Master on Telematics EngineeringAcademic Course 2015-2016

Master Thesis

“Optimization of Resource AllocationNetworks shared by multiple operators”

Author: Pablo Caballero GarcesDirector: Albert Banchs Roca

Leganes, 24th of January 2015

Keywords: IMT-A, 3GPP LTE, Multi-tenancy, RAN Sharing, Fairness, Multi-operator.

Summary: The traditional model of single ownership of the mobile network infrastructure is being challenged by the forecastedmobile data tsunami. In this context, the sharing of network infrastructure among operators has emerged as a way to ensureoperators’ future cost competitiveness. While the elementary concepts related to passive sharing are already exploited, activesharing is raising in importance. The work presented in this paper presents a solution for active RAN sharing in new generationcellular networks that takes into account inter and intra operator fairness and has desirable properties from base station anduser association perspectives. An offline centralized approximation algorithm that provides theoretical performance guaranteesis proposed. Making use of the available real-time neighbour information we designed an approximated distributed algorithmwith an

∑Ni=1 wi log(e) additive ratio. Finally, we confirm the theoretical results by a set of performance evaluations in a

LTE-Advanced system-level simulator.

Resource Allocation for elastic traffic in mobilenetworks shared by multiple operators

Pablo Caballero Garces∗†∗IMDEA Networks Institute, Madrid, Spain †University Carlos III of Madrid, Spain

Abstract—The traditional model of single ownership of themobile network infrastructure is being challenged by the fore-casted mobile data tsunami. In this context, the sharing of net-work infrastructure among operators has emerged as a way toensure operators’ future cost competitiveness. While the elemen-tary concepts related to passive sharing are already exploited,active sharing is raising in importance. The work presented inthis paper presents a solution for active RAN sharing in newgeneration cellular networks that takes into account inter andintra operator fairness and has desirable properties from basestation and user association perspectives. An offline centralizedapproximation algorithm that provides theoretical performanceguarantees is proposed. Making use of the available real-timeneighbour information we designed an approximated distributedalgorithm with an

∑Ni=1 wi log(e) additive ratio. Finally, we con-

firm the theoretical results by a set of performance evaluationsin a LTE-Advanced system-level simulator.

Index Terms—IMT-A, 3GPP LTE, Multi-tenancy, RAN Shar-ing, Fairness, Multi-operator.

I. INTRODUCTION

Mobile networks are a key element of today’s society, en-abling communication, access and information sharing. Allforecasts agree in predicting that the demand for capacitywill grow exponentially over the next years, mainly due tovideo services. However, as cellular networks move from be-ing voice-centric to data-centric, operators’ revenues are notexpected to be able to keep pace with the predicted increase intraffic volume. Such pressure on operators’ return on invest-ment has pushed research efforts towards achieving more cost-efficient mobile network solutions. In this context, networksharing has emerged as a key business model for reducingdeployment and operational costs (CAPEX and OPEX).

Network sharing solutions are already available, standard-ized [2] and partially used in some mobile carrier networks.These solutions can be divided into passive and active networksharing. Passive sharing refers to the reuse of componentssuch as physical sites, tower masts, cabling, cabinets, powersupply, air-conditioning, etc. Active sharing refers to the reuseof backhaul, base stations and antenna systems, the reuse ofthe latter two being labeled as active Radio Access Network(RAN) sharing. According to a market survey [3], mobileinfrastructure sharing has been already deployed by over 65%

The content of this document is part of a collaborative work. I would liketo express thanks to Dr. Albert Banchs, Dr. Xavier Perez-Costa, Dr. Gustavode Veciana, Dr. Jorge Ortin and Dr. Konstantinos Samdanis for their valuablehelp on this work. A previous version of this work was accepted for oralpresentation and publication in VTC-Spring 2015 [1].

of European operators in different ways, and this trend isexpected to expand in the future.

Estimations regarding the expected savings for operators byimplementing active network sharing have been conducted,see e.g., [4]. This study concluded that operators worldwidecould reduce combined OPEX and CAPEX costs in up to $60Billion over a 5 years period through network sharing and atleast 40% of these cost savings are expected to come fromactive network sharing.

The traditional model of single ownership of all networklayers and elements is thus being challenged. While the mostbasic concepts related to Passive network sharing are easier toimplement and have already been partially exploited, Activenetwork sharing is raising in importance to enable substantialand sustainable reduction in network expenses and thus ensureoperators’ future competitiveness. The work presented in thispaper addresses the aforementioned challenges by presenting anetwork model that possess desirable properties for RAN Shar-ing and presents both centralized and distributed solutions withanalytical guarantees that efficiently allocate RAN resourcesfairly among operators.

II. RELATED WORK

Resource allocation optimization in wireless and cellularnetworks has been extensively studied. A common approachfollowed by many proposals is to provide load balancing bymaximizing a certain objective function. The authors of [5]follow a max-min fairness criterion to ensure fairness and loadbalancing through association control. For multirate WLANs,[6] and [7] aim at providing proportional fairness to addressthe resource allocation problem, which is shown to be NP-hard, and propose centralized approximation algorithms thatprovide worst-case guarantee; in particular, [6] designs a 2+ε-approximation algorithm based on a reduction to the Gener-alized Assignment Problem (GAP). This algorithm relies onthe 2-approximation for GAP proposed in [8], which has beenovercome by [9] and recently further improved by [10].

In cellular networks, [11] formulates a proportional fair-ness problem specifically designed for 3G data networks andproposes an heuristic online algorithm to solve the optimalallocation in a network-wide context. In the same context, [12]designs and evaluates a distributed algorithm that converges toa near-optimal solution providing proportional fairness in anheterogeneous cellular networks. In [13], the authors proposeand analyze an iterative distributed α-optimal algorithm foruser association that converges to a globally optimal alloca-tion. All the above focus on a network exploited by a single

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operator, in contrast to our work which addresses Active RANsharing between multiple operators.

Active RAN sharing enables pooling of radio resources en-hancing the overall RAN utilization, while reducing infrastruc-ture investments. 3GPP Services Working Group (WG) SA1studied RAN sharing, analyzing a set of use cases derivingbusiness requirements [14]. The architecture and operationsthat enable different MNOs to share the RAN are specified bythe 3GPP Architecture WG SA2 in [2] detailing two differentapproaches. One refereed to as Multi-Operator Core Network(MOCN), where each operator is sharing eNBs connected tothe core network elements of each MNO using a separate S1interface. In the second one named Gateway Core Network(GWCN), operators share additionally the Mobility Manage-ment Entity (MME). GWCN enables extra cost savings com-pared to MOCN, but at the price of reduced flexibility, i.e. nomobility among different Radio Access Technologies and nointerworking with legacy networks.

Recent efforts have been addressed to design RAN Sharingarchitectures based on Network Virtualization concepts [15]–[18]. Building on eNB virtualization, [15], [16] introduce thenotion of Network Virtualization Substrate (NVS), a virtual-ization technique that provides an interface for operators. Thisinterface allows to reserve wireless resources in slices, mod-ifying the base stations MAC schedulers. The same authorsenhanced NVS with the design of CellSlice [17], which adoptsthe same solution from a gateway-level perspective while en-abling remotely controlled scheduling decisions, thereby sim-plifying near-term adoption. An SDN-based approach to con-trol multi-tenant slices is proposed in [18] and a software de-fined radio access network simulator was designed by authorsof [19]. All these works focus on architectural aspects of RANSharing and do not deal with the design of specific algorithmsto share the resources, in contrast to the approach proposedin this paper which focuses on the algorithmic part. We adoptsuch capacity broker architecture and introduce intelligencein allocating resources and assigning user to particular basestations.

Like in this work, [20], [21] address the design of optimalalgorithms for RAN sharing. In [21], the authors present Ra-dioVisor, which aims at the optimization of the total networkutility by using a max-min fairness criterion. RadioVisor, usingsimulator proposed in [19], proposes an heuristic algorithm toslice time-frequency slots at each base station and share theseamong N network operators while ensuring isolation. In con-trast to RadioVisor, the approach that we propose in this paperrelies on the proportional fairness criterion, which we haveshown to provide many desired features. [20] proposes Net-Share, a network-wide radio resource management frameworkthat provides RAN Sharing. While [20] presents a proportionalfairness formulation similar to the one we propose here, theauthors do not provide a rationale to justify their choice, incontrast to the solid analytical arguments that we provide here.In order to solve the proportional fairness formulation, theauthors use a general solver of non-linear problems, whichincurs a very high computational complexity and can only beuse for small scenarios; in contrast, we propose an efficientalgorithm that provides some performance bound guarantees

and can be used for large scenariosTo the best of our knowledge, this work is the first one

that (i) provides an analytical proof for a set of desirableproperties of the RAN Sharing resource allocation problem,achieving a trade off between optimality and fairness; and(ii) proposes an efficient algorithm that provides performancebound guarantees for this problem.

III. NETWORK MODEL AND PROBLEM FORMULATION

In this section, we describe our assumptions, the utilityfunction of an operator as well as the formulation of ouroptimization problem.

A. Model and notation

Our cellular LTE-Advanced scenario consists on multiplebase stations where there are N users from K different op-erators sharing the network. For simplicity, we assume thattraffic is elastic, and all users are saturated. MORA algo-rithm computes an optimal solution assuming that there is acentralized entity that has all information about the network.Table I presents all the notation symbols and terminology ofour model.

Note that the algorithm proposed here could be combinedwith some other scheme that deals with inelastic traffic. Specif-ically, the technique for inelastic traffic would determine theresources received by each operator to satisfy its inelastictraffic needs, such us the one proposed in our previous work[1], and then the algorithm that we propose here would beused to share the remaining resources among elastic traffic.

TABLE INOTATION

Symbol DefinitionK Number of operatorsvk Share of operator kU Set of usersA Set of base stationsN Number of users: N = |U |M Number of base stations M = |A|Nk Number of users from operator kN Number of users from all operatorswi Weight of user iri Throughput of user ibij Fraction of resource blocks of BS j assigned to user icij Transmission rate of user i in BS jxij Association of user i in BS j

B. Sharing criterion for elastic traffic: Problem formulation

In order to share the resources among users, we need todecide (i) how to associate users to base stations; and (ii)how to share the resources of each base station among itsassociated users. The main goal when doing this is to achieveboth inter- and intra-operator fairness:• Inter-operator fairness: the amount of resources assigned

to each operator should be distributed proportionally theshare of the operator vk, which represents fraction ofnetwork resources to which the operator is entitled.

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• Intra-operator fairness: the resources assigned to anoperator should be fairly distributed among its users ac-cording to the operator’s utility.

Following [22], which addresses a similar problem, we es-tablish that the optimal configuration should be the one thatmaximizes the total welfare of the network, W : this functioncorresponds to the aggregation of the utilities of the differentoperators, where the utility of an operator corresponds in turnto the aggregation of the utility of its users, given by thethroughputs of the users, ri. Thus,

W = Uinter(Uintra(ri))

where we refer to Uinter as the inter-operator utility and toUintra as the the intra-operator utility or (simply) operatorutility.

For the utility of an operator or intra-operator utility, wefollow the proportional fairness criterion, which is a widelyaccepted criterion to provide fairness for elastic traffic,

Uk = Uintra(ri) =

Nk∑i=1

log(ri)

For the inter-operator utility, we sum the individual utilitiesof the different operators, weighted by some weight wk thatrepresents the priority that the users of a given operator aregiven in the resource allocation:

W = Uinter(Uk) =

K∑k=1

wiUk =

K∑k=1

Nk∑i=1

wi log(ri)

In order to set the weights wi, we follow the followingrationale. According to Theorem 1, when two users of differ-ent operators compete for resources in the same base station,they share the base station’s resources proportionally to theirweights. Hence, the total amount of resources received by anoperator will grow proportionally to the sum of the weightsof all its users,

∑i∈Uk wi. If we let the operator’s share vk

represent the contribution of the operator to the overall net-work (like, e.g., the monetary contribution), then the totalamount of resources received by the operator should be givenby vk. Thus, to ensure that resources are fairly shared amongoperators, we enforce that the following equation is satisfied:∑

i∈Uk

wi = vk

The above is interpreted as follows. The vk represents theshare of resources an operator is entitled to. This share isdistributed among the operator’s users, such that the weight ofeach user, wi, represents the fraction of the operator’s shareassigned to the user. Thus, the sum of the wi’s has to addup to the share vk. The operator may assign larger or smallerweights to the user depending on their relative priority. In thecase the operator wants to give all users the same priority, wewill have wi = vk/Nk ∀i ∈ Uk.

With the above, we can formulate our optimization problemas follows:

maximize W =

K∑k=1

Nk∑i=1

vkNk

log(ri) (1a)

subject to:

ri =

M∑j=1

bijxijcij (1b)

M∑j=1

xij = 1,∀i ∈ Nk (1c)

K∑k=1

Nk∑i=1

bij · xij ≤ 1,∀j (1d)

xij ∈ {0, 1} (1e)

bij ≥ 0 (1f)

where the above equations impose the following constraints:• (1b) gives the rate of a user as a function of the transmis-

sion rate and resources blocks assigned at the base stationwhere it is connected; this is given by the base station towhich the user is associated (xij), the fraction of resourceblocks it receives from this base station (bij), and thetransmission rate (cij), which is given by the modulation-coding scheme of the user at this base station.1

• (1c) forces to every user to be connected.• (1d) forces that the fraction of resource blocks used in

a base station cannot exceed 1 (i.e., the total capacity ofthe BS).

• (1e) imposes that the association variable xi,j is binary.• (1f) imposes that the fraction of resource blocks bij is a

positive real number.We refer to the above strategy as the multi-operator resource

allocation (MORA) problem. Next, we prove some propertiesfor this strategy and evaluate the complexity of this problem.

IV. MORA PROPERTIES

In the following we show some desirable properties of theproposed resource allocation strategy, both in terms of overallperformance, fairness and complexity. In terms of fairness, wewould like to show that the proposed allocation is fair bothin the sharing of resources of each base stations as well as inthe way users are allocated to base stations.

A. BS Association

The following theorem shows that the resources of a basestation are fairly shared among the operators using that basestation. In particular, the fraction of resource blocks receivedby each user is proportional to its weight.

Theorem 1. The fraction of resource blocks allocate to useri in base station j with the MORA strategy satisfies:

bij =wixij∑Np=1 wpxpj

,∀i, j (2)

where wi is the weight of user i – this weight depends on theoperator: all the users of operator k have a weight equal towk, i.e., wi = wk = vk/Nk.

1Note that cij is normalized to the total number of resource blocks of thebase station, such that the throughput is given by the fraction of resourceblocks multiplied by cij .

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Proof: The theorem explains how the resources are allo-cated inside the base station and not the way users are allocatedto them. Thus, supposing that we already have an associationfor xij of users x∗ij , using Lagrangian methods, we can showthat the above is satisfied. The lagrangian function for ourproblem is:

L(x, λ) =

N∑i=1

wix∗ij log(ri) + λ(1−

N∑i=1

bij) =

=

N∑i=1

wix∗ij log(bij) + wix

∗ij log(ci) + λ(1−

N∑i=1

bij)

Setting ∂L∂bij

= 0 we get:

wix∗ij

bij− λ = 0,∀i

Then, since∑Ni=1 bij = 1,∀j, the unique optimal transmis-

sion resource allocation stands as follows:

bij =wix

∗ij∑N

p=1 wpx∗pj

,∀i, j

which proves the theorem.It follows from the above theorem that the strategy fol-

lowed by MORA allocates more resources to those opera-tors whose users are at locations in which there is a lowerdemand from other users/operators. We refer to this notion,in which resources are allocated to operators based on thespatial distribution of their users, as spatial fairness. Anotherinherent property resulting from the above theorem relates theoperator’s share: as long as two operators use base stationswith the same level of congestion (i.e., the same demand), thetotal amount of resources that they will receive will be propor-tional to their share. Lastly, the above theorem also indicatesthat only the fraction of users of operators and their shareshave an influence on the amount of resources that receivesfrom a base station and not the absolute number of users, soan operator that multiplies her users will not receive moreresources unless increments her share (vk). All this propertiesare formally proven in the corollaries of the appendix.

B. User association

So far we have looked at the fairness when sharing theresources of a base station among its users. Another desirableproperty from a fairness standpoint is on the user association.

The following theorem confirms the optimality in terms ofoverall performance of the resource allocation strategy pro-posed in the previous section.

Theorem 2. The MORA allocation is Pareto-optimal.

Proof: The proof follows easily from the fact that MORAchooses the configuration {bij , xij} that maximizes

∑k wkUk.

This implies that for any alternative allocation {b′ij , x′ij} forwhich U ′l > Ul for some l, then necessarily we have U ′m <Um. Indeed, if this was not the case than

∑k wkU

′k would be

larger than∑k wkUk, which would contradict our initial state-

ment. This proves that the allocation resulting from MORA isPareto-optimal.

In order to show that the user association provided is sat-isfactory for all operators (and hence fair), we would likethe user association resulting from MORA represents a Nashequilibrium, meaning that an operator cannot obtain any gainby unilaterally deviating from the algorithm and following adifferent user association strategy. While we cannot prove thatthe proposed resource allocation is a Nash equilibrium, the fol-lowing theorem shows that approximates a Nash equilibriumallocation.

Theorem 3. If the resources blocks of all base stations are di-vided among their users according to (2), the user associationresulting from MORA is an ε-approximate Nash equilibrium,with ε = vk log(e).

Proof: Let us look at the maximum gain in utility thatan operator can experience when one of his users, user j, ismoved from one base station to another. The total gain inutility, which can not exceed 0, is given by the gain in utilityby the operator, denoted by ∆Ukj , plus the difference in theutility of the other operators in the previous base station, p,and the new one, n:

∆Ukj +∑i∈n

wi log

( ∑i∈n wi

wk +∑i∈n wi

)+∑i∈p

wi log

(wk +

∑i∈n wi∑

i∈n wi

)≤ 0

where the above has been obtained taking into account thatlog(ri) = log(wi)− log(

∑j wj) + log(cij) and that the terms

log(wi) and log(cij) are constant.The above can be rewritten as

∆Ukj + wk log

( ∑i∈n wi/wk

1 +∑i∈n wi/wk

)∑i∈n wi/wk

+ wk log

(1 +

∑i∈n wi/wk∑

i∈n wi/wk

)∑i∈p wi/wk

≤ 0

Given that (x/(1 + x))x ≤ 1 and ((1 + x)/x)x ≤ e, forx ≥ 0, we have

∆Ukj ≤ wk log(e)

The above gives an upper bound on the increase of utilityfor a user. The increase of the total utility of the operator,∆Uk, is upper bounded by the sum for all users,

∆Uk = Nk∆Ukj ≤ Nkwk log(e) = vk log(e)

which proves the theorem.

C. Complexity & Uniqueness

The MORA problem is a non-linear integer programmingproblem. The following theorem shows that this problem isNP-hard and hence there is no algorithm able to solve theproblem in polynomial time unless P = NP .

Theorem 4. The MORA problem is NP-hard.

Proof: The reduction is via the 3-dimensional matchingproblem which is known to be NP-complete. Recall that the3-dimensional matching problem is stated as follows. Let us

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consider disjoint sets C = {c1, . . . , cn}, D = {d1, . . . , dn}and E = {e1, . . . , en}, and a family F = {F1, . . . , Fm}of triples with |Fi ∩ C| = |Fi ∩ D| = |Fi ∩ E| = 1 fori = 1, . . . ,m, with m ≥ n. The question is whether F containsa matching, i.e., a subfamily F ′ for which |F ′| = n and∪Fi∈F ′Fi = C ∪D ∪ E.

Our reduction is along the lines of [11]. We call the triplesthat contain cj triples of type j. Let fj be the number of triplesof type j for j = 1, . . . , n. BS i corresponds to the triples Fifor i = 1, . . . ,m. We create two types of users, element usersand dummy users. We have 2n element users, correspondingto the 2n elements of D ∪E. There are fj − 1 dummy usersof type j for j = 1, . . . , n. Note that the total number ofdummy users is m−n. Element users can connect to the BSsthat correspond to a triple that contains this element, withan average rate of R. Dummy users of type j can connect(also with an average rate of R) to the BSs that correspondto triples of type j. Element users have a weight w = 1 anddummy users have a weight w = 2. We claim that a matchingexists if and only if the MORA’s optimal objective function isW = 2n log(R/2) + 2(m− n) log(R).

The value of the objective function is bounded above by thefollowing optimization problem:

maximize

2n∑i=1

log(biR) +

m−n∑i=1

2 log(biR) (3)

subject to∑2ni=1 bi+

∑m−ni=1 bi = m, where the first term of the

summation corresponds to the element users and the secondterm to the dummy users, and bi is the fraction of resourceblocks assigned to a user.

By applying the Lagrange multiplier method, it can be easilyseen that the above optimization problem is solved when bi =1/2 for the element users and bi = 1 for the dummy users.This gives an upper bound on W equal to 2n log(R/2) +2(m−n) log(R). This corresponds to a global maximum, andthus any other set of bi values yields a smaller W .

Assume that there is a matching. For each Ti = (cj , dk, el)in the matching, we associate element users dk and el withBS i. For each j, this leaves fj − 1 idle BSs correspondingto tripes of type j that are not in the matching. We associatethe fj − 1 dummy users of type j to these tj − 1 BSs. Thisassignment has an objective function of W = 2n log(R/2) +2(m − n) log(R), which is equal to the upper bound givenabove. In case there is no matching, it is not possible to havethe 2n element users sharing n BSs with bi = 1/2 each, andtherefore we cannot achieve the distribution of bi values thatmaximizes W . According to the above result, this implies thatwe obtain a smaller W value, which proves the theorem.

Other interesting characteristic is that the problem can havemultiple optimal solutions, therefore it’s possible to prove thenon-uniqueness property of the problem, which is formallyexpressed in the following theorem.

Theorem 5. (Non-Uniqueness). MORA problem can havemultiple optimal solutions.

Proof: The proof follows directly analyzing an instance of

the problem where a user u can move from a base station, i.e.n, to another base station p without having an impact on thenetwork utility K, or what is the same, causing that ∆W = 0.Let ∆Uj be the sum of the utilities for every operator in basestation j (∆Uj =

∑k∈K ∆Ukj), then:

∆W =∑j∈M

∆Uj = ∆Un + ∆Up

Equivalent to the proof for Theorem 3 we can present this dif-ferences taking into account that log(ri) = log(wi)−log(

∑j wj)+

log(cij) and that the terms log(wi) and log(cij) are constant,and also supposing that the user can connect to both stationswith the same rate, i.e. (cun = cup), which leads into:

∆W = ∆Un + ∆Up =∑i∈n

wi log

( ∑i∈n wu

wu +∑i∈n wi

)

+∑i∈p

wi log

(wu +

∑i∈p wi∑

i∈p wi

)=

=∑i∈n

wi

[log

(∑i∈n

wi

)− log

(wu +

∑i∈n

wi

)]

+∑i∈p

wi

− log

∑i∈p

wi

+ log

wu +∑i∈p

wi

In the case when

∑i∈n wi =

∑i∈p wi, since wu is the same

for all base stations, ∆W = 0 and there are two possibleassociations of user u that gives the same utility, proving thetheorem.

V. MULTI-OPERATOR RESOURCE ALLOCATION (MORA)CENTRALIZED ALGORITHM

Due to the NP-Hardness of the MORA problem, we pro-pose a polynomial time algorithm similar to the one proposedin [6], which solves this problem while ensuring performanceguarantees with an additive ratio. The proposed algorithm isbased on a discretized relaxation and reduction to General-ized Assignment Problem (GAP), a well know problem. Anschematic view of the algorithm is given in Figure 1, whichincluded a description of the algorithm steps.

1. Discretize scheduling period solving relaxation.y = solve relaxDLP ({cij}, D)

2. Transform into a GAP using (x, b) and u.

3. Solve problem by a rounding techniquex = roundGAP (b, x, u)

4. Redistribute resource blocks percentages

Fig. 1. Approximation algorithm

A. Discretization

In our original MORA problem, we are considering thatthe percentage of resources blocks assigned to a user is acontinuous value between 0 and 1, assumption that doesn’t

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stand in reality since the amount of resource blocks is aninteger number (b ∈ Z). Considering this real world constraint,and in order to reduce the complexity of our problem, wediscretize the scheduling period, moving from a continuouspossible resource blocks distribution to a discrete one and bythat, linearizing the objective function of our problem. Weintroduce a new indicator variable yijτ , which is equal to 1if and only if user i is assigned to access point j, and accesspoint j has allocated τ (out of D) percent of the resourceblocks to user i. The value of τ ranges from 1 to D. Withthis new variable, the formulation of the discretized problemis defined as follows:

maximize

K∑k=1

Nk∑i=1

M∑j=1

D∑τ=1

vkNk

yijτ log( τD· ci,j

)(4a)

subject to:M∑j=1

D∑τ=1

yijτ = 1,∀i ∈ N (4b)

K∑k=1

Nk∑i=1

D∑τ=1

τ

Dyijτ = 1,∀j (4c)

yijτ ∈ {0, 1} (4d)

Let WMORA(x, b) stand for the solution of the original (non-linear, NP-Hard) MORA problem, and WDis(y) the solutionto the above (discretized) algorithm. Then:

Lemma 1. If we choose a big enough D:

WDis(y) ≥WMORA(x, b/(1 + δ))

Proof: If the resolution of the discretized version of theproblem allows to represent every single case of the non-dicretized problem, the loss introduced by the discretizationis 0. To achieve this is neccesary to set a big enough amountof steps for the time scheduling that can be obtained by con-sidering the worst association case.

D = (1 + δ) ·∑Kk=1

∑Nk

i=1 vkmin(vk)

(5)

Although we transformed the objective function into a non-linear function, the problem is combinatorial due to the bi-nary variable yijτ , and hence, the computation is essentiallyimpossible for even a modest-sized cellular network. To over-come this, we can relax the user association by multiple-BS user association to compute an integer relaxation of thelinear problem allowing yijτ take fractional values. This phys-ical relaxation gives an upper bound solution for the problemand it’s then direct that any solution of WDis(y) is feasiblefor the fractional association problem WDis(y) leading intoWDis(y) ≥ WDis(y). It’s as well direct that a fractionalsolution (x′′, b′′) for the original MORA formulation followsWMORA(x′′, b′′) ≥WDis(y)

Lemma 2. Joining previous statements we can ensure that:

WMORA(x′′, b′′) ≥WDis(y) ≥WMORA(x, b/(1 + δ))

B. Reduction to Generalized Assignment Problem

Once we have obtained a solution for the relaxed discretizedversion of the problem, we proceed to reduce our problem to aGAP, by fixing the resource block assignation and utility andonly standing xij as decision variables. Thus, the reductionto GAP it can be easily done by holding resource block andutility by using result obtained from previous step, y.

Set: ∀i ∈ U, j ∈ A:

xij =

D∑τ=1

yijτ (6)

pij =

∑Dτ=1

τD yijτ

xij(7)

uij =vkNk

log (cijbij) (8)

C. Solve problem by rounding technique

Once this transformation is done, our problem fits withthe standard formulation of GAP. It is well known that GAPis NP-Hard [23], but we can use a rounding algorithm totransform our multiple-BS association fractional solution toan single-bs association integer solution. To do this, we usea rounding scheme proposed by [8] in order obtain the sin-gle bs-association solution. This rounding technique is a 2-approximation algorithm for the maximization GAP that takesa fractional assignment x as input, and constructs an integerassignment vector x with three properties: (i) the objectivefunction value for the integer association is greater than theone obtained with the fractional association, (ii) each user i isassigned to one machine and (iii) the load imposed on each

server is at most 2 (Nk∑i=1

bi,j · xi,j ≤ 2) for every base station.

D. Redistribute resource blocks percentage

Once we have the single-bs association solution for ourproblem what left is reassign time assigned to each of theusers b in the way of equation (2). This reassignment give usa solution with the properties described below.

1) The vector x in a feasible (fractional) solution (x′′, b′′)to the formulation MORA is feasible for the formulationGAP. Alternatively, the feasible solution x for GAP to-gether with its vector b is a feasible solution to MORA:

WGAP (x) = WMORA(x′′, b′′) ≥WDis(y)

≥WMORA(x, b/(1 + δ))

2) The solution (x, {t}) may not be feasible, as some ofthe access points may be over-scheduled by a factorof 2. However, if we scale down the time allocationst′ = 1

2 b, we obtain a feasible solution with the followingproperty:

WGAP (x, {b′}) ≥WMORA(x,b

2 · (1 + δ))

Theorem 6. With the proposed centralized algorithm, the totalwelfare of the network satisfies W ≥W ∗MORA

∑Ni=1 wi log(2),

where W ∗MORA is the total welfare of the optimal allocation.

7

Proof: Consider any integral association WMORA(x∗, b∗)that is a feasible solution to 1. Then, for the solution W (x, b)produced by our algorithm it holds that:

W (x, b) ≥WMORA(x∗,b∗

2 +O(δ)) (9)

We can transform this inequality into an additive approxima-tion ratio as follows:

W (x, b) =

K∑k=1

Nk∑i=1

M∑j=1

xij · wi · log(cij bij) ≥

≥K∑

k=1

Nk∑i=1

M∑j=1

x∗ij · wi · log(cijb∗ij2) =

=

K∑k=1

Nk∑i=1

M∑j=1

x∗ij · wi · log(cijb∗ij)−

−K∑

k=1

Nk∑i=1

M∑j=1

x∗ij · wi · log(2) =

= WMORA(x∗, b∗)−

−K∑

k=1

Nk∑i=1

M∑j=1

x∗ij · wi · log(2) =

= WMORA(x∗, b∗)−

N∑i=1

wi log(2)

Last equality holds since∑Nk

i=1

∑Mj=1 xij = 1. Then is proven

that:

W (x, t) ≥WMORA(x∗, b∗)−N∑i=1

wi log(2) (10)

VI. MULTI-OPERATOR RESOURCE ALLOCATION (MORA)DISTRIBUTED ALGORITHM

In LTE-A there is no interface to gather all the data of thenetwork at a centralized location in a timely manner. There areinterfaces to gather information from neighbours. Therefore,an interesting option is to implement an algorithm that opti-mizes performance in real-time is by means of a distributedalgorithm.

In the following, we design an algorithm to maximize thetotal welfare that satisfies W ≥W ∗ −

∑Ni=1 wi log(e), where

W ∗ is the total welfare with the optimal strategy. It is worth-while noting that our centralized algorithm provides a slightlytighter bound, namely W ≥W ∗−

∑Ni=1 wi log(2 + ε). How-

ever, as mentioned above, that algorithm is centralized andmuch more complex to use in real world scenarios.

The proposed algorithm is based on a greedy approach,in which in each iteration a given user i reassociates to thebase station that provides her with the largest ri. The fol-lowing theorem shows that this algorithm satisfies the finiteimprovement property, which states that an arbitrary sequenceof the algorithm executed by each user always converges toan equilibrium in a finite number of steps.

Theorem 7. The proposed distributed algorithm has the finiteimprovement property.

Proof: The throughput of user i, ri, is an increasingfunction of cij/

∑Nl=1 wjxjl. Hence, our algorithm can be

understood as a congestion game in which the load at a basestation is given by the sum of weights of the users at thebase station, lj =

∑j wj , and a user seeks to minimize aij lj ,

where aij = 1/cij . This congestion game falls in the categoryof a singleton weighted congestion game with separable pref-erences, resource-independent weights (the wi values are thesame for all base stations) and linear variable cost (as lj islinear with the wi’s). According to Theorem 4 of [24], such agame has the finite improvement property.

The following theorem shows that the solution at which thedistributed algorithm converges provides a performance that isabove a given lower bound.

Theorem 8. With the proposed distributed algorithm, the totalwelfare of the network satisfies W ≥ W ∗ −

∑Ni=1 wi log(e),

where W ∗ is the total welfare of the optimal allocation.

Proof: Let j be the base station to which user i is asso-ciated with the distributed algorith, and k the base station towhich it is associated with the optimal strategy. Since with thedistributed algorithm we maximize ri, the following holds:

wi log

(wicij∑j∈Jd

wj

)≥ wi log

(wicik∑

k∈Kdwk + wi

)where Jd and Kd are the set of users associated to base stationsj and k, respectively, with the distributed algorithm.

Let Kopt denote the set of users associated to base stationk with the optimal strategy. Since we stated above that wi ∈Kopt, the following holds:

wi log

(wicij∑j∈Jd

wj

)≥ wi log

(wicik∑

k∈Kdwk +

∑k∈Kopt

wk

)If we let r∗i and ri denote the throughput of user i with the

optimal strategy and with the distributed algorithm, respec-tively, from the above it follows that

wi log(r∗i )− wi log(ri) ≤ wi log

(wicik∑k∈Kd

wk

)−

wi log

(wicik∑

k∈Kdwk +

∑k∈Kopt

wk

)=

−wi log

( ∑k∈Kopt

wk∑k∈Kd

wk +∑k∈Kopt

wk

)Summing the above over all users yields

W ∗ −W ≤ −N∑i=1

wi log

( ∑k∈Ki

optwk∑

k∈Kidwk +

∑k∈Ki

optwk

)where Kiopt and Kid are the following set of users. Let usconsider the base station to which user i is associated withthe optimal strategy. Then, Kiopt is the set of users associatedto this base station with the optimal strategy, and Kid is theset of users associated to this base station with the distributedalgorithm.

8

By rearranging terms, we can rewrite the above as

W ∗−W ≤ −M∑j=1

∑k∈Kj

opt

wk log

( ∑k∈Kj

optwk∑

k∈Kjdwk +

∑k∈Kj

optwk

)

where Kjopt is the set of users at base station j with the optimalstrategy, and Kjd is the set of users at base station j with thedistributed algorithm.From the above:

W ∗ −W ≤ −M∑j=1

log

( ∑k∈Kj

optwk∑

k∈Kjdwk +

∑k∈Kj

optwk

)∑k∈Kj

opt

wk

= −M∑j=1

∑k∈Kj

d

wk log

( ∑k∈Kj

optwk/

∑k∈Kj

dwk

1 +∑

k∈Kjopt

wk/∑

k∈Kjdwk

)∑k∈Kj

opt

wk∑k∈Kj

d

wk

Given that (x/(1 + x))x ≥ 1/e for x ≥ 0, we obtain thefollowing bound

W ∗ −W ≤N∑i=1

wi log(e)

which proves the theorem.

VII. PERFORMANCE EVALUATION

In order to evaluate the different algorithms proposed, inthe following we describe the simulation setup used for per-formance benchmarking.

A. Simulation Setup

Mobile Network Model: The mobile network scenario con-sidered, based on the IMT-Advanced evaluation guidelines[25], is depicted in Figure 2. It consists of 19 base stations ina hexagonal cell layout with 3 sector antennas. We focus onthe ’Urban Micro-cell scenario’ to model a dense ’small cell’deployment. The detailed system configuration parameters aresummarized in Table II.

Mobility model: Users mobility follows the SLAW modeldefined in [26], which is a human walk mobility model basedon real GPS-tracking measurements of more than 6 millionUEs. This model allows us to synthetically generate UEs move-ments in our scenario in a realistic way. The configurationparameters used to generate the UEs movement in our exper-iments are given in Table III, and the resulting distributiondepicted in Figure 2.

Signal power and cij estimation: In order to compute cij ,we need to estimate the Signal Interference to Noise Ratio foreach user i to each BS j following the idea proposed by [12],cij is computed as:

cij = f(SINRij)

SINRij =Pjgij∑

k∈A,k 6=j Pjgij + σ2

where Pj is the transmit power of BS j, gij denotes thechannel gain between user i and BS j, which includes pathloss, shadowing and antenna gain, and σ2 denotes the thermal

Parameters ValuesMobile Network 1 Tier, 19 BSs with 3 SectorsVirtual Operators 3IMT-A Scenario Urban Micro (UMi)Intersite distance 200 mUsers Mobility SLAW Model

Carrier Frequency (fc) 2.5 GHzSimulation Bandwidth (BW) 10 MHz

Thermal noise level (σ2) -174 dBm/HzPathloss 36.7 log10(d) + 22.7 + 26 log10(fc)

Antenna tx power (Pj ) 41 dBmAntenna gain 17 dBi

Shadow fading std 4 dBSimulation Time 1 hourWarm-up Time 10 minutes

TABLE IIITU IMT-ADV SIMULATION PARAMETERS CONFIGURATION

−600 −400 −200 0 200 400 600−600

−400

−200

0

200

400

600

Meters

Me

ters

Fig. 2. User distribution models for 550 users. The different colors representusers from three different operators

noise power level. We use a shadowing factor represented bya log-normal function of mean 0 and standard deviation of4 dB. Thus, under the assumption that the user achieves theShannon capacity rate, the value of the rate stands:

cij = f(SINRij) = log2(1 + SINRij) (11)

= log2

(1 +

Pjgij∑k∈A,k 6=j Pjgij + σ2

)

)This value cij represent the rate in bps/Hz achievable by useri in base station j. If we want to compute the throughput inMbps we need to make use of the bandwidth BW :

Throughputij = BW · cij

B. Benchmark Algorithms

Benchmark 1 - Distributed Static Legacy Algorithm : Thefirst algorithm considered in our study for benchmarking pur-poses is the Distributed Static Legacy. With this approach, UEsfrom different mobile network virtual operators are associatedto the base station with the highest SINR. This models legacysystems with no multi-tenant support for UE re-associationwhere the enforcement of the resource allocation per tenantwould be performed at every base station independently atscheduling level (see [16] for an example).

9

ParameterDistance alpha 5Waypoints(wp) 500

Inverse self-similarity of wp 0.95Time (hours) 3

Clustering range (m) 50

TABLE IIISLAW PARAMETERS CONFIGURATION

Benchmark 2 - MORA and Upper bound of MORA : Thesecond algorithm considered in our study for benchmarkingpurposes is MORA. It was previously discussed that this al-gorithm is NP-Hard and posses high computational cost. Forthis reasons all the results for this benchmark algorithm for anumber of users greater than 130 came from an upper boundcomputed by a discretization of the logarithm as done in ourcentralized algorithm.

C. Performance Results

We next present the performance results obtained with theabove simulation setup. We start by analyzing the performancefrom the perspective of the mobile infrastructure provider (e.g.,operator with full management access to the physical infras-tructure), focusing on the overall network performance, eval-uating the utility for the different algorithms as well as thefile transfer delay and throughput improvements due to ouralgorithm. Then, we present the results from the mobile (vir-tual) operators perspective side, looking at the performanceof each operator and showing that proposed algorithms meetthe network sharing agreements. Lastly, we conclude the the-oretical complexity discussion adding some simulation resultsconfirming the theoretical results.

Utility performance: For this experiment, we present thegains in terms of utility that our algorithms provides withrespect to the current Legacy system (Static Slicing). In figure3, we can see that our algorithms both centralized and dis-tributed outperforms the utility achieved by Static slicing fordifferent number of users in the system. It is also noticeablethat both proposed algorithms have a performance close tothe optimal (displayed in green), achieving the centralized thecloser performance. The error bars in this figure represent the95% confidence intervals for the utility in this scenario. 2

Download time: One of the benefits introduced by our modelis the one related with intra-operator fairness. Proportionalfairness permit to users with low rates to receive a highernumber of resources in order to exponentially improve theQoS experienced by them. In the figure 4, we have displayedthe throughput for each of the 50 users of the system when weused Static Slicing (left) and our MORA centralized solution(right).

We can observe that the throughput values are more bal-anced in the case of our centralized solution (right), wherethe users with high rates transfer resource blocks to the users

2Error bars are not displayed in further figures for readability reasons.

50 90 130 170 210 250−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

users

W (

utility

)

Global performance of the different algorithms

Static slicing (Legacy)

MORA Distributed

MORA Centralized

MORA Optimal (Upper bound)

Fig. 3. Algorithm comparison of utility for different number of users in thesystem

with lower rates as is the objective of the proportional fairnessparadigm.

As we pointed out, this have a clear relevance in the per-formance experienced by the users. In the figure 5, we havedisplayed the file transfer delay for a 20 megabytes archive byeach of the 50 users of the system when we used Static Slicing(left) and our MORA centralized solution (right). Again a moreuniform distributed file transfer delay is earned when MORAcentralized algorithm is used in the user association. E.g. whilein the case of static slicing 3 users out of the 50 (labeled 8, 18,22) consume one minute or more, using the MORA algorithmnone of them requires more than 60 seconds to download thefile.

Shares agreements: From the operator perspective, the crit-ical aspect regarding RAN sharing relies on share agreementfairness. Each operator buys a percentage of the total networkshare, defined in our paper as wk, and the operator wouldlike to receive this percentage of resources from the network.Our MORA solution respects the shares of the operators, aswas discussed theoretically, and this is reflected in figure 6.In this figure, a comparison between resources received withthe association

∑i∈k bij (blue) and percentage of network

contracted vk (green) is displayed for an scenario with threeoperators. Three different agreement cases were considered.

1) Uniform share agreement among the 3 operators vk =(1/3, 1/3, 1/3). (left subfigure)

2) Triangular share agreement among the 3 operators vk =(1/5, 3/5, 1/5). (center subfigure)

3) Irregular share agreement among the 3 operators vk =(1/10, 6/10, 3/10). (right subfigure)

In all the three figures, we can see that our formulation respectsthe agreements among the operators. It’s worthwhile to notethat the small deviations are due to the distribution of the usersand for a uniform distribution

∑i∈k bij = vk.

Computational complexity: Lastly, the computational com-plexity of the proposed and benchmark algorithms was eval-

10

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8

user

Th

rou

ghpu

t (b

ps/H

z/s

)

Throughput per user for 50 users with Legacy algorithm

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8

user

Th

rou

ghpu

t (b

ps/H

z/s

)

Throughput per user for 50 users with MORA Centralized

Fig. 4. Comparison of different algorithms throughput per user for 50 users.

0 5 10 15 20 25 30 35 40 45 500

10

20

30

40

50

60

70

user

File

tra

nsfe

r dela

y (

seco

nd

s)

File transfer delay per user for 50 users with Legacy algorithm

0 5 10 15 20 25 30 35 40 45 500

10

20

30

40

50

60

70

File

tra

nsfe

r dela

y (

seco

nd

s)

user

File transfer delay per user for 50 users with MORA Centralized

Fig. 5. Comparison of different algorithms file transfer delay per user for 50 users.

uated for different problem sizes and this comparison is de-picted in figure 7. For the case of MORA optimal the con-sumed time is computed for the discretized version and notfor the non-linear version, which is even more complex. Inthis figure we can see that MORA optimal has an exponen-tial complexity growth due to her integer variables nature.The rest of the algorithms run in polynomial time and it’sclear to see that their complexity is polynomial being theMORA centralized algorithm the more complex followed bythe MORA distributed. The algorithm with less complexitycost is the highest SINR association Static Slicing, as expected.This confirms that both solutions proposed are computationallyfeasible thanks to the polynomial time solvable character.

VIII. CONCLUSIONS AND FUTURE WORK

Multiple Operator Resource Allocation MORA network ac-tive sharing is turning into a key business and a fair ap-proach for this assignment among different tenants needs aconsistent framework. We presented an offline solution thatcopes with most of the desirable properties that this problemsrequires. Our solution takes into account both inter and intraoperator fairness. This give to our solution a set desirableproperties from base station association and user associationperspectives respecting the percentages of the networks con-tracted by each operator and being proportionally fair with

10 20 30 40 50 60 70 80 90 100 1100

2

4

6

8

10

12

users

seconds

Time consumption for resolution

MORA optimal(discretized)

MORA Centralized

MORA Distributed

Static slicing (Legacy)

Fig. 7. Algorithm comparison of convergence time consumption for differentproblem sizes

the users. We proved that MORA is an NP-Hardness prob-lem and accordingly we propose a centralized approximationalgorithm that provides performance guarantees with an ad-ditive ratio of

∑Ni=1 wi log(2) and runs in polynomial time.

In order to be able to optimize performance in real-time, wemake use of the real-time neighbour information provided

11

1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Operator

Perc

en

tage

Regular distribution

1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Operator

Perc

en

tage

Triangular distribution

% of the resources received

% contracted (vk)

1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Operator

Perc

en

tage

Irregular distribution

Fig. 6. Shares vs resources received for three different agreement cases.

by the LTE-Advanced interfaces (e.g. X1 interface) to de-sign a distributed algorithm with a theoretical additive ratioof∑Ni=1 wi log(e). Finally, all the theoretical results detailed

above has been proven by a set of performance evaluationsin a LTE-Advanced system-level simulator. Then, this work isplacing the first stone for efficient Active RAN Sharing. Eventhough there are a set of open issues for this problem suchas the design of an offline optimal algorithm, an analysis ofthe robustness of our solution against any possible operatorcheating decision or a real implementation of the solutionproposed.

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[14] “Study on Radio Access Network (RAN) Sharing Enhancements,” 3GPPTR 22.852, v12.0.0, Jun. 2013.

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[17] R. Kokku, R. Mahindra, H. Zhang, and S. Rangarajan, “Cellslice:Cellular wireless resource slicing for active ran sharing,” in Commu-nication Systems and Networks (COMSNETS), 2013 Fifth InternationalConference on, 2013, pp. 1–10.

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APPENDIX

Corollary 1. Let∑Np=1 wpxpj be the demand at BS j. Then,

those operators that have their users at BS with lower demand

12

receive more resources from the network.

Proof: The proof follows directly from (2). According tothis equation, the throughput received by a user is inverselyproportional the BS’ demand. Therefore, an operator that hasits users associated to BSs with lower demand will receivemore throughput for its users.

Corollary 2. If all the BS used by two operators have thesame level of demand, resources will be distributed amongthem proportionally to their share.

Proof: The proof follows directly from (2). According tothis equation, for uniformly loaded base stations the through-put received by a user given by Kvk/Nk, where K is aconstant. Therefore, the total resources received by an operatorthat has Nk users will be Kvk and hence proportional to vk.

Corollary 3. The amount of resources (resources blocks) thatan operator receives in a base station j only depends on thefraction of total users of operator k at base station j, fkj , andthe share vk.

Proof: This can be proven directly by using theorem 2.If we let nkj be the number of users from operator k in basestation j, since wi = vk/Nk and nkj = fkjNk the amount ofresources (resources blocks) bkj that an operator gets can berepresented as:

bkj =nkjwi

nkjwi +∑q∈j,q /∈k wq

=fkjvk

fkjvk +∑q∈j,q /∈k wq

where it can be directly inferred that the amount of resourceblocks that the operator gets only depends on the relativepercent of users in the bs fkj and the share vk.

13