RAN Slicing for Massive IoT and Bursty URLLC Service Multiplexing: Analysis and Optimization
Peng Yang, Xing Xi, Tony Q. S. Quek, Jingxuan Chen, Xianbin Cao, Dapeng Wu
aa r X i v : . [ c s . N I] A p r RAN Slicing for Massive IoT and BurstyURLLC Service Multiplexing: Analysis andOptimization
Peng Yang, Xing Xi, Tony Q S Quek,
Fellow, IEEE , Jingxuan Chen, XianbinCao,
Senior Member, IEEE , Dapeng Wu,
Fellow, IEEE
Abstract
Future wireless networks are envisioned to serve massive Internet of things (mIoT) via some radioaccess technologies, where the random access channel (RACH) procedure should be exploited for IoTdevices to access the networks. However, modelling of the dynamic process of RACH of massive IoTdevices is challenging. To address this challenge, we first revisit the frame and minislot structure of theradio access network (RAN). Then, we correlate the RACH request of an IoT device with its queuestatus and analyze the evolution of the queue status. Based on the analysis result, we derive the closed-form expression of the random access (RA) success probability of the device. Besides, considering theagreement on converging different services onto a shared infrastructure, we investigate the RAN slicingfor mIoT and bursty ultra-reliable and low latency communications (URLLC) service multiplexing.Specifically, we formulate the RAN slicing problem as an optimization one to maximize the total RAsuccess probabilities of all IoT devices and provide URLLC services for URLLC devices in an energy-efficient way. A slice resource optimization (SRO) algorithm exploiting relaxation and approximationwith provable tightness and error bound is then proposed to mitigate the optimization problem.
Index Terms
Massive IoT, random access channel, bursty URLLC, RAN slicing
P. Yang and T. Q S Quek are with the Information Systems Technology and Design, Singapore University of Technology andDesign, 487372 Singapore.X. Xi, J. Chen, and X. Cao are with the School of Electronic and Information Engineering, Beihang University, Beijing 100083,China, and also with the Key Laboratory of Advanced Technology, Near Space Information System (Beihang University), Ministryof Industry and Information Technology of China, Beijing 100083, China.D. Wu is with the Department of Electrical and Computer Engineering, University of Florida, Gainesville FL 32611 USA.
I. I
NTRODUCTION
With the explosive growth of the Internet of Things (IoT), massive IoT (mIoT) devices, thenumber of which is predicted to reach 20.8 billion by 2020, will access the wireless networks forimplementing advanced applications. These applications include e-health, public safety, smarttraffic, virtual navigation/management, and environment monitoring. To address the IoT market,the third-generation partnership project (3GPP) has identified mIoT as one of the three mainuse cases of 5G and has already initiated several task groups to standardize several solutionsincluding extended coverage GSM (EC-GSM), LTE for machine-type communication (LTE-M),and narrowband IoT (NB-IoT) [1], [2]. For establishing massive connections among wirelessnetworks and mIoT devices, the research on reliable and efficient access mechanisms shouldbe prioritized. In accomplishing the massive connections, when an IoT device wants to transmitsignals in the uplink, it randomly chooses a random access (RA) preamble from an RA preamblepool and transmits it through an RA channel (RACH). If more than one device tries to accessa base station (BS) simultaneously, then interference occurs at the remote radio head (RRH).
A. Prior arts
During the past few years, a rich body of works [3]–[5] on RA mechanisms has been developedto mitigate interference and improve the RA success probability or reduce the access delay of anIoT device. For example, the work in [3] proposed to improve the RA success probability of anIoT device by exploiting a distributed queue mechanism and then proposed an access resourcegrouping mechanism to reduce the access delay caused by the queuing process of the distributedqueue mechanism. The work in [4] proposed a novel scheme to increase RA success probability.First, this scheme increased the number of preambles at the first step of the RA procedureby utilizing a spatial group mechanism. Second, it improved resource utilization through non-orthogonally allocating uplink channel resources at the second step of the RA procedure. Besides,to reduce the access delay, a grant-free non-orthogonal RA system relying on the accurate useractivity detection and channel estimation was proposed in [5].Most of the studies [3]–[5], however, assumed that network resources were reserved for theIoT service and did not study the case of the coexistence of IoT service and many other servicessuch as enhanced mobile broadband (eMBB) and ultra-reliable and low latency communications(URLLC). The research of the coexistence of IoT service and other services is essential asfuture networks are convinced to integrate various services with different latency, reliability, and throughput requirements into a shared physical infrastructure rather than deploying individualnetwork solution for each type of service [6]. What is more, owing to the shared characteristicof network resources, some conclusions obtained in the case of providing sole IoT service maybecome inapplicable if multiple types of services are required to be supported by the networks.Network slicing is considered as a promising technique in future networks to converge variousservices onto a shared physical infrastructure via partitioning the infrastructure into multiplenetwork slices, where a network slice is defined as an end-to-end virtual network runningon the infrastructure, capable of providing a negotiated service quality [7]. Recently, manyslicing frameworks have been developed to provide performance guarantees for IoT or massivemachine-type communications (mMTC) service, eMBB service and URLLC service [8]–[11]. Forexample, instead of slicing the radio access network (RAN) via orthogonal resource allocationamong various services, the work in [8], [9] studied the advantages of allowing for non-orthogonalRAN resources sharing in uplink communications from a set of mMTC, eMBB, and URLLCusers to the same BS. The work in [10] developed a two-level scheduling process to allocatedynamically dedicated bandwidth to each slice based on workload demand and slices’ quality ofservice (QoS) requirement such that flexible resource allocation could be implemented. Besides,in [11], we proposed to orchestrate network resources for a network slicing system to guaranteemore reliable URLLC and energy-efficient eMBB service provisions.
B. Motivation and contributions
Unlike the work in [3]–[5], [8]–[11], this paper simultaneously analyzes the RA process forthe mIoT service and studies the RAN slicing for the mIoT service included service multiplexing.This study is highly challenging because i) performance requirements of a massive number ofIoT devices should be satisfied. Yet, the typical 5G cellular IoT, NB-IoT can admit only 50,000devices per cell [12], and the 5G new radio (NR) technique can only connect a great numberof devices by deploying costly ultra-dense heterogeneous networks; ii) RAN slicing operation(e.g., activating and releasing slices) has to be conducted in a timescale of minutes to hours tokeep pace with the upper layer network slicing. In the process of slicing upper layer networks,some functions (e.g., radio resource control function) and protocols (e.g., RAN protocol stacks)should be activated and configured, which are time-consuming [7]. However, wireless channelsgenerally change in a timescale of millisecond to seconds. The RAN slicing should tackle theissue of operating RAN slicing based on time-varying channels, called a two timescale issue [13]; iii) compared with the resource allocation problem for the eMBB service, the resourceallocation problem for the bursty URLLC service where URLLC packets arrive in burst may bemore challenging due to the stringent high reliability and low latency requirement.These challenges motivate us to investigate the RAN slicing for mIoT and bursty URLLCservice provision to maximize the utility of mIoT slices and that of bursty URLLC slices.The main contributions of this paper can be summarized as the following: 1) We revisit theframe and minislot structure for mIoT transmissions to accommodate more RA requests from amassive number of IoT devices; 2) We adopt a queueing model to capture the IoT packet arrival,accumulation and departure processes and analyze the queue evolution process by employingprobability and stochastic geometry theories. Based on the analysis result, we derive the closed-form expression of the RA success probability of a randomly chosen IoT device; 3) We definemIoT slice utility and bursty URLLC slice utility and formulate the RAN slicing for mIoTand bursty URLLC service multiplexing as a resource optimization problem. The objective ofthe optimization problem is to maximize the total mIoT and URLLC slice utilities, subject tolimited physical resource constraints. The solution of this problem is difficult due to the existenceof indeterministic objective function and thorny non-convex constraints and the requirement oftackling a two timescale issue as well; 4) To mitigate this thorny optimization problem, wepropose a slice resource optimization (SRO) algorithm. In this algorithm, we first exploit asample average approximate (SAA) technique and an alternating direction method of multipliers(ADMM) to tackle the indeterministic objective function and the two timescale issue. Then, asemidefinite relaxation (SDR) scheme joint with a Taylor expansion scheme are leveraged toapproximate the non-convex problem as a convex one. The tightness of the SDR scheme andthe error bound of the Taylor expansion are also analyzed.It is noteworthy that the optimization method is exploited to investigate the service multiplexingof mIoT and URLLC in this paper. As the optimization method has a powerful generalizationability, this paper can be extended to the multiplexing of more types of services.II. S
YSTEM MODEL
We consider a coordinated-multipoint-enabled (CoMP-enabled) RAN slicing system for mIoTand bursty URLLC multiplexing service provision. From the viewpoint of infrastructure compo-sition, the system mainly includes one baseband unit (BBU) pool and multiple RRHs connectingto the BBU via fronthaul links. From the perspective of network slicing, two types of inter-slices, i.e., mIoT slices and URLLC slices, are exploited in this system with S I and S u denoting themIoT slice set and URLLC slice set. We focus on the modelling of uplink IoT data transmission inmIoT slices and the modelling of downlink URLLC data transmission in URLLC slices. IoT de-vices (e.g., water meters and wearable e-health devices) are spatially distributed in R accordingto an independent homogeneous Poisson point process (PPP) Φ s = { u i,s ; s ∈ S I , i = 1 , , . . . } with intensity λ Is , where u i,s is the i -th IoT device’s location in the s -th mIoT slice. Thereare N u URLLC devices (e.g., remote-controlled robot sensors) that are randomly and evenlydistributed in R . The RRHs are spatially distributed in R according to an independent PPP Φ R = { v j ; j = 1 , , . . . } with intensity λ R , where v j represents the location of the j -th RRH.The number and locations of IoT devices and RRHs will be fixed once deployed. Besides, eachRRH is equipped with K antennas, and each device is equipped with a single antenna. The totalnetwork bandwidth W of the system is limited and shared by mIoT slices and URLLC slices. Aflexible frequency division multiple access (FDMA) technique is utilized to achieve the inter-sliceand intra-slice interference isolation [13]. In mIoT network slices, each IoT device is assumedto connect to its geographically closest RRH; thus, the cell area of each RRH constitutes aVoronoi tessellation. Just like [14], the co-channel inter-cell interference received by each RRH isassumed as a part of thermal noise mainly because of the intra-slice (or mIoT slice) interferenceisolation, the long-distance fading, and the severe wall penetration loss. Therefore, we focuson the analysis of the intra-cell interference in mIoT slices. In URLLC network slices, RRHscooperate to transmit signals to a URLLC device to improve its signal-to-noise ratio (SNR).Besides, in view of the architecture, the CoMP-enabled RAN slicing system consists of fourparts including end devices, RAN coordinator (RAN-C), network slice management, and networkproviders. The system time is discretized and partitioned into time slots and minislots with a timeslot consisting of T minislots. At the beginning of each time slot, the RAN-C will decide whetherto accept or reject received network slice requests defined later after checking available resourceinformation (e.g., physical resource blocks (PRBs)) and computing. If a slice request is accepted,network slice management will be responsible for creating or activating corresponding types ofvirtual slices and configuring RAN protocol stacks, the processes of which are time-consumingand usually in a timescale of minutes to hours. Next, if a slice request admission arrives, networkproviders will find the optimal servers and paths to deploy virtual network functions to satisfythe end-to-end QoS requirements of the slice. Meanwhile, at the beginning of each minislot, eachactive IoT device may try to connect to its associated RRH, and RRHs will generate cooperated beamformers pointing at URLLC devices based on sensed channel coefficients.Based on the above mentioned network slice concept, especially from the viewpoint of theslice’s QoS requirement, we can define a mIoT slice request as follows. Definition 1.
A mIoT slice request is defined as a tuple { λ Is , θ ths } for any slice s ∈ S I , where θ ths is the signal-to-interference-plus-noise ratio (SINR) threshold for an RRH to successfully decodepackets (including preamble packets and IoT data packets) sent from an IoT device in s . In this paper, mIoT slice requests are assumed to be always accepted by the RAN-C. Althoughwe give this assumption, it can be released by adding an slice access request indicator, e.g., a s ∈ { , } ; if network resources are adequate and the RAN-C admits the slice request, we let a s = 1 ; otherwise a s = 0 . RRHs will assign IoT devices to different slices according to thereceived SINR. The SINR threshold configured for all IoT devices in a slice is similar. For an IoTdevice in s , if it has the opportunity to send its endogenous arrival packets to the correspondingRRH, then it will randomly select a preamble (e.g., orthogonal Zadoff-hu sequences) from aBBU-maintained preamble pool and transmit the preamble to the RRH. Just like the literature[15], [16], although the whole connection establishment process usually follows an RA four-stepprocedure [17], we assume that a connection between the IoT device and the RRH is set upif the preamble can be successfully transmitted. In other words, the RA success probability isregarded as the probability of successfully transmitting a preamble in this paper. Definition 2.
A bursty URLLC slice request is defined as four tuples { I us , D s , α, β } for slice s ∈S u , where I us is the number of URLLC devices in s , D s is the transmission latency requirementof each URLLC device in s , α and β are the packet blocking probability threshold and thecodeword error decoding probability threshold of each URLLC device, respectively [11]. In this definition, URLLC devices are grouped into |S u | clusters according to the transmissionlatency requirement of each device. As URLLC packets may arrive in burst and network resourcesallocated to URLLC slices may be inadequate, URLLC packets may experience blocking. Thepacket blocking probability threshold is then involved as a QoS requirement of URLLC slices.Besides, owing to the low latency requirement, URLLC packets should be immediately scheduledupon arrival; thus, URLLC slice requests should always be accepted by the RAN-C. Although wehave this assumption, not all devices in URLLC slices can be served owing to limited networkresources, as presented in detail in Section IV. We next analyze the RA success probability. III. A
NALYSIS OF
RA S
UCCESS P ROBABILITY
A. Arrival, accumulation and departure of IoT packets
For a typical IoT device, we leverage a queue maintained in the device to capture thearrival, accumulation and departure of IoT packets. During minislot t , a Poisson distributionwith intensity (or new arrivals) µ w,s ( t ) is exploited to model the random, mutually independentendogenous packet arrivals in an IoT device in slice s . Once arrived, new packets will not besent out immediately in general and will enter a queue in the IoT device, which is modelledas an M/M/k queue with unlimited capacity, to wait for their scheduling. In the
M/M/k queue, packets will be scheduled according to the first-come, first-served (FCFS) basis. Besides,to facilitate the analysis of the queue evolution process, we consider the slotted-ALOHA RAprotocol although there are many other RA protocols such as non-orthogonal and coded RAprotocols. Owing to the RA behavior of the ALOHA protocol, new arrivals during t will onlybe counted at minislot t + 1 . Thus, the accumulated number of packets N a,s ( t ) in the queue ofa randomly selected IoT device in slice s during t is determined by the accumulated number ofpackets and the number of new arrivals during t − and whether the preamble of the device canbe successfully decoded by its associated RRH. The work in [16] presented a queue evolutionmodel based on single packet transmission. We extend [16] to the case of multiple packetstransmission, and (1) shows an evolution model of N a,s ( t ) for all s ∈ S I with N a,s ( t ) = , t = 1[ N w,s ( t − − (RA succeeds) x s ] + , t = 2[ N a,s ( t −
1) + N w,s ( t − − (RA succeeds) x s ] + , t ≥ (1)where N w,s (1) is the number of new arrivals in the st minislot, ( · ) is a function equaling toone if the corresponding RA succeeds; otherwise, ( · ) = 0 . x s = a log (1 + θ ths ) /L packets atthe head of the queue will be popped out if ( · ) = 1 , where a is the size of the tone spacing ina single tone mode [2], L is the IoT packet length; otherwise, they will not. [ x ] + = max( x, .At minislot t , based on the model in (1), for a randomly selected IoT device in slice s ∈ S I ,the probability that its maintained queue is not empty can be defined as P ne,s ( t ) = P { N a,s ( t ) > } (2)(2) explicitly shows that new arrivals at t will not be sent out immediately, which is reflected in(1). (2) is significantly different from the work in [16], which defined the non-empty probability T m = P { N m Cum + N m New > } , where N m Cum was the number of accumulated packets, and N m New denoted the number of new arrivals in the m -th slot. This definition shows that new arrivalsduring the m -th slot have the probability of sending out immediately.Next, we describe the packet departure process combined with a frame and minislot structurefor the mIoT service. As mentioned above, partly due to the limitation on the frame and minislotstructure, NB-IoT and LTE-M can only admit 50,000 devices. For NB-IoT, only one PRB witha bandwidth of KHz in the frequency domain is allocated for the IoT service, and eachphysical channel occupies the whole PRB. For LTE-M, although the physical channels are timeand frequency multiplexed, it only reserves six in-band PRBs with a total bandwidth of . MHzin the frequency domain for the IoT service. Thus, the frame and minislot structure for mIoTtransmissions should be revisited if more RA requests from IoT devices want to be accepted.Fig. 1 depicts a frame and minislot structure for mIoT transmissions. Although it depictssome essential channels, we do not discuss their correlations to the considered problem as thedetailed research on the physical layer supporting the mIoT service is out of our scope. In thisstructure, both the frequency division multiplexing (FDM) scheme and code division multiplexing(CDM) scheme are leveraged to admit more IoT devices in a way of alleviating mutual deviceinterference. Particularly, the FDM scheme can alleviates signal interference through orthogonalfrequency allocation, and the CDM scheme mitigates the co-channel signal interference viareducing the cross-correlation of simultaneous transmissions. For a mIoT slice s ∈ S I , eachsubframe includes F s orthogonal uplink physical RA channels (PRACHs). A single tone modewith a tone spacing of size of a MHz is adopted for each uplink PRACH, which indicates thateach PRACH occupies a PRB. At the beginning of each minislot, an active IoT device, i.e., an IoTdevice whose queue is non-empty, will randomly choose a preamble from a set of non-dedicatedRA preambles of size ξ and transmit the preamble through a randomly selected PRACH. For eachpreamble, it has an equal probability /ξ to be chosen by each IoT device. Similarly, each PRACHhas an equal probability /F s to be selected. Thus, the average number of IoT devices in mIoTslice s ∈ S I choosing the same PRACH and the same preamble is λ Is /ξF s . Notably, a greater ξF s may significantly reduce signal interference experienced at each RRH. Then, a questionshould be tackled: How many PRBs should be reserved for mIoT transmissions?
To improve theresource utilization, the resource allocated to mIoT should be determined based on requirementsof mIoT and other coexistence services. It motivates us to optimize the resources orchestratedfor the mIoT service discussed in Section V, except for the RACH procedure analysis. (cid:53)(cid:3)(cid:39)(cid:3) (cid:19)(cid:3) (cid:20)(cid:3) (cid:21)(cid:3) (cid:22)(cid:3) (cid:23)(cid:3) (cid:24)(cid:3) (cid:25)(cid:3) (cid:26)(cid:3) (cid:27)(cid:3) (cid:28)(cid:3) (cid:19)(cid:3) (cid:20)(cid:3) (cid:21)(cid:3) (cid:22)(cid:3) (cid:23)(cid:3) (cid:24)(cid:3) (cid:25)(cid:3) (cid:26)(cid:3) (cid:27)(cid:3) (cid:28)(cid:3) (cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3) (cid:39)(cid:3) (cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3) (cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3) (cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3) (cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3)(cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3) (cid:39)(cid:3) (cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3) (cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3) (cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3) (cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3) (cid:39)(cid:3) (cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3) (cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3) (cid:39)(cid:3) (cid:53)(cid:3)(cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3) (cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3) (cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3) (cid:39)(cid:3) (cid:53)(cid:3) (cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3) (cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3) (cid:39)(cid:3)(cid:53)(cid:3)(cid:39)(cid:3) (cid:53)(cid:3)(cid:39)(cid:3) (cid:54)(cid:88)(cid:69)(cid:73)(cid:85)(cid:68)(cid:80)(cid:72)(cid:86)(cid:3)(cid:11)(cid:87)(cid:76)(cid:80)(cid:72)(cid:12)(cid:3) (cid:51) (cid:53) (cid:37) (cid:3) (cid:11) (cid:73) (cid:85) (cid:72)(cid:84)(cid:88)(cid:72)(cid:81) (cid:70)(cid:92) (cid:12) (cid:3)
Fig. 1. The frame and minislot structure. ’R’ and ’D’ denote the resource block reserved for preamble and IoT data transmission.The preamble in ’R’ also reflects the usage of a code division multiplexing scheme. PBCH, PSS and SSS represent the PRBsfor physical broadcast channel, primary synchronization signal and secondary synchronization signal transmission, respectively.
B. Access control scheme
In a mIoT network slice, as the slotted-ALOHA protocol allows all active IoT devices torequest for RA at the beginning of each minislot without checking channel statuses, IoT devicesmay simultaneously transmit preambles. It may incur severe slice congestion that may lower theRA success probabilities of IoT devices and degrade the system performance. Access control hasbeen considered as an efficient proposal of alleviating congestion [18], and many access controlschemes such as access class barring (ACB), power ramping and back-off schemes [16] havebeen proposed. As we aim at investigating the performance difference between a network slicingsystem without access control and with access control, we adopt the following two schemes [16]:1)
Unrestricted scheme: each active IoT device requests the RACH at the beginning of minislot t without access restriction; 2) ACB scheme: at the beginning of t , each active IoT device drawsa random number q ∈ [0 , and can request the RACH only when q < P ACB , where P ACB isan ACB factor determined by RRHs based on the slice congestion condition.With the introduced access control schemes, the probability that the RACH requests of arandomly selected IoT device in slice s ∈ S I are not restricted at minislot t is defined as P nr,s ( t ) = P { Unrestricted RACH requests } (3)For all s ∈ S I at any minislot t , we have P nr,s ( t ) = 1 for the unrestricted scheme and P nr,s ( t ) = P ACB for the ACB scheme.
C. Derivation of RA success probability
For an RRH, if its received preamble SINR is no less than a preset SINR threshold, thenthe preamble is considered to be successfully transmitted. Just like [16], we do not investigate the well-investigated preamble collision issue. Owing to the channel deep fading and severeco-channel interference, an IoT device may experience uplink preamble transmission outage. Atminislot t , for a randomly selected active IoT device in s ∈ S I , its RA success probability isdefined as P s ( t ) = P { SIN R s ( t ) ≥ θ ths } ≥ π s (4)where SIN R s ( t ) denotes the preamble SINR experienced at an RRH associating with the IoTdevice, π s denotes a threshold of the required RA success probability.We utilize a power-law path-loss model to calculate the path-loss between an IoT deviceand its RRH in mIoT slices and utilize a truncated channel inversion power control schemeto eliminate the ‘near-far’ effect. In this model, the IoT device’s transmit power decays at therate of r − ϕ with r representing the propagation distance and ϕ denoting the path-loss exponent.In the power control scheme, IoT devices associated with the same RRH compensate for thepath-loss to maintain that the average received signal power of an IoT device at the RRH equalsto a threshold ρ o . Without loss of generality, the cutoff threshold ρ o is set to be the same for allRRHs, and we perform the analysis of RA success probability on an RRH located at the origin.According to Slivnyak’s theorem [19], the analysis holds for a generic RRH located at a genericlocation. For a randomly selected IoT device with a non-empty queue in s ∈ S I , the preambleSINR experienced at the RRH located at the origin can take the form SIN R s ( t ) = ρ o h o / ( σ + I s ( t )) (5)where σ represents the noise power, I s ( t ) denotes intra-cell interference received at the RRH,the useful signal power equals to ρ o h o due to the truncated channel inversion power control[20] with h o denoting the channel power gain between the IoT device and the RRH. Notethat the channel power gain experienced at a generic RRH is related to the spatial locationsof both the RRH and its associated IoT devices. Nevertheless, we drop the spatial indices fornotation lightening. Besides, just like [20], all channel gains are assumed to be known and beindependent of each other, independent of the spatial locations, symmetric and are identicallydistributed (i.i.d.). Considering both the particular IoT device deployment environment and theconvenience of theoretical analysis, the Rayleigh fading is assumed, and the channel power gain h o is assumed to be exponentially distributed with unit mean.The intra-cell interference received at the origin RRH can take the following form I s ( t ) = X m ∈ u Is \{ o } ( p m || d m || − ϕ = ρ o ) ( N a,s ( t ) > ( f m = f o ) ρ o h m (6) where u Is is the set of IoT devices connecting to the origin RRH in s ∈ S I , o is the randomlyselected IoT device associated with the RRH at the origin, p m denotes the transmit power ofthe m -th IoT device, || d m || is the distance between the m -th IoT device and the origin RRH, f o denotes the preamble and channel chosen by the randomly selected IoT device, f o = f m indicatesthat the randomly selected IoT device and the m -th IoT device select the same preamble andchannel. The st ( · ) (from left to right) on the right-hand-side of (6) indicates that the receivedsignal power of an interfering device at the origin RRH equals to ρ o . The nd ( · ) denotes thatthe queue of an interfering device is non-empty. The rd ( · ) indicates that an interfering deviceselects the same preamble and channel as the randomly selected IoT device.Then, for the randomly selected IoT device in s ∈ S I , we can rewrite (4) as follows P s ( t ) = P { h o ≥ θ ths ρ o ( σ + I s ( t )) } ( a ) = E h exp n − θ ths ρ o ( σ + I s ( t )) oi = e − θthsρo σ L I s ( t ) (cid:16) θ ths ρ o (cid:17) (7)where (a) follows from the full probability law over I s ( t ) , and L I s ( t ) ( · ) denotes the Laplacetransform (LT) of the probability density function (PDF) of the random variable I s ( t ) . Note thatthe notation L I s ( t ) ( · ) is a terminology that is a slight abuse of subscript I s ( t ) .The following lemma characterizes the LT of aggregate interference I s ( t ) . Although thework in [16] obtained an expression of the LT of aggregate interference I Intra ( t ) , we derivea significantly different result. In [16], the obtained L I Intra ( t ) ( γ th /ρ ) was a quasi-convex functionover the system bandwidth allocated to IoT devices. In this paper, the obtained L I s ( t ) (cid:0) θ ths /ρ o (cid:1) isthe difference of two quasi-convex functions that significantly increases the difficulty of solvingan optimization problem, which aims at improving IoT devices’ total RA success probabilitiesand is presented in Section V. Lemma 1.
For the origin RRH, based on the packet evolution model in (1), the LT of its receivedaggregate interference from active IoT devices associated with it is given by L I s ( t ) ( ̟ s ) = 1 + ̟ s ρ o (1 + α s ̟ s ρ o / (1 + ̟ s ρ o )) . − ̟ s ρ o (1 + α s ) . (8) where ̟ s = θ ths /ρ o , α s = P nr,s ( t ) P ne,s ( t ) λ Is / (3 . λ R ξF s ) .Proof. See Appendix A.By substituting (8) into (7), we can obtain a mathematical expression of P s ( t ) . The expression,however, is not in the closed-form as it is a function of P ne,s ( t ) the closed-form expression ofwhich is not obtained. Next, we attempt to derive the closed-form expression of P ne,s ( t ) . D. Derivation of non-empty probability
According to the definition of non-empty probability, P ne,s ( t ) is correlated with N a,s ( t ) . Thus,we theoretically analyze P ne,s ( t ) as the following.From (1), we can observe that N a,s (1) = 0 for all s ∈ S I ; thus, at the st minislot, we have P ne,s = P { N a,s > } = 0 (9)where we write x t instead of x ( t ) to lighten the notation.The following lemma presents the closed-form expression of the non-empty probability of arandomly selected IoT device served by the origin RRH when minislot t > . It is noteworthythat the derived expression of P ne,s ( t ) is completely different from that in [16] as this paper and[16] define different non-empty probabilities. Lemma 2.
The number of accumulated packets of a randomly selected IoT device served by theorigin RRH at minislot t > may be approximately Poisson distributed. Therefore, based on themodel in (1), for any mIoT slice s ∈ S I , we approximate the number of accumulated packets N ta,s at t as a Poisson distribution with intensity µ ta,s , which is given by µ ta,s = h µ t − w,s + µ t − a,s − P t − s (cid:16) − e − µ t − w,s − µ t − a,s (cid:17)i + (10) Then, the probability that the queue of the device is non-empty at t can be written as P tne,s = 1 − e − µ ta,s (11) Proof.
See Appendix B.Combined with (7), (8) and (11), the closed-form expression of P s ( t ) ( s ∈ S I ) can be obtained.IV. R ESOURCE ORCHESTRATION FOR BURSTY
URLLC
SLICES
In this section, we aim at addressing the following question:
How to orchestrate networkresources of the CoMP-enabled RAN slicing system to satisfy the QoS requirements of burstyURLLC devices?
During minislot t , a compound Poisson process, where arrivals happen in bursts (or batches,i.e., several arrivals can happen at the same instant) and the inter-batch times are independentand exponentially distributed [21], is utilized to model the number of bursty URLLC packetsaggregated at each RRH. The intensity of the exponential distribution is set to be one batch.The number of new arrivals in each batch is subject to an independent homogeneous Poisson distribution with intensity λ = { λ s ; s ∈ S u } , where λ s denotes the intensity of new arrivals in abatch destined to devices belonging to URLLC slice s . Once arrived, new URLLC arrivals willenter a queue maintained by an RRH to be immediately scheduled. An M/M/W u queueingsystem with limited bandwidth W u is exploited to model the queue. Without loss of anygenerality, we assume that each RRH maintains the same queue due to the exploration ofcooperated transmission. In the queue, a packet destined to URLLC device i ∈ I us , s ∈ S u willbe allocated with a block of network bandwidth ω ui,s ( t ) for a period of time d s ≤ D s at minislot t . Owing to stochastic variations in the bursty packet arrival process, the limited bandwidthmay not be enough to serve new arrivals occasionally. As such, URLLC packet blocking mayhappen. To reduce the probability of URLLC packet blocking, the PRB of the URLLC frame inthe frequency domain should be narrowed while widening it in the time domain [22]. In this way,the number of concurrent transmissions will be increased, and the packet blocking probabilityis reduced. As the tolerable communication latency of a URLLC device i ∈ I us in s ∈ S u is D s ,we can scale up d s and choose d s and ω ui,s ( t ) at minislot t using the following equation d s = D s and ω ui,s ( t ) = b ui,s ( t ) r ui,s ( t ) / ( κD s ) (12)where r ui,s ( t ) denotes channel uses for transmitting a URLLC packet [22], κ is a constantreflecting the number of channel uses per unit time per unit bandwidth of FDMA frame structureand numerology, b ui,s ( t ) ∈ { , } is an indicator that indicates whether the device i in s can beserved at minislot t . As mentioned above, because network resources are limited and sharedby all network slices, not all URLLC devices can be guaranteed to be served at every minislotalthough the RAN-C will always accept the URLLC slice requests. If the QoS requirement of i in s is satisfied at t , then the device i can be served by the slice s , and we let b ui,s ( t ) = 1 ;otherwise, i cannot be served by s , and we let b ui,s ( t ) = 0 . Certainly, we can adjust the slicepriority weight introduced in the following section to guide the resource orchestration for theentire URLLC devices coverage.Based on the result in (12), at minislot t , for a given M/M/W u queue with packet arrivalintensity λ , the minimum upper bound of bandwidth orchestrated for URLLC slices with apacket blocking probability α and a packet queueing probability ς can be given by [11] W u ( r ( t )) ≤ P s ∈S u P i ∈I us λ s b ui,s ( t ) r ui,s ( t ) κ + α − ςας − α s ( P s ∈S u P i ∈I us b ui,s ( t ) λ s D s )( P s ∈S u P i ∈I us λ s b ui,s ( t ) rui,s ( t )2 κ Ds )min s ∈S u { λ s D s } (13) As (13) is correlated with the channel use r ui,s ( t ) , we next discuss how to obtain its expression.For any URLLC slice s ∈ S u , during minislot t , let g ij,s ( t ) ∈ C K be the transmit beamformerpointing at the device i from the j -th RRH and h ij,s ( t ) ∈ C K be the channel coefficient betweenthe i -th URLLC device and the j -th RRH. The channel coefficient may change over minislots.However, it is assumed to be i.i.d. over each minislot and may not significantly change duringeach minislot. Then, the SNR received at device i in s at minislot t can be written as SN R ui,s ( t ) = | X j ∈J h H ij,s ( t ) g ij,s ( t ) | /φσ i,s (14)where J = { , , . . . , J } denotes the set of deployed RRHs, φ > is an SNR loss coefficientowing to imperfect channel status information acquisition [23], σ i,s denotes the noise power. Justlike [13], (14) does not include interference due to the usage of a flexible FDMA mechanism.For URLLC transmission, we resort to the capacity analysis for a finite blocklength channelcoding regime derived in [24] as the blocklength of a URLLC packet is short. For any device i ∈ I us , s ∈ S u , the number of transmitted information bits L ui,s ( t ) at minislot t using r ui,s ( t ) channel uses in an additive white Gaussian noise (AWGN) channel can be accurately correlatedwith the codeword error decoding probability β according to the following equation [24] L ui,s ( t ) ≈ r ui,s ( t ) C ( SN R ui,s ( t )) − Q − ( β ) q r ui,s ( t ) V ( SN R ui,s ( t )) (15)where C ( SN R ui,s ( t )) = log (1 + SN R ui,s ( t )) , V ( SN R ui,s ( t )) = ln (cid:16) − SNR ui,s ( t )) (cid:17) is thechannel dispersion, and Q ( · ) is the Q -function.The complicated expression of V ( SN R ui,s ( t )) in (15) significantly hinders the theoreticalanalysis of network resources orchestrated for URLLC slices. Fortunately, as V ( SN R ui,s ( t )) is upper-bounded by ln , we can obtain the closed-form expression in (16) of the minimumupper bound of r ui,s ( t ) with a codeword error decoding probability β by defining x = q r ui,s ( t ) and solving a quadratic equation with respect to (w.r.t) x . r ui,s ( t ) ≤ L ui,s ( t ) C ( SN R ui,s ( t )) + ( Q − ( β )) C ( SN R ui,s ( t ))) + ( Q − ( β )) C ( SN R ui,s ( t ))) s L ui,s ( t ) C ( SN R ui,s ( t ))( Q − ( β )) (16) It is noteworthy that a URLLC packet will usually be coded before transmission and the generated codeword will betransmitted in the air interface such that the transmission reliability can be improved. V. P
ROBLEM FORMULATION
Based on the above analysis, we next formulate the RAN slicing problem. In mIoT slices, eachRRH may transmit feedback signals to its connected IoT devices for the connection establishmentaccording to the RA four-step procedure [17]. Meanwhile, in URLLC slices, each RRH maytransmit URLLC packets to URLLC devices. As the transmit power E j ( j ∈ J ) of each RRHis limited, we have the following transmit power constraint X s ∈S I (1 + α g ) λ Is λ R ˆ E Ij + X s ∈S u X i ∈I us b ui,s ( t ) g H ij,s ( t ) g ij,s ( t ) ≤ E j (17)where ˆ E Ij is assumed to be a constant and denotes the transmit power of the j -th RRH forconnecting to its associated IoT devices over downlink, α g is a coefficient. As a PPP withintensity λ Is is utilized to model the distribution of IoT devices, the actual number of IoT devicesmay be greater than λ Is once deployed. As a result, the coefficient α g is introduced to reservetransmit power for exceeded IoT devices.In the RAN slicing system, as the total limited network bandwidth W will be shared by mIoTslices and URLLC slices, we have the following bandwidth constraint X s ∈S I (1 + α g ) ω s (¯ t ) + W u ( r ( t )) ≤ W (18)where ω s (¯ t ) correlated with F s by means of F s = ⌊ ω s (¯ t ) /a ⌋ denotes the bandwidth allocated tomIoT slice s ∈ S I . α g ω s (¯ t ) denotes a block of reserved bandwidth for exceeded IoT devices.In (18), F s is an integer, and some integer variable recovery schemes [25] can be leveraged toobtain the suboptimal F s . However, considering the high computational complexity of optimizingan integer variable and the utilization of the scheme of reserving additional bandwidth resources,we directly relax the integer variable into a continuous one, i.e., let F s = ω s (¯ t ) /a . Without lossof any generality, we regard ω s (¯ t ) as an independent variable below. Besides, as at least onePRB should be allocated to each type of mIoT slice s ∈ S I , we have ω s (¯ t ) ≥ a (19)Owing to the exploration of mIoT and bursty URLLC service multiplexing, we should or-chestrate network resources for all mIoT slices and URLLC slices to simultaneously maximizethe slice utilities. For a mIoT slice s ∈ S I , its primary goal is to offload as many packets aspossible from IoT devices. Thus, the number of accumulated packets in each IoT device shouldbe kept at a low level. Considering that a great RA success probability of an IoT device willlead to a low number of accumulated packets, we define the mIoT slice utility as follows. Definition 3.
Over a time slot of duration T , the mIoT slice utility is defined as the time-averageof RA success probabilities of IoT devices in all mIoT slices, which is given by ¯ U I = 1 T X Tt =1 U I ( t ) = 1 T X Tt =1 ˜ P ( t ) (20) where ˜ P ( t ) = P s ∈S I λ Is P s ( t ) P s ∈S I λ Is with the numerator λ Is P s ( t ) represents the expected sum of RAsuccess probabilities of IoT devices in slice s ∈ S I and the denominator P s ∈S I λ Is denoting anormalization coefficient. In (20), λ Is / P s ∈S I λ Is can be regarded as an intra-slice priority coefficient. A mIoT sliceserving more IoT devices will be orchestrated with more network resources.For a URLLC slice s ∈ S u , its primary objective is to maximize the slice gain reflected bythe parameters in the slice request at a low cost. Therefore, we define an energy-efficient utilityfor URLLC slices, as presented below. Definition 4.
Over one time slot of duration T , the bursty URLLC slice utility is defined as thetime-average energy efficiency for serving all URLLC devices, which is given by ¯ U u = T T P t =1 U u ( t ) = T T P t =1 P s ∈S u U us ( D s , g ij,s ( t ))= T T P t =1 P s ∈S u P i ∈I us b ui,s ( t )1 − e − Ds − ηT T P t =1 P s ∈S u P j ∈J P i ∈I us b ui,s ( t ) g H ij,s ( t ) g ij,s ( t ) (21) where η is an energy efficiency coefficient indicating a tradeoff between the URLLC slice gainand the RRH energy consumption. In (21), we characterize the slice gain by T P Tt =1 P s ∈S u P i ∈I us b ui,s ( t )1 − e − Ds as it reflects the latencyrequirements of bursty URLLC slices. Then, during a time slot, the original RAN slicing problemfor mIoT and URLLC service multiplexing can be formulated as follows. maximize { b ui,s ( t ) ,ω s (¯ t ) , g ij,s ( t ) } ¯ U I + ˜ ρ ¯ U u (22a) s . t . b ui,s ( t ) ∈ { , } , ∀ s ∈ S u , i ∈ I us (22b) constraints (4) , (17) − (19) are satisfied . (22c)where ˜ ρ is an inter-slice priority coefficient reflecting the priority of orchestrating networkresources for mIoT slices and URLLC slices.The solution of (22) is quite challenging mainly because i) indeterministic objective function :(22) should be optimized at the beginning of the st minislot. The time-averaged objective function of (22) can only be exactly computed according to the future channel information.Therefore, the value of the objective function is indeterministic at the beginning of the st minislot; ii) two timescale issue : the creation of a network slice is performed at a timescaleof time slot. Thus, the variable ω s (¯ t ) should be determined at the beginning of the time slot ¯ t and kept unchanged over the whole time slot. The channel, however, is time-varying. As aresult, the beamformer g ij,s ( t ) should be optimized at each minislot t . In summary, the variablesin (22) should be optimized at two different timescales; iii) thorny optimization problem : ateach minislot t , the constraint (4) is non-convex over ω s (¯ t ) , and the constraints (17), (18) arenon-convex over g ij,s ( t ) , which together lead to a non-convex problem.Next, we attempt to tackle these challenges by exploiting of an SAA technique [26], anADMM method [27], a relaxation scheme and an approximation scheme.VI. P ROBLEM SOLUTION WITH SYSTEM GENERATED CHANNEL
A. Sample average approximation and alternating direction method of multipliers
As mIoT slices and URLLC slices share the network resources, both ¯ U I and ¯ U u may bedetermined by channel coefficients experienced by URLLC slices. At each minislot t , due to thei.i.d. assumption on the channel coefficients of URLLC slices, we have T X Tt =1 U I ( t ) + 1 T X Tt =1 ˜ ρU u ( t ) ≈ E ˆ h h ˆ U I + ˜ ρ ˆ U u i (23)where ˆ h is the channel samples of URLLC slices collected at the beginning of the time slot ¯ t .Given a collection of channel samples { h m } with h m = [ h , m ; . . . ; h J,sm ; . . . ; h N u J, |S u | m ] and m ∈ M = { , . . . , M } . For notation lightening, we write x m instead of x ( m ) that representsa variable corresponding to the channel sample h m . Just like [11], as constraints (22b) and (22c)construct a nonempty compact set, the conclusion of Proposition 5.1 in [11] is applicable to thispaper by exploiting the SAA technique. The conclusion indicates that if the number of channelsamples M is reasonably large, then M P Mm =1 U Im + ˜ ρM P Mm =1 U um converges to E ˆ h [ ˆ U I + ˆ U u ] uniformly on the nonempty compact set almost surely. In other words, the SAA techniqueenables us to use the channel samples collected at the beginning of a time slot to approximatethe unknown channel coefficients over the time slot.Recall that the variable ω s (¯ t ) will be kept unchanged over the time slot ¯ t and the beamformer g ij,s ( t ) should be calculated at each minislot t , we can further consider (22) as a global consensusproblem [27], which can be effectively mitigated by an ADMM method. In (22), ω s (¯ t ) is a global consensus variable that should be maintained in consensus for all h m ( m ∈ M ), and g ij,sm thatis calculated based on h m is a local variable. The fundamental principle of ADMM is to imposeaugmented penalty terms characterizing global consensus constraints on the objective function ofan optimization problem. In this way, the local variables can be driven into the global consensuswhile still attempting to maximize the objective function. Let G i,sm = g i,sm g H i,sm ∈ R JK × JK , H i,sm = h i,sm h H i,sm ∈ R JK × JK , where g i,sm = [ g i ,sm ; . . . ; g iJ,sm ] ∈ C JK × and h i,sm =[ h i ,sm ; . . . ; h iJ,sm ] ∈ C JK × . By applying the property [28] G i,sm = g i,sm g H i,sm ⇔ G i,sm (cid:23) , rank( G i,sm ) ≤ and utilizing the conclusions of SAA and ADMM, we can approximate (22)as the following problem at the beginning of the time slot ¯ t . minimize { ω sm ,ω s (¯ t ) ,b ui,sm , G i,sm } M X m =1 (cid:20) − U Im M − ˜ ρU um M (cid:21) + M X m =1 X s ∈S I h ψ sm ( ω sm − ω s (¯ t )) + µ k ω sm − ω s (¯ t ) k i| {z } augmented penalty terms (24a) s . t . P sm ≥ π s , ∀ s ∈ S , m ∈ M (24b) X s ∈S I (1 + α g ) λ Is λ R ˆ E Ij + X s ∈S u X i ∈I us b ui,sm tr( Z j G i,sm ) ≤ E j , ∀ j ∈ J , m ∈ M (24c) X s ∈S I (1 + α g ) ω s (¯ t ) + W u ( r m ) ≤ W, m ∈ M (24d) G i,sm (cid:23) , ∀ s ∈ S u , i ∈ I us , m ∈ M (24e) rank( G i,sm ) ≤ , ∀ s ∈ S u , i ∈ I us , m ∈ M (24f) b ui,sm ∈ { , } , ∀ s ∈ S u , i ∈ I us , m ∈ M (24g) constraint (19) is satisfied (24h)where ψ sm is the Lagrangian multiplier, µ is a penalty coefficient, Z j is a square matrix with J × J blocks, and each block in Z j is a K × K matrix. In Z j , the block in the j -th row and j -th column is a K × K identity matrix, and all other blocks are zero matrices.(22) is now reduced to a deterministic single timescale problem (24). What is more, (24) canbe split into M separate problems that can be optimized in parallel as its objective function isseparable. Thus, the following ADMM-based framework from (25) to (27) can be exploited tomitigate (24). n ω ( k +1) sm , b u ( k +1) i,sm , G ( k +1) i,sm o = argmin { ω sm ,b ui,sm , G i,sm } L ( ω sm , G i,sm ) (25a) s . t . for the m − th sample , (24 b ) − (24 g ) are satisfied (25b) for the m − th sample , ω sm ≥ a, ∀ s ∈ S I (25c) ω ( k +1) s (¯ t ) = X Mm =1 ( ω ( k +1) sm + ψ ( k ) sm /µ ) /M, ∀ s ∈ S I (26) ψ ( k +1) sm = ψ ( k ) sm + µ (cid:0) ω ( k +1) sm − ω ( k +1) s (¯ t ) (cid:1) , ∀ s ∈ S I (27)where the augmented partial Lagrangian function ¯ L ( ω sm , G i,sm ) = − U I ( k ) m M − ˜ ρU u ( k ) m M + X s ∈S I h ψ ( k ) sm (cid:0) ω sm − ω ( k ) s (¯ t ) (cid:1) + µ (cid:13)(cid:13) ω sm − ω ( k ) s (¯ t ) (cid:13)(cid:13) i (28)This ADMM-based framework can be executed on multiple processors. Each processor isresponsible for optimizing (25) and calculating (27) with a global value as an input. (26) iscentrally updated in such a way that local variables converge to the global value, which is thesolution of (24). Unfortunately, (25) is a mixed-integer non-convex optimization problem as thereare zero-one variables, continuous variables and non-convex constraints in (25). As a result, theoptimization of (25) is quite difficult. We next discuss how to handle this hard problem. B. Alternative optimization
In this subsection, we exploit an alternative optimization scheme to handle the mixed-integernon-convex optimization problem. Specifically, we first assume that continuous variables areknown and attempt to mitigate a zero-one optimization problem. Given the zero-one variables, wethen try to optimize a non-convex optimization problem. The process is alternatively conducteduntil convergence.
1) URLLC device associations:
Given continuous variables { G ( k ) i,sm , ω ( k ) sm } at the k -th iteration,the association problem of URLLC devices in URLLC slices can take the following form { b u ( k +1) i,sm } = argmin { b ui,sm } − ˜ ρU u ( k ) m /M (29a) s . t . for m, (24 c ) , (24 d ) , (24 g ) are satisfied . (29b)This problem is non-linear and hard to be handled. In theory, an exhaustive algorithm canobtain the optimal solution of (29). The computation complexity of this algorithm is O (2 N u ) that may be impractical in implementation. Therefore, a greedy scheme of the computationalcomplexity O ( N u ) , which is summarized as the following, is proposed to obtain { b u ( k +1) i,sm } .a) initialize two device sets, i.e., candidate device set I u − = I u , association device set I u + = ∅ .b) select the device that maximizes ˜ ρU u ( k ) m /M from I u − , remove it from I u − , and add it to I u + .Given I u + , check the feasibility of (29). If (29) is feasible, then accept the device; otherwise,remove the device from I u + . Continue till I u − = ∅ . maxima inflection point 2inflectionpoint 1 (a) Curve of P sm . (b) nd Taylor expansion.Fig. 2. Curve of P sm and its nd Taylor expansion.
2) Joint bandwidth and beamforming optimization:
Given the obtained b u ( k +1) i,sm , (24) will bereduced to the following joint bandwidth and beamforming problem. n ω ( k +1) sm , G ( k +1) i,sm o = argmin { ω sm , G i,sm } L ( ω sm , G i,sm ) (30a) s . t . for m, (24 b ) − (24 f ) , (25 c ) are satisfied . (30b)In (30), the low-rank constraint (24f) is non-convex, and its objective function is not convexand even not quasi-convex w.r.t. ω sm , the tackling of which is quite tricky. To tackle the non-convex low-rank constraint (24e), we resort to the SDR technique. The primary procedures ofSDR are i) directly drop the low-rank constraint; ii) solve the optimization problem withoutthe low-rank constraint to obtain the solution; iii) if the obtained solution is not rank-one, thensome manipulations such as randomization/scale [29] are needed to perform on it to impose thelow-rank constraint; otherwise, its principal component is the optimal solution to (30).For the tricky objective function, we are reminded of the art of dealing with a non-convexfunction, i.e., study the structure of the function if it is non-convex. A crucial observation is that P sm is quasi-concave w.r.t. ω sm although the objective function is not quasi-convex w.r.t. ω sm .Therefore, we resort to the Taylor expansion to approximate the tricky objection function.The following analysis is based on two facts Fact 1: the value of the objective function of(30) is mainly determined by that of ˜ P ( k ) m (or U I ( k ) m ); Fact 2: the solution ω sm maximizing ˜ P ( k ) m must locate in the range of [ˆ ω lbsm , S ⋆sm ] ∀ s, m shown in Fig. 2, where ˆ ω lbsm = max { ω lbsm , a } with ω lbsm denoting the lower bound of ω sm satisfying the constraint (24b), S ⋆sm is the ω sm maximizing P sm , and the notation P sm | ω sm is utilized to explicitly indicate that P sm is a function of ω sm .Fact 1 holds because the linear terms w.r.t. ω sm will donate little to the objective functionas the consensus constraint is active. Besides, the quadratic terms pull local values towardsthe consensus; thus, they will also donate little to the objective function. Fact 2 holds because the total bandwidth is limited and shared. For example, given a value ω sm, ∈ [ S ⋆sm + δ ω , W ] with δ ω being a small positive constant, there must exist a value ω sm, ∈ [ˆ ω lbsm , S ⋆sm ] such that P sm | ω sm, = P sm | ω sm, . Thus, a small ω sm will be preferred as it indicates that more bandwidthcan be allocated to URLLC slices to further improve the objective function.For all s ∈ S I , it can be proved that P sm is concave in the interval ( a , a ] by evaluating thesecond-order derivative of P sm . Thus, we can use the nd Taylor expansion to approximate P sm in this interval. Considering that P sm is convex in the interval [ˆ ω lbsm , a ] , the st Taylor expansionis always used to obtain the lower bound of P sm . However, this interval is usually rather narrow,and the value of P sm in this interval is much lower than that in the interval ( a , a ] . What ismore, the error bound of the st Taylor expansion is greater than that of the nd expansion.Therefore, we explore the nd Taylor expansion to approximate P sm in the interval [ˆ ω lbsm , a ] .Fig. 2(b) shows an example of the nd Taylor expansion of P sm . Given a local point ω ( k,q ) m atthe q -th iteration, the Taylor expansion of − ˜ P ( k ) m at the local point can be given by − ˜ P ( k ) m ≈ − ˜ P ( k,q ) m − ∇ ˜ P ( k,q ) m ( ω m − ω ( k,q ) m ) T −
12 ( ω m − ω ( k,q ) m ) H ( ω ( k,q ) m )( ω m − ω ( k,q ) m ) T − R ( ω m ) (31)where ω m = [ ω m , . . . , ω |S I | m ] , ∇ ˜ P ( k,q ) m is the gradient of ˜ P ( k ) m over ω m at the local point ω ( k,q ) m with ∂P ( k ) sm ∂ω ( k,q ) sm = λ Is (1 + ̟ s ρ o ) e − ̟ s σ P s ∈S I λ Is " . y sm z s ω . k,q ) sm ( y sm z s + ω ( k,q ) sm ) . − . y sm ω . k,q ) sm ( y sm + ω ( k,q ) sm ) . (32)and H ( ω ( k,q ) m ) is a Hessian matrix with ∂ P ( k ) sm ∂ω k,q ) sm = λ Is (1+ ̟ s ρ o ) e − ̟sσ P s ∈S I λ Is (cid:20) . y sm z s ω . k,q ) sm ( y sm z s + ω ( k,q ) sm ) . − y sm z s ω . k,q ) sm ( y sm z s + ω ( k,q ) sm ) . (cid:21) + λ Is (1+ ̟ s ρ o ) e − ̟sσ P s ∈S I λ Is (cid:20) y sm ω . k,q ) sm ( y sm + ω ( k,q ) sm ) . − . y sm ω . k,q ) sm ( y sm + ω ( k,q ) sm ) . (cid:21) (33) ∂ P ( k ) sm ∂ω ( k,q ) sm ∂ω ( k,q ) s ′ m = 0 , ∀ s = s ′ (34) y sm = aP nr,sm P ne,sm λ Is / (3 . λ R ) , z s = θ ths / (1 + θ ths ) . Besides, we write ω . k,q ) sm rather than ( ω ( k,q ) sm ) . for lightening the notation. Lemma 3.
Let the function ˜ P ( k ) m : R |S I | → R be three times differentiable in a given interval [ˆ ω lbsm , S ⋆sm ] for all s ∈ S I , then the error bound of nd degree Taylor expansion of ˜ P ( k ) m at thelocal point ω ( k,q ) m with ω ( k,q ) sm ∈ [ˆ ω lbsm , S ⋆sm ] is given by R ( ω m ) = 13! (cid:20)X s ∈S I (cid:0) ω sm − ω ( k,q ) sm (cid:1) ∂∂ω ( k,q ) sm (cid:21) max n ˜ P ( k ) m | ˆ ω lbm , ˜ P ( k ) m | S ⋆m o (35) where ˆ ω lbm = [ˆ ω lb m , . . . , ˆ ω lb |S I | m ] and S ⋆m = [ S ⋆ m , . . . , S ⋆ |S I | m ] .Proof. See Appendix C.After conducting the nd Taylor approximation, the objective function becomes a convexfunction. Although the constraint (24b) is P sm ∀ s, m related, we need not to conduct the Taylorapproximation on (24b) as P sm is quasi-concave and unimodal. In fact, the probability constraint(24b) and (25c) are equivalent to the following inequality ˆ ω lbsm ≤ ω sm ≤ ω ubsm (36)where ω ubsm ≤ W represents the upper bound of ω sm satisfying (24b).Next, a low-complexity bisection-search-based scheme, the main procedures of which aredescribed below, is developed to obtain ω lbsm , S ⋆sm , and ω ubsm : a) let the function Q sm = P sm − π s .Perform the bisection search method [30] on Q sm = 0 to obtain ω lbsm and ω ubsm that are the twozero points of Q sm ; b) with the obtained ω lbsm and ω ubsm , find the maximum value S ⋆sm of P sm using the bisection search method again.According to the above analysis, at the q -th iteration, we can rewrite (30) as n ω ( k +1 ,q +1) sm , G ( k +1 ,q +1) i,sm o = argmin { ω sm , G i,sm } ¯ L ( q ) ( ω sm , G i,sm ) (37a) s . t . for m, (24 c ) − (24 e ) , (36) are satisfied . (37b)where ¯ L ( q ) ( ω sm , G i,sm ) = − M ˜ P ( k ) m − ˜ ρU u ( k ) m M + P s ∈S I h ψ ( k,q ) sm ( ω sm − ω ( k,q ) s (¯ t )) + µ k ω sm − ω ( k,q ) s (¯ t ) k i .In (37), the objective function is convex, (24c) is affine, and the constraint (24d) can be provedto be convex w.r.t. both ω sm and G i,sm [11]. Therefore, (37) is a convex problem that can beeffectively mitigated by some standard convex optimization tools such as CVX and MOSEK.Then we can summarize the main steps of mitigating the problem (24) in Algorithm 1. Lemma 4.
For all i ∈ I us , s ∈ S u , and m ∈ M , the obtained power matrix G ( k,q ) i,sm by Algorithm1 at the ( k, q ) -th iteration satisfies the low-rank constraint, i.e., the SDR for the power matrixutilized in Algorithm 1 is tight.Proof. See Appendix D.VII. O
PTIMIZATION OF BEAMFORMING WITH SYSTEM SENSED CHANNELS
In Section VI, we obtained a family of global consensus variables { ω s (¯ t ) } with the systemgenerated channel samples. The time-varying actual channels may require the re-optimization of Algorithm 1
ADMM-based bandwidth allocation algorithm Initialization:
Randomly initialize G (0 , i,s , { ω (0 , s } , let k max = 250 , q max = 250 , q = 0 , k = 0 , and generate channel samples { H i,sm } . repeat repeat Given G ( k,q ) i,sm , ω ( k,q ) sm , call the greedy scheme to obtain b u ( k,q +1) i,sm . Optimize (37) with obtained b u ( k,q +1) i,sm to achieve G ( k,q +1) i,sm and ω ( k,q +1) sm . Update q = q +1 . until Convergence or reach at the maximum iteration times q max . Let ω ( k +1 ,q +1) sm = ω ( k,q +1) sm , update ψ ( k +1) sm , ω ( k +1) s (¯ t ) using (27), (26), and update k = k + 1 . until Convergence or reach at the maximum iteration times k max .beamformers and device associations at each minislot. According to system sensed channels ateach minislot, we next discuss how to calculate beamforms and device associations.At each minislot t , given the global consensus variables { ω s (¯ t ) } , the original problem (22)will be reduced to the following problem maximize { b ui,s ( t ) , G i,s ( t ) } ˜ ρU u ( t ) (38a) s . t . constraints (17) , (18) , (22b) are satisfied . (38b)In (38), the channels are system sensed ones at t . According to the convexity analysis inSection VI, (38) is a mixed-integer non-convex programming problem with positive semidefinitematrices, which is hard to be mitigated. Therefore, the alternative optimization scheme presentedin subsection VI-B can be leveraged to achieve the solutions b ui,s ( t ) and G i,s ( t ) of (38). Lemma4 indicates that the achieved rank( G i,s ( t )) ≤ . Thus, we can obtain the beamformers g i,s ( t ) byperforming the eigendecomposition on G i,s ( t ) . To sum up, over a time slot ¯ t , the slice resourceoptimization algorithm designed for the RAN slicing system can be summarized in Algorithm2. VIII. S IMULATION RESULTS
A. Comparison algorithms and parameter setting
We compare the following three algorithms to verify the effectiveness of the proposed algo-rithm and to explain the impact of access control schemes on the RAN system performance Algorithm 2 slice resource optimization algorithm, SRO Initialization: { H i,s ( t ) } , ∀ i ∈ I u , s ∈ S u , and let P s ∈ [0 , , µ a,s = 0 , ∀ s ∈ S I . Call Algorithm 1 to obtain { ω s (¯ t ) } for all s ∈ S I . for t = 1 : T do Given { ω s (¯ t ) } , mitigate (38) by exploiting the alternative optimization scheme to obtainbeamformers { g i,s ( t ) } and URLLC device associations b ui,s ( t ) for all i ∈ I us , s ∈ S u . end for intuitively i) SRO algorithm that adopts the unrestricted access control scheme; ii) SRO-ACB I algorithm that utilizes the ACB access control scheme with P ACB = 0 . ; iii) SRO-ACB II algorithm that adopts the ACB access control scheme with P ACB = 0 . . However, we do notdiscuss how to select access control schemes and their parameters such as P ACB . We focus oninvestigating whether the conclusions of adopting access control schemes in an individual IoTservice system can still hold in the case of service multiplexing.The parameter setting is as follows: RRHs and IoT devices are deployed following independentPPPs in a one km area. URLLC devices are randomly and uniformly distributed in this area.There are three mIoT slices and two URLLC slices in the RAN slicing system. For the mIoTslices, set µ w,s ( t ) = [1 . , . , . , π s = 0 . , ∀ s, t , ϕ = 4 , L = 2000 bits, σ = − dBm, ρ o = − dBm, ˆ E Ij = 0 . mW, λ R = 3 RRHs/km , λ Is = 18000 IoT devices/km , ∀ s , a = 0 . MHz, the queue serving rate γ ths = a log (1 + θ ths ) , { γ ths } = { . , . , . } Kbits/minislot. Forthe URLLC slices, the transmit antenna gain at each RRH is set to be dB, and a log-normalshadowing path-loss model is used to simulate the path-loss between an RRH and a URLLCdevice with the log-normal shadowing parameter being dB. A path-loss is computed by h (dB) = 128 . . d , where d (in km) is the distance between a device and an RRH.Let L ui,s = 160 bits, σ i,s = − dBm, λ s = λ = 0 . packets/minislot, ∀ i, s , { I us } = { , } devices, and { D s } = { , } milliseconds, E j = 3 W, ∀ j [13]. Other system parameters areshown as follows: J = 3 , K = 2 , ˜ ρ = 1 , η = 100 , T = 60 , W = 60 MHz, M = 100 , φ = 1 . , ξ = 54 , α g = 0 . , κ = 5 . × − , α = 10 − , β = 2 × − , and ς = 2 × − [11]. B. Performance evaluation
To evaluate the comparison algorithms, the following performance indicators are utilized i)RA success probability P s ( t ) ; ii) expected queue length per IoT device at minislot t , E [ Q s ( t )] = Iteration times, k
Fig. 3. The convergence curve of the SRO algorithm. (a) t P s ( t ) SRO (s=1)SRO (s=2)SRO (s=3) (b) t P s ( t ) SRO (s=1)SRO (s=2)SRO (s=3) (c) t E [ Q s ( t ) ] ( pa ck e t s ) SRO (s=1)SRO (s=2)SRO (s=3) (d) t E [ Q s ( t ) ] ( pa ck e t s ) SRO (s=1)SRO (s=2)SRO (s=3)
Fig. 4. Trends of P s ( t ) and E [ Q s ( t )] . µ a,s ( t ) ; iii) total slice utility ¯ U that is the objective function of (22).We first evaluate the convergence mainly determined by that of the ADMM-based frameworkof the proposed SRO algorithm. We then leverage ∆ ω = P s ∈S I | ω ( k +1) s (¯ t ) − ω ( k ) s (¯ t ) | to evaluatethe convergence of the SRO algorithm. Fig. 3 illustrates the algorithm’s convergence. It showsthat SRO can converge after several iterations.We next plot the tendency of the RA success probability P s ( t ) and the corresponding expectedqueue length E [ Q s ( t )] during a time slot in Fig. 4. Fig. 4(a) and 4(c) show the tendency of P s ( t ) and E [ Q s ( t )] in the case of { γ ths } = { . , . , . } Kbits/minislot. Fig. 4(b) and 4(d) depictthe tendency of P s ( t ) and E [ Q s ( t )] in the case { γ ths } = { . , . , . } Kbits/minislot.From Fig. 4, we obtain the following interesting conclusions: the queue of each IoT deviceis not stable when the queue serving rate γ ths is small. In this case, the average queue lengthmonotonously increases over t . On the contrary, the queue of each IoT device is periodicallyflushed when a great queue serving rate is configured. The result that the maintained queue byeach IoT device can be emptied verifies the correctness of the analysis of the RA process.Let the IoT device intensity λ I = [900 n, n, n ] with n ∈ { , , . . . , } . Under theexistence of both mIoT and URLLC slices, we plot trends of the total slice utility ¯ U andbursty URLLC slice utility ¯ U u w.r.t. n in Fig. 5 to understand the impact of the mIoT sliceson the performance of all comparison algorithms. In this figure, B = [ b u , . . . , b u , b u , . . . , b u ] , ω I = [ ω ISRO , ω
IACB I , ω IACB II ] MHz with ω ISRO , ω IACB I and ω IACB II representing the bandwidthallocated to mIoT slices by executing SRO, SRO-ACB I , and SRO-ACB II algorithms, respectively,and ¯ U I = [ ¯ U ISRO , ¯ U IACB I , ¯ U IACB II ] with ¯ U ISRO denoting the achieved mIoT slice utility of SRO.The following observations can be obtained from Fig. 5: i) when n < , all algorithmsalmost obtain the same ¯ U , and the obtained utilities are robust to the average number of IoT
10 15 20 25 n +6.65% (a) total slice utility vs. n .
10 15 20 25 n B=[1 1 1 1 1 1 1 1]B=[1 1 1 1 1 1 1 1] B=[1 1 1 1 1 1 1 1]B=[1 1 1 1 1 1 1 1] (b) bursty URLLC slice utility vs. n .Fig. 5. Trends of the achieved total slice utilities and bursty URLLC slice utilities of all algorithms vs. n . devices; ii) when ≤ n ≤ , the conclusion changes. For the SRO algorithm, its achieved ¯ U decreases with an increasing n due to increasing interference. A great n , however, does not causea significant decrease in the obtained ¯ U by SRO-ACB I and SRO-ACB II . Thanks to the explorationof an access control scheme, both SRO-ACB I and SRO-ACB II can achieve greater ¯ U than SRO.For example, compared with SRO, SRO-ACB II improves ¯ U by . when n = 24 ; iii) when n = 26 , which means that the total average number of IoT devices reaches , devices, theRAN slicing system fails to create and manage mIoT slices as the QoS requirements of mIoTslices serving such a massive average number of devices cannot be simultaneously satisfied. Inthis case, all system resources are allocated to URLLC slices, and the maximum bursty URLLCslice utility is obtained; iv) as mIoT slices and URLLC slices share the system resources, anincreasing n results in a decreasing ¯ U u ; Besides, it is interesting to find that the two access-control-based algorithms may not outperform SRO in terms of obtaining ¯ U u . It indicates thatURLLC slices do not benefit from access control schemes of mIoT slices when changing n ; v)the RAN slicing system can always accommodate the QoS requirements of all URLLC devices.Next, to understand the impact of URLLC slices on the performance of all comparison algo-rithms, we plot the trends of ¯ U and the mIoT slice utilities obtained by all comparison algorithmsw.r.t. URLLC packet arrival rate λ with λ = { . , . , . , . . . , . , . } packets per unit timein Fig. 6. Similarly, the following notations are involved in Fig. 6: ω u = [ ω uSRO , ω uACB I , ω uACB II ] , ¯ U u = [ ¯ U uSRO , ¯ U uACB I , ¯ U uACB II ] with ω uSRO and ¯ U uSRO denoting the bandwidth allocated to URLLCslices and the URLLC slice utility obtained by running the SRO algorithm, respectively.From Fig. 6, we can observe that: i) the obtained ¯ U of all algorithms decrease with λ mainlydue to the decrease of the bursty URLLC slice utility. Two algorithms adopting the access controlscheme always achieve greater utilities ¯ U than SRO. For example, when λ = 5 , compared URLLC packet arrival rate, +29.41% (a) total slice utility vs. λ . URLLC packet arrival rate, +65.94% (b) mIoT slice utility vs. λ .Fig. 6. Trends of the achieved total slice utilities and IoT slice utilities of all algorithms vs. λ .
50 60 70 80 90 100
Bandwidth (MHz) +6.66% +2.41%
Fig. 7. Trend of achieved ¯ U vs. system bandwidth. m Fig. 8. Trend of achieved total slice utility vs. m . with the SRO algorithm, the obtained ¯ U of SRO-ACB II is increased by . ; ii) for allalgorithms, the computed bandwidth for URLLC slices increases with an increasing λ . However,their obtained URLLC slice utilities ¯ U u are reduced owing to the increase of energy consumption;iii) SRO-ACB II may achieve greater ¯ U than SRO-ACB I as a greater ¯ U I is obtained by reducingthe number of interfering IoT devices; iv) the obtained mIoT slice utilities ¯ U I of SRO-ACB I andSRO-ACB II are robust to the URLLC packet arrival rate. The obtained ¯ U I of SRO decreaseswith an increasing λ ; v) an important observation is that the ¯ U I of the access-control-basedSRO-ACB I algorithm is . times that of the SRO algorithm when λ = 5 . It explicitly reflectsthat mIoT slices can still benefit from access control schemes even though λ is changed.Figs. 5 and 6 illustrate the situation of a given total system bandwidth. We next change thetotal bandwidth W and plot its impact on the obtained ¯ U of all algorithms in Fig. 7.The following conclusions can be obtained from Fig. 7: i) when W = 45 MHz, the QoSrequirements of all IoT devices cannot be simultaneously satisfied. As a result, the total bandwidthis allocated to URLLC slices; ii) when W locates in the range of (45 , MHz, the achievedtotal slice utilities ¯ U of SRO and SRO-ACB I increase with W . Owing to the utilization of theaccess control scheme, SRO-ACB I and SRO-ACB II obtain higher ¯ U than SRO. For example, d +2.26% Fig. 9. Trend of achieved total slice utility vs. d . Energy efficiency coefficient, +2.17%
Fig. 10. Trend of achieved total slice utility vs. η . compared with the SRO algorithm, the SRO-ACB II algorithm improves the achieved ¯ U by . when W = 50 MHz; iii) when
W > MHz, all algorithms cannot remarkably improve ¯ U .At last, we discuss other crucial parameters’ impact on the performance of the comparisonalgorithms. We reconfigure { γ ths } of mIoT slices as γ th = 3 . m , γ th = 2 . m and γ th = 1 . m Kbits/minislot with m ∈ { . , . , . . . , . } and { D s } of URLLC slices as D = 0 . d second and D = 0 . d second with d ∈ { , , . . . , } . The impact of QoS requirementsof network slices on the total slice utility is plotted in Figs. 8 and 9. The impact of energyefficiency coefficient η is plotted in Fig. 10. In this figure, we denote the energy consumption ofRRHs of all algorithms by E u = [ E uSRO , E uACB I , E uACB II ] with E uSRO = T P t =1 P s ∈S u P i ∈I us b ui,s tr( G i,s ) .From these figures, the following observations can be achieved: i) the obtained utilities ¯ U of all algorithms decrease with an increasing m . This is because a great m indicates that theaccumulated IoT packets in the queue of each IoT device can be quickly emptied, and thena small P s ( t ) is obtained; ii) a great D s will reduce RRHs’ energy consumption. However, italso reduces the URLLC slice gain. Then, it may be hard to conclude the trend of ¯ U u w.r.t. D s as the energy efficiency coefficient η significantly affects the value of ¯ U u ; iii) it is alsouneasy to conclude the trend of ¯ U u w.r.t. η . An increasing η causes a decrease of RRHs’ energyconsumption. Yet, the value of ¯ U u is determined by the multiplier of η and E u ; iv) the SRO-ACB II algorithm may perform better than the SRO algorithm. However, the performance of theother access-control-based algorithm, SRO-ACB I , is slightly worse than SRO. Besides, it cannotensure that the ¯ U I obtained by the access-control-based algorithms are always higher than thatof SRO. At sometimes, access control schemes may drag down the utility of the mIoT service.To sum up, in the case of service multiplexing, RA control schemes for alleviating signalinterference and enhancing mIoT slice utility may be preferred for mIoT slices. However, considering both the CAPEX and the improvement of slice utility, RA control schemes shouldbe carefully designed and employed because some RA control schemes may worsen the mIoTand total slice utilities. IX. C ONCLUSION
In this paper, we revisited the frame and minislot structure of a RAN slicing system to admitmore IoT devices and proposed a queue evolution model to analyze the RACH of a randomlychosen IoT device. Based on the analysis result, we derived the closed-form expression of the RAsuccess probabilities of the device with an unrestricted access control scheme and an ACB accesscontrol scheme. Next, we formulated the RAN slicing for mIoT and bursty URLLC servicemultiplexing as an optimization problem to optimally orchestrating RAN resources for mIoTslices and URLLC slices, and efficient mechanisms such as SAA and ADMM were exploited tomitigate the optimization problem. Simulation results showed that RA control schemes shouldbe carefully designed and employed in the case of service multiplexing.A
PPENDIX
A. Proof of Lemma 1
The work in [16] adopted a standard stochastic geometry method to derive the LT of theaggregated interference from interfering IoT devices. Different from [16], both the stochasticgeometry method and a gamma-Poisson distribution are exploited to derive the result in thispaper.For the origin RRH, the LT of its aggregate interference from interfering IoT devices in s canbe derived as L I s ( t ) ( ̟ s ) = E I s ( t ) (cid:2) e − ̟ s I s ( t ) (cid:3) = E I s ( t ) " e − ̟ s P m ∈ uIs \{ o } ( p m || d m || − ϕ = ρ o ) ( N a,s ( t ) > ( f m = f o ) ρ o h m ( a ) = E u Is " Q m ∈ u Is \{ o } E h m h e − ̟ s ( P m || u m,s || − ϕ = ρ o ) ( N a,s ( t ) > ( f m = f o ) ρ o h m i ( b ) = ∞ P n =0 P {| Z s | = n } Q m ∈ Z s E h m (cid:2) e − ̟ s ρ o h m (cid:3) ( c ) = P {| Z s | = 0 } + ∞ P n =1 P {| Z s | = n } (cid:16) ̟ s ρ o (cid:17) n ( d ) = ˜ P X s { X s = 1 } + (cid:26) ∞ P n ′ =0 ˜ P X s { X s = n ′ } (cid:16) ̟ s ρ o (cid:17) n ′ − P n ′ =0 ˜ P X s { X s = n ′ } (cid:16) ̟ s ρ o (cid:17) n ′ (cid:27) (1 + ̟ s ρ o ) (39)where ̟ s = θ ths ρ o , Z s denotes the set of interfering IoT devices associated with the origin RRHin mIoT slice s , X s represents the number of active IoT devices associated with the origin RRH in s . According to the conclusion of Lemma 1 in [31], the probability mass function (PMF) ˜ P X s { X s = n ′ } can be written as ˜ P X s { X s = n ′ } = 3 . . Γ( n ′ + 3 . P nr,s ( t ) P ne,s ( t ) λ Is λ R ξF s ) n ′ Γ(3 . n ′ )!( P nr,s ( t ) P ne,s ( t ) λ Is λ R ξF s + 3 . n ′ +3 . (40)with Γ( · ) being the gamma function. Besides, in (39), (a) follows from the i.i.d distribution of h m and its further independence from the Poisson point process Φ s or u Is ; (b) follows from theexpectation of a discrete random variable; (c) follows from the LT over h m ; (d) follows fromthe fact that the number of active IoT device in a specific Voronoi cell is one more than thenumber of active interfering IoT devices in this cell.From (40), we can deduce that X s ( s ∈ S I ) is a gamma-Poisson random variable with X s ∼ gamma − Poisson( α s , . and α s = P nr,s ( t ) P ne,s ( t ) λ Is . λ R ξF s . For a gamma-Poisson random variable X s ∼ gamma − Poisson( α, β ) , the following expression holds: E [ e X s ] = (1 + α − αe ) − β . Thus,we can rewrite (39) as (8). This completes the proof. B. Proof of Lemma 2
The work in [16] derived the expression of the intensity of accumulated packets µ Cum at the nd slot. However, they did not derive the expression of µ tcum with t ≥ , the detail derivationof which was different from that of the expression of µ Cum . Based on the results in [16], wederive the expression of the intensity of accumulated packets µ a,s ( t ) and the expression of thenon-empty probability P ne,s ( t ) for all s ∈ S I , t > .As new endogenous packet arrivals in any IoT device at each minislot t is modelled asa Poisson distribution, the departure process of packets can be regarded as an approximatedthinning process of new arrivals, where the thinning factor is related to the RA success probability.The number of accumulated packets in the queue of any IoT device can then be approximatedas a Poisson distribution with intensity µ ta,s ( s ∈ S I ) after the thinning process in a specificminislot t ( t > ) [16].Thus, we can derive the expression of µ ta,s ( t > ) via combining with the following facts • Fact 1: the accumulated packets during the t − -th minislot will contribute to the accu-mulated packets at the t -th minislot. • Fact 2: the arrival packets during the t − -th minislot will also contribute to the accumulatedpackets in the queue of an IoT device at the t -th minislot. • Fact 3: an IoT device can send packets only if its preamble is successfully transmitted. • Fact 4: at the same minislot, the new packet arrival process and the packet accumulatedprocess are independent.Similar as the Theorem 2 in [16], we can infer that at the nd minislot, for all s ∈ S I , µ a,s depends on the intensity of new packet arrivals µ w,s and the probability P s of a randomlyselected IoT device at the st minislot, which is given by µ a,s = µ w,s − x s P s (cid:16) − e − µ w,s (cid:17) (41)The detailed proof of (41) is omitted for brevity, and a similar proof can be found in Theorem2 in [16].Considering that µ a,s ( t ) is non-negative at each minislot t , we have µ a,s = h µ w,s − x s P s (cid:16) − e − µ w,s (cid:17)i + (42)Then, according to the definition of non-empty probability and the Poisson approximation,the non-empty probability of a randomly selected IoT device in mIoT slice s ∈ S I at the nd minislot can be approximated as P ne,s = 1 − P { N a,s = 0 } = 1 − e − µ a,s (43)At the rd minislot, the intensity of accumulated data packets in the queue of a randomlyselected IoT device can be derived as the following µ a,s = P s (cid:18) ∞ P n =1 (cid:18) [ n − x s ] + n P z =0 P N w,s ( z ) P N a,s ( n − z ) (cid:19)(cid:19) + (1 − P s ) (cid:18) ∞ P n =1 n n P z =0 P N w,s ( z ) P N a,s ( n − z ) (cid:19) ( a ) = P s (cid:20) ∞ P n =1 n P z =0 ( µ w,s ) z e − µ w,s z ! ( µ a,s ) n − z e − µ a,s ( n − z )! × n − x s ∞ P n =1 n P z =0 ( µ w,s ) z e − µ w,s z ! ( µ a,s ) n − z e − µ a,s ( n − z )! (cid:21) + +(1 − P s ) ∞ P n =1 n P z =0 ( µ w,s ) z e − µ w,s z ! ( µ a,s ) n − z e − µ a,s ( n − z )! × n ( b ) = h µ w,s + µ a,s − x s P s (cid:16) − e − µ w,s − µ a,s (cid:17)i + (44)where P N w,s and P N a,s represent the PMFs of new arrival packets and accumulated packetsat the nd minislot, respectively. Besides, (a) follows from the fact: for any two independentPoisson distributions Φ X and Φ X , P X ,X ( X + X = x ) = x P y =0 P X ( X = y ) P X ( X = x − y ) ;(b) holds as Φ X ,X is a two dimensional Poisson distribution with an intensity λ X + λ X , and ∞ P x =1 P X ,X ( X + X = x ) = 1 − P X ,X ( X + X = 0) . Similarly, we have P ne,s = 1 − P { N a,s = 0 } = 1 − e − µ a,s (45)When t > , since the accumulated packets evolution model of the queue of any IoT deviceis the similar as that at t = 3 , we can extend the conclusion obtained at t = 3 to that at t > .Therefore, we can obtain the closed-form expression of µ ta,s for all s ∈ S I at t > with µ ta,s = h µ t − w,s + µ t − a,s − x s P t − s (cid:16) − e − µ t − w,s − µ t − a,s (cid:17)i + (46)and P tne,s = 1 − P { N ta,s = 0 } = 1 − e − µ ta,s (47)This completes the proof. C. Proof of Lemma 4
The nd degree Taylor expansion of ˜ P ( k ) m at the local point ω ( k,q ) m is ˜ P ( k )2 ,m = X j =0 j ! " X s ∈S I (cid:0) ω sm − ω ( k,q ) sm (cid:1) ∂∂ω ( k,q ) sm j ˜ P ( k ) m | ω ( k,q ) m (48)The rd degree Taylor expansion of ˜ P ( k ) m at ω ( k,q ) m must be more accurate than ˜ P ( k )2 ,m with ˜ P ( k )3 ,m = ˜ P ( k )2 ,m + 13! " X s ∈S I (cid:0) ω sm − ω ( k,q ) sm (cid:1) ∂∂ω ( k,q ) sm ˜ P ( k ) m | ω ( k,q ) m (49)Since the error of ˜ P ( k )2 ,m is not greater than the maximum difference between ˜ P ( k )3 ,m and ˜ P ( k )2 ,m ,we have R ( ω m ) = max " X s ∈S I (cid:0) ω sm − ω ( k,q ) sm (cid:1) ∂∂ω ( k,q ) sm ˜ P ( k ) m | ω ( k,q ) m (50)In (50), ω ( k,q ) m is a constant vector, the max operation will not affect the constant vector andthe vector ω m . For any s ∈ S I , the maximum value obtainable by ∂ ˜ P ( k ) m | ω ( k,q ) m ∂ω k,q ) sm will not exceedthe greatest value of that derivative in the interval [ˆ ω lbsm , S ⋆sm ] . Additionally, the maximum valueof ∂ P m | ω ( k,q ) m ∂ω k,q ) sm will generally occur at one of the endpoints of the interval [ˆ ω lbsm , S ⋆sm ] . Therefore,we obtain (35). This completes the proof. D. Proof of Lemma 5
For all i ∈ I u , s ∈ S I , m ∈ M , a feasible way of proving that rank( G i,sm ) ≤ is to utilizethe Lagrange method. However, owing to the complicated expression of W u ( r m ) w.r.t. G i,sm itwill be uneasy to do that. Fortunately, we find that the proof can be conducted if a family ofauxiliary variables is introduced.For the constraint (24d), if we introduce the auxiliary variables { ν i,sm } and let tr( H i,sm G i,sm ) φσ i,s ≥ ν i,sm , ∀ i ∈ I us , s ∈ S u , m ∈ M (51)then (24d) is equivalent to X s ∈S I (1 + α g ) ω sm (¯ t ) + W u ( f m ) ≤ W, and (51) (52)where f m = { f i,sm ; i ∈ I us , s ∈ S u } and f i,sm = L ui,s log (1 + ν i,sm ) + ( Q − ( β )) (1 + ν i,sm ) + ( Q − ( β )) (1 + ν i,sm ) s L ui,s log (1 + ν i,sm )( Q − ( β )) (53)We omit the proof of the equivalence as a similar proof can be found in the proof section ofconstraints’ equivalence in [11].The partial Lagrangian function of (37) can be written as L ( . . . ) = X s ∈S u X i ∈I us " ˜ ρηM tr( G i,sm ) + X j ∈J ¯ λ jm tr( b u ( k,q ) i,sm Z j G i,sm ) − ¯ µ i,sm tr( H i,sm G i,sm ) φσ i,s − ¯ X i,sm ⊙ G i,sm (54)where ¯ λ jm , ¯ µ i,sm , and ¯ X i,sm are Lagrangian multipliers corresponding to constraints (24c), (51)and (24e), ⊙ is the matrix dot product operator. Besides, only terms related to G i,sm are includedin this function for brevity.According to the Karush-Kuhn-Tucker (KKT) conditions, the necessary condition for obtainingthe optimal matrix power at the ( k, q ) -th iteration G ( k,q ) ⋆i,sm is given by ∂L ( ... ) ∂ G ( k,q ) ⋆i,sm = ˜ ρηM I i,sm + P j ∈J ¯ λ jm b u ( k,q ) i,sm Z j − ¯ µ i,sm H i,sm φσ i,s − X i,sm = 0 (55)where I i,sm ∈ R JK × JK is an identity matrix.Then, we can conclude that rank( X i,sm ) ≥ J K − . The reasons are i) ¯ λ jm , b u ( k,q ) i,sm , and ¯ µ i,sm are nonnegative and the matrix I i,sm is full rank; ii) rank( H i,sm ) ≤ .Next, according to the complementary slackness condition, we have X i,sm G ( k,q ) ⋆i,sm = 0 (56) Based on (56) and the rank result of X i,sm , we can conclude that rank( G ( k,q ) ⋆i,sm ) ≤ . Thiscompletes the proof. R EFERENCES [1] Ericsson, “Cellular networks for massive IoT,” Ericsson, Stockholm, Sweden, Tech. Rep. Uen 284 23-3278, Jan. 2016.[2] Nokia, “LTE evolution for IoT connectivity ,” Nokia, Espoo, Finland, Tech. Rep. SR1702006775EN.[3] S. Xing, X. Wen, Z. Lu, Q. Pan, and W. Jing, “A novel distributed queuing-based random access protocol for narrowband-IoT,” in
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