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Ultra-Reliable and Low-Latency Communications Using Proactive Multi-cell Association Chun-Hung Liu, Senior Member, IEEE, Di-Chun Liang, Kwang-Cheng Chen, Fellow, IEEE, and Rung-Hung Gau, Senior Member, IEEE

Abstract—Attaining reliable communications traditionally re- in large networking latency of tens to hundreds of millisec- lies on a closed-loop methodology but inevitably incurs a good onds. This indicates that there exists a tradeoff between high amount of networking latency thanks to complicated feedback reliability and low latency in system network architecture mechanism and signaling storm. Such a closed-loop methodology thus shackles the current cellular network with a tradeoff and subsequent communication protocols of mobile commu- between high reliability and low latency. To completely avoid nication networks. Such a reliability-latency tradeoff problem the latency induced by closed-loop communication, this paper intrinsically impedes the existing cellular systems to extend aims to study how to jointly employ open-loop communication their services in mission-critical communication contexts with and multi-cell association in a heterogeneous network (HetNet) so ultra high reliability and low latency constraints, such as as to achieve ultra-reliable and low-latency communications. We first introduce how mobile users in a HetNet adopt the proposed wireless control and automation in industrial environments, proactive multi-cell association (PMCA) scheme to form their vehicle-to-vehicle communications, and the tactile internet virtual cell that consists of multiple access points (APs) and then which allows controlling both real and virtual objects with analyze the communication reliability and latency performances. real-time haptic feedback [4], [5]. We show that the communication reliability can be significantly improved by the PMCA scheme and maximized by optimizing the The message transmission time for mission-critical applica- densities of the users and the APs. The analyses of the uplink tions needs to be on the order of milliseconds (ms) because and downlink delays are also accomplished, which show that the human reaction time is on the order of tens of milliseconds extremely low latency can be fulfilled in the virtual cell of a single [6] or less toward 1 ms for fully autonomous application user if the PMCA scheme is adopted and the radio resources of scenarios. The end-to-end latency of the LTE systems is each AP are appropriately allocated. usually in the range of 30 ∼ 100 ms, which cannot be further Index Terms—Ultra reliable and low latency communications, reduced because the backbone network of the LTE systems network coverage, open-loop communication, cell association, typically uses a delivery mechanism which is not optimized cellular network, stochastic geometry. for latency-sensitive services. To reduce the end-to-end latency in the cellular systems like LTE, it is necessary to fundamen- I.INTRODUCTION tally change the system architecture relying on the closed- HE international telecommunication union has identi- loop communication and backbone links. The latency of the T fied ultra-reliable low-latency communication (URLLC), backbone link can be significantly reduced by appropriate machine-type communication (mMTC) and enhanced mobile communication architecture and implementation of network broadband (eMBB) as the three pillar services in the fifth protocols to construct the dedicated connection for URLLC generation (5G) mobile communication that aims to provide services [7]. To reduce the latency in the physical layer, good connectivity for many various communication applica- transmission overhead needs to be suppressed by streamlining tions [1]–[3]. Among these three services, URLLC remains the the grant-free transmission mechanism of the physical layer access and allocating resources properly [8]. Nonetheless, arXiv:1903.09739v6 [cs.IT] 22 Jan 2021 most challenging technology due to the need of completely new system design in order to achieve the extremely high reducing the latency in the communication and backbone links system reliability and low latency in 5G cellular systems. is still insufficient to effectively perform ultra low-latency Existing mobile communication systems, such as long-term transmission in the current LTE systems because most of the evolution (LTE) systems and its predecessors, were promi- transmission latency is incurred by the control signaling (e.g., nently designed to achieve the goal of high throughput in grant and pilot signaling usually takes 0.3 ∼ 0.4 ms per mobile communications, yet they can also achieve highly scheduling). Accordingly, the most important and essential reliable communications in the physical layer at the expense means that enables URLLC in 5G heterogeneous cellular net- of complicated closed-loop protocol stack to inevitably result works (HetNets) toward the target latency of one millisecond is to disruptively redesign the transmission protocols in the C.-H. Liu is with the Department of Electrical and Computer Engi- physical layer of HetNets [5], [9]. neering, Mississippi State University, Starkville, MS 39762, USA. (e-mail: [email protected]) K.-C. Chen is with the Department of Electrical Engineering, University of South Florida, Tampa, FL 33620, USA. (e-mail: [email protected]) A. Motivation and Prior Related Work D.-C. Liang and R.-H. Gau are with the Institute of Communica- tions Engineering and Department of Electrical and Computer Engineer- ing, National Chiao Tung University, Hsinchu 30010, Taiwan. (e-mail: To effectively reduce latency in 5G HetNets, the essential [email protected]; [email protected]) approach is to adopt (feedback-free) open-loop communica- 2 tion1 so that no retransmission is needed and receivers can exploited in the network. Accordingly, all the analyses in this save time in performing additional processing and protocol. paper are conducted according to the assumption that all APs Open-loop communication has a distinct advantage to signif- and users are equipped with a single antenna and they thus icantly reduce control signaling overhead for power control accomplish the following contributions: and channel estimation in cellular systems if compared with • To enhance the communication reliability between users closed-loop communication. As such, in this paper we focus and APs, we propose the proactive multi-cell association on how to fulfill URLLC through open-loop communication (PMCA) scheme in which each user proactively asso- in a HetNet owing to the fact that extremely reliable open- ciates with multiple nearby APs and adopts open-loop loop communication is the key to low latency. All the existing communication to send URLLC messages to the APs URLLC works in the literature are hardly dedicated to study- associated with it as well as receive URLLC messages ing the open-loop communication or without retransmission from them. The PMCA scheme is completely different (typically see [10]–[18]). Some of the recent works focus on from any prior association schemes in the literature due how to perform URLLC in wireless systems by employing to multi-cell association and transmission, which was also retransmissions, short packet designs and their corresponding addressed in the recent standard of 3GPP for enhanced estimation algorithms for point-to-point transmission [10]– URLLC as an effective means to enhance communication [13]. Reference [10], for example, studied the energy-latency reliability if compared with retransmission [21]. tradeoff problem in URLLC systems with hybrid automatic • The distribution of the number of the users associating repeat request (HARQ), whereas reference [11] proposed an with an AP in each tier is accurately derived, which is efficient receiver design that is able to exploit useful infor- first found to the best of our knowledge. It importantly mation in the data transmission period so as to improve the reveals that the void AP phenomenon exists in the PMCA reliability of short packet transmission. scheme and needs to be considered in the analysis of There are some of the recent works that investigated re- ultra-reliable communications. source allocation problems in wireless networks under the • The uplink non-collision reliability of a user in the cell of URLLC constraint. In [14], the authors studied how to min- an AP is found for the proactive open-loop communica- imize the required system bandwidth as well as optimize tion and thereby we can characterize the uplink communi- the resource allocation schemes to maximize URLLC loads, cation reliability of a virtual cell for the non-collaborative whereas the problem of optimizing resource allocation in the and collaborative AP cases in a low-complexity form. short block length regime for URLLC was investigated in [15]. • From the analyses of the communication reliability in Reference [16] studied how to jointly optimize uplink and the uplink and downlink, the communication reliability downlink bandwidth configuration and delay components to is shown to be significantly influenced by the densities minimize the total required bandwidth and end-to-end delay. of the users and the APs and the number of the APs in Furthermore, there are few recent works that looked into a virtual cell. The PMCA scheme indeed improves the the URLLC design from the perspective of physical-layer communication reliability of a user and make it achieve system interfaces and wireless channel characteristics. The the target value of 99.999% when appropriately deploying recent work in [19], for example, adopted coding to seamlessly APs for a given user density. distribute coded payload and redundancy data across multiple • The uplink and downlink end-to-end delays between available communication interfaces to offer URLLC without users and their anchor node are modeled and analyzed. intervention in the baseband/PHY layer design. The problem We not only clarify the fundamental interplay among the of how URLLC is affected by wireless channel dynamics and delays, the number of the APs in a virtual cell, user robustness was thoroughly addressed in [20]. and AP densities, but also show that achieving the target Although these aforementioned works and many others in latency of one millisecond is certainly possible as long the literature provide a good study on how to achieve URLLC as the APs are deployed with a sufficient density for a and use it as a constraint to optimize the single-cell perfor- given user density. mance by using the closed-loop communication and retrans- missions, they cannot reveal a good network-wide perspective B. Paper Organization on how interferences from other cells and user/cell association The rest of this paper is organized as follows. In Section schemes impact the URLLC performance of cellular systems. II, we first specify the system architecture of a HetNet for This paper aims to exploit the URLLC performances when URLLC and then introduce the open-loop communication and open-loop communication is adopted in a large-scale cellular propose the PMCA scheme. Section III models and analyzes network. On account of open-loop communication, there is the uplink and downlink communication reliabilities for the no feedback of channel state information in the network so PMCA scheme and some numerical results are provided to that the advantages of multi-antenna transmission cannot be validate the correctness and accuracy of the analytical results. In Section IV, the end-to-end latency problem for the open- 1The open-loop communication mentioned in this paper is a “feedback- loop communication and the PMCA scheme is investigated free” communication technique, i.e., no immediate feedback from the receiver and some numerical results are also presented to evaluate the side. In light of this, it is also a “retransmission-free” communication latency performance of the open-loop communication and the technique because no retransmission happens between a transmitter and its receiver. Note that the open-loop communication may still need some delayed PMCA scheme. Finally, Section V concludes our findings in control signaling in order to successfully perform in a communication system. this paper. 3

(a) (b)

Fig. 1. (a) Two URLLC users and their virtual cell with three APs: The two virtual cells share AP 3 and there is RRU (radio resource unit) allocation collision on User 2. (b) The void cell phenomenon after using the PMCA scheme. Note that all the three APs in a virtual cell connect to the same anchor node.

II.SYSTEM MODELAND ASSUMPTIONS to employ open-loop communication to achieve URLLC in the HetNet. In this paper, we consider an interference-limited planar HetNet in which there are two tiers of APs and the APs in the same tier are of the same type and performance. In particular, A. Open-loop Communication and Proactive Multi-cell Asso- the APs in the mth tier form an independent homogeneous ciation Poisson point process (PPP) of density λm and they can be As mentioned in Section I, closed-loop communication expressed as set Φm given by fundamentally incurs more latency than open-loop communi-

2 cation owing to feedback. This point manifests that open-loop Φm , {Am,i ∈ R : i ∈ N}, m = {1, 2}, (1) communication turns out to be the best solution to reducing latency from the receiver perspective because feedback-related where A denotes AP i in the mth tier and its location. m,i communication latency is completely avoided. However, the Without loss of generality, we assume the first tier consists of reliability performance of wireless communications could be the macrocell APs and the second tier consists of the small seriously weakened due to no feedback transmission in that cell APs. A macrocell AP has a much larger transmit power it cannot be improved by using the hybrid automatic repeat than a small cell AP, whereas the density of the macro AP is request (ARQ), a combination of high-rate forward error- much smaller than that of the small cell APs. To effectively correcting coding and ARQ error control, which is commonly achieve URLLC in the HetNet, an anchor node which governs used in closed-loop communication. This phenomenon reveals a number of nearby APs according to the geographical deploy- that there seemingly exists a tradeoff between latency and ment of the APs is co-located with the edge/fog computing reliability in wireless communications. However, this tradeoff facilities, and a cloud radio access architecture comprised of can be absolutely alleviated or tackled by ultra-reliable open- a core network and a cloud is also employed in the HetNet. loop communication since closed-loop feedback hardly further Macrocell APs and anchor nodes are connected to the core benefits the reliability of a wireless channel with extremely network which helps send complex computing tasks to the high reliability. cloud for further data processing and management. In addition, To create an ultra-reliable open-loop communication context all (URLLC) users in the HetNet also form an independent for the users in the HetNet, the users are suggested to proac- homogeneous PPP of density µ and they are denoted by set tively associate with multiple APs at the same time so that their U as communication reliability can be improved by spatial channel 2 diversity and even boosted whenever the associated multiple U , {Uj ∈ R : j ∈ N}, (2) APs are able to do joint decoding. This proactive multi-cell where Uj stands for user j and its location. An illustration of association approach leads to the concept of the virtual cell of the system model depicted here is shown in Fig. 1 (a), and users, that is, each user seems to form its own virtual cell that the main notations used in this paper are summarized in Table encloses all the APs associated with it [22] and an illustration I. The open-loop communication technique is adopted in the of the virtual cell is shown in Fig. 1(a). Note that all the APs HetNet, i.e., no (immediate) feedback between a AP and a in the same virtual cell are assumed to be connected to the user. In the following, we elaborate on the main idea of how same anchor node for the consideration of modeling simplicity 4

TABLE I NOTATION OF MAIN VARIABLES,SYMBOLS, AND FUNCTIONS

Symbol Meaning Symbol Meaning ul Φm Set of tier-m APs γK Uplink SIR of the kth AP in VK ul dl λm Density of Φm ηK (ηK ) Uplink (Downlink) communication reliability dl Pm Transmit power of the tier-m APs ηk Downlink reliability of the kth AP in VK Am,i AP i in the mth tier and its location 1(E) Indicator function of event E K Number of APs in a virtual cell LZ (s) Laplace transform of random variable Z > 0 VK Virtual cell with K APs Qk Transmit power of the kth AP in VK Vk The kth AP in VK θ SIR threshold for successful decoding U Set of users Hk Downlink channel gain from Vk to typical user µ Density of users hk Uplink channel gain from typical user to Vk Uj User j and its location qj Transmit power of user Uj SK The Kth-truncated shot signal process qk Transmit power of a user in Vk for AP Vk kX − Y k Euclidean distance between nodes X and Y ϑm Probability of an AP in VK from Φm ul dl pm,0 The void probability of Tier-m APs D (D ) Uplink (Downlink) communication delay ul dl α > 2 Pathloss exponent Dba (Dba) Uplink (Downlink) backhaul delay ul ul dl ρ Uplink non-collision reliability of an AP Dtr (Dtr) Uplink (Downlink) transmission delay ul ρK Uplink non-collision reliability of a user ξ Short block-length packet size δ Probability of a user selecting an RRU in VK τ Duration of transmission ωm Tier-m association bias B Bandwidth Om,i Void indicator of AP i in the mth tier  Decoding error probability and how user mobility impacts the URLLC performance of B. The PMCA Scheme and Its Related Statistics a virtual cell due to handover between macro APs is not Assume that each user in the network associates with K considered in this paper. All the radio resources in a virtual cell 2 APs by the following PMCA scheme. Let Vk be defined as can be scheduled and allocated by an anchor node utilizing the ( wm edge/fog computing technology. Thus, letting users form their arg maxm,i:A ∈Φ{ α }, k = 1 m,i kAm,ik Vk , (3) virtual cell (i.e., associate with multiple APs) has an advantage , wm arg maxm,i:A ∈{Φˆ }{ kA kα }, k > 1 in largely reducing control signaling for frequent handovers m,i k−1 m,i between small cell APs, which leads to no handover latency S2 ˆ Sk−1 where k ∈ N+, Φ , m=1 Φm, Φk−1 = Φ \ j=1 Vj, α > 2 in a virtual cell. However, a user should not associate with too is the path-loss exponent, positive constant wm is the tier- many APs at the same time because the signaling overhead m cell association bias, and kX − Y k denotes the Euclidean due to multi-AP synchronization could deteriorate the latency distance between nodes X and Y . For a typical user located at performance of its virtual cell. In addition, the means of short the origin, Vk is thus the kth biased nearest AP of the typical packet transmission will be employed in a virtual cell to further user by averaging out the channel fading gain effect on the shorten the communication latency. user side. More specifically, Vk denotes the kth nearest AP of this typical user if wm = 1 for all m ∈ {1, 2}, whereas Vk becomes the kth strongest AP of the typical user if wm = Pm where Pm is the transmit power of the tier-m APs. The K In the following subsection, we will clarify the fundamental APs associated with the typical user can be expressed as a set interplay among reliability, latency, and multi-cell association given by by formally proposing the PMCA scheme and analyzing its K related statistical properties. For the sake of simplicity, the [ VK , Vk, (4) following analyses are conducted by assuming all users and k=1 APs are equipped with a single antenna rather than multiple which is called the virtual cell of the typical user. antennas. The reasons are specified as follows. As previously According to [24] [25], the distribution of the number of mentioned, open-loop communication that is adopted in a the users associating with an AP is found for the single- virtual cell does not require immediate feedback of channel cell association scheme. The method of deriving it cannot be state information (CSI), and thereby it cannot fully enjoy the directly applied to the case of the PMCA scheme because advantages of multi-antenna transmission, such as diversity the cells of the APs are no longer disjoint in the multi-cell and beamforming. Although space-time coding for multi- association case. Nonetheless, the idea behind the method is antenna transmission can help achieve diversity, it is too fairly helpful for us to derive the distribution of the number complicated to be practically fulfilled in the short packet trans- of the users within the cell of an AP for the PMCA scheme, mission that is needed for ultra-low latency communication. as shown in the following lemma. As such, the single-antenna transmission performance is pretty similar to the multi-antenna transmission performance in a 2For the sake of modeling simplicity, we do not consider the shadowing ef- virtual cell. Moreover, we would like to analyze whether or not fect in (3) since it does not affect the following analyses of the communication the proposed open-loop communication and PMCA scheme reliability in Section III according to the random conservation property found in [23]. Further more, we assume wm and λm for m ∈ {1, 2} are sufficiently can accomplish the practical requirements of URLLC even in large such that each user can detect almost all the APs in the network and the worst-case scenario of single-antenna transmission. associate with at least K APs by using the PMCA scheme in (3). 5

(a) (b)

Fig. 2. Simulation results of pm,n: (a) p1,n for macro cell APs (b) p2,n for small cell APs.

Lemma 1. Suppose each user in the network adopts the that a tier-m AP is not associated with any users, referred to PMCA scheme in (3) to associate with K APs in the network. as the tier-m void probability, can be found as Let N denote the number of the users associating with a m !−ζm,K tier-m AP and its distribution (i.e., pm,n P[Nm = n]) can Kµ , pm,0 = 1 + . (6) be semi-analytically approximated as ζm,K λem !n For a dense cellular network with a moderate user density, this Γ(n + ζ ) Kµ p ≈ m,K void probability may be so large that the void APs could be a m,n n!Γ(ζ ) m,K ζm,K λem considerable amount in the network. For example, we use the !−(n+ζm,K ) network parameters for simulation in Fig. 2 to find the void Kµ × 1 + , (5) probabilities p1,0 = 0.03 and p2,0 = 0.2 for K = 3 and the ζm,K λem void probability of small cell APs is actually not small at all (there are 20% of the small cell APs that are void.). Thus, such R ∞ x−1 −t where Γ(x) , 0 t e dt for x > 0 is the Gamma a void cell phenomenon for the PMCA scheme, as illustrated function, ζm,K > 0 is a positive constant that needs to in Fig. 1 (b), must be considered in the interference model be determined by the real numerical data of pm,n, and [23], [24] when the user density is not very large if compared 2 2 P2 α α λem , i=1 wi λi/wm. with the density of the small cell APs. Proof: See Appendix A. C. The Truncated Shot Signal Process in a Virtual Cell To validate the correctness and accuracy of pm,n in (5), we adopt the network parameters for a two-tier HetNet shown in As the PMCA scheme and the virtual cell of a user Fig. 2 to numerically simulate pm,n for K ∈ {1, 2,..., 5}. introduced in Section II-B, we define the Kth-truncated shot 3 As can be seen in Fig. 2, the simulated results of pm,n signal process of the virtual cell of the typical user as follows : accurately coincide with its corresponding analytical results of K X −α pm,n found in (5). Since the approximated result of pm,n in SK , HkWkkVkk , (7) Lemma 1 is numerically validated, there are some important k=1 implications that can be drawn. First, we can learn that the where Vk ∈ VK is already defined in (3), Hk denotes the fad- average number of the users associating with a tier-m AP is ing channel gain from Vk to the typical user, Wk ∈ {w1, w2} Kµ/λm and this means the average cell size of a tier-m AP e is the cell association bias of AP Vk, and it is a non-negative is K/λm [25], [26]. In other words, the average cell size of an e random variable (RV) associated with Vk. We call SK the AP increases K times as users associate with K APs. Second, Kth-truncated shot signal process because it only captures the for K = 1 users only associate with a single AP so that the cumulative effect at the typical user of the K random shocks cells of the APs do not overlap and the entire network area 3 consists of weighted Voronoi-tessellated cells. For K > 1, the When K goes to infinity, S∞ , limK→∞ SK is traditionally referred to cells of the APs may overlap in part and the cell sizes of the as (complete) Poisson shot noise process [27], [28] since it contains weighted signal powers in a Poisson field of transmitters. Since SK only contains the APs and the numbers of the users associating with the APs signals emitted from the first K weighted nearest transmitters in the network, are no longer completely independent. Third, the probability it is called the Kth-truncated shot signal process. 6

for y, z ∈ R+ is defined as yz Z 1 y `(y, z) , − 1 dt, (9) sinc(z) 0 y + t z

sin(πx) and sinc(x) , πx . For the Laplace transform of SK , it can be explicitly found as

" 2 # K Z ∞ πλse α (πλe) K−1 LSK (s) = exp − y sinc(2/α) (K − 1)! 0     − α 2 × exp πλye 1 + ` sy 2 , dy. (10) α

In addition, the upper bound on P[SK ≥ y] can be found as

K Y   2ky  P[SK ≥ y] ≤ 1 − 1 − L α , (11) Fig. 3. An illustrative example of the Kth truncated shot signal process in Y 2 K(K + 1) a virtual cell for K = 3. In the figure, the typical user form a virtual cell k=1 k by associating with the first three nearest small cell APs using the PMCA V V V scheme in (3), i.e., 1, 2, 3. For this example, we have the 3rd truncated where Yk ∼ Gamma(k, πλe) is a Gamma RV with shape shot signal process S = P3 H kV k−α, which can be referred to as 3 k=1 k k parameter k and rate parameter πλe. the sum of the (random) received powers from all Vk’s when all the small cell APs are transmitting with unit power. Proof: See Appendix B. The above results of Laplace transform in Theorem 1 from the K different random locations (i.e., V1,...,VK ), and indicate that in general the closed-form results of LS−K and −α HkWkkVkk can be viewed as the impulse function of AP LSK are unable to be obtained except in some special cases. α V that gives the H W -weighted attenuation of the transmit For instance, letting s = ϕy 2 for ϕ > 0 and L α (ϕ) can k k k y 2 S−K power of Vk in space. An illustrative example of the Kth be shown as truncated shot signal process in a virtual for K = 3 is visually K Z ∞ (πλe) K−1 −πλye [1+`(ϕ, 2 )] demonstrated in Fig. 3 where the densities of the users, macro L α (ϕ) = y e α dy y 2 S APs, and small cell APs are 50 users/km2, 1.0 APs/km2, and −K (K − 1)! 0 10 APs/km2, respectively. In the figure, the typical user adopts   2 −K = 1 + ` ϕ, . (12) the PMCA scheme in (3) with w1 = w2 = 1 to form its virtual α cell by associating with the first 3 nearest (small cell) APs in the HetNet. Consequently, there is the 3rd-truncated shot Nonetheless, we can still resort to some numerical techniques 3 P −α to evaluate the Laplace transforms of S−K and SK and the signal process in the virtual, i.e., S3 = k=1 HkkVkk , which can be interpreted as the sum of the (desired) received distributions of S−K and SK by numerically evaluating the signal powers from all Vk’s in the virtual cell while all the inverse Laplace transform of S−K and SK . In addition, we small cell APs are transmitting with unit power. are still able to understand the distribution behaviors of SK from the closed-form upper bound on [S ≥ y]. Theorem Let LZ (s) , E[exp(−sZ)] denote the Laplace transform P K of a non-negative RV Z for s > 0 and some statistical results 1 plays an important role in the following analysis of the regarding SK are presented in the following theorem. communication reliability that will be defined in the following section. Theorem 1. Assume all the Hk’s of the Kth-truncated shot signal process in (7) are i.i.d. exponential RVs with unit mean4, i.e., Hk ∼ exp(1). If we define S−K , S∞ − SK and III.COMMUNICATION RELIABILITY ANALYSIS FOR S∞ , limK→∞ SK , then the Laplace transform of S−K can PROACTIVE MULTI-CELL ASSOCIATION be explicitly found as In this section, we would like to exploit the fundamental K Z ∞ (πλe) K−1 performances and limits of the communication reliability of LS−K (s) = y (K − 1)! 0 users in the uplink and the downlink when the PMCA scheme     is employed in the HetNet. We assume that the orthogonal − α 2 × exp −πλye 1 + ` sy 2 , dy, (8) α frequency division multiple access is adopted in the HetNet and the communication reliability analyses are proceeded in 2 2 P2 α α where λe , m=1 wmλm, ϑm , P[Wk = wm] = wmλm/λe is accordance with how the radio resource blocks (RB) in the cell the probability that a user associates with a tier-m AP, `(y, z) of each AP are requested by a user in the uplink and allocated by an AP in the downlink. We will first specify how users 4In this paper, we consider the HetNet is in a rich-scattering environment access the RBs of an AP and then propose and analyze the so that all the fading channel gains of the wireless links in the HetNet can be assumed to be independent due to very weak spatial correlation between uplink communication reliability. Afterwards, we will continue them. to study the communication reliability in the downlink case. 7

7 A. Analysis of Uplink Communication Reliability whenever pm,n decreases by increasing K and/or δ decreases ; thereby, fewer APs in the virtual cell and/or more radio According to the PMCA scheme and the virtual cell of a resources may significantly improve the uplink non-collision user specified in Section II-B, our interest here is to study reliability. Note that the uplink non-collision reliability of a how likely a user is able to successfully access available RBs user may not always increase as K increases since ρul ≈ Kρul of an AP and then send its message to the K APs in its K for ρul  1 and increasing K in this situation may not virtual cell through the open-loop communication. To establish increase ρul because ρul decreases in this case. In addition the uplink access from a user to the K APs, we propose the K to the uplink collision problem happening to APs, whether a following PMCA-based radio resource allocation scheme for user is able to successfully send its messages to at least one AP uplink open-loop communication: in its virtual cell also depends upon all the communication link • To make a user have good uplink connections, the user ul statuses in the virtual cell. Let γk denote the uplink signal- forms its virtual cell by associating with its first K nearest to-interference ratio (SIR) from a typical user located at the APs. Thus, all the cell association biases in (3) are unity, origin to the kth AP in the virtual cell, and it can be expressed i.e., wm = 1 for all m ∈ {1, 2}. as • Each radio RB serves as the basic unit while scheduling h q kV k−α radio resources. Multiple radio RBs in a single time slot γul k k k , (15) k , P h q kV − U k−α are mapped to a single (virtual) radio resource unit (RRU) j:Uj ∈Ua j j k j for transmitting a message. Users are allowed to transmit h one message in each time slot. where k denotes the uplink fading channel gain from the typical user to AP V , q is the transmit power used by the • Due to lack of CSI of each AP in the virtual cell5, a user k k V q U h proactively allocates the radio resource in a distributed typical user for AP k, j is the transmit power of user j, jk U V U ⊆ U manner, that is, it has to randomly select RRUs for the is the uplink fading channel gain from j to k, and a K APs in its virtual cell6. represents the set of the actively transmitting users using the same RRU as the typical user. All uplink fading channel gains Since each user has to randomly select the uplink RRUs are assumed to be i.i.d. exponential RVs with unit mean, i.e., in its virtual cell without considering how other users select hk, hjk ∼ exp(1) for all k, j ∈ {1,...,K}. their uplink RRUs, multiple users in the cell of an AP could According to (15), we can consider two cases of non- select the same RRUs, which leads to transmission collisions collaborative and collaborative APs to define the uplink com- as indicated in Fig. 1(a). If users associated with the same AP munication reliability in a virtual cell. The case of non- select the same RRU, then they fail to upload their message collaborative APs corresponds to the situation in which all to their APs owing to transmission collisions between them. APs in the virtual cell are not perfectly coordinated so that The probability that there is no uplink collision in the virtual they cannot do joint transmission and reception, whereas when cell, referred to as the uplink non-collision reliability, is found all APs in the virtual cell are perfectly coordinated so that they in the following lemma. are able to collaboratively do joint transmission and reception Lemma 2. Suppose a user adopts the PMCA scheme in (3) corresponds to the case of collaborative APs. For the case of to form its virtual cell with K APs. If the probability that the non-collaborative APs, the uplink communication reliability user selects any one of the RRUs for each AP in its virtual is defined as the probability that a message sent by a user cell is δ ∈ (0, 1), then the uplink non-collision reliability of in a virtual cell is successfully received by at least one non- each AP in its virtual cell is found as collision AP in the virtual cell, and it can be expressed as   2 ∞ ul ul1 nc ul X X n−1 ηK , P max {γk (Vk ∈ VK )} ≥ θ , (16) ρ = ϑm pm,n(1 − δ) . (13) k∈{1,...,K} m=1 n=1 where 1(A) is the indicator function that is unity if event A Hence, the uplink non-collision reliability of the user in its is true and zero otherwise, θ > 0 is the SIR threshold for virtual cell is nc successful decoding, and VK ⊆ VK is the subset of the APs K without collision in set V . For the case of collaborative APs, ρul = 1 − 1 − ρul . (14) K K the uplink communication reliability is defined as Proof: See Appendix C. " # Sul Lemma 2 reveals that the uplink non-collision reliability of ul K ηK , P P −α ≥ θ , (17) hjqjkVk − Ujk each AP is mainly influenced by K and δ, e.g., it decreases j:Uj ∈Ua\VK ul P −α ul 5 where S nc hkqkkVkk . Namely, η in (17) is Note that the open-loop communication can be applied to frequency- K , k:Vk∈VK K division duplex (FDD) systems and time-division duplex (TDD) systems. As the probability that a uplink message is successfully received such, it is not necessary to designate the HetNet in the paper to adopt either by at least one non-collision AP in the virtual cell: If there FDD or TDD. Our goal in this paper is to delve how to achieve URLLC in the HetNet under the CSI-free scenario. are at least two non-collision APs in the virtual cell, they can 6In a virtual cell, a user is not suggested to adopt carrier-sense multiple 7 ul access (CSMA) protocols to gain the radio resource since the channel access In general, ϑm does not have a significant impact on ρK in that usually latency induced by CSMA is too large to be satisfied by the ultra-low latency P2  P1 as well as λ1  λ2 and these two condition leads to ϑ1  ϑ2 requirement of URLLC. in most of practical cases 8

ul jointly decode the message. Otherwise, only one non-collision in order to achieve η∞ ≥ 99.999%. In other words, the target AP can decode it.8 reliability 99.999% is not able to be achieved by the PMCA ul The analytical results of (16) and (17) are summarized in scheme if δ > 5%. Note that ηK in (17) is certainly larger ul the following theorem. than that in (16) and the upper bound on ηK in (20) may be greater than the result in (19). In addition, we would like to Theorem 2. Suppose each user employs the PMCA scheme in point out that a virtual cell with K APs can support uplink (3) to form its virtual cell with K APs. If all the K APs in the short packet transmission when ηul is higher than the target virtual are unable to collaborate, the uplink communication K reliability because short packet transmission suffers from the reliability defined in (16) is approximated by problem of degraded transmission reliability and efficiency and  −k K 2 ! such a problem is significantly mitigated by a high value of Y  δθ α (1 − p0)  ηul ≈ 1 − 1 − ρul 1 + , (18) ηul. These aforementioned observations will be numerically K sinc(2/α) K k=1   validated in Section III-C. where p0 , pm,0 that is given in (6) with wm = 1 for m ∈ {1, 2} and ρul is given in (13). For the case of K → ∞, B. Analysis of Downlink Communication Reliability ul ul η∞ , limK→∞ ρK can be approximately found as In this subsection, we would like to study the downlink  ul   communication reliability of users in their virtual cells. Note ul ρ 2 η∞ ≈ 1 − exp − 2 sinc . (19) that the uplink transmission and the downlink transmission in a δθ α α virtual cell are independent and there is thus no order between When all the K APs in the virtual cell are able to collaborate them. To establish the benchmark performance, we assume that to jointly decode the uplink message, ηul in can be upper K (17) the frequency reuse factor in this cellular network is unity (i.e., bounded by all APs share the entire available frequency band) so that we  K   can evaluate the downlink communication reliability in the Y δ(1 − p0) ηul ≤ρul 1 − 1 − 1 + K sinc(2/α) worst-case scenario of interference. We also assume that each k=1 of the downlink RRUs of an AP is uniquely allocated to a user 2 −k  2kθ  α   associating with the AP and users adopt the PMCA scheme × . (20) K(K + 1) to associate with the first K strongest APs (i.e., wm = Pm in (3)). In the virtual cell of the typical user, the SIR of the link Proof: See Appendix D. ul from the kth strongest AP to the typical user is defined as From the results in Theorem 2, the uplink reliability ηK −α in (18) monotonically increases as K increases even though dl HkQkkVkk γk , dl , (22) increasing K makes p0 reduce and it thereupon reduces the P H Q kV k−α + I i:Vi∈VK \Vk i i i K number of the void cells and induces more interference. ul where Hk denotes the downlink fading channel gain from However, ηK suffers from the diminishing returns problem as K increases so that associating with too many APs may Vk to the typical user, Qk ∈ {P1,P2} is the trans- mit power of V , H is also the downlink fading not be an efficient means to significantly improve ηK for a k m,i dl user. In particular, (18) can be used to obtain the following channel gain from Am,i to the typical user, IK , P O H P kA k−α, and O ∈ {0, 1} result m,i:Am,i∈Φ\VK m,i m,i m m,i m,i −K is a Bernoulli RV that is unity if Am,i is not void and zero ul 2 ! 1 − ηK ul δθ α otherwise. All Hi’s and Hm,i’s are assumed to be i.i.d. expo- ul = 1 − ρ 1 + , (21) 1 − ηK−1 sinc(2/α) nential RVs with unit mean. Note that P[Om,i = 1] = 1−pm,0 and it can be found by using (6). The term Idl in (22) is the ul ul K which indicates (1 − ηK )/(1 − ηK−1) ≈ 1 as K  1 interference from all non-void APs that are not in the virtual ul and we thus know ηK /ηK−1 ≈ 1 for large K, i.e., the cell, whereas the term P H Q kV k−α in (22) is i:Vi∈VK \Vk i i i diminishing returns problem occurs. According to (18)-(20), the intra-virtual-cell interference from other K − 1 APs in ul another two efficient approaches to boosting ηK are reducing the virtual cell if the K − 1 APs in the virtual cell are not the probability of scheduling each RRU in each cell and coordinated to avoid using the RRU used by the kth AP. This densely deploying APs in the HetNet, that is, we need small represents the worst case of the downlink SIR of the kth AP δ in that small δ suppresses the magnitude of the interference. in the virtual cell. In this case, the downlink communication ul For instance, if θ = 6 dB, α = 4, then η∞ ≈ 57.9% for reliability of a virtual cell with K APs is defined as9 δ = 0.5, ρul = 0.9 and ηul ≈ 98.96% for δ = 0.1, ρul = 0.95. ∞   Note that ηdl in (19) characterizes the fundamental limit of the dl dl ∞ ηK , P max {γk } ≥ θ , (23) uplink communication reliability if all APs cannot collaborate k∈{1,...,K} in the uplink and it can be used to evaluate whether the PMCA 9 and resource allocation schemes can achieve some target value Due to open-loop communication, each AP does not have channel state ul information and thereby only the multi-AP (multi-channel) diversity can be of ηK . If θ = 6 dB and α = 4, for example, we require δ ≤ 5% exploited while the APs in a virtual cell are performing downlink CoMP. In the case of non-collaborative APs, the downlink transmission in a virtual cell 8For the sake of analytical tractability, we consider that non-coherent signal succeeds as long as at least one AP can successfully transmit to the user in combing happens among all the non-collision APs in the virtual cell even the virtual cell. The downlink communication reliability is thus defined as though such a combining leads to a suboptimal SIR performance. shown in (23). 9

TABLE II NETWORK PARAMETERSFOR SIMULATION [15], [29]

Parameter \ AP Type (Tier m) Macrocell AP (1) Small cell AP (2) Transmit Power Pm (W) 20 5 User Density µ (users/km2) 50 2 AP Density λm (APs/km ) 1.0 [0.2µ, µ] RRU Selection Probability δ 0.05 Path-loss Exponent α 4 Tier-m Association Bias wm (Uplink, Downlink) (1,Pm) Packet Size ξ (bytes) 8, 32, and 64 Duration of Transmission τ (ms) 0.05 Bandwidth B 20 MHz Decoding Error Probability  2 × 10−8   Q−1() SIR Threshold θ exp ξ ln 2 + √G − 1 τB τB

which is the probability that there is at least one AP in the void probability pm,0, yet it also suffers from the diminishing virtual cell that can successfully transmit to the user in the returns problem, like the uplink communication reliability. virtual cell. The tier-m void probability pm,0 also significantly impacts ul When all the K APs in the virtual cell can collaborate ηK when the number of the APs in a virtual cell is not dl to eliminate the intra-virtual-cell interference, the downlink large so that increasing the AP density improves ηK since communication reliability can be simply written as it helps increase pm,0. Moreover, δ can be interpreted as the " # probability that all APs statically allocate their RRUs with PK H Q kV k−α ηdl = k=1 k k k ≥ θ . equal probability and it has to be small in order to achieve K P P O H P kA k−α m,i:Am,i∈Φ\VK m,i m,i m m,i ultra-reliable communications. The result in (26) that does (24) not depend on the densities of the APs and users is the fundamental limit of the downlink communication reliability The explicit results of ηdl defined in (23) and (24) are found K when all non-collaborative APs use different RRUs to transmit in the following theorem. a message to the same user. It reveals whether ultra-reliable Theorem 3. Suppose all APs in a virtual cell are not coor- communications can be attained by the PMCA and resource dinated so that there exists the intra-virtual-cell interference allocation schemes. For example, if θ = 6 dB and α = 4, dl in the virtual cell. The downlink communication reliability in we need δ < 0.05 to achieve ηK ≥ 99.999%, i.e., each the case of non-collaborative APs defined in (23) is explicitly RRU cannot be scheduled with a probability more than 5% in upper bounded by this case. Otherwise, the PMCA scheme cannot successfully achieve the downlink communication reliability of 99.999% no K " 2 #−k Y   2  X matter how many APs are in a virtual cell. Furthermore, ηdl ηdl ≤ 1 − 1 − 1 + δ` θ, ϑ (1 − p ) K K α m 0,m in (27) for the case of collaborative APs is certainly higher k=1 m=1  than that in (25) and it can also provide some insights into , (25) how to schedule resources and deploy APs in the HetNet so dl as to achieve the predesignated target value of ηK . Likewise, a 2 2/α P2 α virtual cell with K AP is able to support downlink short packet where ϑm Pm λm/λe and λe = Pm λm. When K , m=1 transmission when ηdl is extremely high, as the reason already goes to infinity, ηdl lim ηdl can be approximately K ∞ , K→∞ K pointed out in Section III-A. In the following subsection, some found in closed form given by numerical results and discussions will be provided to evaluate " # sinc 2
the performances of the downlink communication reliability ηdl ≈ 1 − exp − α . ∞ 2 (26) for the PMCA scheme. δθ α

dl For the case of collaborative APs, the upper bound on ηK in (24) can be found as C. Numerical Results and Discussions K To validate the analytical results obtained in the previous  "   2 # dl θ 2 X subsections and evaluate the communication reliability per- ηK ≤ 1 − 1 − 1 + δ` α +1 , ϑm(1 − pm,0) K 2 α m=1 formances of open-loop communication and PMCA for short −K packet transmission, some numerical results are provided in . (27) this subsection. The network parameters for simulation is shown in Table II. To clearly show whether or not the target Proof: See Appendix E. communication reliability of 99.999% is attained in the uplink From the results in Theorem 3, we realize that increasing and downlink for different situations of PMCA, the following dl K indeed improves ηK even though it reduces the tier-m figures demonstrate the simulation results of the uplink and 10

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(a) (b)

ul ul ul Fig. 4. Numerical results of the uplink outage 1 − ηK for the case of non-collaborative APs: (a) 1 − ηK versus µ/λ2 for K = 4, (b) 1 − ηK versus K 2 for µ/λ2 = 0.5 (users/small cell AP) and λ2 = 250 (APs/km ).

ul dl downlink outage probabilities (i.e., 1 − ηK and 1 − ηK ) and the received uplink SIRs at different APs are assumed to be the designated outage threshold is thus 1 − 99.999% = 10−5. independent when deriving (18). Figure 4(b) illustrates how Since the simulation of the uplink and downlink outages is the uplink outage probability is suppressed as the number of ul rare-event, it is terminated as the outage event occurs over APs in virtual cell increases. As shown in the figure, 1 − ηK 200 times so that we can obtain much confident statistics. significantly reduces as K increases from 1 to 4, but only the Due to considering short packet transmission, we adopt the case of the blue curves is able to achieve the outage probability maximum achievable rate of short blocklength regime without below the designated outage threshold when K ≥ 4. In light considering channel dispersion found in [15], [29] to infer the of this, users are suggested to associate with more APs and/or outage condition for the HetNet in this paper as follows: reduce their packet size. −1 Q () ξ The simulated results of the uplink outage probabilities for ln(1 + SIRul) − √G ≤ , (28) K τB Bτ the scenario of collaborative APs are shown in Fig. 5 in which the analytical results that are the lower bound on the outage where SIRul equals max {γul1(V ∈ Vnc)} for K k∈{1,...,K} k k K probability are calculated by using (20). Only the results of max {γdl} Q−1(·) uplink and k∈{1,...,K} k for downlink, G stands the black curves in Fig. 5(a) are above the designated outage for the inverse of (Gaussian) Q-function, ξ denotes the size threshold of 10−5 and thereby all the results in Fig. 5(a) are of a short packet, and τ is the duration of transmission. The much better than those in Fig. 4(a). This reveals that we inequality in (28) can be further rewritten as do not need to deploy many APs to significantly decrease Q−1() ξ  1 − ηul when the APs are able to collaborate. In Fig. 5(b), SIRul ≤ exp √G + − 1. (29) K K τB Bτ the simulated outcomes are also much better than those in Fig. 4(b), which indicates that PMCA should be jointly im- Thus, we can get the uplink reliability of short packet trans- ul ul plemented with CoMP so as to better serve the URLLC traffic. mission, i.e., ηK = P[SIRK ≥ θ] by setting the SIR threshold The simulated results of the downlink outage probabilities for  Q−1()  as θ = exp √G + ξ − 1, as shown in Table II. τB Bτ the scenarios of non-collaborative and collaborative APs are Figure 4 shows the numerical results of the uplink outage shown in Figs. 6 and 7, respectively. In general, they have probabilities for the non-collaborative scenario. The analytical similar ascending/descending curve trends if compared with results corresponding to this scenario in the figure are found the results in Figs. 4 and 5, yet there are still some subtle ul by using (18). As shown in Fig. 4(a), 1 − ηK increases discrepancies between them. The downlink outage probability, ul (ηK decreases) as µ/λ2 increases. This is because more co- for instance, is lower than its uplink counterpart because users channel interference is created as µ/λ2 gets larger so that are not interfered by their first K strongest APs owing to more APs are associated with users and become active. The PMCA in the downlink, whereas APs are very likely to be short packet transmission with a longer packet size makes severely interfered by their nearby users in the uplink. Also, the outage probability increase, which can be mitigated by APs can coordinate to allocate their RRUs in the downlink decreasing µ/λ2. Also, all the simulated results are very close with the aid of anchor nodes so that their downlink channel to their corresponding analytical results, which validates that collisions can be largely reduced. As such, the uplink outage is the analytical result in (18) is still fairly accurate even when the main hurdle of achieving URLLC requirements by using 11

10-2 r------,------,------,------=------, ------e-Simulated 1 -771}f(e == 8bytes) ______-Analytical 1- 771}f( e == 8bytes)

...... ◄ ...... --e-Simulated 1- 771}f(e == 32bytes) ------Analytical 1 -771}f( == 32bytes) ------e ------· ------· ------�Simulated 1- 771}f(e == 64bytes) �------...-..=------______-----Analytical1- 771}f( == 64bytes) ...------Designated outage threshold e ------;-...- - - ;• ;• -- ...... -- • • • • • • • I • • • • ;• ------.. . ,- --- I I ,- .... -...... ,­ ----- I I ,,­ ...... •·· ...... ,...... ,,♦ ...... ,,♦ ...-- -� �� ,,♦ ...... ,,♦ , ,,♦ ; 1 o-s ------,,♦ ; f ,,♦ ; , D esig·na t ed outage t • • • • I • • • • • • • • • • ♦ ; ,,, • I • • • • • • • • • • ,,,, , ,, ,,,, ,, ...... --e-Simulated 1- r/il ( == 8bytes) e ...... -Analytical 1-77 4z( e == 8bytes) z --e-Simulated 1-77 4(� == 32bytes) ...... •·· ...... ,. z I I ...... ◄ ...... ___--- Analytical 1-77 ( == 32bytes) ...... •·· ...... ·I· ... 4 e ...... •·· ...... ·I· ...... �Simulated 1-77 z(e == 64bytes) 4 ...... •·· ...... ·I· ...... ----Analytical - z( 64bytes) io-7L------'----� -· ______l_ _TJ_4 _ �______10- 8�--�--�---�--�--�--� 0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3 3.5 4 µ/A2 K

(a) (b)

ul ul ul Fig. 5. Numerical results of the uplink outage 1 − ηK for the scenario of collaborative APs: (a) 1 − ηK versus µ/λ2 for K = 4, (b) 1 − ηK versus K 2 for µ/λ2 = 0.5 (users/small cell AP) and λ2 = 250 (APs/km ).

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7 10- ...... • • • I • • • • • • • • • • • • • • • --e-Simulated 1 - 11tl (e == 8bytes) --e-Simulated 1 - 17f (e == 8bytes) z ...... -Analytical 1 - 11i ( e == 8bytes) -Analytical 1 - 17f ( == 8bytes) z e --&-Simulated 1 - 11i ( e == 32bytes) 8 --e-Simulated 1 - 17f (e == 32bytes) ···· ···· ···· ···· ···· ···· ···· ···· ··· z 10- - - -Analytical 1 - 11i ( e == 32bytes) ---Analytical 1 - 17f ( e == 32bytes) ...... A-Simulated 1 - 11tl (e == 64bytes) ....A-Simulated 1 - 17f (e == 64bytes) dl -----Analytical 1 - rJ4 (e 64bytes) L_ _J__ == _ -----Analytical 1 - 17f ( 64 bytes) 9 ______l______� == 10- 10 9 e 0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3 3.5 4 µ/>..2 K

(a) (b)

dl dl dl Fig. 6. Numerical results of the downlink outage 1 − ηK for the scenario of non-collaborative APs: (a) 1 − ηK versus µ/λ2 for K = 4, (b) 1 − ηK versus 2 K for µ/λ2 = 0.5 (users/small cell AP) and λ2 = 250 (APs/km ). open-loop communication and PMCA and we should use it or not spatially correlated fading notably impacts the analyses to properly determine how densely the macro and small cell of the communication reliability in Sections III-A and III-B APs should be deployed in the HetNet. Moreover, another that are conducted by assuming spatially independent fading important implication that can be learned from Figs. 4(a)- between channels. We consider the distance-based correlated 7(a) is that all the uplink and downlink outage probabilities fading model. Namely, in the expression of the downlink converge to constants as µ/λ ≥ 1, which represents a much SIR in (22), the fading channel gain H is found as H = 2 α k k H1 dk practical scenario of the AP deployment that the density of α + α H where dk is the distance between AP Vk 1+dk 1+dk users is smaller than that of small cell APs. Hence, the target and AP V1 and H ∼ exp(1) is independent of all Hk’s 10 communication reliability in the uplink and downlink can and Hj’s . The simulation results in Fig. 8 (a) and (b) are certainly be achieved in any practical situations of the AP corresponding to those in Fig. 6(a) and those in Fig. 7(a), deployment as long as the number of the collaborating APs 10The channel gain H in interference Idl is also determined based on in a virtual cell and the size of short packet transmission are j K the same model as Hk. For the uplink case, we can also adopt a similar properly chosen. Finally, we would like to illustrate whether distance-based model to characterize the spatially correlated fading between uplink channels. 12

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(a) (b)

dl dl dl Fig. 7. Numerical results of the downlink outage 1 − ηK for the scenario of collaborative APs: (a) 1 − ηK versus µ/λ2 for K = 4, (b) 1 − ηK versus K 2 for µ/λ2 = 0.5 (users/small cell AP) and λ2 = 250 (APs/km ).

dl dl Fig. 8. Numerical results of the downlink outage 1 − ηK when all the APs have spatially correlated fading channels: (a) 1 − ηK versus µ/λ2 for K = 4 dl and non-collaborative APs, (b) 1 − ηK versus µ/λ2 for K = 4 and collaborative APs. respectively. As can be observed in Fig. 8, the outcomes with the anchor node and its associated APs and the communication correlated fading are slightly worse than the analytical results delay between the APs and the user associated with them11. with independent fading so that our analyses are still very The major latency difference between uplink and downlink accurate even though they do not consider spatially fading is that uplink RRU collisions incur additional channel access correlation between channels. This is because the spatially delay. In the following, we will first develop a modeling and fading correlation between the channels of APs are usually analysis approach to the uplink communication delay for the fairly weak because most of APs are deployed far enough PMCA scheme, and then we apply a similar approach to such that their channels are not very likely to fad at the same characterize the downlink communication delay. time.

IV. MODELINGAND ANALYSIS OF COMMUNICATION LATENCY 11Note that our focus in this section is to study the communication delays induced by the PMCA scheme and thus some delays not directly related to The communication latency between an anchor node and a the PMCA scheme, such as signal processing delays on the transmitter and user is mostly contributed by the transmission delay between receiver sides are ignored. 13

ul A. Uplink Communication Delay binomial RV with parameters K and ηK . We thus have the mean of Dul found as follows: The uplink communication delay Dul mainly consists of the ba ul ul Z ∞ channel access delay Dac, uplink transmission delay Dtr , and ul  ul  Dul E[Dba] = P Dba ≥ x dx uplink backhaul delay ba. It can be expressed as 0 ∞ Z K ul ul ul ul = 1 − ηul + ηule−βx
dx. (33) D = Dac + Dtr + Dba. (30) K K 0 ul ul Since there are K APs in a virtual cell and a user has If ηK ≈ 1, we can get E[Dba] ≈ 1/βK. For the case of  ul  to randomly select an RRU for each of the K APs, RRU coordinated APs, P Dba ≥ x = exp(−xβ) because only one collisions could happen in the cells of the K APs. When a message is sent from the virtual cell to the anchor node. The ul ul ul user starts to (randomly) select RRUs in its virtual cell, the mean of Dba is E[Dba] = 1/β and the unit of E[Dba] can channel access delay of the user can be defined as the lapse be set as times/RRU. The mean of the uplink communication of time needed by the user to successfully access at least one delay is readily approximated as ul non-collision RRU in its virtual cell. Recall that ρK in (14) is ( R ∞ ul ul −βx
K the uplink non-collision probability of the user. The mean of 1 − η + η e dx, Non-collab. Dul ≈ 0 K K , Dul is equal to 1/ρul, i.e., [Dul] = 1/ρul, which represents E 1 ac K E ac K β , Collab. the average time for a user to successfully access an RRU 1 1 and then send a message. The uplink transmission delay Dul + + , (34) tr ρul ηul is defined as the time duration between two messages that K K are successfully sent to at least one of the K APs after the which can be employed to evaluate the uplink latency perfor- user successfully accesses at least one non-collision RRU in mance. From the above result, we can see that the mean uplink ul its virtual cell. We assume Dtr is ergodic so that the mean communication latency is inversely proportional to the uplink ul  ul 1 R T ul of Dtr can be found by E Dtr = limT →∞ T 0 Dtr (t)dt, communication reliability, which is also shown in [30] by i.e., using the queuing theory in an autonomous vehicular network.

" T #−1 1    ul X 1 ul B. Downlink Communication Delay E Dtr , lim max{γk (t)} ≥ θ T →∞ T k t=0 In the downlink, since each AP is able to allocate its 1 1 resource to its received message, the downlink communication =  ul
 = ul . (31) 1 maxk{γ } ≥ θ η dl dl E k K delay D mostly consists of the downlink backhaul delay Dba and transmission delay Ddl, i.e., it can be simply written as  ul  ul tr Note that the units of E Dac and E Dtr can be properly dl dl dl transformed to seconds once the time duration (seconds) of D = Dba + Dtr. (35) transmitting a message is determined. The uplink backhaul delay is defined as the minimum The downlink backhaul delay is defined as the maximum time transmission time needed for the APs in a virtual cell to elapsed from the start time of sending a message from the transmit a message to their anchor node so that it can be anchor node to the end time when all K APs in the virtual mathematically expressed as cell receive the message. Suppose the message arrival process at each AP can be modeled as an independent Poisson process Dul min {Dul }, (32) and the downlink backhaul delay can be expressed as ba , nc ba,k k:Vk∈VK ( dl dl maxk∈{1,...,K}{Dba,k}, Collab. APs ul D (36) where D denotes the uplink backhaul delay from the kth ba , dl ba,k mink∈{1,...,K}{Dba,k}, Non-collab. APs, AP to its anchor node. We further assume that the arrival process of the messages from an AP to its anchor node can be dl where Dba,k ∼ exp(β) is the downlink backhaul delay from modeled by an independent Poisson process, which gives rise the anchor node to the kth AP in the virtual cell. In light of to the fact that Dul can be characterized by an exponential dl ba,k this, the distribution of Dba is found as RV with some parameter β. In light of this, the distribution of ul  h dl i Dba in the non-collaborative AP case is  1 − P maxk∈{1,...,K}{Dba,k} ≤ x , [Ddl ≥ x] = P ba h dl i " N #  P mink∈{1,...,K}{Dba,k} ≥ x , Y Dul ≤ x = 1 − Dul ≥ x ( P ba E P ba,k 1 − (1 − e−βx)K , k=1 = e−βKx = 1 − E [exp (−xβN)] ul ul −βx
K = 1 − 1 − ηK + ηK e The downlink transmission delay is the time duration in which the K APs in the virtual cell successfully transmit a message ul since all Dba,k’s are independent and N that denotes the to the user and its mean can be characterized by the downlink dl dl number of the non-collision APs in the virtual cell is a communication reliability, i.e., E[Dtr] = 1/ηK and its bound 14

ul dl ul dl Fig. 9. Numerical results of D + D for showing the latency performances of collaborative APs and non-collaborative APs: (a) P[D + D > 1 ms] ul dl ul dl versus µ/λ2 for K = 5, (b) E[D + D ] versus µ/λ2 for K = 5, (c) P[D + D > 1 ms] versus K for µ/λ2 = 0.2 (users/small cell AP) and 2 ul dl 2 λ2 = 250 (APs/km ), (d) E[D + D ] versus K for µ/λ2 = 0.2 (users/small cell AP) and λ2 = 250 (APs/km ). can be found by using Theorem 3. Accordingly, the mean of V. CONCLUSIONS the downlink communication delay is given by In this paper, we break the longstanding concept that a ( ∞ h K i tradeoff between communication reliability and latency always R 1 − 1 − e−βx
dx, Collab.APs Ddl = 0 exists the in cellular systems and shed the light on the fact E 1 βK , Non-collab. APs that extremely reliable communication is hardly benefited by 1 receiver feedback. Our main contribution is to first claim that + . (37) ηdl ultra-reliable open-loop communication is the key to ultimately K fulfilling the goal of ultra-low latency in the network and  dl Hence, E D increases as K increases in the case of then devise an analytical framework to validate the claim. The collaborative APs, but it decreases as K increases in the case PMCA scheme and corresponding open-loop communication of non-collaborative APs. protocols in a HetNet are proposed and how much the com- munication reliability and latency of a virtual cell achieved by them is analyzed. Our analytical outcomes and numerical C. Numerical Results results show that the proposed open-loop communication and In this subsection, we would like to numerically demonstrate PMAC scheme are able to achieve the target reliability and ul dl how the statistical properties of D + D vary with µ/λ2 latency of URLLC users in a HetNet provided that the APs and K. The network parameters in Table II, URLLC short are sufficiently deployed and the number of the APs in a virtual packets with 32 bytes, and β = 5 messages/ms are adopted cell is properly chosen. for simulation. Figure 9 shows the numerical results of delay outage probability P[Dul + Ddl > 1 ms] and mean delay APPENDIX E[Dul + Ddl]. As can be seen in Figs. 9(a) and 9(b), the PROOFSOF LEMMASAND THEOREMS delay outage probability and mean delay both slightly vary A. Proof of Lemma 1 µ/λ as 2 changes and obviously the latency performance in According to Theorem 1 in [25], we can obtain the follow- the case of non-collaborative APs outperforms that in the case ing result: of collaborative APs. Thus, the latency performance of the   PMCA scheme is mainly dominated by whether or not the −α −α P max wmkAm,ik ≤ x APs in a virtual cell can collaborate since more networking m,i:Am,i∈Φ delay is incurred due to AP coordination. This phenomenon  1  α = P min wmkAm,ik ≥ x can be clearly observed in the numerical results in Figs. 9(c) m,i:Am,i∈Φ and 9(d). As shown in Figs. 9(c) and 9(d), the delay outage 2 ! h i 2 2 X α probability and the mean delay in the case of collaborative = P kAe1k ≥ x = exp −πx wmλm , APs largely degrade as K increases if compared with those in m=1 the case of non-collaborative APs. Therefore, we can draw 1 kA k min w α kA k kA k2 ∼ a conclusion that there exists a trade-off problem between where e1 , m,i:Am,i∈Φ m m,i and e1 P2 2/α the communication reliability and latency, which needs to be exp(π m=1 wm λm) is an exponential random variable P2 2/α 2 aware when the PMCA scheme is employed in practice. with parameter π m=1 wm λm. Let Φe , {Aek ∈ R : 15

P2 2/α d k ∈ N+} be a homogeneous PPP of density m=1 wm λm where = denotes the equivalence in distribution, Φe , {Vej ∈ 2 and Aek is the kth nearest point in Φe to the origin. Also, R : Vej = WjVj,Vj ∈ Φ, j ∈ N+} is a homogeneous PPP of 2 1/α 2 2 2 we define Φem , {Aem,k ∈ R : Aem,k = wm Aek, Aek ∈ density λe, and kVej+ik = kVejk +kVeik for all i, j ∈ N+ and Φe, k ∈ N+} and it is a homogeneous PPP of density λem , i 6= j based on the proof of Proposition 1 in [32]. Therefore, P2 2/α i=1(wi/wm) λi based on the result of Theorem 1 in [23]. the Laplace transform of S−K can be calculated as shown in This means that Φem can be viewed as a sole homogeneous PPP the following:   α  equivalent to the superposition of all independent homoge- !− 2 S2 −s kV k2 neous PPPs in the network (i.e., Φi) when all tier-i APs X ej i=1 LS (s) = exp Hj 1 + 2/α −K E   α 2  in Φi are scaled by (wm/wi) . Thus, (3) can be equivalently kVeK k kVeK k d d j:Vej ∈Φe α expressed as kVkk = kAem,kk, k = 1, 2,..., where = stands   Z ∞  − − α   (a) HsY 2 (1+ r ) 2 K YK for the equivalence in distribution. This result also indicates = EYK exp −πλe EH 1 − e dr that the typical user can be imaged to equivalently associate 0 " " 2 # !# Z ∞   α with the first K nearest APs in Φem. (b) 0 sH 0 = EYK exp −πλYe K P YK r ≤ dr Since the typical user can associate with its first K weighed 1 Z nearest APs in the network, the distance between the typical ( " !#) 2 d (c) s 2 user and its Kth weighed nearest AP is kVK k and kVK k = = EYK exp −πλYe K ` α , , 2 2 2 α kAem,K k where kAem,K k is the sum of K i.i.d. exponential YK random variables (RVs) which have the same distribution as (a) 2 2 = kAem,1k and thus kVK k ∼ Gamma(K, πλem) is a Gamma where follows from the probability generating functional (PGFL) of the homogeneous PPP Φ [33] [34] and Y kV k2, RV with shape K and rate πλem. Let Cm,i denote the cell e k , ek (b) is obtained by using Z ∼ exp(1), and (c) is obtained by area of AP Aem,i in which all users associate with Aem,i when the PMCA scheme is adopted and each user associates with using the derivation technique in the proof of Proposition 2 in [23]. Thus, LS (s) is equal to the result in (8) because its first K nearest APs in Φem. From [23] and [31], we learn −K that the Lebesgue measure of Cm,i, denoted by ν(Cm,i), for Yk ∼ Gamma(k, πλe). K = 1 can be accurately described by a Gamma RV with the (ii) Next, we find the Laplace transform of SK . According to the definition of V in (3), we know that S lim S following pdf for all i ∈ N+: k ∞ , K→∞ K in (7) can be equivalently expressed as (ζλ x)ζ em −ζxλem fν(Cm)(x) ≈ e , for K = 1, d X −α xΓ(ζ) S∞ = Hm,iwmkAm,ik , 7 m,i:Am,i∈Φ where ζ = 2 . Since the mean of ν(Cm,i) for K = 1 is 1/λem 2 and it is also equal to [πkV1k ], the mean of ν(Cm,i) for where all Hm,i’s are i.i.d. RVs with the same distribution E 2 2 as H and Φ S Φ . Thus, L (s) can be found as K > 1 is equal to E[πkVK k ] = K/λem. Accordingly, for k , m=1 m S∞ follows: K > 1, fν(C ) must be equal to m    ζm (ζmxλem/K) X −α −ζmxλem/K L (s) = exp −s H w kA k fν(Cm)(x) ≈ e , for K > 1. S∞ E   m,i m m,i  xΓ(ζm) m,i:Am,i∈Φ Note that All ν(Cm,i)’s have the same distribution and they    (a) may not be independent. Let Φem(Cm,i) denote the number of X −α = E exp −s HkkVekk  users associating with Aem,i so that pm,n for K > 1 can be k:Vek∈Φe expressed as    (b) 2 h 2 i 2 = exp −πλse α E H α Γ 1 − pm,n , P[Nm = n] = P[Φem(Cm,i) = n] α " n # " 2 # (λemν(Cm)µ) πλse α = E exp(−λemν(Cm)µ) , = exp − , n! sinc(2/α) which can be completely carried out by using the above where (a) follows from Theorem 1 [25] and (b) follows from expression of fν(Cm)(x) for K > 1. Thus, the result in (5) the PGFL of the homogeneous PPP Φe. Next, LS−K (s) can be is obtained. derived as shown in the following: B. Proof of Theorem 1    X HkWk L (s) = exp −s (i) According to the definition of S−K , S−K can be S−K E   α  kVkk alternatively written as k:Vk∈V∞\VK ∞    X −α X Hk S−K = HiWikVik = exp −s , E   α  i=K+1 kVekk k:Vek∈Ve∞\VeK d X 2 2 − α = Hj(kVeK k + kVejk ) 2 , where Ve∞ , {Vek : k ∈ N+} is a homogeneous PPP of density j:V ∈Φ ej e λe, Vek is the kth nearest point in Ve∞ to the typical user and 16

2 VeK , {Ve1,..., VeK }. Since kVekk ∼ Gamma(k, πλe) is the inequality, the upper bound on P[SK ≥ y] can be thereupon sum of k i.i.d. exponential RVs with parameter πλe, we can found as follows: get K Y [S ≥ y] ≤ 1 − H W kV k−α ≤ z y    P K P k k k k k=1 X Hk exp −s  K E α h i kVekk Y α k:Vek∈Ve∞\VeK = 1 − P Hk ≤ zkykVekk    k=1 s X Hk K α = exp − α Y h  i E   α 2 2  2 kVeK k (1 + kVekk /kVeK k ) 2 = 1 − E 1 − exp −yzkYk k:Vek∈Ve∞   k=1 ∞ α ! K Z − 2   Y sx Hk (d) Y = exp − f 2 (x)dx α E  2 α  kVeK k = 1 − 1 − L (yzk) . kVekk Y 2 0 (1 + ) 2 k k:Vek∈Ve∞ x k=1  α  ( − 2 ) 2 [Yk] (c) −πλYe K ` sYK , α E 2k Since Yk+1 > Yk , we select zk = PK = K(K+1) to get =EYK e i=1 E[Yi] a tighter lower bound. Then substituting the above zk into the above result in step (d) yields the upper bound on P[SK ≥ y] where (c) is obtained by first finding the PGFL of Ve∞ and we in (11). then follow the derivation steps in the proof of Proposition 4 in [32] to derive function `(·, ·). In addition, we can know the C. Proof of Lemma 2 following: Let Mk denote the number of the users associating with h i −s(SK +S−K ) the kth AP in the virtual cell. The probability of no collisions LS (s) = e ∞ E happening in the cell of the kth AP is δ(1 − δ)Mk−1 if the    probability of a user selecting any one of the RRUs for each X Hk = exp (−sSK ) · exp −s AP is δ. Suppose the radio resource (available bandwidth) of E   α  kVekk k:Vek∈Ve∞\VeK each AP can be divided into R radio resource units so that we 1  have δ = R . Thus, the probability that an AP in the virtual ul = 2 exp (−sSK ) EkVeK k cell does not have collisions, denoted by ρ , can be written as  P −α   −s HkkVekk × e k:Vek∈Ve∞\VeK V E eK R R 1 ul X  Mk−1 X  Mk−1     ρ = δ E (1 − δ) = E (1 − δ) − α 2 R 2 r=1 r=1 = 2 exp −sSK − πλYK ` sY , EYK e K α  Mk−1 = E (1 − δ) , " 2 # α πλse 2 = exp − , Mk−1 P where E[(1 − δ) ] = m=1 P[Vk ∈ Φm]E[(1 − sinc(2/α) M −1 δ) k |Vk ∈ Φm] and we thus have which yields 2 ul X  Nm−1 ρ = ϑmE (1 − δ) " 2 # m=1 πλse α 2 ∞ LSK (s) = exp − sinc(2/α) X X n−1 = ϑm pm,n(1 − δ) ,    α  m=1 n=1 − 2 2 × exp πλYe K ` sY , E K α where ϑm = P[Vk ∈ Φm] and Nm is the number of the users associating with a tier-m AP. Moreover, we know that , and it can be expressed as (10) due to YK ∼ Gamma(K, πλe). the probability that the nearest AP to the user located at the [S ≥ y] y ≥ 0 −α −α (iii) For the upper bound on P K for , we can origin is from the mth tier is P[kAm,∗k ≥ kAi,∗k ] = 2 2 find it by using the following inequality: P[kAm,∗k ≤ kAi,∗k ] for m 6= i in which Ai,∗ is the nearest −1 2 point in Φi to the user. Since c kAm,∗k ∼ exp(cπλm) for " K # " K #  2 2 X X any c > 0, we thus have ϑm = P kAm,∗k ≤ kAk,∗k = P Zk ≥ y = 1 − P Zk ≤ y λm Pm . Substituting the above result of ϑm into the above k=1 k=1 i=1 λi ul K expression of ρ yields the result in (13). In addition, the Y uplink non-collision reliability of the user is the probability ≤ 1 − P [Zk ≤ zky] , k=1 that there is at least one non-collision AP in the virtual and ul ul K it thus can be expressed as ρK = 1 − (1 − ρ ) , which is ul where all Zk’s are non-negative RVs, zk ∈ [0, 1] for all equal to the result in (14) by substituting the result of ρ in PK ul k ∈ {1, 2,...,K} and k=1 zk = 1. By using the above (13) into ρK . 17

ul D. Proof of Theorem 2 The upper bound on ηK in (17) can be thereupon derived as follows: First consider the scenario in which all K APs in the virtual " # P h q kV k−α1(V ∈ Vnc) cell do not collaborate and the transmit power q of users is ul k:Vk∈VK k k k k K ηK =P ≥ θ K q = q = q/K P h q kV − U k−α equally allocated to the APs, i.e., i k for all j:Uj ∈Ua\VK j j k j i ∈ N+ and k ∈ {1,...,K}. If all the non-collision uplink   SIRs in (16) are independent, we have (e) ul X hk X hj =ρ P  α ≥ θ α  kVkk kUjk   k:Vk∈VK j:Uj ∈Ua ul ul1 nc ηK =1 − P max {γk (Vk ∈ VK )} ≤ θ (f)  K    k∈{1,...,K} ul Y πµaYk ≤ ρ 1 − 1 − EYk exp − K sinc(2/α) Y  ul  1 nc k=1 ≈1 − P γk ≤ θ P[ (Vk ∈ VK ) = 1] 2  2kθ  α  k=1 × , 1 nc
K(K + 1) + 1 − P[ (Vk ∈ VK ) = 1] K 1 nc ul (a) Y where (e) follows from P[ (Vk ∈ VK )] = ρ and qj = q/K = 1 − 1 − ρul γul ≥ θ , P k for all j, and (f) follows from the result in (11) for wm = 1. k=1 −k Thence, using EYk [exp(−sYk)] = (1 + s/πλe) for s > 0 ul 1 nc and substituting µa = δ(1 − p0)λe into the above result of ηK where (a) follows from the result of P[ (Vk ∈ VK ) = 1] = P2 P∞ n−1 ul ul yield the result in (20). m=1 ϑm n=1 pm,n(1 − δ) = ρ . Whereas P[γk ≥ θ] can be derived as shown in the following: E. Proof of Theorem 3 " −α # ul hkkVkk (i) Since the intra-virtual-cell interference exits in the virtual P[γk ≥ θ] = P ≥ θ dl P h kV − U k−α cell, we know that γk in (22) can be equivalently expressed j:Uj ∈Ua j k j −α −α as γdl =d HkkVekk ≤ HkkVekk where    k P H kV k−α+Idl Idl i:Ve ∈Ve \Ve i ei K ek (b) hj i K k α X dl Oj Hj = exp −θkVkk P E   α  Iek , j:V ∈Φ\V α , Φe, Vek, and Vek are already defined kUjk ej e ek kVej k j:U ∈U j a in Appendix B and Oj is the Bernoulli random variable " 2 2 !# (c) πθ α kVkk µa associated with Vej and it is unity if Vej is not void and zero = 2 exp − EkVkk sinc(2/α) otherwise. Note that the inequality comes from the fact that dl P −α dl Iek is smaller than i:V ∈V \V HikVeik + IK because it 2 !−k ei eK ek (d) θ α µa does not contain the interference from the first k APs. Thus, = 1 + , n K o ηdl = 1 − Q γdl ≤ θ|V  is upper bounded by sinc(2/α)λe K EVK k=1 P k K

( K " −α #) where (b) follows from hk ∼ exp(1) and the Slivnyak theorem dl Y HkkVekk ηK ≤ 1 − E P ≤ θ Vek saying that the statistical property of a homogeneous PPP Vek dl k=1 Iek evaluated at Vk is the same as that evaluated at the origin K   α P Oj Hj  (a) Y −θkVekk j:V ∈Φ α (or any point in the network) [26], [34], (c) is obtained by = 1 − 1 − E e ej e kVej k , first applying the PGFL of a homogeneous PPP to Ua that is k=1 a homogeneous PPP of density µ = δ(1 − p ) P2 λ = a 0 m=1 m where (a) is obtained by the assumption that all γdl’s are δ(1 − p )λ, and (d) is due to kV k2 ∼ Gamma(k, πλ). Then k 0 e k e independent and H ∼ exp(1). According to the result in substituting the result of [γul ≥ θ] into the result of ηul k P k K (12) and the density of the APs in Φ that uses the same RRU found in (a) leads to the result in (18). Next, we consider the e is δλ, the expectation in step (a) can be simplified as follows: case in which a user can associate with all APs in the network, e   α  !− 2 i.e., K goes to infinity in this case, and we would like to find  2  ul ul X kVejk ρ lim ρ . We use the above results in (a) and (c) exp −θ OjHj ∞ , K→∞ K E  2  ul  kVekk  to approximately express ρ∞ as j:Vej ∈Φe    −k ( ) 2 h 2 i = 1 + ` θ, δ O α . ul Y ul  ul 
E ρ∞ ≈ 1 − E 1 − ρ P γk ≥ θ|Vk α Vk∈Φ dl Thus, we have the upper bound ηK ≤ 1 − ( ∞ 2 )  −k Z πθ α rµa K  h 2 i ul − sinc(2/α) Q 2
α = 1 − exp −πρ λe e dr , k=1 1 − 1 + δ` θ, α E O , which yields 0 the result in (25) because E[O2/α] = P[O = 1] = P2 which is equal to the result in (19) by carrying out the integral m=1 ϑm(1 − p0,m). in the last equality for µa = δλe. (ii) Now consider the case of K → ∞. In this case, all APs ul in the network are associated with the typical user and γk Next we would like to find the bounds on ηK when all the −α dl HkQkkVkk d in (22) can be rewritten as γ = P −α = non-collision APs in the virtual cell are able to collaborate. k HiQikVik i:Vi∈V∞\Vk 18

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