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A two-timescale approach for network slicing in C-RAN. Zhang, He 2018

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A Two-Timescale Approach for Network Slicing in C-RANbyHe ZhangB.E., Xi’an Jiaotong University, 2016A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2018c© He Zhang, 2018iiThe following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the dissertation entitled:Profit Maximization for Network Slicing in Cloud Radio Access Networkssubmitted by He Zhang in partial fulfillment of the requirements forthe degree of Master of Applied Sciencein Electrical and Computer EngineeringExamining Committee:Vincent W. S. Wong, Electrical and Computer Engineering, UBC, VancouverSupervisorVictor C. M. Leung, Electrical and Computer Engineering, UBC, VancouverSupervisory Committee MemberJane Z. Wang, Electrical and Computer Engineering, UBC, VancouverSupervisory Committee MemberiiiAbstractNetwork slicing is a promising technique for cloud radio access networks (C-RANs). Itenables multiple tenants (i.e., service providers) to reserve resources from an infrastructureprovider. However, users’ mobility and traffic variation result in resource demand uncertaintyfor resource reservation. Meanwhile, the inaccurate channel state information (CSI) estimationmay lead to difficulties in guaranteeing the quality of service (QoS). To this end, we proposea two-timescale resource management scheme for network slicing in C-RAN, aiming atmaximizing the profit of a tenant, which is the difference between the revenue from itssubscribers and the resource reservation cost. The proposed scheme is under a hierarchicalcontrol architecture which includes long timescale resource reservation for a slice and shorttimescale intra-slice resource allocation. To handle traffic variation, we utilize the statistics ofusers’ traffic. Moreover, to guarantee the QoS under CSI uncertainty, we apply the uncertaintyset of CSI for resource allocation among users. We formulate the profit maximization as a two-stage stochastic programming problem. In this problem, long timescale resource reservation fora slice is performed in the first stage with only the statistical knowledge of users’ traffic. Giventhe decision in the first stage, short timescale intra-slice resource allocation is performed in thesecond stage, which is adaptive to real-time user arrival and departure. To solve the problem,Abstract ivwe first transform the stochastic programming problem into a deterministic optimizationproblem. We then introduce a maximum interference constraint and transform the QoSconstraint under CSI uncertainty into linear matrix inequalities. We further apply semidefiniterelaxation to transform the problem into a mixed integer nonconvex optimization problem,which can be solved by combining branch-and-bound and primal-relaxed dual techniques.Simulation results show that our proposed scheme can well adapt to traffic variation and CSIuncertainty. It obtains a higher profit when compared with several baseline schemes.vLay SummaryNetwork slicing is a promising technique for the fifth generation (5G) wireless systems. Itallows multiple service providers to run on top of a shared physical network infrastructure.Meanwhile, cloud radio access network (C-RAN) is a centralized, cloud computing-basedarchitecture for radio access networks to support various types of traffic demand in 5G wirelesssystems. However, implementing network slicing in C-RAN is faced with critical challengesdue to time-varying network conditions and inaccurate knowledge of the conditions. In thisthesis, to tackle the aforementioned challenges, we propose a dynamic resource managementscheme for network slicing in C-RAN. Simulation results show that our proposed scheme canachieve a better performance when compared with several baseline schemes.viPrefaceI hereby declare that I am the author of this thesis. This thesis is an original, unpublished workunder the supervision of Professor Vincent W.S. Wong.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Research Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Overview of Network Slicing . . . . . . . . . . . . . . . . . . . . . . 1Table of Contents viii1.1.2 Overview of Cloud Radio Access Networks (C-RANs) . . . . . . . . . 31.1.3 Resource Management for Network Slicing . . . . . . . . . . . . . . . 41.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Network Slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.1 Core Network Slicing . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2 Radio Access Network Slicing . . . . . . . . . . . . . . . . . . . . . . 102.2 C-RAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Network Slicing for C-RAN . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Two-Timescale Resource Management and Profit Maximization . . . . . . . . 133 Two-Timescale Resource Management for Network Slicing in C-RAN . . . . . . 153.1 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . . . 153.1.1 Architecture of Network Slicing in C-RAN . . . . . . . . . . . . . . . 153.1.2 Two-Timescale Framework . . . . . . . . . . . . . . . . . . . . . . . . 16User Traffic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Two-Timescale Resource Management . . . . . . . . . . . . . . . . . 18QoS Requirement under CSI Uncertainty . . . . . . . . . . . . . . . . 20Revenue and Cost of a Tenant . . . . . . . . . . . . . . . . . . . . . . 20Table of Contents ix3.1.3 Two-Stage Stochastic Programming for Profit Maximization . . . . . . 223.2 Solution for the Profit Maximization Problem . . . . . . . . . . . . . . . . . . 233.2.1 Transformation into a Deterministic Problem . . . . . . . . . . . . . . 233.2.2 QoS Constraint Approximation and Semidefinite Relaxation . . . . . . 263.2.3 Primal-Relaxed Dual Technique . . . . . . . . . . . . . . . . . . . . . 313.2.4 Joint Resource Reservation and Allocation Algorithm . . . . . . . . . 353.3 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.1 Simulation Environment and Parameter Setup . . . . . . . . . . . . . . 383.3.2 Algorithm Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.3 Profit Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.4 Resource Reservation and Allocation Performance . . . . . . . . . . . 464 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50A Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57xList of Figures3.1 Two-timescale framework with long timescale resource reservation and shorttimescale intra-slice resource allocation. . . . . . . . . . . . . . . . . . . . . . 173.2 The transformation and relaxation steps taken from problems (3.6) and (3.7) toobtain the solutions of profit maximization problem. . . . . . . . . . . . . . . . 353.3 Convergence of Algorithm 3.1 with different average user arrival rate χ¯(number of users per short timescale slot), rreq = 1.5Mb/s, ε¯2 = 0.05. . . . . . 393.4 Outer iteration convergence of Algorithm 3.2, rreq = 1.5 Mb/s, ε¯2 = 0.05,χ¯ = 2 (number of users per short timescale slot). . . . . . . . . . . . . . . . . 403.5 Outer iteration convergence of Algorithm 3.2, rreq = 1.5 Mb/s, ε¯2 = 0.05,χ¯ = 4 (number of users per short timescale slot). . . . . . . . . . . . . . . . . 413.6 Impact of maximum interference threshold on optimal profit, rreq = 1.5 Mb/s,ε¯2 = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.7 Profit comparison with different average user arrival rate χ¯ (number of usersper short timescale slot), rreq = 1.5Mb/s, ε¯2 = 0.05. . . . . . . . . . . . . . . 433.8 Profit comparison with different QoS requirements rreq (Mb/s), χ¯ = 3 (numberof users per short timescale slot), ε¯2 = 0.05. . . . . . . . . . . . . . . . . . . . 44List of Figures xi3.9 Profit comparison with different average normalized size ε¯2 of CSI uncertaintyset, rreq = 1.5Mb/s, χ¯ = 3 (number of users per short timescale slot). . . . . . 453.10 Total reserved power with different average user arrival rate χ¯ (number ofusers per short timescale slot) and different average normalized size ε¯2 of CSIuncertainty set, rreq = 1.5Mb/s. . . . . . . . . . . . . . . . . . . . . . . . . . 463.11 Power assigned to a user from all RRHs, rreq = 1.5 Mb/s, χ¯ = 3 (number ofusers per short timescale slot), ε¯2 = 0.05. . . . . . . . . . . . . . . . . . . . . 47xiiList of SymbolsXH Conjugate transpose of matrixXTr(X) Trace of matrixXRank(X) Rank of matrixXRm×n Set ofm by n real matricesCm×n Set ofm by n complex matricesHN Set of c by c Hermitian matricesX ≽ 0 MatrixX is positive semidefiniteIn An n by n identity matrix0n An n by 1 all zero vectorOn An n by n all zero matrix⊗ Kronecker productR{x} Real part of the complex number xList of Symbols xiiiW Number of RRHs in the coverage areaB Set of RRHs in the coverage areab An index of an RRH in set Bc Number of sub-channelsl Bandwidth of each sub-channelK Number of long timescale slotsk An index of a long timescale sloti Number of short timescale slots in each long timescale slott An index of a short timescale slotTk Set of indexes of short timescale slots in the long timescale slotkb Number of regions in the coverage aream An index of a regionχkMm Average user arrival rate in regionm in long timescale slot kµkMm Average sojourn time that users stay in region m in longtimescale slot kUtMm Set of users in short timescale slot t in regionmList of Symbols xivUt Set of users in short timescale slot t in the whole coverage areau An index of a userχk Vector of user arrival rate in long timescale slot kµk Vector of average sojourn time in long timescale slot k fordifferent regionsnk Number of reserved sub-channels in long timescale slot kpkMt Amount of power reserved for RRH b ∈ B in long timescale slotkpk Vector of power reserved for all RRHsvtMuMt Beamforming vector from RRH b to user u in short timescaleslot tvtMu Beamforming vector from all RRHs to user u in short timescaleslot tvt Beamforming vector from all RRHs to all users in short timescaleslot th¯tMuMt Mean channel vector between RRH b and user u in shorttimescale slot th¯tMu Mean channel vector for user u in short timescale slot tList of Symbols xvhtMuMt Random channel vector between RRH b and user u in shorttimescale slot thtMu Random channel vector for user u in short timescale slot trtMu Data rate of user u in short timescale slot tσ2 Noise powerRtMu CSI uncertainty set for user u in short timescale slot tεtMu Radius of the uncertainty region of the channel vector htMurreq Required data ratentMu(vt) Revenue of serving user u in short timescale slot t under thebeamforming decision vtα Revenue by offering the service with 1Mb/s data rateβ Penalty of failing to serve user uCk(nkP pk) Reservation cost of a tenant under the resource reservationdecision nk and pkx1 Cost of reserving one sub-channel in a long timescale slotx2 Cost of reserving one Walt of power in a long timescale slotPt Maximum power a tenant can reserve for RRH bList of Symbols xviU seqk Sequence of user sets over short timescale slots in long timescaleslot kl An index of a realization of U seqkL Set of indexes of realizations of U seqkU seqkMl l-th realization of U seqkωl Probability of the occurrence of U seqkMlUtMl User set in short timescale slot t in U seqkMlvseqkMl Beamforming decision for U seqkMl over short timescale slots in longtimescale slot kvtMl Beamforming vector for U seqkMl in short timescale slot tatMlMu Binary admission control variable, indicating whether the ser-vice request of user u in set UtMl is accepted or noth¯tMlMu Mean channel vector for user u in set UtMlhtMlMu Random channel vector for user u in set UtMlRtMlMu CSI uncertainty set for user u in set UtMlε2tMlMu Size of the CSI uncertainty set for user u in set UtMlε˜2tMlMu Normalized size of the CSI uncertainty set for user u in set UtMlList of Symbols xviiφtMlMu An auxiliary variable serving as a lower bound of the signal-to-interference-plus noise ratio for user u in set UtMlI Maximum interference thresholdU relaxtMl Set of users whose atMlMu is relaxedytMlMu Value of atMlMu that has been determinedυtMlMu An auxiliary variable for the linear matrix inequality constraintfor QoS of user u in set UtMlξtMlMu An auxiliary variable for the linear matrix inequality constraintfor QoS of user u in set UtMlQtMlMu An auxiliary matrix for the linear matrix inequality constraint forQoS requirement of user u in set UtMlVtMlMu Beamforming matrix for user u in set UtMlok Sequence of optimization variables in long timescale slot kBt An auxiliary matrix for the power constraint at RRH bxviiiList of AcronymsBBU Baseband UnitBS Base StationCoMP Coordinated MultipointC-RAN Cloud Radio Access NetworkCSI Channel State InformationLMI Linear Matrix InequalitymMTC Massive Machine Type CommunicationsNFV Network Function VirtualizationQoS Quality of ServiceRAN Radio Access NetworkRRH Remote Radio HeadSDN Software-Defined NetworkingList of Acronyms xixSDR Semidefinite RelaxationSINR Signal-to-Interference-plus-Noise RatioURLLC Ultra-Reliable Low-Latency Communications3GPP Third Generation Partnership Project5G Fifth GenerationxxAcknowledgementsI would like to express my deepest appreciation to my supervisor, Prof. Vincent Wong, for hisinvaluable guidance that helped me to shape my research, and for his persistent encouragementand patience that made me fulfill my master program.I want to thank my senior Dr. Yong Zhou and Dr. Hamed Shah-Mansouri, who gave meconstructive comments and help to my research work. I would like to thank my friends in theCommunications Lab: Hao Ma, Zehua Wang, Jun Zhu, Bojiang Ma, Yanan Sun, Manyou Ma,Xueting Yang, Xuan Luo, and Xiuhua Li who provided me with generous support and greatcompany. Finally, I own my heartfelt gratitude to my beloved parents for their love, supportand understanding.xxiDedicationThis thesis is dedicated to my beloved family.1Chapter 1IntroductionIn this chapter, we first introduce the background of network slicing and cloud radio accessnetworks (C-RANs). We then introduce the resource management for network slicing,followed by a discussion on the motivation and contributions of our work. The structure ofthe thesis is shown at the end of this chapter.1.1 Research Background1.1.1 Overview of Network SlicingThe fifth generation (5G) wireless systems are expected to support diverse types of servicesand meet the increasing traffic demands from the end users [1,2]. This scenario leads to highernetwork capital and operating expenditures, as well as higher network resource consumption.To tackle these problems, network slicing is introduced to virtualize the common physicalnetwork into several logical end-to-end networks. Each logical end-to-end network is calleda network slice. As a logical end-to-end network, each slice consists of a part of corenetwork resources, network functions, and radio access network resources. Each slice canbe dynamically created, modified, and released by the centralized controller located at theChapter 1. Introduction 2infrastructure provider. The service provider, which is the owner of each network slice, iscalled a tenant. Based on the network slicing paradigm, each tenant, equipped with a localcontroller, is capable of managing the network slice according to a specific type of serviceand quality of service (QoS) requirements including data rate, latency, reliability, and security.There are several crucial requirements for network slicing. First, slice orchestration requiresa unified and flexible execution environment to run multiple slices. Second, slice isolationrequires separation of resources and independent slice management without interference fromother slices. Third, optimized topology and resource allocation are needed to achieve servicefulfillment assurance.The key enablers for network slicing include software-defined networking (SDN) andnetwork function virtualization (NFV) [3]. The main idea of SDN is to decouple the forwardingprocess of data packets in the data plane from the routing process in the control plane, so thatnetwork management can be performed by a logical network controller. In this way, flexibilityis achieved by allowing simple and efficient network configuration. The OpenFlow [4, 5]standard is one of the first protocols to implement SDN in the core network. For NFV [6], themain idea is to decouple network functions from the physical network equipment and virtualizethese network functions into building blocks that may be chained together to create a specifictype of communication service.Chapter 1. Introduction 31.1.2 Overview of Cloud Radio Access Networks (C-RANs)C-RAN is a novel mobile network architecture for 5G wireless systems [7]. The main ideabehind C-RAN is to detach the radio signal transceiver module and baseband signal processingmodule of conventional base stations (BSs) into two parts. In C-RAN, the baseband signalprocessing module is moved from BSs to a cloud server, which is referred to as a baseband unit(BBU). Multiple BBUs running on a cloud server can form a BBU pool, offering centralizedbaseband signal processing with powerful computation capability. Conventional BSs arereplaced by light and low-cost remote radio heads (RRHs) with radio signal transmission andreception functions. To enhance the capacity of C-RAN, the coordinated multipoint (CoMP)transmission technique is deployed by which multiple RRHs can coordinate together to serveeach user. The group of RRHs serving each user is called an RRH cluster, and the groupingprocess is called user-centric RRH clustering. Furthermore, by implementing multiple antennasat each RRH, the beamforming technique can be deployed to mitigate interference experiencedby each user.There are two fundamental downlink data transmission strategies for C-RAN, i.e., data-sharing strategy and compression strategy [8]. In the data sharing strategy, the BBU poolsends messages of each user directly to multiple RRHs by fronthaul links. The RRHs locallyform the beamforming vector and cooperatively transmit the messages to each user. Inthe compression strategy, the central processor located at the BBU pool is responsible foruser message precoding. Then, a compressed version of the analog beamformed signals isforwarded to RRHs for cooperative transmission.Chapter 1. Introduction 41.1.3 Resource Management for Network SlicingAccording to different types of network resources, network slicing can be categorized intotwo types: core network slicing that partitions network nodes, links or topologies, and radioaccess network (RAN) slicing that partitions baseband resources, BSs, radio resources, andtransmission power [9]. Each tenant estimates the resource demand from its subscribedusers and submits the resource reservation request to the centralized controller. With thereceived resource reservation requests, the centralized controller performs inter-slice resourcevirtualization and assigns the physical resources to each slice. Then, the tenant performsintra-slice resource allocation among its subscribed users. From the perspective of a logicalcentralized controller, inter-slice resource virtualization is performed by the infrastructureprovider, who owns the common physical network. The inter-slice resource virtualization,enabled by SDN and NFV, is responsible of assigning common physical network resourcesto each slice corresponding to the resource reservation request. Meanwhile, since that theresource reservation requests from different tenants may arrive at different time, inter-sliceresource virtualization is also responsible of dynamically scheduling the resources to differenttenants [10]. From the perspective of each tenant, resource reservation process and intra-slice resource allocation process can be jointly considered. The resource reservation decisionmade by the tenant should guarantee sufficient resources for intra-slice resource allocation.Meanwhile, intra-slice resource allocation, performed by the local controller at each tenant,should achieve efficient resource utilization, mitigate interference among users, and guaranteeQoS of users.Chapter 1. Introduction 5With the development of C-RAN, implementing network slicing in C-RAN has nowattracted more attention and is still an open issue. Besides radio resources considered inRAN slicing, resources of RRHs, fronthaul capacity, and BBU pool need to be consideredfor network slicing in C-RAN. Moreover, user-centric RRH clustering and beamforming canbe integrated to enhance network capacity and achieve efficient resource utilization.1.2 MotivationMany research works have been conducted on resource management for core network slicing[10–14]. Compared with core network slicing, RAN slicing is faced with new challenges dueto time-varying channel conditions, user mobility, and interference.Conventional approaches mainly consider inter-slice resource virtualization among tenantsfrom the perspective of a centralized controller to achieve fairness among tenants [15–21]. Toachieve accurate resource demand estimation and efficient resource utilization, some studieshave been conducted on resource reservation for slices and intra-slice resource allocation fromthe perspective of each tenant [22, 23]. However, these works consider the two processes in asingle timescale framework. To achieve real-time adaptation to varying network conditions, theduration of the timescale is designed to be short. In this case, performing resource reservationand intra-slice resource allocation simultaneously may lead to a high computational cost. Totackle this problem, a two-timescale framework can be adopted. In this framework, resourcereservation is performed in a long timescale with the estimated resource demand from the slice,and intra-slice resource allocation is performed in a short timescale to achieve adaptation toChapter 1. Introduction 6real-time network conditions. The two-timescale framework is discussed in several works [24,25]. However, these works neglect the characterization of the profit of each tenant. To achieveprofit maximization, each tenant should control the resource reservation cost and increase therevenue obtained from its subscribed users.In this thesis, we propose a two-timescale resource management scheme for network slicingin C-RAN, aiming at maximizing the profit of the tenant by long timescale resource reservationfor the slice and short timescale intra-slice resource allocation among the subscribed users.We consider two major challenges. First, user traffic varies over time, making it difficult toaccurately estimate the resource demand for resource reservation. Second, due to fast fading,user mobility, coding error, and delay, the uncertainty of channel state information (CSI)of the subscribed users should be considered during intra-slice resource allocation in orderto guarantee the QoS. To tackle these challenges and maximize the profit of the tenant, theinteraction between resource reservation and intra-slice resource allocation is considered. Thelong timescale resource reservation characterizes the statistics of user traffic and ensures thatsufficient resources are reserved for intra-slice resource allocation. Meanwhile, the intra-sliceresource allocation is adaptive to the arbitrary arrival/departure of users while characterizingthe CSI uncertainty to achieve efficient utilization of the reserved resources and guarantee theQoS.Considering that the profit maximization problem involves user traffic variation as wellas the interaction between resource reservation and intra-slice resource allocation, we applytwo-stage stochastic programming to formulate our problem. Stochastic programming is aChapter 1. Introduction 7framework for modeling optimization problems that involve random events [26]. In the two-stage stochastic programming problem, the decision in the first stage is made only with thestatistical knowledge of the random event. Then, based on the decision in the first stage anda realization of the random event, the decision in the second stage is made. The objectiveof the two-stage stochastic programming problem is to maximize the expectation of a certainobjective function over the random event. Therefore, by modeling long timescale resourcereservation as the decision in the first stage and short timescale intra-slice resource allocationas the decision in the second stage, and modeling the user traffic as the random event, weformulate the profit maximization as a two-stage stochastic programming problem. Moreover,since the CSI uncertainty is difficult to be modeled in a probabilistic manner as many factors(e.g., user mobility, coding error, and delay) lead to the uncertainty, for intra-slice resourceallocation in the second stage, we apply the uncertainty set to restrict the realizations of CSI.In this way, the resource allocation decision can be made to guarantee the QoS under the CSIuncertainty.1.3 ContributionsThe main contributions of this thesis are summarized as follows:• We propose a two-timescale resource management scheme to achieve profit maximiza-tion for network slicing in C-RAN. By modeling the problem as a two-stage stochasticprogramming problem, the interaction between resource reservation and intra-sliceChapter 1. Introduction 8resource allocation is achieved, and the user traffic variation is characterized.• We design a profit model for the tenant, which captures the revenue obtained from itssubscribed users and the cost of resource reservation. The revenue is modeled as apiecewise function consisting of a reward obtained by guaranteeing the QoS of users anda penalty due to QoS violation. The cost is modeled as a linear function consisting of thesub-channel and power reservation cost. We characterize the QoS under CSI uncertaintyby applying the CSI uncertainty set.• We transform the stochastic programming problem into a deterministic mixed-integeroptimization problem by introducing a maximum interference threshold and applyingsemidefinite relaxation. We combine branch-and-bound and primal-relaxed dual tech-niques to obtain the suboptimal solution.• We conduct extensive simulations to evaluate the properties and performance of theproposed scheme. Results show that the proposed scheme can achieve a higher profitwhen compared with four other baseline schemes.1.4 Outline of the ThesisThis thesis is organized as follows. In Chapter 2, we introduce the related work. In Chapter 3,we present the two-timescale resource management scheme to achieve profit maximization fornetwork slicing in C-RAN, analyze its properties, and evaluate its performance. Conclusionand future work are given in Chapter 4.9Chapter 2Related Work2.1 Network Slicing2.1.1 Core Network SlicingMany studies have been conducted on core network slicing based on SDN and NFV. Theframework of FlowVisor is introduced in [11], which is implemented as an OpenFlow proxythat intercepts messages between OpenFlow-enabled switches and OpenFlow controllers.FlowVisor is capable of partitioning link bandwidth and flow table in each switch. Many workshave been conducted on the standardization of core network slicing [27,28]. The latest versionof network slicing standard is Release 15 of the 3rd Generation Partnership Project (3GPP)completed in June 2018. Several key concepts, such as network slice, network slice instance,and lifecycle management of network slice instance, are specified.Recently, various dynamic and flexible resource management schemes have been proposedfor core network slicing [10, 12]. Baumgartner et al. in [12] proposed a robust inter-sliceresource allocation scheme with the consideration of survivability to protect network sliceagainst network element (nodes/links) failure. In [10], Sciancalepore et al. applied the Holt-Chapter 2. Related Work 10Winters forecasting method to predict the future traffic of each slice based on historical records.With the predicted traffic information, a slice selection and scheduling scheme was proposed,aiming at improving network utilization. Besides resource management schemes for corenetwork slicing to achieve the resource efficiency and utility maximization, there are also otherworks discussing the reconfiguration of network slicing [13, 14]. Paris et al. in [13] addressedthe problem that frequent flow reconfigurations for network slicing may cause QoS violation.To tackle this problem, they proposed a control policy to minimize the flow allocation costwhile achieving the adaptation to varying network conditions. Pellegrini et al. in [14] proposeda learning algorithm to perform optimal online flow segmentation, which can achieve minimumsignaling over the control channel and can track traffic variations over time.2.1.2 Radio Access Network SlicingMany studies have been conducted on RAN slicing [29]. Early studies on RAN slicing mainlyfocus on the guarantee of slice isolation. Ravi et al. in [15] proposed a time domain resourcepartitioning scheme to assign different slices into different time slots. However, this schemedoes not consider a multi-cell scenario, in which multi-cell interference is a critical problemas slices may share the same spectrum in different cells. To tackle this problem, Gudipati etal. in [16] proposed the concept of SoftRAN, which defines a virtual big-base station thatis comprised of a central controller and a group of geographically close BSs. Besides theresource allocation of time-frequency resource blocks, the authors further proposed a powerallocation scheme, which is performed by a logical centralized controller to mitigate inter-cellChapter 2. Related Work 11interference among users in different slices and guarantee slice isolation. Besides the challengeof slice isolation, partitioning RAN resources into different slices corresponding to different usecases and QoS requirements is also an important problem. Caballero et al. in [17] focused onachieving desirable fairness across network slices and users. They formulated an optimizationproblem for dynamic resource allocation with a weighted proportionally fair objective function.Zhang et al. in [18] proposed a mobility management scheme and a joint power and sub-channel allocation scheme for RAN slicing to enhance resource efficiency. In [19], Xiang et al.designed a hierarchical network slicing architecture, consisting of a centralized orchestrationlayer and a slice instance layer for resource allocation in fog RAN. In [20], Chen et al. proposeda resource pre-allocation scheme for each slice and an intra-slice resource scheduling schemefor users with different priorities to achieve resource efficiency. The aforementioned works onRAN slicing assume that the centralized controller can obtain the perfect knowledge of networkconditions. However, the uncertainty of network conditions may exist. Zheng et al. in [21]proposed a delay-optimal radio resource scheduling scheme with stochastic learning. Theyapplied partially observed Markov decision process to characterize the uncertainty of channelconditions and user traffic for the resource scheduling scheme design.Instead of considering resource management from the perspective of a centralized con-troller, some works have been conducted to design the resource reservation and intra-sliceresource allocation from the perspective of each tenant. For example, in [23], Zhu et al.proposed a hierarchical combinatorial auction mechanism for resource management, in whicheach tenant submits its bid to the centralized controller for a certain amount of resources, andChapter 2. Related Work 12executes an auction to allocate the reserved resources to its subscribed users. Caballero et al.in [22] formulated a network slicing game in which each tenant takes into account the resourcedemand estimation of other tenants to make a resource reservation decision so as to maximizeits user utility. Besides performing resource reservation and intra-slice resource allocation ina single timescale, a two-timescale framework is also considered. In this framework, resourcereservation is performed in a long timescale with the predicted resource demands from the slice,and intra-slice resource allocation is performed in a short timescale to achieve the adaptationto real-time network condition variations. Zhang et al. in [24] proposed a static spectrumreservation and dynamic resource requesting scheme for each tenant to maximize the aggregateutility of users. In [25], Chen et al. designed a resource pre-allocation over a long timescaleand intra-slice resource scheduling over a short timescale for resource efficiency maximization.2.2 C-RANBeamforming and user-centric RRH clustering are two major topics in C-RAN. Shi et al.in [30] proposed a multi-stage scheme for network power minimization. They separated groupsparse beamforming and RRH clustering into different stages. Liu et al. in [31] proposed atwo-timescale RRH clustering and beamforming scheme to achieve the trade-off between theaverage weighted sum rate and implementation cost. User-centric RRH clustering is performedin a long timescale and beamforming is performed in a short timescale. Some other worksdesign joint RRH clustering and beamforming. Dai et al. in [32] proposed a scheme toChapter 2. Related Work 13perform joint sparse beamforming and user-centric RRH clustering by formulating a zero-norm problem, aiming at maximizing the network utility. Wang et al. in [33] proposed a robustbeamforming scheme for C-RAN to maximize the network utility. They characterized the QoSunder CSI uncertainty by applying the uncertainty set and S-procedure.2.3 Network Slicing for C-RANResearch on network slicing for C-RAN is now attracting more attention. Lee et al. in[34] proposed a dynamic end-to-end network slicing scheme for heterogeneous C-RAN,which allocates baseband resources, fronthaul/backhaul capacities, and radio resources tomultiple tenants with different priorities, aiming at achieving high network throughput whileguaranteeing fairness. Costanzo et al. in [35] proposed a prototype for network slicing inC-RAN based on Open Air Interface platform and FlexRAN SDN controller to handle thecreation and configuration of network slices. Ezzaouia et al. in [36] focused on slicing theBBU pool to establish a logical mapping between the BBU pool and RRHs.2.4 Two-Timescale Resource Management and ProfitMaximizationThe two-timescale resource management framework is widely used in 5G wireless systems.Liu et al. in [31] proposed a two-timescale resource management framework for C-RAN,in which RRH clustering is performed in a long timescale and beamforming is performedChapter 2. Related Work 14in a short timescale. Niu et al. in [37] proposed a dynamic resource sharing mechanismamong multiple tenants in C-RAN, in which a global resource allocation process is performedin a long timescale and multiple local resource allocation processes are performed in a shorttimescale. Gao et al. in [38] proposed a two-timescale approach for profit maximization in acloud transcoding system by performing resource provisioning and task scheduling.15Chapter 3Two-Timescale Resource Management forNetwork Slicing in C-RAN3.1 System Model and Problem Formulation3.1.1 Architecture of Network Slicing in C-RANWe consider network slicing in a CoMP based C-RAN system. In this system, multiplebaseband signal processing modules are located at a BBU pool. The RRHs are composedof radio signal transceivers and are connected to the BBU pool via optical fibers. Each RRHis equipped with multiple antennas. Each user is equipped with a single antenna. The CoMPframework enables each user to be served by multiple RRHs, which form an RRH cluster.Meanwhile, the beamforming operation is designed for antennas to mitigate interference. Weapply the data-sharing strategy for downlink data transmission. We denote the set of RRHsin the coverage area as B = {1P 2P . . . P W}. Each RRH is equipped with V antennas. Thereare c sub-channels, each with bandwidth l . Network slicing is implemented in this CoMPbased C-RAN. Each slice corresponds to a virtual network with partial network resources andChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 16network functions. Each slice is owned by a tenant (i.e., service provider) to support a specifictype of service. In this thesis, we assume that each tenant owns a single slice. Our proposedframework can be easily extended to the scenario where each tenant owns multiple slices.We consider resource management for network slicing in C-RAN from the perspectiveof a single tenant. The tenant performs resource reservation for its slice to request radioresources of sub-channels and power from an infrastructure provider, which is the owner ofthe physical infrastructure of C-RAN. The tenant then performs intra-slice resource allocationto allocate the reserved resources to its subscribed users according to channel conditions andQoS requirements of users.3.1.2 Two-Timescale FrameworkAs shown in Fig. 3.1, we divide 24 hours of a day into K long timescale slots (minutes).Each long timescale slot consists of i short timescale slots with the same duration (seconds).Resource reservation is performed over long timescale with the statistical knowledge of usertraffic. Intra-slice resource allocation is performed over short timescale under an arbitraryuser arrival/departure process. The choice of the duration of each long timescale slot shouldguarantee that the statistics of user traffic will not change within the long timescale slot.Meanwhile, since that more frequent submissions of resource reservation requests may lead tohigher computation and reconfiguration cost of the network, the duration of each long timescaleslot should be chosen to avoid high computation and reconfiguration cost. The choice of theduration of each short timescale slot should guarantee that the real-time user traffic variationChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 17Resource reservationIntra-slice resource allocationk = 01 2 Short timescale1TKt = 0 T+1 T+2 KTCost: Revenue:Decision:Decision:Long timescale2T 2T+1 2T+22Figure 3.1: Two-timescale framework with long timescale resource reservation and shorttimescale intra-slice resource allocation.can be captured so that the intra-slice resource allocation can be adaptive to arbitrary userarrival and departure. In this thesis, we assume that the durations of each long timescale slotand short timescale slot are predetermined and do not change over time. The explanations ofnotations in Fig. 3.1 will be given in the following part of this section.User Traffic ModelIn this thesis, we consider the scenario where users arbitrarily arrive and leave the system. Inthe coverage area of C-RAN, different regions may have different statistics of user traffic. Toaddress this issue, we divide the network coverage area into b disjoint regions, according tothe density of user distribution [39]. Within the long timescale slot k = 0P 1P . . . P K − 1, inregion m = 1P . . . Pb , we assume that the arrival of users follows a general distribution withan average user arrival rate of χkMm (number of arrived users per short timescale slot). Theduration that a user stays in region m, called the sojourn time, is a random variable. It followsa general distribution with mean µkMm, the unit of which is a short timescale slot.Chapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 18Within a long timescale slot k, we denote the set of users in short timescale slot t as Ut =⋃Mm=1 UtMm, where UtMm is the set of users in region m and t ∈ Tk = {kiP . . . P (k + 1)i − 1}.Users in set UtMm are assumed to be uniformly distributed in regionm.Based on the user traffic model, the arrival and departure process of users can be depicted.Within the long timescale slot k, in each short timescale slot t, in each region m, there willbe a random number of new user arrivals following the general user arrival distribution withaverage user arrival rate χkMm. Each user stays in the region with a random sojourn time.After the sojourn time, the user will leave the system. Since that general distributions for boththe arrival of users and the sojourn time are assumed, we can design a resource managementscheme that can be applicable for different statistical models of user traffic.Two-Timescale Resource ManagementAt the beginning of a long timescale slot k, the tenant obtains the knowledge of userarrival rate vector χk = (χkM1P . . . P χkMmP . . . P χkM ), the average sojourn time vector µk =(µkM1P . . . P µkMmP . . . P µkM ), and the user set Ukf . The tenant then makes the resourcereservation decision by choosing nk, which is the number of reserved sub-channels, andpk = (pkM1P . . . P pkMT), in which pkMt is the amount of power reserved for RRH b ∈ B.At the beginning of a short timescale slot t ∈ Tk, given the resource reservation decisionnk and pk, and an observation of user set Ut, we design a beamforming scheme. For a useru ∈ Ut, the beamforming decision is denoted as vtMu = [vHtMuM1 · · · vHtMuMT]H ∈ CAT×1, wherevtMuMt ∈ CA×1, b ∈ B, represents the beamforming vector from RRH b to user u for each sub-Chapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 19channel. Furthermore, based on the user location distribution, the mean channel vector of useru ∈ Ut can be estimated as h¯tMu = [h¯HtMuM1 · · · h¯HtMuMT]H ∈ CAT×1, where h¯tMuMt ∈ CA×1, b ∈ B,is the mean channel vector between RRH b and user u. Due to user mobility and fast channelfading, the instantaneous channel vector, denoted as htMu, is a random vector, with mean h¯tMu.Given the beamforming decision vector vt = [vHtM1 · · · vHtM|Ut|]H and channel vector htMu, the datarate of user u in short timescale slot t can be obtained as follows [32]:rtMu = nkl log(1 +|hHtMuvtMu|2∑u′∈Ut\{u} |hHtMuvtMu′|2 + σ2)P (3.1)where σ2 is the noise power. Since the channel vector htMu is a random vector, rtMu is also arandom variable.By designing the sparse beamforming vector vtMuMt for each user u ∈ Ut at each RRH b ∈ B,the tenant can determine the power allocated to user u at RRH b. Meanwhile, the beamformingvector can also indicate the user-centric RRH clustering decision for each user. We note thatwhen vtMuMt = 0AT, user u is not associated with RRH b. When vtMuMt ̸= 0AT, user u is servedby RRH b.In this thesis, we assume that resource reservation and intra-slice resource allocationdecisions made by the tenant will not be affected by the decisions of other tenants. We alsoassume that the infrastructure provider can always satisfy the resource reservation requestsfrom the tenant.Chapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 20QoS Requirement under CSI UncertaintyIn this thesis, the QoS requirement is the required data rate, denoted as rreq. Since only themean channel vector h¯tMu can be obtained, we adopt the uncertainty set to capture the CSIuncertainty. In short timescale slot t ∈ Tk, the CSI uncertainty set of user u ∈ Ut is defined asRtMu , {htMu | (htMu − h¯tMu)H(htMu − h¯tMu) ≤ ε2tMu}P (3.2)where εtMu is the radius of the uncertainty region of the channel vector htMu. We denote ε2tMu asthe size of the CSI uncertainty set RtMu. Then, based on (3.1) and (3.2), the QoS requirementunder the CSI uncertainty can be modeled asrtMu ≥ rreqP htMu ∈ RtMu. (3.3)Inequality (3.3) indicates that rreq should be satisfied for all the realizations of htMu in the CSIuncertainty set RtMu. Therefore, by introducing the CSI uncertainty set, the QoS requirementcan be depicted as deterministic constraint (3.3) without the necessity of knowing the statisticalknowledge of the channel vector.Revenue and Cost of a TenantOne key motivation of the tenant to perform resource reservation and intra-slice resourceallocation is to enhance the revenue obtained from the subscribed users while controlling theresource reservation cost so as to maximize the profit. In this section, we design a revenueChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 21model and a resource reservation cost model for the tenant.In a short timescale slot t ∈ Tk, k = 0P . . . P K − 1, with the knowledge of Ut and vt, therevenue of serving user u ∈ Ut is given asntMu(vt) =p(ε˜tMu)rreqαP rtMu ≥ rreqP htMu ∈ RtMu−βP otherwiseP(3.4)where ε˜2tMu ="2t;uhHt;uht;uis the normalized size of CSI uncertainty setRtMu, p(ε˜tMu) is the probabilitythat the true channel vector is within the CSI uncertainty set, rreqα is the revenue of servinguser u ∈ Ut if perfect CSI information is obtained, in which α is the revenue obtained byoffering the service with 1 Mb/s data rate. We also have β as the penalty of failing to serveuser u. According to revenue function (3.4), higher required data rate rreq results in higherrevenue obtained by the tenant, since that users need to pay more for better service. Meanwhile,satisfying QoS constraint (3.3) is not sufficient to guarantee rtMu ≥ rreq with 100%, since thatthe true realization of channel vector htMu may be out of the CSI uncertainty set. Therefore, weintroduce the probability p(ε˜tMu), which is determined by ε˜tMu of the CSI uncertainty set. Largerε˜tMu may lead to a higher probability that the true realization of channel vector is included in theuncertainty set. Thus, higher probability of QoS guarantee can be achieved for higher revenue.The probability p(ε˜tMu) of user u can be summarized from historical channel vector records ofusers located at the same place of user u.At the beginning of long timescale slot k, given the resource reservation decisions nk andChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 22pk, the cost function can be defined asCk(nkPpk) = x1nk +∑t∈Bx2pkMtP (3.5)where x1 and x2 are the costs of reserving one sub-channel and one Walt of power for one longtimescale slot, respectively.3.1.3 Two-Stage Stochastic Programming for Profit MaximizationThe objective of long timescale resource reservation and short timescale intra-slice resourceallocation is to maximize the expected profit of a tenant in each long timescale slot. For eachlong timescale slot k = 0P 1P . . . P K − 1, we formulate a two-stage stochastic programmingproblem. The first stage decision, i.e., resource reservation, is made at the beginning of the longtimescale slot k, with only the knowledge of the average user arrival rate vector χk, averagesojourn time vector µk, and user set Ukf . We denote U seqk = (Ukf P . . . PU2k513f−1). Then,with the first stage decision and a realization of U seqk , the second stage decision, i.e., intra-sliceresource allocation, is made over short timescale slot t ∈ Tk. The problem is formulated asfollows:maximizenkMpkEU seqk [f(Useqk )]− Ck(nkPpk) (3.6a)subject to nk ∈ {0P . . . P c}P (3.6b)0 ≤ pkMt ≤ PtP b ∈ BP (3.6c)Chapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 23where Pt is the maximum power a tenant can reserve for RRH b ∈ B, f(U seqk ) is the optimalrevenue obtained by the tenant given the knowledge of U seqk , EU seqk [f(Useqk )] is the expectationof f(U seqk ) over all the realizations of U seqk , f(U seqk ) is the optimal value of the following intra-slice resource allocation problem:maximizevtMt∈Tk∑t∈Tk∑u∈UtntMu(vt) (3.7a)subject to nk∑u∈UtTr(vtMuMtvHtMuMt)≤pkMtP b ∈ BP t ∈ Tk. (3.7b)Constraint (3.7b) represents the power constraint given the decisions of nk and pk made in thefirst stage.By solving problem (3.6), the amount of reserved resources and the corresponding costare determined, based on which second stage problem (3.7) determines the optimal revenuef(U seqk ) by making the beamforming decision vt. Therefore, by solving the two-stagestochastic programming problem, the expected profit can be maximized.3.2 Solution for the Profit Maximization Problem3.2.1 Transformation into a Deterministic ProblemThe two-stage stochastic programming problem can not be solved directly due to theexpectation of f(U seqk ) in problem (3.6). Meanwhile, resource reservation in the firststage and intra-slice resource allocation in the second stage build a hierarchical controlChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 24architecture. Therefore, we first transform the two-stage stochastic programming probleminto a deterministic optimization problem [40]. Based on the traffic model in Section 3.1.2,at the beginning of the long timescale slot k, with the knowledge of (χkPµkPUkf ), we canobtain the realizations of the user set sequence U seqk . The l-th (l ∈ L = {1P . . . P a})realization of U seqk is denoted as U seqkMl = (Ukf PUkf51MlP . . . PU2k513f−1Ml). The correspondingprobability of the occurrence of realization U seqkMl is denoted as ωl. The correspondingbeamforming decision sequence is denoted as vseqkMl = (vkfMlP . . . Pv2k513f−1Ml), in which vtMl =[vHtMlM1 · · · vHtMlMu · · · vHtMlM|Ut;l|]H, and vtMlMu = [vHtMlMuM1 · · · vHtMlMuMT]H, u ∈ UtMl, t ∈ Tk, l ∈ L. Then,the two-stage stochastic programming problem can be transformed into the following problem:maximizenkMpkMvseqkL∑l=1ωl∑t∈Tk∑u∈Ut;lntMu(vtMl)− Ck(nkPpk) (3.8a)subject to nk∑u∈Ut;lTr(vtMlMuMtvHtMlMuMt) ≤ pkMtP b ∈ BP t ∈ TkP l ∈ L (3.8b)constraints (3.6b) and (3.6c)Pwhere vseqk = (vseqkM1P . . . PvseqkML).Problem (3.8) cannot be solved directly due to the nonconvexity of ntMu(vtMl). Based on thediscussion in Section 3.1.2, the revenue function (3.4) can be equivalently depicted under a useradmission control scenario. For user u ∈ UtMl, the tenant can obtain the revenue p(ε˜tMu)rreqαfrom the user if the QoS requirement constraint (3.3) is satisfied. The tenant will need to pay apenalty of β if constraint (3.3) is not satisfied. To further save the resources, the tenant will thenChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 25assign no resources to this user, which indicates that the service request of the user is rejected.In this case, we introduce the user admission control variable atMlMu ∈ {0P 1} to indicate whetherthe service request of user u is accepted. Then, for the l-th realization of U seqk , the revenuefunction (3.4) is equivalent ton newtMlMu(atMlMu) = atMlMup(ε˜tMlMu)rreqα− (1− atMlMu)βP (3.9)with QoS constraintnkl log(1 +|hHt;l;uvt;l;u|2∑u′∈Ut;l\{u} |hHt;l;uvt;l;u′ |252)≥ atMlMurreqPhtMlMu ∈ RtMlMuP u ∈ UtMlP t ∈ TkP l ∈ LP (3.10)where htMlMu,RtMlMu, and ε˜2tMlMu are the channel vector, CSI uncertainty set, and its normalized sizefor the l-th realization of U seqk , respectively. Then, we reformulate problem (3.8) as follows:maximizeaseqk MnkMpkMvseqkL∑l=1ωl∑t∈Tk∑u∈Ut;ln newtMlMu(atMlMu)− Ck(nkPpk) (3.11a)subject to atMlMu ∈ {0P 1}P u ∈ UtMlP t ∈ TkP l ∈ L (3.11b)constraints (3.6b), (3.6c), (3.8b), (3.10)Pwhere aseqk = (aseqkM1, . . ., aseqkML), aseqkMl = (akfMl, . . ., a2k513f−1Ml), and atMl = (atMlM1P . . . P atMlM|Ut;l|).Problem (3.11) is a mixed integer optimization problem due to integer variables aseqk andnk. We use branch-and-bound technique [41] to determine the optimal solution of aseqk . WeChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 26first relax each integer variable atMlMu ∈ {0P 1} to 0 ≤ atMlMu ≤ 1, and solve the relaxed problemto obtain nkPpkPvseqk , and relaxed aseqk . We randomly choose a variable atMlMu /∈ {0P 1}, the twonew constraints developed from this variable are atMlMu = 1 and atMlMu = 0, forming two childnodes of the current node. We then proceed to the node with the greatest optimal value andapply the same procedure. If there is an integer solution of aseqk with the greatest optimal valueamong other ending nodes, then the process stops. For the integer variable nk, we relax it to acontinuous variable and obtain the relaxed optimal solution of nk. Then, we simply comparethe optimal profits based on the two integer values of nk that are most close to the relaxedoptimal solution of nk, and pick the optimal integer solution.3.2.2 QoS Constraint Approximation and Semidefinite RelaxationBased on the branch-and-bound technique, we focus on solving problem (3.11) with therelaxation of integer variables at each node. The relaxed optimization problem is still difficultto be solved as QoS constraint (3.10) is nonconvex. To tackle this challenge, we introducea maximum interference threshold to achieve the QoS constraint approximation. The relaxedproblem of (3.11) is formulated as follows:maximizeφseqk Maseqk MnkMpkMvseqkL∑l=1ωl∑t∈Tk∑u∈Ut;ln newtMlMu(atMlMu)− Ck(nkPpk) (3.12a)subject to φtMlMu ≤|hHtMlMuvtMlMu|2I + σ2P htMlMu ∈ RtMlMuP u ∈ UtMlP t ∈ TkP l ∈ L (3.12b)∑u′∈Ut;l\{u}|hHtMlMuvtMlMu′ |2 ≤ IP htMlMu ∈ RtMlMuP u ∈ UtMlP t ∈ TkP l ∈ L (3.12c)Chapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 27nkl log (1 + φtMlMu) ≥ atMlMurreqP u ∈ UtMlP t ∈ TkP l ∈ L (3.12d)0 ≤ atMlMu ≤ 1P u ∈ U relaxtMl P t ∈ TkP l ∈ L (3.12e)atMlMu = ytMlMuP u ∈ UtMl\U relaxtMl P t ∈ TkP l ∈ L (3.12f)0 ≤ nk ≤ cP (3.12g)constraints (3.6c) and (3.8b)Pwhere φtMlMu is an auxiliary variable serving as a lower bound of the signal-to-interference-plus-noise ratio (SINR), φseqk = (φseqkM1P . . . PφseqkML), φseqkMl = (φkfMlP . . . P φ2k513f−1Ml), φtMl =(φtMlM1P . . . P φtMlM|Ut;l|). I is a predefined maximum interference threshold. The optimal solutionof problem (3.12) is required to guarantee that the interference experienced by each user isno larger than threshold I . By introducing φseqk and I , the QoS constraint (3.10) is relaxed asconstraints (3.12b) (3.12c) and (3.12d) [33]. We also have that U relaxtMl ∈ UtMl is the set of userswhose atMlMu is relaxed at the current node. ytMlMu ∈ {0P 1} is the value of atMlMu that has beendetermined at the current node, in which u ∈ UtMl\U relaxtMl , l ∈ L, t ∈ Tk.Due to the CSI uncertainty, constraints (3.12b) and (3.12c) involves infinite number ofconstraints, making it difficult to directly solve problem (3.12). To tackle this problem, weapply S-procedure [42] to transform constraints (3.12b) and (3.12c) into finite number of linearmatrix inequality (LMI) constraints. The S-procedure is introduced in Lemma 3.1:Lemma 3.1. (S-Procedure): Let A1, A2 ∈ HN , d1, d2 ∈ CN×1, and y1 y2 ∈ R. Consider theChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 28following two quadratic functions of vector x ∈ CN×1:f1(x) = xHA1x+ 2R{d1x}+ y1P (3.13)f2(x) = xHA2x+ 2R{d2x}+ y2. (3.14)The implication f1(x) ≤ 0⇒ f2(x) ≤ 0 holds if and only if there exists a θ ≥ 0 such thatθA1 d1dH1 y1−A2 d2dH2 y2 ≽ 0. (3.15)We denote that∆htMlMu = htMlMu− h¯tMlMu. Then, by applying Lemma 3.1 to constraint (3.12b),we obtain the following implication:∆hHtMlMuIAT∆htMlMu + 2R{0H△hu} − ε2tMlMu ≤ 0⇒ −∆hHtMlMu(vtMlMuvHtMlMu)∆hu − 2R{(vtMlMuvHtMlMuhtMlMu)H∆hu}− hHtMlMu(vtMlMuvHtMlMu)htMlMu + φtMlMu(I + σ2) ≤ 0P (3.16)if and only if there exists a υtMlMu ≥ 0 such that the following LMI holds:υtMlMuIAT 0AT0HAT −φtMlMu(I + σ2)− υtMlMuε2tMlMu+ QHtMlMuVtMlMuQtMlMu ≽ 0Pu ∈ UtMlP t ∈ TkP l ∈ LP (3.17)Chapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 29where QtMlMu = [IAT h¯tMlMu], VtMlMu = vtMlMuvHtMlMu, ε2tMlMu is the size of the CSI uncertainty set forthe l-th traffic realization.Similarly, by applying Lemma 3.1 to constraint (3.12c), we obtain the following implica-tion:∆hHtMlMuIAT∆htMlMu + 2R{0H△hu} − ε2tMlMu ≤ 0⇒ ∆hHtMlMu(∑u′∈Ut;l\{u}VtMlMu′)∆htMlMu + 2Re{((∑u′∈Ut;l\{u}VtMlMu′)htMlMu)H∆htMlMu}+ hHtMlMu(∑u′∈Ut;l\{u}VtMlMu′)htMlMu − I ≤ 0P (3.18)if and only if there exists a ξtMlMu ≥ 0 such thatξtMlMuIAT 0AT0HAT I − ξtMlMuε2tMlMu−QHtMlMu (∑u′∈Ut;l\{u}VtMlMu′)QtMlMu ≽ 0Pu ∈ UtMlP t ∈ TkP l ∈ L. (3.19)Then, problem (3.12) is equivalent tominimizeokCk(nkPpk)−L∑l=1ωl∑t∈Tk∑u∈Ut;ln newtMlMu(atMlMu) (3.20a)subject to constraints (3.6c), (3.12d)−(3.12g), (3.17), (3.19),nk|Ut;l|∑u=1Tr(BHtBtVtMlMu) ≤ pkMtP b ∈ BP t ∈ TkP l ∈ L (3.20b)υtMlMu ≥ 0P u ∈ UtMlP t ∈ TkP l ∈ L (3.20c)Chapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 30ξtMlMu ≥ 0P u ∈ UtMlP t ∈ TkP l ∈ L (3.20d)VtMlMu ≽ 0P u ∈ UtMlP t ∈ TkP l ∈ LP (3.20e)Rank(VtMlMu) ≤ 1P u ∈ UtMlP t ∈ TkP l ∈ LP (3.20f)where ok = (φseqk Pυseqk P ξseqk P aseqk P nkPpkPVseqk ), Bt , (0ft−1P 1P0fT−t) ⊗ IA, so that vtMlMuMt =BtvtMlMu and Tr(vtMlMuMtvHtMlMuMt) = Tr(BHtBtVtMlMu). For constraint (3.20f), Rank(VtMlMu) = 0happens when atMlMu = 0, meaning that the service request of user u is rejected and there is noresource assigned to that user.Problem (3.20) is still nonconvex due to constraint (3.20f). We adopt semidefiniterelaxation (SDR) [43] by removing constraint (3.20f) to arrive at a tractable problem, givenas:minimizeokCk(nkPpk)−L∑l=1ωl∑t∈Tk∑u∈Ut;ln newtMlMu(atMlMu)subject to constraints (3.6c), (3.12d)−(3.12g), (3.17), (3.19),(3.20b)−(3.20e).(3.21)For the optimal solution of problem (3.21), if the rank of Hermitian matrix VtMlMu is no largerthan one for all u ∈ UtMl, l ∈ L and t ∈ Tk, then problems (3.20) and (3.21) have the sameoptimal solution and the same optimal objective value. Otherwise, the optimal objective valueof problem (3.20) serves as the lower bound of the optimal objective value of problem (3.21).The tightness of the SDR in problem (3.21) is revealed in the following theorem:Theorem 3.1. Assuming problem (3.21) is feasible, a solution for VtMlMu, ∀ u ∈ UtMl, l ∈ L,Chapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 31t ∈ Tk, can always be constructed to guarantee that the rank of the beamforming matrix is lessthan or equal to one.Proof. Please refer to Appendix A for the proof of Theorem 3.1. Theorem 3.1 illustrates that if the optimal beamforming solution of problem (3.21) doesnot meet constraint (3.20f), we can solve problem (A.1) in Appendix A to obtain the optimalsolution for beamforming matrix, denoted as V¯tMlMu, u ∈ UtMl, t ∈ Tk, l ∈ L, which satisfiesconstraint (3.20f). The rank zero solution of V¯tMlMu indicates that the service request of user uis rejected, i.e., atMlMu = 0. The rank one solution of V¯tMlMu indicates that the service request ofuser u is accepted, i.e., atMlMu = 1. Then, the optimal beamforming vector, denoted as v¯tMlMu, isthe principle eigenvector of matrix V¯tMlMu.3.2.3 Primal-Relaxed Dual TechniqueProblem (3.21) is still difficult to be solved directly due to the nonconvexity of constraints(3.12d) and (3.20b). One observation is that by fixing variables nk and pk, problem (3.21)is convex with respect to variables φseqk , υseqk , ξseqk , aseqk , Vseqk . By fixing variables φseqk , υseqk ,ξseqk , aseqk , Vseqk , problem (3.21) is linear with respect to nk and pk. One classical techniqueto solve this type of optimization problem is the primal-relaxed dual technique [44]. The keyidea of primal-relaxed dual technique is to convert the original problem into primal and relaxeddual subproblems that provide valid upper and lower bounds respectively on the global optimalobjective value.Chapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 32We fix variables φseqk Pυseqk P ξseqk P aseqk PVseqk and solve the primal problem of (3.21) withrespect to nk and pk, which is formulated as follows:minimizenkMpkCk(nkPpk)−L∑l=1ωl∑t∈Tk∑u∈Ut;ln newtMlMu(atMlMu)subject to constraints (3.6c), (3.12d), (3.12g), (3.20b).(3.22)The obtained optimal value is denoted as f upper, serving as an upper bound of problem(3.21). The corresponding solutions of Lagrange multipliers for constraints (3.12d), (3.20b),(3.12g), and (3.6c) are denoted as λ1MtMlMu, λ2MtMlMt, (for all u ∈ UtMl, b ∈ B, t ∈ Tk, l ∈ L), λ=, λ4,λ5Mt, λ6Mt (for all b ∈ B). We use λ as the vector of all Lagrange multipliers.In order to obtain the relaxed dual problem of problem (3.21), we derive the Lagrangian ofproblem (3.21) with constraints (3.12d), (3.20b), (3.12g), and (3.6c), given asa(φseqk P aseqk PVseqk P nkPpkPλ)= x1nk +∑t∈Bx2pkMt −L∑l=1ωl∑t∈Tk∑u∈Ut;l(atMlMup(ε˜tMlMu)rreqα− (1− atMlMu)β)−nkL∑l=1∑t∈Tk∑u∈Ut;lλ1MtMlMul log(1 + φtMlMu) +L∑l=1∑t∈Tk∑u∈Ut;lλ1MtMlMuatMlMurreq−∑t∈BpkMtL∑l=1∑t∈Tkλ2MtMlMt + nkL∑l=1∑t∈Tk∑t∈Bλ2MtMlMt∑u∈Ut;lTr(WHt WtktMlMu)−λ=nk − λ4c + λ4nk −∑t∈Bλ5MtpkMt −∑t∈Bλ6MtPt +∑t∈Bλ6MtpkMt= nkG1(φseqk PVseqk Pλ) +G2(aseqk PpkPλ)P (3.23)Chapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 33whereG1(φseqk PVseqk Pλ) = x1 −L∑l=1∑t∈Tk∑u∈Ut;lλ1MtMlMul log(1 + φtMlMu)+L∑l=1∑t∈Tk∑t∈Bλ2MtMlMt∑u∈Ut;lTr(BHtBtVtMlMu)− λ= + λ4PandG2(aseqk PpkPλ) =∑t∈BpkMt(x2 −L∑l=1∑t∈Tkλ2MtMlMt − λ5Mt + λ6Mt)−L∑l=1ωl∑t∈Tk∑u∈Ut;l(atMlMup(ε˜tMlMu)rreqα− (1− atMlMu)β)+L∑l=1∑t∈Tk∑u∈Ut;lλ1MtMlMuatMlMurreq − λ4c −∑t∈Bλ6MtPt.With the Lagrangian, we further haveinf0≤nk≤Na(φseqk P aseqk PVseqk P nkPpkPλ) = inf0≤nk≤NnkG1(φseqk PVseqk Pλ) +G2(aseqk PpkPλ)=c − δc2G1(φseqk PVseqk Pλ) +G2(aseqk PpkPλ)P (3.24)where δ ∈ {−1P 1} such that δG1(φseqk PVseqk Pλ) ≥ 0. It indicates that whenG1(φseqk PVseqk Pλ) ≤0, δ will be equal to −1, which is equivalent to have nk = c that achieves the minimization ofLagrangian over nk. When G1(φseqk PVseqk Pλ) ≥ 0, δ will be equal to 1, which is equivalent tohave nk = 0 that achieves the minimization of Lagrangian over nk.By fixing Lagrange variables λ and pk, based on the analysis in [44], we obtain the relaxedChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 34dual problem of problem (3.21) as follows:minimizeφseqk Mυseqk Mξseqk Maseqk MVseqk Mc − δc2G1(φseqk PVseqk )+G2(aseqk ) (3.25a)subject to constraints (3.12e), (3.12f), (3.17), (3.19)P (3.20c)−(3.20e)δG1(φseqk PVseqk ) ≥ 0P (3.25b)δ ∈ {−1P 1}. (3.25c)The optimal value of problem (3.25) is denoted as f lower, serving as a lower bound of problem(3.21). We iteratively solve the primal problem (3.22) and the relaxed dual problem (3.25) untilthe gap between the upper and lower bounds is below a predetermined threshold.We present a flow chart to depict the whole process of our problem transformation, asshown in Fig. 3.2. In order to efficiently solve the profit maximization problem, which isoriginally formulated as a two-stage stochastic programming problem, several transformationand approximation steps should be taken. The two-stage stochastic programming problemconsists of problems (3.6) and (3.7). We first transform the problem into deterministicoptimization problem (3.8). Due to the nonconvexity of revenue function (3.4), then wetransform the revenue function into a linear function with QoS constraint (3.10) by introducinga user admission control variable atMlMu for each user. Moreover, problem (3.8) can betransformed into problem (3.11). Due to the nonconvexity of QoS constraint (3.10), weintroduce a maximum interference threshold I to achieve QoS approximation. We also relax theinteger variables to continuous variables and obtain the relaxed optimization problem (3.12).Chapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 35Problems (3.6) (3.7)Problem (3.8) Transformation into a deterministic problemProblem (3.11) Revenue function transformationProblem (3.12) Problem (3.12) Problem (3.12) Problem (3.12) Problem (3.12) Branch-and-bound technique frameworkNode 1Node 2Node 3Node 4Node 5Problem (3.12) Node iQoS approximationProblem (3.20) S-procedureProblem (3.21) SDRProblem (3.25) Problem (3.22) Primal-relaxed dual techniqueFigure 3.2: The transformation and relaxation steps taken from problems (3.6) and (3.7) toobtain the solutions of profit maximization problem.Then, we apply the branch-and-bound technique to obtain the optimal integer solution of atMlMufor each user. In the framework of the branch-and-bound technique, we solve the relaxedoptimization problem (3.12) for each node. To solve this problem, we apply S-procedure andSDR to obtain problem (3.21), which can be solved by applying primal-relaxed dual technique.In Fig. 3.2, a bidirectional arrow represents a transformation into a equivalent problem. Theunidirectional arrows from problem (3.11) to problem (3.12), and from problem (3.20) toproblem (3.21), represent transformations involving approximations.3.2.4 Joint Resource Reservation and Allocation AlgorithmIn this section, we design the algorithm to achieve the two-timescale resource managementfor network slicing in C-RAN, with the objective of maximizing the profit of the tenant. Wefirst design the algorithm to depict the primal-relaxed dual technique for each node, which isChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 36shown in Algorithm 3.1. We then design the global algorithm for resource reservation andAlgorithm 3.1: Primal-Relaxed Dual Technique for Node i1 Input D(i), aseqk (i), and U relaxtMl (i), t ∈ Tk, l ∈ L.2 j := 1.3 Initialize φseqk (iP j)Pυseqk (iP j)P ξseqk (iP j)PVseqk (iP j) subject to constraints (3.17), (3.19),(3.20c)−(3.20e); Set ϵ := 10−=4 aseqk (iP j) := aseqk (i).5 f lower(iP j) := −∞, f upper(iP j) := 0.6 while |fuppor(iP j)− f lowor(iP j)| ≥ ϵ do7 Solve problem (3.22) with fixed φseqk (iP j), υseqk (iP j), ξseqk (iP j), aseqk (iP j),Vseqk (iP j),update nk(iP j + 1), pk(iP j + 1) and f upper(iP j + 1) with the optimal solutions.8 Solve relaxed dual problem (3.25) with fixed pk(iP j) and dual variables obtained inStep 7, with D(i) and U relaxtMl (i), t ∈ Tk, l ∈ L; update φseqk (iP j + 1), υseqk (iP j + 1),ξseqk (iP j + 1), aseqk (iP j + 1),Vseqk (iP j + 1), flower(iP j + 1).9 j := j + 1.10 end11 Return f upper(iP j), f lower(iP j), and optimal solution ok(iP j) := (φseqk (iP j),υseqk (iP j),ξseqk (iP j),Vseqk (iP j), aseqk (iP j), nk(iP j), pk(iP j)).intra-slice resource allocation in long timescale slot k = 0P . . . P K based on the branch-and-bound technique and integrate Algorithm 3.1 in the inner iteration. This algorithm is shown inAlgorithm 3.2.In Algorithms 3.1 and 3.2, we introduce set D(i) at node i to record the determined valueytMlMu and the corresponding index for the user admission control variable atMlMu. In Algorithm3.2, steps 7 − 14 depict the process to calculate the optimal solutions for the two child nodesgenerated from the last chosen node. Steps 15 − 18 depict the process of choosing theending node with the greatest optimal objective value and initializing the two child nodes.Theoretically, the worst case time complexity of Algorithm 3.2 is dominated by the branch-and-bound technique, and is O(2n), where n is the total number of user admission controlChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 37Algorithm 3.2: Global Algorithm for Resource Reservation and Intra-Slice ResourceAllocation in Long Timescale Slot k1 Set i := 1, in which i represents the index of the node of the branch-and-boundtechnique.2 Initialize the admission control decision vector aseqk (i) for the outer iteration byrandomly assigning a value within [0P 1] to each atMlMu(i), u ∈ UtMl, t ∈ Tk, l ∈ L.3 U relaxtMl (i) := UtMl, t ∈ Tk, l ∈ L.4 Initialize D(i) := ∅ to record the set of (ytMlMuP tP lP u) at the current node.5 Initialize Fnode := ∅ to record the optimal values and solutions of ending nodes and theindexes of the nodes./* Outer Iteration: Branch-and-Bound technique */6 while ∃ atMlMu(i) ̸∈ {0P 1}, ∀u, t, l, do7 s := 1.8 while s ≤ 2 do/* Inner Iteration: Primal-Relaxed Dual Technique */9 Perform Algorithm 3.1, with input D(i), aseqk (i), and U relaxtMl (i), t ∈ Tk, l ∈ L.10 f := fupper(i;j)5f lower(i;j)2.11 ok := ok(iP j).12 Fnode := Fnode⋃{(fPokP i)}.13 i := i+ 1, s := s+ 1.14 end15 (f ∗Po∗P i∗) := argminf∗{Fnode}; update aseqk (i) and aseqk (i+ 1) based on o∗;Randomly choose at∗Ml∗Mu∗ /∈ {0P 1} in o∗.16 D(i) := D(i∗)⋃{(0P t∗P l∗P u∗)}, D(i+ 1) := D(i∗)⋃ {(1P t∗P l∗P u∗)}.17 U relaxt∗Ml∗ (i) := U relaxt∗Ml∗ (i∗)\{u∗}, U relaxt∗Ml∗ (i+ 1) := U relaxt∗Ml∗ (i∗)\{u∗}.18 Fnode := Fnode\{(f ∗Po∗P i∗)}.19 end20 Return −f ∗.Chapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 38variables. However, in practice, the algorithm can run fast as only a small number of nodes aresearched before reaching the optimal solutions.3.3 Performance Evaluation3.3.1 Simulation Environment and Parameter SetupThe coverage area of the C-RAN network is 300 × 300 m2. It is divided into nine regions.Each region is 100× 100 m2 with an RRH at its center. Thus, the number of RRHs is 9. EachRRH is equipped with two antennas. The total bandwidth is 20 MHz, which is divided into 20sub-channels. The channel model consists of path loss and small scale fading which followsRayleigh fading. The reference distance for path loss estimation is 2m. The path loss exponentis 3.6. The mean channel vector h¯tMu of user u ∈ Ut in short timescale slot t ∈ Tk is determinedby the path loss. The noise power σ2 is −101 dBm, and the noise of each user follows thezero-mean complex Gaussian distribution with variance σ2. We set the interference thresholdI = 28σ2. The duration of each long timescale slot and short timescale slot are 20 minutesand 5 seconds, respectively. The sub-channel reservation cost x1 is set to be $0.05. The powerreservation cost x2 is set as $0.05. The reward α is $0.005. The penalty β is $0.003. The arrivalprocess of users follows Poisson distribution. The average user arrival rate χkMm, m ∈ M ischosen uniformly within [χ¯−∆χP χ¯+∆χ],∆χ = 1 (number of users per short timescale slot).The sojourn time of users follows the uniform distribution within [2P 10], the unit of which isa short timescale slot. The normalized size ε˜2tMlMu of CSI uncertainty set is chosen uniformlywithin [ε¯2 − ∆ε¯2P ε¯2 + ∆ε¯2], ∆ε¯2 = "22. In our simulation, dividing the coverage area intoChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 39disjoint regions is only for characterizing different statistics of user traffic in different regions.The RRHs located in different regions are still able to coordinate together to serve each user.3.3.2 Algorithm Properties2 4 6 8 10 12 14 16 18 20−60−50−40−30−20−1001020Index of the iteration in Algorithm 3.1Optimalvalue  Upper bound with χ¯ = 2Lower bound with χ¯ = 2Upper bound with χ¯ = 3Lower bound with χ¯ = 3Upper bound with χ¯ = 4Lower bound with χ¯ = 4Figure 3.3: Convergence of Algorithm 3.1 with different average user arrival rate χ¯ (number ofusers per short timescale slot), rreq = 1.5Mb/s, ε¯2 = 0.05.In this section, we evaluate the properties of the proposed algorithm. We first conductsimulations to evaluate the impact of user traffic on the convergence of Algorithms 3.1 and 3.2.The simulation results are shown in Figs. 3.3, 3.4 and 3.5. Fig. 3.3 shows the convergence ofAlgorithm 3.1. Each iteration represents the process of solving problems (3.22) and (3.25) toobtain an upper bound and lower bound. The algorithm converges when the gap between theupper bound and lower bound is smaller than a predetermined threshold. As the average userarrival rate χ¯ (number of arrived users per short timescale slot) increases, the convergence ratebecomes slower. This is because that larger χ¯ leads to a larger number of users in the coverageChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 405 10 15 20 25−6.8−6.78−6.76−6.74−6.72−6.7−6.68−6.66Index of the iteration in Algorithm 3.2OptimalvalueFigure 3.4: Outer iteration convergence of Algorithm 3.2, rreq = 1.5 Mb/s, ε¯2 = 0.05, χ¯ = 2(number of users per short timescale slot).area, thus a larger number of variables to be solved at each iteration in Algorithm 3.1. However,the difference among convergence rates under different χ¯ is negligible. So we can concludethat the user traffic variation only has a minor impact on the convergence of Algorithm 3.1.Figs. 3.4 and 3.5 show the outer iteration convergence of Algorithm 3.2 with the averageuser arrival rate χ¯ = 2 (number of arrived user per short timescale slot) and χ¯ = 4 (number ofarrived user per short timescale slot), respectively. Each iteration consists of obtaining theconverged solution of Algorithm 3.1 at the two child nodes generated from the last node,preceding to the node with the greatest optimal objective value, and generating two new childnodes. The algorithm converges when we obtain an integer solution of aseqk with the greatestoptimal objective value among all ending nodes. In Fig. 3.5, χ¯ = 4, which is larger than χ¯ = 2in Fig. 3.4. So, there is a larger number of users in the system, which leads to a larger numberChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 410 5 10 15 20 25 30 35 40 45 50−12−11.99−11.98−11.97−11.96−11.95Index of the iteration in Algorithm 3.2OptimalvalueFigure 3.5: Outer iteration convergence of Algorithm 3.2, rreq = 1.5 Mb/s, ε¯2 = 0.05, χ¯ = 4(number of users per short timescale slot).of user admission control variables of atMlMu. Therefore, for branch-and-bound technique, ittakes longer time to find integer solutions for all atMlMu, u ∈ UtMl, l ∈ L, t ∈ Tk. In this case, theconvergence rate in Fig. 3.5 is slower than that in Fig. 3.4. However, the convergence is stillfast in practice, compared with the theoretical worst case complexity ofO(2n). This is becausethat at the first iteration of Algorithm 3.2, atMlMu for those users with good channel quality aredirectly assigned to be one. Meanwhile, for those users with really bad channel quality, atMlMuare directly assigned to be zero. Then, the branch-and-bound technique only needs to justifythe optimal integer solutions of atMlMu for a small number of users.As discussed in Section 3.2.2, we can not get the exact optimal solution and the optimalprofit for two reasons. First, we introduced a maximum interference threshold. Second, weapplied semidefinite relaxation. The semidefinite relaxation has been analyzed in AppendixChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 4218 20 22 24 26 28 30 32 34 36 38 405678910111213Maximum interference threshold in multiples of noise, I/σProfitperlongtimescaleslot($)  χ¯ = 4 (number of users per short timescale slot)χ¯ = 3 (number of users per short timescale slot)χ¯ = 2 (number of users per short timescale slot)Figure 3.6: Impact of maximum interference threshold on optimal profit, rreq = 1.5 Mb/s,ε¯2 = 0.05.A. In this part, we focus on evaluating the impact of maximum interference threshold onthe optimal profit. As shown in Fig. 3.6, We find that the optimal profit increases andthen decreases slightly as I increases. Thus, a suitable I can be obtained by running offlinesimulations. We can also find that the changes of the optimal profit with different I/σ2 is notobvious, so the optimal profit is not very sensitive to the choice of I/σ2.3.3.3 Profit ComparisonWe evaluate the performance of the proposed scheme with four other baseline schemes. Inbaseline schemes I and II, we maximize the profit obtained by the tenant. But in baselinescheme I, we only consider the CSI uncertainty. In baseline scheme II, we only consider theuser traffic variation. We use these two baseline schemes to characterize the impact of usertraffic variation and CSI uncertainty on the optimal profit. In baseline schemes III and IV,Chapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 431 2 3 4 5 6 7 8246810121416Average user arrival rate χ¯ (number of users per short timescale slot)Profitperlongtimescaleslot($)  Proposed schemeBaseline scheme IBaseline scheme IIBaseline scheme IIIBaseline scheme IVFigure 3.7: Profit comparison with different average user arrival rate χ¯ (number of users pershort timescale slot), rreq = 1.5Mb/s, ε¯2 = 0.05.both CSI uncertainty and user traffic variation are considered, but the resource managementschemes are slightly different from our proposed scheme. In baseline scheme III, we maximizethe profit of the tenant while accepting all service requests from the users. This scheme isbased on several resource management schemes for C-RAN which do not consider the useradmission control [32,33]. In baseline scheme IV, we maximize the profit of the tenant. In thisscheme, the beamforming and user-centric RRH clustering are separated into two processes.User-centric RRH clustering is performed first for each new arrived user. Then, beamformingis designed for the user according to real-time network conditions.Fig. 3.7 shows that the profit of the proposed scheme is larger than the profit of the fourbaseline schemes under different average user arrival rate χ¯ (number of arrived users per shorttimescale slot). Meanwhile, with the increasing of χ¯, the superiority of the proposed schemeChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 440.5 1 1.5 2 2.5 3 3.50510152025Data rate requirement rreq (Mb/s)Profitperlongtimescaleslot($)  Proposed schemeBaseline scheme IBaseline scheme IIBaseline scheme IIIBaseline scheme IV16 %Figure 3.8: Profit comparison with different QoS requirements rreq (Mb/s), χ¯ = 3 (number ofusers per short timescale slot), ε¯2 = 0.05.in terms of the profit is more obvious compared with the four baseline schemes. It is becausethat higher revenue can be obtained from serving more users and the impact of the trafficvariation and CSI uncertainty become more significant. Moreover, as the average user arrivalrate increases, the increasing rate of the proposed scheme becomes slower. The reason for thisbehavior is that larger χ¯ leads to a larger number of users that may be close to each other in thecoverage area. To mitigate interference, more resources need to be reserved, leading to higherresource reservation cost. We also find that the profit of baseline scheme III is close to the profitof the proposed scheme when χ¯ is no larger than 5. The reason is that when the number of usersin the coverage area is small, the proposed scheme also tends to accept most of the users. Thegap between the proposed scheme and baseline scheme III is due to the fact that the proposedscheme will reject users with really bad channel quality to save resources. The gap betweenChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 450 0.05 0.1 0.15 0.2 0.2545678910Average normalized size ε¯2 of CSI uncertainty setProfitperlongtimescaleslot($)  Proposed schemeBaseline scheme IBaseline scheme IIBaseline scheme IIIBaseline scheme IVFigure 3.9: Profit comparison with different average normalized size ε¯2 of CSI uncertainty set,rreq = 1.5Mb/s, χ¯ = 3 (number of users per short timescale slot).the proposed scheme and baseline scheme IV is due to the fact that our proposed scheme ismore flexible to the network condition variations by designing user-centric RRH clustering andbeamforming simultaneously.Fig. 3.8 shows that the profit of the proposed scheme is larger than the profits of the fourbaseline schemes under different data rate requirement rreq. When rreq is large, the proposedscheme can achieve more than 16% profit improvement compared with the performance of thefour baseline schemes. It is because higher data rate leads to higher revenue per user, makingit more important to consider the user traffic variation and CSI uncertainty to obtain higherrevenue from all users.Fig. 3.9 shows that the proposed scheme achieves a higher profit compared with fourbaseline schemes under different choices of average normalized size ε¯2 of CSI uncertainty set.Chapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 46Meanwhile, for most choices of ε¯2, the proposed scheme can achieve a higher profit comparedwith baseline scheme II, which does not consider the CSI uncertainty. When ε¯2 is close to zero,the CSI uncertainty is not fully considered for QoS guarantee in the proposed scheme. Thus,the gap of the profits between these two schemes is close to zero. With the increase of ε¯2, higherprofit can be obtained by the proposed scheme according to revenue function (3.4). However,when ε¯2 is larger than 0.15, the profit will not increase further. It is because when ε¯2 is large,most of the CSI variations are considered in the CSI uncertainty set, and it is unnecessary tofurther increase ε¯2.3.3.4 Resource Reservation and Allocation Performance0 0.05 0.1 0.15 0.2 0.2544.555.566.577.58Average normalized size ε¯2 of CSI uncertainty setTotalreservedpower(W)  χ¯ = 2, uncertainty scenarioχ¯ = 3, uncertainty scenarioχ¯ = 4, uncertainty scenarioχ¯ = 2, certainty scenarioχ¯ = 3, certainty scenarioχ¯ = 4, certainty scenarioFigure 3.10: Total reserved power with different average user arrival rate χ¯ (number ofusers per short timescale slot) and different average normalized size ε¯2 of CSIuncertainty set, rreq = 1.5Mb/s.In this section, we evaluate the performance of the proposed scheme in terms of the resourceChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 472 4 6 8 10 12 14 16 18 20−300−250−200−150−100−500IterationPowerassignedtoacertaintuser(dBm/Hz)  RRH 1RRH 2RRH 3RRH 4RRH 5RRH 6RRH 7RRH 8RRH 9Figure 3.11: Power assigned to a user from all RRHs, rreq = 1.5Mb/s, χ¯ = 3 (number of usersper short timescale slot), ε¯2 = 0.05.reservation and allocation corresponding to different network conditions. Fig. 3.10 shows thedecision of power reservation under different conditions of user traffic and CSI uncertainty.The total amount of reserved power increases as the average user arrival rate χ¯ increases inorder to guarantee QoS requirements of more users. Meanwhile, the amount of reserved powerincreases as the average normalized size ε¯2 of CSI uncertainty set increases. When ε¯2 is gettingsmaller, the reserved amount of power will converge to the value of the reserved amount ofpower under the CSI certainty scenario. When ε¯2 is large, the increasing rate of the reservedamount of power will become small to avoid high power reservation cost.Fig. 3.11 shows the power allocated to a certain user in a certain short timescale slot from allthe RRHs by intra-slice resource allocation. From this figure, we notice that the power allocatedto the user varies with different RRHs. The RRHs that are close to the user will allocate moreChapter 3. Two-Timescale Resource Management for Network Slicing in C-RAN 48power to the user and the RRHs that are far away from the user will almost allocate no powerto the user to save energy and reduce resource reservation cost. Therefore, the beamformingdesigned in our proposed scheme can help achieve user-centric RRH clustering.49Chapter 4Conclusion and Future WorkIn this chapter, we conclude the thesis by summarizing the research work. We also suggestsome possible extensions for future wok.4.1 ConclusionIn this thesis, we proposed a resource management framework for network slicing in C-RAN from the perspective of each tenant. We maximized the profit of each tenant whilecharacterizing the user traffic variation and CSI uncertainty by proposing a two-timescaleresource management approach. To tackle two key challenges of user traffic variation and CSIuncertainty, we designed long timescale resource reservation with the statistical knowledgeof user traffic, and designed short timescale intra-slice resource allocation, which is adaptiveto arbitrary arrival/departure of users. The uncertainty set was applied for CSI uncertaintycharacterization to achieve robust intra-slice resource allocation for QoS guarantee. Weformulated the profit maximization as a two-stage stochastic programming problem. Wetransformed the stochastic programming problem to a deterministic optimization problem, andperformed QoS constraint approximation and semidefinite relaxation. We then applied branch-and-bound and primal-relaxed dual techniques to solve the problem. We conducted extensivesimulations to evaluate the performance of our scheme. Based on the simulation results, weChapter 4. Conclusion and Future Work 50concluded that the convergence of the proposed scheme can be fast. The proposed scheme canachieve higher profit compared with other baseline schemes.4.2 Future WorkIn terms of future work, possible extensions are as follows:1. In this thesis, we considered the required data rate as the QoS requirement needed to beguaranteed by each tenant. We proposed a profit model based on this QoS requirement.In 5G wireless systems, there are various types of services requiring different QoS.For example, ultra-reliable low-latency communications (URLLC) require low datatransmission delay and high probability of successful data transmission. Massivemachine type communications (mMTC) require efficient connectivity for a massivenumber of devices. Therefore, in our future work, we can consider new metrics fordesigning new QoS models under different types of services. Meanwhile, we canconsider different profit models with revenue and cost functions according to differenttypes of services.2. In this thesis, we considered the profit maximization for a single tenant, and assumed thatthe resource reservation requests from the tenant can always be satisfied. In the futurework, we can consider the scenario where multiple tenants coexist in the network. In thisscenario, due to the limitation of network resources, the resource reservation request fromeach tenant may not always be satisfied. Therefore, we can apply auction and biddingtechniques to formulate a new resource management framework for multiple tenants.Chapter 4. Conclusion and Future Work 513. In this thesis, we assume constant durations of each long timescale slot and each shorttimescale slot. In the future work, we can consider the durations of the long timescaleslot and short timescale slot as two variables to be optimized for achieving real-timeadaptation to network conditions while controlling the computation cost.52Appendix AProof of Theorem 3.1We follow approaches introduced in [45,46] to prove Theorem 3.1. For the optimal solution ofproblem (3.21), if the rank of any optimal beamforming matrix is larger than one, we considerthe following optimization problem:minimizeokx1nk +x2ai∑l∈L∑t∈Tk∑u∈Ut;lTr(VtMlMu)−L∑l=1ωl∑t∈Tk∑u∈Ut;ln newtMlMu(atMlMu)subject to constraints (3.6c), (3.12d)−(3.12g), (3.17), (3.19),(3.20a)−(3.20e).(A.1)Problem (A.1) is a transformation of problem (3.21) by replacing∑t∈B x2pkMt withu2Lf∑l∈L∑t∈Tk∑u∈Ut;l Tr(VtMlMu) in the objective function. In problem (3.21),∑t∈B x2pkMt in theobjective function represents the cost of power reservation, where∑t∈B pkMt serves as theupper bound for intra-slice power allocation in each short timescale slot for each realizationof U seqk . In problem (A.1), 1Lf∑l∈L∑t∈Tk∑u∈Ut;l Tr(VtMlMu) represents the average powerconsumption over short timescale slots, and u2Lf∑l∈L∑t∈Tk∑u∈Ut;lTr(VtMlMu) can be regardedas the corresponding cost. We now prove that by solving problem (A.1), we obtain the optimalbeamforming matrix, the rank of which is guaranteed to be either one or zero. Rank onesolution indicates that the service request of the corresponding user is accepted, i.e., atMlMu = 1.Rank zero solution indicates that the service request of the corresponding user is rejected, i.e.,Appendix A. Proof of Theorem 3.1 53atMlMu = 0.We first obtain the Lagrangian of problem (A.1) but only focus on the terms that are relatedwith out proof. The Lagrangian is given as follows:Γ(Vseqk Pφseqk Pυseqk P ξseqk P aseqk P nkPpkP)=∑l∈L∑t∈Tk∑u∈Ut;lTr(VtMlMu(x2aiIAT + nk∑t∈BϱtMlMtBHtBt −QtMlMuL1MtMlMuQHtMlMu+∑u′∈Ut;l\{u}QtMlMu′L2MtMlMu′QHtMlMu′ − L=MtMlMu+△P(A.2)where △ represents the collection of the terms in the Lagrangian that are not related with ourproof,  contains all the dual variables, ϱtMlMt is the dual variable for constraint (3.20b), L1MtMlMuis the dual matrix for constraint (3.17), L2MtMlMu is the dual matrix for constraint (3.19), L=MtMlMu isthe dual matrix for constraint (3.20e).We focus on the following KKT conditions that are related with our proof:∇Vt;l;uΓ(Vseqk Pφseqk Pυseqk P ξseqk P aseqk P nkPpkP)|okM  = OATP u ∈ UtMlP t ∈ TkP l ∈ L (A.3)L¯=MtMlMuV¯tMlMu = OATP u ∈ UtMlP t ∈ TkP l ∈ L (A.4)(p1MtMlMu(φ¯seqk P υ¯seqk ) +QHtMlMuV¯tMlMuQtMlMu)L¯1MtMlMu = OAT51P u ∈ UtMlP t ∈ TkP l ∈ L (A.5)L¯2MtMlMu ≽ 0P u ∈ UtMlP t ∈ TkP l ∈ L (A.6)V¯tMlMu ≽ 0P u ∈ UtMlP t ∈ TkP l ∈ L (A.7)ϱ¯tMlMu ≥ 0P u ∈ UtMlP t ∈ TkP l ∈ L (A.8)υ¯tMlMu ≥ 0P u ∈ UtMlP t ∈ TkP l ∈ LP (A.9)Appendix A. Proof of Theorem 3.1 54where o¯k , (V¯seqk P φ¯seqk P υ¯seqk P ξ¯seqk P a¯seqk P n¯kP p¯k) and ¯ represents the optimal solutions ofproblem (A.1) and the corresponding optimal dual variables, respectively. We also havep1MtMlMu(φ¯seqk P υ¯seqk ) =υ¯tMlMuIAT 0AT0HAT −φ¯tMlMu(I + σ2)− υ¯tMlMuε2tMlMu . (A.10)By considering (A.3) and (A.4), we haveXtMlMuV¯tMlMu = QtMlMuL¯1MtMlMuQHtMlMuV¯tMlMuP u ∈ UtMlP t ∈ TkP l ∈ LP (A.11)where XtMlMu , u2Lf IAT + n¯k∑t∈B ϱ¯tMlMtBHtBt +∑u′∈Ut;l\{u}QtMlMu′L¯2MtMlMu′QHtMlMu′ . Moreover,since n¯k > 0, ϱ¯tMlMt ≥ 0 andQtMlMu′L¯2MtMlMu′QHtMlMu′ ≽ 0, we haveXtMlMu ≻ 0. Then, we haveRank(V¯tMlMu)= Rank(XtMlMuV¯tMlMu)= Rank(QtMlMuL¯1MtMlMuQHtMlMuV¯tMlMu)≤ min{Rank(QtMlMuL¯1MtMlMuQHtMlMu)PRank(V¯tMlMu)} P u ∈ UtMlP t ∈ TkP l ∈ L. (A.12)To evaluate Rank(QtMlMuL¯1MtMlMuQHtMlMu), we post-multiplyQHtMlMu to (A.5) and obtainptMlMu(φ¯seqk P υ¯seqk )L¯1MtMlMuQHtMlMu +QHtMlMuV¯tMlMuQtMlMuL¯1MtMlMuQHtMlMu = O2AT513×ATPu ∈ UtMlP t ∈ TkP l ∈ L. (A.13)We then pre-multiply the left-hand side of (A.13) by [IAT 0AT]. Considering that QtMlMu =[IAT h¯tMlMu], we haveAppendix A. Proof of Theorem 3.1 55[IAT 0AT]ptMlMu(φ¯seqk P υ¯seqk )L¯1MtMlMuQHtMlMu + [IAT 0AT]QHtMlMuV¯tMlMuQtMlMuL¯1MtMlMuQHtMlMu = OAT⇔ υ¯tMlMu[IAT 0AT]L¯1MtMlMuQHtMlMu + IATV¯tMlMuQtMlMuL¯1MtMlMuQHtMlMu = OAT⇔ υ¯tMlMuQtMlMuL¯1MtMlMuQHtMlMu + V¯tMlMuQtMlMuL¯1MtMlMuQHtMlMu = υ¯tMlMu[OAT h¯tMlMu]⇔ (υ¯tMlMuIAT + V¯tMlMu)QtMlMuL¯1MtMlMuQHtMlMu = υ¯tMlMu[OAT h¯tMlMu]P u ∈ UtMlP t ∈ TkP l ∈ L. (A.14)Based on the derivation in (A.14), we now calculate the rank of matrix QtMlMuL¯1MtMlMuQHtMlMu.Without the loss of generality, we consider two cases of υ¯tMlMu. The first case is that υ¯tMlMu ̸= 0,and the second case is that υ¯tMlMu = 0.Case 1: υ¯tMlMu ̸= 0. According to (A.7) and (A.9), we can claim that the inverse matrix ofυ¯tMlMuIAT + V¯tMlMu exists. Based on the derivation in (A.14), we haveRank(QtMlMuL¯1MtMlMuQHtMlMu)= Rank(υ¯tMlMu(υ¯tMlMuIAT + V¯tMlMu)−1[OAT h¯tMlMu])≤ Rank([OAT h¯tMlMu]) = 1. (A.15)Then we haveRank(V¯tMlMu) ≤ min{Rank(QtMlMuL¯1MtMlMuQHtMlMu)PRank(V¯tMlMu)} ≤ 1Pu ∈ UtMlP t ∈ TkP l ∈ L. (A.16)For user u ∈ UtMl, if the service request is rejected, i.e., a¯tMlMu = 0, according to constraints(3.12d), (3.17) and (3.19), it is possible that Rank(V¯tMlMu) = 0, indicating that no power will beAppendix A. Proof of Theorem 3.1 56assigned to the user. If the service request of user u ∈ UtMl is accepted, i.e., a¯tMlMu = 1, in orderto guarantee QoS requirement (3.12d), Rank(V¯tMlMu) = 1 is required.Case 2: υ¯tMlMu = 0. In this case, we haveV¯tMlMuQtMlMuL¯1MtMlMuQHtMlMu = OAT. (A.17)According to (A.11), we haveV¯tMlMuXtMlMu = OAT. (A.18)Since that XtMlMu ≻ 0, we have V¯tMlMu = OAT. For user u ∈ UtMl, l ∈ L, t ∈ Tk whose servicerequest is accepted, i.e., a¯tMlMu = 1, V¯tMlMu = OAT contradicts the QoS constraint (3.12d). 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