste anager sets a subscription fee of p per-unit tie, he or she only joins the syste if S>p. Therefore, the
5 Randhawa and Kuar: Usage Restriction and Subscription Services Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS 433 INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at nuber of potential subscribers who join the syste at price p per-unit tie and denial probability, Np, can be odeled as Np= n S> + = n F + + p + p In building this odel, we have assued that the subscriber valuations S aong the potential subscribers are not rando draws but are precisely distributed according to the distribution F. That is, as is usually done with deand curves, we assue that for any x>, the fraction of the subscribers who have valuations that exceed x is exactly Fx. In addition to the subscription fee, the syste anager also decides on the capacity level k. Assuing each product costs $c per-unit tie, where c >, the optiization proble of the syste anager can be stated as ax pnp kc p k + + s.t. = dnpk where dnpk is the denial probability as a function of the nuber of subscribers and the capacity level. Note that for any subscription fee and capacity level set, the denial probability ust satisfy an equilibriu condition, which is captured via the constraint = dnpk. The inability to characterize this equilibriu denial probability in a siple for renders this proble, as stated, extreely hard to solve. This otivates us to try to approxiately solve it in the natural asyptotic regie of large nuber of potential subscribers. In particular, we shall let n grow without bound and use the superscript n to ake explicit the dependence of the various paraeters on n, i.e., p n, k n, and n denote the subscription fee, nuber of products, and denial probability, respectively, and N n p n n denotes the nuber of potential subscribers joining the syste. The optiization proble can then be written as ax n p n k n p n N n p n n k n c p n k n + + s.t. n = dn n p n n k n With this setup, we try to find a sequence of subscription fees and nuber of products p n k n optial in the liiting regie as n. We shall begin by providing a noinal solution, and then constructing a refineent that accounts for the variability. The noinal solution optiizes the objective function on the On scale by appealing to the law of large nubers, while use of the central liit theore iplies that the refineent is on the O n scale. The O n scale refineent can be understood by considering a newsvendor odel with deand arising fro n independent, identically distributed sources. In this case, the optial stock level coprises of the ean of the total deand, which is On, and a safety stock against variability, which is soe nuber of standard deviations of the total deand and is O n. A sequence p n k n is said to be a noinal solution to the proble if for all sequences p n k n li inf n n p n k n n li sup n n p n k n n We denote the corresponding noinal profit by, i.e., = li inf n n p n k n /n. The noinal solution is a deterinistic solution, and thus can be refined by studying the variability in the syste. To do so, we introduce the notion of asyptotic optiality at the O n scale. We expect the actual profit in the syste to be less than that expected in the noinal setting. This otivates us to look at the loss in profits due to variability and refine our solution to iniize these losses. That is, for any sequence p n k n, denoting n p n k n n n p n k n /n as the O n scale loss in profit, which is the difference between the noinal profit and that obtained while using the rule p n k n, we define an asyptotically optial sequence as follows. Definition. A sequence p n k n is said to be asyptotically optial if the scaled loss in profit is iniized, that is, for any sequence p n k n li sup n n p n k n li inf n n p n k n It is clear that for a sequence to be asyptotically optial, it ust be a noinal solution.
6 Randhawa and Kuar: Usage Restriction and Subscription Services 434 Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at Noinal Solution Henceforth, we shall assue that =, with the general retrial case discussed in 5. We first loosely illustrate how the noinal solution can be characterized by the solution to a static, deterinistic, optiization proble. Then, we prove a precise stateent characterizing noinal solutions in Proposition. Scaling the objective function by n and letting n grow without bound should result in a deterinistic proble, and, hence, one expects that its solution should correspond to one where =. Further, for =, the capacity level ust equal the offered load in the syste, which is defined as the ean nuber of products that will be in use in steady-state when the syste has infinite capacity. We expect the probability that a subscriber will be using a product in steady-state is / +. Hence, as the nuber of subscribers who join this syste is n Fp n /, where / +, the offered load is n/ + Fp n /. This iplies that if the scaled price and capacity levels converge as p n k n /n p k, then the capacity level k ust equal / + F p/. Thus, we obtain the following static optiization proble, which we call the noinal proble. ax p k 2 + p F s.t. k = p kc F + p The assuptions on F ensure an interior solution. Thus, we use the necessary condition for optiality, that is, the first derivative of the objective function ust be zero, to establish that the optial subscription fee p satisfies F p = f p p + c 2 and the optial capacity level is given by k = / + Fp/. The following result akes the loose reasoning above precise. Proposition. For any sequence p n k n to be a noinal solution, n, p n p, and k n /n k as n. Before scaling, is O/ n, and hence tends to zero as n grows without bound. We shall refine the noinal solution using the right order of in Refining the Noinal Solution We begin by defining the capacity ibalance in the syste as the difference between the nuber of products and the offered load scaled by n. As derived earlier, this offered load is n/ + Fp n /. Thus, for any sequence p n k n, the capacity ibalance is given by p n 3 n = kn n F + The following lea presents a decay condition on the asyptotically optial ibalance. Lea 2. For any asyptotically optial sequence p n k n, li sup n n n <. We now focus on constructing an asyptotically optial sequence p n k n with the associated equilibriu denial probability n. Motivated by the above lea, we restrict our search to n such that li n n n =, where. We choose p n = p + n and k n = k + n n such that p n k n n satisfy 3 and n n. In fact, we shall assue without loss of generality that n n and n n. We can rewrite 3 using the Taylor series expansion of Fp n / about p/ to obtain n = n p f + n + o n 4 and hence = f p + We are now ready to asyptotically characterize the denial probability for our syste. Proposition 2. For a sequence p n k n such that li n n n exists, n n given by [ + p = Fp/ h +f p ] + + Fp/ 5 where h is the hazard rate function associated with the standard noral distribution. Note that in the liit the denial probability solves a fixed-point relation and is a function of only the liiting capacity ibalance. In the following result, we show that the profit is also a function of the capacity ibalance alone, and, hence, pricing and capacity sizing are equivalent levers for the fir at the O n scale.
7 Randhawa and Kuar: Usage Restriction and Subscription Services Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS 435 INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at Proposition 3. For a sequence p n k n such that p n = p + n, k n = k + n n and li n n n exists, the loss in profit function satisfies li n p n k n = c + p 2 f n p Therefore, any sequence p n k n that has the sae liiting ibalance has the sae asyptotic loss in profit. This result iplies that the asyptotically optial sequence is characterized by the value of that iniizes li n n p n k n. Hence, the optial asyptotic capacity ibalance solves in p c + p 2 f / [ + p Fp/ h + f s.t. = ] p + + Fp/ Denoting z h c Fp//p 2 + f p/ pf p/c, the first-order conditions iply that the optial denial probability and capacity ibalance are + = Fp/ hz = z Fp/ + f p p + Checking first-order conditions establishes optiality because h is a convex function cf. Zeltyn and Mandelbau 25, p. 2. We now characterize all asyptotically optial sequences in the following result. Proposition 4. The sequence p + n k + n n, where n /f p/ + n = / n + o/ n, is asyptotically optial. The equivalence of corrections in price and capacity levels suggests that the fir could choose to keep one paraeter at the noinal level and correct the other. This is particularly useful for a DVD rental fir because one expects aking changes to capacity levels to be easier than changing prices. Hence, in this scenario, the fir can set prices at the noinal level 6 and ake corrections in the capacity level in accordance with the equilibriu to achieve the asyptotically optial solution. However, if changing prices is easier than changing capacity, the reverse can be done. Corollary. The sequences p k + / nn and p + / n/f p/ kn are asyptotically optial. This subscriber odel is a natural odel of custoer behavior in a subscription setting under a concurrent contractually enforced usage restriction. When confronted with a different contract without such a restriction, one expects the rational users to odify their behavior. For exaple, when operating in a pay-per-use setting, one would expect the custoers to reduce their frequency of requests because they can have ore than one concurrent rental. This change in behavior ust be factored in before we can set ourselves up to copare subscription and pay-peruse options. This is the subject of the next section. 3. Pay Per Use 3.. A Model of Custoer Behavior Consider the DVD rental world again. Here, Netflix offers solely subscription-based service, while Blockbuster priarily uses pay-per-use pricing. In turn, Blockbuster s custoers ay request additional products, even though they ight already be renting a product, and thus can rent any products concurrently, while Netflix iposes a restriction on the custoer usage, i.e., custoers are only allowed to rent a certain nuber of DVDs at a tie. In this setting, it sees reasonable to assue that custoers have an intrinsic need or desire to watch a certain nuber of DVDs per tie unit, and that they would odify their request rates to achieve this rate depending on the option they sign up for. In this section, our goal is to build a coon custoer odel as the basis of understanding how custoer behavior changes with different options. We will then analyze the fir s decision of offering either a usage-restricting subscription option or a pay-per-use option. We begin by defining our arket. We shall assue that there are n potential custoers, each characterized by. A benefit derived per usage/rental, S, that is rando and distributed according to F.
8 Randhawa and Kuar: Usage Restriction and Subscription Services 436 Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at 2. A usage tie for the rental of each product, which we assue to be exponentially distributed with ean /. 3. A noinal desire of renting x products per tie unit. For exaple, each custoer wants to rent x products per onth. We assue that this desire is constant across custoers. 4. A rando attept/request process that allows the custoer to realize the desired rental level described above. We shall assue the interarrival ties of the requests are exponentially distributed. Any request denied is considered lost, i.e., there are no retrials. Before analyzing the pay-per-use option in detail, we establish that this odel is consistent with the odel proposed in 2. for the subscription option when usage is restricted Subscription Option: Reduction to Subscriber Model of 2 We study the subscription option that restricts a subscriber to renting only one product at a tie. Any request generated while the subscriber is still renting the product is de facto denied, by the ters of the contract. To copensate for this, rational users will adjust their request rate. Let x be the rate at which each subscriber generates requests in this setting. This rate depends on the noinal desire x described; we shall elaborate on this shortly. A straightforward application of the Markovian property of the exponential distribution leads to the following result. Lea 3. Subscribers generating requests according to a renewal process with exponential interarrival ties with ean /x, where requests generated while a product is being rented are denied, can equivalently be odeled according to the on-off odel in 2 of the paper. Hence, the subscribers defined here are exactly those in 2 of the paper, and we can directly use the analysis perfored in that section. Note that as each subscriber wishes to aintain a constant usage level, he or she chooses a request rate x so that the noinal usage under no capacity constraints equals x, i.e., x = x 7 x + For a given x, we can copute the corresponding rate of turning On,, and then perfor the analysis in 2 to copute the optial prices the fir ust set and the corresponding nuber of products. We shall now analyze the pay-per-use option Pay-Per-Use Option Under the pay-per-use option, the fir allows the custoer to rent unliited ultiple products, i.e., a custoer request is always accepted subject to available capacity. The custoer pays per usage in return. Let x denote the rate at which each custoer generates requests in this setting. Further, because custoers wish to aintain a constant usage level, they choose a request rate x so that the noinal usage under no capacity constraints equals x, i.e., x = x 8 We can iic the arguents in 2 to solve the fir s optiization proble when offering a payper-use option. Readers ay choose to skip this section and jup directly to the coparison of the two options in 4. For convenience, we shall drop the dependence of x on x. We now build a deand odel for the pay-per-use custoers. As in 2, let denote the steady-state denial probability. When the syste anager sets a pay-per-use price of p p per use, each custoer joins the syste only if the expected value obtained by joining the syste, which is the probability of obtaining the product ties the value derived fro using the product, S p p is positive, i.e., S>p p. Thus, at a price per usage p p, N p p p = ns > p p custoers join the option. Note that the nuber of pay-per-use custoers joining the syste does not depend on the denial probability. This is because we have iposed no costs on custoers for checking the availability of the products; i.e., if it was costly for custoers to arrive at the fir, there would be an iplicit cost associated with the denial and this would affect the deand. Although the deand is independent of the quality of service, the actual requests accepted still depend on it, and so do the fir s profits. Noting that custoers pay for the products per usage, it is ore convenient to talk of the total arrival rate of custoers, as opposed to the nuber of custoers joining the syste. Note that the custoers requests arrive according to a Poisson process
9 Randhawa and Kuar: Usage Restriction and Subscription Services Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS 437 INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at with the total arrival rate given as a function of the pay-per-use price by p p = N p p p = n Fp p The optiization proble of the syste anager can be stated as ax p p k p 2 + p p p p k p c s.t. = dp p k p where dp p k p is the denial probability as a function of the arrival rate of the custoers and the nuber of products. As before, we shall use the superscript n to ake explicit the dependence of the various paraeters on n, i.e., pp n, kn p, n, and pp n denote the pay-per-use fee, the capacity level, the denial probability, and the rate of arrival of the pay-per-use custoers, respectively. The optiization proble can then be written as ax p n p kn p 2 + n p n p kn p pn p n n p n p kn p c s.t. n = d n p n p kn p With this setup, we try to find a sequence of prices and nuber of products pp n kn p optial in the liiting regie as n Noinal Solution. We begin by solving the noinal proble for this syste analogous to given by ax p p k p 2 + p p Fp p k p c s.t. k p = Fp p It is easy to verify using the first-order conditions that the noinal price p p satisfies the relation Fp p = p p c fp p and the noinal capacity level k p = / Fp p.we then have the following result. Proposition 5. Asequence pp nkn p is a noinal solution if n and pp nkn p /n p p k p as n Refining the Noinal Solution. As before, we define the capacity ibalance in the syste as the difference between the nuber of products and the 9 offered load in the absence of denials scaled by n. Thus, for any sequence pp nkn p, the capacity ibalance is given by n = kn p n Fp n p We now focus on constructing an asyptotically optial sequence pp nkn p with the associated denial probability n. An analog to Lea 2 holds in this setting as well, which otivates us to restrict our search to n such that li n n n =, where. We choose pp n = p p + n p and kn p = k p + n pn such that pp nkn p n satisfy and n p n p. Further, we rewrite using the Taylor series expansion of Fpp n about p p to obtain n = n p fp p + n p + on p The asyptotic denial probability can now be characterized as follows. Proposition 6. For a sequence pp nkn p such that li n n n exists, n n given by = Fp p h 2 Fp p We now show that as in the case of the subscription option, profit is a function of the capacity ibalance alone, and hence pricing and capacity sizing are equivalent levers for the fir at the O n scale. Proposition 7. For a sequence p n p kn p such that pn p = p p + n p, kn p = k + n p n and li n n n exists, the loss in profit function satisfies li n p n p n kn p = c + Fp p p p This result iplies that the asyptotically optial sequence is characterized by the value of that iniizes li n n pp nkn p. Hence, the optial asyptotic capacity ibalance solves in c + Fp p p p s.t. = Fp p h Fp p As before, we use the first-order conditions to characterize the optial solution. Hence, denoting z h c Fp p / Fp p p p, the optial denial probability
10 Randhawa and Kuar: Usage Restriction and Subscription Services 438 Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at and capacity ibalance are given by = Fp p hz Fp p = z 3 We now characterize all asyptotically optial sequences in the following result. Proposition 8. The sequence p p + n p k p + n p n, where n p /f p p + n p = / n + o/ n, is asyptotically optial. Having solved the fir s optiization proble for both the subscription and pay-per-use options, we now turn our attention to their coparison. We will also copare easures like social welfare and consuer surplus in these options. 4. Coparing the Two Options We shall copare the subscription and pay-per-use options in the asyptotic regie of large arkets. Recall that custoers choose their request rates in each scenario to achieve their noinal usage level. Thus, coparing 7 and 8, for a eaningful coparison of the two settings, we ust have = 4 + We begin by coparing the noinal profits in the two options. Because the two settings are statistically equivalent at the noinal level, we expect no difference in their noinal profits. Indeed, coparing 2 and Proposition with 9 and Proposition 5, we obtain the following result. Proposition 9. The noinal profit in the subscription option equals that in the pay-per-use option. Now, we investigate how each option handles the variability in the syste anifested at the O n level. The question of each option s ability to handle variability is oot if one option always has better quality of service at every load. We begin by showing that one option does not doinate the other on quality of service. Given this, we shall copare the profits in the two options. We shall then perfor a finer analysis to ascertain which option results in higher profits. This analysis consists of coparing the Figure Scaled denial probability Coparison of Quality of Service in the Two Options Subscription option Pay-per-use option Capacity ibalance asyptotic loss in profit at the O n level in both options. Finally, we shall copare the consuer surplus and social welfare in these options. 4.. Quality of Service: Neither Option Doinates We copute the asyptotic quality-of-service levels using Propositions 2 and 6 for both options at different capacity ibalance levels and copare the in Figure. We set Fx= e x for x, = 5, and = 5 so that = for these coputations. 4 The figure illustrates that neither option doinates the other. At low capacity ibalance levels, the payper-use option offers a higher quality of service, while the subscription option is better at higher ibalance levels. Figure can be explained loosely as follows. As the nuber of subscribers who are on increases, the nuber of subscribers attepting to obtain products decreases. Consequently, at high capacity levels, the denial probability seen by the subscribers is saller than that seen by the pay-per-use custoer strea whose attept rate is independent of the state of the syste. Siilarly, the denial probability can be explained to be higher in the subscription option at low capacity levels Fir s Profits: Subscription Option Always Doinates We now focus on establishing that the fir s profits are higher in the subscription option. We will noralize = for convenience. We shall copare the asyptotic loss in profit in the two options. Noting that h is a function that does not have a closed
11 Randhawa and Kuar: Usage Restriction and Subscription Services Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS 439 INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at for, proving such a clai in coplete generality is quite difficult. Instead, we prove this result for a specific deand function and will provide nuerical results for a few others. Denoting the optial capacity ibalances in the subscription and pay-per-use options by s and p, respectively, we have the following result. Proposition. For the exponential deand function with Fx= e x for x>, for any constant >, the fir s asyptotic loss in profit is higher in the pay-peruse option. Further, the capacity ibalance is lower in the subscription option, s < p Coparison of Fir s Profits: Nuerical Experients. We now provide nuerical results deonstrating that the profits are higher when the fir offers a subscription option for two deand functions: a linear deand function and a Pareto deand function with constant price elasticity.. Linear deand: To obtain a linear deand function, we choose a unifor distribution for the valuations, i.e., Fx= x for x. We nuerically copute the loss in profit for a large nuber of paraeter values. In particular, we vary both and the cost per product c, while choosing so that =. In each instance, we observed that the loss in profit as well as the capacity ibalance were lower in the subscription option. As a specific exaple, Table copares the optial ibalances and the loss in profits for the two options as the cost per product varies with = 2. We observe that the subscription option has a lower capacity ibalance and realizes a lower loss in profit. 2. Iso-elastic deand: We choose a Pareto distribution for the valuations, i.e., Fx= x 2 for x, which has a price elasticity of 2. Table 2 copares the optial ibalances and the loss in profits for Table Subscription vs. Pay Per Use: Linear Deand Subscription Pay per use c s Loss in profit p Loss in profit Table 2 Subscription vs. Pay Per Use: Pareto Deand Subscription Pay per use s Loss in profit p Loss in profit the two options as varies with the cost per product set at c =. We again choose so that =. We observe that the subscription option has a lower capacity ibalance and realizes a lower loss in profit Coparison of Consuer Surplus and Social Welfare We shall construct exaples for the exponential deand function to show that the consuer surplus and social welfare, which is the su of the consuer surplus and the fir s profit, can be higher or lower in the subscription option as copared to the pay-peruse option. We shall begin by coputing the consuer surplus in the two options. We shall noralize = as before and denote p p s = p p, where p s and p p denote the noinal prices in the subscription and pay-per-use options, respectively. For a syste with n subscribers, the consuer surplus can be written out as CS n s = n Fpdp p n + n / = e p n n n s + n n p p e p n Siilarly, we can write out the consuer surplus for the pay-per-use custoers as Hence, CS n p = n n = e p CSs n li n n p n Fpdp n n n p + n n e p n = li CSp n n n = e p CS Because the fir s profit can be copletely characterized by the value of the capacity ibalance, using Propositions 4 and 8, the fir can choose to
12 Randhawa and Kuar: Usage Restriction and Subscription Services 44 Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS Table 3 Coparison of Consuer Surplus and Social Welfare Table 4 Request Rates Used in the Experients Subscription Pay per use Maxiu products rented out INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at c Loss in CS Loss in SW Loss in CS Loss in SW set any value of n i, i = sp, and adjust the value of n to obtain the sae profit. Hence, denoting the loss in surplus due to variability fro the noinal value as ĈS i = li n ncs CS n i /n, we see that the fir can decrease ĈS i in an unbounded fashion by increasing i arbitrarily. Hence, the custoers obtain a higher consuer surplus when the fir sets lower prices. This result can be explained in a different anner. Note that if the fir sets any particular price, it can adjust the capacity level so that the capacity ibalance, and hence the quality of service, reains unchanged. Thus, reducing prices while aintaining the sae quality of service increases the consuer surplus. To copare the consuer surplus and social welfare in the two options, we shall fix the price levels at the noinal values, i.e., set n i = for i = sp. Table 3 displays the losses in consuer surplus and social welfare for different values of the paraeter c at fixed = 2, prices set at the noinal values, and =. We observe that for low values of c, the loss in these easures are lower in the subscription option, while this is reversed for high values of c Multiple but Liited Concurrent Rentals Thus far, we have considered the two extree cases of allowing just one or unliited concurrent rentals. In reality, Netflix allows subscribers to rent between one and four DVDs concurrently. So we consider what happens if the fir allows subscribers to hold a liited nuber of concurrent rentals, in which the liit is larger than one. If r concurrent rentals are allowed aong n custoers, we need to keep track of an r state Markov chain, and asyptotically, we expect to get an r diensional diffusion process. This is sufficiently coplicated to ake analysis difficult: We choose to perfor a nuerical study to quantify the effect Experient no Pay per use of easing the usage restriction in this anner. As r increases, subscribers would decrease the rate at which they generate requests to eet their noinal desire x. So their noinal usage rate and the consequent noinal profits realized by the fir will not change. Coparing asyptotic profits reduces to coparing asyptotic loss in profit. Using siulations, we copute the optial loss in profit for the cases where the subscribers are allowed to rent up to five products concurrently. The siulations are perfored for a syste with n = potential subscribers, with the ean rental duration of / = 5. We chose to be 2 4 and for r =, and coputed for the other values of r accordingly. Table 4 lists the values of the request rates used in the different settings to ensure the sae noinal usage rate. We use an exponential deand function, i.e., Fx= e x, x and use siulations to estiate the profit for 5 different price values and then pick the optial profit. Appealing to the equivalence of pricing and capacity sizing, we stick with the noinal capacity level. For each price, one needs to estiate the denial probability in an iterative anner because the nuber of subscribers depends on the denial probability and vice versa. We do this by beginning with a denial probability of and then perforing a run of tie units to estiate the denial probability in this syste. We repeat this run 2 ties to estiate with an observed axiu ratio of standard deviation to the ean of %. We then update our estiate of the denial probability and repeat the siulation until there is a difference of or in the nuber of people joining the syste under the old and new denial probability values. The results of this study are illustrated in Table 5. Each nuber displayed in the table corresponds to the loss in profit fro the noinal aount given by the solution to scaled by n. We observe that the scaled loss in profit is increasing as the nuber of concurrent rentals allowed increases.
13 Randhawa and Kuar: Usage Restriction and Subscription Services Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS 44 INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at Table 5 Loss in Profit When Multiple Products Can Be Rented at a Tie Maxiu products rented out Experient no Pay per use Thus, the siulations see to suggest that restricting custoer usage in the subscription option allows the fir to generate higher profits, and the ore restrictive the usage in the subscription option, the higher the profits Iportance of Usage Restriction We highlight that the operational benefit of subscription lies in the anner in which the custoer behavior is restricted by the contract, rather than in the pricing echanis iplied by the contract. To deonstrate this, we shall look at a arket with n potential custoers with a rando valuation S per service. Each custoer s behavior is governed by an underlying Markov chain as in 2 under any pricing contract. Let us first consider the contract wherein the custoers are charged per-unit tie; we shall refer to this as contract A this is the setting in 2. Let denote the rate at which services are copleted as a function of the quality of service ; = / with as defined in Lea. If a fee p A per-unit tie is charged, then a potential subscriber will join the syste if the benefit rate exceeds the cost rate, i.e., S>p A. Thus, the deand function under this contract can be written as N A p A = n Fp A /. Now, let us consider the per-usage pricing contract call it contract B. Again, potential users sign up for this contract if the benefit rate exceeds the cost rate, i.e., they join if S>p B. Thus, the deand function under this contract is given by N B p B = n Fp B. The following result proves that contracts A and B generate the sae aount of profits for the fir. Lea 4. For any arket size n, contracts A and B are revenue equivalent. That is, any profit generated by the fir using one contract can be replicated using the other contract, albeit at a different price. 5. Discussion This paper provides a technique for coputing prices and capacity levels that approxiately axiize a large rental fir s profit. This technique is used to characterize the asyptotically optial solution in subscription and pay-per-use environents, and it is shown that the fir s profit is higher in the subscription setting. This paper considers a special case of the subscriber s retrial rate, i.e., =. The general case is quite difficult to analyze, even asyptotically see 3 of Randhawa and Kuar 27. We shall use nuerical experients to perfor coparative statics on the retrial rate. In particular, as Table 6 deonstrates, for a linear deand function with a cost per product c =, increasing the retrial rate leads to an increase in the fir s profits. Although increasing the retrial rate increases the denial probability, it also enables the subscribers to obtain service quicker, and, hence, increasing the value they obtain by joining the option. It is this latter effect that doinates and allows the fir to extract higher profits. An analytical analysis of this phenoenon is a topic for future work. As the retrial rate increases, we expect the syste to begin to behave ore like a queueing syste, where, upon being denied service, people wait in a queue. Using siulations, we estiate the ratio of the denial rate to the retrial rate and copare it to the expected queue length in a queueing syste. Table 7 displays these estiates along with their 95% confidence intervals. Observe that this ratio is quite close to the expected queue length estiate. This akes a case for approxiating the econoic odel in a queueing syste via a loss odel or vice versa. In this paper we focused on the benefits of a subscription option as opposed to a pay-per-use option. However, a fir ight wish to offer both options in parallel. This case can be handled in a straightforward anner using the tools developed in this paper. Table 6 Fir s Asyptotic Loss in Profit with 95% Confidence Intervals for General Retrial Rates Retrial rate, ± ± ± ± ± ± ± 34 ± ± 8 38 ± ± 566 ± ± ± ± 2
14 Randhawa and Kuar: Usage Restriction and Subscription Services 442 Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at Table 7 Using Increasing Retrial Rates to Model a Queue n = = = Queue 833 ± 94 8 ± ± 6 8 ± ± ± ± ± 29, 6768 ± ± ± ± 455 In fact, we can show that the equivalence of pricing and capacity sizing carries through to this setting as well. As a concluding note, we coent on the distributional assuptions ade in this paper. Although we have ade Markovian assuptions, we contend that all the results for the subscription option hold even for general distributions of on and off ties, as long as the distributions have a density. In fact, these results depend on just the eans of these distributions. Cohen 957 and Randhawa and Kuar 27 proves that the invariant distribution for the nuber of products in use depends only on the first oent of the on and off ties. Regarding the pay-per-use option, we cannot do away with the Poisson arrival process assuption, but the service ties can have an underlying general distribution. This can be proved by using the dependence of the invariant distribution of an M/G/k/k syste on just the first oent of the service distribution see Gross and Harris 998, pp Appendix Before ebarking on the proof of the results in this paper, we shall state soe asyptotic results, proved in Randhawa 26, that will be useful. A.. Asyptotic Results Consider a sequence of systes with the nth syste consisting of an + b n n > subscribers, where a> and b n such that b n b. Let Q n t represent the nuber of products in use at tie t in the syste when there are n subscribers, i.e., Q n t = an+b n n i= subscriber i is on at tie t.we consider capacity levels of the for k n = kn+ n n for soe k + and n such that n. Define q n = Q n /n, the centered and scaled process Q n t = Qn t k n n and q = in k / + a. We then have the following asyptotic results for this syste Lea 2 in Appendix C of Randhawa 26. Lea 5. If q n q a.s., then. q n q. 2. If k = / + a and Q n Q, then Q n Q, where Q is a reflected affine-drift diffusion process with an upper reflecting barrier at. That is, t Qt = Q + Qs + + 2aBt Yt + b ds where B is a standard Brownian otion and Y is the nonnegative, nondecreasing process such that t Qu dy u = and Qt, t and Y =. 3. The invariant distribution of Q n, n, where is the unique invariant distribution of the diffusion process Q. Further, Ɛ nq n Ɛ Q. 4. The density corresponding to is given by ˆpx = exp + x + 2 2a + b [ exp + z + 2 dz] 2a + b x Suppose instead of the subscribers, there is an exogenous strea of custoers who arrive according to a Poisson process with rate p = n + 2 n n, where > and n 2 such that n 2 2. Let Q D be the process that denotes the nuber of products in use by the exogenous custoers. Define q n and Q n as before and q in k /. We then have the following asyptotic results for this syste Lea 2 in Appendix C of Randhawa 26. Lea 6. If q n q a.s., then. q n q. 2. If k = / and Q n Q, then Q n Q, where Q is a reflected affine-drift diffusion process with an upper reflecting barrier at. That is, Qt = Q + 2 t t Qs + ds + 2 Bt Yt where B is a standard Brownian otion and Y is the nonnegative, nondecreasing process such that t Qu dy u = and Qt, t and Y =. 3. The invariant distribution of Q n, n, where is the unique invariant distribution of the diffusion process Q. Further, Ɛ nq n Ɛ Q. 4. The density corresponding to is given by ˆpx = exp x [ exp z dz] x 2 Ared with Leas 5 and 6, we are now ready to prove the results of this paper.
15 Randhawa and Kuar: Usage Restriction and Subscription Services Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS 443 INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at A.2. Proofs A.2.. Proofs of Results in 2. We shall begin by deriving an expression for the denial probability for the general retrial case that will be quite useful in establishing Lea. Let Q t and Q 2 t denote the nuber of subscribers in the on and hold states, respectively, and be the invariant distribution associated with the Markov process Q = Q Q 2. Further, we shall use the notation Ɛ Q i ƐQ i t Q i = X for i = 2 and t, where X is a rando variable distributed according to. Lea 7. For a syste with n subscribers with associated steady-state loss probability, we have = n + Ɛ Q + Ɛ Q 2 n Ɛ Q + Ɛ Q 2 Proof. We can write Q t, for any t>, as follows t Q t = Q + A n Q s Q 2 s ds t t + R Q 2 s ds D Q s ds Yt a.s. 5 where A, R, and D are three independent Poisson processes with unit rate, corresponding to the attepts by subscribers in the off state, retrials by subscribers in the hold state and service copletions by subscribers in the on state, and Y = A A R n Q s Q 2 s ds + R Q s<k n Q s Q 2 s ds Q s<kq 2 s ds Q 2 s ds counts the nuber of denied attepts. Using renewal theory arguents see Proposition 3.3. in Ross 996, we obtain A t li n Q s Q 2 s ds t t Siilarly, we get = n li t t t Q s ds li t t t Q 2 s ds = n Ɛ Q Ɛ Q 2 6 R t li 2s ds = Ɛ t t Q 2 7 D t li s ds = Ɛ t t Q 8 Using 6 8 in 5, we obtain Yt li = n + Ɛ t t Q + Ɛ Q 2 Thus, we obtain Yt = li t A t n Q s Q 2 s ds + R t Q 2s ds = n + Ɛ Q + Ɛ Q 2 n Ɛ Q + Ɛ Q 2 which proves the clai. This result iediately iplies the following. Corollary 2. For a syste with n subscribers and retrial rate =, the associated loss probability is given by = n + Ɛ Q n Ɛ Q where Qt denotes the nuber of subscribers in on state at tie t and is the invariant distribution of Q. Proof of Lea. The ean tie between successful attepts to obtain a product in steady state for a subscriber can be written as t li t No. of successful attepts per subscriber by t nt = li t A t Q s ds = n Ɛ Q 9 We shall now obtain a relation between Ɛ Q and Ɛ Q 2, which will allow us to copute the ean tie between successful attepts as a function of,,, and. For the case k>, we shall first copute the liiting fraction of tie spent by any subscriber in the off state relative to the on state, i.e., Tie spent by subscriber i in off state by t li t Tie spent by subscriber i in on state by t Let T off t and T on t denote the ties spent by subscriber i in the off and on states, respectively. Let on j j off be the sequence of ties spent in the on and off states, respectively, by this subscriber; i.e. on j and j off are sequences of independent, exponentially distributed rando variables with eans / and /, respectively. Let St denote the nuber of services copleted by subscriber i by tie t. Then, we have the following li t St j= j off St j= j on +on St+ St T off t li t T on t li t j= j off +St+ off St j= j on
16 Randhawa and Kuar: Usage Restriction and Subscription Services 444 Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at As each visit to the hold state is for a finite aount of tie, St as t. Thus, we obtain T off t li t T on t = Using this result, we can copute Ɛ Q = + n Ɛ Q 2 2 Observe that this relation holds for the case k = trivially. The result now follows by cobining 2 and Lea 7 in 9. Henceforth, all the proofs shall be for the case = where the hold and off states are indistinguishable. Thus, the syste will be described by Q n, where Q n t is the nuber of subscribers in the on state at tie t when there are a total of n potential subscribers. Let n be the invariant distribution of Q n. Proof of Proposition. Consider any sequence p n k n. Choose a subsequence n such that n, p n p, and k n /n k as n. For this scenario, n p n k n li = p F n n + p kc where use of Corollary 2 iplies F + = F p + + k p k Consider the following optiization proble ax p F p k 2 + p kc + F + p + k s.t. = F + p k 2 We will show that the above proble is axiized at p k and has =, which proves the clai. Fix the nuber of products at k. Reorganizing ters in the expression for in 2, we obtain F + p = k + The optiization proble can thus be rewritten as ax k F k + kc It is easy to see that the above is axiized at =, and hence 2 has the sae solution as, which is p k. We adapt the proof technique in Plabeck and Ward 25 to prove the following result. Proof of Lea 2. We shall prove that for any sequence p n k n with the associated n if li sup n n n =, then li sup n n p n k n = and hence p n k n cannot be asyptotically optial. Given p n k n we can write n p n k n = n p n F n + p n kn n c = p n p n n p F p n n F F n + a = n p n p n n F kn n c + p np n n F F p n n c + p n n F F n + p n + kn c n n + p n p n 22 where the inequality follows because the second ter in a is upper bounded by the solution to the proble p ax p F kc p k 2 + s.t. k = F + p + n which is n c at p = p. We now divide the proof based on two cases. Case : li sup n n n =. Because n, this iplies that the second ter in the lower bound for n p n k n in 22 is nonnegative. Hence, n p n k n n n c and hence li sup n n p n k n =. Case 2: li inf n n n =. Consider a subsequence n such that li n n n =. Using 3, we split each capacity ibalance n into corrections in prices and nuber of products, i.e., set p n = p + n, and k n = k + n n, such that n = n p f + n + o n Using Corollary 2, Taylor s expansion for N n p n n / n = F/ n + /p n about p/, and because Q n k + n n, we have the following relation for n large enough n n F p p n f + o n + o n f p p n / N n p n n Ɛ n Qn
17 Randhawa and Kuar: Usage Restriction and Subscription Services Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS 445 INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at + n F + p + n N n p n n Ɛ n Qn [ p n p f f p n + o n + o n ] [ p + n F ] F + p + o [ ] p = + n f p n + o n [ p F + o] 23 where the second inequality follows as li n p n = p, li n k n /n = k, and li Ɛ n n Qn /n = F + p = k and o represents a function that converges to as n increases without bound. We can siplify 23 to obtain n p p F + f p + o n / n + o n + 24 Because li n n n =, the above iplies li n n = n Using Taylor s expansion in 22, and then using 24 we get li n p n k n n li n n p n c + p + n f li n n c n b = li n n + p + p 2 f n fp/ F p p p n + f + on p p + On + o n / n + o c 2 + where b follows by using 2. Again referring to 2, we ust have p c/ + >. Thus, we obtain li n n p n k n =, which iplies li sup n p n k n = n Proof of Proposition 2. We shall first copute the asyptotic denial probability when the nuber of subscribers joining the syste is independent of the denial probability. Once we have this liit, we shall incorporate the dependence to coplete the proof. Consider a sequence of systes with N n =an + b n n subscribers in the nth syste, where a> and b n such that b n b, and nuber of products given by k n = / + an + n for soe. We can then characterize the asyptotic scaled denial probability for this syste as follows. Lea 8. The scaled denial probability converges as follows. + li n n = n a h + + b a We now let the nuber of subscribers in the nth syste be N n p n n. To coplete the proof, we need to establish that n n converges. To see this convergence, note that N n p n n N n p n, and, hence, perforing a birthdeath chain analysis, we obtain n = dn n p n n k n dn n p n k n. A direct application of Lea 8 allows us to conclude that dn n p n k n n converges as both n n and n n converge. Hence, li sup n n n<. Further, as the relation 5 ust hold for any convergent subsequence of n n, the result follows. Proof of Lea 8. Arguing as in Corollary 2, we can show that n = N n + Ɛ nq n N n Ɛ nq n Hence, using Lea 5c, we have n + 2 n a Ɛ Q + + b Using Lea 5.3, we can copute Ɛ Q+ a + b = + h + + b a Hence, we have + li n n = n a h + b + a Proof of Proposition 3. We can express n p n k n as n p n k n = n p n F n + p n k n /nc = p n n c+ p 2 f n /+o/ n where the second equality follows by application of Taylor s expansion and using 4. Using = li n n n and Proposition 2, we obtain where satisfies 5. li n p n k n = c + p 2 f n p /
18 Randhawa and Kuar: Usage Restriction and Subscription Services 446 Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at A.2.2. Proofs of Results in 3. Proof of Lea 3. The product is always rented for an exponentially distributed aount of tie with ean /. Thus, we only need to show that the request process is identical in the two scenarios. If the custoer is not renting a product, then this equivalence trivially holds. Suppose the custoer is renting the product. Then, as requests generated before the product is returned are denied iediately, the next valid request will be the first request after returning the product. However, a straightforward application of the eoryless property of the exponential distribution iplies that this ust again be exponentially distributed with ean /x. Thus, the result follows. Proofs of Results in 3.2. The key difference between the subscription and pay-per-use options is in the coputation of the denial probability. For the pay-per-use option, the following analog of Lea 7 holds. Lea 9. For a sequence p n p kn with associated loss probability n, we have n = n pp n Ɛ nqn n pp n Proof. We can write the nuber of custoers using the product at tie t, Q n t, for any t>, as follows t Q n t = Q n + A n p n p t Q n s ds Y n t a.s. where A and D are two independent Poisson processes with unit rate, and Y n = A n p n p A n p n p Q n s<k n ds counts the nuber of denied attepts. The result now follows by continuing as in the proof of Lea 7. Using this result and Lea 6, all other results in this section can be proved in a anner siilar to those in 2 and are oitted for brevity. A.2.3. Proofs of Results in 4. Proof of Proposition. We shall use the qualifying subscript s for ters pertaining to the subscription option in 2. The noinal solution is given by p s = p p = / + c/ p. Using this, we characterize the optial capacity ibalances i and denial probabilities i, i = sp for the two options. 6 iplies that s = + e p hz where z = + e p s +e p ps /+. Then using the fact that for a standard noral distribution h x = h 2 x xhx, we can rewrite the optiality condition on z, h z = c/ + c in conjunction with 6 to obtain the following relation s 2 + p + e p + s s = c + c which iplies s = + ep s + 4c + p s c + e p p Perforing a siilar analysis for the pay-per-use option, we first observe that the optiality conditions are identical, i.e., h z = c/ + c. Hence, equating the corresponding z values for the two options, we obtain + e p s + e p p s = e p p + Using the value of s, this can be siplified to obtain p = 2 + p s 2 + p + p 2 + p 4c + p s c + e p Note that this iplies p > s as >. Further, we can derive an analog of 25 to write out p in ters of p to obtain p = ep p 2 + 4c p c e p which can be recast as a function of s alone. Hence, the denial probabilities and the optial capacity ibalance in the pay-per-use syste can be written as a function of s. This enables us to write out the difference in the loss of profits between the two options as P s = c p s + e p p p s p2 s c + e p s s The clai is equivalent to proving Ps >. Noting that coputing the value of s requires us to deal with the hazard rate function, instead, we will prove that P > for any. Via algebraic anipulations, we can show that P> and P= has no real roots. This along with the continuity of P copletes the proof. Proof of Lea 4. Suppose the fir sets a price p A in contract A and stocks k products. Let A be the corresponding quality of service. Then, the fir s profit rate is given by pa A = p A N A p A A ck = n F ck A Now, consider an alternate setting where the fir offers custoers contract B with a per-usage price p B = p A / A per usage. Suppose the fir stocks the sae nuber of products k. Then, in equilibriu, the nuber of custoers who join the syste, and thus the quality-of-service level, will be identical to that in contract A, and the fir s profit rate is given by B = p B B N B p B B ck = p B A N A p A A ck = A
19 Randhawa and Kuar: Usage Restriction and Subscription Services Manufacturing & Service Operations Manageent 3, pp , 28 INFORMS 447 INFORMS holds copyright to this article and distributed this copy as a courtesy to the authors. Additional inforation, including rights and perission policies, is available at The reverse arguent can be constructed by setting p A = p B B, where B is the quality of service under contract B. References Arony, M., C. Maglaras. 24. On custoer contact centers with a call-back option: Custoer decisions, routing rules and syste design. Oper.Res Barro, R., P. M. Roer Ski-lift pricing, with applications to labor and other arkets. Aer.Econo.Rev Borst, S., A. Mandelbau, M. I. Reian. 24. Diensioning large call centers. Oper.Res Buchanan, J. M An econoic theory of clubs. Econoica Cohen, J. W The generalized Engset forulae. Philips Teleco.Rev Cornes, R., T. Sandler The Theory of Externalities, Public Goods, and Club Goods. Cabridge University Press, Cabridge, UK. Gallego, G., G. van Ryzin Optial dynaic pricing of inventories with stochastic deand over finite horizons. Manageent Sci Gross, D., C. Harris Fundaentals of Queueing Theory. John Wiley & Sons, New York. Halfin, S., W. Whitt. 98. Heavy-traffic liits for queues with any exponential servers. Oper.Res Kleinrock, L Queueing Systes, Vol.I: Theory. John Wiley & Sons, New York. Maglaras, C., A. Zeevi. 23. Pricing and capacity sizing for systes with shared resources: Approxiate solutions and scaling relations. Manageent Sci Mendelson, H., S. Whang. 99. Optial incentive-copatible priority pricing for the M/M/ queue. Oper.Res Naor, P The regulation of queue sizes by levying tolls. Econoetrica Paschalidis, I., J. Tsitsiklis. 2. Congestion-dependent pricing of network services. IEEE/ACM Trans.Networking Plabeck, E. L., A. R. Ward. 26. Optial control of a high volue asseble-to-order syste. Math.Oper.Res Randhawa, R. S. 26. Operations with subscribers. Ph.D. thesis, Stanford University, Stanford, CA. Randhawa, R. S., S. Kuar. 27. Multi-server loss systes with subscribers. Working paper. Ross, S. M Stochastic Processes, 2nd ed. John Wiley & Sons, New York. Scotcher, S Two-tier pricing of shared facilities in a freeentry equilibriu. RAND J.Econo Whitt, W. 23. How ultiserver queues scale with growing congestion-dependent deand. Oper.Res Zeltyn, S., A. Mandelbau. 25. Call centers with ipatient custoers: Many-server asyptotics of the M/M/n + G queue. QUESTA 53/