Statistical Multiplexing and Traffic Shaping Games for Network Slicing

Transcription

1 Statistical Multiplexing and Traffic Shaping Games for Network Slicing Jiaxiao Zheng, Palo Caallero, Gustao de Veciana, Seung Jun Baek and Alert Banchs The Uniersity of Texas at Austin, TX Korea Uniersity, Korea Uniersity Carlos III of Madrid & IMDEA Networks Institute, Spain contact: Astract Next generation wireless architectures are expected to enale slices of shared wireless rastructure which are customized to specific moile operators/serices. Gien rastructure costs and the stochastic nature of moile serices spatial loads, it is highly desirale to achiee efficient statistical multiplexing amongst such slices. We study a simple dynamic resource sharing policy which allocates a share of a pool of (distriuted) resources to each slice Share Constrained Proportionally Fair (SCPF). We gie a characterization of SCPF s performance gains oer static slicing and general processor sharing. We show that higher gains are otained when a slice s spatial load is more imalanced than, and/or orthogonal to, the aggregate network load, and that the oerall gain across slices is positie. We then address the associated dimensioning prolem. Under SCPF, traditional network dimensioning translates to a coupled share dimensioning prolem, which characterizes the existence of a feasile share allocation gien slices expected loads and performance requirements. We proide a solution to roust share dimensioning for SCPF-ased network slicing. Slices may wish to unilaterally manage their users performance ia admission control which maximizes their carried loads suject to performance requirements. We show this can e modeled as a traffic shaping game with an achieale Nash equilirium. Under high loads, the equilirium is explicitly characterized, as are the gains in the carried load under SCPF s. static slicing. Detailed simulations of a wireless rastructure supporting multiple slices with heterogeneous moile loads show the fidelity of our models and range of alidity of our high load equilirium analysis. I. INTRODUCTION Next generation wireless systems are expected to emrace SDN/NFV technologies towards realizing slices of shared wireless rastructure which are customized for specific moile serices, e.g., moile roadand, media, OTT serice proiders, and machine-type communications. Customization of network slices may include allocation of (irtualized) resources (communication/computation), per-slice policies, performance monitoring and management, security, accounting, etc. The aility to deploy serice specific slices is iewed, not only as a mean to meet the dierse and sometimes stringent demands of emerging serices, e.g., ehicular, augmented reality, ut also as an approach for rastructure proiders to reduce costs while deeloping new reenue streams. Resource allocation irtualization in this context is more challenging than for traditional cloud computing. Indeed, rather than drawing on a centralized pool of resources, a network slice requires allocations across a distriuted pool of resources, e.g., ase stations. The challenge is thus to promote efficient statistical multiplexing amongst slices oer pools of shared resources. Network slices can e used to enale the sharing of network resources amongst competing (possily irtual) operators. Indeed, the sharing of spectrum and rastructure is iewed as one way of reducing capital/operational costs and is already eing considered y standardization odies, see [], [], which hae specified architectural and technical requirements, ut left the sharing criteria and algorithmic issues open. By aggregating their traffic onto shared resources, it is expected that operators could realize sustantial saings, which might justify/enale new shared inestments in next generation technologies including 5G, mmwae and massie MIMO. The focus of this paper is on resource sharing amongst slices supporting stochastic (moile) loads. A natural approach to sharing is complete partitioning (see, e.g., [6]), which we refer to as static slicing, wherey resources are statically partitioned and allocated to slices, according to a serice leel agreement, irrespectie of slices instantaneous loads. This offers each slice a guaranteed allocation at each ase station, and protection from each other s traffic, ut, as we will see, poor efficiency. Other approaches include full sharing (where all slices are sered on a FCFS asis without resource reseration), general processor sharing [4], which pre-assigns a share to each slice, and allocate resource at each ase station proportionally to the shares among the slices which has actie users there. Instead, we adocate an alternatie approach wherein each slice is pre-assigned a fixed share of the pool of resources, and re-distriutes its share equally amongst its actie customers. In turn, each ase station allocates resources to customers in proportion to their shares. We refer to this sharing model as Share Constrained Proportionally Fair (SCPF) resource allocation. By contrast with static slicing, SCPF is dynamic (since its resource allocations depend on the network state) ut constrained y the network slices pre-assigned shares (which proides a degree of protection amongst slices). Related work. There is an enormous amount of related work on network resource sharing in the engineering, computer science and economics communities. The standard framework used in the design and analysis of communication networks is utility maximization (see e.g., [8] and references there-

2 in) which has led to the design of seeral transport and scheduling mechanisms and criteria, e.g., the often considered proportional fair criterion. The SCPF mechanism, descried aoe, should e iewed as a Fisher market where agents (slices), which are share (udget) constrained, id on network resources, see, e.g., [3], and for applications [5], [0], [5]. The choice to re-distriute a slice s share (udget) equally a- mongst its users, can e iewed as a network mandated policy, ut also emerges naturally as the social optimal, market and Nash equilirium when slices exhiit (price taking) strategic ehaior in optimizing their own utility, see [9]. The noelty of our work lies in considering slice ased sharing, under stochastic loads and in particular studying the expected performance resulting from such SCPF-ased resource allocations among coupled slices. Other researchers who hae considered performance of stochastic networks, e.g., [7], [], and others, hae studied networks where customers are allocated resources (along routes) ased on maximizing a sum of customers utilities. These works focus on network staility for elastic customers, e.g., file transfers. Susequently [8], [7] extended this line of work, to the ealuation of mean file delays, ut only under alanced fair resource allocations (as a proxy for proportional fairness). Our focus here is on SCPF-ased sharing amongst slices with stochastic loads and on inelastic or rate-adaptie customers, e.g., ideo, oice, and more generally customers on properly proisioned networks, whose actiity on the network can e assumed to e independent of their resource allocations. Finally there is much ongoing work on deeloping the network slicing concept, see e.g., [5], [3] and references therein, including deelopment of approaches to network irtualization in RAN architectures, e.g, [], [0], [6], and SDNased implementation, e.g., [6]. This paper focuses on deising good slice-ased resource sharing criteria to e incorporated into such architectures. Contriutions of this paper. This paper makes seeral contriutions centering on a simple and practical resource sharing mechanism: SCPF. First, we consider user performance (it transmission delay) on slices supporting stochastic loads. In particular we deelop expressions for (i) the mean performance seen y a typical user on a network slice; and (ii) the achieale performance gains ersus static slicing (SS) and general processor sharing (GPS). We show that when a slice s load is more imalanced than, and/or orthogonal to, the aggregate network load, one will see higher performance gains. Our analysis proides an insightful picture of the geometry of statistical multiplexing for SCPF-ased network slicing. Second, under SCPF, traditional network dimensioning translates to a coupled share dimensioning prolem, which addresses whether there exist feasile share allocations gien slices expected loads and performance requirements. We proide a solution to roust share dimensioning for SCPF-ased network slicing. Third, we consider decentralized per-slice performance management under SCPF sharing. In particular, we consider admission control aimed at maximizing a slice s carried load suject to a performance constraint. When slices unilaterally optimize their admission control policies, the coupling of their decisions can e iewed as a traffic shaping game, which is shown to hae a Nash equilirium. For a high load regime we explicitly characterize the equilirium and the associated gains in carried load for SCPF ersus static slicing. Finally, we present detailed simulations for a shared distriuted rastructure supporting slices with moility patterns different than that assumed in the theoretical analysis and more practical SINR model. The results match our analysis well, which further supports our conclusions on gains in oth performance and carried loads of SCPF sharing. II. SYSTEM MODEL A. Network Slices, Resources and Moile Serice Traffic We consider a collection of ase stations (sectors) B shared y a set of network slices V, with cardinalities B and V respectiely. For example, V might denote slices supporting different serices or (irtual) moile operators, etc. We enisage each slice proiding a moile serice in the region sered y the ase stations B. Each slice supports a stochastic load of users (deices/customers) with an associated moility/handoff policy. In particular, we assume that exogenous arrials to slice at ase station follow a Poisson process with intensity γ and let γ denote the (column) ector of arrial intensities at each ase station associated with slice, i.e. γ (γ : B). Each slice customer at ase station has an independent sojourn time with mean µ after which it is randomly routed to another ase station or exits the system. As explained elow we assume that such moility patterns do not depend on the resources allocated to users. We let Q (qi,j : i, j B) denote a slice-dependent routing matrix where qi,j is the proaility a slice customer moes from ase station i to j and j B q i,j is the proaility it exits the system. This model induces an oerall traffic intensity for slice across ase stations satisfying flow conseration equations: for all B we hae κ γ + a B κ aq a,, where κ is the traffic intensity of slice on ase station. Accounting for users sojourn times, the mean offered load of slice on ase station is ρ κ µ, and ρ (ρ : B) captures its system load distriution. Letting µ (µ : B), the flow conseration equations can e rewritten in matrix form as: ρ diag(µ )(I (Q ) T ) γ. () If Q is irreducile, I (Q ) T is irreducily diagonally dominant thus always inertile. Otherwise, we can always find a permutation matrix of B, say P to make: P T (I (Q ) T )P A B, A K,

3 where K is the numer of irreducile classes. Moreoer, at least one ase station of each irreducile class has a nonzero exiting proaility, thus A K must e inertile. Then the inertiility of I (Q ) T follows. This model corresponds to a multi-class network of M/GI/ queues (ase stations), where each slice corresponds to a class of customers, see, e.g., [9]. Such networks are known to hae a product-form stationary distriution, i.e., the numers of customers on slice at ase station, denoted y N, are mutually independent and N Poisson(ρ ). Since the sum of independent Poisson random ariales is again Poisson, the total numer of customers on slice is such that N B N Poisson(ρ ) where ρ B ρ. Our network model for the numers of customers and moility across ase stations, assumes that customer sojourn/actiity/moility are independent of the network state and of the resources a customer is allocated. This is reasonale for properly engineered slices where the performance a customer sees does not impact its actiity, e.g., inelastic or rate adaptie applications seeing acceptale performance. This coers a wide range of applications including oice, ideo streaming, IoT monitoring, real-time control, and een, to some degree, elastic we rowsing sessions where users are peak rate constrained and this constraint typically dictates their performance. There are seeral natural generalizations to this model including class-ased routing and user sessions (e.g. we rowsing) which are not always actie at the ase stations they isit, see, e.g., [9]. B. Network Slice Resource Sharing In the sequel we consider a setting where the resources allocated to a slice s customers depend on the oerall network state, i.e., numer of customers each slice has on each ase station, corresponding to the stochastic process descried in Section II-A. Let us consider a snapshot of the system s state and let U, U, U, and U denote sets of actie customers on slice at ase station, at ase station, on slice, and on the oerall network, respectiely. Thus, the cardinalities of these sets correspond to a realization of the system state, i.e., U n and U n, where in a stationary regime n and n are realizations of Poisson random ariales N and N, respectiely. Each ase station is modeled as a finite resource shared y its associated users U. A customer u U can e allocated a fraction f u [0, ] of that resource, e.g., of resource locks in a gien LTE frame, or allocated the resource for a fraction of time, where u U f u. We shall neglect quantization effects. The transmission rate to customer u, denoted y r u, is then gien y r u f u c u where c u denotes the current peak rate for that user. To model customer heterogeneity across slices/ase stations we shall assume c u for a typical customer on slice at ase station is an independent realization of a random ariale, denoted y C, whose distriution may depend on the slice, since slices may support different types of customer deices (e.g., car connectiity s. moile phone) and depend on the ase station, since typical slice users may hae different spatial distriutions with respect to ase station or see different leels of interference. Below we consider three resource allocation schemes; the first two are used as enchmarks, while the third is the one under study in this paper. For all we assume each slice is allocated a share of the network resources s, V such that s > 0 and V s. Definition. Static Slicing (SS): Under SS, slice is allocated a fixed fraction s of each ase station s resources, and each customer u U gets an equal share, i.e., /n, of the slice s resources at ase station. Thus the users transmission rate ru SS is gien y r SS u s n c u. Definition. General Processor Sharing (GPS): [4] Under GPS, each actie slice at ase station such that n > 0 is allocated a fraction of the ase station s resources proportionally to its share s. Thus a user u U sees a transmission rate ru GP S gien y r GP S u n s V s {n >0} c u. () Definition 3. Share Constrained Proportionally Fair (SCPF): Under SCPF each slice re-distriutes its share of the oerall network resources equally amongst its actie customers, which thus get a weight (su-share) w u s n for u U, V. In turn, each ase station allocates resources to customers in proportion to their weights. So a user u U gets a transmission rate ru SCP F gien y r SCP F u w u u U w u c u s n V n s n c u. (3) A simple example illustrating the differences among three schemes is exhiited in Tale I. Suppose there are two ase stations, i.e., B {, }, and two slices V {, } each with an equal share of the network resource. Consider a snapshot of the system where Users u, u are on Slice and u 3, u 4 are on Slice. Also, u, u, and u 3 are at ase station and u 4 is at ase station. Let us assume for simplicity that c u, u U. In this case, under SS at the two users on Slice need to share of the resource while u 3 on Slice is allocated the other, while at, half of the resource is wasted due to the asence of actie users on Slice. By contrast, GPS utilizes all resources at y allocating all of them to u 4, and it makes the same allocation as SS at. Under SCPF, ecause each user is allocated the same weight 4, at, three users are allocated the same rate 3 and at all andwidth is gien to u 4. This example shows how SCPF achiees etter network-wide fairness than GPS and SS, while ensuring that resources are not wasted. Indeed, under SCPF the oerall fraction of resources slice is allocated at ase station is proportional to n n s, i.e., 3

4 TABLE I: Rate allocation under different schemes u User association Rate allocation Slice Slice SS GPS SCPF 4 u 4 u u 4 its share times its relatie numer of users at the ase station. This proides a degree of elasticity to ariations in the slice s spatial loads. Howeer, if a slice has a large numer of customers, its customers weights are proportionally decreased, which protects other slices from such oerloads. Note that SCPF requires minimal ormation exchanges among ase stations and is straightforward to implement, e.g., using SDNlike framework. III. PERFORMANCE EVALUATION In this section we study the expected performance seen y a slice s typical customer. Gien our focus on inelastic/rate adaptie traffic and tractaility, we choose our customer performance metric as the reciprocal transmission rate, referred to as the Bit Transmission Delay (BTD), see, e.g., [30]. This corresponds to the time taken to transmit a it, so lower BTDs indicate higher rates and thus etter performances. BTD is a high-leel metric capturing the instantaneous QoS perceied y a user, e.g., short packet transmission delays are roughly proportional to the BTD. By guaranteeing a good BTD we can guarantee that the user perceied QoS is acceptale all the time, instead of in an aerage sense. Alternatiely, the negatie of the BTD can e iewed as a concae utility function of the rate, which in the literature (see, e.g., []) was referred to as the potential delay utility. Concae utility functions tend to faor allocations that exhiit reduced ariaility in a stochastic setting. Gien the stochastic loads on the network, we shall ealuate the aerage BTD seen y a typical (i.e., randomly selected) customer on a slice, i.e., aeraged oer the stationary distriution of the network state and transmission capacity seen y typical users, e.g., C, at each ase station. Such aerages naturally place higher weights on congested ase stations, where a slice may hae more users, est reflecting the oerall performance customers will see. A. Analysis of BTD Performance Consider a typical customer on slice and let E denote the expectation of the system state as seen y such a customer, i.e., under the Palm distriution [4]. For SCPF, we let R e a random ariale denoting the rate of a typical customer on slice, and R that of such customer on slice at ase station. Similarly, let R,SS, R,SS, R,GP S, and R,GP S denote these quantities under SS and GPS, respectiely. Thus, under SCPF the aerage BTD for a typical slice customer is gien y E [ R ]. The next result characterizes the mean BTD under SCPF, SS, and GPS under our traffic model. We introduce some further notation: Oerall aerage numer of users on slice : ρ E[N ]. Load distriution of slice : ρ (ρ : B), where ρ E[N ]. Relatie load distriution of slice : ρ ( ρ : B), where ρ ρ ρ. Oerall share weighted relatie load distriution: g ( g : B), where g V s ρ. Actie share weighted relatie load distriution: g ( g : B), where g V s ( e ρ ) ρ. If a slice is inactie, i.e., not haing any customer in the network, then its share should e oided. ( e ρ ) is the proaility that slice has at least actie user in the network. Vector of idle shares seen y a typical user on s- lice[ : s ( s : B), where s ] E s {N 0} s e ρ represents how much share is oided seen y slice, due to the asence of actie users at ase station. Also we define S diag( s ). Mean reciprocal capacity of slice : δ (δ : B), where δ E [ C ]. Also we define diag(δ ). We use x, x M x T Mx to denote the weighted inner product of ectors, where M is a diagonal matrix. Also, we use x M x T Mx to denote the weighted norm of a ector, where M is a diagonal matrix. In oth cases, when M is the identity matrix I we simply omit it. In addition, x and x denote the L-norm and L-norm of x, respectiely. Theorem. For network slicing ased on SCPF, the mean BTD for a typical customer on slice is gien y [ ] E R ( ( )) g ρ δ ρ + (ρ + ) s + e ρ ρ. (4) B If ( ρ : V) are fixed, and (ρ : V) are large, then the mean BTD has following asymptotic form: [ ] E R ρ s ρ, g + O() (5) For network slicing ased on SS, the mean BTD for a typical customer on slice is gien y [ ] E R,SS ( ) ρ ρ δ + s. (6) B For network slicing ased on GPS, the mean BTD for a typical customer on slice is gien y [ ] E R,GP S ( ) ρ ρ δ + s ( s ). (7) B Please see appendix for the detailed proof. BTD under all 3 schemes increases with the oerall load ρ and decreases with the share s when ( ρ : V) are fixed. Their dependencies on relatie loads ( ρ : V) are different, implying that they exploit statistical multiplexing differently. 4

5 B. Analysis of Gain Using the results in Theorem one can ealuate the gains in the mean BTD for a typical slice user under SCPF s. SS, defined as, [ ] G SS E R,SS E [ ]. R In general, one would expect G SS since under SCPF typical users should see higher allocated rates and thus lower BTDs. One can erify that is the case when slices hae uniform loads across ase stations ut the general case is more sutle. Similarly, we define the gain of SCPF s. GPS y R,GP S ] [ G GP S E E [ ]. R By taking the ratio of the mean BTD perceied y a typical customer under SS and that under SCPF gien in Theorem, we hae the following corollary. Corollary. The BTD gain of SCPF oer SS for slice is gien y G SS ρ ρ + δ, ρ s δ, ρ s ( (ρ +)e ρ ) ρ +(ρ +) g, ρ (8) For fixed relatie loads ( ρ : V), when slice has a light load, i.e., ρ 0, the gain is greater than and gien y: G SS,L δ, ρ s δ, ρ + g, ρ > Furthermore, G SS is a nonincreasing function of ρ, and if all slices hae high oerall loads, i.e., ρ, V, the gain is gien y: G SS,H ρ g, ρ. The result indicates that when the relatie loads are fixed, the gain decreases with the oerall load ρ, thus if G SS,H > SCPF always proides a gain. Let us consider the heay load gain under the following simplifying assumption. Assumption. Base stations [ are ] said to e homogeneous for slice if for all B: E δ. C Assumption only requires the aerage reciprocal capacity a gien slices customer sees across ase stations is homogenous. In this case, the BTD gain for slice under heay load simplifies to G SS,H ρ g cos(θ( g, ρ )), where θ( g, ρ ) denotes the angle etween the slice s relatie load and the oerall share weighted relatie load on the network. A sufficient condition for gains under high loads is that g ρ. Since g ρ, this follows when the oerall share weighted relatie load on the network is more alanced than that of slice. One would typically expect aggregated traffic to e more alanced than that of indiidual slices. This condition is fairly weak, i.e., it does not depend on where the loads are placed, ut on how alanced they are. The corollary also suggests that gains are higher when cos(θ( g, ρ )) is smaller. In other words, a slice with imalanced relatie loads whose relatie load distriution is orthogonal to the shared weighted aggregate traffic, i.e., cos(θ( g, ρ )) 0, will tend to see higher gains. This is due to that SCPF can achiee sharing elasticity y aligning resource allocations with demands, i.e., load distriutions. Thus when the load distriutions are nearly orthogonal, sharing under SCPF is much etter than that under SS, which is completely inelastic. The simulations in Section VI further explore these oserations. Similarly, for the BTD gain of SCPF oer GPS, we hae following result: Corollary. The BTD gain of SCPF oer GPS for slice is gien y G GP S ρ ( ρ ρ S )+ ρ, s s δ, ρ s ( (ρ +)e ρ ) ρ +(ρ +) g, ρ. (9) For fixed relatie loads ( ρ : V), and fixed oerall loads for other slices (ρ : ), the gain for slice under low oerall load, ρ 0, is gien y: GP S,L G ρ, s s δ, ρ + g, ρ. Furthermore, if all slices hae low load ρ 0, V, then GP S,L G Also, if all slices hae high loads, i.e., ρ, V, the BTD gain oer GPS for slice is gien y: GP S,H G ρ ρ S g, ρ. Please see appendix for detailed proof. Note that when ( ρ : V) are fixed and,, ρ > 0, under heay load, i.e., ρ, V, we hae s 0, thus ρ ρ S ρ, which means GPS otains a similar performance as SS under heay load. Howeer, unlike GP S,L the gain oer SS, G might not e strictly greater than and G GP S might not e monotonic in ρ. One can osere that, different slices can experience different BTD gains, depending on its share and load distriutions. Howeer, to compare the performances of different sharing criteria, a network-wide metric of gain needs to e defined. To e ale to compare scenarios with different load distriutions and shares, it is of particular interest to consider a metric which is roust against changes of load and share. To deise such a metric, we note that users who are perceiing a low aerage capacity from their associated ase stations, and/or users whose allocated shares are small are expected to experience higher BTDs. Thus to achiee the roustness of the metric, let us define the normalized BTD for a typical user on slice at ase station under SCPF as Ē [ R ] δ s ρ E [ R ], (0) 5

6 and thus the normalized BTD for a typical user on slice under SCPF is gien y [ ] Ē R [ ] ρ Ē R. () B [ ] Similarly, one can define Ē, Ē [ ] R,SS R, and [ ],SS Ē R,GP S, Ē [ R,GP S ]. For the oerall performance of the system, let us consider the share weighted sum of the normalized BTD since the system should e tuned to put more emphasis on the slices with higher shares, and define the oerall weighted BTD gain of SCPF oer SS as V G SS all s Ē [ ] R V [,SS ] sē, () R and the oerall weighted BTD gain of SCPF oer GPS as G GP S all V s Ē [ R,GP S ] V sē [ R ], (3) The following results capture the oerall weighted BTD gains. Corollary 3. When ρ, V, the oerall weighted BTD gains of SCPF oer SS and GPS under heay load are gien y G SS,H all and V s ρ V s g, ρ, GGP S,H all G SS,H all GP S,H, Gall. V s ρ I S V s g, ρ, (4) Please see appendix for detailed proof. It is easy to see that if ρ are the same for all V, then oth G SS,H GP S,H all and Gall are when the loads are heay. By contrast, if the relatie loads of different slices are (approximately) all orthogonal, i.e., ρ, ρ 0, and each slice has the same share s V, V, the oerall gain can e as high as V. IV. SHARE DIMENSIONING UNDER SCPF In practice each slice may wish to proide serice guarantees to its customers, i.e., ensure that the mean BTD does not exceed a performance target d. Below we inestigate how to dimension network shares to support slice loads suject to such mean BTD requirements. Henceforth we shall assume the following assumption is in effect. Assumption. The network is said to see high oerall slice loads, if for all V we hae ρ. Consider a network supporting the traffic loads of a single slice, say, so s and g ρ. Note that ρ, δ is the minimum aerage BTD achieale across the network when a slice gets all the ase station resources, so a target requirement satisfies d > ρ, δ. For slice to meet a mean BTD constraint d, it follows from Eq. (5) that: ρ l(d, ρ, δ ) d ρ, δ ρ. We can interpret l(d, ρ, δ ) as the maximal admissile carried load ρ gien a fixed relatie load distriution ρ, BTD requirement d, and mean reciprocal capacities δ. As might e expected, if the relatie load distriution ρ is more alanced (normalized y the mean ase station capacity), i.e., ρ is smaller, or if the BTD constraint is relaxed, i.e., d is higher, or the ase station capacities scale up, i.e., δ is smaller, the slice can carry a higher oerall load ρ. Next, let us consider SCPF ased sharing amongst a set of slices V each with its own BTD requirements. It follows from Eq. (5) that to meet such requirements on each slice the following should hold: for all V s + ρ l(d, ρ, δ ) ρ s u ρ, ρ u ρ. (5) u This can e written as: s h 0, (6) V where we refer to h (h u : u V) as s share coupling ector, gien y { u h u +ρ u ρ u, ρ u l(d u, ρ u,δ u ) ρ u ρ u u u, We can interpret h as the enefit to slice of allocating unit share to itself. When u, h u depends on two factors. +ρ The first u l(d u, ρ u,δ u ) ρ captures the sensitiity of slice u to u the share weighted congestion from other slices. If ρ u is close to its limit l(d u, ρ u, δ u ), its sensitiity is naturally ery high. The second term, ρu, ρ u captures the impact of slice ρ u s load distriution on slice u. u Note that if two slices load distriutions are orthogonal, they do not affect each other. The following result summarizes the aoe analysis. Theorem. There exists a share allocation such that slice loads and BTD constraints ((ρ, ρ, d ) : V) are admissile under SCPF sharing if and only if there exists an s (s : V) such that s, s 0 and s h 0. V Admissiility can then e erified y soling the following maxmin prolem: max { min s h i : s }. (7) s 0 i V If the optimal ojectie function is positie, the traffic pattern is admissile. Moreoer, if there are multiple feasile share allocations, then the optimizer is a roust choice in that it maximizes the minimum share gien to any slice, giing slices margins to tolerate perturations in the slice loads satisfying Eq. (6). If a set of network slice loads and BTD constraints are not admissile, admission control will need to e applied. We discuss this in the next section. 6

7 V. ADMISSION CONTROL AND TRAFFIC SHAPING GAMES A natural approach to managing performance in oerloaded systems is to perform admission control. In the context of slices supporting moile serices where spatial loads may ary sustantially, this may e unaoidale. Below we consider admission control policies that adapt to changes in load. Specifically, an admission control policy for slice is parameterized y a (a : B) [0, ]B where a is the proaility a new customer at ase station is admitted. Such decisions are assumed to e made independently thus admitted customers for slice at ase station still follow a Poisson Process with rate γ a. Based on the flow conseration equation Eq. () one can otain the carried load ρ induced y admission control policy a ia ρ (M ) a diag(µ )(I (Q ) T ) diag(γ )a where M diag(γ ) (I (Q ) T )diag(µ ) is inertile ecause I (Q ) T is irreducily diagonally dominant. By contrast with Section II-A, note that ρ now represents the load after admission control, which may hae a reduced oerall load and possily changed relatie loads across ase stations i.e., shape the traffic on the slice. We also let g e the oerall share weighted relatie loads after admission control, see Section III-A. Note that we hae assumed only exogenous arrials can e locked, thus once a customer is admitted it will not e dropped the intent is to manage performance to maintain serice continuity. Below we consider a setting where slices unilaterally optimize their admission control policies in response to network congestion, rather than a single joint gloal optimization. The intent is to allow slices (which may correspond to competing irtual operators/serices) to optimize their own performance, and/or enale decentralization in settings with SCPF ased sharing. For simplicity we assume that assumption holds true throughout this section, and define the capacity normalized mean BTD requirement d d δ. Suppose each slice optimizes its admission control policy so as to maximize its oerall carried load ρ, i.e., the aerage numer of actie users on the network, suject to a normalized mean BTD constraint d. Under Assumption the optimal policy for slice is the solution to the following optimization prolem: max ρ (8) ρ,ρ s.t. a ρ M ρ, a [0, ] B,, ρ (9) (ρ + ) ( g, ρ (ρ + )e ρ) ρ s d (0) Note that Eq. (9) estalishes a one-to-one mapping etween ( ρ, ρ ) and a. We will use ρ and ρ to parameterize admission control decisions for slice. The BTD constraint in If γ is not strictly positie one can reduce the dimensionality. Eq. (0) follows from Eq. (5). Also note that this admission control policy depends on oth the oerall share weighted loads on the network g, the slice s load and its customer moility patterns (i.e., M ). Unfortunately, for general loads ρ, this prolem is not conex due to the BTD constraint Eq. (0); howeer, for high oerall per slice loads it is easily approximale y a conex function. Under Assumption we hae that + ρ ρ and the left hand side of Eq. (0) ecomes: (ρ + ) s g, ρ ρ ρ s g, ρ (s x ) g, ρ () where we hae defined x (ρ ). Further defining ρ ( ρ : V\{}), Eq. (0) can e replaced y: f ( ρ ; ρ ) g, ρ s ( d )x. () Thus, y defining y ( ρ, x ), which is equialent to ( ρ, ρ ), together with y (y : V\{}), each slice can unilaterally optimize its admission control policy y soling the following prolem: Admission control for slice under SCPF (AC ): Gien other slices admission decisions y, slice determines its admission control policy y ( ρ, x ) y soling min y { x y Y (y ) } (3) where Y (y ) denotes slice s feasile policies and is gien y Y (y ) { y, ρ, 0 M ρ x, f ( ρ ; ρ ) s ( d )x }. (4) Note that AC is coupled to the decisions of other slices through the feasile set Y (y ). Thus, one cannot independently sole each slice s admission control prolem to otain an efficient solution. Furthermore, deising a gloal optimization for all slices rings oth complexity and nonconexity from the BTD constraints. A natural approach requiring minimal communication and cooperation oerhead is to consider a game setup where network slices are players, each seeking to maximize their carried loads (and the corresponding reenue) suject to BTD constraints. We formally define the traffic shaping game for a set of network slices V as follows. We let y (y : V) denote the simultaneous strategies of all slices (gien y the respectie admission control policies). As in AC, each slice picks a feasile strategy, i.e., y Y (y ) to minimize its ojectie function θ (y, y ) x. Note in the sequel we will modify θ (, ) to ensure the games conergence. A Nash equilirium is a simultaneous strategy y such that no slice can unilaterally improe its carried load, i.e., for all V θ (y,, y, ) θ (y, y, ), y Y (y, ). The following result follows from Theorem 3. in [3]. Theorem 3. The traffic shaping game defined aoe has a Nash equilirium. 7

8 Note that at the Nash equilirium, no slice can unilaterally improe its performance. Therefore, finding the Nash equilirium is also a way to achiee fairness under our sharing scheme. In the next susection, we will design an algorithm to achiee such allocation. A. Algorithm In our setting finding the Nash equilirium is not a simple matter. The difficulty arises from the fact that slices strategy spaces depend on other s choices, so oscillation is possile. In the literature such settings are specifically referred to as Generalized Nash Equilirium Prolem (GNEP), see, e.g., [4] and [9]. Howeer, the algorithm proposed in [4] assumed an algorithm capale of soling a penalized unconstrained Nash Equilirium Prolem, which satisfies a set of conditions, and that in [9] relies on the conexity of the joint strategy space. Thus none of them can e directly applied in our setting. Below we propose an algorithm inoling slices and a central entity which is guaranteed to conerge to the equilirium. We summarize the main ideas as follows. To decouple dependencies among strategy spaces, we shall moe slice s BTD constraint into its ojectie function as a penalty term with an associated multiplier λ. Let λ (λ : V). By adjusting the alue of λ according to y at each iteration, one can determine a setting such that, at the induced Nash Equilirium, all slices meet their BTD constraints, and the equilirium is identical to that of the traffic shaping game. In addition, in order to preent oershooting, at each iteration each slice s ojectie function is regularized y the distance to the preious reciprocal carried load x. Specifically, the admission control strategy of slice in response to other slices is now gien as the solution to the following optimization prolem: L ɛ (y; λ ) argmin (y ) Ȳ θ ((y ), y ; λ )+ ɛ (x x ), (5) where we define a BTD penalty function for slice as h (y) f ( ρ ; ρ ) s ( d )x. and the ojectie function for slice is now (different from what is preiously defined): θ (y, y ; λ ) e x + λ [h (y)] +, with [x] + max(0, x). The last term in Eq. (5) seres as a regularization term. The strategy space is now Ȳ {y, ρ, 0 M ρ x } and x is sustituted y e x to ensure strong conexity, which is required for conergence (note that due to the monotonicity, e x and x should result in the same optimizer). We propose to use the inexact line search update introduced in [9]. In order to make sure the iteration is proceeding towards the equilirium, we use Ω ɛ (y; λ) V θ (y, y ; λ ) θ (L ɛ (y; λ), y ; λ ) ɛ (x x ) 0 as a metric, osering that the equilirium is gien y y if and only if Ω ɛ (y ; λ) 0. Therefore we seek to decrease Ω ɛ (y; λ) y a sufficient amount at each iteration. The task executed y each slice is gien in Algorithm, while the central entity, which is responsile for collecting and deliering ormation and updating λ, executes Algorithm. This then follows the algorithm proposed in [4]. Algorithm Algorithm of Slice : Set k 0 and collect ɛ from central entity. : Receie λ (k) and y(k) from central entity. 3: Compute L ɛ (y(k); λ) and transmit it ack to the central entity. Set k k +. Go to step Algorithm Penalized Update in Central Entity : Choose a starting point y(0), λ(0) 0, η (0, ), for V, β, σ (0, ), ɛ > 0 ut small enough (see following theorem for conergence) and set k 0. : If a termination criterion is met then STOP. Otherwise, communicate y(k) together with λ (k) to all slices. 3: All slices compute L ɛ (y(k); λ) and feedack to central entity. 4: Compute t(k) max{β l l 0,,,... } such that if we assume ξ(k) (L ɛ (y(k); λ(k)) : V) y(k): Ω ɛ (y(k)+t(k)ξ(k)) Ω ɛ (y(k)) σ(t(k)) ξ(k). (6) Then set y(k + ) y(k) + t(k)ξ(k). 5: Set I(k) { h (y(k)) > 0}. For eery I(k), if e x(k) > η (λ y h y(k)) ), (7) then λ (k + ) λ (k). Set k k +. Broadcast y(k) and λ(k) to slices and go to step. Theorem 4. Let {y(k)} e the sequence of admission control decisions generated y Algorithm and Algorithm, then eery limit point of this sequence is a Nash equilirium of the traffic shaping game induced y AC. Proof. First we need to erify the Assumption 5. in [9] to guarantee that for a gien λ, step 4 in Algorithm conerges to a Nash equilirium. The non-constant part of Ψ ɛ (y, y ; λ) (defined in [9]) when y is fixed is: ex + λ [h (y)] + ɛ x x. If ɛ is small enough, the concaity of the last term will e canceled out y e x. Then the non-constant part is always conex in y. Hence, the Assumption 5. holds true together with the propositions.(a) - (d) in [9]. Therefore, the proposed algorithm generates Nash equilirium of the game. One can easily erify that the EMFCQ condition gien y Definition.7 in [4] is satisfied. Thus for all, λ gets updated a finite numer of times. According to Theorem.5 in [4], the claim is true. B. Characterization of Traffic Shaping Equilirium Next we study the characteristics of the resulting traffic shaping Nash equilirium. To make this tractale we consider 8

9 networks which are saturated and susequently (in Section VI) proide simulations to ealuate other settings. Assumption 3. (Saturated Regime) Suppose the system is such that for each network slice, the optimal admission control for oth SCPF and SS in response to other slices loads is such that for all V, a. Assumption 3 depends on many factors including the BTD constraints, the moility patterns, and network slices shares, ut it is generally true when the exogenous traffic of all slices at all ase stations γ is high. When this is the case we hae the following result: Theorem 5. Under Assumptions, and 3, the relatie load distriutions at the Nash equilirium of the traffic shaping game ρ ( ρ, : V) are the unique solution to: min ( ρ Γ : V) s ρ + (s ) ρ, (8) where Γ { ρ, ρ, M ρ 0 }, and the associated carried load for slice is ρ, s ( d ) g, ρ,, where g corresponds to the oerall share weighted relatie loads distriutions at the equilirium. See appendix for detailed proof. The first term in the ojectie function in Eq. (8) rewards alancing the oerall share weighted relatie loads on network. The second term rewards a slice for alancing its own relatie loads. The Nash equilirium in the saturated regime is thus a compromise etween those two ojecties while constrained y the network slices moility patterns and feasile admission control policies. Note that as long as ρ > 0, V, B, GPS and SS are approximately the same under heay load. Therefore, we use SS as the enchmark to characterize the carried load at the Nash equilirium under SCPF. Admission control for slice under SS (ACSS ): Under SS slice can determine its optimal admission control y y soling: max ρ,ρ ρ s.t. a ρ M ρ, a [0, ] B, ρ and ρ ρ (s d ). Note slices admission control decisions are clearly decoupled under SS, ut paralleling Theorem 5 we hae following result. Theorem 6. Under Assumptions and 3, the optimal admission control policy under SS are decoupled. The optimal choice for slice, ρ,ss,, is the unique solution to: min ρ Γ ρ, (9) and the associated carried load is gien y ρ,ss,. s d ρ,ss, See appendix for detailed proof. Admission control under SS is defined in the sequel. By comparing Eq. (8) and Eq. (9), one can see that under SS, slices simply seek to alance their own relatie loads on the network. By taking the ratio etween ρ and ρ SS, gien in Theorem 5 and 6, one can show that under Assumptions,, and 3 the gain in carried load for slice is gien y G load ρ, ρ,ss, ρ,ss, g, ρ, s ( d ) s d. (30) The first factor captures a traffic shaping dependent gain for slice. The second factor is a result of statistical multiplexing gains. A simple special case is highlighted in the following corollary. Corollary 4. Under Assumptions, and 3, if user moility patterns are such that B Γ for all V, the gain in the total carried load under the SCPF traffic shaping Nash equilirium s. optimal admission control for SS is gien y: G load s d s s d, V. (3) See appendix for detailed proof. Note that in order for a BTD constraint to e feasile under SS, one must require s d >. It can e seen that the gain exhiited in Corollary 4 can e ery high when s / d., i.e., no actual gain. This result implies that slices with small shares or tight BTD constraints will enefit most from sharing, coinciding with our oserations in Corollary. Furthermore, if s we hae that G load VI. PERFORMANCE EVALUATION RESULTS In this section, we alidate theoretical results in preious sections, and proide quantitatie characterizations ia numerical experiments. We simulated a wireless network shared y multiple slices supporting moile customers following the IMT-Adanced ealuation guidelines [8]. The system consists of 9 ase stations in a hexagonal cell layout with an inter site distance of 00 meters and 3 sector antennas, mimicking a dense small cell deployment. Thus, in this system, B corresponds to 57 sectors. Users associate to the sector offering the strongest SINR, where the downlink SINR is modeled as in [3]: SINR u P G u k B,k P kg uk + σ, where, following [8], the noise σ 04dB, the transmit power P 4dB and the channel gain etween user u and BS sector, denoted y G u, accounts for path loss, shadowing, fast fading and antenna gain. Letting d u denote the current distance in meters from the user u to sector, the path loss is defined as 36.7 log 0 (d u ) log 0 (f c )db, for a carrier frequency f c.5ghz. The antenna gain is set to 7 dbi, shadowing is updated eery second and modeled y a lognormal distriution with standard deiation of 8dB, as in [3]; and fast fading follows a Rayleigh distriution depending on the moile s speed and the angle of incidence. The downlink rate c u currently achieale to user u is ased on discrete set modulation and coding schemes (MCS) and associated SINR 9

10 Scenario Slices Spatial loads ρ g θ( g, ρ ) G SS,H Homogeneous uniform Homogeneous non-uniform Heterogeneous orthogonal Mixed Slices & non-uniform &4 uniform TABLE II: Measured normalized slice and network traffic norms and angles for highest load case of each scenario. Gain in BTD Simulation results Theoretical results Scenario 3 Scenario 4: Slices & Scenario 4: Slices 3&4 Scenario Scenario Offered traffic ( U / B ) Fig. : BTD gain oer SS for our 4 different scenarios. Fig. : Snapshot of users positions per slice and scenario exhiiting the different characteristics of traffic spatial loads. Left to right: Scenarios to Simulation results Theoretical results Scenario 4: Slices & thresholds gien in [3]. This MCS alue is selected ased on the aeraged SINR u, where channel fast fading is aeraged oer a second. We model slices with different spatial loads y modeling different customer moility patterns. Roughly uniform spatial loads are otained y simulating the Random Waypoint model [7], while non-uniform loads otained y simulating the SLAW model []. Instead of the open network assumed in the theoretical analysis, in the simulations we use a closed network where the total numer of users on each slice keeps fixed. Moreoer, the simulated moility models would not induce Markoian motion amongst ase stations assumed in our analysis, yet the analytical results are roust to these assumptions. A. Statistical Multiplexing and BTD Gains We ealuated the BTD gains of SCPF s. oth SS and GPS for four simulation scenarios, each including 4 slices, each with equal shares ut different spatial load patterns. For each scenario, we proide results for simulated BTD gains, and results from our theoretical analysis (Corollary and Corollary ) ased on the empirically otained spatial traffic loads. More detailed ormation regarding simulated scenarios and resulting empirical spatial traffic loads for high load regime are displayed in Tale II and a snapshot of locations for the 4 slices users in a network with a load of 4 users per sector is displayed in Figure. The results gien in Figure show the BTD gains oer SS for each scenario as the oerall network load increases. In Scenario 3, the aggregate network traffic is smoother than the indiidual slice s traffic, and the gains are indeed higher. This is also the case for Slice and in Scenario 4, since these slices loads are more imalanced than the other two slices, they experience higher gains. In Scenario, where slices nonhomogenous spatial loads are aligned, aggregation does not lead to smoothing and the gains are least. Gain in BTD Scenario 3 Scenario Scenario Scenario 4: Slices 3& Offered traffic ( U / B ) Fig. 3: BTD gain oer GPS for our 4 different scenarios. Fig. 4: BTD s. time for a randomly picked user under Scenario 3 Similarly, the results gien in Figure 3 show the BTD gains oer GPS for each scenario as the oerall network load increases. As can e seen, the gain is not necessarily monotonic in the load. In Scenario 4, the Slice and hae significant gains ecause their loads are more imalanced, while Slice 3 and 4 see negatie gains. Howeer, the oerall gain defined in Eq. (3) is still positie, ranging from.6 to.5 for arying oerall load. 0

11 Gain in carried load Arrial rate Slice &3 Slice Slice Slice 3 Aggregate Theoretical Fig. 5: Gain in carried load for arious arrial rates. Sufigure: Balancing in relatie load. As can e seen in Figures and 3 the simulated and theoretical gains (dashed lines) of Corollary are an excellent match. The theoretical model has een calirated to the mean reciprocal capacities seen y slice customers (i.e., δ s) and the measured induced loads resulting from the slice moility patterns. In addition to performance aeraged oer time, to illustrate the dynamic of the BTD perceied y a typical user, we plot the BTD s. time for a randomly picked user on Slice in Scenario, as shown in Fig. 4, where the left/right part is under light/heay load regime, respectiely. Note that under heay load, GPS and SS are approximately the same. SCPF outperforms for most of the time. Under light load, the mean BTD under SCPF is 4.044, while that under SS (GPS) is (5.86), respectiely. The standard deiation of BTD under SS (GPS) is 3.90 (3.0449), and SCPF reduces it to.93. Similar phenomenon is osered under heay load, when oth SS and GPS proide mean BTD of 9.65 and associated standard deiation of 3.79, SCPF reduce them to 6.79 and 3.45, respectiely. Therefore, SCPF can effectiely improe the perceied BTD and also smooth the user perceied QoS. B. Traffic Shaping Equilirium and Carried Load Gains In order to study the equiliria reached y the traffic shaping game, we measured the underlying user moility patterns in Section VI-A, and modeled it ia a random routing matrix. We further assumed uniform intensity of arrials rates at all ase stations and uniform exit proailities of 0.. The mean holding time at each ase station was again calirated with the simulations in Section VI-A. We considered a traffic shaping game for a network shared y 3 slices, where Slice has uniform spatial loads and Slice and 3 hae different non-uniform spatial loads. All slices hae equal shares and their capacity normalized BTD requirements are set to d 0, d, d 3 5 respectiely. The Nash equilirium was soled ia the algorithm included in Section V-A. The conergence is reached within 3 rounds of iterations under the parameters η β 0.5, V, σ 0., ɛ 0.0. The results shown in Figure 5 exhiit dashed lines corresponding to the theoretical carried load gains in the saturated regime. As can e seen, these coincide with the Nash equiliria of the simulated traffic shaping games for high arrial rates. For lower arrial rates the gains can e much higher, e.g., almost a factor of.6, for slices with non-uniform moility patterns. This was to e expected since for lower loads we expect higher statistical multiplexing gains from sharing, and thus relatiely higher carried loads to e admitted. For ery low loads, as expected, there are no gains since all traffic can e admitted and BTD constraints are met. Also shown in Figure 5(sufigure) is the degree to which the relatie loads of slices, and the weighted aggregate traffic on the network g are alanced, as measured y, as the arrial rates on the network increase. As expected, ased on Theorem 5, as arrials increase relatie loads of slices and the network ecome more alanced, showing the compromise the traffic shaping game is making, alancing slices relatie loads and that of the oerall network. VII. CONCLUSIONS This paper has thoroughly explored a relatiely simple and natural approach for resource sharing amongst network slices SCPF which corresponds to socially optimal allocations in a Fisher market. Our analysis of performance in settings where slices support stochastic loads proides explicit formulas for (i) the performance gains one can expect oer SS and GPS, (ii) how to dimension slice shares to meet performance ojecties, and (iii) how to go aout performance management through admission control. If dynamic resource sharing amongst network slices is to e adopted, the aility to realize disciplined engineering and performance prediction will e the key. Our analysis of SCPF seems to meet these requirements and at the same time reeals some intriguing insights regarding the load interactions in such sharing models, in particular the impact of relatie load distriutions on statistical multiplexing, and the role of traffic shaping in optimizing admission control. Finally, we note that our approach to admission control in an SCPF shared system is noel in that each slice exploits knowledge of its customers moility patterns to optimize its carried load and assure serice continuity. REFERENCES [] Telecommunication management; Network sharing; Concepts and requirements. 3GPP TS 3.30,.0.0, Dec. 04. [] Serice aspects; Serice principles. 3GPP TS.0, 4.0.0, Jun. 05. [3] Eoled Uniersal Terrestrial Radio Access (E-UTRA); Physical layer procedures. 3GPP TS 36.3,.5.0, Rel., Mar. 05. [4] F. Baccelli and P. Brémaud. Palm proailities and stationary queues, olume 4. Springer Science & Business Media, 0. [5] A. Banchs. User fair queuing: fair allocation of andwidth for users. In IEEE INFOCOM, olume 3, pages IEEE, 00. [6] C. J. Bernardos et al. An architecture for software defined wireless networking. IEEE Wireless Communications, (3):5 6, 04. [7] T. Bonald and L. Massoulié. Impact of fairness on Internet performance. In ACM SIGMETRICS, olume 9, pages 8 9, 00. [8] T. Bonald and A. Proutiere. Insensitie andwidth sharing in data networks. Queueing systems, 44():69 00, 003. [9] S. Brânzei, Y. Chen, X. Deng, A. Filos-Ratsikas, S. K. S. Frederiksen, and J. Zhang. The fisher market game: Equilirium and welfare. 04.