SUPPORTING ultra-reliable and low-latency communications

Transcription

1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 1, JANUARY Cross-Layer Optimization for Ultra-Reliable and Low-Lateny Radio Aess Networs Changyang She, Member, IEEE, Chenyang Yang, Senior Member, IEEE, and Tony Q. S. Que, Senior Member, IEEE Abstrat In this paper, we propose a framewor for rosslayer optimization to ensure ultra-high reliability and ultra-low lateny in radio aess networs, where both transmission delay and queueing delay are onsidered. With short transmission time, the blolength of hannel odes is finite, and the Shannon apaity annot be used to haraterize the maximal ahievable rate with given transmission error probability. With randomly arrived paets, some paets may violate the queueing delay. Moreover, sine the queueing delay is shorter than the hannel oherene time in typial senarios, the required transmit power to guarantee the queueing delay and transmission error probability will beome unbounded even with spatial diversity. To ensure the required quality-of-servie (QoS) with finite transmit power, a proative paet dropping mehanism is introdued. Then, the overall paet loss probability inludes transmission error probability, queueing delay violation probability, andpaet dropping probability. We optimize the paet dropping poliy, power alloation poliy, and bandwidth alloation poliy to minimize the transmit power under the QoS onstraint. The optimal solution is obtained, whih depends on both hannel and queue state information. Simulation and numerial results validate our analysis, and show that setting the three paet loss probabilities as equal auses marginal power loss. Index Terms Ultra-low lateny, ultra-high reliability, rosslayer optimization, radio aess networs. I. INTRODUCTION SUPPORTING ultra-reliable and low-lateny ommuniations (URLLC) has beome one of the major goals in the fifth generation (5G) ellular networs 2]. Ensuring suh a Manusript reeived June 25, 2016; revised January 6, 2017, April 12, 2017, and July 21, 2017; aepted Otober 2, Date of publiation Otober 17, 2017; date of urrent version January 8, The wor of C. She was supported in part by the National Natural Siene Foundation of China under Grant , in part by MOE ARF Tier 2 under Grant MOE2015-T , and in part by the SUTD-ZJU Researh Collaboration under Grant SUTD-ZJU/RES/01/2016. The wor of C. Yang was supported by the National Natural Siene Foundation of China under Grant The wor of T. Q. S. Que was supported in part by the MOE ARF Tier 2 under Grant MOE2015-T and in part by the SUTD-ZJU Researh Collaboration under Grant SUTD-ZJU/RES/01/2016. This paper was presented in part at the 2016 IEEE Global Communiations Conferene 1]. The assoiate editor oordinating the review of this paper and approving it for publiation was Q. Li. (Corresponding author: Chenyang Yang.) C. She was with the Shool of Eletronis and Information Engineering, Beihang University, Beijing , China. He is now with the Information Systems Tehnology and Design Pillar, Singapore University of Tehnology and Design, Singapore ( shehangyang@gmail.om). C. Yang is with the Shool of Eletronis and Information Engineering, Beihang University, Beijing , China ( yyang@buaa.edu.n). T. Q. S. Que is with the Information Systems Tehnology and Design Pillar, Singapore University of Tehnology and Design, Singapore ( tonyque@sutd.edu.sg). Color versions of one or more of the figures in this paper are available online at Digital Objet Identifier /TWC stringent quality-of-servie (QoS) enables various appliations suh as ontrol of exoseletons for patients, remote driving, free-viewpoint video, and synhronization of suppliers in a smart grid in tatile internet 3], and autonomous vehiles and fatory automation in ultra-reliable mahine-typeommuniations (MTC) 4], despite that not all appliations of tatile internet and MTC require both ultra-high reliability and ultra-low lateny. Sine tatile internet and MTC are primarily applied for mission ritial appliations, the message suh as touh and ontrol information is usually onveyed in short paets, and the reliability is refleted by paet loss probability 2]. The traffi supported by URLLC distinguishes from traditional real-time servie in both QoS requirement and paet size. For human-oriented appliations, the requirements on delay and reliability are medium. For example, in the long term evolution (LTE) systems, the maximal queueing delay and its violation probability for VoIP are respetively 50 ms and in radio aess networs, and the minimal paet size is 1500 bytes 5]. For ontrol-oriented appliations suh as vehile ollision avoidane or fatory automation, the endto-end (E2E) or round-trip delay is around 1 ms, the overall paet loss probability is ], 6], and the paet size is 20 bytes or even smaller 2]. LTE systems were designed for human-oriented appliations, where the E2E delay inludes uplin (UL) and downlin (DL) transmission delay, oding and proessing delay, queueing delay, and routing delay in bahaul and ore networs 7]. The radio resoures are alloated in every transmit time interval (TTI), whih is set to be 1 ms 8]. This means that the paets need to wait in the buffer of base station (BS) more than 1 ms before transmission. Therefore, even if other delay omponents in bahaul and ore networs are redued with new networ arhitetures 9], LTE systems annot ensure the E2E or round-trip lateny of 1 ms. A. Related Wor While reduing lateny in wireless networs is hallenging, further ensuring high reliability maes the problem more intriate. To redue the delay aused by transmission and signalling 10], a short frame struture was introdued in 11], and the TTI was set idential to the frame duration. To ensure high reliability of transmission with short frame, proper hannel oding with finite blolength is important. Fortunately, the results in 12] indiate that it is possible to guarantee very low transmission error probability with short blolength hannel odes, at the expense of ahievable rate redution IEEE. Personal use is permitted, but republiation/redistribution requires IEEE permission. See for more information.

2 128 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 1, JANUARY 2018 By using pratial oding shemes lie Polar odes 13], the delays aused by transmission, signal proessing and oding an be redued. Exploiting diversity among multiple lins has long been used as an effetive way to improve the suessful transmission probability in wireless ommuniations. To support the high reliability over fading hannels, various diversity tehniques have been investigated, say frequeny diversity and marosopi diversity in single antenna systems 14], 15] and spatial diversity in multi-antenna systems 16]. Simulation results using pratial modulation and oding shemes in 17] and 18] show that the required transmit power to ensure given transmission delay and reliability an be rapidly redued when the number of antennas at a BS inreases. In all these wors, only transmission delay and transmission error probability are taen into aount in the QoS requirement. In pratie, sine the paets arrive at the buffer of the BS randomly, there is a queue at the BS. To ontrol the delay and paet loss aused by both queueing and transmission, ross-layer optimization should be onsidered 1]. Similar to the real-time servie suh as VoIP, the required queueing performane of URLLC an be modeled as statistial queueing requirement, haraterized by the maximal queueing delay and a small delay violation probability. By using effetive bandwidth 19] and effetive apaity 20] to analyze performane of tatile internet under the statistial queueing requirement, the tradeoff among queueing delay, queueing delay violation probability and throughput was studied in 21], and UL and DL resoure alloation was jointly optimized to ahieve the E2E delay requirement in 22]. In both wors, the Shannon apaity is applied to derive the effetive apaity. However, with short transmission delay requirement, hannel oding is performed with a finite blo of symbols, with whih the Shannon apaity is not ahievable. In fat, the results obtained by using networ alulus in 23] show that if Shannon apaity is used to approximate the ahievable rate of short blolength odes for designing resoure alloation, the queueing delay and delay violation probability annot be guaranteed. Based on the ahievable rate of a single antenna system with finite blolength hannel odes derived in 12], queueing delay/length was analyzed in 24] and 25]. For appliations with medium delay and reliability requirements, the throughput subjet to statistial queueing onstraints was studied in 24], where the effetive apaity was derived by using the ahievable rate with finite blolength hannel odes, and an automati repeat-request (ARQ) mehanism was employed to improve reliability. An energy-effiient paet sheduling poliy was optimized in 25] to ensure a strit deadline by assuming paet arrival time and instantaneous hannel gains nown aprior, while the deadline violation probability under the transmit power onstraint was not studied. B. Major Challenges and Our Contributions Supporting URLLC leads to the following hallenges in radio resoure alloation. First, the required queueing delay and transmission delay are shorter than hannel oherene time in typial senarios of URLLC. 1 This results in the following problems. (1) ARQ mehanism an no longer be used to improve reliability. This is beause retransmitting a paet in subsequent frames not only introdues extra transmission delay but also an hardly improve the suessful transmission probability when the hannels in multiple frames stay in deep fading. (2) Time diversity annot be exploited to enhane reliability, and frequeny diversity may not be salable to the large number of nodes. Moreover, whether spatial diversity an guarantee the reliability is unnown. (3) The studies in 26] show that when the average delay approahes the hannel oherene time, the average transmit power ould beome infinity, beause transmitting paets during deep fading leads to unbounded transmit power. Hene, how to ensure both the ultra-low delay and the ultra-high reliability with finite transmit power is unlear. Seond, the blolength of hannel odes is finite. The maximal ahievable rate in finite blolength regime is neither onvex nor onave in radio resoures suh as transmit power and bandwidth 12], 27]. As a result, finding optimal resoure alloation poliy for URLLC is muh more hallenging than that for traditional ommuniations, where Shannon apaity is a good approximation of ahievable rate and is jointly onave in transmit power and bandwidth. Third, effetive bandwidth is a powerful tool for designing resoure alloation to satisfy the statistial queueing requirement of real-time servie 19]. Sine the distribution of queueing delay is obtained based on large deviation priniple, the effetive bandwidth an be used when the delay bound is large and the delay violation probability is small 28]. Therefore, using effetive bandwidth for URLLC seems problemati. In this paper, we propose a ross-layer optimization framewor for URLLC. While tehnial hallenges in ahieving ultra-low E2E/round-trip delay exist at various levels, we only onsider transmission delay and queueing delay in radio aess networs, and fous on DL transmission. The major ontributions of this wor are summarized as follows: We show that only exploiting spatial diversity annot ensure the ultra-low lateny and ultra-high reliability with finite transmit power over fading hannels. To ensure the QoS with finite transmit power, we propose a proative paet dropping mehanism. We establish a framewor for ross-layer optimization to guarantee the low delay and high reliability, whih inludes a resoure alloation poliy and the proative paet dropping poliy depending on both hannel and queue state information. By assuming frequeny-flat fading hannel model, we first optimize the power alloation and paet dropping poliies in a single-user senario, and then extend to the multi-user senario by further optimizing bandwidth alloation among users. Moreover, how to apply the framewor to frequeny-seletive hannel is also disussed. We validate that even when the delay bound is extremely short, the upper bound of the omplementary 1 In this senario, effetive apaity an no longer be applied.

3 SHE et al.: CROSS-LAYER OPTIMIZATION FOR ULTRA-RELIABLE AND LOW-LATENCY RADIO ACCESS NETWORKS 129 Fig. 2. Relation of the required delay bound, hannel oherene time, frame duration and TTI. The UL transmission delay is equal to T f,andthesameto the DL transmission delay. Fig. 1. System model. umulative distributed funtion (CCDF) of queueing delay derived from effetive bandwidth still wors for Poisson proess and Interrupted Poisson Proess (IPP), whihismoreburstythanpoissonproess,andswithed Poisson Proess (SPP), whih is an autoorrelated two-phase Marov Modulated Poisson Proess 29]. We onsider the transmission error probability with finite blolength hannel oding, the queueing delay violation probability,andtheproative paet dropping probability in the overall reliability. By simulation and numerial results, we show that setting paet loss probabilities equal is a near optimal solution in terms of minimizing transmit power. The rest of this paper is organized as follows. Setion II desribes system model and QoS requirement. Setion III shows how to represent queueing delay onstraint with effetive bandwidth. Setion IV introdues the paet dropping poliy, and the framewor for ross-layer optimization. Setion V illustrates how to apply the framewor to frequeny-seletive hannel. Simulation and numerial results are provided in Setion VI to validate our analysis and to show the optimal solution. Setion VII onludes the paper. II. SYSTEM MODEL AND QOS REQUIREMENT Consider a frequeny division duplex ellular system, 2 where eah BS with N t antennas serves K + M single-antenna nodes. The nodes are divided into two types. The first type of nodes are K users, whih need to upload paets and download paets from the BS. The seond type of nodes are M sensors, whih only upload paets. In the ases without the need to distinguish between users and sensors, we refer both as nodes. Time is disretized into frames. Eah frame onsists of a data transmission phase and a phase to transmit ontrol signaling (e.g., pilot for hannel estimation). We onsider frequeny reuse among adjaent ells and orthogonal frequeny division multiple aess (OFDMA) to avoid interferene. All nodes in a ell upload their messages with short paets to the BS. The BS proesses the reeived messages from the nodes, and then transmits the relevant messages to the target users. For example, nodes 2, K +1, and K +2 lie in the area of interest with respet to (w.r.t.) user 1, as shown in Fig. 1, and 2 Our studies an be easily extended into time division duplex system, whih is with different short frame struture 11]. the BS only transmits the messages from nodes 2, K + 1, and K +2 to user 1. Suh system model an be applied in analyzing E2E delay in loal ommuniation senarios, where all nodes are assoiated to adjaent BSs that are onneted with eah other by fiber bahaul. The delay in fiber bahaul is muh less than 1 ms 30], and hene the delay in radio aess networ dominates the E2E delay. For other ommuniation senarios (e.g., remote ontrol), the delay omponents in bahaul and ore networs should be taen into aount, yet our model an still be used to analyze the delay in radio aess 2]. Moreover, the model aptures one of the ey features of ultra-reliable MTC 4]: a paet generated by one node may be required by multiple users, and one user may also require paets generated by multiple nodes. Hene the model is representative for URLLC, although it annot over all appliation senarios. 3 All the notations to be used throughout the paper are summarized in Table I. A. QoS Requirement The QoS requirement of eah user is haraterized by the E2E delay and overall loss probability for eah paet 2], 4]. In the onsidered radio aess networ, the E2E delay bound, denoted as D max, inludes UL and DL transmission delay and queueing delay. We only onsider one-way delay requirement. By setting D max less than half of round-trip delay, our study an be diretly extended to the appliations with requirement on round-trip delay. To ensure ultra-low transmission delay, we onsider the short frame struture proposed in 10], where the TTI is equal to the frame duration T f, eah onsisting of a duration for data transmission φ and a duration for ontrol signalling, as shown in Fig. 2. Owing to the required short delay, T f D max, and retransmission mehanism is unable to be used. Both UL transmission and DL transmission of eah short paet are finished within one frame, respetively. If a paet is not transmitted error-free in one frame, then the paet will be lost. Beause only a few symbols an be transmitted within φ, the transmission error is not zero with finite blolength hannel odes among these symbols. Sine UL transmission has been studied in 32], we fous on the DL transmission in this wor. Then, the overall reliability for eah user, denoted 3 Diret transmission between nodes (i.e., devie-to-devie (D2D) ommuniation mode) an help redue delay with only one hop transmission. However, in D2D mode, the interferene beomes more omplex than the entralized ommuniations 31]. How to use D2D mode for URLLC deserves further study but is beyond the sope of this wor.

4 130 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 1, JANUARY 2018 TABLE I SUMMARY OF NOTATIONS as ε D, is the overall paet loss probability minus the UL transmission error probability. Denote the DL transmission error probability (i.e. the blo error probability 27]) for the th user as ε. Sine the UL and DL transmissions need two frames, the queueing delay for every paet should be bounded as Dmax q D max 2T f. If the queueing delay bound is not satisfied, then a paet will beome useless and has to be dropped. Denote the reative paet dropping probability due to queueing delay violation as ε q. As detailed later, to satisfy the requirement imposed on the queueing delay for eah paet (Dmax,ε q q ) and ε to the th user, the required transmit power may beome unbounded in deep fading. To guarantee QoS with finite transmit power, we proatively drop several paets in the queue under deep fading and ontrol the overall reliability. Denote the proative paet dropping probability for the th user as ε h. Then, the overall reliability for the th user an be haraterized by the overall paet loss probability, whih is 1 (1 ε )(1 εq )(1 εh ) ε + εq + εh ε D, (1) where the approximation is aurate sine ε, εq,andεh are extremely small. B. Channel Model We onsider blo fading, where the hannel remains onstant within a oherene interval and varies independently among intervals. Denote the hannel oherene time as T. Sine the required delay bound D max is very short, it is reasonable to assume that T > D max > Dmax, q asshown in Fig In the following, we onsider suh a representative senario for typial appliations of URLLC, whih is more hallenging than the other ase with T Dmax. q SineT f should be less than D max and the hannel oding is performed within φ of eah frame, suh a hannel (i.e., T f < T )is referred to as quasi-stati fading hannel as in 27]. Denote the average hannel gain of the th user as μ, and the orresponding hannel vetor in a ertain oherene interval as h CN(0, 1) C Nt 1 with independent and identially distributed (i.i.d.) zero mean and unit variane Gaussian elements. Denote the size of eah paet as u bits. Aording to the Shannon apaity formula with infinite blolength oding, when μ and h are perfetly nown at the BS, the maximal number of paets that an be transmitted to the th user in the nth frame an be expressed as 1 + μ P (n)g N 0 BN s (n) = φbn u ln 2 ln (paets), (2) where P (n) is the transmit power alloated to the th user in the nth frame, g = h H h, N 0 is the single-sided noise spetral density, B is the separation among subarriers, N is number of subarriers alloated to the th user, and ] H denotes the onjugate transpose. When the bandwidth alloated to the th user, W = BN, is smaller than oherene bandwidth, the hannel is flat fading and the hannel gains over N subarriers are approximately idential. We first onsider flat 4 For instane, for users with veloities less than 120 m/h in a vehile ommuniation system operating in arrier frequeny of 2 GHz, the hannel oherene time is larger than 1 ms, whih exeeds the delay bound of eah paet. For other appliations lie smart fatory, the veloities of sensors are slow or even zero, and hene T 1ms. ]

5 SHE et al.: CROSS-LAYER OPTIMIZATION FOR ULTRA-RELIABLE AND LOW-LATENCY RADIO ACCESS NETWORKS 131 nearby nodes of the th user is A ={ +1,..., +m}. Then, the number of paets waited in the queue for the th user at the beginning of the (n + 1)th frame an be expressed as Q (n + 1) = max {Q (n) s (n), 0} + i A a i (n), (5) Fig. 3. Queueing model at the BS. fading hannel, whih is appliable for many senarios of tatile internet and utra-reliable MTC where the number of users is large. We then disuss how to apply the proposed framewor to frequeny-seletive hannels in Setion V. The number of symbols transmitted in one frame (also referred to as the blolength of hannel oding) for the th user, n s, is determined by the bandwidth and duration, i.e. n s = φw. To ensure the ultra-low lateny, the transmission duration φ is very short. Considering that the bandwidth for eah user is limited, n s is far from infinite, and hene s (n) is not ahievable. The maximal ahievable rate with finite blolength oding is with very ompliated expression 27]. By using the normal approximation in 27], the maximal number of paets that an be transmitted to the th user in the nth frame an be aurately approximated as s (n) φbn u ln 2 { ln 1 + μ P (n)g N 0 BN ] V } φbn fq 1 (ε ) (paets), (3) where fq 1 (x) is the inverse of Q-funtion, and V is given by 27] 1 V = 1 ] 2. (4) 1 + μ P (n)g N 0 BN (3) is obtained for interferene-free systems, whih is valid for the onsidered OFDMA (and also for time division multiple aess or spae division multiple aess with zero-foring beamforming). To onsider other multiple aess tehniques where interferene annot be ompletely avoided, the ahievable rate with finite blolength in interferene hannels should be used, whih however is not available in the literature until now. As shown in 23], if (2) is used to design resoure alloation with finite blolength oding, then the queueing delay and the queueing delay violation probability will be underestimated. As a result, the alloated resoure is insuffiient for ensuring the queueing performane. This indiates that to guarantee ultra-low lateny and ultra-high reliability, (3) should be applied. C. Queueing Model In the nth frame, the th user requests the paets uploaded from its nearby nodes. The indies of the nodes that lie in theareaofinterestw.r.t.theth user onstitute a set A with ardinality A. As illustrated in Fig. 3, the index set of the where a i (n), i A is the number of paets uploaded to the BS from the ith nearby node of the th user. We onsider the senario that the inter-arrival time between paets ould be shorter than D q max (otherwise the queueing delay is zero), whih happens when the paets for a target user are randomly uploaded from multiple nearby nodes, i.e. A > 1. At the first glane, suh a senario seems to our with a low probability. However, to ensure the ultra-high reliability of ε D = 0.001% %, the senario of nonzero queueing delay is not negligible. Denote the number of paets departed from the th queue in the nth frame as b (n). If all the paets in the queue an be ompletely transmitted in the nth frame, then b (n) = Q (n). Otherwise,b (n) = s (n). Hene, we have b (n) = min {Q (n), s (n)}. (6) Using (5) and (6), the evolution of the queue length an be desribed as follows, Q (n + 1) Q (n) = i A a i (n) b (n). (7) III. ENSURING THE QUEUEING DELAY REQUIREMENT In this setion we employ effetive bandwidth to represent the queueing delay requirement. We validate that effetive bandwidth an be applied in the short delay regime for Poisson arrival proess, and then extend the disussion to IPP and SPP. A. Representing Queueing Delay Constraint With Effetive Bandwidth For stationary paets arrival proess { a i (n), n = i A 1, 2,...}, the effetive bandwidth is defined as 19] E B (θ 1 N ) = lim ln N NT f θ E exp θ a i (n) n=1 i A (paets/s), (8) where θ is the QoS exponent for the th user. A larger value of θ indiates a smaller queueing delay bound with given queueing delay violation probability. Remar 1: When the queueing delay bound is not longer than the hannel oherene time, the servie proess is onstant within the delay bound with given resoures suh as transmit power and bandwidth, and the power alloation over fading hannel is hannel inversion in order to guarantee queueing delay 33]. This is also true when ahievable rate in (3) is applied, as explained in what follows. To satisfy the queueing delay requirement of the th user (D q max,ε q ) in fading hannels, the onstant servie rate should be no less than the effetive bandwidth of the arrival proess of the user. By setting s (n) in (3) equal to E B (θ ), P (n)g is onstant,

6 132 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 1, JANUARY 2018 i.e., the power alloation is hannel inversion, whih is not always feasible in pratial fading hannels. We will show how to handle this issue in the next setion. When the th user is served with a onstant rate equal to E B (θ ), the steady state queueing delay violation probability an be approximated as 20] Pr{D ( ) >Dmax q } η exp{ θ E B (θ )Dmax q }, (9) where η is the buffer non-empty probability and the approximation is aurate when Dmax q (i.e. queue length is large enough) 19]. Sine η 1, we have Pr{D ( ) >Dmax q } exp{ θ E B (θ )Dmax q } PUB D. (10) If the upper bound in (10) satisfies PD UB = exp{ θ E B (θ )Dmax q }=εq, (11) then the queueing delay requirement (Dmax,ε q q ) an be satisfied. In other words, if the number of paets transmitted in every frame to the th user is a onstant that satisfies s (n) = T f E B (θ )(paets), (12) then (Dmax,ε q q ) an be ensured 19]. When the th queue is served by the onstant servie proess {s (n), n = 1, 2,...} that satisfies (12), the departure proess in (6) beomes b (n) = min{q (n), T f E B (θ )} (paets). (13) If the departure proess {b (n), n = 1, 2,...} satisfies (13), then (Dmax,ε q q ) an be guaranteed. Satisfying (13) does not require onstant servie proess. For example, when Q (n) = 0, the buffer is empty, then no servie is needed. B. Validating the Upper Bound PD UB in (10) With Representative Arrival Proesses 1) Representative Arrival Proesses: The aggregation of paets that are independently generated by A nodes lie in the onerned area w.r.t the th user (i.e. a i (n) in (5)) an i A be modeled as a Poisson proess in vehile ommuniation and other MTC appliations 34], 35]. Denote the average paet rate of the th Poisson proess as λ. Sine the features of traffi, say burstiness and autoorrelation, have large impat on the delay performane of queueing systems 29], 36], and the effetive bandwidth for real-world arrival proesses is hard to obtain, we also onsider another two representative traffi models. As shown in 37], the event-driven paet arrivals in vehiular ommuniation networs an be modelled as a bursty proess, IPP. When no event happens, no sensor sends paets to the BS. When an event happens (e.g., a sudden brae) and deteted by nearby sensors, the sensors send the paets to the BS. IPP has two states. In the OFF state, no paet arrives. In the ON state, paets arrive at the buffer of the BS aording to a Poisson proess with average paet rate λ on paets/frame. The durations that the proess stays in OFF" and ON" states are exponential distributed with mean values of α 1 and β 1 frames, respetively. Both Poisson proess and IPP are renewal proesses, whih annot haraterize the autoorrelation of a traffi. In 37], SPP is used to model the aggregation of event-driven paets and periodi paets in vehile ommuniation networs. Similar to IPP, SPP has two states, where the durations that a SPP stays in the first state and the seond state are exponential distributed with mean values of αi 1 and αii 1 frames, respetively. In the two states, paets arrive at the buffer of the BS aording to Poisson proesses with average paet rates λ I and λii paets/frame, respetively. Therefore, a SPP is determined by parameters (λ I,λII,α I,α II ). The effetive bandwidths of Poisson proess, IPP and SPP are provided in Appendix VII. 2) Validating the Upper Bound: The approximation in (9) is aurate when the delay bound is suffiiently large and ε q is very small 19], 28]. However, it is unlear how large Dmax q needs to be for an aurate approximation. One possible reason is that it is very diffiult to obtain an aurate distribution of the queueing delay. In fat, what really onerned here is whether the upper bound in (10) is appliable to our problem. If PD UB is indeed an upper bound of Pr{D ( ) >Dmax}, q then a transmit poliy optimized under the onstraint in (12) or (13) an satisfy the queueing delay requirement. In what follows, we derive the queueing delay distribution for Poisson proess, whih an be used to validate the upper bound in short Dmax q regime numerially. For arrival proesses that are more bursty than Poisson proess, the upper bound in (10) is appliable 38]. When a Poisson arrival proess is served by a onstant servie proess {s (n), n = 1, 2,...}, the well-nown M/D/1 queueing model an be applied 36]. For a disrete state M/D/1 queue, the CCDF of the steady state queue length an be expressed as Pr{Q ( ) >L} =1 L π l,where l=1 π l = Pr{Q ( ) = l} is the probability that there are l paets in the queue, i.e., π 0 = 1 ξ,π 1 = (1 ξ )(e ξ 1), π l = (1 ξ ) ] l 1 elξ + e jξ ( 1) l j ( jξ ) l j (l j)! + ( jξ ) l j 1 (l j 1)!, j=1 (l 2), (14) with ξ = λ /s (n) 36]. For a Poisson arrival proess served by a onstant servie rate T 1 f s (n) = E B(θ ), Pr{D ( ) >Dmax q }=Pr{Q ( ) >E B (θ )Dmax q }. (15) Then, from (14), the CCDF of the queueing delay an be derived as L Pr{D ( ) >T f L/s (n)} =Pr{Q ( ) >L} =1 π l, (16) whih is too ompliated to obtain a losed-form onstraint on queueing delay due to expressions of π l in (14). Nonetheless, (16) an be used to validate the upper bound PD UB in (10) numerially. l=0

7 SHE et al.: CROSS-LAYER OPTIMIZATION FOR ULTRA-RELIABLE AND LOW-LATENCY RADIO ACCESS NETWORKS 133 IV. A FRAMEWORK FOR CROSS-LAYER TRANSMISSION OPTIMIZATION In this setion, we first show that the required transmit power to guarantee the queueing delay and transmission error probability requirement for some paets may beome unbounded for any given bandwidth and N t,owing to D q max < T. To guarantee the QoS in terms of D q max and ε D with finite transmit power, we then propose a proative paet dropping mehanism. Finally, we propose a framewor to optimize ross-layer transmission strategy, whih inludes resoure alloation and paet dropping poliies depending on both hannel information and queue length. A. Proative Paet Dropping and Power Alloation We onsider the ase where Q (n) T f E B(θ ), then b (n) = T f E B(θ ). If a transmit power an guarantee suh a departure rate, then for the other ase where Q (n) <T f E B(θ), b (n) <T f E B(θ ) an also be supported, i.e., (Dmax,ε q q D ) an be satisfied aording to (13). Substituting s (n) in (3) into (12), we an obtain the required SNR γ to ensure (Dmax,ε q q ) and ε for all paets to the th user using the following equation, ln (1 + γ ) T fu ln 2 φbn E B (θ V ) + φbn fq 1 ( ) ε. (17) Sine h C N t is with i.i.d. elements, the hannel gain g = h H h follows Wishart distribution 39], whose probability density funtion is f g (x) = 1 (N t 1)! x Nt 1 e x.inthe onsidered typial appliation senario with Dmax q < T, some paets to be transmitted within the delay bound may experiene deep fading with hannel gain g arbitrarily lose to zero. 5 Then, the required transmit power to ahieve γ in the nth frame, P (n) N 0 BN γ μ g, is unbounded. This means that s (n) annot exeed E B(θ ) with finite transmit power if the nth frame is in a oherene interval subjet to deep fading, even when there is spatial diversity. In other words, for the paets in suh an interval, ε q + ε will exeed ε D will happen if P (n) is finite. To satisfy the QoS requirement with a finite transmit power, we introdue a proative paet dropping mehanism. By proative", we mean that a paet will be intentionally disarded even when its queueing delay has not exeeded Dmax q in the ase ε q +ε >ε D, and then the total number of paets proatively and reatively dropped 6 is judiiously ontrolled to ensure the overall reliability for eah user. The rational behind suh a mehanism lies in the fat that we only need to ensure the overall paet loss probability ε D no matter how the paets are lost. Denote the maximal transmit power of the BS as P max. We disard some paets before transmission in deep fading hannels when the required SNR γ annot be ahieved 5 This is true also for other hannel distribution, say Naagami-m fading, whih is a general model of wireless hannels 40]. 6 By reative", we mean that a paet is lost when D q max is violated or a oding blo is not deoded suessfully. K with P (n) P max. However, we an hardly ontrol the =1 paet dropping probability of eah user from K =1 N 0 BN γ μ g P max sine the required total transmit power depends on the hannel gains of multiple users. To ontrol the paet dropping probability of eah user, we introdue the maximal transmit power that an be alloated to the th user P th. When the required transmit power is higher than P th, the BS transmits paets to the th user with power P th and drop several paets in the nth frame. Then, the total transmit power of the BS is bounded by K =1 P th. To ensure (Dmax,ε q q ) and ε,thepower alloation poliy should depend on both hannel gain and queueing length, whih is, P (n) P th, = N 0 BN γ if Q (n) T f E B(θ ), g < N 0 BN γ μ P th, μ g, if Q (n) T f E B (θ ), g > N 0 BN γ μ P th. (18) In the ase Q (n) < T f E B(θ ), P (n) should satisfy s (n) = Q (n) when s th > Q (n) or P (n) = P th when s th Q (n), wheres th is the number of paets that an be transmitted in the nth frame with P (n) = P th.fromthe approximation in (3), we obtain s th as s th φbn u ln 2 { ln 1 + μ P th g N 0 BN ] V φbn fq 1 ( ) } ε. (19) When g < N 0 BN γ in the nth frame, s th μ P th < T f E B(θ ). Sine b (n) = min{q (n), T f E B(θ )} needs to be satisfied to ensure (Dmax,ε q q ), the BS has to disard some paets waiting in the queue. Denote the number of paets dropped in the nth frame as b d (n) = max{b (n) s th, 0}. Then, the proative paet dropping poliy is b d (n)= max ( T f E B(θ ) s th, 0), if Q (n) T f E B(θ ), max ( Q (n) s th, 0), if Q (n) <T f E B(θ ). (20) This poliy is implemented as follows. If Q (n) T f E B(θ ) and g < N 0 BN γ,thenp th μ P th is used to transmit paets and b d (n) paets that annot be onveyed within the nth frame with P th are dropped, where P th and b d (n) will be optimized in the next subsetion. Sine the BS simply disards some paets from the buffer if the hannel gain is low, suh a poliy only introdues negligible proessing delay due to several operations of omparison.

8 134 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 1, JANUARY 2018 Similar to the delivery ratio in 41], we define the paet dropping probability as ε h lim N N b d (n) n=1 N n=1 i A a i (n) = Ebd (n)] E{ i A a i (n)}, (21) where the seond equality is obtained under the assumption that the queueing system is ergodi, and the average on nominator is taen over both hannel gain and queue length. Based on the analysis in Appendix VII-C, the paet dropping probability an be approximated by ε h 0 N0 BN γ μ P th ln 1 ( 1 + μ P th g N 0 BN ln (1 + γ ) ) f g (g) dg. (22) B. A Framewor for Cross-Layer Transmission Optimization With the proative paet dropping mehanism, the total transmit power is bounded by K P th. To find the minimal resoures required to ensure the QoS, we optimize the =1 ross-layer transmission strategy, whih inludes a transmit power alloation poliy P (n) and a proative paet dropping poliy b d (n) for single user senario and also inludes a bandwidth alloation poliy for multi-user senario, to minimize K P th with given total bandwidth of the system. =1 Aording to (18), P (n) depends on the values of γ and P th. Given the values of γ and ε h, the minimal value of P th an be obtained from (22) by letting the equality hold. Moreover, the required SNR γ is determined by ε and εq aording to (17). Therefore, the power alloation poliy and the minimal P th are uniquely determined by the values of ε, ε q and εh. Aording to (20), the number of paets to be dropped b d (n) depends on sth, whih an be obtained from (19) after P th and ε are obtained. This indiates that to optimize the power alloation poliy and paet dropping poliy that minimize P th, we only =1 need to ontrol ε q, ε,andεh. For easy exposition, we first onsider single user ase, and then extend to multi-user senario. 1) Single-User Senario: When K = 1, the index an be omitted for notational simpliity. We onsider the ase that Q(n) > 0. For Q(n) = 0, no power is alloated, i.e., P(n) = 0. The values of ε, ε q,andε h that minimize P th an be obtained from the following problem, min P th (23) ε q,ε,ε h ( ) N 0 BN γ ln 1 + μpth g s.t. ε h αp = th N 0 BN 1 f g (g) dg, (23a) ln (1 + γ ) 0 K ln (1 + γ ) = T fu ln 2 φbn E B V (θ) + φbn f Q 1 ( ε ), (23b) ε + ε q + ε h ε D and ε,ε q,ε h R +, (23) where onstraint (23a) and onstraint (23b) are the single-user ase of (17) and (22), respetively, E B (θ) depends on the soure as well as (Dmax,ε q q ),andr + represents the positive real number. 7 In the following, we propose a two-step method to find the optimal solution of problem (23). In the first step, ε0 h (0,ε D) is fixed. Given ε0 h, Pth in the right hand side of (23a) inreases with γ. Hene, minimizing P th is equivalent to minimizing γ. For Poisson proess, the optimal values of ε and ε q that minimize the required γ an be obtained by solving the following problem, T f u ln 2 ln (1/ε q ) min ] + ε q,ε φbn Dmax q ln 1 + T f ln(1/ε q ) D q maxλ V φbn f Q 1 ( ε ) (24) s.t. ε + ε q ε D ε0 h, (24a) where the effetive bandwidth in (A.2) is used to derive the objetive funtion. As proved in Appendix VII-C, the objetive funtion in (24) is stritly onvex in ε and ε q, and hene the problem is onvex. To ensure the stringent QoS requirement, the required SNR γ is high, in this ase V 1asshownin(4). Then, there is a unique solution of ε and ε q that minimizes γ. Denote the minimal SNR obtained from problem (24) as γ. Sine the right hand side of (23a) dereases with P th, for given ε0 h and γ,thevalueofp th an be obtained numerially via binary searhing 42] as a funtion of ε0 h, denoted as Pth (ε0 h). In the seond step, we find the optimal ε0 h (0,ε D) that minimizes P th (ε0 h ). Sine there is no losed-form expression of P th (ε0 h ), exhaustive searhing is needed to obtain the optimal ε0 h in general. However, numerial results indiate that P th (ε0 h) first dereases and then inreases with εh 0. With this property, we an find the optimal solution of ε0 h and the required transmit power to ensure ε D via the exat linear searh method 42]. As proved in Appendix VII-C, the solution obtained from the two-step method is the global optimal solution of problem (23) if the solutions of both steps are global optimal. Impat of traffi feature: To show the impat of burstiness on the ross-layer optimization, we onsider IPP with fixed average paet rate in two asymptoti ases, i.e. C 2 1and C 2,whereC 2 is the variane oeffiient that an be used to haraterize burstiness 29]. To show the impat of α α+β λon, burstiness, we eep the average paet rate of IPP, as a onstant. Then, the average paet rate an be expressed as 1+δ λon,andc2 = 1 + 2δλon 29], where δ = β/α. (1+δ) 2 α When α, C 2 1, the effetive bandwidth of the IPP an be expressed as E B ( (θ) = T λon f θ(1+δ) e θ 1 ),whih is the same as the effetive bandwidth of a Poisson proess 7 The distribution of hannel gain f g (g) depends on the number of antennas N t. Therefore, the optimal solution of problem (23a) will depend on N t. We will illustrate the impat of N t via numerial results in the next setion.

9 SHE et al.: CROSS-LAYER OPTIMIZATION FOR ULTRA-RELIABLE AND LOW-LATENCY RADIO ACCESS NETWORKS 135 with average paet rate 1+δ λon.whenα 0, C2, the effetive bandwidth of the IPP an be expressed as E B ( (θ) = T λon f θ e θ 1 ), whih is the same as the effetive bandwidth of a Poisson proess with average paet rate λ on. To show the impat of autoorrelation, we onsider a SPP with parameters (λ I,λ II,α I,α II ),whereλ I 0,λ on ], λ II = λ on, α I = α and α II = β. An upper bound of the effetive bandwidth of it an be obtained by substituting λ = λ on into (A.1). Therefore, the effetive bandwidth of SPP is less than that of a Poisson proess with average paet rate max{λ I,λ II }. Remar 2: For IPP, when C 2 inreases from 1 to, the effetive bandwidth (i.e. the required onstant servie rate) inreases 1 + δ times. For SPP, the required onstant servie rate does not exeed the upper bound, whih equals to the effetive bandwidth of a Poisson proess with average paet rate max{λ I,λ II }. This indiates that the servie rate requirement is still finite for IPP with C 2 or for SPP with any values of α I and α II. Therefore, the burstiness and autoorrelation will not hange the proposed framewor. 2) Multi-User Senario: In this ase, we jointly optimize N, ε, εq,andεh, with whih we an obtain the optimal rosslayer strategy inluding bandwidth alloation, power alloation and paet dropping poliies. The optimization problem in the multi-user senario is formulated as min P tot N,εq,ε,εh =1,2,...,K s.t. ε h = 0 N0 BN γ μ P th K =1 P th (25) ln 1 ( 1 + μ P th g N 0 BN ln (1 + γ ) ) ln (1 + γ ) = T fu ln 2 φbn E B (θ V ) + φbn ε + εq + εh ε D and ε,εq,εh R+, K N N max, N Z+, = 1,...,K, f g (g) dg, fq 1 ( ) ε, (25a) (25b) (25) (25d) where Nmax is the maximal number of subarriers for DL transmission. 8 Sine N is integer, this is a mixed-integer programming problem. Given the values of N, = 1,...,K, the problem an be deomposed into K single-user problems similar to (23), whih an be solved by the two-step method. Then, the power alloation poliy among subsequent TTIs and the paet dropping poliy an be obtained similarly to those in the singleuser senario, i.e., (18) and (20). We refer to the K single-user problems as subproblem I. Sine binary searh and exat linear 8 By solving problem (25), the bandwidth (i.e., the number of subarriers) alloation is obtained. With onstraint (25d), the total number of subarriers alloated to all the K users is less than the maximal number of subarriers of the system. Therefore, we an always find a subarrier alloation poliy, with whih eah subarrier is only alloated to one user. TABLE II SUBCARRIER ALLOCATION ALGORITHM searh methods are applied in solving subproblem I, the omplexity of the two-step method is O(log 2 ( ε D ) log h 2 ( ε D )). 9 The omplexity of problem (25) is determined by the integer programming that optimizes N, = 1,...,K with given ε,εq,εh to minimize the objetive funtion in (25). We refer this integer programming as subproblem II. Sine N 1, the remaining number of subarriers is N max K. To solve problem (25), we need to alloate the remaining subarriers to K users. Thus, subproblem II inludes around K N max K feasible solutions. To redue omplexity, a heuristi algorithm is proposed, as listed in Table II. The basi idea is similar to the steepest desent method 42]. The subarrier alloation algorithm inludes Nmax K steps. In eah step, one subarrier is alloated to one of the K users that leads to the steepest total transmit power desent. The proposed algorithm only needs to solve subproblem I for K (Nmax K ) times, and hene the omplexity is O ( K (Nmax K )). Further onsidering the omplexity of the two-step method for solving subproblem ( I, the overall omplexity of the ) proposed algorithm is O K (Nmax K ) log 2 ( ε D ) log h 2 ( ε D ). V. APPLYING THE FRAMEWORK TO FREQUENCY-SELECTIVE CHANNEL If the number of users is not very large, the bandwidth alloated to a user (say W = BN in problem (25)) ould be larger than the oherene bandwidth. In this setion, we show how to apply the framewor to frequeny-seletive hannel. We divide the bandwidth alloated to the th user into N s subhannels, where eah subhannel onsists of multiple subarriers. The bandwidth of eah subhannel is W that is less than the oherene bandwidth. Then, the subarriers within eah subhannel subjet to flat fading, while the subhannels 9 The omplexity of a searhing algorithm depends on the stopping riterion. Here, the iterations stop if ε h(i) εh (i +1) < h or ε (i) ε (i +1) < is satisfied, where ε h(i) and ε (i) are the results obtained after i iterations.

10 136 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 1, JANUARY 2018 Fig. 4. Illustration of two hannel oding shemes, where four paets need to be transmitted in a frame and W = 2 W. subjet to frequeny-seletive fading. To study the delay and reliability performane, we first need to find the ahievable rate with finite blolength. As shown in Appendix VII-C, the number of paet that an be transmitted in one frame an be obtained as, s fs φw u ln 2 N s j=1 ln 1 + μ ] P j (n)g j N 0 W ( ) ε V fq 1 φw (paets), (26) where P j (n) is the transmit power alloated to the jth subhannel of the th user in the nth frame, g j is the instantaneous hannel gain on the jth subhannel of the N th user, and V = N s s 1 ] 2. Sine j=1 1+ μ P j (n)g j N 0 W the hannel gains ould be arbitrarily lose to zero, the required transmit power to guarantee queueing delay is also unbounded. The paet rate in (26) an be ahieved if all the paets in a frame are oded in one blo with length W N sφ (alled the optimal oding sheme), as illustrated in Fig. 4(a). By substituting (26) into (12), we annot obtain the required SNR to ensure (Dmax,ε q q ) and ε as that in (17). This is beause eah hannel oding blo onsists of paets transmitted over multiple subhannels with different instantaneous hannel gains. As a result, it is very hallenging to derive and optimize the proative paet dropping probability that guarantees the QoS. To overome this diffiulty, we onsider a suboptimal oding sheme that the paets to be transmitted on different subhannels are oded independently. As illustrated in Fig. 4(b), the blolength of the suboptimal oding sheme is W φ. With shorter blolength, the suboptimal oding sheme an support lower paet rate for a given ε, thus the required resoures with the suboptimal hannel oding sheme are higher than that with the optimal sheme in order to ahieve the same QoS 43]. Nonetheless, with the optimal sheme, if a blo is not deoded without error, then all the paets transmitted in one frame will be lost. By ontrast, with the suboptimal sheme, if the paets in one blo is not deoded suessfully, the paets in other blos an still be deoded orretly. This suggests that the paet transmission errors with the suboptimal sheme is less busty than those with the optimal sheme. 10 When the number of paets transmitted over eah subhannel is E B(θ )/N s, the onstraints on proative paet dropping probability, queueing delay violation probability and transmission error probability an be obtained by replaing BN and E B(θ ) in (25a) and (25b) with W and E B(θ )/N s, respetively. In this way, the proposed framewor an be applied over frequeny-seletive hannel. In what follows, we analyze the rate loss. With the suboptimal sheme, the number of paets that an be transmitted over the N s s fs φw N s u ln 2 subhannels an be expressed as follows, ln 1 + μ ] P j (n)g j Ṽ fs j f 1 ( ) Q ε N 0 W φw (paets), (27) j=1 where the number of paets transmitted in eah subhannel is obtained by replaing bandwith BN in (3) with W,and hene Ṽj fs 1 = 1 ] 2. From (26) and (27), we an 1+ μ P j (n)g j N 0 W derive the gap between s fs φw s fs sfs u ln 2 and sfs as, N s Ṽ fs j=1 j V fq 1 ( ) ε, whih shows that s fs sfs O(N s N s), 11 and thus the gap between s fs and s fs inreases with N s. From (27), we have s fs O(N s ), hene (sfs sfs )/ sfs O(1). This means that the normalized rate loss (s fs sfs )/ sfs approahes to a onstant when N s is large. VI. SIMULATION AND NUMERICAL RESULTS In this setion, we first validate that the effetive bandwidth an be used as a tool to optimize resoure alloation in short delay regime for Poisson proess, IPP and SPP. Then, we show the optimal values of ε q, ε and ε h, and the required maximal transmit power for both Poisson proess and IPP. 12 Next, we ompare the required transmit power of the proposed algorithm with the global optimal poliy obtained by exhaustive searhing. A single-bs senario is onsidered in the sequel. The users are uniformly distributed with distanes from the BS as 50 m 200 m. The arrival proess of eah user is modeled as Poisson proess, IPP, or SPP with average rate 1000 paets/s, i.e., eah user requests the safety messages from 50 nearby sensors, and eah sensor uploads paets to the BS with average rate 20 paets/s 37]. Other parameters are listed in Table III, unless otherwise speified. 10 Some appliations lie safe messages transmission in vehile networs may prefer suh suboptimal sheme, whih is also appliable for flat fading hannels. 11 Here y(n s) O ( x(n s)) means y(n s)/x(ns ) approahes to a onstant when N s is large. 12 The optimal values of ε q, ε and εh and the required transmit power for SPP are similar to that for IPP, and hene the results for SPP are omitted for oniseness.

11 SHE et al.: CROSS-LAYER OPTIMIZATION FOR ULTRA-RELIABLE AND LOW-LATENCY RADIO ACCESS NETWORKS 137 TABLE III PARAMETERS 6], 37] Fig. 6. Single-user senario, where user-bs distane is 200 m, N = 4, B = 0.15 MHz, and α = β. Fig. 5. Validating the upper bound in (10). The CCDFs of queue length and queueing delay for the paets to the th user are shown in Fig. 5, where (15) is used to translate the CCDF of the queueing delay into the CCDF of queue length. To obtain the upper bound in (10), Pr{D ( ) > D th } exp{ θ E B (θ )D th } is omputed by hanging D th from 0 to Dmax. q The CCDFs of queueing delay are obtained via Monte Carlo simulation by generating arrival proess and servie proess during frames. Numerial results in Fig. 5(a) indiate that for Poisson proess, the upper bound derived by effetive bandwidth wors when the maximal queue length is short. Simulation results in Fig. 5(b) show that the upper bound also wors for IPP and SPP. In fat, it has been observed in 44] that effetive bandwidth an be used for resoure alloation under statistial queueing delay requirement when Dmax q is small, if the TTI is muh shorter than the delay bound. The optimal solution of problem (23) and the required maximal transmit power for both Poisson and IPP are shown in Fig. 6. The results in Fig. 6(a) show that ε, εq and εh are in the same order of magnitude with different values of N t. In fat, similar to ε h, when either ε or εq is set as zero, the required transmit power will beome infinite, beause E B(θ ) when ε q = 0 (as an be learly seem from (A.2)) and fq 1 (x) (and hene s (n) in (3)

12 138 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 1, JANUARY 2018 TABLE IV REQUIRED TRANSMIT POWER, Nmax = 16, B = 0.15 MHZ, AND N t = 8 Fig. 7. Number of dropped paets over Naagami-m fading hannel, where Nmax = 1024 and Pmax = 46 dbm. approahes infinity) when ε = 0. This implies that the optimal probabilities will also be in the same order when other system parameters hange. On the other hand, Fig. 6(b) shows that ompared with ε = εq = εh, the required maximal transmit power only redues 2 5% with the optimized ε, εq and ε h when N t 8. This implies that dividing the required paet loss probability equally to the three probabilities will ause minor performane loss. Moreover, the optimal queueing delay violation probability for IPP is higher than that for Poisson proess. This indiates that bursty arrival proesses lead to higher queueing delay violation probability. Furthermore, P th dereases extremely fast as N t inreases. This agrees with the intuition: inreasing the number of transmit antennas is an effiient way to redue the required maximal transmit power thans to the spatial diversity. The required K obtained by the proposed algorithm =1 P th and the global optimal solution with exhaustive searhing are provided in Table IV. The results illustrate that the proposed algorithm is near-optimal. Beause the omplexity of exhaustive searh method is extremely high when N max and K are large, we only provide results with small values of N max and K. The number of dropped paets is determined by the distribution of hannel gain, whih depends on the propagation environments and N t as well. In Fig. 7, we provide the number of dropped paets over Naagami-m fading hannel with different values of m and N t. We onsider the worst ase that all the users are loated at the edge of the ell (i.e., user-bs distane is 200 m). Sine the average hannel gains of all the users are the same, the total bandwidth and transmit power are equally alloated to all the users. Then, N = N max /K and P th = P max /K.Wesetε = ε q = ε D /3. ε h is alulated from (22), where f g (g) = (mg)m 1 (m 1)! exp ( mg) when N t = 1 and m > 1 40] and f g (g) = (N 1 t 1)! gnt 1 e g when N t > 1 and m = 1 39]. All the results in the figure are obtained under onstraint ε h ε D/3. The results when N t = 1and m = 1 are not shown, beause onstraint ε h ε D/3 annot be satisfied under the transmit power onstraint. The number of dropped paets in transmitting paets with proative paet dropping poliy is ε h. To show the performane gain of proative paet dropping, we also provide the results for an intuitive paet dropping poliy, whih simply drops all the paets to the th user when g < N 0 BN γ. We an μ P th see that proative paet dropping poliy an help redue the number of dropped paets. VII. CONCLUSIONS In this paper, we studied how to optimize resoure alloation to guarantee ultra-low lateny and ultra-high reliability for radio aess networs in typial appliation senarios where the required delay is shorter than hannel oherene time. Both queuing delay and transmission delay were onsidered in the lateny, and the transmission error probability, queueing delay violation probability, and paet dropping probability were taen into aount in the reliability. We first showed that the required transmit power to ensure the QoS is unbounded when queueing delay bound is shorter than hannel oherene time. To satisfy the QoS requirement with finite transmit power, a proative paet dropping mehanism was proposed. A framewor for optimizing resoure alloation to ensure the stringent QoS was established, where a queue state and hannel state information dependent transmit power alloation and paet dropping poliies were optimized for single user ase, and bandwidth alloation was further optimized for multi-user senario, to minimize the required maximal transmit power of the BS. How to apply the proposed framewor to frequeny-seletive hannel was also addressed. Simulation results validated that effetive bandwidth an be used to optimize resoure alloation for Poisson proess, IPP and SPP, whih are representative traffi models to haraterizing performane of a system with queueing. Numerial results showed that the transmission error probability, queueing delay violation probability, and paet dropping probability are in the same order of magnitude, and setting the three paet loss probabilities equal will ause minor power loss. APPENDIX A EFFECTIVE BANDWIDTH OF SEVERAL RELEVANT ARRIVAL PROCESSES A. Poisson Arrival Proess The effetive bandwidth of Poisson proess is given by E B (θ ) = λ ( e θ 1 ) (paets/s). (A.1) T f θ Substituting (A.1) into (11), we an] obtain the required QoS exponent θ = ln T f ln(1/ε q ) λ D q max + 1. Then, (A.1) an be

13 SHE et al.: CROSS-LAYER OPTIMIZATION FOR ULTRA-RELIABLE AND LOW-LATENCY RADIO ACCESS NETWORKS 139 re-expressed as a funtion of (D q max,ε q ) as E B (θ ) = ln(1/ε q ) ] (paets/s). (A.2) Dmax q T ln f ln(1/ε q ) λ Dmax q + 1 B. IPP The effetive bandwidth of the IPP an be expressed as 45] E B (θ ) = 2θ T f (paet/s), (A.3) where ( e θ 1 ) λ on (α + β)] + (e θ 1 ) λ on (α + β) ] 2 ( + 4α e θ 1 ) λ on. Substituting (A.3) into (11), the QoS exponent θ an be obtained from = 2T f ln ε q Dmax q numerially. C. SPP Deriving the effetive bandwidth of autoorrelated proesses is muh harder than that of renewal proesses. To overome this diffiulty, we provide an upper bound of the effetive bandwidth of SPP. Without loss of generality, we assume λ I λii. Consider a Poisson proess with average arrival rate λ II, the arrival rate in the first state of SPP is less than that of the Poisson proess. Thus, the effetive bandwidth of the SPP is less than that of the Poisson proess, whih an be obtained by substituting λ = λ II into (A.1). APPENDIX B UPPER BOUND OF THE PACKET DROPPING PROBABILITY Proof: To derive ε h, we introdue an upper bound of b d (n) as follows, { b U (n) = max ( T f E B(θ ) s th, 0), if Q (n) > 0, 0, if Q (n) = 0, onsidering that b U (n) = bd (n) when Q (n) T f E B(θ) or Q (n) = 0, and b U (n) > bd (n) when 0 < Q (n) < T f E B(θ ). Then, we an derive an upper bound of Eb d (n)] as Eb U (n)] =η N 0 BN γ μ P th 0 (T f E B (θ ) s th ) f g(g)dg. Substituting Eb U (n)] into (21), we obtain an upper bound of the paet dropping probability as ε h 0 N0 BN γ μ P th 1 ] s th T f E B(θ f g (g) dg, ) (B.1) where η = Pr{Q (n) > 0} = E{ a i (n)}/es (n)] = i A E{ a i (n)}/t f E B(θ )] is applied. i A By substituting s th in (19) and onsidering (17), we have ( ) 1 + μ P thg N 0 BN V φbn fq 1 (ε ) s th ln T f E B (θ ) ln (1 + γ ) V φbn fq 1 (ε ). (B.2) Beause a paet is dropped only if it will be transmitted in deep fading, i.e. g 0, V in (4) approahes 0, and then (B.2) an be further aurately approximated by s th ln T f E B (θ ) ( 1 + μ P th g N 0 BN ln (1 + γ ) ). (B.3) Substituting (B.3) into (B.1), we obtain the approximation in (22). APPENDIX C PROOF OF THE CONVEXITY OF THE OBJECTIVE FUNCTION IN (24) Proof: For the Q-funtion f Q (x) = ( ) 1 2π x exp τ 2 2 dτ, wehave fq (x) = 1 e x2 /2 < 0, 2π and fq (x) = x x2 e /2 > 0whenx > 0. Thus, f 2π Q (x) is an dereasing and stritly onvex funtion when x > 0, i.e. f Q (x) < 0.5. Sine the inverse funtion of a dereasing and stritly onvex funtion is also stritly onvex 42], fq 1 (ε ) is stritly onvex when ε < 0.5 (whihistruefor any appliation). Hene, the seond term of (24) is stritly onvex. To prove that the first term of (24a) is stritly onvex, we first derive its seond order derivative. Denote y = ln (ε q ) and z = T f Dmaxλ q > 0. After removing the nonrelevant onstants, the first term of (24a) an be expressed y as f (y) = ln(1+zy), and its seond order derivative is derived as ( d 2 )( ) f dy 2 ( )( df d 2 ) y dy 2 dε q + dy d(ε q ) 2. (C.1) d 2 f d(ε q ) 2 = After some regular derivations, we an obtain that dy dε q = 1 ε q, d 2 y d(ε q ) 2 = ( 1 ε q ) 2, (C.2) df (1 + zy) ln (1 + zy) zy = dy ln (1 + zy)] 2 (1 + zy), (C.3) d 2 ( f dy 2 = 2z2 y 2z + z 2 y ) ln (1 + zy) ln (1 + zy)] 3 (1 + zy) 2. (C.4) After substituting (C.2), (C.3) and (C.4) into (C.1), we an finally obtain that d 2 f { d(ε q ) 2 = (1 + zy) 2 ln (1 + zy)] 2 ( 2z + zy + z 2 y + z 2 y 2) } ln (1 + zy) + 2z 2 y { ln (1 + zy)] 3 (1 + zy) 2( ε q) } 2 1. (C.5)

14 140 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 1, JANUARY 2018 Sine the denominator is positive, we only need to show the numerator is positive. Denote the numerator of (C.5) as f mun (x, z), wherex = yz. Then, we have f mun (x, z) = (1 + x) 2 ln(1 + x)] 2 (x + x 2 ) ln(1 + x) (2 + x) ln(1 + x) 2x] z. (C.6) For ε q < 10 5, whih is true for appliations with ultrahigh reliability requirement, y > ln ( 10 5) > 10, and then x > 10 z. Moreover,(2+x) ln(1+x) 2x > 0, x > 0. Then, we an obtain a lower bound of f mun (x, z) as follows, f LB (x) = (1 + x) 2 ln(1 + x)] 2 (x + x 2 ) ln(1 + x) (2 + x) ln(1 + x) 2x] x/10. (C.7) When x = 0, f LB (x) = 0. To prove f LB (x) >0, x > 0, we substitute ν = x + 1into(C.7)andprove flb (ν) > 0, ν >1. It is not hard to derive that flb (ν)= 20ν2 (ln ν) 2 +(10ν 2ν 2 ) ln ν+(3ν 11)(ν 1). 10ν (C.8) Denote the numerator of (C.8) as f LBnum (ν), whih equals zero when ν = 0. Besides, flbnum (ν) = 40ν(ln ν)2 + ( ν)ln ν + 4(ν 1) >0, ν >1. As a result, f LB (ν) > 0, and hene f LB(x) inreases with x. Therefore, we have f LB (x) >0, x > 0. This ompletes the proof. APPENDIX D PROOF OF THE OPTIMALITY OF THE TWO-STEP METHOD Proof: Denote an arbitrary feasible solution of problem (23) and the related transmit power as ( ε q, ε, ε h ) and P max, respetively. Given ε h, we an obtain the global minimal transmit power P max ( ε h ) P max by solving problem (24a), whih is for Poisson arrival proess. In the seond step, the global optimal ε h is obtained suh that P max P max ( ε h ). Therefore, P max P max. APPENDIX E ACHIEVABLE RATE OVER FREQUENCY- SELECTIVE CHANNEL Denote the hannel vetor on the jth subhannel of the th user as h j C Nt 1. 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Gursoy, Wireless throughput and energy effiieny with random arrivals and statistial queuing onstraints, IEEE Trans. Inf. Theory, vol. 62, no. 3, pp , Mar Changyang She (S 12 M 17) reeived the B.Eng. degree from the Honors College (formerly Shool of Advaned Engineering), Beihang University, Beijing, China, in 2012, and the Ph.D. degree from the Shool of Eletronis and Information Engineering, BUAA, in Sine 2017, he has been a Post- Dotoral Researh Fellow, Singapore University of Tehnology and Design. His researh interests lie in the areas of ultra-reliable and low-lateny ommuniations, tatile internet, big data for resoure alloation in wireless networs, and energy effiient transmission in 5G ommuniation systems. Chenyang Yang (M 99 SM 08) reeived the Ph.D. degree from Beihang University, Beijing, China, in Sine 1999, she has been a Full Professor with the Shool of Eletroni and Information Engineering, Beihang University. She has authored over 200 international journal and onferene papers and filed over 70 patents in the fields of energy effiient transmission, oordinated multi-point, interferene management, ognitive radio, and relay. Her reent researh interests inlude loal ahing, URLLC, and wireless big data. She was nominated as an Outstanding Young Professor of Beijing in 1995 and was supported by the First Teahing and Researh Award Program for Outstanding Young Teahers of Higher Eduation Institutions by the Ministry of Eduation from 1999 to She was the Chair of the Beijing Chapter of the IEEE Communiations Soiety from 2008 to 2012 and the Membership Development Committee Chair of the Asia Paifi Board of the IEEE Communiations Soiety from 2011 to She has served as a tehnial program ommittee member for numerous IEEE onferenes. She has served as an Assoiate Editor for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and the Guest Editor for the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS and the IEEE JOURNAL ON SELECTED TOPICS IN SIGNAL PROCESSING. She is urrently an Assoiate Editor-in-Chief of the Chinese Journal of Communiations and the Chinese Journal of Signal Proessing. Tony Q. S. Que (S 98 M 08 SM 12) reeived the B.E. and M.E. degrees in eletrial and eletronis engineering from the Toyo Institute of Tehnology, respetively, and the Ph.D. degree in eletrial engineering and omputer siene from MIT. Currently, he is a tenured Assoiate Professor with the Singapore University of Tehnology and Design (SUTD). He also serves as the Assoiate Head of ISTD Pillar and the Deputy Diretor of the SUTD-ZJU IDEA. His main researh interests are the appliation of mathematial, optimization, and statistial theories to wireless ommuniation, networing, signal proessing, and resoure alloation problems. Speifi urrent researh topis inlude networ intelligene, wireless seurity, Internet-of-Things, and big data proessing. Dr. Que has been atively involved in organizing and hairing sessions, and has served as a member of the tehnial program ommittee as well as a symposium hair in a number of international onferenes. He is urrently an eleted member of the IEEE Signal Proessing Soiety SPCOM Tehnial Committee. He was an Exeutive Editorial Committee Member for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, an Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS, and an Editor for the IEEE WIRELESS COMMUNICATIONS LETTERS. He is a o-author of the boo Small Cell Networs: Deployment, PHY Tehniques, and Resoure Alloation (Cambridge University Press, 2013) and the boo Cloud Radio Aess Networs: Priniples, Tehnologies, and Appliations (Cambridge University Press, 2017). Dr. Que was honored with the 2008 Philip Yeo Prize for Outstanding Ahievement in Researh, the IEEE Globeom 2010 Best Paper Award, the 2012 IEEE William R. Bennett Prize, the IEEE SPAWC 2013 Best Student Paper Award, the IEEE WCSP 2014 Best Paper Award, the 2015 SUTD Outstanding Eduation Awards Exellene in Researh, the 2016 Thomson Reuters Highly Cited Researher, the 2016 IEEE Signal Proessing Soiety Young Author Best Paper Award, and the 2017 IEEE Communiations Soiety Asia Paifi Outstanding Paper Award.