Allowing Multi-Hops in Cellular Networks: an Economic Analysis∗ Patrick Maillé GET/ENST Bretagne 2, rue de la Ch^ ataigneraie - CS 17607 35576 Cesson Sévigné Cedex, France
[email protected] ABSTRACT Extending the coverage of cellular networks in an ad hoc fashion appears as a promising solution for the future. This paper considers the economic aspects of such an hybrid network, especially focusing on incentives: what subscription discount should be granted to users who accept to transfer traffic with respect to users who refuse it? Does offering such an option improve the net benefit of a wireless Internet access provider? Based on models for user preferences, we establish that allowing multi-hops in cellular networks can actually be profitable to the provider, especially in densed regions, since it permits to install fewer access points to cover a region, and that economy in terms of installation and maintenance costs exceeds the eventual loss in revenue due to fee discounts.
Categories and Subject Descriptors C.2.1 [Computer-Communication Networks]: Network Architecture and Design—Wireless communication
General Terms Economics, Design, Performance
Keywords Ad hoc and hybrid networks, Connectivity, Pricing
1. INTRODUCTION Wireless networks are more and more used to access the Internet: WiFi hotspots can now be found in many different places, like cafes, restaurants, coffeeshops, libraries, airports, hotels, ... . Whereas Internet access is freely provided in some of those commercial places as an additionary service, offering access can also be a service in itself, that users ∗This work has been partially supported by the GET project EcoMESH (see http://ecomesh.objectis.net/)
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have to pay for: a price per unit of time is then paid to the owner of the wireless network (a WiFi access point owner, or a GPRS/UMTS operator). We consider here such an access provider, that covers a given area1 . In order to satisfy his clients, the provider needs to provide a certain quality of service, that can be quantified by the mean connectivity for a user, i.e. the probability that the user can effectively access the Internet. The number and positions of access points and the connection price per unit of time therefore have to be carefully chosen in order to maximize revenue. Indeed, a trade-off appears between the installation and maintenance costs of access points to offer a satisfactory quality of service, and the income associated to the price paid by customers. Actually, a way to improve the connectivity without installing more access points would be to use the clients themselves as relay nodes until an access point is reached. Pure ad hoc networks, and more recently hybrid networks (cellular network with eventually multi-hops to reach the base station) have therefore received a lot of attention from the networking community [4, 5, 9]. We focus here on the incentive problems that are raised by such networks: selfish users are not willing to transfer the traffic of the other nodes, since it implies transmission costs in terms of battery, and users may fear security problems like viruses carried by that traffic. To compensate for that reticence to transfer, some papers study incentive mechanisms based on reputation [8], but most of them consider pricing as a solution to subsidy intermediate nodes [1, 2, 3, 7]. In this paper, we also investigate a solution based on pricing to incentivize intermediate nodes to transfer the traffic of others to an access point. Whereas the mechanisms in [1, 2, 3] are quite complicated and difficult to apply in practice since the decision of a node to transfer or not is per flow and depends on the subsidy offered, we consider here a much simpler model, where nodes decide, upon connecting to a hotzone, to transfer or not all the traffic they will be submitted (the time scale here is therefore the connection duration of the considered user). If the client decides not to serve as a relay then he will be charged a given price per unit of time to access the Internet; if he accepts to transfer packets from other nodes then he will pay a lower price. To our knowledge, such a simple scheme for hybrid networks has not been proposed and studied yet in the literature. 1 One can think of classical examples such as an airport or a railway station, but in the future, with the appearance of new embedded services, it might also be interesting to cover roads, freeways, or other areas.
We study that system in the framework of non-cooperative game theory [6], i.e. we assume that each player (user or provider) will at so as to maximize selfishly his own utility. More precisely, we analyse the game as a leader-follower (or Stackelberg) game (see [6] for a formal definition), where the access provider can predict the reactions of the users, and consequently adapts his behavior (here, subscription prices and distance between the access points) to the most profitable situation for him. Based on theoretical results about connectivity in hybrid networks and on a model of user preferences, we prove in this paper through numerical calculations that such a mechanism can be profitable to a wireless access provider, since offering the users the opportunity to relay traffic and pay less can actually increase the provider’s benefit, through a diminution of the number of access points to install. Moreover, this revenue increase is not done at the expense of user satisfaction, since when the aversion to transfer traffic is not too high, the choices made by a revenue-maximizing provider lead to a situation that has the following properties:
should not be an interesting solution in the 2-dimension case, since for low-density zones nodes are not likely to find relaying neighbors so that access points should cover the whole region, and for high-density zones each node is connected to any other node in a pure ad hoc fashion, thus only one access point is needed for the whole network.
• the price per unit of time for a transferring node is lower than the price chosen by the provider when no multi-hops are allowed,
In this subsection, we define a model to represent the preferences of users who may access the Internet through the (possibly multi-hop) wireless network. We assume that users have quasi-linear utility functions. More precisely, the utility Ui of a user i is the difference between the price θi (Pc ) he is willing to pay to obtain a certain connectivity Pc , and the sum of the costs that are:
• some zones with a low density of potential clients, that were not covered by the provider because of their negative budget balance (income minus costs of access points) are now covered. Those results suggest that allowing multi-hops in cellular networks is an economically interesting solution that benefits to all the actors of the non-cooperative game played by the provider and the clients. The paper is organized as follows. Section 2 presents the model we consider throughout all this paper as concerns the network and the preferences of users. In Section 3 we study the decisions made by an access provider who does not allow multi-hops (pure cellular network). Section 4 considers the case when several hops may be necessary to reach an access point, and users who accept to relay traffic are offered a discount on their subscription fee. The maximum benefit for the provider is computed in such a context and compared to the pure cellular one. Section 5 presents our conclusions, and suggests directions for future work.
2. NETWORK MODEL IN DIMENSION 1 In this section, we describe how we model the (eventually multi-hop) cellular network considered here. The network is made of access points2 and of nodes (the potential clients). Notice that this paper only studies the 1D-case, i.e. we assume that the network spans along a line, like for example a road or a railway platform. Such a network is considered mainly for two reasons. First, as we will see in the following, we have an analytical expression for the connectivity in a one-dimensional network. Second, it is argued and illustrated numerically in [4] that multi-hop cellular networks
2.1
2.2
User preferences
• the subscription fee; • the mean transferring cost (if any), that represents the battery power spent to transfer traffic if user i has to transfer traffic from other users (We denote ci the cost for user i to transfer a packet, and n ¯ the mean number of packets a node has to transfer per unit of time, so that the transferring cost equals ci n ¯ .); • the aversion to transfer Ai , that stems from the reticence to carry the traffic from other nodes for reasons of security. Notice that all those values are per unit of time. Remark: a more general model would include the aversion to transfer and the cost of battery into the willingness-topay function θi . However, it seems natural to separate those variables, since the associated costs can be interpreted as monetary costs: Ai can be seen as the amount that user i would like to be paid to overcome his reticence to serve as a relay. Likewise, ci × n ¯ corresponds to the minimal subsidy to give user i to compensate for his battery expenses for transferring traffic. Actually, it seems to us that the reticence due to security issues has much more importance than battery costs. We therefore neglect ci in the following. With those definitions and assuming that θi (0) = 0, the utility for a user i equals 8 > > > > >
> > > θi (Pc ) − p − Ai > :
if user i pays p and has to transfer traffic. (1)
2.3 Model for the distribution of preferences among users We describe here the assumptions we make as concerns the repartition of user preferences. Those assumptions are needed for our analytical and numerical study, and are summarized in Assumption 1. Assumption 1. • Each user i is willing to pay a fixed amount θ¯i for the service if the connectivity is higher than a certain threshold Pi , and 0 otherwise. In other terms, the willingness-to-pay function θi is such that θi (Pc ) = θ¯i 11Pc ≥Pi , where θ¯i ≥ 0 and Pi ∈ [0, 1], as illustrated in Figure 1. • The values of the criteria θ¯i , Pi and Ai are independently distributed among users. • Aversions to transfer Ai (resp. willingness-to-pay values θ¯i ) are distributed according to an exponential distribution with parameter µA (resp. µθ ).
3.1
Mean connectivity
We decide not to take interference into account in this paper: each node is assumed to have a coverage radius that we denote r, which means that a node is directly connected to another node or to an access point if and only if the distance between them is lower than r. When no multi-hops are allowed, the mean connectivity (that we define as the probability that a given node can access the Internet via an access point), is simply the probability that the node distance to an access point be less than r. The (cellular) mean connectivity Pccell therefore equals the proportion of the network area that is covered by an access point, that is 2r if L ≥ 2r L (2) Pccell = 1 if L < 2r In the following, we will always consider that L ≥ 2r, since L < 2r gives the same connectivity than L = 2r, but implies larger costs in terms of installation and maintenance of the access points. The mean connectivity in that case is illustrated in Figure 2. Access point
Access point
L
• Connectivity thresholds Pi are distributed according to a uniform distribution over the interval [Pmin , Pmax ]. r
Some of those assumptions may not seem realistic. However we make them in order to facilitate the interpretation of results and to enable some analytical conclusions. Considering more complex utility functions can be the subject of a future work.
Valuation for the connectivity
θi(P˜c)
θ¯ i
Figure 2: Mean connectivity in the cellular case: Pccell = 2r . L We denote p the subscription price a user has to pay per unit of time to access the Internet. The access provider therefore has to decide on two parameters: that subscription price p and the distance L between access points. To make his decision, the provider will first study the effect of those parameters on user behavior.
3.2
0
Pi 1 P˜c Perceived mean connectivity
User reactions
Users are assumed to act so as to maximize their utility. Therefore in a simple cellular network, a user i chooses to connect if and only if his willingness-to-pay θi (Pccell ) for the offered connectivity exceeds the subscription price p. Under Assumption 1, the proportion of users that will choose to subscribe is easy to compute: according to (1) and (2), that proportion, that we denote β cell , depends on the distance L between two access points and on the subscription price, and equals
β cell (L, p) = Figure 1: Form of the willingness-to-pay function θi (Pc )
3. PURE CELLULAR NETWORK (WITHOUT MULTI-HOPS): REVENUE MAXIMIZATION In this section, we study the choices made by the provider if no multi-hops are allowed. To access the Internet in such a (pure cellular) network, a user must be directly connected to an access point.
r
3.3
min(Pmax , 2r ) − Pmin L Pmax − Pmin
+
e−µθ p
(3)
Benefit maximization
The provider will act so as to maximize his net benefit, that is the difference between the subscription fees received and the installation and maintenance costs. Expressing that benefit B cell per unit of time and per unit of distance of the network, we obtain Cap B cell (L, p) = pβ cell (L, p)d − , (4) L where Cap denotes the installation and maintenance cost per unit of time for an access point. The following proposition describes the optimal decisions from the point of view of the provider.
12
Proposition 1. Under Assumption 1, the optimal strategy for a revenue-maximizing provider is to satisfy the most demanding users if the area is sufficiently densed, and not to cover the area at all if the user density is too low. More precisely,
Maximum revenue Optimal subscription price Optimal MESH distance Acceptance rate
10
8
• if d ≥
ues
Pmax µθ exp(1) 2r
then the provider chooses the val-
2r Loptimal = Pmax poptimal = µ1θ
6
4
θ exp(1) • if d < Pmax µ2r then the provider does not cover the area: Loptimal = +∞, since providing access to the Internet would always yield a non-positive benefit.
2
Proof. Relation (2) implies that the mean connectivity should either be 0, or in the interval [Pmin , Pmax ] to maximize the network benefit. Indeed, 2r L
• if 2r < Pmin then β cell (L, p) = 0; there are no subL scription incomes and the best strategy is not to cover the area, i.e. to take L = +∞, 2r does not modify > Pmax then choosing L = Pmax • if 2r L receipts, but reduces expenses and therefore increases net benefit.
Therefore, a maximum and strictly positive net benefit can only be attained when 2r ∈ [Pmin , Pmax ]. In that case (4) L becomes B cell (L, p) = pe−µθ p d
2r/L − Pmin Cap − , Pmax − Pmin L
which is maximized in p for p = µ1θ . The maximum net benefit with a strictly positive connectivity is consequently B cell (L, poptimal ) = =
1 L
2r/L − Pmin e−1 Cap − ×d× µθ Pmax − Pmin L
e−1 d × Pmin 2dre−1 − Cap − . µθ (Pmax − Pmin ) µθ (Pmax − Pmin )
For that last quantity to be strictly positive, it is necessary −1 that µθ (P2dre ≥ Cap . In that case the benefit is maxmax −Pmin ) 2r (since imized for the smallest value of L, that is L = Pmax 2r ≤ P ). max L To sum up, if the provider would effectively decide to cover the region, then he would choose the parameters p = µ1θ and 2r , and his benefit equals L = Pmax Pmax Cap de−1 − . µθ 2r
(5)
To make his decision, the provider then has to check whether that maximum benefit is positive, which is the case for sufµ P C exp(1) . In ficiently dense regions, such that d ≥ θ max 2rap that case a proportion β = exp(−1) of the potential clients subscribe to the service. If the quantity in (5) is negative, then it implies that no positive benefit can be expected from covering that region, and the provider chooses L = +∞. Figure 3 illustrates those decisions and plots the corresponding values of the provider benefit when the user density varies, for fixed values of µθ and Cap .
0 0
2
4
6
8
10
User density d r
Figure 3: Optimal decisions from the point of view of the provider, and associated revenue as a function of the user density, for µ1θ = 5, Pmax = 1, and Cap = 15.
4.
MULTI-HOP CELLULAR NETWORK: EQUILIBRIA OF USER CHOICES AND REVENUE MAXIMIZATION
In this section, we introduce the possibility for a client to accept to transfer traffic from other nodes, which enables him to connect at a lower price p. The client can choose not to serve as a relay node, and in that case pays the fee p. As in the previous section, we study the Stackelberg game that appears between the provider and his potential clients. The access provider is the leader, since he plays first: he announces the distance L between two access points, and the two subscription fees p and p. Based on those data, the potential users make their choice: a proportion that we denote α chooses the “transfer” subscription, a proportion β prefers the “no transfer” one, and a proportion 1 − α − β does not connect at all. Remark: the provider must ensure that users who subscribed to the “transfer” option effectively relay the traffic from other users. Verifying that users behave according to their subscription is beyond the scope of this paper, but must be done in order for the mechanism to work. The interesting aspect of the game we are defining is the positive externality due to the fact that users who choose the “transfer” option help improve the mean connectivity. We are therefore faced with a fixed point problem: an increase in the proportion α of relaying nodes changes the mean connectivity, which in turn has an effect on α.
4.1
Connectivity and mean connectivity
Since multi-hops are allowed, a node may have to use other nodes in an ad hoc fashion to reach an access point. Then if a node distance to the closest access point is larger than r, the connectivity depends on whether there exists a set of successive nodes (that accept to transfer traffic), distant of less that r, that enable the traffic to reach an access point. In a recent work, Dousse et al. obtain analytical expressions for the probability P¯c (x) that a node located at distance x < L to an access point can reach it, eventually with several hops,
L Relaying
Relaying
Relaying
user user 111111111111 000000000000 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 Access 000000000000 111111111111 000000000000 111111111111 point
user 111111111111 000000000000 000000000000 111111111111 Access 000000000000 111111111111 point
x
L−x
Figure 4: Connection to one of the two closest access points.
when relaying nodes are distributed according to a Poisson process with intensity λ on a one-dimension network [4]:
case when 2r < L < 3r, an analytical expression can be derived:
8 1 if 0 ≤ x ≤ r, > > < i −λr P⌊x/r⌋ (−λe (x−ir)) P¯c (x) = i=0 i! > i > −λr : (x−(i+1)r)) −λr P⌊x/r⌋−1 (−λe
if x ≥ r. i=0 i! (6) A node located at distance x to the access point on his left is at distance L − x to the access point on his right. The connectivity for such a node is therefore the probability of being connected to at least one of those two access points, i.e. P¯c (x) + P¯c (L − x) − P¯c (x)P¯c (L − x) because of the independance between the two connection probabilities, due to the Poisson distribution assumption. That connectivity, illustrated in Figure 4, is represented as a function of x in Figure 5 for x ∈ [0, L] when L = 5r, for several values of λ × r. Consequently, for a given node, the mean probability
Pc
=
1 − e−2λr
1−
2
−e
+λ L
2
2r 2r + λL 1 − L L
4r3 r 2r2 1 − + 2 − 6 L L 3L3
Mean connectivity
0.99
λr=4 0.98
Connectivity
,
λr=5 λr=4
0.99
λr=5
λr=3.5
0.97
!
but for other values of L we use numerical calculations in the following. Figure 6 shows the evolution of the mean connectivity Pc as a function of the distance L, for different values of the relaying nodes density λ (notice that multiplying L and r by a given value, and dividing λ by that value is equivalent to changing the measure unit: therefore the mean connectivity depends only on L/r and λr). 1
1
2
0.96
0.98
0.97
λr=3
0.96
0.95
0.94 0.95 0.93
0.92
λr=3
0.93 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
of being connected to the Internet is 1 L
Z
L
P¯c (x) + P¯c (L − x) − P¯c (x)P¯c (L − x)dx,
(7)
0
or by symmetry Pc =
2 L
Z
L/2
2.5
3
3.5
4
4.5
5
5
Figure 5: Connectivity for a node, depending on its position with respect to the access point on its left, for L = 5r.
∆
2
Distance L between two access points (multiples of r)
Distance x to the access point on the left (multiples of r)
Pc =
λr=2
λr=1
0.94
P¯c (x) + P¯c (L − x) − P¯c (x)P¯c (L − x)dx.
0
That connectivity, taking into account the expression (6) for P¯c (x), is not easy to compute in general. For the particular
Figure 6: Mean connectivity Pc for a node, as a function of λr and L/r.
4.2
User behavior
When the option of serving as a relay exists, user i has to choose between three actions (“tranfer”,“no transfer”, no connection), and will choose the most profitable one for him: • the value Ai of his aversion directly determines the preference of the user between both subscriptions (if Ai > p − p then user i prefers the “no transfer” option to the “transfer” one if it exists, and inversely), • if θi (Pc ) − min(p, p − Ai ) > 0 then the user will choose to subscribe to his prefered service, otherwise he will decide not to connect.
Those choices are illustrated in Figure 7, depending on the values of the valuation θi (P˜c ) for the mean perceived3 connectivity P˜c and the aversion Ai .
p¯
p
111111111111111111 000000000000000000 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 “No transfer” 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 00000000000000000000000000 11111111111111111111111111 000000000000000000 111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111
“Transfer”
Valuation for the connectivity
θi(P˜c)
11111111111111111111111111 00000000000000000000000000 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111
No subscription
p¯ − p Aversion to transfer
A
expressed as follows: µA e−µθ p − e−µA (p−p)−µθ p αeq = µA + µθ [min(Pmax , Pc (αeq )) − Pmin ]+ (8) Pmax − Pmin + [min(Pmax , Pc (αeq )) − Pmin ] = e−µA (p−p)−µθ p (9) Pmax − Pmin ×
βeq
The right-hand side of (8) gives the proportion of potential clients that choose the “transfer” option when a proportion αeq of users choose that option. αeq is therefore a fixed point of the application α-reply: [0, 1] → [0, 1] that describes the reaction to a perceived α. That application is illustrated in Figure 8 for different values of the “transfer” subscription price p, all other values being fixed. Under Assumption 1, that α-reply application is continuous and therefore a fixed point always exists. 0.8
0.7
p=1 0.6
p=2
0.5
α reply
Figure 7: Choices for user i depending on the preference criteria Ai (reticence to transfer traffic) and θi (P˜c ) (willingness-to-pay for the perceived connectivity).
p=3
0.4
0.3
p=4
4.3 Prediction of the equilibrium reached for given actions of the provider Since the provider plays first, he needs to be able to foresee the benefit he will obtain, in order to efficiently choose the parameters L, p and p. To do so, the reactions of the potential users to given values of those parameters must be studied. In this paper, we do not take into account the phase during which users adjust their choice to the choices of the others. We are only interested here in the equilibria (in terms of user decisions) attained, i.e. the proportions αeq and βeq that verify the following properties: • αeq and βeq do not change over time, • the mean connectivity P˜c perceived by each user equals the real connectivity Pc (αeq ) given by (6) and (7) with λ = αeq d. Those two points imply that the proportion αeq of users choosing the “transfer” subscription corresponds to the proportion of users for which that option is the best one when the mean connectivity equals Pc (αeq ). Likewise, βeq should correspond to the proportion of users choosing the “no transfer” subscription when faced to prices p, p and connectivity Pc (αeq ). Under Assumption 1, those equilibrium conditions can be 3 It is possible that the user does not exactly know the mean connectivity Pc , since that value depends on the proportion of users accepting to transfert traffic. That is the reason why we denote P˜c the connectivity user i thinks he will benefit.
0.2
p=5 0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
α : proportion of relaying nodes
0.7
0.8
Figure 8: α as a reply of a perceived α, and existence of fixed points for different values of p, and for L/r = 5, dr = 8, p = 5, 1/µA = 1, 1/µθ = 5, Pmin = 0.8, and Pmax = 1. As Figure 8 illustrates, α = 0 is always a fixed point of the α-reply application, and it is the only one when p is too high, in which case either the economy on the subscription price is not enough to compensate for the aversion to transfer, or the offered connectivity is too low. On the contrary, when p is low enough, then two more equilibrium points exists. Actually the smallest of those two points is an instable equilibrium, since deviating on the left would lead to the point α = 0, and deviating on the right would lead to the largest equilibrium point, that is a stable one. On the example of Figure 8, such an equilibrium exists for p = 1 and p = 2 with respective values αeq = 0.67 and αeq = 0.53.
4.4
Maximizing the provider’s benefit
We can now study the behavior of the access provider, who is able to predict the situation that will occur once the values of L, p, and p are chosen. Notice that we implicitly assume here that the provider knows the distribution of preferences among the potential users, and the density of the zone he is considering. We do not enter in that paper into the processes
necessary to obtain that information, and eventually leave those considerations for future work. To determine the best values (for him) of L, p, and p, the provider can compute the benefit he will obtain after an equilibrium is reached for different values of the vector (L, p, p), and keep the most profitable ones. More precisely, faced with a given (L, p, p), the users react and reach an equilibrium (αeq (L, p, p), βeq (L, p, p)) as described by (8) and (9). When a stable and strictly positive αeq (L, p, p) exists, we assume that the provider is able to drive the users to that equilibrium rather than to the situation where αeq = 0 (for example by offering free access for transfering nodes during a limited period of time, which leads to a satisfactory connectivity and incentivizes to keep the “transfer” option). The total benefit (per unit of time and per distance unit) for the provider at equilibrium for a fixed (L, p, p) is then
6
5
4
3
2
1
0
Cap B (L, p, p) = pd · αeq (L, p, p) + pd · βeq (L, p, p) − L (10) The form (7) of the mean connectivity is too complex to allow an analytical determination of the equilibrium αeq as a function of the subscription prices and the distance between access points. We therefore only provide numerical results in this subsection. Figure 9 illustrates the variations of the provider net benefit with p and L for p = 5. Such computations enable the
Benefit L/r p p α β
0
1
2
3
hybrid
4
5
6
7
8
9
10
7
8
9
10
user density d r 6
Benefit L/r p p α β
5
4
3
2
8 6 4
1
2 0
0
−2
0
1
2
3
4
5
6
user density d r
−4
Figure 10: Optimal prices p, p and distance L between access points, corresponding subscription proportions αeq and βeq , and provider benefit when user density varies for a mean aversion Amean = 1 (top) and Amean = 0.5 (bottom), and for Pmin = 0.8, Pmax = 1, 1/µθ = 5, and Cap = 15.
−6 2.5 3 3.5
p
4 4.5 5 2
3
4
5
6
7
8
9
10
Distance L
Figure 9: Net benefit B hybrid of the provider as given by (10) as a function of the distance L between two access points and the “transfer” fee p, for p = 5, d × r = 8, Pmin = 0.8, Pmax = 1, Amean = .5, θ¯mean = 5, and Cap = 15 provider to calculate the most profitable values of L and p when p is fixed. Then maximizing the benefit over p leads to the optimal vector (L, p, p). Figures 10 illustrates the optimal decisions of the provider and the corresponding benefit when the user density varies, for two values of the mean user aversion to transfer Amean = 1 . µA The optimal decisions shown in Figure 10 (top) for lowdensity regions (rd ≤ 6.5) are exactly the same as in the cellular case: below a certain threshold, covering the region
is not profitable to the provider, and above that threshold the best strategy is to make people subscribe only to the “no transfer” option. This is due to the fact that users are reticent to serve as relay nodes, so the discount in the subscription price to incentivize some users to choose the “transfer” subscription implies a loss of revenue that is not compensated by the economy on access points. It is therefore better in that case for the provider to choose p = p. For dense regions (rd ≥ 6.5), then it becomes interesting for the provider that some users serve as relay nodes. Notice that in that zone the price p remains at about 1/µθ = 5, whereas the “transfer” fee p falls to a value close to 3.5 . The optimal choices from the point of view of the provider are quite different when users are less reticent to transfer traffic. In that case, the situation where a pure cellular coverage is chosen does not occur anymore: as soon as the provider can expect a strictly positive benefit, then he
chooses to offer the two options “transfer” and “no transfer” with prices p and p respectively around 3.5 and 5. Figure 11 summarizes the benefit curves for the seller without allowing multi-hops, and when that option is offered for different values of the mean reticence to transfer Amean . It appears that when user aversion is quite low, Optimal benefits 14
Amean=0.2
12
A =0.5 mean A =0.75 Amean =1 mean
10
No multi−hops
8
6
4
2
0
3
4
5
6
7
8
9
10
11
User density d r
Figure 11: Maximal benefit for a cellular network, and for an hybrid network, with Amean = 0.2, 0.5, 0.75, 1 (the other parameters are the ones used for Figure 10).
then the provider revenue can be increased by offering a (judiciously computed) discount for transferring nodes. The improvement in the benefit increases as the mean aversion decreases. Figure 11 also exhibits an interesting phenomenon: some relatively low-densed regions that would not be covered in a pure cellular fashion (since it would bring a negative benefit), could actually be covered if users are not much reticent to transfer traffic. This is the case for regions where user density is between rd = 3.8 and rd = 4.1.
5. CONCLUSIONS AND PERSPECTIVES In this paper, we studied hybrid networks (or multi-hop cellular networks) from an economic perspective, especially focusing on the point of view of the provider. We highlighted that proposing two connection fees to the users, depending whether they accept or not to transfer traffic for the others, can be interesting for the access provider in terms of revenue. The “transfer” fee is lower than the “no transfer” one, but the economy in terms of the number of access points to install can exceed the loss of benefit due to that discount. This holds in particular when users are not much reticent to transfer traffic. Moreover, with that option, some regions that were not worth covering in a pure cellular fashion can bring the provider a positive benefit if the possibility to act as a relay node is offered. We therefore consider that offering users the possibility of serving as a relays in cellular networks is a solution that should seriously be considered by wireless access providers when dimensioning their networks.
A lot of future work could be done to reach more precise results. A first direction could be to study the 2-dimension case, but carrying out an analytical study should be much more difficult than in the 1-dimension case. Also, more complex models as concerns user preferences could be considered, for example taking into account the battery costs, which are difficult to compute since the number of packets a node will have to transmit depends on many parameters. It would also be interesting to consider not only connectivity in user preferences, but also available transmission rates. The results in that case might be less optimistic than what we obtain in this paper, since transmission rates drop as the number of intermediate nodes increases, or when the number of nodes associated with the base station increases. Likewise, interference between users can be taken into account. As concerns the provider decisions, the model can also be complemented. It would for example be interesting to take into account the uncertainty of the provider knowledge about the distribution of user preferences. A last point that could deserve some attention is the introduction of other possibility for users, associated with other subscription fees: a user could choose no to serve as a relay, but also not to benefit from the relays offered by the others, or a user may wish to serve as a relay just to earn money without using the access for himself. Such options might lead to interesting, although more and more complicated, non-cooperative games.
6.
REFERENCES
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