Cross-layer Optimization for Ultra-reliable and Low-latency Radio Access Networks

Transcription

1 Cross-layer Optimization for Ultra-reliable and Low-lateny Radio Aess Networs Changyang She, Chenyang Yang and Tony Q.S. Que arxiv: v2 s.it] 7 Ot 2017 Abstrat In is paper, we propose a framewor for rosslayer optimization to ensure ultra-high reliability and ultra-low lateny in radio aess networs, where bo transmission delay and queueing delay are onsidered. Wi short transmission time, e bloleng of hannel odes is finite, and e Shannon Capaity an not be used to haraterize e maximal ahievable rate wi given transmission error probability. Wi randomly arrived paets, some paets may violate e queueing delay. Moreover, sine e queueing delay is shorter an e hannel oherene time in typial senarios, e required transmit power to guarantee e queueing delay and transmission error probability will beome unbounded even wi spatial diversity. To ensure e required quality-of-servie (QoS) wi finite transmit power, a proative paet dropping mehanism is introdued. Then, e overall paet loss probability inludes transmission error probability, queueing delay violation probability, and paet dropping probability. We optimize e paet dropping poliy, power alloation poliy, and bandwid alloation poliy to minimize e transmit power under e QoS onstraint. The optimal solution is obtained, whih depends on bo hannel and queue state information. Simulation and numerial results validate our analysis, and show at setting e ree 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 e major goals in e fif generation (5G) ellular networs 2]. Ensuring suh a 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 Manusript reeived June 25, 2016; revised January 06, 2017, April 12, 2017 and July 21, 2017; aepted Ot 2, The assoiate editor oordinating e review of is paper and approving it for publiation was Q. Li. This paper was presented in part at e 2016 IEEE Global Communiations Conferene 1]. C. She was wi e Shool of Eletronis and Information Engineering, Beihang University, Beijing , China. He is now wi e Information Systems Tehnology and Design Pillar, Singapore University of Tehnology and Design, 8 Somapah Road, Singapore ( shehangyang@gmail.om). C. Yang is wi e Shool of Eletronis and Information Engineering, Beihang University, Beijing , China ( yyang@buaa.edu.n). C. She and C. Yang s wor was supported in part by National Natural Siene Foundation of China (NSFC) under Grant T. Q. S. Que is wi e Information Systems Tehnology and Design Pillar, Singapore University of Tehnology and Design, 8 Somapah Road, Singapore ( tonyque@sutd.edu.sg). C. She and T. Q. S. Que s wor was supported in part by was supported in part by e MOE ARF Tier 2 under Grant MOE2015-T and e SUTD-ZJU Researh Collaboration under Grant SUTD-ZJU/RES/01/2016. a smart grid in tatile internet 3], and autonomous vehiles and fatory automation in ultra-reliable mahine-typeommuniations (MTC) 4], despite at not all appliations of tatile internet and MTC require bo ultra-high reliability and ultra-low lateny. Sine tatile internet and MTC are primarily applied for mission ritial appliations, e message suh as touh and ontrol information is usually onveyed in short paets, and e reliability is refleted by paet loss probability 2]. The traffi supported by URLLC distinguishes from traditional real-time servie in bo QoS requirement and paet size. For human-oriented appliations, e requirements on delay and reliability are medium. For example, in e long term evolution (LTE) systems, e maximal queueing delay and its violation probability for VoIP are respetively 50 ms and in radio aess networs, and e minimal paet size is 1500 bytes 5]. For ontrol-oriented appliations suh as vehile ollision avoidane or fatory automation, e endto-end (E2E) or round-trip delay is around 1 ms, e overall paet loss probability is , 6], and e paet size is 20 bytes or even smaller 2]. LTE systems were designed for human-oriented appliations, where e 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 at e paets need to wait in e buffer of base station (BS) more an 1 ms before transmission. Therefore, even if oer delay omponents in bahaul and ore networs are redued wi new networ arhitetures 9], LTE systems annot ensure e E2E or round-trip lateny of 1 ms. A. Related Wor While reduing lateny in wireless networs is hallenging, furer ensuring high reliability maes e problem more intriate. To redue e delay aused by transmission and signalling 10], a short frame struture was introdued in 11], and e TTI was set idential to e frame duration. To ensure high reliability of transmission wi short frame, proper hannel oding wi finite bloleng is important. Fortunately, e results in 12] indiate at it is possible to guarantee very low transmission error probability wi short bloleng hannel odes, at e expense of ahievable rate redution. By using pratial oding shemes lie Polar odes

2 13], e 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 e suessful transmission probability in wireless ommuniations. To support e 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, 18] show at e required transmit power to ensure given transmission delay and reliability an be rapidly redued when e number of antennas at a BS inreases. In all ese wors, only transmission delay and transmission error probability are taen into aount in e QoS requirement. In pratie, sine e paets arrive at e buffer of e BS randomly, ere is a queue at e BS. To ontrol e delay and paet loss aused by bo queueing and transmission, ross-layer optimization should be onsidered 1]. Similar to e real-time servie suh as VoIP, e required queueing performane of URLLC an be modeled as statistial queueing requirement, haraterized by e maximal queueing delay and a small delay violation probability. By using effetive bandwid 19] and effetive apaity 20] to analyze performane of tatile internet under e statistial queueing requirement, e tradeoff among queueing delay, queueing delay violation probability and roughput was studied in 21], and UL and DL resoure alloation was jointly optimized to ahieve e E2E delay requirement in 22]. In bo wors, e Shannon apaity is applied to derive e effetive apaity. However, wi short transmission delay requirement, hannel oding is performed wi a finite blo of symbols, wi whih e Shannon apaity is not ahievable. In fat, e results obtained by using networ alulus in 23] show at if Shannon apaity is used to approximate e ahievable rate of short bloleng odes for designing resoure alloation, e queueing delay and delay violation probability annot be guaranteed. Based on e ahievable rate of a single antenna system wi finite bloleng hannel odes derived in 12], queueing delay/leng was analyzed in 24, 25]. For appliations wi medium delay and reliability requirements, e roughput subjet to statistial queueing onstraints was studied in 24], where e effetive apaity was derived by using e ahievable rate wi finite bloleng 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 a prior, while e deadline violation probability under e transmit power onstraint was not studied. B. Major Challenges and Our Contributions Supporting URLLC leads to e following hallenges in radio resoure alloation. First, e required queueing delay and transmission delay are shorter an hannel oherene time in typial senarios of URLLC. 1 This results in e 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 e suessful transmission probability when e 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 e large number of nodes. Moreover, wheer spatial diversity an guarantee e reliability is unnown. (3) The studies in 26] show at when e average delay approahes e hannel oherene time, e average transmit power ould beome infinity, beause transmitting paets during deep fading leads to unbounded transmit power. Hene, how to ensure bo e ultra-low delay and e ultra-high reliability wi finite transmit power is unlear. Seond, e bloleng of hannel odes is finite. The maximal ahievable rate in finite bloleng regime is neier onvex nor onave in radio resoures suh as transmit power and bandwid 12, 27]. As a result, finding optimal resoure alloation poliy for URLLC is muh more hallenging an at for traditional ommuniations, where Shannon apaity is a good approximation of ahievable rate and is jointly onave in transmit power and bandwid. Third, effetive bandwid is a powerful tool for designing resoure alloation to satisfy e statistial queueing requirement of real-time servie 19]. Sine e distribution of queueing delay is obtained based on large deviation priniple, e effetive bandwid an be used when e delay bound is large and e delay violation probability is small 28]. Therefore, using effetive bandwid for URLLC seems problemati. In is 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 is wor are summarized as follows: We show at only exploiting spatial diversity annot ensure e ultra-low lateny and ultra-high reliability wi finite transmit power over fading hannels. To ensure e QoS wi finite transmit power, we propose a proative paet dropping mehanism. We establish a framewor for ross-layer optimization to guarantee e low delay and high reliability, whih inludes a resoure alloation poliy and e proative paet dropping poliy depending on bo hannel and queue state information. By assuming frequeny-flat fading hannel model, we first optimize e power alloation and paet dropping poliies in a single-user senario, and en extend to e multi-user senario by furer optimizing bandwid alloation among users. Moreover, how 1 In is senario, effetive apaity an no longer be applied.

3 to apply e framewor to frequeny-seletive hannel is also disussed. We validate at even when e delay bound is extremely short, e upper bound of e omplementary umulative distributed funtion (CCDF) of queueing delay derived from effetive bandwid still wors for Poisson proess and Interrupted Poisson Proess (IPP), whih is more bursty an Poisson proess, and Swied Poisson Proess (SPP), whih is an autoorrelated two-phase Marov Modulated Poisson Proess 29]. We onsider e transmission error probability wi finite bloleng hannel oding, e queueing delay violation probability, and e proative paet dropping probability in e overall reliability. By simulation and numerial results, we show at setting paet loss probabilities equal is a near optimal solution in terms of minimizing transmit power. The rest of is paper is organized as follows. Setion II desribes system model and QoS requirement. Setion III shows how to represent queueing delay onstraint wi effetive bandwid. Setion IV introdues e paet dropping poliy, and e framewor for ross-layer optimization. Setion V illustrates how to apply e framewor to frequeny-seletive hannel. Simulation and numerial results are provided in Setion VI to validate our analysis and to show e optimal solution. Setion VII onludes e paper. II. SYSTEM MODEL AND QOS REQUIREMENT Consider a frequeny division duplex ellular system, 2 where eah BS wi 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 e BS. The seond type of nodes are M sensors, whih only upload paets. In e ases wiout e need to distinguish between users and sensors, we refer bo 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 orogonal frequeny division multiple aess (OFDMA) to avoid interferene. Type I Type II UL DL node K+1 node K+M node node K node 1 K+2 node 2 Area of interest w.r.t. node 1 Fig. 1. System model. 2 Our studies an be easily extended into time division duplex system, whih is wi different short frame struture 11]. All nodes in a ell upload eir messages wi short paets to e BS. The BS proesses e reeived messages from e nodes, and en transmits e relevant messages to e target users. For example, nodes 2, K + 1, and K + 2 lie in e area of interest wi respet to (w.r.t.) user 1, as shown in Fig. 1, and e BS only transmits e 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 at are onneted wi eah oer by fiber bahaul. The delay in fiber bahaul is muh less an 1 ms 30], and hene e delay in radio aess networ dominates e E2E delay. For oer ommuniation senarios (e.g., remote ontrol), e delay omponents in bahaul and ore networs should be taen into aount, yet our model an still be used to analyze e delay in radio aess 2]. Moreover, e model aptures one of e 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 e model is representative for URLLC, alough it annot over all appliation senarios. 3 All e notations to be used roughout e paper are summarized in Table I. A. QoS Requirement The QoS requirement of eah user is haraterized by e E2E delay and overall loss probability for eah paet 2, 4]. In e onsidered radio aess networ, e 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 an half of round-trip delay, our study an be diretly extended to e appliations wi requirement on round-trip delay. To ensure ultra-low transmission delay, we onsider e short frame struture proposed in 10], where e TTI is equal to e 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 e required short delay, T f D max, and retransmission mehanism is unable to be used. Bo UL transmission and DL transmission of eah short paet are finished wiin one frame, respetively. If a paet is not transmitted error-free in one frame, en e paet will be lost. Beause only a few symbols an be transmitted wiin φ, e transmission error is not zero wi finite bloleng hannel odes among ese symbols. Sine UL transmission has been studied in 32], we fous on e DL transmission in is wor. Then, e overall reliability for eah user, denoted as ε D, is e overall paet loss probability minus e UL transmission error probability. Denote e DL transmission error probability (i.e. e blo error probability 27]) for e user as ε. Sine e UL and DL transmissions need two frames, e queueing delay for every paet should be bounded as 3 Diret transmission between nodes (i.e., devie-to-devie (D2D) ommuniation mode) an help redue delay wi only one hop transmission. However, in D2D mode, e interferene beomes more omplex an e entralized ommuniations 31]. How to use D2D mode for URLLC deserves furer study but is beyond e sope of is wor.

4 TABLE I SUMMARY OF NOTATIONS K number of users M number of sensors T hannel oherene time T f duration of one frame D max required delay bound in radio aess networ D q max queueing delay bound φ duration for data transmission in eah frame N t number of antennas at e BS ε q queueing delay violation probability of e user ε q transmission error probability of e user ε h proative paet dropping probability of e user ε D overall paet loss probability N s number of subhannels alloated to e user N number of subarriers alloated to e user W bandwid of eah subhannel B bandwid of eah subarrier n s bloleng of hannel oding of e user W total bandwid alloated to e user s (n) ahievable rate wi finite bloleng of e user in e n frame s (n) apaity of e user in e n frame h hannel vetor of e user µ average hannel gain of e user g normalized instantaneous hannel power gain of e user P (n) transmit power alloated to e user in e n frame N 0 single-sided noise spetral density u number of bits in one paet f 1 Q (x) inverse of Q-funtion fg(x) probability density funtion of normalized instantaneous hannel gain A a set onsists of e indies of e nodes at lie in e area of interest w.r.t. e user a i(n) e number of paets uploaded to e BS from e i node b (n) number of paets departed from e queue in e n frame Q (n) queue leng of e user in e n frame E B (θ ) effetive bandwid of e arrival proess to e user θ e QoS exponent of e user P UB D upper bound of queueing delay violation probability of e queue π l probability at ere are l paets in e queue λ average paet rate of e Poisson proess λ on average paet rate in e ON state of e IPP α 1 average duration of OFF state of IPP β 1 average duration of ON state of IPP α 1 I average duration of e first state of SPP α 1 II average duration of e seond state of SPP λ I average paet rate in e first state of e SPP λ II average paet rate in e seond state of e SPP ξ ratio of average arrival rate to servie rate of e queue γ required SNR of e user η buffer non-empty probability of e queue P maximal transmit power at an be alloated to e user Dmax q D max 2T f. If e queueing delay bound is not satisfied, en a paet will beome useless and has to be dropped. Denote e reative paet dropping probability due to queueing delay violation as ε q. As detailed later, to satisfy e requirement imposed on e queueing delay for eah paet (Dmax, q ε q ) and ε to e user, e required transmit power may beome unbounded in deep fading. To guarantee QoS wi finite transmit power, we proatively drop several paets in e queue under deep fading and ontrol e overall reliability. Denote e proative paet dropping probability for e user as ε h. Then, e overall reliability for e user an be haraterized by e overall paet loss probability, whih is 1 (1 ε )(1 ε q )(1 εh ) ε + ε q + εh ε D, (1) where e approximation is aurate sine ε, εq, and εh are extremely small. B. Channel Model We onsider blo fading, where e hannel remains onstant wiin a oherene interval and varies independently among intervals. Denote e hannel oherene time as T. Sine e required delay bound D max is very short, it is reasonable to assume at T > D max > Dmax, q as shown in Fig In e following, we onsider suh a representative 4 For instane, for users wi veloities less an 120 m/h in a vehile ommuniation system operating in arrier frequeny of 2 GHz, e hannel oherene time is larger an 1 ms, whih exeeds e delay bound of eah paet. For oer appliations lie smart fatory, e veloities of sensors are slow or even zero, and hene T 1 ms. senario for typial appliations of URLLC, whih is more hallenging an e oer ase wi T Dmax. q Sine T f should be less an D max and e hannel oding is performed wiin φ of eah frame, suh a hannel (i.e., T f < T ) is referred to as quasi-stati fading hannel as in 27]. Duration of one frame (i.e. TTI) T f Control signaling UL delay Coherene time of hannel q Dmax Required delay bound T Data transmission DL delay Dmax Fig. 2. Relation of e required delay bound, hannel oherene time, frame duration and TTI. The UL transmission delay is equal to T f, and e same to e DL transmission delay. Denote e average hannel gain of e user as µ, and e orresponding hannel vetor in a ertain oherene interval as h CN (0, 1) C Nt 1 wi independent and identially distributed (i.i.d.) zero mean and unit variane Gaussian elements. Denote e size of eah paet as u bits. Aording to e Shannon apaity formula wi infinite bloleng oding, when µ and h are perfetly nown at e BS, e maximal number of paets at an be transmitted to e user in e n frame an be expressed as s (n) = φbn u ln 2 ln 1 + µ ] P (n)g N 0 BN (paets), (2) where P (n) is e transmit power alloated to e user

5 in e n frame, g = h H h, N 0 is e single-sided noise spetral density, B is e separation among subarriers, N is number of subarriers alloated to e user, and ] H denotes e onjugate transpose. When e bandwid alloated to e user, W = BN, is smaller an oherene bandwid, e hannel is flat fading and e hannel gains over N subarriers are approximately idential. We first onsider flat fading hannel, whih is appliable for many senarios of tatile internet and utra-reliable MTC where e number of users is large. We en disuss how to apply e proposed framewor to frequeny-seletive hannels in Setion V. The number of symbols transmitted in one frame (also referred to as e bloleng of hannel oding) for e user, n s, is determined by e bandwid and duration, i.e. n s = φw. To ensure e ultra-low lateny, e transmission duration φ is very short. Considering at e bandwid for eah user is limited, n s is far from infinite, and hene s (n) is not ahievable. The maximal ahievable rate wi finite bloleng oding is wi very ompliated expression 27]. By using e normal approximation in 27], e maximal number of paets at an be transmitted to e user in e n frame an be aurately approximated as s (n) φbn u ln 2 { ln 1 + µ ] P (n)g N 0 BN V φbn f 1 Q (ε ) } (paets), (3) where f 1 Q (x) is e inverse of Q-funtion, and V is given by 27] V = 1 1 ] 2. (4) 1 + µ P (n)g N 0BN (3) is obtained for interferene-free systems, whih is valid for e onsidered OFDMA (and also for time division multiple aess or spae division multiple aess wi zero-foring beamforming). To onsider oer multiple aess tehniques where interferene annot be ompletely avoided, e ahievable rate wi finite bloleng in interferene hannels should be used, whih however is not available in e literature until now. As shown in 23], if (2) is used to design resoure alloation wi finite bloleng oding, en e queueing delay and e queueing delay violation probability will be underestimated. As a result, e alloated resoure is insuffiient for ensuring e queueing performane. This indiates at to guarantee ultra-low lateny and ultra-high reliability, (3) should be applied. C. Queueing Model In e n frame, e user requests e paets uploaded from its nearby nodes. The indies of e nodes at lie in e area of interest w.r.t. e user onstitute a set A wi ardinality A. As illustrated in Fig. 3, e index set of e nearby nodes of e user is A = {+1,, +m}. Then, e number of paets waited in e queue for e user at e beginning of e (n + 1) frame an be expressed as Q (n + 1) = max {Q (n) s (n), 0} + i A a i (n), (5) where a i (n), i A is e number of paets uploaded to e BS from e i nearby node of e user. We onsider e senario at e inter-arrival time between paets ould be shorter an D q max (oerwise e queueing delay is zero), whih happens when e paets for a target user are randomly uploaded from multiple nearby nodes, i.e. A > 1. At e first glane, suh a senario seems to our wi a low probability. However, to ensure e ultrahigh reliability of ε D = 0.001% %, e senario of non-zero queueing delay is not negligible. Denote e number of paets departed from e queue in e n frame as b (n). If all e paets in e queue an be ompletely transmitted in e n frame, en b (n) = Q (n). Oerwise, b (n) = s (n). Hene, we have Node Node +1 Node +m a a a b (n) = min {Q (n), s (n)}. (6) 1 n m n UL n Q1 n Q n Buffers in BS Fig. 3. Queueing model at e BS. s1 s n n DL User 1 User Using (5) and (6), e evolution of e queue leng 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 is setion we employ effetive bandwid to represent e queueing delay requirement. We validate at effetive bandwid an be applied in e short delay regime for Poisson arrival proess, and en extend e disussion to IPP and SPP. A. Representing Queueing Delay Constraint wi Effetive Bandwid For stationary paets arrival proess { a i (n), n = i A 1, 2, }, e effetive bandwid is defined as 19] E B 1 (θ ) = lim ln N NT f θ { E exp ( N θ n=1 )]} a i (n) i A (paets/s), (8)

6 where θ is e QoS exponent for e user. A larger value of θ indiates a smaller queueing delay bound wi given queueing delay violation probability. Remar 1: When e queueing delay bound is not longer an e hannel oherene time, e servie proess is onstant wiin e delay bound wi given resoures suh as transmit power and bandwid, and e 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 e queueing delay requirement of e user (Dmax, q ε q ) in fading hannels, e onstant servie rate should be no less an e effetive bandwid of e arrival proess of e user. By setting s (n) in (3) equal to E B(θ ), P (n)g is onstant, i.e., e power alloation is hannel inversion, whih is not always feasible in pratial fading hannels. We will show how to handle is issue in e next setion. When e user is served wi a onstant rate equal to E B(θ ), e steady state queueing delay violation probability an be approximated as 20] Pr{D ( ) > D q max} η exp{ θ E B (θ )D q max}, (9) where η is e buffer non-empty probability and e approximation is aurate when D q max (i.e. queue leng is large enough) 19]. Sine η 1, we have Pr{D ( ) > D q max} exp{ θ E B (θ )D q max} P UB D. (10) If e upper bound in (10) satisfies P UB D = exp{ θ E B (θ )D q max} = ε q, (11) en e queueing delay requirement (Dmax, q ε q ) an be satisfied. In oer words, if e number of paets transmitted in every frame to e user is a onstant at satisfies s (n) = T f E B (θ ) (paets), (12) en (Dmax, q ε q ) an be ensured 19]. When e queue is served by e onstant servie proess {s (n), n = 1, 2, } at satisfies (12), e departure proess in (6) beomes b (n) = min{q (n), T f E B (θ )} (paets). (13) If e departure proess {b (n), n = 1, 2, } satisfies (13), en (Dmax, q ε q ) an be guaranteed. Satisfying (13) does not require onstant servie proess. For example, when Q (n) = 0, e buffer is empty, en no servie is needed. in (10) wi Represen- B. Validating e Upper Bound PD UB tative Arrival Proesses 1) Representative arrival proesses: The aggregation of paets at are independently generated by A nodes lie in e onerned area w.r.t e user (i.e. a i (n) in (5)) an i A be modeled as a Poisson proess in vehile ommuniation and oer MTC appliations 34, 35]. Denote e average paet rate of e Poisson proess as λ. Sine e features of traffi, say burstiness and autoorrelation, have large impat on e delay performane of queueing systems 29, 36], and e effetive bandwid for real-world arrival proesses is hard to obtain, we also onsider anoer two representative traffi models. As shown in 37], e 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 e BS. When an event happens (e.g., a sudden brae) and deteted by nearby sensors, e sensors send e paets to e BS. IPP has two states. In e OFF state, no paet arrives. In e ON state, paets arrive at e buffer of e BS aording to a Poisson proess wi average paet rate paets/frame. The durations at e proess stays in OFF and ON states are exponential distributed wi mean values of α 1 and β 1 frames, respetively. Bo Poisson proess and IPP are renewal proesses, whih annot haraterize e autoorrelation of a traffi. In 37], SPP is used to model e aggregation of event-driven paets and periodi paets in vehile ommuniation networs. Similar to IPP, SPP has two states, where e durations at a SPP stays in e first state and e seond state are exponential λ on distributed wi mean values of α 1 I and α 1 II frames, respetively. In e two states, paets arrive at e buffer of e BS aording to Poisson proesses wi 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 bandwids of Poisson proess, IPP and SPP are provided in Appendix A. 2) Validating e upper bound: The approximation in (9) is aurate when e 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 at it is very diffiult to obtain an aurate distribution of e queueing delay. In fat, what really onerned here is wheer e upper bound in (10) is appliable to our problem. If PD UB is indeed an upper bound of Pr{D ( ) > Dmax}, q en a transmit poliy optimized under e onstraint in (12) or (13) an satisfy e queueing delay requirement. In what follows, we derive e queueing delay distribution for Poisson proess, whih an be used to validate e upper bound in short Dmax q regime numerially. For arrival proesses at are more bursty an Poisson proess, e upper bound in (10) is appliable 38]. When a Poisson arrival proess is served by a onstant servie proess {s (n), n = 1, 2, }, e well-nown M/D/1 queueing model an be applied 36]. For a disrete state M/D/1 queue, e CCDF of e steady state queue leng an be expressed as Pr{Q ( ) > L} = 1 L π l, where π l = Pr{Q ( ) = l} is e probability at ere are l paets in e queue, i.e., π 0 = 1 ξ, π 1 = (1 ξ )(e ξ 1), π l = (1 ξ ) l=1

7 l 1 elξ + j=1 ] e jξ ( 1) l j (jξ ) l j (l j)! + (jξ ) l j 1 (l j 1)!, (l 2), (14) wi ξ = λ /s (n) 36]. For a Poisson arrival proess served by a onstant servie rate 1 T f s (n) = E B (θ ), Pr{D ( ) > D q max} = Pr{Q ( ) > E B (θ )D q max}. (15) Then, from (14), e CCDF of e queueing delay an be derived as L Pr{D ( ) > T f L/s (n)} = Pr{Q ( ) > L} = 1 π l, l=0 (16) whih is too ompliated to obtain a losed-form onstraint on queueing delay due to expressions of π l in (14). Noneeless, (16) an be used to validate e upper bound PD UB in (10) numerially. IV. A FRAMEWORK FOR CROSS-LAYER TRANSMISSION OPTIMIZATION In is setion, we first show at e required transmit power to guarantee e queueing delay and transmission error probability requirement for some paets may beome unbounded for any given bandwid and N t, owing to D q max < T. To guarantee e QoS in terms of D q max and ε D wi finite transmit power, we en 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 bo hannel information and queue leng. A. Proative Paet Dropping and Power Alloation We onsider e ase where Q (n) T f E B(θ ), en b (n) = T f E B(θ ). If a transmit power an guarantee suh a departure rate, en for e oer ase where Q (n) < T f E B(θ), b (n) < T f E B(θ ) an also be supported, i.e., (Dmax, q ε q D ) an be satisfied aording to (13). Substituting s (n) in (3) into (12), we an obtain e required SNR γ to ensure (Dmax, q ε q ) and ε for all paets to e user using e following equation, ln (1 + γ ) T fu ln 2 φbn E B (θ ) + V φbn f 1 Q (ε ). (17) Sine h C Nt is wi i.i.d. elements, e hannel gain g = h H h follows Wishart distribution 39], whose 1 probability density funtion is f g (x) = (N t 1)! xnt 1 e x. In e onsidered typial appliation senario wi Dmax q < T, some paets to be transmitted wiin e delay bound may experiene deep fading wi hannel gain g arbitrarily lose to zero. 5 Then, e required transmit power to ahieve γ in e n frame, P (n) N0BN γ µ g, is unbounded. This means 5 This is true also for oer hannel distribution, say Naagami-m fading, whih is a general model of wireless hannels 40]. at s (n) annot exeed E B(θ ) wi finite transmit power if e n frame is in a oherene interval subjet to deep fading, even when ere is spatial diversity. In oer words, for e paets in suh an interval, ε q + ε will exeed ε D will happen if P (n) is finite. To satisfy e QoS requirement wi a finite transmit power, we introdue a proative paet dropping mehanism. By proative, we mean at a paet will be intentionally disarded even when its queueing delay has not exeeded Dmax q in e ase ε q + ε > ε D, and en e total number of paets proatively and reatively dropped 6 is judiiously ontrolled to ensure e overall reliability for eah user. The rational behind suh a mehanism lies in e fat at we only need to ensure e overall paet loss probability ε D no matter how e paets are lost. Denote e maximal transmit power of e BS as P max. We disard some paets before transmission in deep fading hannels when e required SNR γ annot be ahieved wi K P (n) P max. However, we an hardly ontrol e =1 paet dropping probability of eah user from K =1 N 0BN γ µ g P max sine e required total transmit power depends on e hannel gains of multiple users. To ontrol e paet dropping probability of eah user, we introdue e maximal transmit power at an be alloated to e user P. When e required transmit power is higher an P, e BS transmits paets to e user wi power P and drop several paets in e n frame. Then, e total transmit power of e BS is bounded by K =1 P. To ensure (Dmax, q ε q ) and ε, e power alloation poliy should depend on bo hannel gain and queueing leng, whih is, P (n) = P, N 0BN γ if Q (n) T f E B(θ ), g < N0BN γ, µ P µ g, if Q (n) T f E B(θ ), g > N0BN γ µ P. (18) In e ase Q (n) < T f E B(θ ), P (n) should satisfy s (n) = Q (n) when s > Q (n) or P (n) = P when s Q (n), where s is e number of paets at an be transmitted in e n frame wi P (n) = P. From e approximation in (3), we obtain s as { s φbn ln 1 + µ P g ] } V u ln 2 N 0 BN φbn f 1 Q (ε ). (19) When g < N0BN γ µ P in e n frame, s < T fe B (θ ). Sine b (n) = min{q (n), T f E B(θ )} needs to be satisfied to ensure (Dmax, q ε q ), e BS has to disard some paets 6 By reative, we mean at a paet is lost when D q max is violated or a oding blo is not deoded suessfully.

8 waiting in e queue. Denote e number of paets dropped in e n frame as b d (n) = max{b (n) s, 0}. Then, e proative paet dropping poliy is { ( b d max Tf E (n) = B(θ ) s, 0), if Q (n) T f E B(θ ), max ( Q (n) s, 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 < N0BN γ, en P µ P is used to transmit paets and b d (n) paets at annot be onveyed wiin e n frame wi P are dropped, where P and bd (n) will be optimized in e next subsetion. Sine e BS simply disards some paets from e buffer if e hannel gain is low, suh a poliy only introdues negligible proessing delay due to several operations of omparison. Similar to e delivery ratio in 41], we define e paet dropping probability as ε h lim N N b d (n) n=1 Eb d = (n)] N E{ a i (n)}, (21) a i (n) i A n=1 i A where e seond equality is obtained under e assumption at e queueing system is ergodi, and e average on nominator is taen over bo hannel gain and queue leng. Based on e analysis in Appendix B, e paet dropping probability an be approximated by ) (1 + µ P ε h N 0 BN γ µ P 0 g N 0BN ln 1 ln (1 + γ ) f g (g) dg. (22) B. A Framewor for Cross-layer Transmission Optimization Wi e proative paet dropping mehanism, e total transmit power is bounded by K =1 P. To find e minimal resoures required to ensure e QoS, we optimize e rosslayer 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 bandwid alloation poliy for multi-user senario, to minimize K =1 P wi given total bandwid of e system. Aording to (18), P (n) depends on e values of γ and P. Given e values of γ and ε h, e minimal value of P an be obtained from (22) by letting e equality hold. Moreover, e required SNR γ is determined by ε and εq aording to (17). Therefore, e power alloation poliy and e minimal P are uniquely determined by e values of ε, ε q and εh. Aording to (20), e number of paets to be dropped b d P (n) depends on s, whih an be obtained from (19) after and ε are obtained. This indiates at to optimize e power alloation poliy and paet dropping poliy at minimize K =1 P, we only need to ontrol ε q, ε, and εh. For easy exposition, we first onsider single user ase, and en extend to multi-user senario. 1) Single-user Senario: When K = 1, e index an be omitted for notational simpliity. We onsider e ase at Q(n) > 0. For Q(n) = 0, no power is alloated, i.e., P (n) = 0. The values of ε, ε q, and ε h at minimize P an be obtained from e following problem, min P (23) ε q,ε,ε h ( ) N 0 BN γ ln 1 + µp g s.t. ε h αp = N 0BN 1 f g (g) dg, ln (1 + γ) 0 (23a) ln (1 + γ) = T fu ln 2 V φbn EB (θ) + φbn f 1 Q (ε ), ε + ε q + ε h ε D and ε, ε q, ε h R +, (23b) (23) where onstraint (23a) and onstraint (23b) are e single-user ase of (17) and (22), respetively, E B (θ) depends on e soure as well as (Dmax, q ε q ), and R + represents e positive real number. 7 In e following, we propose a two-step meod to find e optimal solution of problem (23). In e first step, ε h 0 (0, ε D ) is fixed. Given ε h 0, P in e right hand side of (23a) inreases wi γ. Hene, minimizing P is equivalent to minimizing γ. For Poisson proess, e optimal values of ε and ε q at minimize e required γ an be obtained by solving e 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 1 Q (ε ) (24) s.t. ε + ε q ε D ε h 0, (24a) where e effetive bandwid in (A.2) is used to derive e objetive funtion. As proved in Appendix C, e objetive funtion in (24) is stritly onvex in ε and ε q, and hene e problem is onvex. To ensure e stringent QoS requirement, e required SNR γ is high, in is ase V 1 as shown in (4). Then, ere is a unique solution of ε and ε q at minimizes γ. Denote e minimal SNR obtained from problem (24) as γ. Sine e right hand side of (23a) dereases wi P, for given ε h 0 and γ, e value of P an be obtained numerially via binary searhing 42] as a funtion of ε h 0, denoted as P (ε h 0). In e seond step, we find e optimal ε h 0 (0, ε D ) at minimizes P (ε h 0). Sine ere is no losed-form expression of P (ε h 0), exhaustive searhing is needed to obtain e 7 The distribution of hannel gain f g (g) depends on e number of antennas N t. Therefore, e optimal solution of problem (23) will depend on N t. We will illustrate e impat of N t via numerial results in e next setion.

9 optimal ε h 0 in general. However, numerial results indiate at P (ε h 0) first dereases and en inreases wi ε h 0. Wi is property, we an find e optimal solution of ε h 0 and e required transmit power to ensure ε D via e exat linear searh meod 42]. As proved in Appendix D, e solution obtained from e two-step meod is e global optimal solution of problem (23) if e solutions of bo steps are global optimal. Impat of traffi feature: To show e impat of burstiness on e ross-layer optimization, we onsider IPP wi fixed average paet rate in two asymptoti ases, i.e. C 2 1 and C 2, where C 2 is e variane oeffiient at an be used to haraterize burstiness 29]. To show e impat of α burstiness, we eep e average paet rate of IPP, α+β λon, as a onstant. Then, e average paet rate an be expressed as λon 1+δ, and C2 = 1 + 2δλon (1+δ) 2 α 29], where δ = β/α. When α, C 2 1, e effetive bandwid ( of e IPP an be expressed as E B (θ) = λon T f θ(1+δ) e θ 1 ), whih is e same as e effetive bandwid of a Poisson proess wi average paet rate λon 1+δ. When α 0, C2, e effetive ( bandwid of e IPP an be expressed as E B (θ) = λ on T f θ e θ 1 ), whih is e same as e effetive bandwid of a Poisson proess wi average paet rate λ on. To show e impat of autoorrelation, we onsider a SPP wi parameters (λ I, λ II, α I, α II ), where λ I 0, λ on ], λ II = λ on, α I = α and α II = β. An upper bound of e effetive bandwid of it an be obtained by substituting λ = λ on into (A.1). Therefore, e effetive bandwid of SPP is less an at of a Poisson proess wi average paet rate max{λ I, λ II }. Remar 2: For IPP, when C 2 inreases from 1 to, e effetive bandwid (i.e. e required onstant servie rate) inreases 1 + δ times. For SPP, e required onstant servie rate does not exeed e upper bound, whih equals to e effetive bandwid of a Poisson proess wi average paet rate max{λ I, λ II }. This indiates at e servie rate requirement is still finite for IPP wi C 2 or for SPP wi any values of α I and α II. Therefore, e burstiness and autoorrelation will not hange e proposed framewor. 2) Multi-user Senario: In is ase, we jointly optimize N, ε, εq, and εh, wi whih we an obtain e optimal ross-layer strategy inluding bandwid alloation, power alloation and paet dropping poliies. The optimization problem in e multi-user senario is formulated as min P tot N,εq,ε,εh =1,2,,K s.t. ε h = N 0 BN γ µ P 0 K =1 ln (1 + γ ) = T fu ln 2 φbn E B (θ ) + P (25) ) ln (1 + µ P g N 0BN 1 f g (g) dg, ln (1 + γ ) V φbn ε + ε q + εh ε D and ε, ε q, εh R +, (25a) f 1 Q (ε ), (25b) (25) K N Nmax, N Z +, = 1,, K, (25d) where Nmax is e maximal number of subarriers for DL transmission. 8 Sine N is integer, is is a mixed-integer programming problem. Given e values of N, = 1,, K, e problem an be deomposed into K single-user problems similar to (23), whih an be solved by e two-step meod. Then, e power alloation poliy among subsequent TTIs and e paet dropping poliy an be obtained similarly to ose in e singleuser senario, i.e., (18) and (20). We refer to e K singleuser problems as subproblem I. Sine binary searh and exat linear searh meods are applied in solving subproblem I, e omplexity of e two-step meod is O(log 2 ( εd ) log h 2 ( εd )). 9 The omplexity of problem (25) is determined by e integer programming at optimizes N, = 1,, K wi given ε, εq, εh to minimize e objetive funtion in (25). We refer is integer programming as subproblem II. Sine N 1, e remaining number of subarriers is N max K. To solve problem (25), we need to alloate e remaining subarriers to K users. Thus, subproblem II inludes around K N max K feasible solutions. To redue omplexity, a heuristi algorim is proposed, as listed in Table II. The basi idea is similar to e steepest desent meod 42]. The subarrier alloation algorim inludes Nmax K steps. In eah step, one subarrier is alloated to one of e K users at leads to e steepest total transmit power desent. The proposed algorim only needs to solve subproblem I for K(Nmax K) times, and hene e omplexity is O (K(Nmax K)). Furer onsidering e omplexity of e two-step meod for solving subproblem I, e overall omplexity of e proposed algorim is O ( K(Nmax K) log 2 ( εd ) log h 2 ( εd ) ). V. APPLYING THE FRAMEWORK TO FREQUENCY-SELECTIVE CHANNEL If e number of users is not very large, e bandwid alloated to a user (say W = BN in problem (25)) ould be larger an e oherene bandwid. In is setion, we show how to apply e framewor to frequeny-seletive hannel. We divide e bandwid alloated to e user into N s subhannels, where eah subhannel onsists of multiple subarriers. The bandwid of eah subhannel is W at is less an e oherene bandwid. Then, e subarriers wiin eah subhannel subjet to flat fading, while e subhannels subjet to frequeny-seletive fading. To study e delay and reliability performane, we first need to find e ahievable rate wi finite bloleng. As shown in Appendix E, e 8 By solving problem (25), e bandwid (i.e., e number of subarriers) alloation is obtained. Wi onstraint (25d), e total number of subarriers alloated to all e K users is less an e maximal number of subarriers of e system. Therefore, we an always find a subarrier alloation poliy, wi whih eah subarrier is only alloated to one user. 9 The omplexity of a searhing algorim depends on e stopping riterion. Here, e 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 e results obtained after i iterations.

10 TABLE II SUBCARRIER ALLOCATION ALGORITHM Input: Number of users K, total number of subarrers Nmax, duration for data transmission in eah DL frame φ, paet size u, noise spetral density N 0, number of transmit antennas N t, average hannel gains of users µ, = 1,, K. Output: Subarrier alloation N, = 1,, K. 1: Set N(0) := 1, = 1,, K. Set l := 1. 2: Solve subproblem I wi N(0) = 1, and obtain e total transmit power P tot (0). 3: while l Nmax K do 4: Set ˆ := 1 5: while ˆ K do 6: N ˆ(l) := N ˆ(l 1) + 1; N(l) := N(l 1), ˆ. 7: Solve subproblem I wi N(l), tot and obtain ˆP ˆ (l). 8: ˆ := ˆ : end while 10: := arg min ˆP ˆ tot (l). ˆ 11: N (l) := N (l 1) + 1; N (l) := N (l 1),. 12: l := l : end while 14: return N = N (l 1), = 1,, K. number of paet at an be transmitted in one frame an be obtained as, s fs φw N s ln 1 + µ ] P j (n)g j V f 1 Q u ln 2 N 0 W φw (ε ) j=1 (paets), (26) where P j (n) is e transmit power alloated to e j subhannel of e user in e n frame, g j is e instantaneous hannel gain on e j subhannel of e s user, and V = N s N 1 ] 2. Sine e hannel j=1 1+ µ P j (n)g j N 0 W gains ould be arbitrarily lose to zero, e required transmit power to guarantee queueing delay is also unbounded. The paet rate in (26) an be ahieved if all e paets in a frame are oded in one blo wi leng W N sφ (alled e optimal oding sheme), as illustrated in Fig. 4(a). By substituting (26) into (12), we annot obtain e required SNR to ensure (Dmax, q ε q ) and ε as at in (17). This is beause eah hannel oding blo onsists of paets transmitted over multiple subhannels wi different instantaneous hannel gains. As a result, it is very hallenging to derive and optimize e proative paet dropping probability at guarantees e QoS. To overome is diffiulty, we onsider a suboptimal oding sheme at e paets to be transmitted on different subhannels are oded independently. As illustrated in Fig. 4(b), e bloleng of e suboptimal oding sheme is W φ. Wi shorter bloleng, e suboptimal oding sheme an support lower paet rate for a given ε, us e required resoures wi e suboptimal hannel oding sheme are higher an at wi e optimal sheme in order to ahieve e same QoS 43]. Noneeless, wi e optimal sheme, if a blo is not deoded wiout error, en all e paets transmitted in one frame will be lost. By ontrast, wi e suboptimal sheme, if e paets in one blo is not deoded suessfully, e paets in oer blos an still be deoded orretly. This suggests at e paet transmission errors wi e suboptimal sheme is less busty an ose wi e optimal sheme. 10 Frequeny 1 blo wi leng 2W 2W (a) 4 paets Optimal sheme Time Frequeny 2 blos wi leng W W W (b) 2 paets 2 paets Time Suboptimal sheme Fig. 4. Illustration of two hannel oding shemes, where four paets need to be transmitted in a frame and W = 2W. When e number of paets transmitted over eah subhannel is E B(θ )/N s, e onstraints on proative paet dropping probability, queueing delay violation probability and transmission error probability an be obtained by replaing BN and EB (θ ) in (25a) and (25b) wi W and E B(θ )/N s, respetively. In is way, e proposed framewor an be applied over frequeny-seletive hannel. In what follows, we analyze e rate loss. Wi e suboptimal sheme, e number of paets at an be transmitted over e 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 N 0 W φw (ε ) j=1 (paets), (27) where e number of paets transmitted in eah subhannel is obtained by replaing bandwi BN in (3) wi W, and hene Ṽ j fs = 1 1 ] 2. From (26) and (27), we 1+ µ P j (n)g j N 0 W an derive e gap between s fs N φw s s fs s fs u ln 2 j=1 and sfs as, Ṽ fs j V f 1 Q (ε ), whih shows at s fs sfs s O(N N s), 11 and us e gap between s fs s and sfs inreases wi N. From (27), we have s fs s O(N ), hene (sfs sfs )/ sfs O(1). This means at e normalized rate loss (s fs sfs )/ sfs approahes to a onstant when N s is large. 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 onstant when N s ) O ( x(n s)) means y(n s is large. )/x(n s ) approahes to a

11 VI. SIMULATION AND NUMERICAL RESULTS In is setion, we first validate at e effetive bandwid an be used as a tool to optimize resoure alloation in short delay regime for Poisson proess, IPP and SPP. Then, we show e optimal values of ε q, ε and εh, and e required maximal transmit power for bo Poisson proess and IPP. 12 Next, we ompare e required transmit power of e proposed algorim wi e global optimal poliy obtained by exhaustive searhing. A single-bs senario is onsidered in e sequel. The users are uniformly distributed wi distanes from e BS as 50 m 200 m. The arrival proess of eah user is modeled as Poisson proess, IPP, or SPP wi average rate 1000 paets/s, i.e., eah user requests e safety messages from 50 nearby sensors, and eah sensor uploads paets to e BS wi average rate 20 paets/s 37]. Oer parameters are listed in Table III, unless oerwise speified. TABLE III PARAMETERS 6, 37] Overall reliability requirement ε D % E2E delay requirement D max Queueing delay requirement Dmax q Duration of eah frame (equals to TTI) Duration of data transmission in one frame φ Single-sided noise spetral density N 0 Paet size u 1 ms 0.8 ms 0.1 ms 0.06 ms 173 dbm/hz 20 bytes Pa loss model 10 lg(µ ) lg(d ) Average duration of OFF or ON 1 s (i.e state α 1 or β 1 frames) The CCDFs of queue leng and queueing delay for e paets to e user are shown in Fig. 5, where (15) is used to translate e CCDF of e queueing delay into e CCDF of queue leng. To obtain e upper bound in (10), Pr{D ( ) > D } exp{ θ E B(θ )D } is omputed by hanging D 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 at for Poisson proess, e upper bound derived by effetive bandwid wors when e maximal queue leng is short. Simulation results in Fig. 5(b) show at e upper bound also wors for IPP and SPP. In fat, it has been observed in 44] at effetive bandwid an be used for resoure alloation under statistial queueing delay requirement when Dmax q is small, if e TTI is muh shorter an e delay bound. The optimal solution of problem (23) and e required maximal transmit power for bo Poisson and IPP are shown in Fig. 6. The results in Fig. 6(a) show at ε, εq and εh are in e same order of magnitude wi different values of N t. In fat, similar to ε h, when eier ε or εq is set 12 The optimal values of ε q, ε and εh and e required transmit power for SPP are similar to at for IPP, and hene e results for SPP are omitted for oniseness. CCDF of queue leng Pr{Q ( )>L} Pr{D ( )>D } Upper bound, Q max = 10 M/D/1, Q max = 10 Upper bound, Q max = 5 M/D/1, Q max = 5 Upper bound, Q max = 3 M/D/1, Q max = 3 Q max Queue leng L (paets) (a) Poisson arrivals, where ε q = ε q exp θ E B (θ )D ] Queueing delay D (ms) ε q Poisson, λ = 0.1 paet/frame IPP λ on =2λ, α 1 =β 1 =10 4 frames SPP (0.5λ,1.5λ,α,β) D q max (b) Poisson arrival, IPP and SPP, where C 2 = 1001 for e IPP. Fig. 5. Validating e upper bound in (10). as zero, e required transmit power will beome infinite, beause E B(θ ) when ε q = 0 (as an be learly seem from (A.2)) and f 1 Q (x) (and hene s (n) in (3) approahes infinity) when ε = 0. This implies at e optimal probabilities will also be in e same order when oer system parameters hange. On e oer hand, Fig. 6(b) shows at ompared wi ε = εq = εh, e required maximal transmit power only redues 2 5% wi e optimized ε, ε q and εh when N t 8. This implies at dividing e required paet loss probability equally to e ree probabilities will ause minor performane loss. Moreover, e optimal queueing delay violation probability for IPP is higher an at for Poisson proess. This indiates at bursty arrival proesses lead to higher queueing delay violation probability. Furermore, P dereases extremely fast as N t inreases. This agrees wi e intuition: inreasing e number of transmit antennas is an effiient way to redue e required maximal transmit power ans to e spatial diversity.