Spectrum Reservation Options for Mobile Virtual Network Operators

Spectrum Reservation Options for Mobile Virtual Network Operators Loretta Mastroeni

Maurizio Naldi, Senior Member, IEEE

Department of Economics University of Rome Roma Tre, Rome, Italy e-mail: [email protected]

Department of Computer Science, Systems and Manufacturing University of Rome at Tor Vergata, Rome, Italy e-mail: [email protected].

Abstract—Mobile Virtual Network Operators (MVNO) operate without owning spectrum usage rights, by leasing spectrum from Mobile Network Operators owning a license. Spectrum leasing may occur through a reservation process that assigns the MVNO the right, but not the obligation, to lease spectrum at a later time. Reservation contracts may be signed with strict guarantees, that provide MVNOs with the certainty of obtaining the needed spectrum, or soft guarantees, which allow the MNO to adopt an overbooking policy and refusing to lease the spectrum, compensating the refused MVNO through the payment of a penalty. In this paper the two kinds of reservation contracts are compared from the viewpoint of the MVNO, by considering the profitability of the two alternatives. The cash flows associated to the two kinds of reservation contracts are determined and employed to compare the alternatives. The probability of overbooking and the expiry time of reservations appear as the major decision factors, while the penalty value has a negligible influence. It is shown that the reservation contract with soft guarantees is to be prefereed for larger values of the expiry time and for the lower values of the overbooking probability. Index Terms—Pricing, MVNO, Mobile Networks, Spectrum trading, Network Economics.

I. I NTRODUCTION Spectrum needed for mobile services is a scarce resource and is typically awarded through competitive mechanisms, such as beauty contests (comparative selections mainly based on technical merits) and auctions. Winners of such contests are assigned licenses that give them exclusive usage rights on portions of spectrum. The number of licenses awarded is usually small, so that the market is represented by a small number of network operators and may show significant market power situations as well as inefficiencies (should the licensed operators not make full use of the spectrum assigned to them) and barriers to entry. A possible solution to make it easier for new operators to enter the market is to allow service providers not to need their own infrastructure but use that of licensed network operators to provide services to end customers [1]. This new breed of service providers is represented by Mobile Virtual Network Operators (MVNO). The MVNO business has been shown to be potentially profitable [2]. Partnerships with licensed network operators, hereafter referred to as Mobile Network Operators (MNO), may be beneficial to both [3], though competition for end customers is always present [4]. In any case the MVNO has to acquire temporary spectrum usage rights from the MNO through spectrum trading. Though

many ways may be envisaged to implement spectrum trading (a bibliographic list on the subject is contained in [5]), we may classify them according to the timescale on which they act: real time markets (spot markets) and reservation-based markets. In this paper we focus on latter category, which allows MVNOs to book spectrum, in return for a payment for the reservation followed by the payment for the usage rights should the MVNO decide to actually proceed with the purchase. However, in the case of reservation, the contract between the MVNO and the MNO may include the possibility that the MNO practices overbooking, i.e., allows reservations for a quantity of spectrum larger than what it actually owns usage rights for [6]. In this case the reservation is not accompanied by strict guarantees on its completion (the MNO will be charged a penalty to be paid to the MVNO). The MVNO faces then two alternative types of reservation contracts and has to decide which to opt for. The decision can be based on the profitability of the two alternatives. In this paper we provide a comparison of the two reservation contracts from the viewpoint of the MVNO. Our main contribution here is the identification and evaluation of the cash flows associated to the two reservation contracts, the subsequent derivation of the profit expression for each of the two cases, and the identification of the most relevant factors for the MVNO’s choice among the two types of contract. Our analysis in typical scenarios shows that contracts allowing for overbooking may be profitable for the MVNO when the overbooking probability is low, while the penalty has a very limited influence on the profitability. After describing the reservation alternatives and the cash flows between the MVNO and the MNO in Section II, we report the procedures that can be employed to price the reservation itself in Section III. In Section IV we derive analytical expressions for the profits obtained by the MVNO in the two types of contracts, which we employ in Section V to provide examples for a typical scenario and identify which are the most relevant parameters for the choice of the MVNO. II. S PECTRUM R ESERVATION In Dynamic Spectrum Allocation (DSA) mechanisms have to be put into place to assign and price spectrum on a quite smaller time scale than that adopted to assign licenses. Most studies on DSA have focussed on real-time allocation through

the creation of spot markets, as envisaged in [7]. In that context prices can be set either by unilateral decisions of the seller (e.g., by announcements) or through mechanisms taking into account the willingness to pay of prospective buyers (such as auctions). The two approaches to pricing are also named respectively primary user driven and secondary user driven in [8]. Examples of the first approach are investigated in [9], [10]. Auctions have instead been studied in [11], [12], [13]. However, spot markets imply a large degree of uncertainty as to the availability of radio resources for secondary users, who are then prevented to carry out minimal planning activities. In the absence of planning secondary users may find it difficult to accomodate the traffic demand of their own end customers. A possible remedy to the shortcomings of spot markets is to introduce reservation mechanisms, by which secondary users can book the spectrum they need and know in time whether they will be granted such spectrum and hence be able to provide service to their own customers. Such reservation mechanisms would therefore allow to smooth out uncertainties in the availability of spectrum resources for secondary users. Reservation mechanisms are well known and adopted to solve the contention issue in multiple access systems (see, e.g., [14] or Chapter 11 in [15]), though in the absence of an associated economic transaction. However, reservations provide their holders with a right to use that resource in the future and therefore represent valuable assets, which should be conferred a price. Reservation mechanisms for spectrum have been proposed in [16] and [6]. In [16] a market composed of a number of network providers is envisaged, where each provider owns an amount of bandwidth capacity and is available to temporarily lease it to other network providers. Demand and offer meet when a network provider has some excess capacity (i.e., its capacity is larger than what its end customers currently ask for) and another network provider has an excess demand. The envisaged market is then horizontal, i.e., any player can alternatively take the role of seller or buyer. Instead in [6] a structured market is considered, where a primary user (i.e., a company owning usage rights on some portions of spectrum) temporarily leases those rights to a secondary user (i.e., another provider using that spectrum to provide services to its own end customers). In both cases a market is envisaged where spectrum is exchanged through a reservation mechanism, i.e., a prospective buyer reserves for itself an amount of bandwidth by acquiring the right to buy it at a later time (the expiry time T ) at a specified price (the exercise price). Now a payment is associated to the reservation. If, when the time comes, the prospective buyer decides to actually buy the spectrum portion, it pays the seller the exercise price, otherwise the right expires and nothing else is due by the prospective buyer. The flows of money between the two parties are shown in Fig. 1, where the dashed line used for the esercise payment stresses the fact that the prospective buyer is not compelled to buy the spectrum channel; on the other hand, in this basic reservation mechanism the seller gives the prospective buyer the certainty to obtain the reserved spectrum when the time comes. Here

we consider a context where secondary users are represented by MVNOs that rent spectrum and network portions from a mobile network provider (MNO) playing the role of primary user. A reservation may concern either a single frequency (channel) or a block of channels. Hereafter we assume that channels are always traded individually, so that a reservation over a block of Nch channels is carried out as Nch single channel reservations. Pays the reservation price at time 0 MVNO

MNO

Pays the spectrum price at time T

Fig. 1. Money exchanges between the MVNO and the MNO in the basic reservation mechanism (no overbooking)

In the basic reservation scheme the seller receives the reservation payment but faces the possibility of receiving nothing else if the prospective buyer lets the reservation expire. In this case the seller ends up with a small amount of money (as we expect the reservation payment to be) and an unsold spectrum block. Since this state of things may be unsatisfactory for the seller, an overbooking strategy has been investigated in [6] as a hedging device against the risk of remaining with unsold spectrum channels. In the overbooking strategy the owner of usage rights accepts more reservations that it can handle, i.e., a number of reservations larger than the number of spectrum channels it owns and can actually sell. If, when the expiry time T comes, the number of claimed channels is not larger than the number of sellable channels, the seller has no difficulty in satisfying all the buyers and give them the spectrum channels they required. But, if the number of reservations that turn into actual purchase requests is larger than the number of sellable channels, then the requests in excess cannot be satisfied, and the seller has to compensate the unsatisfied prospective buyers. We assume here that this compensation is monetary, though other kinds of compensation can be envisaged. An important issue here is the amount of money that has to be paid back to the prospective buyer to compensate it for the failed purchase (the penalty). On reasonable grounds, we can assume that this amount lies somewhere between the reservation price and the value of spectrum channel (which in turn we can assume to be linked to the profits it can generate when used by the prospective buyer), i.e., between the actual loss and the potential profit. In [6] the compensation has been set equal to a multiple of the reservation price. Here we follow the same assumption. The flows of money between the MVNO and the MNO is now modified as in Fig. 2, where now the dashed lines represent two alternative flows: if, when the expiry time comes, the prospective buyer decides to buy the spectrum channel, it either pays the exercise price or receives the penalty as a compensation. In order to distinguish this reservation schene from the basic one described above, we refer to

it as reservation with overbooking or with soft guarantees (as opposed to the basic mechanism which provides strict guarantees). If the prospective buyer reserves Nch spectrum channels and decides to buy them all, under the soft guarantee scheme it may end up with a number of channels M < Nch . Since we may assume that the prospective buyer selected the number of channels on the basis of the expected traffic demand, that number Nch is deemed adequate to provide the target Quality of Service (QoS), embodied by the target value of the blocking probability. A consequence of the reduced assignment is that the MVNO will be forced to channel its traffic on a number of channels lower than Nch , therefore inadequate to guarantee its target QoS, i.e., leading to a largerthan-target blocking probability and a lower carried traffic. Since revenues are directly related to the carried traffic, this amounts to a reduction of revenues. On the other hand, the possibility of overbooking makes the reservation less valuable, so that the MVNO expects to pay a lower price for the reservation with respect to the no overbooking case. Pays the reservation price at time 0

MVNO

Pays the spectrum price at time T

MNO

Receives the penalty at time T

Fig. 2. Money exchanges between the MVNO and the MNO in the reservation mechanism with overbooking

Hence, the prospective buyer of spectrum usage rights may face two different flavours of reservation contracts: one with strict guarantees (no overbooking) and the other with soft guarantees (with overbooking). The two types of contracts leads to different flows of payments and revenues, so that it is not straightforward to identify the best choice for the MVNO. III. R ESERVATION P RICING In Section II we have seen that the reservation contract is associated to different cash flows depending on the tightness of the guarantees. The identification and evaluation of those cash flows is crucial to the comparison of the two alternative reservation contracts. While we may envisage straightforward ways to evaluate the traffic-generated revenues, setting the reservation price is a more complex task. In addition, dealing with the overbooking case is even more complex. In [16] standard tools of financial options have been employed to define a pricing methodology for the reservation in the case of no overbooking. In [17] a new method has been proposed to assess the value of the reservation when overbooking is considered. Here we review the notion of financial option and report the pricing procedures for both types of reservation contracts. We will employ these procedures in Section V to compare the profitability of the two schemes. Options are a well known device in financial markets. An option is a contract between two parties, a buyer and a seller,

that gives the buyer the right (but not the obligation) to buy or to sell a particular asset (the underlying asset) at (or within) a later time at an agreed price (the strike price or exercise price) [18]. Options that must be exercised at a precise time are named European options, while contracts that provide a time framework to exercise the option rather than a strict date are named American options. In return for granting the option, the seller collects a payment (the premium) from the buyer. Many complex types of option can be devised: the two basic ones are often named (plain vanilla) call and put options. A call option gives the buyer the right to buy the underlying asset, while a put option gives the buyer of the option the right to sell the underlying asset. When the time comes to exercise the option, the buyer may choose to exercise this right (and the seller of the option is then obliged to sell or buy the asset at the strike price) or not to exercise the right and let it expire. The underlying asset may be a piece of property, or shares of stock, or some other security. When the underlying asset is an investment project rather than securities, the term real option is used. In that case the present value of the underlying asset is the Net Present Value (NPV) of the project in the absence of the flexibility embodied by the option. This assumption is known as the Marketed Asset Disclaimer (MAD) [19]. The correspondence between financial and real options is complete if we consider the investment needed for the project as the equivalent of the exercise price, and the time at which the investment decision will have to be taken as the expiry time of the option. Real options have been considered for investments in mobile networks infrastructures [20], [21], [22], [23] and licenses [24]. The real options framework has been considered for spectrum trading in [16] and [6]. In our spectrum trading case the asset is the spectrum channel (i.e., its usage right), and the exercise price is then the price at which the MVNO will be able to buy the spectrum channel from the MNO, and the expiry time is the time at which the spectrum consumer will decide whether to actually buy the channel. The value of the underlying asset is the amount of revenues that the MVNO will get from its end customers for the duration of the temporary lease. The exercise price, as well as the revenues, are expected to be directly related to the duration of the temporary lease: the longer this duration the larger both the revenues and the exercise price. However, we will not refer explicitly to the duration of the lease but consider it embodied into those quantities. A. No Overbooking In the absence of overbooking the reservation contract is the equivalent of a plain vanilla European call option, as shown in both [16] and [6]. For this model the Black-Scholes formula is widely established as a means to assess the value of the option [18]. This formula is based on the following assumptions (which we rephrase in the context of real options, though they were formulated for financial options): • The value S of the spectrum channel follows a lognormal random walk (a.k.a. a Geometric Brownian motion), with

a mean drift µ and a volatility σ, i.e., dS = µSdt+σSdz, where z is a Wiener process (this hypothesis is widely adopted in the context of investments in the wireless industry, see, e.g., [22], [23]); • The risk-free rate of interest r is constant; • The value S0 of the spectrum channel at the time of the option purchase is the Net Present Value (NPV) of the investment in the spectrum channel, i.e., the present value of the net profits to be obtained by the MVNO during the temporary use of the spectrum channel; • There are no transaction costs associated to the channel purchase. Under these hypotheses the correct price for the option, when the exercise price is Z, is Vnovb = S0 G(d1 ) − Z exp(−rT )G(d2 ),

(1)

where G(·) is the cumulative standard normal distribution, and d1 and d2 are the following quantities d1 =

− ln(S0 /Z) + (r + σ 2 /2)T √ , σ T

(2)

− ln(S0 /Z) + (r − σ 2 /2)T √ . (3) σ T An alternative pricing tool, based on a discrete approximation of the lognormal random walk employed in the BlackScholes formula and thus leading to practically the same values, is the Cox-Ross-Rubinstein (CRR) method [25], also known as the multiplicative binomial model. In the CRR approach a binomial tree is built to model the evolution of the value of the underlying asset till the expiry time. The option price is set as the result of a procedure that traverses that tree twice, first forward and then backward. A full description of CRR is provided in [18]; here we skip a number of details, focussing on the most relevant expressions. We indicate the time at which the option has to be underwritten (i.e., bought) by 0; the option’s expiry date is then T . The time to expiry is divided into N time intervals of duration δT = T /N . We expect the approximation to be better as the subdivision gets finer, i.e., as the number of intervals grows. We index the intervals by i = 0, 1, . . . , N , so that the value of the investment into the channel at the end of the j-th interval is S (j) . As already stated, the initial value of the project S (0) = S0 is the Net Present Value of the project in the absence of any flexibility, i.e., in the absence of the call option. While we suppose to know S (0) (though it is actually estimated), the values S (i) , i > 0, are random quantities, described by a binomial process. Over any single time interval we suppose that the project value can move from its previous value to one of two alternative values  uS (i) (i+1) S = (4) dS (i) d2 =

where the √ two up and down factors u and d are respectively u = 1 + σ δT and d = 1/u, and σ is the standard deviation of the value of the investment in the spectrum channel (a.k.a. the volatility).

Since over the time T we have N such time intervals, we end up with 2N possible values for the investment into the spectrum channel at the option expiry date. In order to account for all the possible values we indicate by S (i,j) , i = 0, 1, . . . , N and j = 0, 1, . . . , i, the j + 1-th smallest value of the project at the i-th time interval (then S (i,0) is the lowest possible value at the i-th stage). Since we have u = 1/d (and therefore ud = 1), the tree is recombining and the possible alternative values of the investment at the expiry date are actually N + 1. Since we have S (i,j) = S (0) uj di−j , at the expiry date (i = N ) we are able to compare the values of the investment at that date with the exercise price Z of the option. When the investment’s value lies above the exercise price, the prospective buyer finds it worth buying the spectrum channel, i.e., exercising the call option. If the investment’s value lies below the exercise price, the option is not worth exercising and its value is 0. We are then able to associate a decision (concerning the exercise of the option) to each of the investment’s possible outcomes, and to compute the possible values of the option at the expiry date, which are W (N,j) = max(S (N,j) − Z, 0). The forward traversal of the event tree maps therefore the present value of the investment S 0 into N + 1 possible option values at expiry W (N,j) , j = 0, 1, . . . , N . The present value of the option (which sets its price) is obtained by traversing the binomial tree backward and repeatedly using the expression W (i−1,j) =

p∗ W (iji+1) + (1 − p∗ )W (i,j) , 1 + rδT

(5)

where

√ 1 r δT p = + (6) 2 2σ is the so-called risk-neutral probability. In fact, the resulting option value can be seen as the expected value of the option under the risk-neutral probability, discounted at the risk-free rate r. The correct price Vnovb for the option is then its present value Vnovb = W 0,0 . (7) ∗

B. Overbooking The basic CRR approach provides the value of the option when no overbooking is adopted. The presence of overbooking reduces the value of the option, since the option holder does not have any guarantee that it will get the spectrum channel in the end. Options for which the seller does not guarantee the possibility of exercising them are known in the financial literature as naked or vulnerable options. Though pricing methods for such options have been considered in the literature (see [26] and [27]), those methods cannot be applied in this context since they assume different payoffs (i.e., different compensations for the event that the option cannot be exercised) in the case of vulnerability, namely the value of the company’s assets in [26] or a fraction of the value of the option at expiry in [27], rather than a multiple of the option price as in our case.

In [17] the basic CRR method has been modified to take into account the possibility that the option cannot be exercised at expiry because of overbooking. In this paper we adopt that method, described hereafter, to otbtain the price Vovb of the option under overbooking. For this purpose we introduce the probability Povb that overbooking results in more requests to exercise the options than available channels, so that the option cannot be exercised. This event takes place because a large number of option holders decide to exercise their option at the same time. Such decisions are typically correlated, since they are influenced by the overall demand for mobile services at that time. A normal copula model has been proposed in [6] to model such correlation, but in this paper we do not enter into the details of that model and simply consider it to be an exogeneous quantity. For each non-zero value of the investment at expiry, the option value at expiry is then either the penalty X = αVovb , α > 0, (with probability Povb ) or the no-overbooking result max(S (N,j) − E, 0) (with probability 1 − Povb ). The value of the option at expiry may then be set as the expected value h i W (N,j) = Povb αVovb + (1 − Povb )(S (N,j) − E) IS (N,j) −E>0 , (8) where we employ the indicator function Ix , which is equal to 1 if the subscript condition x is satisfied and 0 otherwise. But in expr. (8), the value of the option at expiry depends on the option price Vovb , i.e., the present value of the option W (0,0) . The self-dependence character of expr. (8) can be overcome by adopting an iterative procedure, i.e., by cycling through the following steps: 1) Traversing back the binomial tree, through the repeated use of expr. (5), starting with the option value at expiry, given by expr. (8), to get the option price and then the penalty; 2) Putting the option price back into expr. (8) to get the option value at expiry. We obtain therefore a sequence {v0 , v1 , . . .} of approximations to the actual option price, starting with the option price v0 = Vnovb in the absence of vulnerability. The full algorithm is summarized as Algorithm 1. IV. P ROFITS OF R ESERVATION A LTERNATIVES FOR THE MVNO In Section II we have described the two types of reservation contracts, either with strict guarantees or with soft guarantees. The two mechanisms are associated to different flows of money between the MVNO and the MNO. In both cases the MVNO received an additional flow of revenues from its end customers. In this section we consider an MVNO facing the choice between those two mechanisms and examine the profitability conditions that drive the choice between one of the two. In the following two subsections we describe in detail the cash flows associated to the two reservation alternatives, considering the case of a single MVNO asking for Nch spectrum channels. A. No Overbooking In the no overbooking case the revenues and expenses for the MVNO are the flows of money between the MVNO and

Algorithm 1 (Estimation of option price) 1: Compute value v0 = Vnovb for option price under no vulnerability through CRR 2: Set threshold T olerance for accuracy in the estimation of option price 3: Set U pdate ← v0 4: Set the index for the number of iterations i = 0 5: while U pdate > T olerance do 6: Update the number of iterations i ← i + 1 7: Compute penalty X ← αvi−1 8: Compute the value of the option at expiry through expr. (8) with V = vi−1 9: Update the present value of the option vi through the backward tree in CRR 10: Compute the variation in the estimate of the present value of the option U pdate ← |vi − vi−1 | 11: end while

the MNO (described in Fig. 1) plus the traffic-related revenues: 1) Expense to buy the options (towards the MNO); 2) Expense to exercise the options (towards the MNO); 3) Traffic-generated revenues (from the end customers). The MVNO pays the MNO the full option price, as derived in Section III. If we indicate the full unitary option price by Vnovb , the MVNO pays then Ynovb,opt = Nch Vnovb .

(9)

In addition the MVNO pays the full exercise price for a total expense Ynovb,ch = Nch Z.

(10)

These two negative cash flows are countered by the revenues represented by the random variable Rnovb , which, under the CRR approach, takes the values corresponding to the leaves of the binomial tree described in Section III-A, i.e., the values RNch ,k = S0 uk dN −k . We consider just the set {k : RNch ,k ≥ Z}, since the option would not be exercised otherwise. As shown in [25], the revenues
k followNa−kbinomial distribution √ P[Rnovb = RNch ,k ] = N , where k p (1 − p) p = 12 + 12 σµ δT . The expected traffic-generated revenues for the set of Nch channels are then E[Rnovb ] = Nch

N   X N k p (1 − p)N −k S0 uk dN −k , (11) k ∗

k=k

where k ∗ = min {k : RNch ,k ≥ Z}. The overall expected profit is then Qnovb = E[Rnovb ] − Ynovb,opt − Ynovb,ch " N   # X N k N −k k N −k = Nch p (1 − p) S0 u d − Vnovb − Z . k k=k∗ (12)

B. Overbooking In the case of overbooking the cash flows exchanged by the MVNO are the following: 1) Expense to buy the options (towards the MNO); 2) Expense to exercise the options (towards the MNO); 3) Traffic-generated revenues (from the end customers); 4) Penalty for the undelivered channels (from the MNO). The MVNO has then two positive cash flows (the last two in the above list) and two negative ones. While the penalty only occurs in the case of overbooking, the remaining three ones are present in the no overbooking case too. However, their values are reduced with respect to that case. Hereafter we examine first separately all this profit/loss components. The expense for the options is reduced since the unitary option price is lower in the case of overbooking, as derived in Section III-B: Yovb,opt = Nch Vovb . (13) As to the expense to actually buy the channels, the MVNO receives a lower number of channels than what it would ask for. Namely, if it would ask to exercise the option for Nch channels the probability that it receives C channels (C ∈ {0, 1, . . . , Nch } is   Nch ch −i P[C = i] = (1 − Povb )i PN . (14) ovb i The expected expense Yovb,ch for the channels is then Yovb,ch =

Nch X

iP[C = i]Z = Nch (1 − Povb )Z.

(15)

i=0

The MVNO receives a penalty proportional to the option price and to the number of unobtained slots (the penalty for each unobtained slot is X = αVovb as stated in Section III-A). Hence, the expected revenues from penalties are Rpenalty =

Nch X (Nch − i)P[C = i]X = Nch Povb αVovb .

(16)

i=0

Finally, the MVNO receives a flow of revenues from its end customers. In the no overbooking contract the revenues distribute according to the possible end values of the underlying process (the lognormal random walk or its discrete counterpart, as shown in Section III). However, when the reservation contract allows overbooking, the MVNO may receive a number of channels lower than what it had asked for, hence the traffic distributes over a lower number of channels. The reduced number of channels will bring an increased congestion loss, so that the carried traffic (generating revenues) is correspondingly reduced. There is therefore a reduction of revenues with respect to the no overbooking case. In order to quantify this reduction we need to evaluate the congestion loss. Here we make the simplifying assumption that we have a single value of congestion loss for each assigned number of channels, and that the reduction factor for revenues is equal to the reduction factor of the carried traffic. If we indicate by B = B(C, A) the congestion loss when the traffic A is offered

to C channels, and by RC,k the revenues when the MVNO receives C channels (see Section IV-A), we have A(1 − B(C, A)) RC,k = . RNch ,k A(1 − B(Nch , A))

(17)

In order to evaluate the congestion loss, though more accurate models exist to evaluate the performance of a mobile network (see, e.g, [28]), we make the simplifying assumption that we can adopt the M/M/C/0 queuing model (as done, e.g., in [9]), so that we can employ the well known Erlang-B formula in the recursive version [29] B(M, A) =

AB(M − 1, A) , M + AB(M − 1, A)

(18)

where the offered traffic A is computed so to produce the target loss B(Nch , A) = Bnom (e.g., 1%) when we have M = Nch , i.e., no overbooking. The overall expected traffic-generated revenues are then Nch N   X X N k P[C = i] E[Rovb ] = p (1 − p)n−k RC,k . (19) k ∗ i=0 k=k

The overall expected profit is then Qovb = E[Rovb ] + Rpenalty − Yovb,opt − Yovb,ch  Nch  X Nch ch −i = (1 − Povb )i PN × ovb i i=0 N   X N k p (1 − p)n−k RC,k + k ∗

(20)

k=k

Nch Povb αVovb − Nch Vovb − Nch (1 − Povb )Z. V. N UMERICAL R ESULTS In Section IV we have derived the expressions of the net profits for the MVNO in the two types of reservation contracts: with strict guarantees (no overbooking) and with soft guarantees (with overbooking). In this section we exploit those expressions for a number of cases with two aims: 1) to examine whether the reservation with overbooking may be profitable for the MVNO; 2) to identify the most relevant parameters in that kind of contract. Throughout this section we adopt the following parameter values: • Expected revenues S0 = 100 • Exercise price Z = 100 • Yearly Volatility σ = 20% • Yearly Risk-free rate r = 1.5% • Yearly Drift µ = 3.5% • Target Congestion Loss Bnom = 1% The expected revenues and the exercise price are set as reference values (not representative of a real situation), since the resulting profits can be expressed as percentage values. Instead the parameters concerning the business trend during the option lifetime (i.e., the volatility, the risk-free rate, and the drift) are given typical values. As to the volatility, it can be estimated from historical movements of telecom shares on the stock market. However, estimates vary significantly

5 channels 10 channels

Relative profit

1.04

As shown in Fig. 4, plot for Nch = 10 and Povb = 0.12, the profits grows approximately linearly for both types of contracts. In the case of the no overbooking contract the growth factor is roughly four for an eightfold increase in the expiry time (from 1 week to nearly 2 months); for the overbooking case the profits are quite negligible when the expiry time is 1 week, but grow very quickly to reach nearly 40 over eight weeks. The expiry time is likewise relevant to No Overbooking Overbooking

40

Profit

30

20

10

0 0

1

2

3

4

5

6

7

8

9

Expiry time (no. of weeks)

Fig. 4.

Impact of expiry time on the profit

the choice between the two contracts, as shown in Fig. 5, plot for Povb = 0.12. The contract with soft guarantees is preferable for larger expiry times (at least 6 weeks for the 10 channels case). Again, for MVNOs with a smaller demand (a smaller number of channels) the range of profitability of the overbooking contract is wider: for Nch = 5 the minimum expiry time to make overbooking preferable is just 4 weeks instead of 6. 1.2

5 channels 10 channels

1

Relative profit

according to the particular phase of the economic cycle. A brief review is contained in [24]. The value we have considered here can be deemed as conservative on the low side. The riskfree rate and the drift are again fairly tipical of the current economic situation and practically identical to those adopted in [23], where r = 1.37% and µ = 3.43%. Finally, the target congestion loss is a typical value in network dimensioning practices. We now consider the impact of the following parameters on the profitability of the reservation contract: • the Penalty/Option price ratio α; • the Expiry Time T ; • the Overbooking Probability Povb . In this paper the overbooking probability is considered as an exogeneous factor, over which neither the MNO nor the individual MNO have any influence. In fact, it results from the concurrent actions of many MVNOs, hencw, it cannot be considered as a control parameter by either the MNO or the MVNO. Instead, both the penalty/Option price ratio and the expiry time represent control parameters for the MVNO. In fact, though they are expected to be set by the MNO as part of its bundle offer, the MVNO can exploit the results shown in the following to evaluate whether the values set by the MNO are acceptable, i.e., lead to profitability for the MVNO itself. In order to analyse the profitability of the reservation contract with overbooking vs the contract with no overbooking we compute the ratio H = Qovb /Qnovb , which we name the relative profit. If H > 1 the reservation contract with overbooking is more profitable to the MVNO. The impact of the penalty is represented in the graph shown in Fig. 3. Here the expiry time is 1 month and the overbooking probability is 0.12. Increasing the penalty from 1 (just reimbursing the price paid for the option) to 20 has a very limited effect on the profit (a few percentage points). The contract with soft guarantees appears to be preferable in the case of a small number of channels, since reducing Nch from 10 to 5 pushes the relative profit beyond the unitary threshold.

0.8

0.6

0.4

0.2

1.02

0

1

1

2

3

4

5

6

7

8

Expiry time (weeks) 0.98

Fig. 5.

0.96

Impact of the expiry time on the relative profit

0.94 5

10

15

20

Penalty/Option price

Fig. 3.

Impact of penalty on the relative profit

Instead, the effect of the expiry time is much more relevant.

Finally we examine the impact of the overbooking probability. The relative profit is shown in Fig. 6, where the expiry time is 1 month. The overbooking probability appears to be the dominant factor in the relative profitability of the two kinds of contract. In contrast to the previous cases, the trend here is not monotonic. Though, as expected, the overbooking contract is preferable for the smaller values of the overbooking

probability, there is an optimal value of Povb that maximizes the MVNO’s profits. That optimal value is approximately Povb = 0.05 for both cases (Nch = 5, 10) Both the maximum profit and the profitability range are larger for the smaller number of channels. 5 channels 10 channels

1.3

Relative profit

1.2 1.1 1 0.9 0.8 0.7 0.02

0.04

0.06

0.08

0.1

0.12

0.14

Overbooking Probability

Fig. 6.

Impact of the overbooking probability on the relative profit

VI. C ONCLUSION We have examined two kinds of reservation contracts for an MVNO wishing to buy spectrum usage rights. The first kind of contract provides the MVNO with the certainty to obtain the frequency channels it will ask for (within the limitation of its reservation). The second kind of contract allows the seller to apply an overbooking policy and not to be obliged to provide the frequency channels asked for by the MVNO, though the latter will be compensated through a penalty. We have derived analytical expressions for the profits obtained by the MVNO under the two contracts. Among the driving factors examined, the overbooking probability and the expiry time have a significant influence on the best choice for the MVNO. Contracts with soft guarantees are preferable for longer expiry times and lower values of the overbooking probability. Instead the amount of the penalty has a very limited influence on the profitability of the overbooking contract. In the case of MVNOs with a smaller demand (i.e., asking for a smaller number of channels) contracts with soft guarantees exhibit a wider range of profitability. In all cases there appears to be a significant range of usefulness for MVNOs to adopt contracts with soft guarantees, though these were introduced as a protection device for the MNO possessing the usage rights on spectrum. R EFERENCES [1] G. Pogorel, “Regulation and Competition,” Communications & Strategies, vol. 65, pp. 169–183, 2007. [2] B. Olsen, D. Katsianis, D. Varoutas, K. Stordahl, J. Harno, N. Elnegaard, I. Welling, F. Loizillon, T. Monath, and P. Cadro, “Technoeconomic evaluation of the major telecommunication investment options for european players,” Network, IEEE, vol. 20, no. 4, pp. 6 –15, July-August 2006. [3] A. Banerjee and C. M. Dippon, “Voluntary relationships among mobile network operators and mobile virtual network operators: An economic explanation,” Information Economics and Policy, vol. 21, no. 1, pp. 72– 84, 2009.

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