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StatisticalMultiplexingandTrafficShapingGamesforNetworkSlicing

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Statistical Multiplexing and Traffic Shaping Gamesfor Network Slicing

Jiaxiao Zheng∗, Pablo Caballero∗, Gustavo de Veciana∗, Seung Jun Baek† and Albert Banchs‡∗The University of Texas at Austin, TX

†Korea University, Korea‡University Carlos III of Madrid & IMDEA Networks Institute, Spain

contact: gustavo@ece.utexas.edu

Abstract—Next generation wireless architectures are expectedto enable slices of shared wireless infrastructure which arecustomized to specific mobile operators/services. Given infras-tructure costs and the stochastic nature of mobile services’spatial loads, it is highly desirable to achieve efficient statisticalmultiplexing amongst network slices. We study a simple dynamicresource sharing policy which allocates a ‘share’ of a pool of(distributed) resources to each slice– Share Constrained Propor-tionally Fair (SCPF). We give a characterization of the achievableperformance gains over static slicing, showing higher gains whena slice’s spatial load is more ‘imbalanced’ than, and/or ‘orthog-onal’ to, the aggregate network load. Under SCPF, traditionalnetwork dimensioning translates to a coupled share dimensioningproblem, addressing the existence of a feasible share allocationgiven slices’ expected loads and performance requirements. Weprovide a solution to robust share dimensioning for SCPF-basednetwork slicing. Slices may wish to unilaterally manage theirusers’ performance via admission control which maximizes theircarried loads subject to performance requirements. We show thiscan be modeled as a “traffic shaping” game with an achievableNash equilibrium. Under high loads the equilibrium is explicitlycharacterized, as are the gains in the carried load under SCPFvs. static slicing. Detailed simulations of a wireless infrastructuresupporting multiple slices with heterogeneous mobile loads showthe fidelity of our models and range of validity of our high loadequilibrium analysis.

I. INTRODUCTION

Next generation wireless systems are expected to embraceSDN/NFV technologies towards realizing slices of sharedwireless infrastructure which are customized for specific mo-bile services e.g., mobile broadband, media, OTT serviceproviders, and machine-type communications. Customizationof network slices may include allocation of (virtualized) re-sources (communication/computation), per-slice policies, per-formance monitoring and management, security, accounting,etc. The ability to deploy service specific slices is viewed, notonly as means to meet the diverse and sometimes stringentdemands of emerging services, e.g., vehicular, augmented real-ity, but also an approach for infrastructure providers to reducecosts while developing revenue streams. Resource allocationin this context is more challenging than for traditional cloudcomputing. Indeed, rather than drawing on a centralized poolof resources, a network slice requires allocations across adistributed pool of resources, e.g., base stations. The challengeis thus to promote efficient statistical multiplexing amongstslices over pools of shared resources.

Network slices could also be used to enable the sharingof network resources amongst competing (possibly virtual)operators. Indeed, sharing of spectrum and infrastructure isviewed as one way of reducing capital/operational costs andis already being considered by standardization bodies, see[1], [2], which have specified architectural and technical re-quirements, but left the sharing criteria and algorithmic issuesopen. By aggregating their traffic onto shared resources, it isexpected that operators could realize substantial savings, thatmight justify/enable new shared investments in next generationtechnologies including 5G, mmWave, massive MIMO.

The focus of this paper will be resource sharing amongstslices supporting stochastic (mobile) loads. A natural approachto sharing is static slicing, whereby resources are staticallypartitioned and allocated to slices. This offers each slice aguaranteed allocation at each base station, and protectionfrom each other’s traffic, but, as we will see, poor efficiency.Instead, we consider, an alternative wherein each slice ispre-assigned a fixed share of the pool of resources, and re-distributes its share equally amongst its active customers. Inturn, each base station allocates resources to customers inproportion to their shares. We refer to this sharing modelas Share Constrained Proportionally Fair (SCPF) resourceallocation. By contrast with static slicing, SCPF is dynamic(since its resource allocations depend on the network state) butconstrained by the network slices’ pre-assigned shares (whichprovides a degree of protection amongst slices).

Related work. There is an enormous amount of related workon network resource sharing in the engineering, computerscience and economics communities. The standard frameworkused in the design and analysis of communication networksis utility maximization (see e.g., [23] and references there-in) which has led to the design of several transport andscheduling mechanisms and criteria, e.g., the often consideredproportional fair criterion. The SCPF mechanism, describedabove, should be viewed as a Fisher market where agents(slices), which are share (budget) constrained, bid on networkresources, see, e.g., [20] and for applications [5], [14]. Thechoice to re-distribute a slice’s shares (budget) equally a-mongst its users, can be viewed as a network mandated policy,but also emerges naturally as the social optimal, market andNash equilibrium when slices exhibit (price taking) strategicbehavior in optimizing their own utility, see [9].

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The novelty of our work lies in considering slice basedsharing, under stochastic loads and in particular studyingthe expected performance resulting from such SCPF-basedcoupling slices’ customer allocations. Other researchers whohave considered performance of stochastic networks, e.g., [7],[11] and others, have studied networks where customers areallocated resources (along routes) based on maximizing a sumof customers utilities. These works focus on network stabilityfor ‘elastic’ customers, e.g., file transfers. Subsequently [8],[22] extended this line of work, to the evaluation of meanfile delays, but only under balanced fair resource allocations(as a proxy for proportional fairness). Our focus here ison SCPF-based sharing amongst slices with stochastic loadsand on “inelastic” or “rate-adaptive” customers, e.g., video,voice, and more generally customers on properly provisionednetworks, whose activity on the network can be assumed tobe independent of their resource allocations.

Finally there is much ongoing work on developing thenetwork slicing concept, see e.g., [21], [28] and referencestherein, including development of approaches to network vir-tualization in RAN architectures, e.g, [10], [18], and SDN-based implementation, e.g., [6]. This paper focuses on devisinggood slice-based resource sharing criteria to be incorporatedinto such architectures.

Contributions of this paper. This paper makes several con-tributions centering on a simple and practical resource sharingmechanism: SCPF. First, we consider user performance (bittransmission delay) on slices supporting stochastic loads. Inparticular we develop expressions for (i) the mean perfor-mance seen by a typical user on a network slice; and (ii)the achievable performance gains versus static slicing. Weshow that when a slice’s load is more ‘imbalanced’ than,and/or ‘orthogonal’ to, the aggregate network load, one willsee higher performance gains. Our analysis provides an in-sightful picture of the “geometry” of statistical multiplexingfor SCPF-based network slicing. Second, under SCPF, tra-ditional network dimensioning translates to a coupled sharedimensioning problem, which addresses whether there existfeasible share allocations given slices’ expected loads andperformance requirements. We provide a solution to robustshare dimensioning for SCPF-based network slicing. Third,we consider decentralized per-slice performance managementunder SCPF sharing. In particular, we consider admissioncontrol aimed at maximizing a slice’s carried load subjectto a performance constraint. When slice unilaterally optimizetheir admission control policies, the coupling of their decisionscan be viewed as a “traffic shaping” game, which is shownto have a Nash equilibrium. For a high load regime weexplicitly characterize the equilibrium and the associated gainsin carried load for SCPF vs static slicing. Finally, we presentdetailed simulations for a shared distributed infrastructuresupporting slices with different mobility patterns which matchour analysis well, and further support our conclusions on gainsin both performance and carried loads of SCPF sharing.

II. SYSTEM MODEL

A. Network Slices, Resources and Mobile Service Traffic

We consider a collection of base stations (sectors) B sharedby a set of network slices V , with cardinalities B and Vrespectively. For example, V might denote slices supportingdifferent services or (virtual) mobile operators etc.

We envisage each slice v as supporting a mobile servicein the region served by the base stations B. Each slicesupports a stochastic load of users (devices/customers) with anassociated mobility/handoff policy. In particular, we assumethat exogenous arrivals to slice v at base station b follow aPoisson process with intensity γvb and γv = (γvb : b ∈ B)T .Each slice v customer at base station b has an independentsojourn time with mean µvb after which it is randomly routedto another base station or exits the system. As explained belowwe assume that such mobility patterns do not depend on theresources allocated to users. We let Qv = (qvi,j : i, j ∈ B)denote a slice-dependent routing matrix where qvi,j is theprobability a slice v customer moves from base station i toj and 1−∑j∈B q

vi,j is the probability it exits the system. This

model induces an overall traffic intensity for slice v acrossbase stations satisfying flow conservation equations: for allb ∈ B we have

κvb = γvb +∑

a∈Bκvaq

va,b,

where κvb is the traffic intensity of slice v on base station b.Accounting for users’ sojourn times, the mean offered load ofslice v on base station b is ρvb = κvbµ

vb , and ρv , (ρvb : b ∈ B)T

captures its overall system load. Letting µv = (µvb : b ∈ B)T ,the flow conservation equations can be rewritten in matrixform as:

ρv = diag(µv)(I − (Qv)T )−1γv. (1)

If Qv is irreducible, I − (Qv)T is irreducibly diagonallydominant thus always invertible. Otherwise, we can alwaysfind a permutation matrix of B, say P to make:

P T (I − (Qv)T )P =

A1 B1,2 . . .

. . ....AK

,

where K is the number of irreducible classes. Moreover, atleast one base station of each irreducible class has a nonzeroexiting probability, thus AK must be invertible. Then theinvertibility of I − (Qv)T follows.

This model corresponds to a multi-class network ofM/GI/∞ queues (base stations), where each slice corre-sponds to a class of customers, see, e.g., [17]. Such networksare known to have a product-form stationary distribution, i.e.,the numbers of customers on slice v at base station b denotedby Nv

b are mutually independent and Nvb ∼ Poisson(ρvb ). Since

the sum of independent Poisson random variables is againPoisson, the total number of customers on slice v is such thatNv =

∑b∈BN

vb ∼ Poisson(ρv) where ρv =

∑b∈B ρ

vb .

2

Our network model for the numbers of customers andmobility across base stations, assumes customer sojourn-s/activity/mobility are independent of the network state and ofthe resources a customer is allocated. This is reasonable forproperly engineered slices where the performance a customersees does not impact its activity, e.g., inelastic or rate adaptiveapplications seeing acceptable performance. This covers awide range of applications including voice, video streaming,IoT monitoring, real-time control, and even web browsingsessions experiencing good performance. This model howeveris not appropriate for customers sensitive to file downloaddelays, e.g., who might leave the system earlier if allocatedmore resources.

There are several natural generalizations to this modelincluding class-based routing and user sessions (e.g. webbrowsing) which are not always active at the base stationsthey visit, see e.g., [17].

B. Network Slice Resource Sharing

In the sequel we consider a setting where the resourcesallocated to a slice’s customers depend on the overall networkstate, i.e., number of customers each slice has on each basestation, corresponding to the stochastic process described inSection II-A. Let us consider a snapshot of the system’s stateand let Uvb ,Ub,Uv and U denote sets of active customers onslice v at base station b, at base station b, on slice v andon the overall network respectively. Thus, the cardinalities ofthese sets correspond to a realization of the system ‘state’, i.e.,|Uvb | = nvb and |Uv| = nv , where in a stationary regime nv

and nvb are realizations of Poisson random variables Nv andNvb , respectively.Each base station b is modeled as a finite resource shared

by its associated users Ub. A customer u ∈ Ub can be allocateda fraction fu ∈ [0, 1] of that resource, e.g., of resource blocksin a given LTE frame, or allocated the resource for a fractionof time, where

∑u∈Ub fu = 1. We shall neglect quantization

effects. The transmission rate to customer u, denoted by ru,is then given by ru = fucu where cu denotes the currentpeak rate for that user. To model customer heterogeneity acrossslices/base stations we shall assume cu for a typical customeron slice v at base station b is an independent realization ofa random variable with the same distribution as Cvb . It maydepend on the slice, since slices may support different typesof customer devices (e.g., car connectivity vs mobile phone)and depend on the base station, since typical slice v users mayhave different spatial distributions with respect to base stationb or see different levels of interference.

Below we consider two resource allocation schemes. Forboth we assume each slices is allocated a ‘share’ of the net-work resources sv, v ∈ V such that sv > 0 and

∑v∈V sv = 1.

Definition 1. Static Slicing (SS): Under SS, slice v is allo-cated a fixed fraction sv of each base station b’s resources,and each customer u ∈ Uvb gets an equal share, i.e., 1/nvb ,

of the slice v’s resources at base station b. Thus the userstransmission rate rSSu is given by

rSSu =svnvbcu.

Definition 2. Share Constrained Proportionally Fair(SCPF): Under SCPF each slice re-distributes its share ofthe overall network resources equally amongst its activecustomers, which thus get a sub-share (weight) wu = sv

nvfor

u ∈ Uv,∀v ∈ V. In turn, each base station allocates resourcesto customers in proportion to their weights. So a user u ∈ Uvbgets a transmission rate rSCPFu given by

rSCPFu =wu∑

u′∈Ub wu′cu =

svnv

∑v′∈V

nv′b sv′nv′

cu. (2)

Thus under SCPF the overall fraction of resources slicev is allocated at a base station b is proportional to nv

b

nv sv ,i.e., its share and its relative number of users at the basestation. This provides a degree of elasticity to variations in theslice’s spatial loads. However, if a slice has a large numberof customers, its customers’ weights are proportionally de-creased, which protects other slices. In addition to being quitesimple to implement, as mentioned in Section I SCPF resourceallocations are socially optimal, and correspond to market andNash equilibria for certain types of budget-constrained FisherMarkets.

Such an allocation (with equal user weight) is simple toimplement and can be shown to be socially optimal.

Theorem 1. When slices make equal weight assignmentsto their customers SCPF resource allocations are sociallyoptimal in that they maximize the following surrogate networkutility:

maxw0

∑

v∈V

sv|Uv|

∑

u∈Uv

log(ru)

s.t. sv =∑

u∈Uv

wu ∀v ∈ V

ru =wu∑

u′∈Ub wu′cu ∀u ∈ U . (3)

Alternatively, the above optimization (and thus resourceallocation) can be recognized as the Eisenberg Gale programcharacterizing the equilibrium of a Fisher market and thus theallocations reached under a variety of response dynamics, seee.g., [27].

III. PERFORMANCE EVALUATION

In this section we study the expected performance seen bya slice’s typical customer. Given our focus on inelastic/rateadaptive traffic and tractability, we choose our customer per-formance metric as the reciprocal transmission rate, referredto as the Bit Transmission Delay (BTD), see e.g., [25]. Thiscorresponds to the time taken to transmit a ‘bit’, so lowerBTDs indicate higher rates and thus better performance. Shortpacket transmission delays are roughly proportional to theBTD. Alternatively the negative of the BTD can be viewed as

3

a concave utility function of the rate, which in the literaturewas referred as the potential delay utility. Given the stochasticloads on the network, we shall evaluate the average BTD seenby a typical (i.e., randomly selected) customer on a slice,i.e., averaged over the stationary distribution of the networkstate and transmission capacity seen by typical users, e.g., Cvb ,at each base station. Such averages naturally place a higherweight on congested base stations, where a slice may havemore users, best reflecting the overall performance customerswill see.

A. Analysis of BTD PerformanceConsider a typical customer on slice v and let Ev denote

the expectation of the system state as seen by such a customer,i.e., the Palm distribution [4]. For SCPF, we let Rv be arandom variable denoting the rate of a typical customer onslice v, and Rvb that of such customer of slice v at base stationb. Similarly, let Rv,SS and Rv,SSb denote these quantitiesunder static slicing. Thus, under SCPF the average BTD fora typical slice v customer is given by Ev[ 1

Rv ]. The nextresult characterizes the mean BTD under SCPF and SS underour traffic model. We introduce some further notation: thenormalized load distribution of slice v is ρv = (ρvb : b ∈ B)T

where ρvb , ρvbρv ; the overall share weighted normalized load

distribution is g = (gb : b ∈ B)T where gb ,∑v∈V svρ

vb ;

and g′ = (g′b : b ∈ B)T where g′b ,∑v∈V sv(1 − e−ρ

v

)ρvb ;and the mean reciprocal resource capacity for slice v isδv = (δvb : b ∈ B)T where δvb , Ev[ 1

Cvb

].

Theorem 2. For network slicing based on SCPF, the meanBTD for a typical customer on slice v is given by

Ev[

1

Rv

]=

∑

b∈Bρvbδ

vb (1− ρvb +

(ρv + 1)

svgb (4)

− (ρv + 1)

sv

∑

v′ 6=vsv′e

−ρv′ ρv′b ).

Note that when ∀v ∈ V, ρv → ∞, the mean BTD hasfollowing asymptotic form:

Ev[

1

Rv

]=∑

b∈Bρvbδ

vb

(1− ρvb +

(ρv + 1)

svgb

). (5)

For network slicing based on SS, the mean BTD for a typicalcustomer on slice v is given by

Ev[

1

Rv,SS

]=∑

b∈Bρvbδ

vb

(ρvb + 1

sv

). (6)

Proof. Recall that Poisson arrivals see time averages, i.e., seethe remaining users in the product-form stationary distribution,given in Section II-A. Thus the distribution as seen by a typicaluser on slice v at base station b is the same as the product-form distribution plus an additional customer on slice v atbase station b. Using this fact and SCPF resource allocationsas given by Eq. (2), the BTD of a typical slice v user at basestation b can be expressed as follows:

Ev[

1

Rvb

]

= Ev[

1

Cvb

]E

sv

Nvb +1

Nv+1 +∑v′ 6=v

sv′Nv′b

Nv′ 1Nv′>0sv

(Nv+1)

= δvbE

(Nv

b + 1) +Nv + 1

sv

∑

v′ 6=v

sv′Nv′b

Nv′1Nv′>0

= δvb (ρvb + 1 +ρv + 1

sv

∑

v′ 6=vsv′(1− eρ

v′)ρv′b )

= δvb (1− ρvb +(ρv + 1)

svgb −

(ρv + 1)

sv

∑

v′ 6=vsv′e

−ρv′ ρv′b ),

where the last equality follows by noticing that (i) Nv is

independent of Nv′b and Nv′ and (ii) E[

Nv′b

Nv′ 1Nv′>0] =

P (Nv′ > 0)E[Nv′

b

Nv′ |Nv′ > 0] =ρv′

b

ρv′P (Nv′ > 0). The latter

result is given by the following lemma:

Lemma 1. If N1, N2, . . . , Nn are independent Poisson ran-dom variables, such that Ni ∼ Poisson(ρi), i = 1, 2, . . . , n.Then for all i we have that:

E

Ni∑n

j=1Nj|n∑

j=1

Nj > 0

=

ρi∑nj=1 ρj

Proof. First we note that if X1, X2, . . . , XL are exchangeablerandom variables, e.g. i.i.d. Poisson random variables, wehave:

1 = E

[∑Li=1Xi∑Li=1Xi

|L∑

i=1

Xi > 0

]

= LE

[X1∑Li=1Xi

|L∑

i=1

Xi > 0

]

⇒ E

[X1∑Li=1Xi

|L∑

i=1

Xi > 0

]=

1

L,

where the second equality follows by symmetry.Now suppose for some ε small enough, for all i, we have

that mi = ρiε is integer valued. Let Xi,j , i = 1, . . . , n, j =

1, . . . ,mi be i.i.d. Poisson random variables with parameter ε.Since Poisson random variables are infinitely divisible we havethat Ni ∼

∑mi

j=1Xi,j . It follows from the previous symmetryargument that:

E

Ni∑n

j=1Nj|n∑

j=1

Nj > 0

= E

∑mi

j=1Xi,j∑ni=1

∑mi

j=1Xi,j|n∑

j=1

Nj > 0

= miE

Xi,1∑n

i=1

∑mi

j=1Xi,j|n∑

j=1

Nj > 0

=mj∑nj=1mj

=ρi∑nj=1 ρj

.

4

The above equality also follows when ρ = (ρi, i = 1, . . . , n)takes rational value. Since the conditional expectation will bea continuous function of the parameter vector ρ, the equalityfollows more generally for the case where ρ is a real valuedvector.

Under static slicing we have that

Ev[

1

Rv,SSb

]= Ev

[1

Cvb

]E[

(Nvb + 1)

sv

]= δvb

ρvb + 1

sv.

The theorem follows by taking an weighted average acrossbase stations – weighted by the fraction of customers at eachbase station, i.e., ρvb .

B. Analysis of Gain

Using the results in Theorem 2 one can evaluate the gainsin the mean BTD for a typical slice v user under SCPF versusSS, i.e.,

Gv =Ev[

1Rv,SS

]

Ev[

1Rv

] .

In general, one would expect Gv ≥ 1 since under SCPF typicalusers should see higher allocated rates and thus lower BTDs.One can verify that is the case when slices have uniform loadsacross base stations but the general case is more subtle. Forsimplicity from here on in this paper we focus on the casewhere the following additional assumption is in effect:

Assumption 1. Base stations are said to be homogeneous forslice v if for all b ∈ B: Ev

[1Cv

b

]= δv.

Note that Assumption 1 only requires the average reciprocalcapacity a given slices’ customer sees across base stations ishomogenous.

Corollary 1. Under Assumption 1, the BTD gain of SCPFover SS for slice v is given by

Gv =ρv‖ρv‖22 + 1

sv(1− (1− e−ρv − ρve−ρv )‖ρv‖22) + (ρv + 1)g′,T ρv

For fixed relative loads ρv and g, the gain for low overallload (ρv → 0) is positive and given by:

GLv =1

sv + g′,T ρv≥ 1

where g′ is the effective shared weighted relative load ofother slices on network. Furthermore, Gv is a nonincreasingfunction of ρv , and for high loads (∀v, ρv →∞) is given by:

GHv =‖ρv‖2‖g‖2

× 1

cos(θ(g, ρv)),

where θ(g, ρv) denotes the angle between the slice’s relativeloads and the overall share weighted relative loads on thenetwork.

Please see appendix for detailed proof. The result indicatesthat for slice v with fixed relative loads ρv , the gains decreasein the overall load ρv , thus if GHv > 1 SCPF always provides

a gain. A sufficient condition for gains under high loads isthat ‖g‖2 ≤ ‖ρv‖2. Since ‖g‖1 = ‖ρv‖1 = 1, this followswhen the overall share weighted relative load on the network ismore balanced than that of slice v. One would typically expectaggregated traffic to be more balanced than that of individualslices. This condition is fairly weak, i.e., it does not depend onwhere the loads are placed, but on how balanced they are. Thecorollary also suggests gains are higher when cos(θ(g, ρv))is small. In other words, a slice with imbalanced normalizedloads whose relative load distribution is ‘orthogonal’ to theshared weighted aggregate traffic, i.e., θ(g, ρv) ≈ 0, will tendto see higher gains. The simulations in Section V furtherexplore these observations.

IV. PERFORMANCE MANAGEMENT

In practice each slice v ∈ V may wish to provide serviceguarantees to its customers, i.e., ensure that the mean BTDdoes not exceed a performance target dv . Below we investigatehow to dimension network shares to support slice loads subjectto such mean BTD requirements. In the following part weassume the following assumption is in effect.

Assumption 2. All slice v ∈ V have heavy load. That is,

∀v ∈ V, ρv 1.

A. Share Dimensioning under SCPF

Consider a network supporting the traffic loads of a singleslice, say v, so sv = 1 and g = ρv and let dv , dv/δvdenote slice v’s normalized BTD constraint. Note that δv isthe minimum BTD achievable when a user gets all the basestation resources, so a target requirement satisfies dv > δv andso dv > 1. For slice v to meet a mean BTD constraint dv , itfollows from Eq. (5) that :

ρv ≤ l(dv, ρv) ,dv − 1

‖ρv‖22.

We can interpret l(dv, ρv) as the maximal admissible carriedload ρv given a fixed relative load distribution ρv and require-ment dv . As might be expected, if the relative load distributionρv is more balanced, i.e., ‖ρv‖22 is smaller, or if the BTDconstraint is relaxed, i.e., dv is higher, the slice can carry ahigher overall load ρv .

Next, let us consider SCPF based sharing amongst a setof slices V each with its own BTD requirements. It followsfrom Eq. (5) that to meet such requirements on each slice thefollowing should hold: for all v ∈ V

sv ≥1 + ρv

l(dv, ρv)− ρv∑

u6=vsu‖ρu‖2‖ρv‖2

cos(θ(ρu, ρv)). (7)

This can be written as:∑

v∈Vsvh

v 0, (8)

5

where we refer to hv = (hvu : u ∈ V)T as v’s share couplingvector, given by

hvu =

1 v = u

− 1+ρu

l(du,ρu)−ρu‖ρv‖2‖ρu‖2 cos(θ(ρu, ρv)) v 6= u.

We can interpret hvv = 1 as the benefit to slice v of allocatingunit share to it. When v 6= u, hvu depends on two factors.The first 1+ρu

l(du,ρu)−ρu captures the sensitivity of slice u to the‘share weighted congestion’ from other slices. If ρu is closeto its limit l(du, ρu), its sensitivity is naturally very high. Thesecond term, ‖ρ

v‖2‖ρu‖2 cos(θ(ρu, ρv)) captures the impact of slice

v’s load distribution on slice u. Note that if two slices loaddistributions are orthogonal, they do not affect each other.

The following result summarizes the above analysis.

Theorem 3. There exists a share allocation such that sliceloads and BTD constraints ((ρv, ρv, dv) : v ∈ V) areadmissible under SCPF sharing if and only if there exists ans = (sv : v ∈ V)T such that ‖s‖1 = 1, s 0 and

∑

v∈Vsvh

v 0.

Admissibility can then be verified by solving the followingmaxmin problem:

maxs0 min

i

∑

v∈Vsvh

vi : ‖s‖1 = 1 . (9)

If the optimal objective function is positive, the traffic patternis admissible. Moreover, if there are multiple feasible shareallocations, then the optimizer is a ‘robust’ choice in that itmaximizes the minimum share given to any slice, giving slicesmargins to tolerate perturbations in the slice loads satisfyingEq. (8).

If a set of network slice loads and BTD constraints are notfeasible, admission control will need to be applied. We discussthis in the next section.

B. Admission Control and Traffic Shaping Games

A natural approach to managing performance in overloadedsystems is to perform admission control. In the context ofslices supporting mobile services where spatial loads may varysubstantially, this may be unavoidable. Below we consider ad-mission control policies that adapt to changes in load. Specif-ically an admission control policy for slice v is parameterizedby av = (avb : b ∈ B)T ∈ [0, 1]B where avb is the probability anew customer at base station b is admitted. Such decisions areassumed to be made independently thus admitted customersfor slice v at base station b still follow a Poisson Process withrate γvb a

vb . Based on the flow conservation equation Eq. (1) one

can obtain the carried load ρv induced by admission controlpolicy av via

ρv = (Mv)−1av = diag(µv)(I − (Qv)T )−1diag(γv)av

where Mv , diag(γv)−1(I−(Qv)T )diag(µv)−1 is invertiblebecause I − (Qv)T is irreducibly diagonally dominant.1 By

1If γv is not strictly positive one can reduce the dimensionality.

contrast with Section II-A, note that ρv now represents theload after admission control, which may have a reduced overallload and possibly changed relative loads across base stations– i.e., shape the traffic on the slice. We also let g be theoverall share weighted relative loads after admission control,see Section III-A. Note that we have assumed only exogenousarrivals can be blocked, thus once a customer is admitted itwill not be dropped –the intent is to manage performance tomaintain service continuity.

Below we consider a setting where slices unilaterally opti-mize their admission control policies in response to networkcongestion, rather than a single joint global optimization. Theintent is to allow slices (which may correspond to competingvirtual operator/services) to optimize their own performance,and/or enable decentralization in settings with SCPF basedsharing.

Suppose each slice v optimizes its admission control policyso as to maximize its overall carried load ρv , i.e., the averagenumber of active users on the network, subject to a mean BTDconstraint dv . Under Assumption 1 the optimal policy for slicev is the solution to the following optimization problem:

maxρv,ρv

ρv (10)

s.t. av = ρvMvρv, av ∈ [0, 1]B , 1T ρv = 1 (11)(ρv + 1)

svgT ρv − ‖ρv‖22 ≤ dv − 1. (12)

Note that Eq. (11) establishes a one-to-one mapping be-tween (ρv, ρv) and av . We will use ρv and ρv to parameterizeadmission control decisions for slice v. The BTD constraint inEq. (12) follows from Eq. (5). Also note that this admissioncontrol policy depends on both the overall share weightedloads on the network g, the slice’s load and its customermobility patterns (i.e., Mv). Unfortunately this problem isnot convex due to the BTD constraint Eq. (12); however, forhigh overall per slice loads it is easily shown to be convex,hence for simplicity we will make the following additionalassumption.

Assumption 3. The network is said to see high overall sliceloads, if for all v ∈ V we have ρv 1.

Under Assumption 3 we have that 1 + ρv ≈ ρv and the lefthand side of Eq. (12) becomes:

(ρv + 1)

svgT ρv − ‖ρv‖22 ≈

ρv

svgT ρv = (svxv)

−1gT ρv (13)

where we have defined xv , (ρv)−1. Further defining ρ−v ,(ρv

′: v′ ∈ V\v), Eq. (12) can be replaced by:

fv(ρv; ρ−v) , gT ρv ≤ sv(dv − 1)xv. (14)

Thus, defining yv , (ρv, xv), which is equivalent to (ρv, ρv),together with y−v , (yv

′: v′ ∈ V\v), under Assumption 3

each slice can unilaterally optimize its admission controlpolicy by solving the following problem:

6

Admission control for slice v under SCPF(ACv): Givenother slices’ admission decisions y−v , slice v determines itsadmission control policy yv = (ρv, xv) by solving

minyv xv | yv ∈ Y v(y−v) (15)

where Y v(y−v) denotes slice v’s feasible policies and is givenby

Y v(y−v) , yv | 1T ρv = 1, 0 Mvρv xv1,fv(ρ

v; ρ−v) ≤ sv(dv − 1)xv . (16)

Note that ACv is coupled to the decisions of other slicesthrough the constraint set Yv(y−v). Thus, one cannot indepen-dently solve each slice’s admission control problem to obtainan efficient solution. Furthermore, devising a global optimiza-tion for all slices brings both complexity and nonconvexityfrom the BTD constraints. A natural approach requiring mini-mal communication and cooperation overhead is to consider agame setup where network slices are players, each seeking tomaximize their carried load (and the corresponding revenue)subject to BTD constraints.

We formally define the traffic shaping game for a set ofnetwork slices V as follows. We let y , (yv : v ∈ V) denotethe simultaneous strategies of all slices (given by the respectiveadmission control policies). As in ACv , each slice v picks afeasible strategy, i.e., yv ∈ Y v(y−v) to maximize its objectivefunction θv(yv,y−v) , xv . Note in the sequel we will modifyθv(·, ·) to ensure the games convergence. A Nash equilibriumis a simultaneous strategy y∗ such that no slice can unilaterallyimprove its carried load, i.e., for all v ∈ V

θv(yv,∗,y−v,∗) ≤ θv(yv,y−v,∗) ∀yv ∈ Y v(y−v,∗).

The following result follows from Theorem 3.1 in [12].

Theorem 4. The traffic shaping game defined above has aNash equilibrium.

C. Algorithm

Given the existence of a Nash equilibrium, the next questionis whether one can devise an iterative mechanism to find it. Inour setting this is not a simple matter. The difficulty arises fromthe fact that slices’ strategy spaces depend on other’s choices,and the overall feasible joint strategies y lie in a nonconvexset, so oscillation is possible. In the literature such settingsare specifically referred to as Generalized Nash EquilibriumProblem (GNEP), see e.g., [13] and [24]. Below we proposean algorithm involving slices and a central entity which isguaranteed to converge to the equilibrium.

We summarize the main ideas as follows. To decoupledependencies among strategy spaces, we shall move slice v’sBTD constraint into its objective function as a penalty termwith an associated multiplier λv . Let λ = (λ1, λ2, . . . , λV ).By adjusting the value of λ according to y at each iteration,one can determine a setting such that, at the induced NashEquilibrium, all slices meet their BTD constraints, and theequilibrium is identical to that of the traffic shaping game. Inaddition, in order to prevent overshooting, at each iteration

each slice’s objective function is regularized by the distanceto the previous reciprocal carried load xv.

Specifically, the admission control strategy of slice v inresponse to other slices is now given as the solution to thefollowing optimization problem:

Lvε (y;λv) = argmin(yv)′∈Y v

θv((yv)′,y−v;λv)+

ε

2(xv−x′v)2, (17)

where we define a BTD penalty function for slice v as

hv(y) , fv(ρv; ρ−v)− sv(dv − 1)xv.

and the objective function for slice v is now (differentfrom what is previously defined): θv(yv,y−v;λv) , exv +λv[hv(y)]+, with (x)+ , max(0, x). The last term in Eq. (17)serves as a regularization term. The strategy space is nowY v , yv|1T ρv = 1, 0 Mvρv xv1 and xv issubstituted by exv to ensure strong convexity, which is requiredfor convergence (note that due to the monotonicity, exv andxv should result in the same optimizer).

We propose to use the inexact line search update introducedin [24]. In order to make sure the iteration is proceedingtowards the equilibrium, we use

Ωε(y;λ) ,∑

v∈Vθv(y

v,y−v;λv)− θv(Lvε (y;λ),y−v;λv)

− ε

2(xv − x′v)2 ≥ 0

as a metric, observing that the equilibrium is given by y∗ if andonly if Ωε(y

∗;λ) = 0. Therefore we seek to decrease Ωε(y;λ)by a sufficient amount at each iteration. The task executed byeach slice v is given in Algorithm 1, while the central entity,which is responsible for collecting and delivering informationand updating λ, executes Algorithm 2. This then follows thealgorithm proposed in [13].

Algorithm 1 Algorithm of Slice v

1: Set k ← 0 and collect ε from central entity.2: Receive λv(k) and y(k) from central entity.3: Compute Lvε (y(k);λ) and transmit it back to the central

entity. Set k ← k + 1. Go to step 2

Theorem 5. Let y(k) be the sequence of admission controldecisions generated by Algorithm 1 and Algorithm 2, thenevery limit point of this sequence is a Nash equilibrium of thetraffic shaping game induced by ACv .

Proof. First we need to verify the Assumption 5.1 in [24] toguarantee that for a given λ, step 4 in Algorithm 2 convergesto a Nash equilibrium. The non-constant part of Ψε(y,y

′;λ)when y′ is fixed is:

∑v e

xv +λv[hv(y)]+− ε2‖xv−x′v‖2. If ε

is small enough, the concavity of the last term will be canceledout by exv . Then the non-constant part is always convexin y. Hence, the Assumption 5.1 holds true together withthe propositions 2.1(a) - (d) in [24]. Therefore, the proposedalgorithm generates Nash equilibrium of the game.

7

Algorithm 2 Penalized Update in Central Entity

1: Choose a starting point y(0), λ(0) 0, ηv ∈ (0, 1),for v ∈ V , β, σ ∈ (0, 1), ε > 0 but small enough (seefollowing theorem for convergence) and set k ← 0.

2: If a termination criterion is met then STOP. Otherwise,communicate y(k) together with λv(k) to all slices.

3: All slices compute Lvε (y(k);λ) and feedback to centralentity.

4: Compute t(k) = maxβl|l = 0, 1, 2, . . . such that if weassume ξ(k) = (Lvε (y(k);λ(k)) : v ∈ V)− y(k):

Ωε(y(k)+t(k)ξ(k)) ≤ Ωε(y(k))−σ(t(k))2‖ξ(k)‖. (18)

Then set y(k + 1) = y(k) + t(k)ξ(k).5: Set I(k) = v|hv(y(k)) > 0. For every v ∈ I(k), if

exv(k) > ηv(λv‖∇yvhvy(k))‖), (19)

then λv(k+1)← 2λv(k). Set k ← k+1. Broadcast y(k)and λ(k) to slices and go to step 2.

One can easily verify that the EMFCQ condition given byDefinition 2.7 in [13] is satisfied. Thus for all v the λv getsupdated a finite number of times. According to Theorem 2.5in [13], the claim is true.

D. Characterization of Traffic Shaping Equilibrium

Next we study the characteristics of the resulting trafficshaping Nash equilibrium. To make this tractable we considernetworks which are saturated and subsequently (Section V)provide simulations to evaluate other settings.

Assumption 4. (Saturated Regime) Suppose the system is suchthat for each network slice, the optimal admission control forboth SCPF and SS2 in response to other slices’ load is suchthat for all v ∈ V , av ≺ 1.

Assumption 4 depends on many factors including the BTDconstraint, the mobility pattern and network slices’ shares, butit is generally true when the exogenous traffic of all slices atall base stations γvb is high. When this is the case we have thefollowing:

Theorem 6. Under Assumptions 1, 3 and 4, the relative loaddistributions at the Nash equilibrium of the traffic shapinggame ρ∗ , (ρv,∗ : v ∈ V) are the unique solution to:

min(ρv∈Γv :v∈V)

‖∑

v

svρv‖22 +

∑

v

s2v‖ρv‖22, (20)

where Γv , ρv | 1T ρv = 1,Mvρv 0 , and theassociated carried load for slice v is ρ∗v = sv(dv−1)

g∗,T ρv,∗ , whereg∗ corresponds to the overall share weighted relative loadsdistributions at the equilibrium.

See appendix for detailed proof.The first term in the objective function in Eq. (20) rewards

balancing the overall share weighted relative loads on network.

2Admission control under SS is defined in the sequel.

The second term rewards a slice for balancing its own relativeloads. The Nash equilibrium in the saturated regime is thus acompromise between these two objectives while constrainedby the network slices mobility patterns and feasible admissioncontrol policies.Admission control for slice v under SS (ACSSv): UnderSS slice v can determine its optimal admission control yv bysolving:

maxρv,ρv

ρv

s.t. av = ρvMvρv, av ∈ [0, 1]B

1T ρv = 1 and ρv‖ρv‖22 ≤ (svdv − 1).

Note slice admission control decisions are clearly decoupledunder SS, but paralleling Theorem 6 we have following result.

Theorem 7. Under Assumptions 1 and 4, the optimal admis-sion control policy under SS are decoupled. The optimal choicefor slice v ρv,SS,∗ is the unique solution to:

minρv∈Γv

‖ρv‖22, (21)

and the associated carried load is given by ρSS,∗v = sv dv−1‖ρv,SS,∗‖22

.

By comparing Eq. (20) and Eq. (21), one can see that underSS, slices simply seek to balance their own relative loads onthe network. By taking the ratio between ρ∗v and ρSS,∗v , onecan show that under Assumptions 1, 3 and 4 the gain in carriedload for slice v is given by

Gloadv , ρ∗v

ρSS,∗v

=‖ρv,SS,∗‖22g∗,T ρv,∗

× sv(dv − 1)

svdv − 1. (22)

The first factor captures a traffic shaping dependent gain forslice v. The second factor is a result of statistical multiplexinggains. A simple special case is highlighted in the followingcorollary.

Corollary 2. Under Assumptions 1, 3 and 4, if user mobilitypatterns are such that 1

B1 ∈ Γv for all v ∈ V , the gain inthe total carried load under the SCPF traffic shaping Nashequilibrium vs. optimal admission control for SS is given by:

Gloadv =

svdv − svsvdv − 1

≥ 1, ∀v ∈ V. (23)

See appendix for detailed proof.Note that in order for a BTD constraint to be feasible under

SS, one must require svdv > 1. It can be seen that the gainexhibited in Corollary 2 can be very high when sv ↓ 1/dv .Furthermore if sv ↑ 1 we have that Gload

v ↓ 1, i.e., no actualgain. This result implies that slices with small shares or tightBTD constraints will benefit most from sharing, coincidingwith our observations in Corollary 1.

V. PERFORMANCE EVALUATION

We simulated a wireless network shared by multiple slicessupporting mobile customers following the IMT-Advancedevaluation guidelines [16]. The system consists of 19 base

8

Scenario: Slices Spatial loads ‖ρv‖2 ‖g‖2 θ(g, ρv) GHv

1 Homogeneous uniform. 0.27 0.27 7.09 1.0 %

2 Homogeneous non-uniform 0.32 0.32 6.18 1.0 %

3 Heterogeneous orthogonal 0.36 0.26 41.78 83.3 %

4 Mixed Slices 1&2 non-uniform 0.36 0.23 25.52 70.4 %

3&4 uniform 0.19 0.23 48.00 23.7 %

TABLE I: Measured normalized slice and network trafficnorms and angles for highest load case of each scenario.

stations in a hexagonal cell layout with an intersite distanceof 200 meters and 3 sector antennas, mimicking a dense ‘smallcell’ deployment. Thus, in this system, B corresponds to 57sectors. Users associate to the sector offering the strongestSINR , where the downlink SINR is modeled as in [26]:

SINRub =PbGub∑

k∈B,k 6=b PkGuk + σ2,

where, following [16], the noise σ2 = −104dB, the transmitpower Pb = 41dB and the channel gain between user u and BSsector b, denoted by Gub, accounts for path loss, shadowing,fast fading and antenna gain. Letting du,b denote the currentdistance in meters from the user u to sector b, the path lossis defined as 36.7 log10(dub) + 22.7 + 26 log10(fc)dB, for acarrier frequency fc = 2.5GHz. The antenna gain is set to 17dBi, shadowing is updated every second and modeled by a log-normal distribution with standard deviation of 8dB, as in [26];and fast fading follows a Rayleigh distribution depending onthe mobile’s speed and the angle of incidence. The downlinkrate cu currently achievable to user u is based on discrete setmodulation and coding schemes (MCS) and associated SINRthresholds given in [3]. This MCS value is selected based onthe averaged SINRub, where channel fast fading is averagedover a second.

We model slices’ with different spatial loads by modelingdifferent customer mobility patterns. Roughly uniform spatialloads are obtained by simulating the Random Waypoint model[15], while non-uniform loads obtained by simulating theSLAW model [19]. These mobility models would not induceMarkovian motion amongst base stations assumed in our anal-ysis, yet the analytical results are robust to these assumptions.

A. Statistical Multiplexing and BTD Gains

We evaluated the BTD gains of SCPF vs SS for foursimulation scenarios, each including 4 slices, each with equalshares but different spatial load patterns. For each scenario,we provide results for simulated BTD gains, and results fromour theoretical analysis (Corollary 1) based on the empiri-cally obtained spatial traffic loads. More detailed informationregarding simulated scenarios and resulting empirical spatialtraffic loads for high load regime are displayed in Table I anda snapshot of locations for the 4 slices’ users in a networkwith a load of 4 users per sector is displayed in Figure 1.

The results given in Figure 2 show the BTD gains for eachscenario as the overall network load increases. In Scenario 3,

Fig. 1: Snapshot of users positions per slice and scenarioexhibiting the different characteristics of traffic spatial loads.Left to right: Scenarios 1 to 4.

Offered traffic (|U|/|B|)

4 8 12 16 24 30

Gainin

BTD

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2Simulation resultsTheoretical results

Scenario 4: Slices 1&2

Scenario 3

Scenario 2

Scenario 4: Slices 3&4

Scenario 1

Fig. 2: BTD gain for our 4 different scenarios simulation.

the aggregate network traffic is “smoother” than the individualslice’s traffic, and the gains are indeed higher. This is alsothe case for Slice 1 and 2 in Scenario 4, since these slicesloads are more “imbalanced” than the other two slices, theyexperience higher gains. In Scenario 2, where slices non-homogenous spatial loads are ‘aligned’, aggregation does notlead to smoothing and the gains are least.

As can be seen in Figure 2 the simulated and theoreticalgains (dashed lines) of Corollary 1 are an excellent match. Thetheoretical model has been calibrated to the mean reciprocalcapacities seen by slice customers (i.e., γvb ’s) and the measuredinduced loads resulting from the slice mobility patterns.

B. Traffic Shaping Equilibrium and Carried Load Gains

In order to study the equilibria reached by the traffic shapinggame, we measured the underlying user mobility patterns inSection V-A, and modeled it via a random routing matrix.We further assumed uniform intensity of arrivals rates atall base stations and uniform exit probabilities of 0.1. Themean holding time at each base station was again calibratedwith the simulations in Section V-A. We considered a trafficshaping game for a network shared by 3 slices, where Slice1 has uniform spatial loads and Slice 2 and 3 have differ-ent non-uniform spatial loads. All slices have equal sharesand their capacity normalized BTD requirements are set tod1 = 10, d2 = 12, d3 = 15 respectively. The Nash equilibriumwas solved via the algorithm included in Section IV-C. The

9

Arrival rate0 0.1 0.3 0.5 0.7 1 1.2 1.5

Gai

n in

car

ried

load

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0 0.3 0.5 0.7 1 1.2 1.5

0.12

0.14

0.16

0.18

0.2

0.22More

balanced

Slice 1

Slice 2

Slice 3

Aggregate

Theoretical

Slices 2 & 3

Aggregate

Slice 1

Fig. 3: Gain in carried load for various arrival rates.Subfigure: Balancing in relative load.

convergence is reached within 3 rounds of iterations under theparameters ηv = β = 0.5,∀v ∈ V , σ = 0.1, ε = 0.01. Theresults shown in Figure 3 exhibit dashed lines correspondingto the theoretical carried load gains in the saturated regime.As can be seen, these coincide with the Nash equilibria ofthe simulated traffic shaping games for high arrival rates. Forlower arrival rates the gains can be much higher, e.g., almost1.6x, for slices with non-uniform mobility patterns. This was tobe expected since for lower loads we expect higher statisticalmultiplexing gains from sharing, and thus relatively highercarried loads to be admitted. For very low loads, as expected,there are no gains since all traffic can be admitted and BTDconstraints are met.

Also shown in Figure 3(subfigure) is the degree to which therelative loads of slices ρv , and the weighted aggregate trafficon the network g are balanced, as measured by || · ||2, as thearrival rates on the network increase. As expected, based onTheorem 6, as arrivals increase relative loads of slices and thenetwork become more balanced, showing the compromise thetraffic shaping game is making, balancing slices relative loadsand that of the overall network.

VI. CONCLUSIONS

This paper has thoroughly explored a relatively simple andnatural approach for resource sharing amongst network slices– SCPF – which corresponds to socially optimal allocations ina Fisher market. Our analysis of performance in settings whereslices support stochastic loads provides explicit formulas for(i) the performance gains one can expect over static slicing,(ii) how to dimension slice shares to meet performanceobjectives, and (iii) how to go about performance manage-ment through admission control. If dynamic resource sharingamongst network slices is to be adopted, the ability to realizedisciplined engineering and performance prediction will be thekey. Our analysis of SCPF seems to meet these requirementsand at the same time reveals some intriguing insights regarding

the load interactions in such sharing models, in particular theimpact of relative load distributions on statistical multiplexing,and the role of traffic shaping in optimizing admission control.Finally, we note that our approach to admission control inan SCPF shared system is novel in that each slice exploitsknowledge of its customers’ mobility patterns to optimize itscarried load and assure service continuity.

VII. APPENDIX

A. Proof of Theorem 1

A w induces a probability distribution at resource b as:

pb(w) = (pu(w) , wu∑u′∈Ub wu′

: u ∈ Ub),

Suppose w∗ = (w∗u = sv|Uv| : u ∈ U), where u ∈ Uv . Let us

define the objective function of problem (1) as U(w). For anarbitrary w we have:

U(w∗)− U(w) =∑

v∈V

∑

u∈Uv

sv|Uv| (log(

w∗ucu∑u′∈Ub w

∗u′

)

− log(wucu∑u′∈Ub wu′

))

=∑

b∈B(∑

u′∈Ubw∗u′)

∑

v∈V

∑

u∈Uv

pu(w∗) log(pu(w∗)pu(w)

)

=∑

b∈B(∑

u′∈Ubw∗u′)D(pb(w

∗)||pb(w))

≥ 0,

where D(pb(w∗)||pb(w)) denotes the Kullback-Leibler diver-

gence between pb(w∗) and pb(w). The second equality holdstrue by repartitioning users by the base stations they are associ-ated with. The last inequality comes from the nonnegativity ofthe Kullback-Leibler divergence. Thus w∗ has a higher utilitythan any other arbitrary weight allocation thus is optimal.

B. Proof of Corollary 1

Using the result in Theorem 2 under Assumption 1 we havethat

Ev[

1

Rv

]= δv[1−(1−e−ρv−ρve−ρv )‖ρv‖22+

(ρv + 1)

svg′,T ρv]

(24)Similarly for static slicing we have that

Ev[

1

Rv,SS

]=δvsv

(ρv‖ρv‖22 + 1). (25)

Taking the ratio of the overall mean BTDs gives the gain forslice v:

Gv =ρv‖ρv‖22 + 1

sv(1− (1− e−ρv − ρve−ρv )‖ρv‖22) + (ρv + 1)g′,T ρv(26)

By plugging in ρv = 0, it is easy to see that:

GLv =1

sv + g′,T ρv≥ 1

(gv)T ρv + sv=

1

(1− sv)( gv

1−sv )T ρv + sv.

10

Because ‖ gv

1−sv ‖1 = 1, ( gv

1−sv )T ρv ≤ 1. Therefore GLv ≥ 1.

Let’s say g′−v =∑v′ 6=v sv′(1−e−ρ

v′)ρ+svρ. Equation (26)

can be written as:

Gv =‖ρv‖22g′,T−v ρv

+

1− [‖ρv‖22 + (1− ‖ρv‖22)sv‖ρv‖22g′,T−v ρ

v]

(ρv + 1)g′,T−v ρv + sv(1− ‖ρv‖22). (27)

Note that sv‖ρv‖22g′,T−v ρ

v≤ 1, thus the numerator of the second

term is nonnegative. Therefore Gv is decreasing in ρv . Whenρv → ∞,∀v ∈ V , the second term in Eq. (27) vanishes, andg′−v → g. Then GHv is given by:

GHv =‖ρv‖22gT ρv

C. Proof of Theorem 6

Under the saturated regime, BTD constraint of each v ∈ Vmust be binding because we are blocking traffics. Thus wehave:

xv =fv(ρ

v; ρ−v)

sv(dv − 1), (28)

Also Assumption 4 guarantees that for all v ∈ V , Mvρv xv1 is not binding then the ACv can be reformulated as:

minρv∈Γv

svfv(ρv; ρ−v), (29)

A constant factor sv in the objective function has no impactto the minimizer. We can show that the optimality conditionof problem (20) is the same as each slice v optimizing its ownproblem given by Eq. (29). Dividing the objective function by2, the Lagrangian of problem (20) is:

L(ρ; ζ,χ) =1

2(‖∑

v

svρv‖22 +

∑

v

s2v‖ρv‖22)

+∑

v∈Vζv(1

T ρv − 1)−∑

v∈V(χv)TMvρ

v, (30)

where ρ , (ρv : v ∈ V)T , and dual variables ζ , (ζv :v ∈ V)T with χ , (χv : v ∈ V)T . According to the KKTcondition, the solution ρ∗ must be such that, for all v ∈ V:

∇ρvL(ρ∗; ζ∗,χ∗) = s2vρ

v,∗ + svg + ζ∗v1− (Mv)Tχv,∗

= 0, (31)

and χv,∗ 0, ρv,∗ ∈ Γv. The Lagrangian of problem (29) is:

Lv(ρv; ζv,χ

v) =gT (svρv)

+ ζv(1T ρv − 1)− (χv)TMvρ

v. (32)

If slice v’s relative load ρv,∗ optimizes problem (29) givenother slices’ ρ−v,∗, following KKT condition should be met:

∇ρv,∗Lv(ρv,∗; ζ∗v ,χ

v,∗) = s2vρ

v,∗ + svg

+ ζ∗v1− (M−1v )Tχv,∗

= 0, (33)

and χv,∗ 0, ρv,∗ ∈ Γv, which is exactly the same as theKKT condition of problem (20). Therefore, problem (20) issolved at the Nash equilibrium. Moreover, we could computethe total carried load of slice v by setting

fv(ρv,∗; ρ−v,∗) = sv(dv − 1)/ρ∗v,

which gives us ρ∗v = sv(dv−1)g∗,T ρv,∗ .

D. Proof of Theorem 7

Under the saturated regime, BTD constraint of each v ∈ Vmust be binding. Thus we have:

ρv =svdv − 1

‖ρv‖22.

Moreover, the constraint av 1 should be satisfied with strictinequality, thus the optimal policy ρv,SS,∗ under SS is givenby:

minρv

‖ρv‖22s.t. Mvρv 0

1T ρv = 1.

Then the optimal load is obtained by plugging in the BTDconstraint.

E. Proof of Corollary 2

Under SS, it is obvious that the optimal relative loaddistribution of slice v is

ρv,SS,∗ =1

B1

Plugging it in the BTD constraint one can get ρSS,∗v =B(svdv − 1), thus ρv,SS,∗ = (svdv − 1)1.

Dividing the objective function by sv and discarding therouting constraints, the Lagrangian of Eq. (29) is:

(gv)T ρv + ν(1T ρv − 1),

where ν is the dual variable. Solving its KKT condition wehave:

ρv,∗ = − 1

2sv(ν1 +

∑

v′ 6=vsv′ ρ

v′). (34)

Substituting in 1T ρv,∗ = 1 we have ν = − sv+1B .

When 1B1 ∈ Γv,∀v ∈ V , Eq. (34) implies that if all

other slices v′ 6= v pick their relative loads as 1B1, then

(∑v′ 6=v sv′ ρ

v′) ‖ 1, meaning that this is the Nash equilibriumof the game. Note that since ρv,∗ is feasible and optimal fora relaxed feasible set, it will still be optimal if we put backthe routing constraints. Thus we have that:

g∗,T ρv,∗ =1

B.

Then the carried load gain is obtained by plugging in Eq. (22).

11

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