A Ruin Theoretic Design Approach for Wireless Cellular Network Sharing with Facilities
aa r X i v : . [ c s . C Y ] M a y A Ruin Theoretic Design Approach for WirelessCellular Network Sharing with Facilities
Malcolm Egan , Gareth W. Peters , Ido Nevat and Iain B. Collings Faculty of Electrical Engineering, Czech Technical University in Prague, Czech Republic. Department of Statistical Sciences, University College London (UCL), London, England. Institute for Infocomm Research, Singapore. Department of Engineering, Macquarie University.
Abstract —With the rise of cheap small-cells in wireless cellularnetworks, there are new opportunities for third party providers toservice local regions via sharing arrangements with traditionaloperators. In fact, such arrangements are highly desirable forlarge facilities—such as stadiums, universities, and mines—asthey already need to cover property costs, and often have fibrebackhaul and efficient power infrastructure. In this paper, wepropose a new network sharing arrangement between largefacilities and traditional operators. Our facility network sharingarrangement consists of two aspects: leasing of core networkaccess and spectrum from traditional operators; and serviceagreements with users. Importantly, our incorporation of a userservice agreement into the arrangement means that resourceallocation must account for financial as well as physical resourceconstraints. This introduces a new non-trivial dimension intowireless network resource allocation, which requires a newevaluation framework—the data rate is no longer the onlymain performance metric. Moreover, despite clear economicincentives to adopt network sharing for facilities, a businesscase is lacking. As such, we develop a general socio-technicalevaluation framework based on ruin-theory, where the key metricfor the sharing arrangement is the probability that the facilityhas less than zero revenue surplus. We then use our framework toevaluate our facility network sharing arrangement, which offersguidance for leasing and service agreement negotiations, as wellas design of the wireless network architecture, taking into accountnetwork revenue streams.
Index Terms —network sharing, facilities, ruin theory
I. I
NTRODUCTION
In traditional wireless networks, expensive infrastructureand the rapid adoption of new radio technologies resultedin small numbers of new market entrants. More recently,sharing of the radio access network (RAN) has been adoptedto reduce capital expenditures (including infrastructure) of newoperators, while still offering wide coverage and high qualityof service (QoS). This trend has encouraged new operatorsto enter the market and sophisticated arrangements betweeninfrastructure owners and operators are being considered.There are many possible network sharing arrangementsfor wireless cellular network infrastructure, including mastsharing, full RAN sharing, roaming, or core-network sharing.At present, the most common sharing arrangements haveinvolved only the RAN [1], which allows operators to poolresources and can increase the capacity available for operatorsto service regions that already have high base station (BS) density or to improve coverage to under-served areas indeveloping regions. A common RAN sharing arrangement isbased on companies that only offer base stations to operators(known as tower companies), which are now a large part of thewireless industry; particularly in India and the United States.In other regions, established operators have heavily investedin infrastructure and negotiated BS sharing arrangements withnew entrants to the market, such as in Sweden. Ultimately,these approaches can be viewed as the first steps towardsthe dynamic market-based “networks without borders” visionwhere resources including spectrum, RANs and core networksare pooled, with contributors ranging from individuals totraditional operators [2].In parallel with the adoption of network sharing, small-celltechnology (also known as femto, micro, pico, or metro-cellsdepending on the provider and transmitting capabilities) isrevolutionizing the wireless industry with cheap, low-powerbase stations [3]. By exploiting small-cells, wireless networkscan offer high data rates with a small footprint via denseplacement, which reduces the distance between the small-celland the user—the most effective means of increasing networkcapacity [4].Despite the success of current network sharing arrangementsand small-cells, there remain complex issues to be resolved.In particular, property must be leased to anchor small-cells,and backhaul must be installed in order to implement networkMIMO (including CoMP) as well as inter-cell interferencecoordination (eICIC) [5]—key techniques for the effectiveoperation of small-cell networks. Moreover, operational ex-penditures (OpEx)—largely due to small-cell maintenance—are growing with the increasing number of small-cells in thenetwork. This increase in OpEx will continue until effectiveself-organizing network (SON) technologies [6] are success-fully implemented.A particularly challenging scenario for operators is largefacilities with high data rate demands, even when standardnetwork sharing arrangements are employed. Common exam-ples of these facilities are universities, stadiums, conventioncenters, utilities, mines, and high density urban residences.The challenge arises due to the high cost in leasing small-celllocations, cost of leasing high bandwidth backhaul (despitethe fact that the backhaul is often available within thesefacilities via fibre links), the often unusual characteristics ofuser demands—such as high upload rates in stadiums [7]— and large variations in data rate demand over time. Moreover,large facilities may desire to only charge low rates to users inorder to improve the operation of the facility; in stark contrastwith standard wireless service arrangements. This can occureither because the users are employees and the mobile serviceis paid for by the facility, or a cheap mobile service is used asan attraction to the facility; similar to how WiFi is currentlyoffered free of charge to users in many convention centers andhotels.
A. A New Socio-Technical Network Sharing Approach
There is strong motivation for large facilities to offer cheapmobile services and to develop their own capability to do so;namely, the low cost of small-cells, and the availability of highbandwidth backhaul. As such, there is a genuine need for analternative network sharing arrangement that can exploit theunique characteristics of large facilities.In this paper, we propose a new network sharing arrange-ment for facilities. Our facility network sharing arrangementis based on a network consisting of small-cells operated byand within the facility, which contrasts with tower companiesthat own multiple small-cells and BSs, which are leased tooperators.There are two key aspects to our facility network sharingarrangement: the core network leasing agreement betweentraditional operators and the facility; and the service agreementbetween the facility and users. The purpose of the core networkleasing agreement is to provide the facility with data to serveusers that are subscribed to traditional operators as well asspectrum localized to the facility. This agreement involves afee that the facility pays to the traditional operator, in exchangefor core network access and spectrum. On the other hand,the service agreement is the mechanism by which the facilityobtains revenue. In particular, the facility charges users forits services; depending on the resources required to ensurereliable transmission, and also pricing parameters designed tocompensate for the leasing fee for core network access.In concept, our facility network sharing arrangement bearssimilarities to the “local network operated by an independentactor” classification for indoor cellular networks proposed in[8]. More specifically, the approach in [8] introduced thenotion of third party operators to service users inside largebuildings, which negotiate with traditional operators for dataaccess. Our facility network sharing arrangement differs in twokey aspects: (i) we are not limited to indoor operation, whichis achieved using a general stochastic geometry model that canin principle be extended to include small-cell cooperation viaeICIC techniques; and (ii) we propose a specific agreementstructure between operators, users, and facilities. In particular,core network access is provided via a new leasing arrangementthat can in principle be formulated as a contractual obligation,which we detail in Section III.While facilities have currently not been considered withina network sharing arrangement, there are six key economicincentives, subject to the presence of appropriate antitrustregulation: (1) the facility already owns or leases the property,which means that there is zero sunk cost for positioning small-cells; (2) the facility’s high bandwidth fibre network can be exploited at no additional cost to provide high rates throughadvanced eICIC; (3) competition will be increased betweentraditional operators in the region through core network accesscosts (in contrast with RAN discrimination); (4) users willbe charged only at the facility’s incremental costs as thefacility is either covering the costs itself (e.g., in mines orutilities) or the service is offered as an attraction (e.g. inhotels or convention centers); (5) facilities can offer moreefficient power sources as they also must power the facility(e.g., through large-scale solar cell sources); and (6) SONinnovation is promoted as the facilities chase a reduced OpEx.An incremental approach to SON is also possible since thefacilities will have a relatively small number of small-cells,which means that there are fewer complexities compared withthe large-scale RANs of traditional operators.These benefits clearly suggest that there is a significantpotential for network sharing arrangements involving facilities.
B. Evaluation of Facility Network Sharing Arrangements
Although there are clear economic incentives for the adop-tion of facility network sharing arrangements, a business casebased on a quantitative framework is lacking. We addressthis issue by evaluating facility network sharing using a newframework, which incorporates both the wireless communica-tions network, as well as the service and leasing agreementsbetween users, the facility, and traditional wireless operatorsthat own the core networks. In contrast with standard analyticalframeworks in the wireless communications literature, ourframework models the facility as a socio-technical system . Thismeans that the conclusions arising from our framework lieclose to those used directly in real-world practice. Moreover,our framework is general, which means that it can be adaptedto a range of wireless settings, where resource sharing is usedand financial sustainability is a key design criterion.Profit-based approaches for resource allocation in cellularnetworks have been considered in [9]–[14]. Early work oncapacity pricing adapted a real options framework to copewith uncertainties [14]. In [9], [13], cognitive cellular net-works and small-cell networks, respectively, were modeledvia Stackelberg games. The approach in [13] employed sub-sidies to incentivize closed access small-cell access points toshare allocated spectrum. The approaches in [11], [12] alsoexclusively focused on spectrum allocation. In [10] the BSdensity and spectrum usage were optimized to maximize thenet profit; however, this analysis was based solely on long-term average rates. The drawback of such an approach isthat it does not consider any assessment of the feasibility ofthe business model nor optimal solutions obtained from theperspective of a financial risk management analysis. As such, itis not easy to assess the long-term profitability of the operator.This is important as network design that takes profitabilityinto account will differ from standard designs under purelytechnical constraints and generally lead to improved operatorlongevity. Key parameters such as compounded interest rates,initial capital, expenditures, and future investments also can-not be incorporated into the standard approaches. Moreover,previous work does not account for the effect of fluctuationsin the number of users and their cellular data demands.
Jointly addressing a number of the key financial andtechnical factors that arise in real-world cellular networks(including features such as channel gains, power control,path loss and the duration of user connections), is crucial toreliably predicting and optimizing the financial sustainabilityof network sharing arrangements. However, this requires amore sophisticated socio-technical framework. In this paper,we propose a new quantitative framework to evaluate facilitynetwork sharing arrangements based on a reformulation of ruintheory [15] to facilitate a study of business models, where thekey performance metric is the probability that the owner ofthe facility has a negative revenue surplus within a period of n months—leading to financial insolvency. We emphasize thatthe probability of these events, termed ruin, is dependent onboth economic and technical factors, which include: the initialcapital of the facility; the compound interest rate (compoundedmonthly); link parameters, such as channel gains, powercontrol and path loss; duration of user connections; and thenumber of users in the network. C. Key Contributions
We summarize our three key contributions as follows.1) We propose a new network sharing arrangement betweentraditional operators, facilities, and users. The arrange-ments accounts for both the expenditures and revenue forthe facility, which arise due to the cost of leasing accessto traditional operators’ core networks (expenditures), andthe income from servicing users normally subscribed totraditional operators (revenue). Importantly, the servicecharges incurred by each user depend on the physicalresources required to ensure reliable transmission. Assuch, the service charges are strongly coupled to thecapabilities of the RAN.2) We develop a quantitative ruin theory-based framework toevaluate facility network sharing. The evaluation is basedon the probability of ruin; that is, the probability thatthere is a negative revenue surplus within a period of n months (a standard metric for financial risk analysis [15]).In order to obtain the probability of ruin, we:a) Derive the moments of the revenue from user servicecharges, based on a practical wireless heterogeneousnetwork model. In particular, we exploit stochasticgeometry techniques [16] to model the placement ofbase stations and users.b) Use the moments of the revenue time-varying stochas-tic process to compute the probability distributionof the net profit derived by the facility operator byincorporating revenue from the service of each user.c) Compute the probability of ruin via the probability dis-tribution of the net profit and efficient recursions (basedon linear difference equations), which are motivatedby results in insurance theory [17], [18]. Importantly,we extend the applicability of the standard recursionsaccount for idiosyncratic (from the perspective of in-surance) aspects of wireless communication networks. Historically, ruin theory originally arose in the study of solvency require-ments for insurance [15].
3) We evaluate facility network sharing based on numericalevaluation of the probability of ruin via our framework.First, we demonstrate that facility network sharing canbe profitable. Second, we provide insights to guide thedesign of wireless networks to account for revenue andexpenditures, in addition to ensuring reliable transmis-sion. II. W
IRELESS N ETWORK M ODEL
In this section, we detail the wireless networking aspects ofour model for the facility (financial aspects will be discussedin Section III). The key aspects that we consider are theplacement of the small-cells, which small-cell a given userwill connect to, and the physical layer transmission model.We note that our modeling assumptions are based on previousapproaches to wireless cellular networks (see e.g., [19]) andthe only significant difference is that the scale of facilities aretypically smaller than those considered for standard networks.Our assumptions for small-cell placement and the userselection protocol are as follows:
A1:
The facility’s small-cells are arranged according to ahomogeneous spatial Poisson point process (PPP) withintensity β . This means that in each realization there isa different number of small-cells, which provides insightinto the behavior of the network irrespective of how thesmall-cells are positioned. A2:
All users serviced by the facility:(i) connect to the nearest facility small-cell;(ii) are arranged according to an independent stationarypoint process, not necessarily Poisson.The key consequence of assumptions ( A1 ) and ( A2 ) is thatthe distribution of the distance R of a small-cell and any userit services is given by [19, Section III-A] Pr( Z ≤ z ) = e − βπz πzβ. (1)We now detail our assumptions for the physical layertransmission model. A3:
The facility’s small-cell network operates in discrete timewith block fading. Each time slot (also corresponding toa fading block), has a duration (coherence time) of T seconds. A4:
Each small-cell interferes with the others. Due to thePoisson interference assumption ( A1 ), the interference is M/M shot noise [19]. As such, the received signal inthe l -th time slot ( l = 1 , , . . . ) of a user’s connection isgiven by y l = h l q r − αU P x l + n l + z l , (2)where h l ∼ CN (0 , are the independent fading coeffi-cients for time slots l , and r U is the distance betweenthe user and its small-cell (the closest one, by ( A2 )),distributed according to (1). P is the power level thesmall-cell transmits at; not necessarily constant. α is thepath loss exponent, x l ∼ CN (0 , is the (Gaussian) datasymbol, and n l ∼ CN (0 , σ ) is additive white Gaussiannoise (AWGN), with noise variance σ . z l is the M/M shot noise.
Based on assumption ( A4 ), the instantaneous signal-to-interference and noise ratio (SINR) in the l -th time slot ofa given user’s connection is given by γ l = | h l | r − αU P σ + I l , (3)where I l is the interference power in the l -th time slot ofthe connection. In our analysis, we focus on the interferencelimited scenario where σ = 0 .The achievable data rate in each time slot under Gaussiansignaling for a user in the l -th slot of her connection is givenby R max l = B log (1 + γ l ) , (4)where B is the bandwidth (a constant) and γ l is given by (3).III. P ROPOSED N ETWORK S HARING A RRANGEMENT
In this section, we introduce our network sharing arrange-ment between the facility, users, and traditional operators(which have their own core networks). It is important toconsider traditional operators as their core networks are a keysource of data for the facility’s small-cell network as wellas spectrum. Although it is also possible to obtain data fromthe internet (although not necessarily calls), when there are asignificant number of users this can consume a large amount ofbandwidth and may also require special agreements with ISPs.As such, we focus on the scenario where traditional operatorsprovide data via their core networks.Our network sharing arrangement consists of two compo-nents: a leasing agreement with traditional operators; and aservice agreement with users, which is illustrated in Fig. 1.The agreements determine financial exchanges (contractualobligations) between the facility, users, and the traditionaloperators. Moreover, these exchanges induce revenue andexpenditure processes that are stochastic due to uncertaintiesin the resources required to provide the service (due to thestochastic nature of the wireless channel) and the randomduration of the users’ connections. To this end, we formalizethese processes which forms the basis of our evaluationframework in Section IV.
Users Facility Micronetwork Traditional OperatorCore Networks
Service Agreement LeasingAgreement
Fig. 1. Network sharing arrangement diagram.
A. Leasing and Service Agreements
A leasing agreement between the facility and a traditionaloperator allows the facility to access the data required by usersit seeks to service. In the leasing agreement, the facility pays afee to the traditional operator in return for access to their corenetwork and spectrum localized to the facility. A key featureof our leasing agreement is that the facility pays a fee perconnection . This means that the facility only pays for accessit actually requires, which is consistent with the broader visionof dynamic wireless network sharing proposed in [2].We assume that the fee is only determined by the traditionaloperator that is providing the service; that is, it is independentof other factors such as the duration of the connection. Asusers are typically subscribed to a single operator, this meansthat the fee for each user may be different. Our leasingagreement is summarized as follows.Our proposed core network leasing agreement betweentraditional operators and the facility consists of the con-nection fee for the j -th user. The connection fee is givenby C ( τ i,j , k i,j ) = C ∗ ( k i,j ) , (5)where C ∗ ( k i,j ) is the connection fee for users subscribedto operator k i,j .Observe that the leasing agreement does not depend on theduration of each connection. As each core network typicallyhas a large number of connections at any given time, it islikely that the distribution of the number and duration ofeach connection is known to the corresponding traditionaloperator. Moreover, the infrastructure required to support agiven connection is also known due to the stability of thewired links (as opposed to the wireless scenario). In this case,the operator can obtain good estimates of equilibrium pricesrequired to ensure that profit targets are met, which meansthat the duration of each connection does not play a key role.However, when there is a scarcity of access to the core networkthen it is necessary to consider connection durations in the fee,which may be achieved via a market mechanism such as anauction.We now consider the service agreement between the facilityand each user. The basis of the service agreement is that thereis a cost to the facility to provide users with the wirelessservice. The cost depends on four key factors: (i) the traditionalwireless operator that the user has a subscription; (ii) theduration of the users connection to the facility; (iii) the qualityof the wireless service requested (i.e., the rate at which theuser requests to be serviced); and (iv) the SINR of the linkbetween the user and the facility small-cell that it is servicedby. Importantly, all of these factors are random quantities.
The service agreement corresponds to the revenue thefacility receives in exchange for servicing each user, whichcan be obtained from two sources. The first source is directlyfrom the users; that is, the users pay the facility for the useof the service. In this case, the users pay the facility for every connection they require. The second source is the facility itself.In this case, the facility subsidizes the users and the agreementcorresponds to the funds the facility allocates to cover the costof servicing each user, which is applicable when the facility isproviding the service for its employees or when the service isan attraction (e.g., for hotels or stadiums). Again, this is on aper connection basis, which means that users (or the facility)only pay for the connections that they actually use.We assume that the service provider offers Q products ofvarying QoS. In practice, the user selects an application that itseeks to use—such as voice, video, or data—and the serviceprovider offers a QoS product that can support the application. Remark 1 (User indexing notation) . In a given time slot,multiple users are likely to conclude their connection. To referto a given user, we index it by the pair ( i, j ) , where i isthe time slot that the user concludes its connection and j isthe index of the user in the set of users that conclude theirconnection in time slot i . For example, when a user is the -thuser that terminates its connection in time slot , then the useris indexed as (5 , . In order to provide increasing levels of QoS over thewireless link, the facility is required to employ an increas-ing number of physical resources; for instance, additionalbandwidth, power, or infrastructure (e.g. relays or distributedantennas). It is important to note that a different number ofphysical resources are usually required in each time slot toaccount for channel fading. As such, the facility varies theSINR in each time slot by a scaling factor c i,j,l (correspondingto the l -th time slot of the connection for the ( i, j ) -th user) toachieve R ( q ) i,j = B log (1 + γ i,j,l c i,j,l ) , (6)where R ( q ) i,j corresponds to the rate required to support QoSproduct q ∈ { , , . . . , Q } for user ( i, j ) and c i,j,l encapsulatesthe additional physical resources required to support QoSproduct q . Observe that (6) follows directly from (4) with γ l replaced with γ i,j,l , which clarifies the particular user that isserviced. While it might seem more natural to directly scalethe rate (i.e. define R ( q ) = Bc ′ i,j,l log(1 + γ i,j,l ) ), it is infact simpler to use our formulation. We will show this inSection IV, where we develop our evaluation framework basedon ruin theory. It is also worthwhile to note that the twoformulations are identical with the appropriate definition of c ′ i,j,l .In practice, the facility’s physical resources are limited. Assuch, the scaling factor c i,j,l is upper bounded by a constant.We also introduce a lower bound on c i,j,l so that the facilityis guaranteed a minimum income, even when the channelis good. This means that facility can ensure that it can paythe core network leasing fee, detailed in (8). Taking theseconsiderations into account, we define c i,j,l as c i,j,l = max ( min ( R ( q ) i,j /B − γ i,j,l , c max ) , c min ) . (7)It is also necessary to account for the duration of each user’sconnection. In particular, we make the following assumptions: A5:
Let τ i,j be the duration (in integer time slots) of theconnection of the j -th user to end its connection inthe i -th time slot; i.e user ( i, j ) . Each duration τ i,j isindependent and identically distributed. The durations τ i,j has CDF F τ i,j with support { , , . . . , i } .We now detail our proposed service agreement.Our proposed service agreement consists of the chargeto the j -th user to end its connection in the i -th time slot(user ( i, j )), which is given by v ( τ i,j , c i,j ) = τ i,j X l =1 c i,j,l T ρ, (8)where c i,j = [ c i,j, , . . . , c i,j,τ i,j ] T is the vector of scalingfactors (each element corresponding to a different timeslot) to satisfy (6) and ρ is the premium rate; that is,the income received from user ( i, j ) by the facility pertime slot with a unit scaling factor. We assume that thepremium rate ρ is constant, irrespective of the operatorthat user ( i, j ) is subscribed to. B. Summary of the Proposed Network Sharing Arrangement
From the perspective of the facility, the leasing agreementcorresponds to an expenditure process and the service agree-ment corresponds to an income process. As such, our networksharing arrangement can be formalized as a revenue surplusprocess, which is defined in terms of the expenditure andincome processes.An important feature of the revenue surplus process is thatinterest is compounded. This means that the total revenuesurplus is dependent on not only the current surplus, revenueand costs, but also the interest rate. Moreover, the time intervalbetween interest compounding is typically different to theduration of a time slot. The number of users with connectionsending in compound interest interval m is given in ( A6 ). A6:
The number of users N n that have their connection endin compound interest interval [ n − , n ) of duration κT ( κ ∈ N ) is geometrically distributed as Pr( N i = u ) = (1 − w N ) u w N . (9)We first formalize the facility expenditure process, whichis based on the core network leasing agreement with thetraditional operators. Definition 1 (Facility Expenditure Process) . Consider the ( i, j ) -th user associated with the k -th wireless operator, whichhas:(i) a random connection duration τ i,j time intervals, dis-tributed according to τ i,j ∼ F τ i,j (detailed in ( A6 ));(ii) a requested rate product R ( q ) i,j with q ∈ { , , . . . , Q } ,distributed according to R ( q ) i,j ∼ F R i,j for any discretedistribution on { , , . . . , Q } ;(iii) and required resources c i,j = [ c i,j, , . . . , c i,j,τ i,j ] T , whichare i.i.d random variables defined in (7). Then the total expenditure of the facility in the compoundinterest interval [ n − , n ) is given by E n = N n X j =1 C ( τ m,j , k m,j ) , (10) where ( n − κT ≤ m < nκT are the time slots that endin interval [ n − , n ) , and N n is geometrically distributed (asdetailed in ( A6 )) and C ( τ n,j , k n,j ) is defined in (8). Next, we formalize the facility income process, which isbased on the service agreement with the users.
Definition 2 (Facility Income Process) . Consider the ( i, j ) -thuser associated with the k -th wireless operator, which satisfiesthe hypotheses in Definition 1. Then, the proposed charge tothe ( i, j ) -th user (corresponding to the facility income) forconnecting to the facility for a duration τ i,j is given by thecompound random sum v ( τ i,j , c i,j ) = τ i,j X l =1 c i,j,l T ρ. (11)
The total income generated in the period [ n − , n ) is givenby I n = N n X j =1 v ( τ m,j , c m,j ) , (12) where ( n − κT ≤ m < nκT are the time slots that endin interval [ n − , n ) , and N n is geometrically distributed, asdetailed in ( A7 ). Finally, the revenue surplus process for the facility is definedas follows. This can be viewed as the accumulation of theinitial capital and the difference between the income andexpenditure processes, taking into account compound interest.
Definition 3.
The revenue surplus is the current micro-networkprofit in the l -th time slot generated by serving users, afteraccounting for interest and the cost of accessing the operators’core networks. This is given by S l ( u ) = u (1 + r ) l + l X i =1 (1 + r ) l − i N i X j =1 (cid:2) v ( τ m i ,j , c m i ,j ) − C ( τ m i ,j , k m i ,j ) (cid:3) , (13) where iκT ≤ m i < ( i + 1) κT , u is the facility’s initial capitaland r is the compound interest rate (compounded at intervalsof κT ). We also define S net ( i ) = N i X j =1 [ v ( τ m,j , c m,j ) − C ( τ m,j , k m,j )] , (14) as the net profit in compound interval i , where iκT ≤ m < ( i + 1) κT . It is important to note that there is potentially a significantdifference in the time scales at which users are priced (orderof minutes) and that the revenue surplus is calculated (orderof months). IV. E
VALUATION F RAMEWORK : A R
UIN T HEORY A PPROACH
In this section, we detail our framework to evaluate ourfacility network sharing arrangement, which will be evaluatedin Section V. Our framework is based on ruin theory [15],where the key performance metric is the probability that thefacility has a negative revenue surplus within n months, knownas the probability of ruin. This is an important metric as thefacility can only operate while it has the resources to do so.In fact, knowing whether these financial resources are likelyto be available is important in the decision of whether or notto invest in the facility, how to structure products, and howmuch to charge for services.Unfortunately, it is not straightforward to directly computethe probability of ruin. The main reason for this is that theincome and expenditure processes (defined in Section III-B)involve a number of random variables, which result in randomsums without closed-form distributions. Due to these difficul-ties, we instead focus on obtaining an accurate approximationof the probability of ruin.Fortunately, we are able to leverage techniques from ruintheory to compute an accurate approximation for the proba-bility of facility ruin. However, it is important to note thatmodifications to the standard theory are required, due tothe fact that the parameters of facilities yield non-standarddistributions.In this section, there are four subsections: (A) the definitionand overview of the key steps to compute the facility ruinprobability; (B) the first step: the approximation of the incomedistribution, which is based on an orthogonal polynomialrepresentation; (C) the second step: the recursion for the netprofit distribution; and (D) The third step: the recursion forthe probability of ruin. A. Ruin Probability Definition and Overview
Intuitively, the probability of ruin is the probability that therevenue surplus is negative before a period of l time periods(e.g., months). First, we define the stopping time known asthe time of ruin, followed by the ruin probability. Definition 4 (Ruin Time) . The time of ruin is the first timethat the revenue surplus is negative, i.e. L R = inf { l : S l ( u ) < } , (15) where u is the initial capital. Note that we allow for thepossibility that u < . Definition 5 (Probability of Ruin) . The probability of ruinbefore a period of l time periods is then ψ l ( u ) = Pr( L R ≤ l ) , (16) and the probability of survival is φ l ( u ) = 1 − ψ l ( u ) . (17)In order to compute the ruin probability; the steps aredetailed in Algorithm 1. Algorithm 1
Ruin Probability Computation1. Derive the orthogonal polynomial basis expansion for theincome PDF in Section IV-B.2. Evaluate the distribution for the net profit in time slot l in Section IV-C. This is achieved by first discretizing theincome PDF from Section IV-B, then using a recursion fromactuarial science to compute the compound distribution.3. Evaluate the probability of ruin via another recursion inSection IV-D (different to the recursion in Stage 2), whichcircumvents the difficulty of directly computing the proba-bility of ruin. B. Orthogonal Polynomial Representation of the Income PDF
The first stage of computing the probability of ruin is tocompute the PDF of the income from a user with connectionending in time slot i , which from Definition 2 is given by V i = v ( τ i,j , c i,j ) = τ i,j X l =1 c i,j,l T ρ, (18)which is a random sum due to τ i,j and c i,j,l (given in (7)). Remark 2.
In general, the distribution of τ i,j depends on i . For short compounding intervals this is to ensure that theconnection duration is not longer than the time the system hasbeen running. In other cases, the distribution may vary dueto seasonal usage trends and events, such as holidays. Thishas the important consequence that the moments and hencedistribution of V i depend on the time slot i that user ( i, j ) ’sconnection ends. It is clear from the fact that V i is a random sum andthe form of c i,j,l (see (7)) that the distribution of V i is notreadily obtained in closed-form. As such, we instead adopt aprincipled approach to approximating V i , which is based onan Askey-orthogonal polynomial expansion. In particular, weuse the Jacobi polynomials since the support of the income V i is bounded on [ − v i, min , v i, max ] . This follows from the factthat both τ i,j and c i,j,l are bounded.It is important to note that the Jacobi polynomials are onlyorthogonal on [ − , . Hence, we need to transform V i sothat it has also has support [ − , . This is achieved via thetransformation W i = 2( V i + v i, min ) v i, max + v i, min − , (19)where v i, min = − c min T ρ and v i, max = ic max T ρ , from (18).The distribution of W i via the Jacobi polynomial represen-tation is then given by [20] f W i ( x ) ≈ K ( x ) d X m =0 a m P ( a,b ) m ( x ) , (20)where d is the order of the approximation, P ( a,b ) m ( x ) is the m -th Jacobi poynomial with parameters a, b , a m is given by a n = B ( a, b )(2 n + a + b − n + a + b − n !Γ( n + a )Γ( n + b ) × Z − f W i ( x ) P ( a,b ) n ( x ) dx = b n Z − f W i ( x ) P ( a,b ) n ( x ) dx, (21)and K ( x ) is given by K ( x ) = (1 + x ) a − (1 − x ) b − B ( a, b )2 a + b − , (22)where B ( x, y ) = Γ( x )Γ( y ) / Γ( x + y ) . (23)We note that the coefficients a n minimize the mean squareerror between the approximation and f W i ( x ) as d → ∞ . Remark 3.
From extensive numerical experiments, we havefound that the choice a = b = 1 (corresponding to Legendrepolynomials) yields the most accurate approximations for arange of moments (corresponding to different network setups). As P ( a,b ) n is a polynomial, we can write a n as a n = b n Z − f W i ( x ) n X s =0 ζ n,s x s = b n n X s =0 ζ n,s E [ W si ] , (24)where ζ n,s corresponds to the coefficient of the s -th order termof P ( a,b ) n . This means that the approximation is completelycharacterized by the raw moments of W i . Observe that themoments of W i are related to the moments of V i via E [ W si ] = E (cid:20)(cid:18) V i + v i, min ) v i, max + v i, min − (cid:19) s (cid:21) , (25)which can be readily evaluated (given the moments of V i ) viathe binomial expansion. Remark 4.
To obtain the moments of V i (and hence the mo-ments of W i ), we compute and then differentiate the momentgenerating function, which allows us to use the probabilitygenerating functional of the PPP; surpassing the need to ex-plicitly derive the distribution of the interference. The momentsand details of the derivations are given in Appendix A. The distribution of V i is then obtained from the distributionof W i via the transformation f V i ( x ) = 2 v i, max + v i, min f W i (cid:18) x + v i, min ) v i, max + v i, min − (cid:19) , (26)where f W i ( x ) is given by (20). We summarize the procedurein Algorithm 2. C. Recursion for the Net Profit PDF
The second stage is to compute the PDF of the net profit.Recall from Definition 3, that the net profit for time slot i isgiven by S net ( i ) = N i X j =1 [ v ( τ i,j , c i,j ) − C ( τ i,j , k i,j )] . (27) Algorithm 2
Orthogonal Polynomial Basis Expansion of theNet Profit PDF1. Compute the moments of V i using (60) and (61) in Ap-pendix A.2. Transform V i to the random variable W i with support [ − , via (19).3. Compute the basis expansion coefficients a n via (24).4. Compute the basis expansion of f W i ( x ) using (20).5. Transform f W i ( x ) via (26) to obtain f V i ( x ) .It is important to note that S net ( i ) has a different distributionfor each i (see Remark 2). Moreover, observe that the distri-bution of C ( τ i,j , k i,j ) is discrete and the approximation of thedistribution of v ( τ i,j , c i,j ) from Section IV-B is continuous(with bounded support). Also note that S net ( i ) is a randomsum (in both the summands and number of summands).As such, it is not straightforward to efficiently compute thedistribution of S net ( i ) .Fortunately, linear recursions have been developed to eval-uate distributions for sums closely related to S net ( i ) . In orderto apply these recursive approaches, we first need to findthe distribution of v ( τ i,j , c i,j ) after it has been discretized.The discretization of v ( τ i,j , c i,j ) can be obtained via standardapproaches such as the Lloyd algorithm [21]; although itis important to carefully consider any negative terms in thedistribution of v ( τ i,j , c i,j ) that might occur due to the basisexpansion approximation. We denote the discretization as ˆ v ( τ i,j , c i,j ) .To obtain the PDF of the net profit, denoted by h Z j , wenow embed the discretized income into the lattice ∆ Z bychoosing ∆ > sufficiently small and rounding the supportof ˆ v ( τ i,j , c i,j ) . We then obtain the distribution h Z j ( k ) = Pr( Z j = k ) , (28)of the random quantity Z j = [ˆ v ( τ i,j , c i,j ) − C ( τ i,j , k i,j )] , (29)by convolving the (discrete) distributions of ˆ V = ˆ v ( τ i,j , c i,j ) and C ( τ i,j , k i,j ) , which both have discrete support on S ⊂ ∆ Z , with | S | < ∞ .At this point, we obtain the approximate distribution of S net ( i ) . This is based on the following linear recurrence basedon [17, Eq. (1.7)], f S net ( i ) (( n + 1)∆)= w N − w N h Z i (0) × ∞ X k = −∞ , k = − ( n + 1) h Z i (( k + 1)∆) f S net ( i ) (( n − k )∆) . (30)We note that similar difference equations can be derivedfor distributions of N i , other than geometric; in fact, it isstraightforward to extend to any distribution in the Katz family(i.e., binomial, Poisson, and negative binomial distributions).Further details may be found in [17, Eq. (1.3)] and [22, Eq.(1)]. Remark 5.
We emphasize that the support of S net ( i ) is over Z , not just N . This means that the standard recursion in [22](known as the Panjer recursion) is not applicable; the moregeneral approach in [17] is required. In practice, the sum (30) is in fact bounded (due to the finiteresolution of the discretization step), which gives f S net ( i ) (( n + 1)∆)= w N − w N h Z i (0) × k max X k = k min , k = − ( n + 1) h Z i (( k + 1)∆) f S net ( i ) (( n − k )∆) , (31)where k min = min Supp( S net ( i )) − and k max =max Supp( S net ( i )) − , depend on the discretization resolu-tion.In order to solve the linear recurrence relation in (31), weuse a convex reformulation in terms of a least squares problem, min f : f T f =1 k Af k , (32)where A consists of the coefficients in terms of h Z i from (31)and f is the distribution of S net ( i ) . We note that in principleother objectives can be used in (32), such as those discussedin [17]; however, we have found via numerical experimentsthat the least squares objective is suitable for the purpose ofapproximating the ruin probability. D. Recursion for the Probability of Ruin
The third (and final) stage of the calculation is to computethe probability of ruin. This is achieved by collecting thedistributions of S net ( l ) and substituting into the new ruinprobability recursion, which we derive in this section.It is important to note that the ruin probability (definedin Definition 5) is the probability that the revenue surplus isnegative at any l ≤ L . Formally, the ruin probability can bewritten as ψ L ( u ) = Pr( S ( u ) < ∪ · · · ∪ S L ( u ) < . (33)It is helpful to write the ruin probability in terms of the survivalprobability (see Definition 5), which is given by φ L ( u ) = 1 − ψ L ( u ) = Pr( S ≥ ∩ · · · ∩ S L ≥ . (34)We now give the recursion for the probability of ruin interms of the distributions of S net ( l ) , l = 1 , , . . . , L . First,define G l ( y ) = Pr( S net ( l ) ≤ y ) . (35)We then have the following theorem, which generalizes theruin probability recursion in [18], [23]. The set
Supp( S net ( i )) corresponds to the support of S net ( i ) . Theorem 1.
The survival probability after L time periods(e.g., months) is given by φ L ( u ) = φ L − ( u )(1 − G L (0))+ Z −∞ φ L − (cid:18) u + y (1 + r ) L (cid:19) dG L ( y ) , ∀ L ≥ , (36) where φ ( u ) = 1 − G ( − u (1 + r )) . (37) The ruin probability is then obtained via ψ L ( u ) = 1 − φ L ( u ) .Proof: See Appendix B.We note that the integral in Theorem 1 is a Stieltjes integral.This is important as G l , l = 1 , , . . . , L is in fact a sequenceof distributions, all with discrete and bounded support. Assuch, the Stieltjes integral is a finite sum and can be efficientlyevaluated numerically.V. N UMERICAL AND S IMULATION R ESULTS
With our evaluation framework based on ruin-theory inhand, we now turn to evaluating facility network sharing. Weobtain pricing schemes that yield a probability of ruin lessthan , under practical operating conditions. Along theway, we present important tradeoffs between physical layerresources such as pathloss, and financial constraints such asinitial capital, interest rate (compounded monthly) and pricing.Before evaluating the ruin probability using the techniqueswe have developed in Section IV, we first present a simplifiedanalysis based purely on the expected revenue surplus (seeDefinition 3). This analysis provides initial insights into thefinancial aspects of the problem, which may be unfamiliar tothe wireless communication community. That is, we consider E [ S net ( l )] = u (1 + r ) l + l X i =1 (1 + r ) l − i E [ N ] ( E [ V ] − E [ C ]) , (38)where we assume that the connection duration time τ i,j isconstant, which means that the revenue per user V i has ex-pectation E [ V ] for all i . The condition on the initial capital toensure positive average revenue surplus (i.e., E [ S net ( l )] ≥ )at a given time n for interest rate r is then given by u ≥ E [ N ] ( E [ C ] − E [ V ]) (cid:18) (1 + r ) n − r (1 + r ) n (cid:19) . (39)To illustrate this condition, Fig. 2 plots the expected revenue E [ V ] versus the bound on u ; each curve corresponding to adifferent time slot n . First, observe the initial capital needsto be non-zero to ensure a positive expected revenue surplus when the costs exceed the revenue; i.e., E [ C ] > E [ V ] . Second,observe that there is a diminishing increase in the requiredinitial capital as the number of time steps is increased; thiscan be seen by comparing the gaps between the curves, fora fixed E [ V ] . Moreover, as the bound on u in (39) scaleslinearly with E [ N ] , the initial capital increases significantly asthe number of connected users increase, when E [ C ] > E [ V ] .These are general trends; however, it is important to note the Expected Revenue Surplus E[V] I n i t i a l C ap i t a l B ound Time steps (in months)n = 1,2,3,...
Fig. 2. Plot of bound on initial capital u to ensure E [ S net ( l )] ≥ , where r = 0 . , E [ N ] = 100 , E [ C ] = 0 . . limitations of any analysis based on expected revenue surplus.In particular, short-term behavior of the revenue surplus is notaccounted for, which has the consequence that fluctuationsin user demand or available network resources can cause therevenue surplus to drop below zero—leading to ruin. As such,it is necessary to use our framework based on ruin theoryto account for these fluctuations and ultimately reduce thefinancial risk to the facility.To account for fluctuations in the revenue surplus, we nowturn to our evaluation framework based on ruin theory. Weconsider the network setup in Table I and define A d =2 R (1) /B − , where R (1) is the rate product on offer. TABLE IS
UMMARY OF NETWORK PARAMETERS .Parameter
K σ P = P I w N Q T ρ C κT
Value . month We first consider the moments of the revenue E [ V ] . Thesemoments are ultimately used to compute the probability ofruin. They are also of interest in their own right. We showthis next by examining the role of the key wireless networkand financial parameters α , β , c min , and c max .In Table II, we compare the small-cell density β with thefirst moment of V , E [ V ] , obtained numerically via (61) andvia Monte Carlo simulation. We point out that the momentsobtained numerically via (61) are in good agreement with themoments obtained via Monte Carlo simulation. Importantly,the moment E [ V ] is constant irrespective of the small-celldensity for both the numerical and Monte Carlo approaches.This suggests that the small-cell density does not play animportant role in networks well-modeled by PPPs. We believethe reason for this is that as the density of small-cells increases,there is an increase in nearby interfering small-cells, which isbalanced by a closer servicing small-cell. Note that the small-cell density has a similar effect of outage probability in the TABLE IIT
ABLE SHOWING EFFECT OF VARYING SMALL - CELL DENSITY β , FOR DIFFERENT PATHLOSS EXPONENT α USING NUMERICAL (N UM .) AND M ONTE C ARLO (M.C.)
APPROACHES . E
VALUATION PERFORMED WITH c min = 0 . , c max = 100 , A d = 100 . β E [ V ] Num., α = 3 E [ V ] M.C., α = 3 E [ V ] Num., α = 4 E [ V ] M.C., α = 4 low noise region, as shown in [19]. Pathloss Exponent α E x pe c t ed R e v enue E [ V ] Numerical (Based on (63))Monte CarloA d = 50, 100, 200 Fig. 3. Plot of expected revenue E [ V ] versus pathloss exponent α , withvarying A d = 2 R (1) /B − . Parameters are β = 0 . , c min = 0 . , c max =100 , A d = 100 . Next, we consider the effect of the pathloss on the revenue.Fig. 3 plots the pathloss exponent versus the expected revenue E [ V ] , which shows excellent agreement between our numeri-cal result (based on (61)) and Monte Carlo results. The trendillustrated by the figure is that the revenue decreases as thepathloss exponent increases, irrespective of A d (correspondingto different rate products R (1) on offer). This is due to the factthat it is easier to service users with a high pathloss exponentas it means that there are often lower interference levels.Following Algorithm 1, the next step to compute the ruinprobability is to obtain the PDF of the revenue V via basisexpansion. To illustrate this step, the basis expansion approx-imation for the CDF of V is plotted in Fig. 4. We observethat our numerical approximation is in good agreement forrevenue less than , and then has small oscillations for highrevenue values. It is important to note that tail oscillationsare common when using the Jacobi polynomial representation,and must be carefully accounted for. In the setup for Fig. 4,the approximation is good; however, as c min is increased or c max decreased, the discontinuity affects the quality of theapproximation. As such, additional moments (i.e., E [ V ] ,..)are required to smooth the oscillations, which can be obtainedeasily from our analysis in Appendix A. We also note that theparameters a, b in the Jacobi polynomials (see Section IV-B)strongly influence the approximation; extensive numerical v C u m u l a t i v e d i s t r i bu t i on f un c t i on o f V Monte CarloNumerical (Based on (28))
Fig. 4. Plot of basis expansion approximation for the CDF of V using fourmoments. Parameters are α = 4 , β = 0 . , c min = 0 . , c max = 1000 , A d = 100 . experiments suggest that a = b = 1 is a robust choice.Table III shows the ruin probability after months versusthe initial capital u , for varying A d (reflecting different rateproducts on offer). As expected from Fig. 2, in order to obtaina low probability of ruin, the initial capital needs to be chosencarefully. This is reflected in both the numerical and MonteCarlo results. We note that for low ( u = 100 ) and high ( u ≥ ) initial capital, the numerical and Monte Carlo approachesare in agreement. However, for u ≈ , there is a differenceof approximately . . This is due to the discretization stepdetailed in Section IV-C. Importantly, to ensure the probabilityof ruin is less than approximately , an initial capital of u > is required, with the parameters used in Table III. TABLE IIIR
UIN PROBABILITY AFTER MONTHS WITH VARYING INITIAL CAPITAL u .P ARAMETERS ARE α = 4 , β = 0 . , c min = 0 . , c max = 1000 , A d = 100 , AND r = 0 . .Initial Capital u
100 150 200 250 300
Numerical (Using Theorem 1) .
33 0 .
09 0 .
01 0 0
Monte Carlo .
33 0 .
22 0 .
11 0 .
05 0 . VI. C
ONCLUSIONS
Due to the recent availability of cheap small-cells and theunique operating requirements of facilities, there is a needfor alternative network sharing arrangements. To this end, we proposed a facility network sharing arrangement, which isbased on: a leasing agreement between traditional operatorsand the facility; and a service agreement between users andthe facility. Unlike traditional operators, the local scale offacilities means that technical design at the level of the networkarchitecture is intimately connected with the profitability;instead of loosely coupled.In order to evaluate our facility network sharing arrange-ment, we adopted a socio-technical system design approach.As such, our key performance metric is the ruin probability—leading to a new ruin-based evaluation framework. To evaluatethe ruin probability, we proposed a numerical approximationmethod, which is shown to be in good agreement with MonteCarlo simulation. Our approximation method includes two keynovel aspects: computation of the moments of the revenuevia stochastic geometry techniques; and a new recursionfor the ruin probability, which is tailored for the wirelesscommunication setting. Our numerical results suggested thatthere are in fact concrete conditions where profitable operationof facilities is possible, with sufficient initial capital.A CKNOWLEDGEMENTS
This publication was supported by the European socialfund within the framework of realizing the project “Sup-port of inter-sectoral mobility and quality enhancement ofresearch teams at Czech Technical University in Prague”,CZ.1.07/2.3.00/30.0034. Period of the projects realization1.12.2012 30.6.201. A
PPENDIX
AIn this appendix, we derive the moments of the income fora user with connection ending in time slot i , which is usedto compute the ruin probability. Importantly, our analysis canbe straightforwardly extended to higher moments. Recall thatthe income from a single user (a typical node, by Slivnyak’stheorem for homogeneous Poisson point processes) is givenby v ( τ i,j , c i,j ) = τ i,j X l =1 c i,j,l T ρ, (40)where the scaling factor is c i,j,l = max ( min ( (cid:0) R i,j /B − (cid:1) P I I i,j,l | h i,j,l | d − αi,j P , c max ) , c min ) . (41)We also define: A = P I (2 Ri,j/B − P d − αi,j ; H l = | h i,j,l | ; τ = τ i,j ;and V = v ( τ i,j , c i,j ) .In order to compute the four moments, we require themoment generating function (MGF) of V . In turn, the MGFis obtained from the Laplace transform L X ( t ) , which is given by L V ( t ) = E " exp − t τ X l =1 max (cid:26) min (cid:26) AI l H l , c max (cid:27) , c min (cid:27) T ρ ! = E τ (cid:20) E A (cid:20)(cid:18) E H l ,I l (cid:20) exp (cid:18) − t × max (cid:26) min (cid:26) AI l H l , c max (cid:27) , c min (cid:27) T ρ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) A (cid:21)(cid:19) τ (cid:12)(cid:12)(cid:12)(cid:12) τ (cid:21)(cid:21) . (42) Note that R i,j and τ have discrete support, so these expec-tations are sums we evaluate last. We now evaluate the innerexpectation over the interference I and the channel gain H l .Observe that the inner expectation can be written as E ( t ) = E H l ,I l (cid:20) exp (cid:18) − t max (cid:26) min (cid:26) AI l H l , c max (cid:27) , c min (cid:27) × T ρ ) (cid:12)(cid:12)(cid:12)(cid:12) A, τ (cid:21) = E ( t ) + E ( t ) + E ( t ) , (43)where E ( t ) = E (cid:20) e − tT ρc min (cid:12)(cid:12)(cid:12)(cid:12) AI l H l < c min (cid:21) Pr (cid:18) AI l H l < c min (cid:19) = e − tT ρc min Pr (cid:18) AI l H l < c min (cid:19) ,E ( t ) = e − tT ρc max Pr (cid:18) AI l H l > c max (cid:19) ,E ( t ) = E (cid:20) e − tT ρ AIlHl (cid:12)(cid:12)(cid:12)(cid:12) c min ≤ AI l H l ≤ c max (cid:21) × Pr (cid:18) c min ≤ AI l H l ≤ c max (cid:19) = Z c max c min e − tT ρw f W ( w ) dw, (44)with W = AI l H l . (45)To compute E ( t ) , observe that Pr( W ≤ w ) = E I l h e − AIlw i ⇒ f W ( w ) = E I l (cid:20) AI l w e − AIlw (cid:21) . (46)Hence, E ( t ) = E I l (cid:20)Z c max c min e − tT ρw AI l w e − AIlw dw (cid:21) . (47)Next, we integrate by parts to obtain E ( t ) = E ( t ) + E ( t ) + E ( t )= e − tT ρc max + tT ρ Z c min1 c max u e − tT ρ/u E I l (cid:2) e − AI l u (cid:3) du. (48) In order to compute E ( t ) , we require E I [ e − AI l u ] , which isgiven by E I l (cid:2) e − AI l u (cid:3) ( a ) = E X , { g m } Y m ∈X \{ b } e − Aug m r − αm ( b ) = E X Y m ∈X \{ b } E g h e − Augr − αm i ( c ) = exp (cid:18) − πβ Z ∞ r U (cid:16) − E g h e − Augz − α i(cid:17) zdz (cid:19) , (49)where: ( a ) follows from H l ∼ exp(1) ; ( b ) follows from thefact that { g m } is independent of the spatial point process; and ( c ) follows from the probability generating functional of thePPP .Continuing, we have E I l (cid:2) e − AI l u (cid:3) = exp − πβ Z ∞ α Z r − αU (cid:0) − e − Agyu (cid:1) y − α − dye − g dg ! , (50)which follows from the change of variables y = z − α ⇒ z = y − /α ⇒ dz = − α y − α − dy .Now consider the inner integral in (50), given by F = Z r − αU (cid:0) − e − Agyu (cid:1) y − α − dy = − α (cid:16) − e − Agr − αU u (cid:17) r U + αAgu F , (51)where F = Z r − αU e − Agyu y − α dy. (52)Integrating by parts again we obtain, F = (cid:18) Agu (cid:19) − α +1 γ (cid:18) − α , Agur − αU (cid:19) , (53)where γ ( s, x ) = R x t s − e − t dt .Next, we compute the outer integral in (50), which is givenby G = Z ∞ F e − g dg = G + G , (54)where G = − αr U Z ∞ (cid:16) − e − Agur − αU (cid:17) e − g dg,G = α (cid:18) Au (cid:19) − α Z ∞ g α e − g γ (cid:18) − α , Agur − αU (cid:19) dg. (55) The probability generating functional property of the PPP gives E [ Q x ∈X f ( x )] = e − β R R (1 − f ( x )) dx . We now require the identity from [24, Eq. 6.4552], Z ∞ x µ − e − βx γ ( ν, αx ) dx = α ν Γ( µ + ν ) ν ( α + β ) µ + ν × F (cid:18) , µ + ν ; ν + 1; αα + β (cid:19) . (56)Identifying terms, we obtain G = α (cid:18) Au (cid:19) − α (cid:0) Aur − αU (cid:1) − α (cid:0) − α (cid:1) (cid:0) Aur − αU + 1 (cid:1) × F (cid:18) ,
2; 2 − α ; Aur − αU Aur − αU (cid:19) . (57)To compute E [ e − AI l u ] , substitute G and G into (50). Thisresult is then used to compute the Laplace transform and obtainthe moments via E [ V ni ] = ( − n d n L V i ( t ) dt n | t =0 . (58)All that remains is to explicitly compute the moments. Wedefine the following terms: E ( k )0 ( t ) is the k -the derivative of E ( t ) ; E ( k )0 is the k -the derivative of E ( t ) evaluated at t =0 ; and L ( k )0 is the k -th derivative of L X ( t ) conditioned on d, R i,j , τ evaluated at t = 0 . Note that E (0) = 1 .Using (48), we have E ( t ) = e − tT ρc max + tT ρ Z c min1 c max u e − tT ρ/u E I l (cid:2) e − AI l u (cid:3) du,E ( s ) ( t ) = ( − T ρc max ) s e − tT ρc max + s ( − s +1 ( T ρ ) s Z c min1 c max u s +1 e − tT ρ/u E I l (cid:2) e − AI l u (cid:3) du − t ( T ρ ) s +1 Z c min1 c max u s +2 e − tT ρ/u E I l (cid:2) e − AI l u (cid:3) du, (59)where E ( s ) ( t ) is the s -th derivative of E ( t ) and E [ e − AI l u ] isgiven by (50).Now, let denote the indicator function. The momentsconditioned on d i,j , R i,j , τ i,j can now be readily obtained. Weillustrate with the first conditional moment. L (1)0 = τ E (1)0 τ> . (60)Finally, the moments for the revenue from a user withconnection ending in time slot i are given by E [ V ki ] = n X l =1 Q X k =1 Pr( R i,j = k )Pr( τ = l ) Z ∞ L ( k )0 e − πβz πβzdz, (61)which can be efficiently evaluated numerically. A PPENDIX B Proof of Theorem 1:
First (from (34)), observe that thesurvival probability can be written as φ n ( u ) = Pr u (1 + r ) n + S net ( n ) + n − X i =1 S net ( i )(1 + r ) n − i ≥ ,. . . , u (1 + r ) + S net (1) ≥ . (62)Also recall that G n ( y ) = Pr( S net ( n ) ≤ y ) . (63)Now, using (34) and (63) yields φ ( u ) = Pr( S net (1) ≥− u (1 + r )) = 1 − G ( − u (1 + r )) . We then have for n ≥ φ n ( u ) = Pr (cid:18) u (1 + r ) n − + S net ( n )1 + r + n − X i =1 S net ( i )(1 + r ) n − − i ≥ , . . . ! = φ n − ( u )(1 − G n (0))+ Z −∞ φ n − (cid:18) u + y (1 + r ) n (cid:19) dG n ( y ) , (64)which follows after considering ( u (1 + r ) n − + S net ( n )1 + r + n − X i =1 S net ( i )(1 + r ) n − − i ≥ ∩ u (1 + r ) n − + n − X i =1 S net ( i )(1 + r ) n − − i ≥ ) . (65)R EFERENCES[1] GSMA,
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