Admission Control and Resource Reservation for Prioritized ...

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Admission Control and Resource Reservationfor Prioritized Slice Requests with Guaranteed SLA

under UncertaintiesQuang-Trung Luu, Sylvaine Kerboeuf, and Michel Kieffer

Abstract—Network slicing has emerged as a key conceptin 5G systems, allowing Mobile Network Operators (MNOs)to build isolated logical networks (slices) on top of sharedinfrastructure networks managed by Infrastructure Providers(InP). Network slicing requires the assignment of infrastructurenetwork resources to virtual network components at slice ac-tivation time and the adjustment of resources for slices underoperation. Performing these operations just-in-time, on a best-effort basis, comes with no guarantee on the availability of enoughinfrastructure resources to meet slice requirements.

This paper proposes a prioritized admission control mecha-nism for concurrent slices based on an infrastructure resourcereservation approach. The reservation accounts for the dynamicnature of slice requests while being robust to uncertainties inslice resource demands. Adopting the perspective of an InP,reservation schemes are proposed that maximize the numberof slices for which infrastructure resources can be grantedwhile minimizing the costs charged to the MNOs. This requiresthe solution of a max-min optimization problem with a non-linear cost function and non-linear constraints induced by therobustness to uncertainties of demands and the limitation ofthe impact of reservation on background services. The cost andthe constraints are linearized and several reduced-complexitystrategies are proposed to solve the slice admission controland resource reservation problem. Simulations show that theproportion of admitted slices of different priority levels can beadjusted by a differentiated selection of the delay between thereception and the processing instants of a slice resource request.

Index Terms—Network slicing, resource reservation, priori-tized slice processing, slice admission control, uncertainty, wire-less network virtualization, 5G

I. INTRODUCTION

IN the fifth-generation communication systems [1, 2], Net-work slicing (NS) aims at replacing the traditional one-

size-fits-all network architecture. NS may address divergingrequirements imposed by verticals [3] while reducing oper-ational costs [4, 5], thanks to its ability to provide highernetwork flexibility. NS exploits network virtualization to elas-tically allocate and reallocate infrastructure resources tailoredto the time-varying needs of various applications [6, 7]. With

Quang-Trung Luu is with Nokia Bell Labs, 91620 Nozay, France,and also with Laboratoire des Signaux et Systèmes, Paris-Saclay Uni-versity - CNRS - CentraleSupélec, 91192 Gif-sur-Yvette, France (e-mail:[email protected]).

S. Kerboeuf is with NSSR Lab, Nokia Bell Labs, 91629 Nozay, France(e-mail: sylvaine.kerboeuf@ nokia-bell-labs.com).

M. Kieffer is with Laboratoire des Signaux et Systèmes, Paris-SaclayUniversity - CNRS - CentraleSupélec, 91192 Gif-sur-Yvette, France (e-mail:[email protected]).

NS, multiple slices, i.e., customized, isolated, and service-dedicated end-to-end logical networks, can be established andoperated simultaneously on a shared physical infrastructurenetwork, provided by one or several Infrastructure Providers(InPs) [8].

Several authors, see, e.g., [9–12], have recently consideredthe resource allocation problem raised by network slicing. Thisproblem involves efficiently assigning infrastructure networkresources to virtual network components at (or just before)slice activation time and dynamically adjusting these resourcesfor slices under operation to maximize resource utilization andminimize operating costs. With such just-in-time slice man-agement, it is difficult to guarantee the availability of enoughinfrastructure resources at the deployment time and during thelifetime of a slice. Slice admission control mechanisms havetherefore been proposed to prioritize, accept, possibly delay,or even reject demands for slices [13–18].

Network slicing with guaranteed satisfaction of some Ser-vice Level Agreement (SLA) is facilitated by adopting aninfrastructure resource reservation approach, as specified by[19, 20], rather than the just-in-time slice resource man-agement approach considered, e.g., in [21, 22]. Resourcereservation aims at determining whether enough infrastructureresources are (or will be) available to satisfy a slice resourcerequest (feasibility check). The actual resource allocation maybe done later, once the reservation is successful and the slicerequest has been granted.

Nevertheless, slice resource reservation raises several chal-lenging problems: Slice requests are submitted by ServiceProviders (SPs) at different time instants, with various activa-tion delays, life durations, and user demands fluctuating withtime. The variety of services supported by slices induces verydifferent Quality of Service (QoS) requirements [7]. Moreover,various constraints may be imposed by the different networksegments on which slices have to be deployed [12, 14]. Forexample, coverage constraints are imposed by slices involvingthe radio access network [23]. Additionally, several sourcesof uncertainty have to be considered in a reservation ap-proach, e.g., the number of slice users, the hardly predictableuser locations [24], and the time-varying per-user resourcerequirements. Consequently, enough infrastructure resourcesshould be reserved for each slice to guarantee an adequateQoS specified in the SLA and provide robustness againstuncertainties. Too many infrastructure resources should not bereserved too, in order to reduce costs and leave resources toconcurrent slices.

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Contributions. This work considers a model where SPs submitslice service requests (possibly largely before their activation)to some Mobile Network Operator (MNO). The MNO and theInP are considered as two separate entities, possibly belongingto the same company. The MNO evaluates the amount ofresources required to operate each slice efficiently and submitsslice resource reservation requests to an InP. The InP has todetermine whether it is able to book, as much in advanceas possible, enough infrastructure resources to ensure thatthe MNO will have access to enough and properly locatedinfrastructure resources with service characteristics as statedin some SLA. The first contribution of this paper is a sliceadmission control and resource reservation framework able toprovide a probabilistic guarantee related to SLA satisfaction.Overbooking of resources by the InP is allowed, as in [14],but the probability of SLA non-satisfaction is bounded, with acontrolled bound. The proposed method to reserve infrastruc-ture resources for concurrent slices accounts for the dynamicnature of slice requests (including their arrival, activation, anddeactivation times while being robust against the uncertaintiesin the number of users and the amount of resource employedby a typical user). We adopt the perspective of an InP andpropose an approach where the InP tries to find the resourcereservation scheme which maximizes the amount of slicesfor which the reservation is successful while minimizingthe resource operation costs charged to the MNOs. The InPdecides then to accept or reject each slice resource request.

The processing of a slice request submitted largely beforeits activation can be anticipated by the MNO who may checkits feasibility in advance with one or several InPs. The secondcontribution of this paper is a process that anticipates more orless this processing depending on the priority level of the slicerequests. Slices can be admitted, possibly largely before theiractivation, when enough infrastructure resources are reservedfor meeting their QoS requirements. When this condition is notsatisfied, the slice resource request is not granted and MNOsmay address their slice resource request to alternative InPs.The proposed approach is consistent with the 3GPP viewsof the management aspects of network slicing [19, 20]. Theproposed slice reservation and admission control takes place inthe network environment preparation task of the preparationphase. In this phase, the design and capacity planning ofnetwork slices, the on-boarding and evaluation of requirednetwork functions, and the reservation of infrastructure re-sources have to be done before the creation and activation ofnetwork slice instances, which belong to the commissioningand operation phases (see Figure 1).

The third contribution of this paper is to formulate the pro-cessing of concurrent slice resource reservation requests andtheir admission control as a max-min optimization problem.Its solution provides the InP maximum earnings for grantedslice requests and is also appropriate for the MNO in terms ofcharged reservation costs. Reservation and adaptation costs areintroduced to charge variations in slice resource demands. Anonlinear objective function is then obtained. The robustnessto uncertainties in the demands and the limitation of the impactof reservation on background services is considered as in [22].This introduces nonlinear constraints in the max-min opti-

mization problem. After linearizing the cost and some of theconstraints, several reduced-complexity reservation strategiesare proposed to solve the problem of slice resource reservationwith dynamic resource demands.

Preparation

On-boarding

Neworkenvironmentpreparation

Design

Network Slice Instance Life-Cycle

Creation

Commissioning

Operation

Activation

Supervision Reporting

Modification

Deactivation

Decommissioning

Termination

Fig. 1. 3GPP view on network slicing managements aspects [19].

The remainder of this paper is organized as follows. Sec-tion II presents some related work. In Section III, we describethe problem statement, in which the system model is detailed.Section IV introduces the proposed approaches to efficientlyreserve resources for concurrent slices, while being robust tothe dynamic nature of slice requests and to the uncertainties re-lated to infrastructure and slice parameters. Numerical resultsare then provided in Section VI to evaluate the performanceof the proposed reservation and admission control approaches.Finally, Section VII draws some conclusions and perspectives.

II. RELATED WORK

In network slicing, a slice is composed of one or multipleService Function Chains (SFCs) of different types. An SFCconsists of an ordered set of interconnected Virtual NetworkFunctions (VNFs) describing the processing applied to dataflows related to a given service.

Many papers have addressed the problem of slice/SFC re-source allocation with uncertain or time-varying requirementsand available physical resources, see, e.g., [9, 10, 25–28].Conservative strategies allocating resources considering worst-case peak traffic conditions [9, 10] are costly and usuallylead to an inefficient utilization of resources. In [26–28], anadjustable safety factor is considered to give a probabilisticguarantee of resource availability, e.g., ensuring that everyslice benefits from sufficient infrastructure resources with acertain probability. Inspired by the approach in [28], [22] hasextended the approach introduced in [29] to account for theimpact of resource reservation on background services.

The dynamic nature of slice/SFC requests is taken into ac-count in [21, 30–32]. In [30], a dynamic resource allocation forSFCs is investigated. The deployment of newly arrived SFCsand readjustment of in-service SFCs are taken into account. AnInteger Linear Program (ILP) formulation is used to addressthe dynamic deployment problem, aiming at minimizing thecost of VNF deployment and migration. A pre-calculation ofall possible routing paths has to be performed in advance,which requires some computational effort before using thedeployment algorithm. In [21], the adaptive adjustment ofallocated resources of each slice is enabled after each decisiontime period (slicing time). An hybrid slice reconfigurationframework is introduced in [31]. The slice can be reconfiguredeither within small time intervals for individual slices, or

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within large time intervals to readjust resource allocationof multiple slices. A deep-learning approach is adopted in[32] for dynamic slice resource allocation, with the aim tomaximize the long-term revenue of the network provider.Uncertainties related to the slice requests and occupationtime are considered. Nevertheless, slices are regarded as awhole, i.e., not made up of multiple elements (e.g., VNFs),which somewhat over-simplifies the problem of slice resourceallocation.

Slice Admission control (SAC) mechanisms have beendeveloped recently [13–18] to address issues related to theunavailability of enough resources to satisfy all slice requests.A yield-driven approach is proposed in [14], assuming thatthe MNO manages the infrastructure resources and decides toaccept or reject slice requests in order to maximize the revenueobtained from the SPs. Resource overbooking is allowed anda penalty in case of non-satisfaction is considered in theoptimization process. Nevertheless, there is no control onthe level of satisfaction of the SLA requirements imposedby the service provider due to the overbooking possibility.Consequently, a slice request may be granted with a penaltyvery close to the revenue provided by the slice. Slice requestsin [14] are assumed to be processed as they arrive, withoutconsidering any explicit prioritization between requests. Pri-oritized processing results only implicitly from the differencesin the revenue and penalty associated to the various requests.

In [15], SAC is formulated as a boolean linear programand a two-step sub-optimal algorithm based on variants ofthe knapsack problem is proposed to alleviate the complexity.Admission is done for slices with highest profit consideringfirst the RAN and aggregated core network resources. In thesecond step, the core network resources are considered withoutany aggregation to determine whether a slice deployment ispossible.

In [17], SAC and resource allocation are performed jointly,to minimize the power consumption of cloud nodes and net-work bandwidth of the infrastructure provider. Transmissiondelay is accounted in the slice SLA. Some elastic variablesare introduced in an ILP formulation to extend the boundson some constraints. They help determining when resourcesmay be lacking, in which case slices are rejected startingfrom those with the highest requirements in terms of resource.Nevertheless, the dynamics of slice requests (time of arrival,slice duration) and the variation of slice resource demandsduring their life time are not considered in [15] and [17].

The dynamics of slice requests is considered by [18] inthe SAC problem. If not accepted, a request is queued forbeing potentially served later. The case of impatient tenants,who may leave their queues before being served, is takeninto account. Nevertheless, neither the dynamics of resourcedemands within each slice, nor the activation time of a sliceare accounted for. Moreover, infrastructure resources of eachtype are fully aggregated. As opposed to [17] and to ourwork, none of the details about the structure of the slice andof the infrastructure are taken into account in the resourcemodel. Consequently, the proposed mechanism does not allowto reserve nor allocate resource to the slice in addition toadmission control.

Online SAC is considered in [13] and [16] leveraging onmachine learning approaches. The aim is to maximize the rev-enue of the InP while guaranteeing the SLAs of the admittedslices. Both papers focus on radio resources of base stations.In [13], two different types of slices are considered to accountfor elastic and inelastic traffic. An admissibility region isdetermined first, indicating the maximum number of slices thatthe system can support without breaking the SLAs. Both worksformalize the admission control problem into a semi MDP andderived the optimal policy obtained when the request arrivalparameters are known. The approach has a high computationcost and is off-line (requires system parameters to be knowna priori). An alternative Q-learning approach is proposed in[13] to adapt to changing environments while achieving closeto optimal performance. In [16], a deep reinforcement learningmethod is developed to overcome the scalability issue of theQ-learning approach.

These works consider tenants submitting slice requestsfor an immediate deployment, contrary to our work, whereslice requests are assumed to be submitted for an immediateor future deployment, which permits the development of aresource reservation strategy.

III. PROBLEM STATEMENT AND NOTATIONS

A typical network slicing system involves several entities:one or several InPs, MNOs, and SPs (also known as slicetenants), as depicted in Figure 2 [4]. InPs own and managethe wireless and wired infrastructure such as the cell sites,the fronthaul and backhaul networks, and data centers. Sec-tion III-A details the considered model for the infrastructurenetwork. MNOs lease resources from InPs to set up andmanage slices. SPs then exploit the slices supplied by MNOsand provide their customers with the required services runningwithin the slices. An SP forwards a slice service bookingrequest to an MNO within an SLA denoted SM-SLA in whatfollows.

SP InPInPSP

users

...SFC 1SFC 2

......

...

SFC 1

SFC 2...

SM-SLA MI-SLA

MNO

compute of slice resourcereservationresponse to resource

reservation request

compute of slice SFC embedding

... ...

slice service booking request1 slice resource

reservation request2

34response toslice servicebooking request

5

6

slice deployment and activation

7SFCs (to beexploited)8service request9

assigning users to SFCs to provide services

10

Fig. 2. Network slicing entities and their SLA-based relationships.

The SM-SLA describes at a high level of abstractioncharacteristics of the service with the desired QoS. These

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characteristics may be time-varying due, e.g., to user mobility,see Section III-B1. In this paper, one considers SM-SLAscomposed of: (i) the target number of users/devices to besupported by the slice, (ii) a description of the characteristicsof the service and of the way typical user/device employs it,and (iii) a target Service Satisfaction Probability (SSP) psp.Since the target number of users is usually uncertain, it isdescribed by a random variable with a known probabilitymass function (pmf). The MNO has to provide a slice ableto serve user demands with a probability of at least psp. Inaddition, several time intervals may be considered in the SM-SLA, intervals over each of which the target number of users,the service characteristics, and the probability of satisfactionare assumed invariant. They may vary from a time intervalto the next one. These time intervals translate, e.g., day andnight variations of user demands and last between tens ofminutes to hours. It is the responsibility of the SP and MNO toproperly scale the requirements expressed in the SM-SLA byconsidering, for example, similar services deployed in the past.Following the 3GPP approach (cf. Figure 4.8.1 of [19]), theMNO is in charge of the slice admission control via assessingthe feasibility of the SP’s request.

The MNO translates the SP high-level demands into SFCsable to fulfill the service requirements. Based on the character-istics of the service and of its usage, the MNO describes howa given user/device consumes the slice (SFCs) resources. Tocharacterize the variability over time and among users of thesedemands, we assume that the MNO considers a probabilisticdescription of the consumption of slice resources by a typicaluser.

In what follows, one assumes that the MNO and the InPare two distinct entities (possibly belonging to a single stake-holder but having their domain responsibilities, like, e.g., twobusiness divisions of the same organization). In this case, theMNO submits resource reservation requests to one or severalInPs upon the arrival of slice booking requests. Section III-Bprovides a model of the requests for slice resource reservationsent by the MNO to an InP and of their associated costs.Each resource reservation request contains the description ofthe resource demand characteristics (the SSP constraint istranslated into deterministic requirements, see Section IV-E)as well as the slice priority class as part of an SLA betweenthem (MI-SLA), see Section III-B2.

Each InP, considering the various slice resource reservationrequests received during some time interval, tries to maximizesthe number of slices for which the reservation can be satisfied.Costs induced by the variation with the time of the resourcereservation request are taken into account by the InP, seeSection III-B3. A resource reservation request for a slice isconsidered satisfied when i) enough resources are availableto meet a target resource requirements and ii) the ImpactProbability (IP) on other best-effort services running on theinfrastructure network remains below some threshold pim, seeSection III-C. The slice priority level is taken into accountwhen processing the reservation requests. The InP answerspositively or negatively to a reservation request. In the lattercase, the MNO may contact alternative InPs, or, when no InPhas enough available resources, the MNO may reject the slice

service booking request from the SP or ask the SP to updateits slice service request (SM-SLA negotiation).

Table I summarizes the main notations introduced in thispaper.

TABLE ITABLE OF MAIN NOTATIONS

INFRASTRUCTURE NETWORK

G Infrastructure network graph, G = (N , E)N Set of infrastructure nodesE Set of infrastructure linksΥ Set of node resource types, Υ = {c,m,w}

an (i) Available resource of type n ∈ Υ at node iab (ij) Available bandwidth of link ijcn (i) Per-unit cost of resource of type n ∈ Υ for node icb (ij) Per-unit cost for link ijcf (i) Fixed cost for using node ica (i) Reservation adaptation cost at node i

TEMPORAL NOTATIONS

Pk Processing time interval in time slot kT Duration of a time slotεT Processing duration (of Pk)

SLICE REQUESTS AND RESOURCE DEMANDS

Gs SFC graph of slice s, Gs = (Ns, Es)Ns Set of virtual network functionsEs Set of virtual linksP cs Priority class

Ps,k Priority level at time kKs Slice active interval, Ks =

[kons , k

offs

]rs Vector of resource demands of an SFCUs,k Vector of resource demands of a typical

user in time slot kRs,k Vector of aggregate resource demands

in time slot kBk Vector of resources consumed by background

services in time slot kU , R, B Mean value of U , R, BU , R, B Standard deviation of U , R, Bpsps

Required service satisfaction probabilitypim Slice impact probability threshold

(w.r.t. background services)Sk Slices requests received before (k + 1)T − εTRk Slices requests processed during Pk

A. Network Model

In this paper, to simplify presentation, one considers an in-frastructure network owned by a single InP. The infrastructurenetwork managed by the considered InP is represented by adirected graph G = (N , E).N is the set of infrastructure nodesand E is the set of infrastructure links, which correspond tothe wired connections between and within nodes (loop-backlinks) of the infrastructure network.

Each infrastructure node i ∈ N is characterized by its com-puting ac (i), memory am (i), and wireless aw (i) resources.For each node i, the InP charges the MNO a fixed cost cf (i)for node disposal (paid for each slice using node i), and per-unit variable costs cn(i), n ∈ Υ = {c,m,w}, which dependlinearly on the amount of resources provided by that node.

Similarly, each infrastructure link ij ∈ E connecting node ito j is characterized by its bandwidth ab (ij), and an associatedper-unit bandwidth cost cb(ij). Several distinct VNFs of thesame slice may be deployed on a given infrastructure node.When communication between these VNFs is required, an

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internal (loop-back) infrastructure link ii ∈ E can be usedat each node i ∈ N , as in [33], in the case of interconnectedvirtual machines (VMs) deployed on the same host. In thatcase, the InP charges the MNO per-unit bandwidth cost cb (ii).

B. Slice Resource Reservation Requests and Adaptation Costs

1) Request Arrivals: One considers that time is slotted intoslots of constant duration T (typically of few tens of minutes),which represents the time unit considered for the slice resourcereservation duration in the booking calendar. The slot of indexk ∈ N lasts over the time interval [kT, (k + 1)T [. Oneconsiders that the slice lifetime spans over one or severaltime slots of duration T . Resources have to be reserved soas to be compliant with the variations of the number of usersand of their demands during the slice lifetime. The servicecharacteristics are assumed stable over each time slot, andmay vary from one time slot to the next.

Let ts be the time instant at which the reservation request fora slice s is received by the InP. This slice is also characterizedby the index kon

s of the time slot at the beginning of which ithas to be activated (put into service), and the index koff

s of thetime slot at the end of which it has to be deactivated. Thus,the slice s is active over the time interval

[kons T,

(koffs + 1

)T[.

Figure 3 depicts an example of arrivals of slice resourcereservation requests, as well as the time slots over which thecorresponding services have to be active.

1T 2T 3T 4T 5T 6T 7T 8T 9T 10T 11T 12Ttime instant

0

0.5

1

slic

e 1

0

0.5

1

slic

e 2

0

0.5

1

slic

e 3

Fig. 3. Arrivals of slice resource reservation requests as a function of time;Black arrows represent the arrival times ts of each request; The types of slicesare illustrated by different plot line styles; The slice resource demands evolvewith time; Peak demands have been normalized.

2) Slice Resource Demand: A demand for resources for aslice s is defined on the basis of the translation of the SM-SLAbetween an SP and an MNO. A priority class P c

s determinesthe priority level with which the resource demand has to beprocessed. As in [34], the type of service provided by slice sis used to identify the type of SFC, i.e., the ordered set ofVNFs involved in s. We consider that a slice is devoted toa single type of service supplied by a given type of SFC.Several SFCs of the same type may have to be deployed soas to satisfy user demands within the slice. The topology ofeach SFC of slice s is represented by a graph Gs = (Ns, Es)representing the VNFs and their interconnections. Each virtualnode v ∈ Ns represents a VNF, and each virtual link vw ∈ Esrepresents the connection between virtual nodes v and w.

Based on Gs, one introduces the vectors rs, Us,k, and Rs,k,respectively representing the resource demands of a single

SFC (SFC-RD), of a typical user (U-RD) during time slot k,and the aggregate resource demand of the users of slice s(S-RD) during time slot k.

The deterministic SFC-RD vector

rs = [rs,n (v) , rs,b (vw)]>n∈Υ,(v,vw)∈Gs (1)

gathers the computing (rs,c (v)), memory (rs,m (v)), wireless(rs,w (v)), and bandwidth (rs,b (vw)) resource requirementsof the VNFs v ∈ Ns and the virtual links vw ∈ Es of asingle SFC. The vector rs is assumed to be time invariant,as it characterizes the resources which need to be allocatedto run an instance of the considered SFC. In the consideredreservation context, rs also represents the maximum amountof reserved resources that will be made available to theconsidered SFC.

Each user of slice s is assumed to consume a randomamount of the resources of an SFC of that slice. The randomvector

Us,k = [Us,n,k (v) , Us,b,k (vw)]>n∈Υ,(v,vw)∈Gs (2)

of U-RD represents the resource demands of a single user ofslice s during time slot k. Us,n,k (v), n ∈ Υ, and Us,b,k (vw)are the random amounts of employed resources of VNF v andof virtual link vw. In addition, the resources consumed byvarious users are represented by independently and identicallydistributed random vectors. Minor variations of the user re-source demand within time slot k are accounted for by theprobability distribution characterizing Us,k.

The random S-RD vector

Rs,k = [Rs,n,k (v) , Rs,b,k (vw)]>n∈Υ,(v,vw)∈Gs (3)

gathers Rs,n,k (v), n ∈ Υ, and Rs,b,k (vw), the aggregateamount of resources employed by a random number Ns,kof independent users of slice s during time slot k. Minorvariations of the number of users within time slot k are aswell captured by the probability mass function characterizingNs,k.

The probability distributions characterizing Us,k, Ns,k, andconsequently Rs,k depend on the time slot index k, to repre-sent possible large changes in the U-RD or in the number ofusers of slice s in successive time slots.

One considers, for a typical user and during a given timeslot k, that for each virtual node v ∈ Ns, the resource demandsof different types n ∈ Υ are correlated. A correlation may alsoexist between the demands for resources of the same typeamong virtual nodes. Finally, the resulting traffic demandsbetween nodes is usually also correlated with the resourcedemands for a given virtual node, as reported in [35]. Consid-ering the U-RD vector Us,k, one assumes that the elementsUs,n,k (v), ∀n ∈ Υ, and Us,b,k (vw) are normally distributedduring the time slot k. Us,k thus follows a multivariate normaldistribution with probability density

f(x;µs,k,Γs,k

)=

(2π)− 1

2 card(Us,k) |Γs,k|−12 e−

12 (x−µs,k)

>Γ−1

s,k(x−µs,k), (4)

where card (Us,k) is the number of elements of Us,k,

µs,k =[Us,n,k (v) , Us,b,k (vw)

]>(v,vw)∈Gs,n∈Υ

(5)

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is its mean value and and Γs,k is its covariance matrix, withdiagonal elements

diag (Γs,k) =[U2s,n,k (v) , U2

s,b,k (vw)]>

(v,vw)∈Gs,n∈Υ, (6)

and off-diagonal elements representing the correlation betweendifferent types of resource demands. One has thus

Us,n,k(v) ∼N(Us,n,k (v) , U2

s,n,k (v)), n ∈ Υ, and

Us,b,k (vw) ∼N(Us,b,k (vw) , U2

s,b,k (vw)). (7)

The probability that the number of users Ns,k to be sup-ported by slice s in the k-th time slot is equal to η is

ps,k,η = Pr (Ns,k = η) , η ∈ N. (8)

The amount of resources of the VNF v and of the vir-tual link vw consumed by different users is representedby independently and identically distributed copies of Us,k.Consequently, the joint distribution of the aggregate amountUs,η,k of resources consumed by η independent users isf(x, ηµs,k, η

2Γs,k). The total amount of resources employed

by a random number Ns,k of independent users, Rs,k =

Us,Ns,k,k = (Rs,n,k (v) , Rs,b,k (vw))>n∈Υ,(v,vw)∈Gs , is dis-

tributed according to

g(x,µs,k,Γs,k

)=

∞∑η=0

ps,k,ηf(x, ηµs,k, η

2Γs,k). (9)

3) Adaptation Costs to Request Variations: During thelifetime of a slice, the amount of required slice resourcesmay evolve from one time slot to another. An increase of therequired resources may impact the resource allocation schemeby requiring more infrastructure resources to be allocated.Compared to a situation where the resource allocation is staticfor the whole lifespan of a slice, this induces more operationsto be performed on the network infrastructure (assignmentor re-assignment of resources, launching virtual machines orcontainers on which VNFs will be operated) and results inadditional costs to the InP. The anticipation of those allocationadaptation costs are then considered when processing theslice resource reservation request. A cost ca (i) for each unitincrease of the amount of instances (i.e., virtual machines orcontainers) of a VNF between two time slots is assumed to becharged by the InP to the MNO. Resource release costs areassumed to be incorporated within ca (i).

As will be seen in Section IV-F, this cost reduces SFCmigrations within a given slice between consecutive time slots.

C. Resource Consumption of Background Services

In a given time slot k, we assume that infrastructureresources are partly consumed by best-effort background ser-vices for which no resource reservation has been performed.One denotes

Bk = [Bn,k (i) , Bb,k (ij)]>(i,ij)∈G,n∈Υ (10)

the vector gathering all resources consumed by backgroundservices during time slot k. The elements Bc,k (i), Bm,k (i),Bw,k (i), ∀i ∈ N , and Bb,k (ij), ∀ij ∈ E of Bk are random

variables representing the aggregate amount of computing,memory, wireless, and bandwidth resources consumed bythese best-effort services. As in [22], each of those variablesis assumed to be uncorrelated and Gaussian distributed,

Bn,k (i) ∼N(Bn,k (i) , B2

n,k (i)),∀i ∈ N ,∀n ∈ Υ, and

Bb,k (ij) ∼N(Bb,k (ij) , B2

b,k (ij)),∀ij ∈ E . (11)

Consequently, during each time slot k, Bk is distributedaccording to f

(x;µB,k,ΓB,k

), with

µB,k =[Bn,k (i) , Bb,k (ij)

]>(i,ij)∈G,n∈Υ

, (12)

ΓB,k = diag[B2n,k (i) , B2

b,k (ij)]

(i,ij)∈G,n∈Υ. (13)

The evolution of resources consumed by background servicesover time slots may be predicted by the InP from pastobservations, see, e.g., [36]. The smaller variations within eachtime slot are taken into account in the probability distribution.

IV. SLICE RESOURCE RESERVATION APPROACHES

Taking the InP perspective, slice resource reservation aimsat booking, somewhat in advance, enough infrastructure re-sources to ensure that the MNOs will be able to provide sliceswith characteristics as stated in the SM-SLA. For that purpose,the InP has to identify i) the infrastructure nodes which willprovide resources for the future deployment of VNFs and ii)the links able to transmit data between these nodes/VNFs,while respecting the structure of SFCs. This correspond tothe network environment preparation block represented inFigure 1. Within some time slot over which a slice s is active,the slice resource reservation can be represented by a mappingbetween the infrastructure graph G and the S-RD graph Gs asillustrated in Figure 4. In this example, the slice s consists ofseveral linear SFCs of the same type for which resources havebeen reserved from some infrastructure network.

virtual graph

infrastructure graph

Fig. 4. Reservation of infrastructure resources for slice s: Resources fromthe infrastructure node i1 and the aggregate resources from the infrastructurenode pairs (i2, i4) and (i3, i5) are respectively reserved for the virtualnodes v1, v2, and v3. Correspondingly, resources of the infrastructure linkpairs (i1i2, i1i4) and (i2i3, i4i5) (highlighted by the bold lines) are reservedfor the virtual links v1v2 and v2v3.

The mapping between G and Gs may evolve between suc-cessive time intervals due to the evolution of the characteristicsof the MI-SLA for slice s, to the arrival of new slice resourcereservation requests, and to resources released by terminatedslices.

PREPRINT 7

A. Prioritized Processing of Resource Reservation Requests

Several strategies for processing resource reservation re-quests can be considered to account for their dynamicity. Afirst approach consists in processing the resource reservationrequests as soon as they are submitted by an MNO. The advan-tage is to immediately indicate to the MNO whether enoughinfrastructure resources are available to satisfy the request. Asecond approach is to wait some time and process severalrequests simultaneously. This second approach, considered inthis paper, helps process the resource reservation requests sincethe InP has a better view of concurrent requests. Additionally,by a slice-priority differentiated processing delay of the slicereservation request a compromise between response delay tothe MNO and optimization of the reservation of resources bythe InP can be found.

When processing a new request, already granted resourcesrequests may be adjusted. This update possibility gives moredegrees of freedom to the InP to satisfy new requests, butcomes at the price of higher computational complexity. Up-dates must be done while satisfying previous requests whichhave been indicated to the MNOs as granted. In this paper,we have chosen not to change any assignment of previouslysuccessfully processed slice requests.

Independently of the chosen strategy, the InP has to accountfor the time required for processing the resource reservationand performing the slice deployment and activation (lastingfew minutes, as indicated in [37]). Consequently, resourcereservation requests for slices to be activated at (k + 1)Tshould reach the InP before (k + 1)T − εT , where εT ,ε ∈ ]0, 1[, is an upper bound of the time required for the sliceresource request processing operations, the slice activation andupdates.

Let Sk be the set of slices whose resource reservationrequests have been received before (k + 1)T − εT . A flagfs ∈ {0, 1} indicates for each slice s ∈ Sk whether the requesthas been processed (fs = 1) (granted or denied) or is still tobe processed (fs = 0).

In what follows, we consider two classes of slices, namelyPremium and Standard. The priority level is indicated bythe MNO to the InP in the MI-SLA of the slice. Eachslice request, when received for the first time in the intervalTk = [kT − εT, (k + 1)T − εT [, gets fs = 0, and is assigneda priority level Ps,k ∈ R depending on its class

Ps,k =

Pmax for Premium slices,0 for Standard slice, if kon

s > k + 1,

Pmax − 1 for Standard slice, if kons = k + 1.

(14)

Standard slice requests, which have to be activated in the nexttime slot, get thus a higher priority level. Then, only sliceswhose priority level is above a certain threshold

Pthres = α (Pmax − 1) , with α ∈ [0, 1] , (15)

are processed in the time interval Pk =[(k + 1)T − εT, (k + 1)T [ of duration εT . The set ofslices whose resource request has to be processed during thetime interval Pk is

Rk , {s ∈ Sk : fs = 0, Ps,k > Pthres} . (16)

Once the resource request of a slice in Rk is processed, its flagis set to fs = 1. All standard slice requests with Ps,k < Pthres(pending requests) are delayed and may be processed in thenext time interval Pk+1. Their priority is updated as

Ps,k+1 =

{min {Ps,k + ∆P, Pmax − 1} if kon

s > k + 2,

Pmax − 1 if kons = k + 2,

(17)

where ∆P > 0 is some priority increment. When several slicesof equal priority have to be processed in a given time slot,a possible choice, adopted in this paper, is to process firstthose who have to be activated first, then those who have beensubmitted first. Premium slices are always processed first. Theprocessing delay of Standard slice requests depends thus onα and ∆P . Deferring more the processing of Standard slicerequests gives more chance to satisfy Premium slice requests.

When α = 0, whatever the value of ∆P , all slicesresource reservation requests received in the time interval Tkare processed, starting from the Premium slices, with the riskof having no resources available for Premium slice requestsreceived in the few next time slots. This corresponds to theas-they-arrive processing approach, considered, e.g., in [14].

When α = 1 and ∆P = 0, the processing of Standardslice resource reservation requests is delayed until the timeslot preceding their activation, leaving a maximum amountof resources available for Premium slice resource reservationrequests. Standard slice requests are always processed just-in-time, while processing of Premium requests is anticipated.

Figure 5 illustrates a scenario taking place during theprocessing time interval Pk when the processing of Standardslice requests is maximally delayed (α = 1 and ∆P = 0).The three slice requests s1, s2, and s3 in Sk are assumed stillto be processed. The slice request s4 arrives within Pk andwill thus be considered in Pk+1. Among the slices s1, s2, ands3, only s3 is Premium (the time instant at which the resourcereservation request is submitted is indicated by a solid arrow),and is therefore processed in Pk. The slice requests s1 ands2 are Standard (reservation request time instants indicatedby dashed arrows). They have to be active in the time slotsKs1 =

[kons1 , k

offs1

]and Ks2 =

[kons2 , k

offs2

]. Since kon

s1 = k + 1and kon

s2 = k + 2, only s1 is processed in Pk. Finally, the setof slice resource reservation requests to be processed in Pk isRk = {s1, s3} (highlighted by red arrows).

...

...

... ...

slice set standardpremium

kT (k+1)T (k+2)Tk k+1 k+2

timeinstants

time slots

Fig. 5. Time slots, arrival times of the slice resource reservation requests,and time intervals during which the reservation is processed.

B. Decision Variables

Processing a resource reservation request for some slice s ∈Rk amounts to defining a mapping κs,` between the graphsG = (N , E) and Gs = (Ns, Es) for each time slot ` ∈ Ks ,

PREPRINT 8

[kons , k

offs

]during which the slice s is active. This mapping

describes i) the number κs,` (i, v) ∈ N of VNF instances oftype v ∈ Ns for which node i ∈ N will reserve resources, andii) the number κs,` (ij, vw) ∈ N of links vw ∈ Es betweenVNF instances for which the InP will reserve resources onthe infrastructure link ij ∈ E , both in time slot `. The amountof resource of type n ∈ Υ reserved by node i for a VNFinstance of type v is κs,` (i, v) rs,n(v). The bandwidth reservedon link ij to support the traffic between two virtual nodes oftype v and w is represented κs,` (ij, vw) rs,b(vw).

The mapping κs,` is thus defined as

κs,` =

{κs,` (i, v) ,κs,` (ij, vw)

}(i,ij)∈G,(v,vw)∈Gs

(18)

for each ` ∈ Ks. By convention, κs,` = 0 when ` /∈ Ks.Moreover, one introduces κs = {κs,` : ` ∈ Ks} to gather allassignments performed for the slice s.

Enough infrastructure resources are not always available fora given slice s ∈ Rk. The binary decision variable ds indicateswhether all conditions are met to satisfy the resource reserva-tion request for slice s and consequently whether resourcesare actually reserved for slice s (ds = 1) or not (ds = 0).These conditions are detailed in the following sections.

Consequently, the set of variables which have to be assignedby the InP in the processing time interval Pk are

dRk= {ds : s ∈ Rk} , and (19)

κRk= {κs,` : s ∈ Rk, ` ∈ Ks} . (20)

The vector dRkindicates which slice resource reservation

requests in Rk have been granted, and κk describes the wayresources have been reserved by the InP.

C. Constraints

During the processing time interval Pk of time slot k, theInP has to account for all resource reservation requests ofslices s ∈ Sk-1 which have been previously processed, i.e.,with fs = 1. The set of these slices is denoted as

Spk-1 = {s ∈ Sk-1 : fs = 1} . (21)

Moreover, the mappings κs,` for all slices s ∈ Rk have tosatisfy some constraints to ensure that i) enough resources arereserved to properly deploy the SFCs and ii) the SSP psp

sis

reached. These constraints have to be satisfied for all time slotsduring which the slice is active. The InP has also to keep theimpact probability on background services below pim. Theseconstraints are described in what follows.

The total amount of resources reserved (and allocated forslices in service) by each infrastructure node i ∈ N and eachinfrastructure link ij ∈ E for all slices s ∈ Rk has to be lessthan their available resources, see Section III-A. Consequently,the following constraints have to be satisfied, for each ` =mins∈Rk

{kons } > k, . . . ,maxs∈Rk

{koffs

},∑

s∈Rk

∑v∈Ns

κs,` (i, v) rs,n (v) 6 an (i)−∑s∈Sp

k-1

∑v∈Ns

κs,` (i, v) rs,n (v) ,∀i ∈ N , n ∈ Υ, (22)

∑s∈Rk

∑vw∈Es

κs,` (ij, vw) rs,b (vw) 6 ab (ij)−∑s∈Sp

k-1

∑vw∈Es

κs,` (ij, vw) rs,b (vw) ,∀ij ∈ E . (23)

In (22) and (23), the right-hand sides of the inequalitiesrepresent the remaining part of the resources once previousresource reservation requests have been processed. Whenupdates for granted slice requests are allowed, κs,`, s ∈ Sp

k-1are considered as variables, but not ds, s ∈ Sp

k-1, since thestatus of successfully processed resource reservation requestsshould not be changed. In what follows, one considers thatsuch updates are not allowed.

The inequalities (22) and (23) may be more compactlywritten for ` > k as follows∑

s∈Rk∪Spk-1

∑v∈Ns

κs,` (i, v) rs,n (v) 6 an (i) , (24)

∑s∈Rk∪Sp

k-1

∑vw∈Es

κs,` (ij, vw) rs,b (vw) 6 ab (ij) . (25)

The conditions (22) and (24) are equivalent, as κs,` = 0 when` /∈ Ks. The same is also true for conditions (23) and (25).

For each virtual link vw ∈ Es, resources on a sequenceof infrastructure links must be reserved between each pair ofinfrastructure nodes that have reserved resources to the virtualnodes v and w. This leads to the following flow conservationconstraint, for each ` ∈ Ks, s ∈ Rk, i ∈ N , and vw ∈ Es,∑j∈N

[κs,` (ij, vw)− κs,` (ji, vw)] =(rs,b(vw)∑

vu∈Esrs,b(vu)

)κs,` (i, v)−

(rs,b(vw)∑

uw∈Esrs,b(uw)

)κs,` (i, w) ,

(26)

which is similar to that introduced in [34]. More specifically,(26) imposes that the reserved bandwidth is commensuratewith the reserved node resources. This allows the MNO to findan appropriate VNF embedding and chaining solution. Finally,when the SFC embedding is performed, each SFC node willbe mapped onto one single node, and each SFC link will bemapped onto one single path. SFCs are not allowed to bemapped towards multiple physical paths.

D. Demand Satisfaction and Impact Probabilities

An assignment κs,`, ` ∈ Ks of a given slice s ∈ Rk, whichsatisfies (22)–(26), has to ensure a SSP above psp

sfor all time

slots ` ∈ Ks during which slice s is active. As introduced inSection III, the SSP for a slice is the probability that resourcesreserved for the deployment of the SFCs of that slice will meetthe demand of users of the service. This leads to the followingconditions, for all ` ∈ Ks and s ∈ Rk,

ps,` (κs,`, ds) > psps, (27)

PREPRINT 9

where

ps,` (κs,`, ds) =

Pr{ ∑i∈N

κs,` (i, v) rs,n (v) > dsRs,n,` (v) ,∀v ∈ Ns, n ∈ Υ,∑ij∈E

κs,` (ij, vw) rs,b (vw) > dsRs,b,` (vw) ,∀vw ∈ Es}.

(28)

The variable ds in (28) is introduced to cancel the SSPconstraint when the request for a slice s ∈ Rk is not granted(ds = 0). In this case, as described later in Section IV-F, theminimization of the cost function yields κs,` = 0, ∀` ∈ Ks,which satisfies (28), thus avoiding the reservation of anyresource. The evaluation of (28) is detailed in Section C.

The constraints (22) and (23) ensure that the resourcesreserved for the slices s ∈ Rk are less than the availableresources in each infrastructure node i ∈ N and infrastructurelink ij ∈ E . Nevertheless, for some s ∈ Rk, the assignmentsκs,`, ` ∈ Ks evaluated in the processing time interval Pk, tak-ing into account all previously processed reservation requests(described by κSk−1

= {κs,` : s ∈ Sk−1, ` ∈ Ks}), may besuch that in time slot ` > kon

s , not enough resources are left forthe background best-effort services described in Section III-C,and may then significantly affect such services. Consequently,in the processing time interval Pk, when evaluating κRk

, oneshould have, for all ` > k,

pimn,` (κRk

, i) 6 pim,∀n ∈ Υ,∀i ∈ N , (29)

pimb,` (κRk

, ij) 6 pim,∀ij ∈ E , (30)

where

pimn,` (κRk

, i) = Pr{ ∑s∈Rk∪Sp

k-1

∑v∈Ns

κs,` (i, v) rs,n (v)

> an (i)−Bn,` (i)}, (31)

pimb,` (κRk

, ij) = Pr{ ∑s∈Rk∪Sp

k-1

∑vw∈Es

κs,` (ij, vw) rs,b (vw)

> ab (ij)−Bb,` (ij)}. (32)

As indicated before, the evaluations of (31) and (32) for all ` >k involve κSp

k-1which have already been evaluated in previous

processing time intervals and are considered as constants inthe current processing time interval, i.e., Pk. The dependencyin κSp

k-1of pim

n,` and pimb,` is omitted to lighten notations.

The constraints (29) ensure that the reserved resourceshave a limited impact on background services at each nodei ∈ N . The constraints (30) have the same role for theinfrastructure links. The value of pim is chosen by the InPto provide sufficient resources for the background servicesat every infrastructure nodes and links. A small value ofpim leads to a small impact of reserved resources for sliceson background services, but makes the resource reservationproblem more difficult compared to the case of pim close toone, where the impact on background service is less taken intoaccount. The InP, by adjusting the impact probability thresholdpim, can trade the revenues provided by slices and thoseprovided by the background services. For example, choosingpim = 1 may leave no resources for background services.

E. Relaxation of Probabilistic Constraints

This section introduces the relaxation of the probabilisticconstraints (27), (29), and (30). These constraints are nonlineardue to the need to evaluate ps,` (κs,`, ds), pim

n,` (κRk, i), and

pimb,` (κRk

, ij) with (28), (31), and (32). Using the approachintroduced in [22], the MNO translates the SSP constraint (27),for all s ∈ Rk and ` ∈ Ks, into the following lineardeterministic constraints∑

i∈Nκs,` (i, v) rs,n (v) > dsRs,n,` (v, γs,`) ,∀v, n, (33)∑

ij∈Eκs,` (ij, vw) rs,b (vw) > dsRs,b,` (vw, γs,`) ,∀vw, (34)

where

Rs,n,` (v, γs,`) = Rs,n,` (v) + γs,`Rs,n,` (v) , (35)

Rs,b,` (vw, γs,`) = Rs,b,` (vw) + γs,`Rs,b,` (vw) , (36)

are the target aggregate user demands, depending on someparameter γs,` > 0. Rs,n,` (v) and Rs,n,` (v) are the meanand standard deviation of Rs,n,` (v), while Rs,b,` (vw) andRs,b,` (vw) are the mean and standard deviation of Rs,b,` (vw).These quantities are evaluated by the MNO. Appendix Adetails this evaluation when when the number of users Ns,` ofslice s at time slot ` is described by a binomial distribution.Appendix B describes the choice of γs,` such that the satisfac-tion of (33, 34) implies that of (27). This way, the MNO cancontrol the bound of the probability of SLA non-satisfaction.The quantities Rs,n,` and Rs,b,` are transmitted by the MNOto the InP as part of the MI-SLA.

Similarly, the InP translates the IP constraints (29, 30),∀ (i, ij) ∈ G and ∀n ∈ Υ, into∑s∈Rk∪Sp

k-1,v∈Ns

κs,` (i, v) rs,n (v) 6 an (i)− Bn,` (i, γB,`) , (37)

∑s∈Rk∪Sp

k-1,vw∈Es

κs,` (ij, vw) rs,b (vw) 6 ab (ij)− Bb,` (ij, γB,`) ,

(38)

for ` > k, where

Bn,` (i, γB,`) = Bn,` (i) + γB,`Bn,` (i) , (39)

Bb,` (ij, γB,`) = Bb,` (ij) + γB,`Bb,` (ij) (40)

are the considered target level of background service demands.The parameter γB,` > 0 has to be chosen such that thesatisfaction of the constraints (37, 38) implies the satisfactionof the IP constraints (29, 30), see Appendix C for more details.Moreover, if the constraints (37, 38) are satisfied by someassignment κRk

, then the conditions (24) and (25) are alsosatisfied.

F. Costs and Incomes

Consider the processing time interval Pk during whicha resource reservation scheme for all slices s ∈ Rk hasto be evaluated. This amounts at evaluating dRk

and theassignments κRk

.

PREPRINT 10

The costs charged by the InP to the MNO for a reservationscheme for slice s ∈ Rk described by κs,` in time slot ` arespread between node and bandwidth resource reservation costs

Cr (κs,`) =∑i∈N

∑v∈Ns

∑n∈Υ

κs,` (i, v) rn (v) cn (i)

+∑ij∈E

∑vw∈Es

κs,` (ij, vw) rb (vw) cb (ij) (41)

as well as fixed node disposal costs

Cf (κs,`) =∑i∈N

κs,` (i) cf (i) (42)

for the infrastructure nodes used, where

κs,` (i) =

{1 if

∑v∈Ns

κs,` (i, v) > 0

0 otherwise,(43)

indicates whether node i is used by slice s in time slot `.Additionally, when the amount of reserved resources for

slice s increases during two consecutive time slots, resourcevariation adaptation costs are also charged by the InP to theMNO

Ca (κs,`,κs,`−1)

=∑i∈N

∑v∈Ns

max {κs,` (i, v)− κs,`-1 (i, v) , 0} ca (i) , (44)

see Section III-B3.Once a reservation request for a slice s ∈ Rk has been

granted by the InP, the MNO will be able to deploy the slice(see the commissioning and operation blocks of Figure 1)and receives from the SP some income Is depending on thecomplexity and of the load of the slice.

G. Optimization Problem

For a given assignment κRk, the earnings of the InP are the

costs charged to the MNOs

EInPk (κRk

) =∑s∈Rk

∑`∈Ks

(Cr (κs,`) + Cf (κs,`)

+Ca (κs,`,κs,`−1)) . (45)

The InP may be interested in an assignment κRkthat max-

imizes EInPk (κRk

). Nevertheless, such assignment would bedetrimental for the earnings of the MNOs expressed as

EMNOk (κRk

) =∑s∈Rk

dsIs − EInPk (κRk

) . (46)

Consequently, MNOs may not be interested by InPs applyingan optimization strategy trying to maximizing EInP

k (κRk).

Alternatively, the InP may try to find an assignment whichmaximizes EMNO

k (κRk). This approach reduces the per-slice

income for the InP, but allows more slice resource reservationrequests to be granted. Nevertheless, InPs are usually unawareof the income Is obtained by the MNOs from the SP, thereforeEMNOk (κRk

) cannot be evaluated by the InP.Consequently, we will consider an approach where the

InP tries, for a given dRk(which determines the number

of accepted slices), to find an assignment κRkminimizing

the reservation costs charged to the MNO. Moreover, the

InP also tries to maximize, with respect to dRk, its earning

minκRkEInPk (κRk

). This approach leads to a max-min op-timization problem. Its solution provides the InP maximumearnings for granting slice requests and is also appropriate forthe MNO in terms of charged reservation costs. Moreover, thisapproach potentially saves infrastructure resources to satisfyfuture slice resource reservation requests.

Consequently, the joint reservation of resources for allslices s ∈ Rk during the processing time interval Pk isformulated as Problem 1.

Problem 1: Max-Min Joint Slice Resource Reservation

maxdRk

minκRk

EInPk (κRk

)

s.t.∀` > k :∑i∈N

κs,` (i, v) rs,n (v) > dsRs,n,` (v, γs,`) ,∀n ∈ Υ, v ∈ Ns,∑ij∈E

κs,` (ij, vw) rs,b (vw) > dsRs,b,` (vw, γs,`) ,∀vw ∈ Es,

and s.t. ∀` > k,∀i ∈ N :∑s∈Rk∪Sp

k-1,v∈Ns

κs,` (i, v) rs,n (v) 6 an (i)− Bn,` (i, γB,`) ,

and s.t. ∀` > k,∀ij ∈ E :∑s∈Rk∪Sp

k-1,vw∈Es


∑j∈N


vu∈Esrs,b(vu)

)κs,` (i, v)−

(rs,b(vw)∑

uw∈Esrs,b(uw)

)κs,` (i, w) ,

∀s ∈ Rk, i ∈ N ,∀vw ∈ Es.

The max-min optimization makes Problem 1 difficult tosolve. This motivates us to develop some efficient heuristicsin Section V.

V. SLICE RESOURCE RESERVATION ALGORITHMS

In this section, two heuristics are introduced to provideapproximate solutions to Problem 1, performing either jointor sequential slice resource reservation.

A. Linearization of the Cost Function

In (45), the term Ca (κs,`,κs,`−1) makes the objectivefunction nonlinear. To address this issue, consider the set ofvariables

ys = {ys,` (i, v) : ` ∈ Ks, i ∈ N , v ∈ Ns} (47)

for each s ∈ Rk, yRk= {ys : s ∈ Rk}, and reformulate the

objective function (45) as

EInPk (κRk

,yRk) =

∑s∈Rk

∑`∈Ks

(Cr (κs,`) + Cf (κs,`)

+∑i∈N

∑v∈Ns

ys,` (i, v) ca (i))

(48)

PREPRINT 11

with the additional constraints, to be satisfied for all s ∈ Rk,` ∈ Ks, i ∈ N , and v ∈ Ns

ys,` (i, v) > κs,` (i, v)− κs,`-1 (i, v) , (49)ys,` (i, v) > 0. (50)

For a given value of dRk, the objective function has now to

be minimized with respect to κs,`, s ∈ Rk, ` ∈ Ks, and yRk.

Moreover, the evaluation of Cf (κs,`) involves κs,` (i)defined in (43). The variable κs,` (i) can be related to∑vκs,` (i, v) using the following linear inequality constraints∑

v

κs,` (i, v) > 0, (51)

κs,` (i)N >∑v

κs,` (i, v) , (52)

where N is an upper bound on the number of VNF instancesof all types for which resources may be reserved by a giveninfrastructure node.

B. Relaxed Joint Max-Min Optimization Problem

Even with the results of Section V-A, the solution of Prob-lem 1 requires addressing a constrained max-min optimizationproblem, which is still quite complex. To address this issue, fora fixed value of dRk

, we introduce the following optimizationproblem.

Problem 2: Joint Slice Resource Reservation Given dRk

minκRk

,yRk

EInPk (κRk

,yRk)

s.t.∀s ∈ Rk,∀` > k :∑i∈N



ys,` (i, v) > κs,` (i, v)− κs,`-1 (i, v) ,∀i ∈ N , v ∈ Ns,ys,` (i, v) > 0,∀i ∈ N , v ∈ Ns,

and s.t. ∀` > k,∀i ∈ N :∑s∈Rk∪Sp

k-1,v∈Ns

κs,` (i, v) rs,n (v) 6 an (i)− Bn,` (i, γB,`) ,

and s.t. ∀` > k,∀ij ∈ E :∑s∈Rk∪Sp

k-1,vw∈Es


∑j∈N


vu∈Esrs,b(vu)

)κs,` (i, v)−

(rs,b(vw)∑

uw∈Esrs,b(uw)

)κs,` (i, w) ,

∀s ∈ Rk, i ∈ N ,∀vw ∈ Es.

One considers then a greedy solution approach to Prob-lem 1, where the slices to be processed inRk = {s1, . . . , sRk

}

are assumed to be ordered, see Section IV-A. When severalMNOs have submitted slice resource reservation requests inthe same time slot, the InP may also prioritize MNOs.

C. Relaxed Single Slice Max-Min Optimization Problem

The number of variables involved in Problem 2 introducedin Section V-B may become relatively large when the resourcereservation is performed simultaneously for several slices. Forthis reason, we introduce a reduced-complexity version ofProblem 2, where the resource reservation is performed sliceby slice. One focuses on a slice s ∈ Rk which resource reser-vation request has to be processed. Some resource reservationrequests for slices s′ ∈ Rk, s′ 6= s may have been previouslyprocessed, in which case, when the request is granted, ds′ = 1and κs′ 6= 0 and when it is not granted, ds′ = 0 and κs′ = 0.For not yet processed requests of slices s′ ∈ Rk, s′ 6= s,one considers ds′ = 0 and κs′ = 0. With these assumptions,reserving resources for slice s ∈ Rk requires the solution ofProblem 3.

Problem 3: Single Slice Resource Reservation

maxds

minκs,ys

EInPk (κs,ys)

s.t. ∀` ∈ Ks :∑i∈N



ys,` (i, v) > κs,` (i, v)− κs,`-1 (i, v) ,∀i ∈ N , v ∈ Ns,ys,` (i, v) > 0,∀i ∈ N , v ∈ Ns,

and s.t. ∀` ∈ Ks,∀i ∈ N :∑s′∈Rk∪Sp

k-1,v∈Ns

κs′,` (i, v) rs′,n (v) 6 an (i)− Bn,` (i, γB,`) ,

and s.t. ∀` ∈ Ks,∀ij ∈ E :∑s′∈Rk∪Sp

k-1,vw∈Es

κs′,` (ij, vw) rs′,b (vw) 6 ab (ij)− Bb,` (ij, γB,`) ,

∑j∈N


vu∈Esrs,b(vu)

)κs,` (i, v)−

(rs,b(vw)∑

uw∈Esrs,b(uw)

)κs,` (i, w) ,

∀i ∈ N , vw ∈ Es.

Assuming again that the slice resource reservation requestsare ordered, see Section IV-A, one may get a second greedyreservation algorithm where slice resource reservation requestsare processed slice by slice solving Problem 3 for each slice.The highest priority slice is processed first. The lower priorityslices are then processed, whatever the reservation result ofa higher priority slice. Even if high-priority slices may havetheir resource reservation request rejected, lower-priority slicerequests may be granted for slices with smaller resourcerequirements.

PREPRINT 12

D. Slice Resource Reservation Approaches

For the suboptimal algorithms introduced in Sections V-Band V-C, two Prioritized slice resource Reservation (PR)variants are considered, depending on whether slices resourcereservation requests are processed jointly (J-PR) or sequen-tially (S-PR).

1) Joint Approach: In the J-PR approach, all slices in Rkare processed jointly. This is done by solving Problem 2,starting by fixing dRk

= (d1, . . . , dRk) = (1, . . . , 1), i.e., we

try initially to satisfy all slice resource reservation requests. Ifthe reservation is successful, the algorithm stops. If no solutionis returned, the resource reservation request of the slice withlowest priority is not granted, i.e., dRk

= 0. Problem 2 issolved again considering dRk

= (1, . . . , 1, 0). If there is stillno solution, the resource reservation request for the slice withsecond lowest priority is not granted, and so forth. If morethan two slice requests have the same lowest priority, thelast arrived one is not granted. The first part of Algorithm 1(Lines 4–13) summarizes the J-PR approach.

2) Sequential Approach: In the S-PR approach, slice re-source reservation requests in Rk are processed sequentially.This is done by solving Problem 3, for each slice s ∈ Rk,ds ∈ {0, 1}, starting from that with highest priority. Themaximization of the cost function considers ds = 1. If thefollowing minimization problem admits a solution, one keepsds = 1. If the minimization problem admits no solution,ds = 0 and κs = 0 is the solution.

The second part of Algorithm 1 (Lines 14–17) summarizesthe S-PR approach. Note that, S-PR when α = 0 implementsa first-arrived first-served processing policy.

Algorithm 1: Prioritized Slice Resource Reservation

1 foreach processing time interval Pk do2 Get sorted slice request set Rk from MNO;3 switch reservation_variant do4 case J-PR (joint prioritized reservation) do5 Initialize dRk

= (1, . . . , 1);6 i = |Rk|;7 while i > 0 do8 Solve Problem 2 to get κRk

;9 if no feasible solution found then

10 di = 0;11 i = i− 1;12 else13 break;

14 case S-PR (sequential prioritized reservation) do15 Initialize ds = 0, κs = 0, ∀s ∈ Rk;16 foreach s ∈ Rk do17 Solve Problem 3 for slice s to get ds and κs;

18 # Update flag of processed slice requests19 Set fs = 1, ∀s ∈ Rk;

3) Complexity Analysis: Each variant in Algorithm 1requires the solution of one or several ILPs, whose complexityincreases exponentially with the number of variables in theworst case. The J-PR variant considers a single ILP involving∑s∈Rk

∑`∈Ks

(|N |+ 2 |N | |Ns|+ |E| |Es|) variables and∑s∈Rk

∑`∈Ks

(|N | |Es|+ |Ns| |Υ|+ |Es|+ 2 |N | |Ns|) +∑`∈Ks

(|N |+ |E|) constraints. The S-PR variant instead

splits the task into |Rk| subproblems, each of whichinvolves

∑`∈Ks

(|N |+ 2 |N | |Ns|+ |E| |Es|) variables and∑`∈Ks

(|N | |Es|+ |Ns| |Υ|+ |Es|+ 2 |N | |Ns|+ |N |+ |E|)constraints. Therefore, each subproblem in the sequentialapproach implies |Rk| times less variables than the jointvariant. Consequently, due to the exponential complexityof the NP-hard ILP, the sequential approach may provide asolution faster than the joint variant. In Section VI, the twoproposed variants are compared via numerical simulations.

VI. EVALUATION

This section presents simulations to evaluate the perfor-mance of the two reservation algorithms, J-PR and S-PR,described in Section V. The simulation setup is describedin Section VI-A. All simulations described in Section VI-Bare performed with the CPLEX MILP solver interfaced withMATLAB. Our work focuses on the slice admission controland resource reservation mechanisms, both taking place beforeany slice deployment and activation. Consequently, in thesimulation, aspects related to user admission control, radiocoverage/interference, or packet queuing and propagation de-lays are not considered.

A. Simulation Conditions

1) Infrastructure Topology: The considered infrastructurenetwork is represented by the binary fat tree topology de-picted in Figure 6, taken from [38, 39]. The leaf nodesrepresent the far-edge hosts of Radio Unit (RU)/DistributedUnit (DU). The other nodes represent the edge, regional,and central data centers. Infrastructure nodes provide a givenamount of computing, storage, and possibly wireless resources(ac, am, aw), expressed in number of CPUs, Gbytes, and Gbps,depending on the layer they are located. Infrastructure linksprovide a given bandwidth ab, expressed in Gbps. The per-unit resource cost (cn (i) and cb (ij)), fixed cost cf (i), andreservation adaptation cost ca (i) are respectively 1, 10, and20, ∀ (i, ij) ∈ G.

Central (64, 240, 0)

Regional (16, 32, 0)

Edge (4, 4, 0)

RRH (2, 1.25, 10)

100

10

10

Fig. 6. Description of the binary fat-tree infrastructure network consideredin the simulations; Nodes provide a given amount of computing ac, memoryam, and wireless aw resources expressed in number of used CPUs, Gbytes,and Gbps; Links are able to transmit data at a rate ab expressed in Gbps.

2) Slice Resource Demand (S-RD): The number of usersof a slice s is assumed to follow a binomial distribution ofparameter ps,k, see [40, 41]. Considering a Gaussian distri-bution for the individual user resource demands, the resultingresource demand for the slice follows a distribution close toa log-normal distribution, as observed in [42]. One considerstwo patterns to represent the evolution with time of ps,k, which

PREPRINT 13

impacts the evolution of the slice resource demands. The first,illustrated in Figure 7a, corresponds to a constant demandps,k = 1 during the whole lifetime of the slice. The second,shown in Figure 7b, describes a slice whose resource demandevolves from one time slot to the next.

Time slotoffon

(a)

offon

Time slot

(b)

Fig. 7. Probability pattern of service usage: (a) constant over a time intervaland (b) piece-wise constant.

Three types of slices are considered.• Slices of type 1 aim to provide an HD video streaming

service at an average rate of 6 Mbps for VIP users,e.g., in a stadium. The number of users of a slice s oftype 1 follows a binomial distribution B (500, ps,k). Thefunction architecture of slices of type 1 is composed of 3VNFs: a virtual Video Optimization Controller (vVOC),a virtual Gateway (vGW), and a virtual Base Band Unit(vBBU). The required SSP for type 1-slices is psp

s= 0.99;

• Slices of type 2 are dedicated to provide an SD videostreaming service at an average rate of 4 Mbps. Thenumber of users of a slice s of type 2 follows a binomialdistribution B (2000, ps,k). The function architecture ofslices of type 2 is similar to that of type 1. The requiredSSP for type 2-slices is psp

s= 0.95;

• Slices of type 3 aim to provide a video surveillance andtraffic monitoring service at an average rate of 2 Mbpsfor 200 cameras, e.g., installed along a highway. Thedemand pattern of such type of slice is thus always thatof Figure 7a. The third slice type consists of five virtualfunctions: a vBBU, a vGW, a virtual Traffic Monitor(vTM), a vVOC, and a virtual Intrusion Detection Preven-tion System (vIDPS). The required SSP for type 3-slicesis psp

s= 0.9.

The slice type is chosen uniformly at random. For slices oftype 1 and 2, the demand pattern is also chosen uniformly atrandom.

A normalized unit duration time slot is considered, i.e.,T = 1. The processing duration is chosen as εT = 0.1T . Thenumber of reservation request arrivals in each time slot obeysa Poisson distribution Pois (µ) of parameter µ = 2. The arrivaltime of each slice request is uniformly distributed within eachtime interval Tk. The activation delay (i.e., kon

s − ks) followsthe uniform distribution U (1, 6) and the lifetime follows theuniform distribution U (1, 3).

As detailed in Section III-B2, the resource requirementsfor the various SFCs that will have to be deployed withina slice are aggregated within an S-RD graph that mimicsthe SFC-RD graph. S-RD nodes and links are characterizedby the aggregated resources needed to support the targetednumber of users. Details of each resource type as well asthe associated U-RD, SFC-RD, and S-RD graph are given in

Table II. Numerical values in Table II have been adapted from[43]. Each slice request is randomly assigned one type amongthese three types.

3) Background Services: At each infrastructure node i ∈N and link ij ∈ E and for all time slots k, we as-sume that the resources consumed by best-effort back-ground services follow a normal distribution with meanand standard deviation equal to respectively 20% and 5%of the available resources at a node and at a link, i.e.,{Bn,k (i) , Bn,k (i)

}= {0.2an (i) , 0.05an (i)} ∀i ∈ N , ∀n ∈

Υ and{Bb,k (ij) , Bb,k (ij)

}= {0.2ab (ij) , 0.05ab (ij)},

∀ij ∈ E . The reservation impact probability threshold pim isset to 0.1.

B. Results

The performance of the reservation variants (J-PR and S-PR)is evaluated considering the following metrics: slice requestacceptance rate, per-slice reservation cost, average responsedelay (i.e., time between the time instant the request arrivesand the time instant at which it is processed), average numberof adjusted VNF instances per slice, and average computingtime for each slice resource reservation request.

1) Resource Reservation for a Single Slice: The first sim-ulation aims at illustrating the impact of the adaptation costsdescribed in Section III-B3, on the adjustments of the reservedresources between consecutive time slots. A single slice sis considered. Consequently, the J-PR and S-PR reservationvariants yield the same assignment κs,`.

offon

(a) κs,` (i, v) when ca = 0.

offon

(b) κs,` (i, v) when ca > 0.

Fig. 8. Evolution of the assignment κs,` (i, v) for a single slice (for eachmatrix, rows correspond to i, columns to v) when (a) ca = 0 and (b) ca > 0;the matrix entries with κs,` (i, v)− κs,`-1 (i, v) > 0 are highlighted in red,whereas entries with κs,` (i, v) − κs,`-1 (i, v) 6 0 and κs,` (i, v) > 0 arein green.

Figure 8b illustrates the evolution with the time index ` ofκs,` for a slice s of type 1, characterized by an activationduration of three time slots and a demand pattern of type 2(increasing for the second time slot and decreasing for thethird one). In Figure 8b, the entries for which ys,` (i, v) =max {κs,` (i, v)− κs,`-1 (i, v) , 0} > 0 are highlighted in red,indicating an increase of the reserved resources for slice s

PREPRINT 14

TABLE IIPARAMETERS OF U-RD, SFC-RD, AND S-RD

Type 1: HD video streaming at 6 Mbps, psps

= 0.99

Node (Us,c, Us,c) (Us,m, Us,m) (Us,w, Us,w) (rc, rm, rw) Link (Us,b, Us,b) rs,b

vVOC (5.4, 0.54) e-3 (1.5, 0.15) e-2 — (0.29, 0.81, 0) vVOC→vGW (4, 0.4) e-3 0.22vGW (9.0, 0.90) e-4 (5.0, 0.50) e-4 — (0.05, 0.03, 0) vGW→vBBU (4, 0.4) e-3 0.22vBBU (8.0, 0.80) e-4 (5.0, 0.50) e-4 (4, 0.4) e-3 (0.04, 0.03, 0.2)

Type 2: SD video streaming at 4 Mbps, psps

= 0.95


vVOC (1.1, 0.11) e-3 (7.5, 0.75) e-3 — (0.17, 1.20, 0) vVOC→vGW (2, 0.2) e-3 0.32vGW (1.8, 0.18) e-4 (2.5, 0.25) e-4 — (0.03, 0.04, 0) vGW→vBBU (2, 0.2) e-3 0.32vBBU (0.8, 0.08) e-4 (2.5, 0.25) e-4 (2, 0.2) e-3 (0.01, 0.04, 0.3)

Type 3: Video surveillance and traffic monitoring at 2 Mbps, psps

= 0.9


vBBU (2.0, 0.20) e-4 (1.3, 0.13) e-4 (1, 0.1) e-3 (0.4, 0.25, 2) e-2 vBBU→vGW (1, 0.1) e-3 0.02vGW (9.0, 0.90) e-4 (1.3, 0.13) e-4 — (0.018, 0.003, 0) vGW→vTM (1, 0.1) e-3 0.02vTM (1.1, 0.11) e-3 (1.3, 0.13) e-4 — (0.266, 0.003, 0) vTM→vVOC (1, 0.1) e-3 0.02vVOC (5.4, 0.54) e-3 (3.8, 0.38) e-3 — (0.108, 0.080, 0) vVOC→vIDPS (1, 0.1) e-3 0.02vIDPS (1.1, 0.11) e-2 (1.3, 0.13) e-4 — (0.214, 0.003, 0)

during consecutive time slots. Comparing Figure 8a, whereca = 0 and Figure 8b, where ca > 0, one observes thatthe number of adjustments of node assignment κs,` (i, v) isreduced when ca > 0, as expected.

2) Resource Reservation for Multiple Slices: In this sim-ulation, 1000 slice requests are generated among which 250are tagged as Premium uniformly at random. Four choices areconsidered for the parameters α and ∆P , all with Pmax = 3,see Section IV-A. These choices impact the processing strategyof Premium and Standard slice requests. When (α,∆P ) =(0.5, 0), Premium requests are processed immediately andStandard requests are processed in the time slot preceding theiractivation time slot. When α = 0, whatever the value of ∆P ,Premium and Standard requests are processed immediately,starting with the Premium requests. With (α,∆P ) = (0.5, 0.5)and (α,∆P ) = (0.5, 1), intermediate processing delays areobtained for Standard slices. We also evaluate the J-PR andS-PR variants with slice resource reservation requests pro-cessed just before slice activation (just-in-time processing).This approach is close to that considered, e.g., in [14]. Nev-ertheless, Premium slice requests are still processed first.

Figure 9 compares the performance of the J-PR and S-PRreservation variants considering the four slice requests pro-cessing strategies induced by the choices of (α,∆P ) and thejust-in-time approach.

The average response delay (expressed as a multiple of T )for each slice request is shown in Figure 9a. The J-PR and S-PRvariants share the same prioritized processing policy, therefore,both variants provide the same result in terms of responsedelay. When α = 0, all requests are processed immediately,independently of their priority. The observed delay is onlydue to the processing which takes place at the end of eachtime slot during the processing time interval of duration εT .When α = 0.5, the processing delay remains constant forPremium slices and increases when ∆P decreases for Standardslices. As expected, the delay is maximum for the just-in-time

processing.

Figure 9b illustrates the acceptance rate for the various pro-cessing strategies. Processing the slices jointly yields a slightlyhigher acceptance rate compared to a sequential approach. Theacceptance rate of Premium slice requests is higher than thatof Standard ones. The difference decreases when the averageprocessing delay of Standard slice requests decreases. Thedifference is minimum when Standard slices are processed justafter Premium slices in the same processing slot, i.e., when(α,∆P ) = (0, ·) or with the just-in-time approach. Selectingthe processing strategy allows one to adjust the acceptancerate difference between Premium slices and Standard slices.In practice, the income associated to both types of slices, aswell as the share among these types of slice requests may helpthe MNO in determining the best value of the pair (α,∆P ),e.g., that maximizes its expected income.

Figure 9c illustrates the average number of adjustmentsof node assignments ys,` (i, v) per slice and per time slot.A joint approach is again more efficient than a sequentialapproach. Moreover, when the processing delay of Standardslices decreases, the number of adjustments for Standardslices decreases too, while the average number of adjustmentsof node assignments increases for Premium slices. This isexplained by the fact that delaying more the processing ofStandard slices facilitates finding assignments with feweradjustments during the lifetime of Premium slices. The priceto be paid is more adjustments for Standard slices. The numberof adjustments is maximum with the just-in-time approach.

Figure 9d shows the average per-slice reservation costcharged by the InP to the MNO. Joint resources reservationleads to lower costs compared to a sequential reservation.The reservation costs increase for Premium slices when theprocessing of Standard slices is less delayed. For Standardand Premium slices, the reservation costs are consistent withthe evolution of the average number of adjustments of nodeassignments observed in Figure 9c. The costs for the just-in-

PREPRINT 15

(0.5, 0) (0.5, 0.5) (0.5, 1) (0, ·) just-in-time(α,∆P )

0.0

2.5T

5.0T

Avg

.re

spon

sede

lay J-PR (Prem.)

S-PR (Prem.)

J-PR (Std.)

S-PR (Std.)

(a)

(0.5, 0) (0.5, 0.5) (0.5, 1) (0, ·) just-in-time(α,∆P )

0.0

0.2

0.4

0.6

0.8

1.0

Acc

epta

nce

rate

J-PR (Prem.)

S-PR (Prem.)

J-PR (Std.)

S-PR (Std.)

(b)

(0.5, 0) (0.5, 0.5) (0.5, 1) (0, ·) just-in-time(α,∆P )

16

18

20

22

Avg

.ad

just

edin

stan

ces

J-PR (Prem.)

S-PR (Prem.)

J-PR (Std.)

S-PR (Std.)

(c)

(0.5, 0) (0.5, 0.5) (0.5, 1) (0, ·) just-in-time(α,∆P )

400

450

500

Per

-slic

eto

tal

cost

J-PR (Prem.)

S-PR (Prem.)

J-PR (Std.)

S-PR (Std.)

(d)

(0.5, 0) (0.5, 0.5) (0.5, 1) (0, ·) just-in-time(α,∆P )

0

5

10

15

20

Com

puti

ngti

me

(s) J-PR S-PR

(e)

Fig. 9. Performance comparison of the different processing strategies (α,∆P ) with the J-PR and S-PR variants, in terms of (a) average response delay(expressed as a multiple of T ), (b) acceptance rate of slice requests, (b) average adjusted instances, (d) average cost per slice, and (e) computing time.

time approach are similar to those obtained when the slicesrequests are processed as they arrive, when (α,∆P ) = (0, ·).

Figure 9e shows that the computing times are independentof the processing strategy of Premium and Standard slicerequests. As expected the S-PR variant is less time-consumingthan the J-PR variant, due to the reduced number of variablesinvolved. The computing time is significantly less than thetypical time slot duration T when it is of few tens of minutes.As discussed in Section V-D3, there is a linear relationbetween the number of nodes and links of the infrastructurenetwork and the number of variables and constraints of themax-min optimization problem. The computational complexityis exponential in the worst case in the number of variables. Thefact that reservation slot duration is of a few tens of minutes al-lows InPs to deal with moderate-sized infrastructure networks.Nevertheless, for large networks, additional heuristics have tobe developed to be able to employ the proposed approach.

VII. CONCLUSIONS AND PERSPECTIVES

This paper considers a network slicing scenario with slicerequests characterized by variable delays between their sub-mission and activation and by different priority levels (e.g.,Premium and Standard). Considering these hypotheses, weintroduce a prioritized slice resource reservation and admissioncontrol mechanism. Resources required for admitted slices arereserved, and admission decisions are provided with a responsedelay depending on the slice priority and on time left beforeits activation.

Adopting the perspective of the InPs, slice resource reser-vation and admission control is formulated as a max-minoptimization problem. The InP aims to maximize the numberof admitted slices, i.e., slices for which enough resources canbe reserved, while minimizing the cost charged to the MNOs.Uncertainties in the slice resource demands and the presenceof background service sharing the infrastructure are takeninto account. Two reduced-complexity resource reservation

variants, namely J-PR and S-PR, are proposed to solve themax-min problem.

Numerical results show that the proportion of admittedslices can be adjusted depending on the difference in theprocessing delay between Premium and Standard slices. Whenthe delay difference increases, resource reservation requests forPremium slices are granted significantly more frequently, withfewer adjustments with time in the reservation scheme. Thisdirectly impacts the reservation costs, which are reduced forPremium slices compared to Standard slices when the delaydifference is large. We also illustrate the benefits in termsof satisfaction differentiation for Premium slice reservationrequests provided by an anticipated processing, compared toa just-in-time processing.

The approach presented in this paper may be incorporated inmore realistic simulation environments such as that proposedin [14] to confirm the additional flexibility provided to MNOby differentiated processing of resource reservation requests.

In the simulations presented in this work, only resourcereservation requests for finite-duration slices have been con-sidered. Periodic resource reservation requests or requestsfor slices of very long duration may be considered withoutchanging the approach. Nevertheless, this would significantlyincrease the number of variables to store the reservationdecisions for such requests and would require additionaldevelopments to the proposed approach.

Another possible extension of this work is to enable anadjustment in future time slots of the resource reservationscheme for slices that have been admitted, as long as theyare not yet activated. In a given time slot, when Premiumslices are processed, often several assignments lead to the samecosts. As seen in [44], accounting for known slice requeststhat will have to be processed in future time slots, may helpin the selection among assignments of Premium slices requestswith the same costs, to finally reduce the adaptation costof future reservation assignments. Radio coverage constraints

PREPRINT 16

may also be considered in the resource reservation process,using an approach inspired, e.g., from [34]. Accounting foruser mobility would also require a model of the mobilitypatterns of typical slice users.

APPENDIX ADERIVATION OF MEAN AND VARIANCE OF S-RD

Consider an active slice s within time slot ` ∈ Ks. Thenumber of users Ns,` of slice s and the resource demandsUs,n,` (v) and Us,b,` (vw), v ∈ Ns, vw ∈ Es, n ∈ Υ, of eachuser of this slice are assumed as independently distributed.Denoting E (Ns,`) = Ns,`, Var (Ns,`) = N2

s,`, E [Us,n,` (v)] =

Us,n,` (v), and Var (Us,n,` (v)) = U2s,n,` (v), the mean value

and variance of Rs,n,` (v), can be evaluated, ∀n ∈ Υ and∀v ∈ Ns, as

Rs,n,` (v) =E (Ns,Ùs,n,` (v)) = Ns,Ùs,n,` (v) , (53)

R2s,n,` (v) =N

2

s,Ù2s,n,` (v) + U

2

s,n,` (v) N2s,`

+ N2s,Ù

2s,n,` (v) , (54)

see [45]. Similarly, for all vw ∈ Es, one obtains

Rs,b,` (vw) = Ns,Ùs,b,` (vw) , (55)

R2s,b,` (vw) = N

2

s,Ù2s,b,` (vw) + U

2

s,b,` (vw) N2s,`

+ N2s,Ù

2s,b,` (vw) . (56)

APPENDIX BRELAXATION OF THE SSP CONSTRAINT

For a given slice s ∈ Rk, , the MNO has to determine foreach time slot ` ∈ Ks the smallest value of γs,` such that thesatisfaction of (33), (34) implies that of (28).

Consider the following probability

ps,` (γs,`) = Pr{Rs,n,` (v, γs,`) > Rs,n,` (v) ,∀v, n,Rs,b,` (vw, γs,`) > Rs,b,` (vw) ,∀vw

},

(57)which only involves the slice resource demands as well asRs,n,` (v, γs,`) and Rs,b,` (vw, γs,`) introduced in (35) and(36). For a given value of γs,`, if κs,` is such that (33), (34)are satisfied, then, from (57), one has

ps,` (κs,`, ds) > ps,` (γs,`) . (58)

Consequently, choosing γs,` such that ps,` (γs,`) > psps

impliesthe satisfaction of (28).

Using (8) and (9), one has

ps,` (γs,`) =

ms,`∑η=1

pη

∫R(γs,`)

f(x, ηµ, η2Γ

)dx, (59)

where R (γs,`) ={

x ∈ Rns |x 6 R (γs,`)}

with

R (γs,`) =(Rs,c,` (v1, γs,`) , Rs,m,` (v1, γs,`) , . . .

. . . , Rs,b,` (v1v2, γs,`) , . . .)>

(60)

of size ns = |Υ| |Ns|+ |Es|. If the pmf pη of the number Ns,òf users is known, the value of γs,` such that ps,` (γs,`) = psp

s

can be obtained by bisection search methods, see, e.g., [46].The multidimensional integral in (59) can be evaluated, e.g.,using a quasi-Monte Carlo integration algorithm presented in[47].

The evaluation of γs,k can be performed as soon as thereservation request for slice s is received, prior to the opti-mization of the reservation.

APPENDIX CRELAXATION OF THE IP CONSTRAINT

For a given slice s ∈ Rk and an assignment κs,`, ` ∈ Ksthat satisfies (37, 38) and (22, 23, 26), the IP defined in (31)can be evaluated as, ∀i ∈ N , ∀n ∈ Υ,

pimn,` (i) = Pr

{Bn,` (i) > Bn,` (i, γB,`)

}= 1−

∫ Bn,`(i,γB,`)

−∞f(x;Bn,` (i) , B2

n,` (i))

dx

= 1− Φ (γB,`) , (61)

where Φ is the cumulative distribution function (CDF) of thezero-mean, unit-variance normal distribution. Similarly, the IPdefined in (32) can also be evaluated ∀ij ∈ E as,

pims,b,` (ij) = Pr

{Bb,` (ij) > Bb,` (ij, γB,`)

}= 1− Φ (γB,`) . (62)

Both (61) and (62) are independent of κs,`, ∀s ∈ Rk. Tosatisfy the impact constraints imposed by (31, 32), γB,` has tobe chosen such that 1− Φ (γB,`) 6 pim, i.e., such that

γB,` > Φ−1(1− pim) . (63)

Since the larger γB,`, the more difficult the satisfaction of (37)and (38), the optimal γB,` would be γB,` = Φ−1

(1− pim

).

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J. Folgueira, “Network Slicing for 5G with SDN/NFV: Concepts, Architectures, andChallenges,” IEEE Communications Magazine, vol. 55, no. 5, pp. 80–87, 2017.

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Quang-Trung Luu is currently a postdoctoral fel-low at the Laboratory for Analysis and Architectureof Systems (LAAS), French National Centre forScientific Research (CNRS), Toulouse, France. Hereceived a Ph.D in networking and telecommunica-tions from Paris-Saclay University, France in 2021.During the Ph.D, he was also a research engineerat Nokia Bell Labs, France. His research focuses onthe optimization of resource management in com-munication networks, in particular on key enablingtechnologies for next-generation mobile systems.

Sylvaine Kerboeuf received the M.S. degree inphysics and the Ph.D. degree in solid-state physicsfrom the University of Paris-Sud, Orsay, France,in 1991 and 1994, respectively, and the Ph.D. de-gree in superconductivity from the Centre Nationald’Etude des Télécommunications, France Telecom,Paris, France. She joined the Research and Innova-tion Department, Alcatel-Lucent Bell Labs, Nozay,France, where she was involved in research projectson optoelectronics for several years. In 2004, shejoined a project involved in radio access networks

and focusing on fourth generation discontinuous networks and on cachingtechnology. She is currently a Senior Researcher in the Wireless Programwith Nokia Bell Labs. Her current research interests include software definednetwork architecture, network slicing and end-to-end orchestration of micro-services for 5G networks.

Michel Kieffer (M’02, SM’07) received the Ph.D.degree in control theory from the University of ParisXI, Orsay, France, in 1999. He is a Full Professorin signal processing for communications with theUniversity of Paris-Sud and a Researcher with theLaboratoire des Signaux et Systèmes (L2S), Gif-sur-Yvette, France. Since 2009, he is a part-timeInvited Professor with the Laboratoire Traitementet Communication de l’Information, Télécom Paris-Tech, Paris, France. He is coauthor of more than 150contributions in journals, conference proceedings, or

books. He is one of the coauthors of the books Applied Interval Analysis(Springer-Verlag, 2001) (this book was translated in Russian in 2005) andJoint Source-Channel Decoding: A Cross-Layer Perspective With Applicationsin Video Broadcasting (Academic, 2009). His research interests are in sig-nal processing for multimedia, communications, and networking; distributedsource coding; network coding; joint source-channel coding and decodingtechniques; and joint source-network coding. Applications are mainly in thereliable delivery of multimedia contents over wireless channels. He is alsointerested in guaranteed and robust parameter and state bounding for systemsdescribed by nonlinear models in a bounded-error context. Prof. Kieffer wasa junior member of the Institut Universitaire de France from 2011 to 2016.He serves as an Associate Editor of SIGNAL PROCESSING since 2008 and ofthe IEEE TRANSACTIONS ON COMMUNICATIONS from 2012 to 2016.

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