A New Pricing Moel for Competitive Telecommunications Services Using Congestion Discounts N. Keon an G. Ananalingam Department of Systems Engineering University of Pennsylvania Philaelphia, PA 19104-6315 July 2000 Revise: June 2001 This research was partially fune by a grant from the National Science Founation NCR-9612781 an forms part of the PhD issertation of the first author. We than Roch Guerin, Nelson Dorny, an Yannis Korilis for constructive criticisms on earlier rafts of this paper. We also than the referees of this journal for insightful comments that improve the quality of the paper. We remain responsible for any remaining errors.
Abstract In this paper, we present a new moel for using prices as a way to shift traffic from congeste pea perios to non-pea perios in telecommunications networs, an hence balance the loa an also ensure that almost no one is turne away (or bloce ) from being provie service. We use the offer of congestion iscounts to customers who have the choice of accepting these rebates an returning uring a subsequent non-pea perio, or who can reject the offer an obtain services right away. We moel the problem as a mathematical program in which the networ provier tries to reuce cost by minimizing total iscounts offere but at the same time ensuring that almost all (i.e. 99%) of those requesting services are serve. We apply this moel to various scenarios an show that, except uring the situations of extreme persistence of high traffic volume, the scheme woul lea to zero blocing an an increase in revenue over the noniscounting case.
1 Introuction In this paper, we propose an analyze a pricing mechanism that coul be use in telecommunications networs for connection-oriente services with guarantee quality of service (QoS). Even with the expansion of high spee networs, new services such as vieo-on-eman, graphics an real-time auio an vieo, have emerge to consume the available banwith of existing networs uring pea perios. In the future, it is expecte that public an private networs with large banwiths will be available to consumers with guarantee QoS. Methos for allocating banwith among iverse users have become an important research topic. Using economic incentives to control users behavior, such as with pricing schemes, appears to be an effective approach to prouce fair an efficient use of resources. In aition, pricing is an effective means of controlling the flow into the networ, an thus managing congestion. A networ with guarantee QoS must use a call amission policy to ensure sufficient resources are available to each connection. This results in the rejection of some connection requests. For these types of networs, the proportion of bloce connection requests is an important measure of networ performance. In a systems sense, effective flow an congestion control throughout the networ coul minimize connection blocing. In this paper, we present an aaptive price iscounting scheme that coul be use as an efficient form of flow control. The basis of the iscounting scheme is the allocation of connections across several time perios base on iniviual users valuations of the service, an the provision of a choice to users who willingly accept iscounts (or rebates) for postponing service in place of immeiate service. We implement the scheme for a single service, an examine how the iscount offere can be aapte to eman fluctuations, an changes in the flow of connection requests to the networ. We are eveloping a pricing policy to cope with fluctuating eman over a relatively short perio such as a few hours. In connection-oriente networs, with guarantee QoS, only a fixe number of users for any service can be accommoate simultaneously, each with his or her own connection. Fluctuations in eman for connection-oriente services coul therefore become a critical problem unless vast capacity is installe. Having large capacity coul result in it being grossly uner-utilize in most perios. A metho to provie users an incentive to istribute eman evenly in the aggregate is therefore esirable. Time of ay price scheules may be aequate in certain marets for certain services. However, even in such cases eman forecasting errors may require a real-time control mechanism to avoi blocing a high percentage of connection requests in a busy perio. We present a pricing mechanism to cope with eman 1
fluctuations that cannot be easily preicte. We will implement the scheme uner uncertain information, requiring no avance nowlege of the eman curve or users preferences. 1.1 Our Moel When eman excees capacity at a particular price (i.e. when the networ is congeste), the service provier is face with one of two choices: Either bloc the new user, i.e. o not allow the user access to the networ, who will liely go to another service provier, or else provie an incentive for the user to return at a time when the networ is not congeste. In much of the telecommunications literature, access control an blocing is use as a mechanism for flow an congestion control. In our moel, we use iscounts (or rebates) as incentive prices to shift user eman to another perio an hence also provie congestion control. The main focus in this paper is to moel this process, an to erive optimal iscount rates. Obtaining iscount rates is complex because there is uncertainty about the level of eman an also as to what proportion of the users will accept the iscounts to shift eman. In our paper, we assume to be woring in a regime where the prevailing price for the service uring any perio of time is a parameter fixe outsie the eman regulation problem, at least in the short-term. This is consistent with the view that the user who is bloce can always obtain service at the prevailing price from another service provier. The inability of any particular service provier to change actual prices can be foun in situations of perfect competition, or in a situation where there is a monopolist who is price regulate. A perfect competition moel is appropriate where there are many competing service proviers, each offering an ientical service, with no barriers to entry, an users have the ability to change from one service provier to another. In this case, no eviation from the maretclearing price is possible. Even in cases where there are small numbers of competitors but no significant barriers to entry, a competitive price as escribe above may exist, base solely on the threat of new competitors entering the maret. At the other extreme of the competitive setting, a monopolist can observe effects of prices on aggregate eman since a monopolist controls the entire maret. Nonetheless, monopolists often must sell at a price set outsie their control, if regulators eem the monopolist s uncontrolle behavior to be harmful to the public. This was the case in longistance telephone service in the Unite States prior to eregulation. We expect to see an environment between these two extremes in future marets for consumer telecommunications services. Such an environment coul be escribe as monopolistic competition, if there are many firms, or an oligopolistic maret, where there are only a few large 2
competitors. In the former case, the goo or service is ifferentiate across competitors, resulting in some bran loyalty an allowing some marginal ifferentiation in prices among competitors for similar services. Nonetheless, such competitors o not have complete freeom to set prices, which are often set accoring to mareting consierations, an a broa ifferentiation in prices among proviers is not liely. Strategic behavior by other competitors, which may occur within monopolistic competition but is more liely in the case of an oligopoly, maes large eviations from the prices of competitors very unliely, especially if a service provier is a maret follower in a setting with few firms. 1.2 Relate Literature This paper eals with the issue of anticipating an avoiing pea traffic in telecommunications networs. Pea-loa pricing has been extensively stuie in both the economics an electricity pricing literature (See for example [22][27][28]). Our paper is relate to this literature but is part of an emerging literature specifically concerne with communications networs. Much of the wor on pricing for pacet-switche networs offering best-effort service has focuse on so-calle incentive compatible pricing. [14] [15]. It has also been shown through simulations that priority pricing improve networ performance when there was either single or multiple service classes [2]. It has also been shown by offering a number of routes, with a corresponing set of relative iscount rates, that a networ can elicit users to select routes for ata traffic accoring to the esire operating point of the networ provier [10]. The optimal iscount rates iscusse in [10] can be foun using an aaptive rule on-line, an are consistent with congestion pricing. Finally, in [4], the authors show that maring iniviual pacets at congeste resources allows the networ to estimate the shaow prices at iniviual resources in a networ, accoring to moels presente in [9]. Pricing has also been offere as a means of flow control for available bit rate service in ATM [3]. A ynamic pricing mechanism was propose in [16]. This aaptive pricing scheme assumes no nowlege of the eman function on the part of the networ or the iniviual users. The scheme oes not always converge, ue to errors in users expectations an errors in price estimates, but exponential smoothing of prices an eman estimates across perios ensures convergence for the M/M/1 queue. Dynamic priority pricing has also been stuie extensively by Gupta, et al. [6][7]. They have use an innovative approach base on ynamic programming to compare ynamic pricing with fixe prices. Given the computational intractability of this moel, they use simulations to perform their assignments. 3
Other authors have consiere the pricing problem in the context of networs offering Quality of Service (QoS) guarantees. The pricing ecision for a single lin point-to-point integrate services networ was formulate as a constraine optimal control problem an a threestage solution proceure was evelope to calculate a price scheule in [25]. A negotiation base framewor for allocating networ resources, using effective banwith as a base for pricing was propose in [8]. In another approach to aressing the QoS issue, some authors have propose offering networ resources such as banwith an buffer space irectly to users as part of a biing process in [12], an subject to announce prices in [21]. In such schemes, users coul achieve a esire QoS by irectly purchasing access to either reserve or share resources in the networ. While much of the pricing literature assumes users will ivulge their valuations of service in a biing process, it seems more realistic to assume that networ service proviers will serve the eman at a single price face by all users for the same service. However, there may be limite ability to set prices, as maret forces ictate prices in a competitive setting. In this paper, we propose a simple pricing scheme that coul be use only when networ congestion seems imminent. Users are offere iscounts (or rebates) to postpone their eman for service to a less congeste perio. Discounts can be ajuste, uner varying eman, to control the flow of connection requests to the networ. 1.3 Organization of the Paper The paper is organize as follows: In section 2, we explain the price iscount moel for shifting eman from high eman perios to low eman perios using iscounts offere to users. We inclue a moel for the response of users to the iscount offere by the service provier, which will enable us to estimate the proportion of users who actually accept the price iscount. In section 3, we present the service moel, i.e. the queuing moel at the switch which represents the ecision to either serve or not serve the connection request. We provie the necessary efinitions of blocing probability an the maximum arrival rate tolerable for any blocing specification, using the queuing moel. This also enables us to set capacity limits for the system. We erive the optimal iscounts in section 4. We present some examples, which emonstrate the effectiveness of the scheme in controlling the flow of requests to the service provier in section 5. Finally section 6 contains concluing remars, an the appenices inclue the proofs as well as etaile simulation results. 4
2 The Price Discounting Moel 2.1 Overview We consier a case where the price for a service is etermine outsie the problem an fixe. The service provier can only serve a certain number of connections at one time an woul prefer to shift some eman from the higher eman perios to lower eman perios in orer to limit the number of customers who are refuse service, i.e. bloce. In orer to shift eman, the service provier offers price iscounts (i.e. rebates) to the users if they will postpone the fulfillment of the service by one perio. Some users will accept the price iscounts an obtain their service in the next perio. Some users will reject the offere iscount an insist on being serve right away. Clearly the service provier woul prefer that all excess customers shift their eman to non-congeste perios so that none are bloce. However, just in case this oes not happen, the service provier must choose a reasonable number, calle the blocing probability for the proportion of requests for service that may be bloce. The provier wishes to satisfy this limit on blocing in every perio. In the next section, we will examine the service moel where we escribe the arrival process, queuing moel, an the service process in more etail. In this section, we will escribe the iscounting moel in more etail an provie an expression for the proportion of users who accept the price iscount or rebate. 2.2 Shifting Deman Between Perios In each perio, there is a maximum feasible rate at which requests arrive (see (17) in section 3.2), for which the probability of blocing requests is below a limit prescribe by the service provier. The eman shifting we wish to accomplish is illustrate in Figure 1. Arrival Rate of Requests λ * Deman Fluctuations Shifting of Requests Net Deman 1 2 3 4 5 6 Time Perio 1 2 3 4 5 6 Time Perio 1 2 3 4 5 6 Time Perio Figure 1. Deman shifting across perios. In Figure 1, we illustrate a case where users are ase to elay service over one perio. During perios 1 an 2, the rate of requests excees the feasible threshol of arrivals. Requests elaye from perios 1, 2, 3 an 4 are serve uring perios 2, 3, 4 an 5 respectively. The 5
elaye requests arriving from perios 1 an 2, necessitate further elaye requests from other users in perios 2 through 4. Figure 1 is a conceptual illustration an is not intene to be an accurate portrayal of the unerlying queuing process. We propose a strategy of offering price iscounts or rebates to users to shift some eman. The iscounts are offere as an incentive to users to elay consumption of the service. Only users who are sufficiently compensate by the iscounts for their inconvenience will elay their consumption. When eman excees the maximum feasible level, the service provier sells the service to iniviual users accoring to the following sequence: An iniviual user requests service at the price, which is nown publicly an fixe. A right to immeiate service is sol to the user. The sale is bining for both the user an the provier. When congestion is imminent, a iscount is offere to the iniviual user privately. The user chooses whether to relinquish the right to immeiate service in perio, in exchange for a right to service at any time in perio + 1, at the iscounte price. In our moel, we have assume that the users will be elaye at most one perio. Clearly if there is a very high arrival rate at any perio, then elaying the excess arrivals by only one perio will wor only if the arrival rate in the next few perios is not too great. There are two implicit assumptions about the one-perio elay. First, the competitive prices (see section 1.1) will tae into account traffic flow an will be large enough to prevent the persistence of excess eman. Secon, the efinition of perio will epen on whether or not there is pea traffic. Clearly, we are trying to move pea traffic to non-pea perios. The length of a perio is an implementation issue of our price iscounting strategy, an we will choose the length of a perio base on how long the pea lasts. The structure of the transaction is outline in Figure 2 below: 6
Ranom arrival of requests Arrival rate too high? No Yes Discount transaction Does user accept iscount? No Amit Call? No Bloc Request Yes Yes Delay user User receives service Delaye users from previous perio Figure 2. Congestion avoiance transaction using iscounts The iscounts are offere in avance of call amission so that the lielihoo of blocing is restricte. The goal of the system is to have a sufficient proportion of users accept the iscount offere so that the sum total of current arrival who reject the iscount an the returning users who have previously accepte iscounts is limite to a level where service can be provie with acceptably high probability, e.g. 99% liely service will be provie, even in a stochastic setting where the possibility of blocing always exists. 2.3 Iniviual User Optimization Behavior The iscount, in return for elaye use of the service, is offere to every user requesting service in a perio. Some users will accept the iscount an postpone their requests, an others will refuse the iscount an use the service immeiately. If too many users request immeiate service an the provier has to bloc some of the users, they have the recourse to go to alternative 7
service proviers. Clearly, users who contract for service at a particular price may not tae inly to being bloce, an may choose legal recourse. We assume that this oes not happen. We wish to investigate the proportion of users who will choose to elay consumption of the service. In fact, the amount of the price iscount has to be carefully chosen so that blocing is ept below a prescribe level at a minimum cost. We mae the following assumptions on the behavior of iniviual users: Users are unable to observe the overall level of eman, an there is no collusion among users, i.e. an iniviual user is uncertain whether a iscount will be offere. Users will arrive base on the total price charge for service for that perio alone; they o not see the price less the expecte iscount when they arrive. Note that this is similar to the papers by Menelson an co-authors who have moele arrival rate for computer an/or communications as a function of price [17][18][16]. The total price has two parts: competitive maret price plus the opportunity cost of being bloce. Whether or not there is elay of service is entirely uner the control of the user an epens on their willingness-to-pay (WTP). Thus, when they arrive they nee not be concerne about the possibility of a elay. The service provier is temporally ris neutral, an treats all revenues the same. The inconvenience ue to elay is ientical for all users. Users elaye in perio are free to scheule the return for any time in the perio + 1. At the prevailing price, the iniviual user solves a simple optimization problem: Max u( WTP p) u { 0, 1} (1) 1if the user requests service u = (2) 0 if the user oes not request service where, WTP = the iniviual user's willingness to pay for sevice p = the total price = sum of competitive pricean opportunity cost of being bloce The optimal solution for the iniviual at the first stage is clearly: u * 1if WTP p = (3) 0 if WTP < p All users who have requeste service an agree to pay the announce price have acquire a right to the service, with a positive value equal to the iniviual consumer s surplus. 8
V = WTP p (4) where, V = value of iniviual right to service The iscount offere is a bunle, comprise of a fee an a right to the service after one perio, offere in exchange for the user to relinquish his or her purchase right to service in the current time perio. The iscount an future right is weighe against the value of the right alreay purchase. In orer to ecie whether or not to accept the iscount, the user has to solve the following optimization problem: Max ( 1 u ) V + u ( β V ) { 0,1} u + (5) 1if the user accepts the iscount an the right to future service u = (6) 0 if the user exercises the right to service immeiately where, = the price iscount (i.e. rebate) offere β = iscount factor reflecting the iniviual's time preference for use of the service Substituting for V using (4), we get the iniviual user s optimal solution to be: 1if WTP p + * 1 β u = (7) 0 if WTP > p + 1 β 2.4 Aggregate User Behavior: Proportion of Discounts Accepte We now efine a cumulative istribution function for willingness-to-pay (WTP). The probability that an iniviual has a WTP less than the price, p, is given by: F WTP ( p) = P( WTP p) (8) namely: The function F WTP (p), must satisfy the simple properties of any istribution function, F WTP (p) 1 as p F WTP (p) 0, for all p F WTP (p) is monotonically increasing. 9
The service provier treats each user ientically an is intereste in the probability that a user will accept the iscount for postponing service, after purchasing the service at the public price. Theorem: Let the istribution function of a user s willingness-to-pay, WTP, be given by F WTP (p), an the time value of consumption between perios is given by β, 0 < β < 1, i.e. the value of consumption from one perio to the next ecreases from V to βv. Of the users who contract for service at price p, the proportion which will accept a iscount,, an a elay of service by one perio is given by: F P( A) = WTP p + F 1 β 1 F ( p) WTP Proof: See Appenix A.1. WTP ( p) (9) Lemma: For a uniform istribution of users WTP, on the interval [a,b], a p b, provie b 1, where b = 1/((1-β)(b-p)), the probability of an iniviual user rawn at ranom, accepting a iscount to elay service by one perio is proportional to the iscount offere: ( A) b P = (10) Proof: See Appenix A.1. 3 The Service Moel 3.1 Moeling the Connection Service Even if the eman, λ, an the istribution function for WTP are nown exactly an use to set price iscounts, the actual number of arrivals an proportion of users who accept the iscounts are stochastic an may be ifferent from the expecte values, which we use as the planning variables. Thus, some of the users who o not accept the iscounts may have to be bloce. In orer to state the optimization moel for the service provier, we nee to erive the constraint for the specification of the blocing probability. The blocing probability epens on the service moel that we esign for the service provier. In this section, we moel a generic telecommunications service, provie to a group of users through a single switch. The service offere is the use of a connection with guarantee quality of service (QoS). The service provier interacts with the users through the switch. Both pricing ecisions an whether or not to amit the 10
user into the networ is one at the switch. Once the user is amitte, the QoS is guarantee. Given available banwith, the number of connection that can be serve at any perio of time with guarantee QoS is limite, an given by c, as in Figure 3. Request Source Request Source. Request Source Switch Number of Connections c Remainer Remainer of of Networ Networ Incluing Incluing Connection Connection Enpoints Enpoints Figure 3. Service moel. We assume no scarcity of banwith between the switch an the users, or between the switch an the enpoints of the requeste connections, i.e. the problem is a bottlenec at the local switch. Note that either the switch or the connection enpoints in Figure 3 woul typically be referre to as servers. However, in the queuing moel below, the servers are the c connections allowe through the switch an not the connection enpoints, from which users may be retrieving ata. We have chosen this language to avoi confusion. This moel coul escribe an ISP or perhaps a wireless voice or ata service, where the bottlenec is the number of channels that can be supporte in a particular cell, given the available spectrum. We assume arrivals of user requests for service occur with exponential inter-arrival times. Requests, which arrive when the system is full, are enie access an lost to the system. Guarantee QoS is offere by assigning a fixe amount of banwith to each connection. Without loss of generality, we assume that each connection requires the same amount of banwith. Thus, the maximum number of connections is a fixe integer, c, given by iviing the total amount of available banwith by the banwith require per connection. We assume no particular istribution on the holing time for the iniviual connections an that each connection is inepenent. This service can be moele as an M/GI/c/c queue. For a escription of the properties of this queuing moel, see [23]. For this system, the istribution governing the number of users in the system is the truncate Poisson istribution: ρ n! P[ N = n] =,0 n c c i= 0 n i ρ i! (11) 11
ρ = λt (12) where, are also nown: N = number of ongoing connections ρ = traffic intensity λ = arrival rate of connection requests T = expecte holing time of a connection M/GI/c/c queue. The expecte number of ongoing connections, E[N], an the blocing probability, B(ρ,c), E [ N] = λe[ T ]( 1 B( ρ, c) ) (13) ρ c! B( ρ, c) (14) = c i= 0 c i ρ i! The Erlang Loss Formula, (14), gives the probability of blocing a request for the 3.2 Maximum Acceptable Arrival Rate, λ * Given the service moel escribe above an its relaxation, we can now estimate the maximum acceptable arrival rate which epens on the service provier specifie limit on the acceptable blocing probability, P b : B( ρ, c) Pb (15) The probability of blocing, B(λ,E[T],c) (14), is an increasing function of the arrival rate, λ, through the traffic intensity, ρ = λt. We can calculate a maximum acceptable arrival rate λ *, in orer to satisfy (15), by setting the right han sie of the Erlang Loss Formula, (14) equal to P b. c i=0 c * ρ c! ρ = ρ such that = P (16) i ρ i! b The maximum feasible arrival rate, λ *, is simply the maximum traffic intensity, ρ *, ivie by the average holing time, T: * * ρ λ = (17) T 12
4 The Optimal Price Discount In this section, we present the optimization moel that is solve by the service provier in orer to etermine optimal iscounts. We will first summarize the important conclusions of the analysis in the previous sections. In section 3, we iscusse the case where the service provier treats call-blocing probability in a given perio as a constraint (15) which etermines the maximum feasible rate of requests, λ *, (17). In section 2, we erive an expression for the proportion of users accepting a iscount to efer service by one perio; we showe that this proportion was relate to the price iscount through the constant b in the case of uniform istribution of user valuations, (10). We now require two further assumptions beyon those in sections 2 an 3: The holing times of the connections are relatively short compare to the scale of the time perios which exhibit pea an off-pea eman (the provier chooses the time perios for the moel when implementing the propose iscount pricing scheme), e.g. if connections exhibit an average holing time of 5 minutes the pea perio may be roughly 1 hour an the corresponing elay of service will be 1 hour. Delaye users from perio arrive uring perio + 1, with exponential inter-arrival times. Thus, the elaye arrivals in aition to the unerlying eman for the perio are the sum of two Poisson processes an are in aggregate a Poisson process. The net arrival rate of connection requests uring a perio is the arrival rate uner the maret price, less some proportion of users who accept the iscount, plus elaye arrivals of users who ha accepte a previous iscount offer: λ = λ ( 1 b ) + λ (18) where, λ = arrival rate of requests in perio, of users who previously accepte iscounts = arrival rate of new requests for immeiate service λ λ = net arrival rate of requests for immeiate service, after iscounts offere The service provier observes a Poisson process of arrivals, with a rate given by (18). The expecte arrival rate of elaye requests in any perio is a function of the iscount offere in the previous perio, (19). Delaye users cannot be elaye again. Only the first time arrivals, λ, are offere the iscount,. Note that the arrival rate of elaye requests is etermine by the iscount offere in the previous perio, -1, an the resulting acceptance probability of users, given in (10). Multiplying 13
the acceptance probability by the arrivals of new requests in the previous perio, λ -1, we obtain the expecte arrivals of elaye requests: = λ 1b 1 1 λ (19) The objective function of the service provier is to minimize the total iscounts pai to users. Minimizing the iscounts pai to users reflects the view that prices an eman are outsie the irect control of the service provier, who s primary goal is therefore to offer satisfactory service at all times by regulating blocing in all perios. Note that expecte revenue before the iscounts are offere is etermine by the price (p), the arrival rate (λ), the expecte holing time of a connection (T) an the proportion of requests actually bloce (B), i.e. expecte revenue per unit of time before iscounts = (1-B)λpT. While λ, p an T are constants, the proportion actually bloce, B, is a complex nonlinear function that epens on a number of factors incluing λ, capacity c, etc., an once the iscounting scheme is in place, the iscount offere, also etermines B. To eep the analytics simple an the potential implementation realistic, we will focus only on reucing the cost of proviing iscounts. User acceptance of iscounts is one perio by perio, an we formulate the moel in a multi-perio setting. The service provier minimizes the total expecte costs of the iscounts offere subject to the maximum feasible arrival rate. Therefore, in a given perio, the optimization problem with regars to selecting the iscounts is: N (P-iscount) Min λ T b (20) = 1 subject to, b 1 1 N (21) * λ ( 1 b ) + λ λ 1 N (22) 0 p 1 N (23) The objective, (20), is to minimize the expecte value of the iscounts pai to users, as this quantity represents lost revenue. Note that iscounts are offere to users before the system blocs any users. The first constraint, (21), restricts us to limit the expecte acceptance rate of the offere iscount to less than or equal to 100%. Obviously, one cannot elay more than 100% of new connection requests. The secon constraint, (22), restricts the net arrival rate in each perio, λ, to less than or equal to the maximum acceptable rate, λ * in each perio. Finally, the iscount is restricte to a non-negative range boune by the price, (23); one cannot offer a iscount more than the price itself. 14
Note that the for the overall optimal iscount problem, (P-iscount), the iscount for any perio,, is a function of the iscounts offere in previous perios, -1, -2, However, we prove that the optimal solution for the problem (P-iscount) can be obtaine from a set of optimization problems, one for each perio, = {,1,2, }, given by: (P-) Min λ T b (24) subject to, b 1 (25) * λ ( 1 b ) + λ λ (26) 0 p (27) Theorem: Suppose a feasible optimal solution exists for (P-iscount) an is given by the vector * iscount. Let the feasible optimal solutions for (P-) be given by the scalar * ( = 1,2,,, ). Then * iscount. = { * 1, * 2,, *, }. Proof: See Appenix A.2. Before we give the optimal price iscounts for the problem (P-), it shoul be note that in some instances, the problem coul be infeasible. Given that the price iscount cannot excee the price itself, there may be situations in which insufficient users are shifte to other perios, an the call blocing specification set by the service provier is too low to be achieve. In such cases, we suggest the best one can o is to offer a sufficient price iscount to elay as many users as possible, subject to the requirement that the price iscount be less than the price. Thus, the complete statement of the optimal price iscount is as follows: * 0 * 1 λ λ = 1 b λ 1 Min p, b * 1 λ λ if 1 0 b λ * 1 λ λ < < 1 if 0 1 Min p, b λ b * 1 λ λ 1 if 1 Min p, b λ b (28) (29) (30) The expressions containe in (28), (29) an (30) can be unerstoo as follows: If the net arrival rate at a particular price (which is set exogenously) is less than the maximum allowable arrival rate ictate by the esigne blocing probability an given by conition (28), the ecision is not to offer a iscount, i.e. = 0. 15
If all the constraints in problem (P-) are satisfie, given by the conitions in (29), then the price iscount is simply set at the lowest feasible value. Thus, the lowest price iscount is calculate using constraint (26), which has to be bining (see the proof in Appenix A.2). The situation where the feasible region of problem (P-) is empty is given by the conition in (30). In this case, the lowest price iscount offere will be ictate by either the exogenously set price, p, or the logical constraint of not being able to shift more than 100% of new arrivals, given by constraint (25). Clearly, the price iscount cannot excee the price itself. Thus, if p 1/b, the optimal iscount woul be p, an less than 100% of the new users will be iverte. Conversely, if 1/b p, then 100% of the new users will be iverte by proviing an optimal iscount, 1/b, less than the price. We offer this part of the optimal solution only as the best of a ba situation, since in fact no feasible, much less optimal, solution exists for the situation just escribe. If the calculation of solutions for the problems (P-) is performe on-line, the number of elaye users returning is historical information in any perio. Thus, we can use an exact calculation of the elaye arrival rate: n 1 λ = (31) t n 1 = number of users who accepte iscounts in perio -1 t = length of time elapse over one perio We use the exact number of users who accept elays in the previous perio, n -1, in (19) for the net arrival rate calculation. Note that E[n -1 ] = b -1-1. 5 Simulation We now present an implementation of the iscounting scheme, applie to a few examples, similar to the situation illustrate in Figure 1, at the beginning of the paper. 16
5.1 Implementation In orer to implement the aaptive iscounting scheme, we must set a traffic intensity, ρ *, which is etermine by the Erlang Loss formula for a given blocing specification. We will consier 10 possible values, corresponing to maximum blocing probabilities of 1% to 10%: Parameter Value Comment c 250 Max. number of connections ρ * 228.3 Max. traffic intensity for P b = 1% 235.8 Max. traffic intensity for P b = 2% 241.4 Max. traffic intensity for P b = 3% 246.2 Max. traffic intensity for P b = 4% 250.5 Max. traffic intensity for P b = 5% 254.6 Max. traffic intensity for P b = 6% 258.4 Max. traffic intensity for P b = 7% 262.2 Max. traffic intensity for P b = 8% 265.9 Max. traffic intensity for P b = 9% 269.6 Max. traffic intensity for P b = 10% Table 1. Range of maximum traffic intensities consiere in simulation experiments. We choose the traffic intensity ρ * in Table 1 accoring to the Erlang loss formula, (16). The maximum arrival rate, λ * is calculate using as a function of ρ *, λ * = ρ * /T. The offere iscounts in the simulations, given in expressions (28) (30), are in turn etermine as a function of λ *. 5.2 Example Problems We will consier four scenarios with variable levels of eman, λ(), an time-value of consumption, β(). The istribution of users valuations of immeiate service, F WTP, is unnown to the service provier an the iniviual users, but is assume to be a uniform ranom variable, istribute on the interval (0,40), throughout the simulations. The fixe price is assume to be $0.10 per minute (the uniform F WTP in the preceing sentence is enominate in cents) per connection an the holing times are exponentially istribute with a mean holing time of 5 minutes for all four scenarios. We always use a capacity, c, of 250 connections. The first three scenarios range from an easy problem, with a short-term pea an subsequent low eman, to a ifficult problem, with persistently excess eman followe by eman that is near the maximum permitte level, λ *. The fourth scenario is ranomly generate. The problem scenarios are summarize below: 17
Scenario Short Pea Moerate Persistence Extreme Persistence Ranom Time (hours) Arrival Rate per Minute (λ) Time Valuation of Consumption (β) Proportional Acceptance Rate of Discounts (b ) 0 2 55.0 0.5 0.067 2 4 35.0 0.4 0.056 > 4 35.0 0.3 0.047 0 2 50.0 0.5 0.067 2 4 55.0 0.4 0.056 > 4 35.0 0.3 0.047 0 2 50.0 0.5 0.067 2 4 55.0 0.4 0.056 4 6 45.0 0.3 0.047 > 6 35.0 0.3 0.047 0-1 54.77 0.5 0.067 1-2 54.21 0.5 0.056 2-3 44.19 0.4 0.047 3-4 54.51 0.4 0.047 4-5 49.19 0.3 0.047 5-6 46.42 0.3 0.047 >6 35 0.3 0.047 Table 2. Four example eman scenarios. The eman scenarios presente in Table 2, inclue three scenarios esigne to illustrate the potential for shifting eman between perios, as well as a ranomly generate scenario. In the Short Pea scenario, the pea perio lasts for 2 hours an is followe by low eman. The Moerate Persistence scenario has a slightly excessive eman for the initial 2 hours, followe by a high pea of 2 hours, but we can easily accommoate elaye users requests after 4 hours, when eman falls. The Extreme Persistence scenario again has a slightly excessive eman for the initial 2 hours, followe by a high pea of 2 hours. The eman from hours 4-6 is near pea capacity an leaves little room to accommoate the elaye users, requiring elaying more users again into the low eman perios after 6 hours. Finally, the Ranom scenario is ranomly generate for hours 0 through 6 to provie an example of an unpreictable series of changes in eman. In aition to the changing eman, β ecreases from 0.5 to 0.3 in each scenario, as users place more importance on consuming in the current perio as the simulation progresses, reflecting an en effect, such as users may not wish to elay consumption towars the en of a ay. 18
5.3 An Illustrative Example First, we inclue the sample paths for a number of variables from a single simulation run. We will use the Extreme Persistence eman scenario containe in Table 2. This example is inclue to illustrate the system performance uner the iscounting scheme. It is first interesting to observe the optimal iscount offers. 0.10 Offere Discounts 0.08 Value of Discounts 0.06 0.04 0.02 0.00 0 2 4 6 8 10 Time (minutes) Figure 4. Optimal iscount offers. In Figure 4, we see that the price iscounts offere change over time, increasing between hours 2 an 4, an ecreasing between hours 5 an 7. This is ue to the return of elaye users from the earlier perio in each of these two-hour perios. For instance users elaye between hours 2 an 3 return between hours 3 an 4. This persistence of the pea requires even more users be elaye uring the secon hour of the pea. 19
Blocing Performance 1500 Cumulative Number of Bloce Requests 1250 1000 750 500 250 0 0 2 4 6 8 10 Time (hours) With Discounts Without Discounts Figure 5. Comparison of Call-Blocing (Extreme Persistence Case) Using the price-iscounting scheme, it turns out that very few users are bloce even in the Extreme Persistence case (Figure 5). Between hours 0 an 6 without iscounting, 6.86% of requests are bloce versus 0.84% with the iscounting scheme activate. The target blocing rate in this example was set at 1%. Finally we examine the change in the system occupancy uner the iscounting scheme. The results are shown in Figure 6: Change in System Occupancy Number of Connections (5 Perio Moving Average) 240 220 200 180 160 140 120 100 Without Discounts With Discounts 0 2 4 6 8 10 Time (hours) Figure 6. Comparison of number of connections with an without price iscounts. In Figure 6, we observe that without the price iscounts, the system operates at a slightly higher occupancy until roughly hour 6. As eman subsies, in the later perios, the number of 20
connections ecreases accoringly. The iscounting scheme successfully shifts eman from the high eman perios to the low eman perios. Connections are slightly reuce from hours 0 to 4, but are significantly higher than without iscounting in the latter portion of the simulation. The aggregate effect on system performance is to accommoate more eman an observe a more consistent level of eman over time. 5.4 Effects of Price-Discounting on Performance We simulate a number of problems for each of the eman scenarios in Table 2. We varie the prescribe blocing probability from 1% to 10% for each of the problems above, running 1000 simulations of each again to get average performance measures. For each eman scenario we simulate the case with no iscounting as a measure of baseline performance. Every simulation was initialize with a six-hour perio of simulate time at the maximum arrival rate of requests per minute (e.g. for 1% cases the simulations were initialize with an arrival rate of 228.3/5 = 45.66). This initialization begins each simulation with the system operating in a state exhibiting the maximum tolerate blocing. As state before, we use a capacity of 250 an expecte holing time of 5 minutes for all simulations. For each eman scenario uner all 10 blocing specifications, we present the improvement in blocing performance, the total iscounts pai out to users as a percentage of total revenue an the net revenue effect. The results are illustrate in Figure 7 - Figure 10 below: Performance Effects with Price Discounting (Short Pea) 30.0% 25.0% Performance Effects 20.0% 15.0% 10.0% 5.0% 0.0% -5.0% -10.0% -15.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% Prescribe Maximum Blocing Revenue Pai Out as Discounts Net Revenue Change Figure 7. Performance of Price Discounting for Short Pea Deman Scenario. 21
As Figure 6 shows, when the blocing specification is ease from 1% to 10%, the total iscounts pai out as a percentage of revenue ecrease significantly, to almost zero uner the 10% specification. For the short-live pea, the iscounting scheme is essentially self-financing, with almost no ecrease in net revenue uner any specification an small increases in net revenue in the relatively easier cases where the blocing specification is over 2%. This is because the aitional revenue from serving users that woul otherwise be bloce offsets the cost of proviing the iscount as an incentive to users to elay consumption. Figure 7 emonstrates that the service provier is better off with a iscounting scheme in place than when a simple flat price is use, because better service is being offere with little or no costs to the service provier. Performance Effects with Price Discounting (Moerate Persistence Pea) 30.0% 25.0% Performance Effects 20.0% 15.0% 10.0% 5.0% 0.0% -5.0% -10.0% -15.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% Prescribe Maximum Blocing Revenue Pai Out as Discounts Net Revenue Change Figure 8. Performance of Price Discounting for Moerate Persistence Deman Scenario. With the moerate persistence of the pea perios, the increase revenue erive through reuce blocing offsets more than half of the iscounts pai out in the stringent blocing specification cases (1% an 2%), an the scheme becomes self-financing, even yieling small improvements in net revenue at less stringent blocing specifications. Keep in min the relative magnitue of the excess eman in quite large (~ 20% in excess of the maximum arrival rate). The higher revenue observe in the non-iscounting cases is not consiere acceptable by the service provier, who wishes to regulate blocing. Inee in the higher cost cases (1% an 2% blocing specifications) the blocing is reuce roughly 10%, which is a large improvement in performance at a reasonable cost. 22
The extreme persistence eman is the most ifficult case consiere. For low blocing specifications, revenue is severely ecrease by the nee to elay many users an hence offer large iscounts to accommoate them later. For blocing rates roughly 4% an higher, the scheme is essentially revenue neutral. In such a case of extreme persistence of the pea eman, the problem is really that the capacity of the system is not sufficient for the level of eman over several perios. In this case the long-term solution is to either increase capacity or raise the price to restrict eman, if this is possible. However, in the short-term, the price-iscounting scheme offers a metho to satisfy blocing criteria. Pe rformance Effec ts Performance Effects with Price Discounting (Extre me Persistence Pea) 30.0% 25.0% 20.0% 15.0% 10.0% 5.0% 0.0% -5.0% -10.0% -15.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% Prescribe Maximum Blocing Revenue Pai Out as Discounts Net Revenue Change Figure 9. Performance of Price Discounting for Extreme Persistence Deman Scenario. 23
Performance Effects with Price Discounting (Ranom) 30.0% Pe rformance Effec ts 25.0% 20.0% 15.0% 10.0% 5.0% 0.0% -5.0% -10.0% -15.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% Prescribe Maximum Blocing Revenue Pai Out as Discounts Net Revenue Change Figure 10. Performance of Price Discounting for Ranom Deman Scenario. The ranom eman scenario is inclue as an example of a volatile eman scenario, where unpreictable changes occur from one perio to the next. The eman range consiere was uniformly istribute between 45 an 55 requests per minute for the first six perios. Again we observe a significant control effort mae through the iscounts when the blocing criteria is stringent, i.e. 1 or 2%. However, the scheme oes become self-financing as the blocing objective becomes more realistic given the capacity an the unerlying eman profile. Furthermore, esire improvement in blocing performance is achieve in each case, an in the 1 an 2% cases the eman profile consiere represents a large excess eman (up to 20% in excess of capacity) The simulation results inicate three important results: The iscounting scheme can be use to amit more connections in every case, i.e. bloc fewer connections, as we expect. The blocing specification was satisfie in every case simulate. This represents higher maret share in a competitive maret. Revenue increases uner the iscounting scheme where the persistence an magnitue of pea eman is reasonable. In cases of extreme persistence it may happen that revenue is penalize by offering iscounts in orer to limit the call-blocing to a prescribe level, P b. However, even in this case less connections are bloce, an the higher blocing observe without the iscounting scheme is strictly consiere an infeasible outcome, accoring to the provier s prescribe blocing. 24
We have not aresse blocing of the returning users explicitly. Delaye users who are then enie service are liely to require significant compensation. We suggest a small number of slots coul be hel in reserve by the provier to prevent such unfortunate occurrences. 6 Concluing Remars We have presente a metho for calculating optimal price iscounts, which are use to shift eman from congeste to uncongeste perios in a telecommunications system. We evelop a user moel of behavior so we may preict the proportion of users who will accept a price iscount an elay use of the service. For problems where the peas o not persist significantly, the iscounting scheme actually increases revenue. In more ifficult cases where eman persists over a long perio, many users must be elaye to amit relatively few aitional requests. In these cases, it may not be possible to increase revenue by the priceiscounting scheme. However, we have also shown how to trae-off blocing specification with a revenue enhancing iscounting scheme. The iscounting scheme also provies an alternative to capacity expansion, when capacity is sufficient in all but a few perios. The results presente here reflect the assumption that the problem parameters can be irectly observe by the service provier an then use to calculate the iscounts. Implementation uner uncertainty, where the problem parameters are not nown to the service provier, is the subject of further research with the price-iscounting scheme, an will be presente in a subsequent publication. 25
A.1 Appenix: Proof of Discount Acceptance Theorem Users ecie to purchase service accoring to the ecision escribe by (3) in section 2.3. If a user, who has purchase service, elays consumption between one an three perios, the value of the service is reuce by a factor, β, so that the iniviual user surplus is now β(wtp - p). Customers rawn at ranom from the population, are characterize by a willingness-to-pay (WTP), with cumulative istribution function F WTP (p). The probability of the user accepting a iscount,, in return for elaying consumption epens on the user s ecision, given in (7) in section 2.3: P( A) = P WTP p + WTP > p 1 β P WTP p + I WTP > p P( A) 1 β = P( WTP > p) P p WTP p + P( A) 1 β = 1 P( WTP p) (32) (33) (34) F P( A) = WTP p + F 1 β 1 F ( p) WTP WTP ( p) This proves the theorem. To prove the Lemma, consier the case where WTP is uniformly istribute over the interval [a,b], a p b: 0, p < a p a F WTP ( p) =, a p b (36) b a 1 p > b Note that the formulation of problem (P-iscount) uses the efinition b = 1/((1-β)(b-p)) an requires that b 1, maing it easy to show that p + /(1-β) b. This is a technical requirement for the valiity of the Lemma. Thus, the probability of accepting the iscount is foun to be proportional to the iscount offere: p + a 1 β p a P( A) = b a b a (37) p a 1 b a (35) P( A) = ( 1 β )( b p) (38) 26
A.2 Appenix: Proof of Single Perio Formulation (P-) Theorem We assume a feasible solutions exist for both problems (P-iscount) an (P-). Throughout the appenix we use the substitution λ = λ -1 b -1-1 to mae the relationships between perios clear. First, we consier the set of single perio problems (P-), restate for convenience: (P-) Min λ T b (39) subject to, 1 (40) b * λ ( 1 b ) + λ b λ (41) 1 1 1 0 (42) p (43) Recall that that there is one problem (P-) for each perio, an the problems must be solve in orer for = 1, = 2,, = N. Therefore, in each perio, the iscount from the previous perio, -1, is a problem parameter an no longer a ecision variable. Consier the following solution for problem (P-): * 1 λ λ 1b 1 1 * 1 = if λ + λ 1 1 1 > λ λ b 1 N (44) b * = 0 if λ + λ 1b 1 1 λ 1 N (45) The expression we obtain for the optimal solution to, given in (44) (45), comes from constraint (41), which requires that the iscount offer,, be use to regulate new arrivals, λ, so that the maximum acceptable arrival rate, λ *, is not exceee. Intuitively, the solution is easy to see. If total arrivals are below the prescribe limit the service proviers oes not offer a iscount leaving system performance unchange at no cost. If total arrivals excee the prescribe limit, then the minimum iscount that achieves a feasible solution is offere, satisfying the constraints an minimizing the objective, i.e. the expecte payout of iscounts. The Karush-Kuhn-Tucer conitions, assuming feasibility, are: () 1 + v ( λ b ) + w ( 1) + z () 1 0 2 λ T b + u = (46) 1 u = 0 (47) b * v ( λ ( 1 b ) + λ b λ ) 0 (48) 1 1 1 = w = 0 (49) 27