I Introduction
With the explosive growth of the Internet of Things (IoT), massive IoT (mIoT) devices, the number of which is predicted to reach 20.8 billion by 2020, will access to the wireless networks for implementing advanced applications, such as ehealth, public safety, smart traffic, virtual navigation/management, remote maintenance and control, and environment monitoring. To address the IoT market, the thirdgeneration partnership project (3GPP) has identified mIoT as one of the three main use cases of 5G and has already initiated several task groups to standardize several solutions including extended coverage GSM (ECGSM), LTE for machinetype communication (LTEM), and narrowband IoT (NBIoT) [1, 2].
For establishing massive connections among the wireless networks and mIoT devices, the investigation of reliable and efficient access mechanisms should be prioritized. In accomplishing the massive connections, when an active IoT device wants to transmit signal in the uplink, it randomly chooses a random access (RA) preamble from an RA preamble pool and transmits it through an RA channel (RACH). If more than one device tries to access to a base station (BS) simultaneously, then interference occurs at the RRH. During the past few years, a rich body of works on RA mechanisms has been developed [3, 4, 5, 6, 7, 8, 9, 10, 11] to mitigate interference and improve the RA success probability or reduce the access delay of an IoT device.
Most of the studies [3, 4, 5, 6, 7, 8, 9, 10, 11], however, assumed that the whole network resources were reserved for the IoT service and did not investigate the case of the coexistence of IoT service and many other services such as enhanced mobile broadband (eMBB) and ultrareliable and low latency communications (URLLC). The research of the coexistence of IoT service and other services is essential as future networks are convinced to converge variety of services with different latency, reliability, and throughput requirements onto a shared physical infrastructure rather than deploying individual network solution for each service [12]. What is more, owing to the shared characteristic of network resources, some conclusions obtained in the case of providing sole IoT service may become inapplicable if multiple types of services are required to be supported by the networks.
Network slicing is considered as a promising technology in future networks for providing scalability and flexibility in allocating network resources to various services. Recently, many network slicing frameworks have been developed to provide performance guarantees to IoT or massive machinetype communications (mMTC) service, eMBB service, and URLLC service [13, 14, 15, 16, 17, 18, 19].
Different from previous works, this paper investigates the mIoT and bursty URLLC service multiplexing via slicing the radio access network (RAN). This study is highly challenging because i) performance requirements of a massive number of IoT devices should be satisfied. Yet, the typical 5G cellular IoT, NBIoT can admit only 50,000 devices per cell [20]; ii) RAN slicing operation (e.g., creating, activating, and releasing slices) has to be conducted in a timescale of minutes to hours to keep pace with the upper layer slicing. However, the wireless channel generally changes in a timescale of millisecond to seconds. Results of the RAN slicing operation are desired to be achieved based on the timevarying channel. Thus, the RAN slicing should tackle a two timescale issue [21].
These challenges motivate us to investigate the RAN slicing for mIoT and bursty URLLC service 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:

We revisit the frame and minislot structure for mIoT transmission to accommodate more RA requests from a massive number of IoT devices.

We adopt a queueing model to track the IoT packet arrival, accumulate and departure processes and analyze the queue evolution process by employing probability and stochastic geometry theories. Based on the analysis result, we derive the closedform expression of the RA success probability of a randomly chosen IoT device.

We define mIoT slice utility and bursty URLLC slice utility and formulate the RAN slicing for mIoT and bursty URLLC service multiplexing as a resource optimization problem. The objective of the optimization problem is to maximize the total mIoT and URLLC slice utilities, subject to limited physical resource constraints. The mitigation of this problem is difficult due to the existence of indeterministic objective function and thorny nonconvex constraints and the requirement of tackling a two timescale issue as well.

To mitigate this thorny optimization problem, we propose a slice resource optimization (SRO) algorithm. In this algorithm, we first exploit a sample average approximate (SAA) technique and an alternating direction method of multipliers (ADMM) to tackle the indeterministic objective function and the two timescale issue. Then, a semidefinite relaxation (SDR) scheme joint with a Taylor expansion scheme are leveraged to approximate the nonconvex problem as a convex one. The tightness of the SDR scheme and the error bound of the Taylor expansion are also analyzed.
The remaining of this paper is organized as follows. We review the prior arts in Section II. In Section III, we describe our system model and formulate the service multiplexing problem in Section IV. The problemmitigating algorithm is presented in Sections V and VI. In Section VII, we give the simulation results and conclude this paper in Section VIII.
Ii Prior arts
Recently, many researches have been conducted to increase RA success probabilities and/or reduce the access delay of mIoT devices. They can be generally classified into two groups: traffic detection and estimation based algorithms and algorithms without traffic detection and estimation.
The fundamental idea of the traffic detection and estimation based algorithms is to design an RA algorithm based on the detected and/or estimated users’ activity and traffic congestion situation and so on. For example, to reduce the access delay, a grantfree nonorthogonal RA system relying on the accurate user activity detection and channel estimation was proposed in [6]. A trafficaware spatiotemporal model for the contentionbased RA analysis is conducted for mIoT networks in [7]. With the spatiotemporal model, a hybrid power ramping and backoff RA scheme was then developed to improve the RA success probability. Besides, an extended pseudoBayesian backlog estimation scheme was exploited in [11] to estimate the number of backlogged nodes to attempt access. A versatile access control mechanism was then designed to reduce the access delay based on the estimation results.
For algorithms without detection and estimation, they design RA schemes without detecting users’ activity or estimating the statistical characteristic of traffic. For instance, the work in [3] proposed to improve the RA success probability of an IoT device by exploiting a distributed queue mechanism and then proposed an access resource grouping mechanism to reduce the access delay caused by the queuing process of the distributed queue mechanism. To increase RA success probability, the work in [4]
proposed to increase the number of preambles at the first step of the RA procedure by utilizing a spatial group mechanism and improve resource utilization through nonorthogonally allocating uplink channel resources at the second step of the RA procedure. Additionally, without knowing the statistical characteristic of traffic, a reinforcement learningbased algorithm was proposed in
[5] to determine the uplink resource configuration for RA such that the average number of served IoT devices was maximized while ensuring a high RA success probability.Except for the IoT service, future networks are envisioned to simultaneously support different services and applications with significantly different requirements on reliability, latency and bandwidth. As a result, researchers are now paying more attention to the service multiplexing of IoT/mMTC and many other services such as eMBB and URLLC. For example, instead of slicing the RAN via orthogonal resource allocation among different services, the work in [13, 14] studied the potential advantages of allowing for nonorthogonal RAN resources sharing in uplink communications from a collection of mMTC, eMBB, and URLLC devices to the same BS. The work in [15] developed a twolevel scheduling process to allocate dynamically dedicated bandwidth to each network slice according to workload demand and slices’ quality of service (QoS) requirement such that flexible resource allocation could be implemented. The work in [16] proposed to maintain slicespecific radio resource control elements with which the RAN protocol stacks and different slices were configured. Besides, the work in [17] aimed to optimize the virtual network functions and infrastructure resources such as the system bandwidth to implement slice recovery and reconfiguration for mMTC and eMBB service provision. The work in [18] proposed to maintain slice isolation between mMTC and eMBB slices and meet the performance requirements of these slices through limiting and dynamically updating the amount of resources allocated to each slice and monitoring the resource usage of each slice. After representing the slice performance requirements as the required amount of resources per deadline interval, an idea of earliestdeadline and firstscheduling was exploited in [19] to allocate radio resources to mMTC, eMBB, and URLLC slices effectively.
Iii System model
We consider a coordinatedmultipointenabled RAN slicing system for mIoT and bursty URLLC multiplexing service provision. From the viewpoint of the infrastructure composition, the system mainly includes one baseband unit (BBU) pool and multiple remote radio heads (RRHs) that connect to the BBU via fronthaul links. From the perspective of network slicing, two types of interslices, i.e., mIoT slices and URLLC slices, are exploited in this system with and representing the sets of mIoT slices and URLLC slices, respectively. We focus on the modelling of uplink IoT data transmission in mIoT slices and the modelling of downlink URLLC data transmission in URLLC slices. The IoT devices are spatially distributed in according to an independent homogeneous Poisson point process (PPP) with intensity , where denotes the location of the th IoT device in the th mIoT slice. There are also URLLC devices that are randomly and evenly distributed in . The RRHs are spatially distributed in according to an independent PPP with intensity , where represents the location of the th RRH. The number and locations of IoT devices and RRHs will be fixed once deployed. Besides, each RRH is equipped with antennas, and each device is equipped with a single antenna. In IoT network slices, each IoT device is assumed to connect to its geographically closest RRH [7]
; thus, the cell area of each RRH constitutes a Voronoi tessellation. In URLLC network slices, RRHs cooperate to transmit signals to a URLLC device to improve its signaltonoise ratio (SNR). A flexible frequency division multiple access (FDMA) technique is utilized to achieve the interslice and intraslice interference isolation
[21].The system time is discretized and partitioned into time slots and minislots with a time slot consisting of minislots. On the one hand, at the beginning of each time slot, a RAN slicing coordinator [22] will decide whether to accept or reject received network slice requests which will be defined in the following subsections. Once a slice request is accepted, a network slice management will be responsible for activating or creating a virtual slice that is well resourceconfigured to satisfied the QoS requirements of devices in the slice [22]. The slice configuration process is time costly and will generally be conducted in a timescale of minutes to hours [21]. On the other hand, at the beginning of each minislot, each active IoT device may try to connect to its associated RRH, and RRHs will generate cooperated beamformers based on sensed channel coefficients.
Iiia mIoT slice model
By referring to the concept of a network slice [16], especially from the viewpoint of the QoS requirement of a slice, we can define a mIoT slice request as follows.
Definition 1 (mIoT slice request).
A mIoT slice request is defined as a tuple for any slice , where is the requirement of data transfer rate from an IoT device in to its associated RRH, denotes the number of accumulated packets in a queue of an IoT device in .
In this paper, all mIoT slice requests are always accepted by the RAN slicing coordinator. IoT devices with the same data transfer rate are assigned to the similar slice. For an IoT device in , if it has the opportunity to send its endogenous arrival packets to the corresponding RRH, then it will randomly select a preamble (e.g., orthogonal Zadoff CChu sequences) from a BBUmaintained preamble pool and transmit the preamble to the RRH at the data rate . Just like the literature [23, 24], if the RRH can successfully decode the preamble, then a connection between the IoT device and the RRH are considered to be set up although the whole connection establishment process usually follows an RA fourstep procedure [8]. In other word, the RA success probability is regarded as the probability of successfully transmitting a preamble in this paper. Next, we will analyze an IoT device queue evolution model, with the analysis of which the RA success probability of the IoT device will be derived.
IiiA1 Queue evolution model
The queue evolution process consists of the packet arrival process, packet accumulate process, and packet departure process.
During minislot
, a Poisson distribution with intensity (or arrival rate)
is exploited to model the random, mutually independent endogenous packet arrivals in an IoT device in slice . Then during minislot with a duration , the arrival intensity of new packets can be expressed as . Once arrived, new packets will not be sent out immediately in general and will enter a queue, which is modelled as an queue with unlimited capacity, to wait for their scheduling. In the queue, packets will be scheduled according to the firstcome, firstserved (FCFS) basis. The unlimited queue capacity indicates that the age of information [25, 26] of new arrivals will not be considered, and packets will not be dropped before sending out. Besides, owing to the RA behavior of a slottedALOHA protocol, new arrivals during will only be counted at minislot . Thus, the accumulated number of packets of a randomly selected IoT device in slice at is determined by the accumulated number of packets and the number of new arrivals at and whether the preamble of the device can be successfully decoded by its associated RRH. Table I shows the evolution of accumulated packets in an IoT device. In this table, packets at the head of the queue will be popped out if the corresponding RA succeeds, where denotes the IoT packet length; otherwise, they will be kept in the queue and wait for the opportunity of retransmission at the next minislot. The operation .Value  Success  Failure 

  
+   +  
 +     +  
With the evolution of accumulated packets, we can define the nonempty probability of the queue of an IoT device in as the following.
Definition 2 (Nonempty probability).
At minislot , for a randomly selected IoT device in slice , the probability that its queue is not empty can be defined as
(1) 
(1) implicitly reflects that new arrival packets at will not be sent out immediately. According to the evolution of , it can be observed that
is determined by the probability distribution of
and the RA success probability. Since these probabilities and their correlations are unknown, the derivation of the explicit expression of is difficult.Next, we describe the packet departure process combined with a frame and minislot structure for mIoT packets transmission. As mentioned above, partly because of the limitation on the frame and minislot structure, NBIoT and LTEM can only admit 50,000 devices. For NBIoT, only one physical resource block (PRB) with a bandwidth of
KHz in the frequency domain is allocated for IoT transmission, and each physical channel occupies the whole PRB. For LTEM, although the physical channels are time and frequency multiplexed, it only reserves six inband PRBs with a total bandwidth of
MHz in the frequency domain for IoT data transmission. Therefore, the frame and minislot structure for mIoT transmission should be revisited if more RA requests from IoT devices want to be accepted.Fig. 1 depicts a frame and minislot structure for mIoT transmission in each mIoT slice^{1}^{1}1We do not show all channels in this figure as the detailed research of the physical layer supporting the mIoT service is out of the scope of this paper.. In this structure, both the frequency division multiplexing scheme and code division multiplexing scheme are leveraged to admit more IoT devices in the way of alleviating the mutual device interference. Particularly, the frequency division multiplexing scheme alleviates signal interference through orthogonal frequency allocation, and the code division multiplexing scheme mitigates the cochannel signal interference via reducing the crosscorrelation of simultaneous transmissions. The combination of the two schemes may significantly mitigate interference experienced at an RRH. In this way, the QoS requirements of more IoT devices may be satisfied, and the RAN slicing system may support more IoT devices. For a mIoT slice , each subframe includes orthogonal uplink physical RA channels (PRACHs). A single tone mode with a tone spacing of size of MHz is adopted for each uplink PRACH, which indicates that each PRACH occupies a PRB. At the beginning of each minislot, an active IoT device, i.e., the device’s queue is nonempty, will randomly choose a preamble from a set of nondedicated RA preambles of size and transmit the preamble through a randomly selected PRACH. For each preamble, it has an equal probability to be chosen by each IoT device. Similarly, each PRACH has an equal probability to be selected. Thus, the average number of IoT devices in mIoT slice choosing the same PRACH and the same preamble is . Notably, a greater may significantly reduce signal interference experienced at each RRH.
Then, the following question should be tackled: how many PRBs should be reserved for mIoT transmission? To improve the resource utilization, the resource allocated to mIoT should be determined according to the requirements of mIoT and other coexistence services. It motivates us to optimize the resources orchestrated for the mIoT service except for analyzing the RACH procedure of IoT devices. The optimization procedure will be discussed in detail in the next section.
IiiA2 Access control scheme
In a mIoT network slice, as the slottedALOHA protocol allows all active IoT devices to request for RA at the beginning of each minislot without checking the status of channels, IoT devices may simultaneously transmit preambles. It may incur severe slice congestion that may lower the RA success probabilities of IoT devices and degrade the system performance. Access control has been considered as an efficient proposal of alleviating congestion, and many access control schemes have been proposed [7, 27]. In this paper, we aim at illustrating the performance difference between a network slicing system without access control and with access control. Therefore, we adopt the following two schemes [7]:

Unrestricted scheme: each active IoT device requests the RACH at the beginning of minislot without access restriction. If mIoT slices are not crowded or in a lightcrowded condition, then this scheme may quickly flush queues of IoT devices. However, if a heavycrowded condition is encountered, then this scheme may result in a high packet queueing delay.

Access class barring (ACB) scheme: at the beginning of , each active IoT device throws a random number and can request the RACH only if , where is an ACB factor determined by RRHs based on the slice congestion condition. The ACB scheme can relieve slice congestion to some extent by reducing RACH requests of active IoT devices.
With the introduced access control schemes, we can define the nonrestriction probability of a randomly selected IoT device in as follows.
Definition 3 (Nonrestriction probability).
At minislot , for a randomly selected IoT device in slice , the probability that its RACH request is not restricted is defined as
(2) 
For all at any minislot , we have for the unrestricted scheme and for the ACB scheme.
IiiA3 Analysis of RA success probability
For an RRH, two significant reasons may lead to an error preamble decoding i) the achieved preamble transfer rate at the RRH is less than a preset threshold; ii) the RRH simultaneously decodes at least two similar cochannel preambles, and thus preamble collision occurs. The research of the mitigation of preamble collision has been well conducted in [11, 28]. Just like [29], we focus on the exploration of enabling successful single preamble transmission that is discussed in detail as follows.
We utilize a powerlaw pathloss model to calculate the pathloss between an IoT device and its RRH in mIoT slices and utilize a truncated channel inversion power control scheme to eliminate the ’nearfar’ effect. In the powerlaw pathloss model, the IoT device transmit power decays at the rate of with representing the propagation distance and denoting the pathloss exponent. In the power control scheme, IoT devices associated with the same RRH compensate for the pathloss to maintain the average received signal power at the RRH equal to a threshold . Without loss of generality, the cutoff threshold is set to be the same for all RRHs. Owing to the channel deep fading, severe cochannel interference, and insufficient transmit power, an IoT device may experience uplink preamble transmission outage. The following definition describes the definition of the probability that a randomly selected IoT device can successfully transmit a chosen preamble to its corresponding RRH.
Definition 4 (RA success probability).
At minislot , for a randomly selected active IoT device in slice , its RA success probability is defined as
(3) 
where denotes the achieved preamble transfer rate at the IoT device’s associated RRH and is the signaltointerferenceplusnoise ratio (SINR).
Then, for any active IoT device in , its QoS requirement is given by
(4) 
where denotes a threshold of the required RA success probability.
This definition shows that the QoS requirement of each active IoT device in should be satisfied if the slice request of is accepted. The definition also states that is correlated with the nonempty probability . Recall that the RA success probability of an IoT device impacts its nonempty probability, we can know that the RA success probability and the nonempty probability are intertwined. Additionally, is a function of complicated cochannel interference. Thus, it is hard to obtain the closedform expression of .
Without any loss in generality, we perform the analysis of RA success probability on an RRH located at the origin. According to Slivnyak’s theorem [30], the analysis holds for a generic RRH located at a generic location. For a randomly selected IoT device with nonempty queue in , the theoretical preamble transfer rate experienced at the RRH located at the origin can take the form
(5) 
where represents the noise power, denotes signal interference received at the RRH, the useful signal power equals to due to the truncated channel inversion power control^{2}^{2}2Owing to the truncated channel inversion power control, not all of the IoT devices in mIoT slices can communicate in the uplink when the cutoff threshold is relatively high [31]. However, this paper assumes that the transmit power of each IoT device is large enough such that the IoT device will not experience preamble outage resulting from the insufficient power. [31] with denoting the channel power gain between the IoT device and the RRH. It is noteworthy that the channel power gain experienced at a generic RRH is related to the spatial locations of both the RRH and its associated IoT devices. Nevertheless, we drop the spatial indices for notation lightening. Besides, just like [31], all channel gains are assumed to be known and be independent of each other, independent of the spatial locations, symmetric and are identically distributed (i.i.d.). Considering both the particular IoT device deployment environment and the convenience of theoretical analysis, the Rayleigh fading is assumed, and the channel power gain
is assumed to be exponentially distributed with unit mean.
Based on the following five facts, we next present the analytical expression of signal interference

Fact 1: the average signal received from any single IoT device belonging to intercells is strictly less than .

Fact 2: the average interference signal received from any single interfering IoT device associated with the origin RRH strictly equals to .

Fact 3: IoT devices choosing the same cochannel preamble as the randomly selected IoT device may become an interfering IoT device.

Fact 4: at each minislot, IoT devices with nonempty queue may become interfering IoT devices.

Fact 5: IoT devices in difference slices may not mutually interfere.
Note that Fact 1 and Fact 2 are direct consequences of the deviceRRH association policy and power control scheme. Fact 5 holds due to the exploration of intraslice isolation. Therefore, the aggregate interference received at the origin RRH can take the following form
(6) 
where represents the randomly selected IoT device associated with the RRH at the origin, represents the transmit power of the th IoT device, is the distance between the th IoT device and the origin RRH, denotes the preamble and channel chosen by the randomly selected IoT device, indicates that the randomly selected IoT device and the th IoT device select the same preamble and channel. is the indicator function that equals to one if the statement is ture; otherwise, it equals to zero. Just like [32], in (6), cochannel intercell interference is assumed as a part of thermal noise mainly because of the severe wall penetration loss.
Then, for the randomly selected IoT device in , we can rewrite (5) as the following form with (6)
(7) 
where . (a) follows from the full probability law over , and
denotes the Laplace transform (LT) of the probability density function (PDF) of the random variable
. Note that the notation is a terminology that is a slight abuse of subscript .The following lemma characterizes the LT of aggregate interference .
Lemma 1.
For the origin RRH, the LT of its received aggregate interference from active IoT devices associated with it is given by
(8) 
where , , for all .
Proof.
Please refer to Appendix A. ∎
With the conclusion in Lemma 1, we can then obtain the mathematical expression of the RA success probability of a randomly selected IoT device at in the following corollary.
Corollary 1.
For a randomly selected IoT device in a mIoT slice , its RA success probability at minislot is given by
(9) 
Although Corollary 1 presents a mathematical expression of , the expression is not in the closedform as it is a function of the closedform expression of which is not obtained. Next, we derive the closedform expression of .
IiiA4 Analysis of nonempty probability
According to the definition of nonempty probability, is correlated with the number of accumulated packets of the randomly selected IoT device in mIoT slice . Thus, we theoretically analyze the nonempty probability of the randomly selected IoT device as the following.
As the number of the accumulated packets in the queue of a randomly selected IoT device in slice at the minislot is empty, its nonempty probability at the minislot can take the form
(10) 
where we write instead of to lighten the notation.
The following lemma presents the closedform expression of the nonempty probability of a randomly selected IoT device served by the origin RRH when minislot .
Lemma 2.
The number of accumulated packets of a randomly selected IoT device served by the origin RRH at minislot may be approximately Poisson distributed. Therefore, for any mIoT slice , we approximate the number of accumulated packets at minislot as a Poisson distribution with intensity , which is given by
(11) 
Then, the nonempty probability of the device at minislot can be written as
(12) 
Proof.
Please refer to Appendix B. ∎
IiiB Bursty URLLC slice model
Similar to the definition of a mIoT slice request, a bursty URLLC slice request can be defined as the following.
Definition 5 (Bursty URLLC slice request).
A bursty URLLC slice request is defined as four tuples for slice , where denotes the number of URLLC devices in , denotes the transmission latency requirement of each URLLC device in , and represent the packet blocking probability and the codeword error decoding probability of each URLLC device, respectively.
In this definition, URLLC devices are grouped into clusters according to the transmission latency requirement of each device. Owing to the low latency requirement URLLC packets should be immediately scheduled upon arrival; thus, all URLLC slice requests will always be accepted by the RAN coordinator. Except for the low packet error decoding probability that has been emphasized for URLLC transmission in a plenty of works [21], this paper attempts to orchestrate slice resources to reduce the packet blocking probability for bursty URLLC transmission. This is because the bursty characteristic of URLLC traffic [22] may lead to the packet blocking in URLLC slices, which may significantly reduce the reliability of URLLC transmission. Therefore, the indicators and are involved to reflect the ultrareliable requirement of URLLC transmission jointly.
Then, we address the following question: how to orchestrate slice resources for reducing packet blocking probability and codeword error decoding probability?
IiiB1 Reduction of packet blocking probability
As mentioned above, the bursty feature of URLLC traffic is the crucial factor that leads to the URLLC packet blocking for URLLC transmission. Therefore, we next model the bursty URLLC traffic based on which we discuss how to orchestrate slice resources to alleviate the impact of bursty URLLC traffic.
During minislot , an independent homogeneous Poisson distribution with intensity is utilized to model the number of bursty URLLC packets aggregated at RRHs, where denotes the intensity of new arrivals destined to devices belonging to URLLC slice .
Once arrived, new URLLC arrivals will enter a queue maintained by an RRH to be immediately scheduled. An queueing system with limited bandwidth is exploited to model the queue. Without loss of any generality, we assume that each RRH maintains the same queue due to the exploration of cooperated transmission. In the queue, a packet destined to URLLC device , will be allocated with a block of system bandwidth for a period of time at . Owing to stochastic variations in the bursty packet arrival process, the limited bandwidth may not be enough to serve new arrivals occasionally. As such, URLLC packet blocking may happen. To reduce the probability of URLLC packet blocking, the redesign of URLLC frame and minislot structure may be required.
At minislot , let denote the packet blocking probability experienced at an RRH, where and . The Theorem 1 in [33] provides us with a clue of redesigning the URLLC frame and minislot structure in the timefrequency plane for bursty URLLC traffic transmission. This theorem indicates that if we narrow the PRB of the URLLC frame in the frequency domain while widening it in the time domain, then the number of concurrent transmissions will be increased. As a result, the packet blocking probability is reduced.
Therefore, for a URLLC packet destined to device , , we should scale up and choose and at using the following equation
(13) 
where denotes channel uses for transmitting a URLLC packet, is a constant reflecting the number of channel uses per unit time per unit bandwidth of FDMA frame structure and numerology, is an indicator variable reflecting whether the QoS requirement of device in slice can be satisfied at . As network resources are limited and shared by all network slices, not all URLLC devices can be guaranteed to be served at every minislot. Certainly, we can adjust the slice priority weight that will be introduced in the following section to guide the resource orchestration for enforcing the entire URLLC devices coverage.
Based on the result in (13) and the conclusion of the Lemma 3.2 in our previous work [22], we can derive the minimum upper bound of bandwidth orchestrated for URLLC slices in the following lemma.
Lemma 3.
At minislot , for a given queue with packet arrival intensity and a family of packet transmit rates , let denote the minimum upper bound of bandwidth orchestrated for URLLC slices such that and , is of the order of , where represents the queueing probability, and . If , then we have
(14) 
Proof.
We omit the proof here as the similar proof can be found in the proof section of Lemma 3.2 in [22]. ∎
IiiB2 Reduction of codeword error decoding probability
The crucial factor that impacts the codeword error decoding probability is the network capacity. Next, we discuss the relationship between the network capacity and codeword error decoding probability.
For any URLLC slice , during minislot , let be the original data symbol destined to a URLLC device with , be the transmit beamformer pointing at the device from the th RRH and be the channel coefficient between the th URLLC device and the th RRH. The channel coefficient may change over minislots. However, it is assumed to be i.i.d. over each minislot and remain unchanged during each minislot. Then, the received signal at device in during minislot is given by
(15) 
where the first term is the useful signal for and is the additive white Gaussian noise (AWGN) experienced at . Similar to [21], interference signal is not involved in (15) due to the utilization of a flexible FDMA mechanism. Then the SNR received at device in at minislot can be written as
(16) 
where is an SNR loss coefficient. The perception of perfect channel status information (CSI) or accurate channel coefficients requires the information exchange between an RRH and its associated device before data transmission, the process of which is generally time consuming. URLLC packets, however, have a stringent latency requirement. As a result, perfect CSI or accurate channel parameters may be unavailable for URLLC transmission, which may incur the SNR loss. The coefficient is then utilized to characterize the SNR loss [34].
Shannon capacity formula is created under a crucial assumption of transmitting a block with long enough blocklength. However, URLLC packets are typically very short to satisfy the ultralow latency requirement. Thus, the famous Shannon capacity formula cannot be utilized to model the URLLC transmission data rate and capture the corresponding codeword error decoding probability. For URLLC transmission, we resort to the capacity analysis for a finite blocklength channel coding regime derived in [35]. For any device , , the number of transmitted information bits at minislot using channel uses in AWGN channel can be accurately correlated with the codeword error decoding probability according to the following equation
(17) 
where is the AWGN channel capacity under infinite blocklength assumption, is the channel dispersion, is the function. It is noteworthy that a URLLC packet will usually be coded before transmission and the generated codeword will be transmitted in the air interface such that the transmission reliability can be improved.
Iv Problem formulation
This section aims to formulate the problem of RAN slicing for mIoT and bursty URLLC service multiplexing based on the above models.
In mIoT slices, each RRH may transmit feedback signal to its connected IoT devices for the connection establishment according to an RA fourstep procedure [8]. Meanwhile, in URLLC slices, each RRH may transmit URLLC packets to URLLC devices. As the transmit power of each RRH is limited, we have the following transmit power constraint
(19) 
where is assumed to be a constant and denotes the transmit power of the th RRH for connecting to its associated IoT devices over downlink, represents a safety margin coefficient. As a PPP with intensity is utilized to model the distribution of IoT devices, the actual number of IoT devices may be greater than once deployed. As a result, the coefficient is introduced to reserve transmit power for exceeded IoT devices.
In the RAN slicing system, as the total limited system bandwidth will be shared by mIoT slices and URLLC slices, we have the following bandwidth constraint
(20) 
where denotes the bandwidth allocated to mIoT slice that is correlated with by means of , and denotes a block of reserved bandwidth.
In (20), is an integer, and some integer variable recovery schemes [36] can be leveraged to obtain the suboptimal . However, considering the high computational complexity of optimizing an integer variable and the utilization of the scheme of spectrum safety margin, we directly relax the integer variable into a continuous one, i.e., let . Without loss of any generality, we regard as an independent variable below. Besides, as at least one PRB should be allocated to mIoT slices, we have
(21) 
Owing to the exploration of mIoT and bursty URLLC service multiplexing, we should orchestrate network resources for all mIoT slices and URLLC slices to simultaneously maximize the utilities of mIoT slices and URLLC slices.
For a mIoT slice , its primary goal is to offload as many data packets as possible from IoT devices. In this way, the number of accumulated packets in each IoT device should be kept at a low level. Considering that a great RA success probability of an IoT device will lead to a low number of accumulated packets in its queue, we define the utility of a mIoT slice as the following.
Definition 6 (mIoT slice utility).
Over a time slot of duration , the mIoT slice utility is defined as the timeaverage of RA success probabilities of IoT devices in all mIoT slices, which is given by
(22) 
where with the numerator representing the expected sum of RA success probabilities of IoT devices in slice and the denominator denoting a normalization coefficient.
In (22), can be regarded as an intraslice priority coefficient. A mIoT slice serving more IoT devices will be orchestrated with more network resources.
For a URLLC slice , its primary objective is to maximize the slice gain that is reflected by the parameters in the slice request at a low cost. Therefore, we define an energyefficient utility for URLLC slices, as presented below.
Definition 7 (Bursty URLLC slice utility).
Over a time slot of duration , the bursty URLLC slice utility is defined as the timeaverage energy efficiency for serving all URLLC devices, which is given by
(23) 
where is an energy efficiency coefficient reflecting a tradeoff between the URLLC slice gain and the RRH energy consumption.
Then, over a time slot of duration , the RAN slicing problem for mIoT and URLLC service multiplexing can be formulated as follows
(24a)  
(24b)  
(24c) 
where is an interslice priority coefficient reflecting the priority of orchestrating network resources for mIoT slices and URLLC slices.
The mitigation of (24) is quite challenging mainly because

indeterministic objective function: (24) should be optimized at the beginning of the minislot. The timeaveraged objective function of (24) 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 minislot.

two timescale issue: the creation of a network slice is performed at a timescale of time slot. Thus, the variable should be determined at the beginning of the time slot and kept unchanged over the whole time slot. The channel, however, is timevarying. As a result, the beamformer should be optimized at each minislot. In summary, the variables in (24) should be optimized at two different timescales.
V Problem solution with system generated channel
This section aims to tackle these challenges by exploiting of an SAA technique [37], an ADMM method [38], a semidefinite relaxation scheme and a Taylor expansion scheme.
Va Sample average approximation and alternating direction method of multipliers
As mIoT slices and URLLC slices share the network resources, both and may be determined by channel coefficients experienced by URLLC slices. At each minislot , due to the i.i.d. assumption on the channel coefficients of URLLC slices, we have
(25) 
where represents the channel samples of URLLC slices collected at the beginning of the time slot .
Given a collection of channel samples with and , Just like [22], as constraints (24b) and (24c) construct a nonempty compact set, the conclusion of Proposition 5.1 in [22] is applicable to this paper by exploiting the SAA technique. The conclusion indicates that if the number of channel samples is reasonably large, then converges to uniformly on the nonempty compact set almost surely. In other words, the SAA technique enables us to use the channel samples collected at the beginning of a time slot to approximate the unknown channel coefficients over the time slot. For notation lightening, we write instead of that represents a variable corresponding to the channel sample .
Recall that the variable will be kept unchanged over the time slot and the beamformer should be calculated at each minislot , we can further consider (24) as a global consensus problem, which can be effectively mitigated by an ADMM method. In this problem, is a global consensus variable that should be maintained in consensus for all , and that is calculated based on is a local variable.
The fundamental principle of an ADMM method is to impose augmented penalty terms characterizing global consensus constraints on the objective function of an optimization problem. In this way, the local variables can be driven into the global consensus while still attempting to maximize the objective function. Let , , where and . By applying the matrix property and utilizing the conclusions of SAA and ADMM, we can approximate (24) as the following problem at the beginning of the time slot
(26a)  
(26b)  
(26c)  
(26d)  
(26e)  
(26f)  
(26g) 
where is the Lagrangian multiplier, is a penalty coefficient, is a square matrix with blocks, and each block in is a matrix. In , the block in the th row and th column is a identity matrix, and all other blocks are zero matrices.
(24) is now reduced to a deterministic single timescale problem (26). What is more, (26) can be split into separate problems that can be optimized in parallel as its objective function is separable. Thus, the following ADMMbased framework from (27) to (29) can be exploited to mitigate (26)
(27a)  
(27b) 
(28) 
(29) 
where the augmented partial Lagrangian function
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