arXiv:1409.4033v2 [cs.CY] 25 May 2015 osriergosta led aehg aesain(BS) station base high have already pool that opera to regions for have operators available service capacity allows arrangements to the which increase sharing can [1], and RAN resources common the most only involved the sharin core-network m present, or including roaming, At sharing, infrastructure, RAN network full sharing, cellular wireless for considered. between being arrangements are operators operators sophisticated new and and encouraged owners market infrastructure has the trend enter qual This to high (QoS). and coverage service wide of offering recently, adopted still been More while has infrastructure operators, entrants. (including (RAN) expenditures resulted network capital market reduce access technologies to new radio radio the of of new sharing numbers of small adoption in rapid the and sdsg ftewrls ewr rhtcue aiginto streams. taking revenue architecture, a network network negotiations, wireless the agreement of service design o as and which leasing arrangement, faci for sharing framework the guidance our network use that facility then probability We our surplus. socio-technic evaluate the revenue zero general is than less arrangement a busin has sharing develop me a key the we the facilities, where for ruin-theory, such, only on for based economic As framework the sharing evaluation lacking. clear network longer is despite new adopt case no Moreover, a to is metric. incentives requires rate performance i which data main dimension allocation, framework—the non-trivial resource evaluation new network res resource a physical that wireless introduces as well means This as financial constraints. arrangement for servic account the must and a allocation into of networ operators; incorporation agreement core our traditional service Importantly, of users. from leasing with spectrum agreements large aspects: and two between access of networ arrangement facility consists sharing Our arrangement paper, operators. network traditional fibr this and have new In facilities often infrastructure. a and power costs, propose mine efficient property cover and and to backhaul desirable universities, need highly stadiums, already are they as tradit arrangements facilities—such with such large arrangements fact, sharing In via operators. regions provi party local third for service opportunities new are there networks, hr r aypsil ewr hrn arrangements sharing network possible many are There ne Terms Index ntaiinlwrls ewrs xesv infrastructu expensive networks, wireless traditional In Abstract unTertcDsg prahfrWireless for Approach Design Theoretic Ruin A Wt h ieo ha ml-el nwrls cellular wireless in small-cells cheap of rise the —With ewr hrn,fclte,ri theory ruin facilities, sharing, network — ellrNtokSaigwt Facilities with Sharing Network Cellular .I I. acl Egan Malcolm NTRODUCTION 2 1 eateto ttsia cecs nvriyCleeLo College University Sciences, Statistical of Department aut fEetia niern,CehTcnclUniver Technical Czech Engineering, Electrical of Faculty 1 4 aehW Peters W. Gareth , eateto niern,McureUniversity. Macquarie Engineering, of Department 3 nttt o noomRsac,Singapore. Research, Infocomm for Institute sharing k fnew of ) account esto ders well s ource s—as ional ffers user tors tric lity nto for ast ess we ity re to g. al k e e 2 d Nevat Ido , r old ihcnrbtr agn rmidvdasto individuals from ranging [2]. contributors operators traditional with towards networks pooled, steps core and first are RANs visio spectrum, the including borders” Ultimately, resources without as where Sweden. “networks viewed in market-based be dynamic as the can such market, approaches the these w to arrangements inves entrants sharing heavily new BS negotiated have and Sta operators infrastructure United established in the regions, and other the India of In in part large particularly operator a industry; to now in are wireless stations which base companies), areas offer tower as is only under-served (known arrangement that companies to sharing on RAN coverage based common A improve regions. to developing or density a fe ihdt ae ihasalfopitvadense network via increasing of footprint [4]. means effective capacity small small-ce most the a user—the between the with distance and the rates reduces which data placement, net high wireless offer small-cells, low-pow exploiting i can cheap, By capabilities) [3]. with stations transmitting industry base wireless and the metro-cell provider or revolutionizing the pico, micro, on femto, depending as known (also technology srdmnssc shg podrtsi tdus[7]— stadiums these in within rates upload available characterist high unusual often as often is demands—such the links), user backhaul (despit fibre the backhaul via facilities that bandwidth small-ce high fact leasing leasing in the residenc cost of high urban cost the to density locations, due high arises conve challenge and stadiums, The mines, universities, are utilities, standard facilities centers, when these exam- even Common of employed. ples demands, are rate arrangements sharing data network high with facilities succes effectiv are until [6] implemented. continue fully technologies the will (SON) in OpEx network small-cells in self-organizing of increase number This increasing network. the maintenance— with small-cell e growing to are operational effective due Moreover, (OpEx)—largely the networks. penditures interference for small-cell inter-cell of techniques as operation [5]—key well (eICIC) as network coordination implement CoMP) small-cell to anchor order (including in to MIMO installed leased be resolved must be be backhaul must to and property issues particular, complex remain In there small-cells, and nprle ihteaoto fntoksaig small-cel sharing, network of adoption the with parallel In ept h ucs fcretntoksaigarrangement sharing network current of success the Despite atclrycalnigseai o prtr slarge is operators for scenario challenging particularly A dn(C) odn England. London, (UCL), ndon iyi rge zc Republic. Czech Prague, in sity 3 n anB Collings B. Iain and 4 works c of ics ntion tes. ted es. ith x- er s- s, ll ll n e e s s s s l . 2 and large variations in data rate demand over time. Moreover, exploited at no additional cost to provide high rates through large facilities may desire to only charge low rates to users in advanced eICIC; (3) competition will be increased between order to improve the operation of the facility; in stark contrast traditional operators in the region through core network access with standard wireless service arrangements. This can occur costs (in contrast with RAN discrimination); (4) users will either because the users are employees and the mobile service be charged only at the facility’s incremental costs as the is paid for by the facility, or a cheap mobile service is used as facility is either covering the costs itself (e.g., in mines or an attraction to the facility; similar to how WiFi is currently utilities) or the service is offered as an attraction (e.g. in offered free of charge to users in many convention centers and hotels or convention centers); (5) facilities can offer more hotels. efficient power sources as they also must power the facility (e.g., through large-scale solar cell sources); and (6) SON A. A New Socio-Technical Network Sharing Approach innovation is promoted as the facilities chase a reduced OpEx. There is strong motivation for large facilities to offer cheap An incremental approach to SON is also possible since the mobile services and to develop their own capability to do so; facilities will have a relatively small number of small-cells, namely, the low cost of small-cells, and the availability of high which means that there are fewer complexities compared with bandwidth backhaul. As such, there is a genuine need for an the large-scale RANs of traditional operators. alternative network sharing arrangement that can exploit the These benefits clearly suggest that there is a significant unique characteristics of large facilities. potential for network sharing arrangements involving facilities. In this paper, we propose a new network sharing arrange- ment for facilities. Our facility network sharing arrangement B. Evaluation of Facility Network Sharing Arrangements is based on a network consisting of small-cells operated by Although there are clear economic incentives for the adop- and within the facility, which contrasts with tower companies tion of facility network sharing arrangements, a business case that own multiple small-cells and BSs, which are leased to based on a quantitative framework is lacking. We address operators. this issue by evaluating facility network sharing using a new There are two key aspects to our facility network sharing framework, which incorporates both the wireless communica- arrangement: the core network leasing agreement between tions network, as well as the service and leasing agreements traditional operators and the facility; and the service agreement between users, the facility, and traditional wireless operators between the facility and users. The purpose of the core network that own the core networks. In contrast with standard analytical leasing agreement is to provide the facility with data to serve frameworks in the wireless communications literature, our users that are subscribed to traditional operators as well as framework models the facility as a socio-technical system. This spectrum localized to the facility. This agreement involves a means that the conclusions arising from our framework lie fee that the facility pays to the traditional operator, in exchange close to those used directly in real-world practice. Moreover, for core network access and spectrum. On the other hand, our framework is general, which means that it can be adapted the service agreement is the mechanism by which the facility to a range of wireless settings, where resource sharing is used obtains revenue. In particular, the facility charges users for and financial sustainability is a key design criterion. its services; depending on the resources required to ensure Profit-based approaches for resource allocation in cellular reliable transmission, and also pricing parameters designed to networks have been considered in [9]–[14]. Early work on compensate for the leasing fee for core network access. capacity pricing adapted a real options framework to cope In concept, our facility network sharing arrangement bears with uncertainties [14]. In [9], [13], cognitive cellular net- similarities to the “local network operated by an independent works and small-cell networks, respectively, were modeled actor” classification for indoor cellular networks proposed in via Stackelberg games. The approach in [13] employed sub- [8]. More specifically, the approach in [8] introduced the sidies to incentivize closed access small-cell access points to notion of third party operators to service users inside large share allocated spectrum. The approaches in [11], [12] also buildings, which negotiate with traditional operators for data exclusively focused on spectrum allocation. In [10] the BS access. Our facility network sharing arrangement differs in two density and spectrum usage were optimized to maximize the key aspects: (i) we are not limited to indoor operation, which net profit; however, this analysis was based solely on long- is achieved using a general stochastic geometry model that can term average rates. The drawback of such an approach is in principle be extended to include small-cell cooperation via that it does not consider any assessment of the feasibility of eICIC techniques; and (ii) we propose a specific agreement the business model nor optimal solutions obtained from the structure between operators, users, and facilities. In particular, perspective of a financial risk management analysis. As such, it core network access is provided via a new leasing arrangement is not easy to assess the long-term profitability of the operator. that can in principle be formulated as a contractual obligation, This is important as network design that takes profitability which we detail in Section III. into account will differ from standard designs under purely While facilities have currently not been considered within technical constraints and generally lead to improved operator a network sharing arrangement, there are six key economic longevity. Key parameters such as compounded interest rates, incentives, subject to the presence of appropriate antitrust initial capital, expenditures, and future investments also can- regulation: (1) the facility already owns or leases the property, not be incorporated into the standard approaches. Moreover, which means that there is zero sunk cost for positioning small- previous work does not account for the effect of fluctuations cells; (2) the facility’s high bandwidth fibre network can be in the number of users and their cellular data demands. 3
Jointly addressing a number of the key financial and 3) We evaluate facility network sharing based on numerical technical factors that arise in real-world cellular networks evaluation of the probability of ruin via our framework. (including features such as channel gains, power control, First, we demonstrate that facility network sharing can path loss and the duration of user connections), is crucial to be profitable. Second, we provide insights to guide the reliably predicting and optimizing the financial sustainability design of wireless networks to account for revenue and of network sharing arrangements. However, this requires a expenditures, in addition to ensuring reliable transmis- more sophisticated socio-technical framework. In this paper, sion. we propose a new quantitative framework to evaluate facility network sharing arrangements based on a reformulation of ruin II. WIRELESS NETWORK MODEL theory1 [15] to facilitate a study of business models, where the In this section, we detail the wireless networking aspects of key performance metric is the probability that the owner of our model for the facility (financial aspects will be discussed the facility has a negative revenue surplus within a period of in Section III). The key aspects that we consider are the n months—leading to financial insolvency. We emphasize that placement of the small-cells, which small-cell a given user the probability of these events, termed ruin, is dependent on will connect to, and the physical layer transmission model. both economic and technical factors, which include: the initial We note that our modeling assumptions are based on previous capital of the facility; the compound interest rate (compounded approaches to wireless cellular networks (see e.g., [19]) and monthly); link parameters, such as channel gains, power the only significant difference is that the scale of facilities are control and path loss; duration of user connections; and the typically smaller than those considered for standard networks. number of users in the network. Our assumptions for small-cell placement and the user selection protocol are as follows: C. Key Contributions A1: The facility’s small-cells are arranged according to a We summarize our three key contributions as follows. homogeneous spatial Poisson point process (PPP) with 1) We propose a new network sharing arrangement between intensity β. This means that in each realization there is traditional operators, facilities, and users. The arrange- a different number of small-cells, which provides insight ments accounts for both the expenditures and revenue for into the behavior of the network irrespective of how the the facility, which arise due to the cost of leasing access small-cells are positioned. to traditional operators’ core networks (expenditures), and A2: All users serviced by the facility: the income from servicing users normally subscribed to (i) connect to the nearest facility small-cell; traditional operators (revenue). Importantly, the service (ii) are arranged according to an independent stationary charges incurred by each user depend on the physical point process, not necessarily Poisson. resources required to ensure reliable transmission. As The key consequence of assumptions (A1) and (A2) is that such, the service charges are strongly coupled to the the distribution of the distance R of a small-cell and any user capabilities of the RAN. it services is given by [19, Section III-A] 2) We develop a quantitative ruin theory-based framework to −βπz2 evaluate facility network sharing. The evaluation is based Pr(Z ≤ z)= e 2πzβ. (1) on the probability of ruin; that is, the probability that We now detail our assumptions for the physical layer there is a negative revenue surplus within a period of n transmission model. months (a standard metric for financial risk analysis [15]). A3: The facility’s small-cell network operates in discrete time In order to obtain the probability of ruin, we: with block fading. Each time slot (also corresponding to a) Derive the moments of the revenue from user service a fading block), has a duration (coherence time) of T charges, based on a practical wireless heterogeneous seconds. network model. In particular, we exploit stochastic A4: Each small-cell interferes with the others. Due to the geometry techniques [16] to model the placement of Poisson interference assumption (A1), the interference is base stations and users. M/M shot noise [19]. As such, the received signal in b) Use the moments of the revenue time-varying stochas- the l-th time slot (l =1, 2,...) of a user’s connection is tic process to compute the probability distribution given by of the net profit derived by the facility operator by −α incorporating revenue from the service of each user. yl = hl rU P0xl + nl + zl, (2) c) Compute the probability of ruin via the probability dis- q where hl ∼ CN (0, 1) are the independent fading coeffi- tribution of the net profit and efficient recursions (based cients for time slots l, and rU is the distance between on linear difference equations), which are motivated the user and its small-cell (the closest one, by (A2)), by results in insurance theory [17], [18]. Importantly, distributed according to (1). P0 is the power level the we extend the applicability of the standard recursions small-cell transmits at; not necessarily constant. α is the account for idiosyncratic (from the perspective of in- path loss exponent, xl ∼ CN (0, 1) is the (Gaussian) data surance) aspects of wireless communication networks. 2 symbol, and nl ∼ CN (0, σ ) is additive white Gaussian 2 1Historically, ruin theory originally arose in the study of solvency require- noise (AWGN), with noise variance σ . zl is the M/M ments for insurance [15]. shot noise. 4
Based on assumption (A4), the instantaneous signal-to- A. Leasing and Service Agreements interference and noise ratio (SINR) in the l-th time slot of a given user’s connection is given by A leasing agreement between the facility and a traditional operator allows the facility to access the data required by users 2 −α |hl| r P0 it seeks to service. In the leasing agreement, the facility pays a U (3) γl = 2 , σ + Il fee to the traditional operator in return for access to their core network and spectrum localized to the facility. A key feature where Il is the interference power in the l-th time slot of the connection. In our analysis, we focus on the interference of our leasing agreement is that the facility pays a fee per limited scenario where σ2 =0. connection. This means that the facility only pays for access The achievable data rate in each time slot under Gaussian it actually requires, which is consistent with the broader vision signaling for a user in the l-th slot of her connection is given of dynamic wireless network sharing proposed in [2]. by We assume that the fee is only determined by the traditional operator that is providing the service; that is, it is independent max Rl = B log2(1 + γl), (4) of other factors such as the duration of the connection. As users are typically subscribed to a single operator, this means where B is the bandwidth (a constant) and γl is given by (3). that the fee for each user may be different. Our leasing agreement is summarized as follows. III. PROPOSED NETWORK SHARING ARRANGEMENT In this section, we introduce our network sharing arrange- ment between the facility, users, and traditional operators Our proposed core network leasing agreement between traditional operators and the facility consists of the con- (which have their own core networks). It is important to consider traditional operators as their core networks are a key nection fee for the j-th user. The connection fee is given by source of data for the facility’s small-cell network as well ∗ as spectrum. Although it is also possible to obtain data from C(τi,j , ki,j )= C (ki,j ), (5) the internet (although not necessarily calls), when there are a ∗ significant number of users this can consume a large amount of where C (ki,j ) is the connection fee for users subscribed bandwidth and may also require special agreements with ISPs. to operator ki,j . As such, we focus on the scenario where traditional operators provide data via their core networks. Observe that the leasing agreement does not depend on the Our network sharing arrangement consists of two compo- duration of each connection. As each core network typically nents: a leasing agreement with traditional operators; and a has a large number of connections at any given time, it is service agreement with users, which is illustrated in Fig. 1. likely that the distribution of the number and duration of The agreements determine financial exchanges (contractual each connection is known to the corresponding traditional obligations) between the facility, users, and the traditional operator. Moreover, the infrastructure required to support a operators. Moreover, these exchanges induce revenue and given connection is also known due to the stability of the expenditure processes that are stochastic due to uncertainties wired links (as opposed to the wireless scenario). In this case, in the resources required to provide the service (due to the the operator can obtain good estimates of equilibrium prices stochastic nature of the wireless channel) and the random required to ensure that profit targets are met, which means duration of the users’ connections. To this end, we formalize that the duration of each connection does not play a key role. these processes which forms the basis of our evaluation However, when there is a scarcity of access to the core network framework in Section IV. then it is necessary to consider connection durations in the fee, Users Facility Micronetwork Traditional which may be achieved via a market mechanism such as an Operator auction. Core Networks We now consider the service agreement between the facility and each user. The basis of the service agreement is that there is a cost to the facility to provide users with the wireless service. The cost depends on four key factors: (i) the traditional wireless operator that the user has a subscription; (ii) the duration of the users connection to the facility; (iii) the quality of the wireless service requested (i.e., the rate at which the user requests to be serviced); and (iv) the SINR of the link between the user and the facility small-cell that it is serviced by. Importantly, all of these factors are random quantities. The service agreement corresponds to the revenue the Service Leasing Agreement Agreement facility receives in exchange for servicing each user, which can be obtained from two sources. The first source is directly Fig. 1. Network sharing arrangement diagram. from the users; that is, the users pay the facility for the use of the service. In this case, the users pay the facility for every 5
connection they require. The second source is the facility itself. A5: Let τi,j be the duration (in integer time slots) of the In this case, the facility subsidizes the users and the agreement connection of the j-th user to end its connection in corresponds to the funds the facility allocates to cover the cost the i-th time slot; i.e user (i, j). Each duration τi,j is of servicing each user, which is applicable when the facility is independent and identically distributed. The durations τi,j providing the service for its employees or when the service is has CDF Fτi,j with support {1, 2,...,i}. an attraction (e.g., for hotels or stadiums). Again, this is on a We now detail our proposed service agreement. per connection basis, which means that users (or the facility) only pay for the connections that they actually use. Our proposed service agreement consists of the charge We assume that the service provider offers Q products of to the -th user to end its connection in the -th time slot varying QoS. In practice, the user selects an application that it j i (user )), which is given by seeks to use—such as voice, video, or data—and the service (i, j provider offers a QoS product that can support the application. τi,j v(τi,j , ci,j )= ci,j,lT ρ, (8) Remark 1 (User indexing notation). In a given time slot, l=1 multiple users are likely to conclude their connection. To refer X where c = [c ,...,c ]T is the vector of scaling to a given user, we index it by the pair (i, j), where i is i,j i,j,1 i,j,τi,j the time slot that the user concludes its connection and j is factors (each element corresponding to a different time the index of the user in the set of users that conclude their slot) to satisfy (6) and ρ is the premium rate; that is, connection in time slot i. For example, when a user is the 4-th the income received from user (i, j) by the facility per user that terminates its connection in time slot 5, then the user time slot with a unit scaling factor. We assume that the is indexed as (5, 4). premium rate ρ is constant, irrespective of the operator that user (i, j) is subscribed to. In order to provide increasing levels of QoS over the wireless link, the facility is required to employ an increas- ing number of physical resources; for instance, additional bandwidth, power, or infrastructure (e.g. relays or distributed B. Summary of the Proposed Network Sharing Arrangement antennas). It is important to note that a different number of From the perspective of the facility, the leasing agreement physical resources are usually required in each time slot to corresponds to an expenditure process and the service agree- account for channel fading. As such, the facility varies the ment corresponds to an income process. As such, our network SINR in each time slot by a scaling factor ci,j,l (corresponding sharing arrangement can be formalized as a revenue surplus to the l-th time slot of the connection for the (i, j)-th user) to process, which is defined in terms of the expenditure and achieve income processes. (q) An important feature of the revenue surplus process is that R = B log (1 + γ c ) , (6) i,j i,j,l i,j,l interest is compounded. This means that the total revenue where R(q) corresponds to the rate required to support QoS surplus is dependent on not only the current surplus, revenue i,j and costs, but also the interest rate. Moreover, the time interval product q ∈{1, 2,...,Q} for user (i, j) and ci,j,l encapsulates the additional physical resources required to support QoS between interest compounding is typically different to the duration of a time slot. The number of users with connections product q. Observe that (6) follows directly from (4) with γl ending in compound interest interval m is given in (A6). replaced with γi,j,l, which clarifies the particular user that is serviced. While it might seem more natural to directly scale A6: The number of users Nn that have their connection end (q) ′ the rate (i.e. define R = Bci,j,l log(1 + γi,j,l)), it is in in compound interest interval [n − 1,n) of duration κT fact simpler to use our formulation. We will show this in (κ ∈ N) is geometrically distributed as
Section IV, where we develop our evaluation framework based u on ruin theory. It is also worthwhile to note that the two Pr(Ni = u)=(1 − wN ) wN . (9) formulations are identical with the appropriate definition of We first formalize the facility expenditure process, which ′ ci,j,l. is based on the core network leasing agreement with the In practice, the facility’s physical resources are limited. As traditional operators. such, the scaling factor ci,j,l is upper bounded by a constant. We also introduce a lower bound on ci,j,l so that the facility Definition 1 (Facility Expenditure Process). Consider the is guaranteed a minimum income, even when the channel (i, j)-th user associated with the k-th wireless operator, which is good. This means that facility can ensure that it can pay has: the core network leasing fee, detailed in (8). Taking these (i) a random connection duration τi,j time intervals, dis- considerations into account, we define c as i,j,l tributed according to τi,j ∼ Fτi,j (detailed in (A6)); (q) (q) (ii) a requested rate product with , Ri,j /B Ri,j q ∈ {1, 2,...,Q} 2 − 1 (q) ci,j,l = max min ,cmax ,cmin . (7) distributed according to R ∼ FR for any discrete γi,j,l i,j i,j ( ( ) ) distribution on {1, 2,...,Q}; c T It is also necessary to account for the duration of each user’s (iii) and required resources i,j = [ci,j,1,...,ci,j,τi,j ] , which connection. In particular, we make the following assumptions: are i.i.d random variables defined in (7). 6
Then the total expenditure of the facility in the compound IV. EVALUATION FRAMEWORK: A RUIN THEORY interest interval [n − 1,n) is given by APPROACH
Nn In this section, we detail our framework to evaluate our En = C(τm,j , km,j), (10) facility network sharing arrangement, which will be evaluated j=1 in Section V. Our framework is based on ruin theory [15], X where the key performance metric is the probability that the where (n − 1)κT ≤ m < nκT are the time slots that end facility has a negative revenue surplus within months, known in interval [n − 1,n), and N is geometrically distributed (as n n as the probability of ruin. This is an important metric as the detailed in (A6)) and C(τ , k ) is defined in (8). n,j n,j facility can only operate while it has the resources to do so. Next, we formalize the facility income process, which is In fact, knowing whether these financial resources are likely based on the service agreement with the users. to be available is important in the decision of whether or not to invest in the facility, how to structure products, and how Definition 2 (Facility Income Process). Consider the (i, j)-th much to charge for services. user associated with the k-th wireless operator, which satisfies Unfortunately, it is not straightforward to directly compute the hypotheses in Definition 1. Then, the proposed charge to the probability of ruin. The main reason for this is that the the (i, j)-th user (corresponding to the facility income) for income and expenditure processes (defined in Section III-B) connecting to the facility for a duration τ is given by the i,j involve a number of random variables, which result in random compound random sum sums without closed-form distributions. Due to these difficul- τi,j ties, we instead focus on obtaining an accurate approximation c v(τi,j , i,j )= ci,j,lT ρ. (11) of the probability of ruin. Xl=1 Fortunately, we are able to leverage techniques from ruin The total income generated in the period [n − 1,n) is given theory to compute an accurate approximation for the proba- by bility of facility ruin. However, it is important to note that modifications to the standard theory are required, due to N n the fact that the parameters of facilities yield non-standard I = v(τ , c ), (12) n m,j m,j distributions. j=1 X In this section, there are four subsections: (A) the definition where (n − 1)κT ≤ m < nκT are the time slots that end and overview of the key steps to compute the facility ruin in interval [n − 1,n), and Nn is geometrically distributed, as probability; (B) the first step: the approximation of the income detailed in (A7). distribution, which is based on an orthogonal polynomial Finally, the revenue surplus process for the facility is defined representation; (C) the second step: the recursion for the net as follows. This can be viewed as the accumulation of the profit distribution; and (D) The third step: the recursion for initial capital and the difference between the income and the probability of ruin. expenditure processes, taking into account compound interest. Definition 3. The revenue surplus is the current micro-network A. Ruin Probability Definition and Overview profit in the l-th time slot generated by serving users, after Intuitively, the probability of ruin is the probability that the accounting for interest and the cost of accessing the operators’ revenue surplus is negative before a period of l time periods core networks. This is given by (e.g., months). First, we define the stopping time known as the time of ruin, followed by the ruin probability. l Ni l l−i c Sl(u)= u(1 + r) + (1 + r) v(τmi,j , mi,j ) Definition 4 (Ruin Time). The time of ruin is the first time i=1 j=1 X X that the revenue surplus is negative, i.e. −C(τmi,j , kmi,j ) , (13) LR = inf{l : Sl(u) < 0}, (15) where iκT ≤ mi < (i+1)κT, u is the facility’s initial capital and r is the compound interest rate (compounded at intervals where u is the initial capital. Note that we allow for the of κT ). We also define possibility that u< 0.
Ni Definition 5 (Probability of Ruin). The probability of ruin Snet(i)= [v(τm,j , cm,j ) − C(τm,j , km,j )] , (14) before a period of l time periods is then j=1 X ψl(u) = Pr(LR ≤ l), (16) as the net profit in compound interval i, where iκT ≤ m < (i + 1)κT . and the probability of survival is It is important to note that there is potentially a significant φl(u)=1 − ψl(u). (17) difference in the time scales at which users are priced (order of minutes) and that the revenue surplus is calculated (order In order to compute the ruin probability; the steps are of months). detailed in Algorithm 1. 7
Algorithm 1 Ruin Probability Computation m-th Jacobi poynomial with parameters a,b, am is given by 1. Derive the orthogonal polynomial basis expansion for the B(a,b)(2n + a + b − 1)Γ(n + a + b − 1)n! income PDF in Section IV-B. a = n Γ(n + a)Γ(n + b) 2. Evaluate the distribution for the net profit in time slot l 1 in Section IV-C. This is achieved by first discretizing the (a,b) × fWi (x)Pn (x)dx income PDF from Section IV-B, then using a recursion from Z−1 actuarial science to compute the compound distribution. 1 (a,b) (21) 3. Evaluate the probability of ruin via another recursion in = bn fWi (x)Pn (x)dx, −1 Section IV-D (different to the recursion in Stage 2), which Z circumvents the difficulty of directly computing the proba- and K(x) is given by bility of ruin. (1 + x)a−1(1 − x)b−1 K(x)= , (22) B(a,b)2a+b−1 where B. Orthogonal Polynomial Representation of the Income PDF B(x, y)=Γ(x)Γ(y)/Γ(x + y). (23)
The first stage of computing the probability of ruin is to We note that the coefficients an minimize the mean square compute the PDF of the income from a user with connection error between the approximation and fWi (x) as d → ∞. ending in time slot i, which from Definition 2 is given by Remark 3. From extensive numerical experiments, we have
τi,j found that the choice a = b = 1 (corresponding to Legendre Vi = v(τi,j , ci,j )= ci,j,lT ρ, (18) polynomials) yields the most accurate approximations for a range of moments (corresponding to different network setups). Xl=1 (a,b) As Pn is a polynomial, we can write an as which is a random sum due to τi,j and ci,j,l (given in (7)). 1 n n Remark 2. In general, the distribution of depends on s E s τi,j an = bn fWi (x) ζn,sx = bn ζn,s [Wi ], (24) i. For short compounding intervals this is to ensure that the −1 s=0 s=0 Z X X connection duration is not longer than the time the system has where ζn,s corresponds to the coefficient of the s-th order term been running. In other cases, the distribution may vary due (a,b) of Pn . This means that the approximation is completely to seasonal usage trends and events, such as holidays. This characterized by the raw moments of Wi. Observe that the has the important consequence that the moments and hence moments of Wi are related to the moments of Vi via distribution of Vi depend on the time slot i that user (i, j)’s s connection ends. E s E 2(Vi + vi,min) [Wi ]= − 1 , (25) vi,max + vi,min It is clear from the fact that V is a random sum and i which can be readily evaluated (given the moments of V ) via the form of c (see (7)) that the distribution of V is not i i,j,l i the binomial expansion. readily obtained in closed-form. As such, we instead adopt a principled approach to approximating Vi, which is based on Remark 4. To obtain the moments of Vi (and hence the mo- an Askey-orthogonal polynomial expansion. In particular, we ments of Wi), we compute and then differentiate the moment use the Jacobi polynomials since the support of the income generating function, which allows us to use the probability Vi is bounded on [−vi,min, vi,max]. This follows from the fact generating functional of the PPP; surpassing the need to ex- that both τi,j and ci,j,l are bounded. plicitly derive the distribution of the interference. The moments It is important to note that the Jacobi polynomials are only and details of the derivations are given in Appendix A. orthogonal on [−1, 1]. Hence, we need to transform Vi so The distribution of Vi is then obtained from the distribution that it has also has support [−1, 1]. This is achieved via the of Wi via the transformation transformation 2 2(x + vi,min) fVi (x)= fWi − 1 , (26) 2(Vi + vi,min) vi,max + vi,min vi,max + vi,min Wi = − 1, (19) vi,max + vi,min where fWi (x) is given by (20). We summarize the procedure in Algorithm 2. where vi,min = −cminTρ and vi,max = icmaxTρ, from (18). The distribution of Wi via the Jacobi polynomial represen- C. Recursion for the Net Profit PDF tation is then given by [20] The second stage is to compute the PDF of the net profit.
d Recall from Definition 3, that the net profit for time slot i is (a,b) given by fWi (x) ≈ K(x) amPm (x), (20) m=0 Ni X Snet(i)= [v(τi,j , ci,j ) − C(τi,j , ki,j )] . (27) (a,b) j=1 where d is the order of the approximation, Pm (x) is the X 8
Algorithm 2 Orthogonal Polynomial Basis Expansion of the Remark 5. We emphasize that the support of Snet(i) is over Net Profit PDF Z, not just N. This means that the standard recursion in [22] 1. Compute the moments of Vi using (60) and (61) in Ap- (known as the Panjer recursion) is not applicable; the more pendix A. general approach in [17] is required. 2. Transform Vi to the random variable Wi with support [−1, 1] via (19). In practice, the sum (30) is in fact bounded (due to the finite 3. Compute the basis expansion coefficients an via (24). resolution of the discretization step), which gives 4. Compute the basis expansion of fW (x) using (20). i f ((n + 1)∆) 5. Transform f (x) via (26) to obtain f (x). Snet(i) Wi Vi w = N 1 − wN hZi (0) kmax It is important to note that S (i) has a different distribution net × (n + 1)h ((k + 1)∆)f ((n − k)∆), for each i (see Remark 2). Moreover, observe that the distri- Zi Snet(i) k=kmin, k6=−1 bution of C(τi,j , ki,j ) is discrete and the approximation of the X (31) distribution of v(τi,j , ci,j ) from Section IV-B is continuous 3 (with bounded support). Also note that Snet(i) is a random where kmin = min Supp(Snet(i)) − 1 and kmax = sum (in both the summands and number of summands). max Supp(Snet(i)) − 1, depend on the discretization resolu- As such, it is not straightforward to efficiently compute the tion. distribution of Snet(i). In order to solve the linear recurrence relation in (31), we Fortunately, linear recursions have been developed to eval- use a convex reformulation in terms of a least squares problem, uate distributions for sums closely related to Snet(i). In order to apply these recursive approaches, we first need to find min kAfk2, (32) f: f T f=1 the distribution of v(τi,j , ci,j ) after it has been discretized. The discretization of v(τ , c ) can be obtained via standard A i,j i,j where consists of the coefficients in terms of hZi from (31) approaches such as the Lloyd algorithm [21]; although it and f is the distribution of Snet(i). We note that in principle is important to carefully consider any negative terms in the other objectives can be used in (32), such as those discussed distribution of v(τi,j , ci,j ) that might occur due to the basis in [17]; however, we have found via numerical experiments expansion approximation. We denote the discretization as that the least squares objective is suitable for the purpose of vˆ(τi,j , ci,j ). approximating the ruin probability.
To obtain the PDF of the net profit, denoted by hZj , we now embed the discretized income into the lattice ∆Z by choosing ∆ > 0 sufficiently small and rounding the support D. Recursion for the Probability of Ruin of c . We then obtain the distribution vˆ(τi,j , i,j ) The third (and final) stage of the calculation is to compute the probability of ruin. This is achieved by collecting the hZj (k) = Pr(Zj = k), (28) distributions of Snet(l) and substituting into the new ruin of the random quantity probability recursion, which we derive in this section.
Zj = [ˆv(τi,j , ci,j ) − C(τi,j , ki,j )] , (29) It is important to note that the ruin probability (defined in Definition 5) is the probability that the revenue surplus is ˆ c by convolving the (discrete) distributions of V =v ˆ(τi,j , i,j ) negative at any l ≤ L. Formally, the ruin probability can be and C(τi,j , ki,j ), which both have discrete support on S ⊂ written as ∆Z, with |S| < ∞. At this point, we obtain the approximate distribution of ψL(u) = Pr(S1(u) < 0 ∪···∪ SL(u) < 0). (33) Snet(i). This is based on the following linear recurrence based on [17, Eq. (1.7)], It is helpful to write the ruin probability in terms of the survival probability (see Definition 5), which is given by fSnet(i)((n + 1)∆) w = N φL(u)=1 − ψL(u) = Pr(S1 ≥ 0 ∩···∩ SL ≥ 0). (34) 1 − wN hZi (0) ∞ We now give the recursion for the probability of ruin in
× (n + 1)hZi ((k + 1)∆)fSnet(i)((n − k)∆). terms of the distributions of Snet(l), l = 1, 2,...,L. First, k=−∞, k6=−1 define X (30) We note that similar difference equations can be derived Gl(y) = Pr(Snet(l) ≤ y). (35) for distributions of N , other than geometric; in fact, it is i We then have the following theorem, which generalizes the straightforward to extend to any distribution in the Katz family ruin probability recursion in [18], [23]. (i.e., binomial, Poisson, and negative binomial distributions). Further details may be found in [17, Eq. (1.3)] and [22, Eq. 3 (1)]. The set Supp(Snet(i)) corresponds to the support of Snet(i). 9
Theorem 1. The survival probability after L time periods (e.g., months) is given by 70
φL(u)= φL−1(u)(1 − GL(0)) 60 0 y + φ u + dG (y), ∀L ≥ 2, L−1 (1 + r)L L 50 Z−∞ (36) Time steps (in months) 40 n = 1,2,3,... where
φ1(u)=1 − G1(−u(1 + r)). (37) 30
The ruin probability is then obtained via ψL(u)=1 − φL(u). Initial Capital Bound 20 Proof: See Appendix B. We note that the integral in Theorem 1 is a Stieltjes integral. 10 This is important as Gl, l =1, 2,...,L is in fact a sequence of distributions, all with discrete and bounded support. As 0 such, the Stieltjes integral is a finite sum and can be efficiently 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Expected Revenue Surplus E[V] evaluated numerically.
Fig. 2. Plot of bound on initial capital u to ensure E[Snet(l)] ≥ 0, where V. NUMERICAL AND SIMULATION RESULTS r = 0.05, E[N] = 100, E[C] = 0.1. With our evaluation framework based on ruin-theory in hand, we now turn to evaluating facility network sharing. We obtain pricing schemes that yield a probability of ruin less limitations of any analysis based on expected revenue surplus. In particular, short-term behavior of the revenue surplus is not than 10%, under practical operating conditions. Along the way, we present important tradeoffs between physical layer accounted for, which has the consequence that fluctuations resources such as pathloss, and financial constraints such as in user demand or available network resources can cause the initial capital, interest rate (compounded monthly) and pricing. revenue surplus to drop below zero—leading to ruin. As such, Before evaluating the ruin probability using the techniques it is necessary to use our framework based on ruin theory we have developed in Section IV, we first present a simplified to account for these fluctuations and ultimately reduce the analysis based purely on the expected revenue surplus (see financial risk to the facility. Definition 3). This analysis provides initial insights into the To account for fluctuations in the revenue surplus, we now financial aspects of the problem, which may be unfamiliar to turn to our evaluation framework based on ruin theory. We consider the network setup in Table I and define Ad = the wireless communication community. That is, we consider (1) 2R /B − 1, where R(1) is the rate product on offer. l l l−i E[Snet(l)] = u(1 + r) + (1 + r) E[N] (E[V ] − E[C]) , TABLE I i=1 SUMMARYOFNETWORKPARAMETERS. X 2 (38) Parameter K σ P0 = PI wN Q T ρ C κT Value 1 0 1 0.2 1 1 100 1 month where we assume that the connection duration time τi,j is constant, which means that the revenue per user Vi has ex- pectation E[V ] for all i. The condition on the initial capital to We first consider the moments of the revenue E[V ]. These ensure positive average revenue surplus (i.e., E[Snet(l)] ≥ 0) moments are ultimately used to compute the probability of at a given time n for interest rate r is then given by ruin. They are also of interest in their own right. We show this next by examining the role of the key wireless network (1 + r)n − 1 u ≥ E[N] (E[C] − E[V ]) . (39) and financial parameters α, β, cmin, and cmax. r(1 + r)n In Table II, we compare the small-cell density β with the To illustrate this condition, Fig. 2 plots the expected revenue first moment of V , E[V ], obtained numerically via (61) and E[V ] versus the bound on u; each curve corresponding to a via Monte Carlo simulation. We point out that the moments different time slot n. First, observe the initial capital needs obtained numerically via (61) are in good agreement with the to be non-zero to ensure a positive expected revenue surplus moments obtained via Monte Carlo simulation. Importantly, when the costs exceed the revenue; i.e., E[C] > E[V ]. Second, the moment E[V ] is constant irrespective of the small-cell observe that there is a diminishing increase in the required density for both the numerical and Monte Carlo approaches. initial capital as the number of time steps is increased; this This suggests that the small-cell density does not play an can be seen by comparing the gaps between the curves, for important role in networks well-modeled by PPPs. We believe a fixed E[V ]. Moreover, as the bound on u in (39) scales the reason for this is that as the density of small-cells increases, linearly with E[N], the initial capital increases significantly as there is an increase in nearby interfering small-cells, which is the number of connected users increase, when E[C] > E[V ]. balanced by a closer servicing small-cell. Note that the small- These are general trends; however, it is important to note the cell density has a similar effect of outage probability in the 10
TABLE II TABLE SHOWING EFFECT OF VARYING SMALL-CELL DENSITY β, FOR DIFFERENT PATHLOSS EXPONENT α USING NUMERICAL (NUM.) AND MONTE CARLO (M.C.) APPROACHES.EVALUATION PERFORMED WITH cmin = 0.1, cmax = 100, Ad = 100. β E[V ] Num., α = 3 E[V ] M.C., α = 3 E[V ] Num., α = 4 E[V ] M.C., α = 4 0.01 76.5 75.5 60.5 60.2 0.1 76.5 77.0 60.5 60.9 1 76.5 76.4 60.5 59.8
low noise region, as shown in [19]. 1
0.9 90 0.8 80 0.7
70 0.6 Monte Carlo Numerical (Based on (28)) 60 0.5
50 0.4
0.3 40 A = 50, 100, 200 d 0.2 30 Cumulative distribution function of V
Expected Revenue E[V] 0.1 20 0 Numerical (Based on (63)) 0 200 400 600 800 1000 10 Monte Carlo v 0 3 3.5 4 4.5 5 Pathloss Exponent α Fig. 4. Plot of basis expansion approximation for the CDF of V using four moments. Parameters are α = 4, β = 0.1, cmin = 0.001, cmax = 1000, Ad = 100. Fig. 3. Plot of expected revenue E[V ] versus pathloss exponent α, with R(1) /B varying Ad = 2 − 1. Parameters are β = 0.1, cmin = 0.1, cmax = 100, Ad = 100. experiments suggest that a = b =1 is a robust choice. Table III shows the ruin probability after 5 months versus Next, we consider the effect of the pathloss on the revenue. the initial capital u, for varying Ad (reflecting different rate Fig. 3 plots the pathloss exponent versus the expected revenue products on offer). As expected from Fig. 2, in order to obtain E[V ], which shows excellent agreement between our numeri- a low probability of ruin, the initial capital needs to be chosen cal result (based on (61)) and Monte Carlo results. The trend carefully. This is reflected in both the numerical and Monte illustrated by the figure is that the revenue decreases as the Carlo results. We note that for low (u = 100) and high (u ≥ pathloss exponent increases, irrespective of Ad (corresponding 250) initial capital, the numerical and Monte Carlo approaches to different rate products R(1) on offer). This is due to the fact are in agreement. However, for u ≈ 200, there is a difference that it is easier to service users with a high pathloss exponent of approximately 0.1. This is due to the discretization step as it means that there are often lower interference levels. detailed in Section IV-C. Importantly, to ensure the probability Following Algorithm 1, the next step to compute the ruin of ruin is less than approximately 10%, an initial capital of probability is to obtain the PDF of the revenue V via basis u> 250 is required, with the parameters used in Table III. expansion. To illustrate this step, the basis expansion approx- imation for the CDF of V is plotted in Fig. 4. We observe TABLE III RUIN PROBABILITY AFTER 5 MONTHS WITH VARYING INITIAL CAPITAL u. that our numerical approximation is in good agreement for PARAMETERS ARE α = 4, β = 0.1, cmin = 0.001, cmax = 1000, revenue less than 200, and then has small oscillations for high Ad = 100, AND r = 0.05. revenue values. It is important to note that tail oscillations Initial Capital u 100 150 200 250 300 are common when using the Jacobi polynomial representation, Numerical (Using Theorem 1) 0.33 0.09 0.01 0 0 and must be carefully accounted for. In the setup for Fig. 4, Monte Carlo 0.33 0.22 0.11 0.05 0.02 the approximation is good; however, as cmin is increased or cmax decreased, the discontinuity affects the quality of the approximation. As such, additional moments (i.e., E[V 5],..) are required to smooth the oscillations, which can be obtained VI. CONCLUSIONS easily from our analysis in Appendix A. We also note that the Due to the recent availability of cheap small-cells and the parameters a,b in the Jacobi polynomials (see Section IV-B) unique operating requirements of facilities, there is a need strongly influence the approximation; extensive numerical for alternative network sharing arrangements. To this end, we 11 proposed a facility network sharing arrangement, which is by based on: a leasing agreement between traditional operators τ and the facility; and a service agreement between users and AIl LV (t)= E exp −t max min , cmax , cmin Tρ the facility. Unlike traditional operators, the local scale of Hl " l=1 !# facilities means that technical design at the level of the network X E E E −t architecture is intimately connected with the profitability; = τ A Hl,Il exp instead of loosely coupled. AI τ × max min l , c , c Tρ A τ . In order to evaluate our facility network sharing arrange- H max min l ment, we adopted a socio-technical system design approach. (42)
As such, our key performance metric is the ruin probability— leading to a new ruin-based evaluation framework. To evaluate Note that Ri,j and τ have discrete support, so these expec- the ruin probability, we proposed a numerical approximation tations are sums we evaluate last. We now evaluate the inner method, which is shown to be in good agreement with Monte expectation over the interference I and the channel gain Hl. Carlo simulation. Our approximation method includes two key Observe that the inner expectation can be written as novel aspects: computation of the moments of the revenue AI via stochastic geometry techniques; and a new recursion E(t)= E exp −t max min l ,c ,c Hl,Il H max min for the ruin probability, which is tailored for the wireless l communication setting. Our numerical results suggested that × Tρ) A, τ there are in fact concrete conditions where profitable operation of facilities is possible, with sufficient initial capital. = E1(t)+ E2(t)+ E3(t), (43)
where ACKNOWLEDGEMENTS −tT ρcmin AIl AIl E1(t)= E e
AIl × Pr cmin ≤ ≤ cmax APPENDIX A Hl cmax −tT ρw In this appendix, we derive the moments of the income for = e fW (w)dw, (44) c a user with connection ending in time slot i, which is used Z min to compute the ruin probability. Importantly, our analysis can with be straightforwardly extended to higher moments. Recall that AI the income from a single user (a typical node, by Slivnyak’s W = l . (45) theorem for homogeneous Poisson point processes) is given Hl by To compute E3(t), observe that
τi,j AI AI − l AIl − l v(τ , c )= c T ρ, (40) Pr(W ≤ w)= EI e w ⇒ fW (w)= EI e w . i,j i,j i,j,l l l w2 l=1 X h i (46) where the scaling factor is Hence,
R /B cmax 2 i,j − 1 P I AIl AIl I i,j,l E −tT ρw − w c = max min ,c ,c . E3(t)= Il e e dw . (47) i,j,l 2 −α max min w2 ( ( |hi,j,l| di,j P0 ) ) Zcmin (41) Next, we integrate by parts to obtain
Ri,j /B PI (2 −1) 2 We also define: A = −α ; Hl = |hi,j,l| ; τ = τi,j ; E(t)= E1(t)+ E2(t)+ E3(t) P0d i,j 1 and V = v(τi,j , ci,j ). cmin 1 −tT ρcmax −tT ρ/uE −AIlu = e + tTρ 2 e Il e du. In order to compute the four moments, we require the 1 u moment generating function (MGF) of V . In turn, the MGF Z cmax (48) is obtained from the Laplace transform LX (t), which is given 12
−AI u In order to compute E(t), we require EI [e l ], which is We now require the identity from [24, Eq. 6.4552], given by ∞ ν µ−1 −βx α Γ(µ + ν) x e γ(ν, αx)dx = µ+ν (a) −α 0 ν(α + β) E e−AIlu = E e−Augmrm Z Il X ,{gm} α m∈X \{b0} × 2F1 1,µ + ν; ν + 1; . Y α + β − (56) (b) −Augr α = E E e m X g m∈XY \{b0} h i Identifying terms, we obtain ∞ − (c) −Augz α = exp −2πβ 1 − Eg e zdz , 2 1− 2 − α −α α rU α 1 AurU Z h i G2 = (49) 2 Au 2 −α 2 1 − α AurU +1 where: follows from ; follows from the −α (a) Hl ∼ exp(1) (b) 2 Aur U × 2F1 1, 2;2 − ; . (57) fact that {gm} is independent of the spatial point process; and −α α 1+ AurU (c) follows from the probability generating functional of the 4 PPP . −AI u To compute E[e l ], substitute G1 and G2 into (50). This Continuing, we have result is then used to compute the Laplace transform and obtain the moments via E −AIlu Il e − ∞ r α n U n n d LVi (t) 1 −Agyu − 2 −1 −g E (58) = exp −2πβ 1 − e y α dye dg , [Vi ] = (−1) n |t=0. α dt Z0 Z0 ! (50) All that remains is to explicitly compute the moments. We (k) −α define the following terms: E0 (t) is the k-the derivative of which follows from the change of variables y = z ⇒ z = (k) 1 −1/α 1 − α −1 E(t); E0 is the k-the derivative of E(t) evaluated at t = y ⇒ dz = − α y dy. (k) Now consider the inner integral in (50), given by 0; and L0 is the k-th derivative of LX (t) conditioned on d, Ri,j , τ evaluated at t =0. Note that E(0) = 1. −α rU −Agyu − 2 −1 Using (48), we have F = 1 − e y α dy 0 1 Z c −α min α αAgu −tT ρcmax 1 −tT ρ/u −AIlu −AgrU u 2 E = − 1 − e rU + F1, (51) E(t)= e + tTρ 2 e Il e du, 1 u 2 2 c Z max (s) s −tT ρcmax where E (t) = (−Tρcmax) e − 1 α c rU min 1 −Agyu − 2 s+1 s −tT ρ/uE −AIlu F = e y α dy. (52) + s(−1) (Tρ) s+1 e Il e du 1 1 u 0 Z cmax Z 1 cmin 1 Integrating by parts again we obtain, s+1 −tT ρ/uE −AIlu − t(Tρ) s+2 e Il e du, 1 u − 2 +1 Z cmax 1 α 2 F = γ 1 − , Agur−α , (53) (59) 1 Agu α U where E(s)(t) is the s-th derivative of E(t) and E[e−AIlu] is where γ(s, x)= x ts−1e−tdt. 0 given by (50). Next, we compute the outer integral in (50), which is given R Now, let 1 denote the indicator function. The moments by conditioned on di,j , Ri,j , τi,j can now be readily obtained. We ∞ −g illustrate with the first conditional moment. G = Fe dg = G1 + G2, (54) Z0 (1) (1) L = τE 1τ>0. (60) where 0 0 2 ∞ αr −α Finally, the moments for the revenue from a user with U −AgurU −g G1 = − 1 − e e dg, connection ending in time slot are given by 2 0 i Z 2 − α ∞ α 1 2 2 Q α −g −α n ∞ G2 = g e γ 1 − , Agur dg. 2 2 Au α U E[V k]= Pr(R = k)Pr(τ = l) L(k)e−πβz 2πβzdz, Z0 i i,j 0 (55) l=1 k=1 Z0 X X (61) 4The probability generating functional property of the PPP gives − − E β RR2 (1 f(x))dx [Qx∈X f(x)] = e . which can be efficiently evaluated numerically. 13
APPENDIX B [12] R. Berry, M. Honig, T. Nguyen, V. Subramanian, H. Zhou, and R. Vohra, “On the nature of revenue-sharing contracts to incentivize spectrum- Proof of Theorem 1: First (from (34)), observe that the sharing,” in Proc. IEEE INFOCOM, 2013. survival probability can be written as [13] Y. Yang and T. Quek, “Optimal subsidies for shared small cell networks– a social network perspective,” IEEE Journal on Selected Topics in Signal n−1 Processing, vol. 8, no. 4, pp. 690–702, 2014. n n−i φn(u)=Pr u(1 + r) + Snet(n)+ Snet(i)(1 + r) ≥ 0[14], Y. d’Halluin, P. Forsyth, and K. Vetzal, “Managing capacity for telecom- i=1 munications networks under uncertainty,” IEEE/ACM Transactions on X Networking, vol. 10, no. 4, pp. 579–588, 2002. ...,u(1 + r)+ Snet(1) ≥ 0) . (62) [15] S. Asmussen and H. Albrecher, Ruin Probabilities. World Scientific, 2010. Also recall that [16] H. ElSawy, E. Hossain, and M. Haenggi, “Stochastic geometry for modeling, analysis, and design of multi-tier and cognitive cellular Gn(y) = Pr(Snet(n) ≤ y). (63) wireless networks: a survey,” vol. 15, no. 3, pp. 996–1019, 2013. [17] W. H¨urlimann, “Negative claim amounts, Bessel functions, linear pro- Now, using (34) and (63) yields φ1(u) = Pr(Snet(1) ≥ gramming and Miller’s algorithm,” Insurance: Mathematics and Eco- −u(1 + r)) =1 − G1(−u(1 + r)). We then have for n ≥ 2 nomics, vol. 19, pp. 9–20, 1991. [18] L. Sun and H. Yang, “Ruin theory in a discrete time risk model with interest income,” British Actuarial Journal, vol. 9, no. 3, pp. 637–652, n−1 Snet(n) φ (u)=Pr u(1 + r) + 2003. n 1+ r [19] J. Andrews, F. Baccelli, and R. Ganti, “A tractable approach to coverage n−1 and rate in cellular networks,” IEEE Transactions on Communications, n−1−i vol. 59, no. 11, pp. 3122–3134, 2011. + Snet(i)(1 + r) ≥ 0,... [20] M. Cruz, G. Peters, and P. Shevshenko, Fundamental Aspects of Oper- i=1 ! X ational Risk and Insurance Analytics. Wiley, 2014. = φn−1(u)(1 − Gn(0)) [21] A. Gersho and R. Gray, Vector Quantization and Signal Compression. Springer, 1992. 0 y + φ u + dG (y), (64) [22] H. Panjer, “Recursive evaluation of a family of compound distributions,” n−1 n n Astin Bulletin, vol. 12, pp. 22–26, 1981. −∞ (1 + r) Z [23] F. De Vylder and M. Goovaerts, “Recursive calculation of finite-time which follows after considering ruin probabilities,” Insurance: Mathematics and Economics, vol. 7, pp. 1–7, 1988. n−1 [24] I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products. Snet(n) u(1 + r)n−1 + + S (i)(1 + r)n−1−i ≥ 0 Academic Press, Seventh ed., 2007. 1+ r net ( i=1 n−1 X n−1 n−1−i ∩u(1 + r) + Snet(i)(1 + r) ≥ 0 . (65) i=1 ) X
REFERENCES [1] GSMA, Mobile Infrastructure Sharing. gsma.com, 2012. [2] L. Doyle, J. Kibilda, T. Forde, and L. DaSilva, “Spectrum without bounds, networks without borders,” Proceedings of the IEEE, vol. 102, no. 3, pp. 351–365, 2014. [3] A. Ghosh, N. Mangalvedhe, R. Ratasuk, B. Mondal, M. Cudak, E. Visotsky, T. Thomas, J. Andrews, P. Xia, H. Jo, H. Dhillon, and T. Novlan, “Heterogeneous cellular networks: from theory to practice,” IEEE Communications Magazine, vol. 50, no. 6, pp. 54–64, 2012. [4] J. Andrews, “Seven ways that HetNets are a cellular paradigm,” IEEE Communications Magazine, vol. 51, no. 3, pp. 136–144, 2013. [5] D. Lopez-Perez, I. Guvenc, G. de la Roche, and M. Kountouris, “En- hanced intercell interference coordination challenges in heterogeneous networks,” IEEE Wireless Communications Magazine, vol. 18, no. 3, pp. 22–30, 2011. [6] H. ElSawy, E. Hossain, and D. Kim, “HetNets with cognitive small cells: user offloading and distributed channel access techniques,” IEEE Communications Magazine, vol. 51, no. 6, pp. 28–36, 2013. [7] D. Calin, A. Kaya, B. Kim, K. Yang, and S. Yiu, “On the high capacity lightRadioTM metro cell design for stadiums,” Bell Labs Technical Journal, vol. 18, no. 2, pp. 77–97, 2013. [8] M. Markendahl and A. Ghanbari, “Shared smallcell networks multi- operator or third party solutions - or both?,” in Proc. International Symposium on Modeling & Optimization in Mobile, Ad Hoc & Wireless Networks, 2013. [9] L. Duan, J. Huang, and B. Shou, “Investment and pricing with spectrum uncertainty: a cognitive operator’s perspective,” IEEE Transactions on Mobile Computing, vol. 10, no. 11, pp. 1590–1604, 2011. [10] J. Park, S. Kim, and J. Zander, “Asymptotic behavior of ultra-dense cel- lular networks and its economic impact,” in Submitted to GLOBECOM 2014, 2014. [11] R. Berry, M. Honig, and R. Vohra, “Spectrum markets: motivation, challenges and implications,” IEEE Communications Magazine, vol. 48, no. 11, pp. 146–155, 2010.