Cross-layer Optimization for Ultra-reliable and Low-latency Radio Access Networks
CCross-layer Optimization for Ultra-reliable andLow-latency Radio Access Networks
Changyang She, Chenyang Yang and Tony Q.S. Quek
Abstract —In this paper, we propose a framework for cross-layer optimization to ensure ultra-high reliability and ultra-lowlatency in radio access networks, where both transmission delayand queueing delay are considered. With short transmission time,the blocklength of channel codes is finite, and the ShannonCapacity can not be used to characterize the maximal achievablerate with given transmission error probability. With randomlyarrived packets, some packets may violate the queueing delay.Moreover, since the queueing delay is shorter than the channelcoherence time in typical scenarios, the required transmit powerto guarantee the queueing delay and transmission error prob-ability will become unbounded even with spatial diversity. Toensure the required quality-of-service (QoS) with finite transmitpower, a proactive packet dropping mechanism is introduced.Then, the overall packet loss probability includes transmissionerror probability , queueing delay violation probability , and packetdropping probability . We optimize the packet dropping policy,power allocation policy, and bandwidth allocation policy tominimize the transmit power under the QoS constraint. Theoptimal solution is obtained, which depends on both channeland queue state information. Simulation and numerical resultsvalidate our analysis, and show that setting the three packet lossprobabilities as equal causes marginal power loss. Index Terms —Ultra-low latency, ultra-high reliability, cross-layer optimization, radio access networks
I. I
NTRODUCTION
Supporting ultra-reliable and low-latency communications(URLLC) has become one of the major goals in the fifthgeneration (5G) cellular networks [2]. Ensuring such a strin-gent quality-of-service (QoS) enables various applicationssuch as control of exoskeletons for patients, remote driving,free-viewpoint video, and synchronization of suppliers in
Manuscript received June 25, 2016; revised January 06, 2017, April12, 2017 and July 21, 2017; accepted Oct 2, 2017. The associate editorcoordinating the review of this paper and approving it for publication wasQ. Li.This paper was presented in part at the 2016 IEEE Global CommunicationsConference [1].C. She was with the School of Electronics and Information Engi-neering, Beihang University, Beijing 100191, China. He is now withthe Information Systems Technology and Design Pillar, Singapore Uni-versity of Technology and Design, 8 Somapah Road, Singapore 487372(email:[email protected]).C. Yang is with the School of Electronics and Information Engineering,Beihang University, Beijing 100191, China (email:[email protected]).C. She and C. Yang’s work was supported in part by National NaturalScience Foundation of China (NSFC) under Grant 61671036.T. Q. S. Quek is with the Information Systems Technology and DesignPillar, Singapore University of Technology and Design, 8 Somapah Road,Singapore 487372 (e-mail: [email protected]).C. She and T. Q. S. Quek’s work was supported in part by was supportedin part by the MOE ARF Tier 2 under Grant MOE2015-T2-2-104 and theSUTD-ZJU Research Collaboration under Grant SUTD-ZJU/RES/01/2016. a smart grid in tactile internet [3], and autonomous vehi-cles and factory automation in ultra-reliable machine-type-communications (MTC) [4], despite that not all applicationsof tactile internet and MTC require both ultra-high reliabilityand ultra-low latency.Since tactile internet and MTC are primarily applied formission critical applications, the message such as “touch”and control information is usually conveyed in short packets,and the reliability is reflected by packet loss probability [2].The traffic supported by URLLC distinguishes from traditionalreal-time service in both QoS requirement and packet size.For human-oriented applications, the requirements on delayand reliability are medium. For example, in the long termevolution (LTE) systems, the maximal queueing delay andits violation probability for VoIP are respectively ms and × − in radio access networks, and the minimal packetsize is 1500 bytes [5]. For control-oriented applications suchas vehicle collision avoidance or factory automation, the end-to-end (E2E) or round-trip delay is around 1 ms, the overallpacket loss probability is − ∼ − [3, 6], and the packetsize is 20 bytes or even smaller [2].LTE systems were designed for human-oriented applica-tions, where the E2E delay includes uplink (UL) and downlink(DL) transmission delay, coding and processing delay, queue-ing delay, and routing delay in backhaul and core networks[7]. The radio resources are allocated in every transmit timeinterval (TTI), which is set to be ms [8]. This means that thepackets need to wait in the buffer of base station (BS) morethan ms before transmission. Therefore, even if other delaycomponents in backhaul and core networks are reduced withnew network architectures [9], LTE systems cannot ensure theE2E or round-trip latency of 1 ms. A. Related Work
While reducing latency in wireless networks is challenging,further ensuring high reliability makes the problem moreintricate. To reduce the delay caused by transmission andsignalling [10], a short frame structure was introduced in[11], and the TTI was set identical to the frame duration.To ensure high reliability of transmission with short frame,proper channel coding with finite blocklength is important.Fortunately, the results in [12] indicate that it is possible toguarantee very low transmission error probability with shortblocklength channel codes, at the expense of achievable ratereduction. By using practical coding schemes like Polar codes a r X i v : . [ c s . I T ] O c t a prior , while the deadline violation probability underthe transmit power constraint was not studied. B. Major Challenges and Our Contributions
Supporting URLLC leads to the following challenges inradio resource allocation. First, the required queueing delay and transmission delayare shorter than channel coherence time in typical scenariosof URLLC. This results in the following problems. (1) ARQmechanism can no longer be used to improve reliability.This is because retransmitting a packet in subsequent framesnot only introduces extra transmission delay but also canhardly improve the successful transmission probability whenthe channels in multiple frames stay in deep fading. (2)Time diversity cannot be exploited to enhance reliability, andfrequency diversity may not be scalable to the large number ofnodes. Moreover, whether spatial diversity can guarantee thereliability is unknown. (3) The studies in [26] show that whenthe average delay approaches the channel coherence time,the average transmit power could become infinity, becausetransmitting packets during deep fading leads to unboundedtransmit power. Hence, how to ensure both the ultra-low delayand the ultra-high reliability with finite transmit power isunclear.Second, the blocklength of channel codes is finite. Themaximal achievable rate in finite blocklength regime is neitherconvex nor concave in radio resources such as transmit powerand bandwidth [12, 27]. As a result, finding optimal resourceallocation policy for URLLC is much more challenging thanthat for traditional communications, where Shannon capacity isa good approximation of achievable rate and is jointly concavein transmit power and bandwidth.Third, effective bandwidth is a powerful tool for designingresource allocation to satisfy the statistical queueing require-ment of real-time service [19]. Since the distribution of queue-ing delay is obtained based on large deviation principle, theeffective bandwidth can be used when the delay bound is largeand the delay violation probability is small [28]. Therefore,using effective bandwidth for URLLC seems problematic.In this paper, we propose a cross-layer optimization frame-work for URLLC. While technical challenges in achievingultra-low E2E/round-trip delay exist at various levels, weonly consider transmission delay and queueing delay in radioaccess networks, and focus on DL transmission. The majorcontributions of this work are summarized as follows: • We show that only exploiting spatial diversity cannotensure the ultra-low latency and ultra-high reliability withfinite transmit power over fading channels. To ensure theQoS with finite transmit power, we propose a proactivepacket dropping mechanism. • We establish a framework for cross-layer optimizationto guarantee the low delay and high reliability, whichincludes a resource allocation policy and the proactivepacket dropping policy depending on both channel andqueue state information. By assuming frequency-flat fad-ing channel model, we first optimize the power allocationand packet dropping policies in a single-user scenario,and then extend to the multi-user scenario by further opti-mizing bandwidth allocation among users. Moreover, how In this scenario, effective capacity can no longer be applied. o apply the framework to frequency-selective channel isalso discussed. • We validate that even when the delay bound is extremelyshort, the upper bound of the complementary cumulativedistributed function (CCDF) of queueing delay derivedfrom effective bandwidth still works for Poisson processand Interrupted Poisson Process (IPP), which is morebursty than Poisson process, and Switched Poisson Pro-cess (SPP), which is an autocorrelated two-phase MarkovModulated Poisson Process [29]. • We consider the transmission error probability with finiteblocklength channel coding, the queueing delay violationprobability , and the proactive packet dropping probability in the overall reliability. By simulation and numericalresults, we show that setting packet loss probabilitiesequal is a near optimal solution in terms of minimizingtransmit power.The rest of this paper is organized as follows. Section II de-scribes system model and QoS requirement. Section III showshow to represent queueing delay constraint with effectivebandwidth. Section IV introduces the packet dropping policy,and the framework for cross-layer optimization. Section Villustrates how to apply the framework to frequency-selectivechannel. Simulation and numerical results are provided inSection VI to validate our analysis and to show the optimalsolution. Section VII concludes the paper.II. S
YSTEM M ODEL AND Q O S R
EQUIREMENT
Consider a frequency division duplex cellular system, where each BS with N t antennas serves K + M single-antennanodes. The nodes are divided into two types. The first typeof nodes are K users, which need to upload packets anddownload packets from the BS. The second type of nodes are M sensors, which only upload packets. In the cases without theneed to distinguish between users and sensors, we refer bothas nodes. Time is discretized into frames. Each frame consistsof a data transmission phase and a phase to transmit controlsignaling (e.g., pilot for channel estimation). We consider fre-quency reuse among adjacent cells and orthogonal frequencydivision multiple access (OFDMA) to avoid interference. ULDLType IType IIArea of interest w.r.t. node 1node 1node K +1 node K +2 node 2 node K node K + M ... Fig. 1. System model. Our studies can be easily extended into time division duplex system, whichis with different short frame structure [11].
All nodes in a cell upload their messages with short packetsto the BS. The BS processes the received messages from thenodes, and then transmits the relevant messages to the targetusers. For example, nodes , K + 1 , and K + 2 lie in thearea of interest with respect to (w.r.t.) user , as shown inFig. 1, and the BS only transmits the messages from nodes , K + 1 , and K + 2 to user . Such system model canbe applied in analyzing E2E delay in local communicationscenarios, where all nodes are associated to adjacent BSs thatare connected with each other by fiber backhaul. The delayin fiber backhaul is much less than ms [30], and hencethe delay in radio access network dominates the E2E delay.For other communication scenarios (e.g., remote control), thedelay components in backhaul and core networks should betaken into account, yet our model can still be used to analyzethe delay in radio access [2]. Moreover, the model capturesone of the key features of ultra-reliable MTC [4]: a packetgenerated by one node may be required by multiple users,and one user may also require packets generated by multiplenodes. Hence the model is representative for URLLC, althoughit cannot cover all application scenarios. All the notations tobe used throughout the paper are summarized in Table I.
A. QoS Requirement
The QoS requirement of each user is characterized by theE2E delay and overall loss probability for each packet [2, 4].In the considered radio access network, the E2E delay bound,denoted as D max , includes UL and DL transmission delay andqueueing delay. We only consider one-way delay requirement.By setting D max less than half of round-trip delay, our studycan be directly extended to the applications with requirementon round-trip delay.To ensure ultra-low transmission delay, we consider theshort frame structure proposed in [10], where the TTI is equalto the frame duration T f , each consisting of a duration for datatransmission φ and a duration for control signalling, as shownin Fig. 2. Owing to the required short delay, T f (cid:28) D max ,and retransmission mechanism is unable to be used. BothUL transmission and DL transmission of each short packetare finished within one frame, respectively. If a packet is nottransmitted error-free in one frame, then the packet will be lost.Because only a few symbols can be transmitted within φ , thetransmission error is not zero with finite blocklength channelcodes among these symbols. Since UL transmission has beenstudied in [32], we focus on the DL transmission in this work.Then, the overall reliability for each user, denoted as ε D , isthe overall packet loss probability minus the UL transmissionerror probability. Denote the DL transmission error probability(i.e. the block error probability [27]) for the k th user as ε ck .Since the UL and DL transmissions need two frames,the queueing delay for every packet should be bounded as Direct transmission between nodes (i.e., device-to-device (D2D) communi-cation mode) can help reduce delay with only one hop transmission. However,in D2D mode, the interference becomes more complex than the centralizedcommunications [31]. How to use D2D mode for URLLC deserves furtherstudy but is beyond the scope of this work.ABLE IS
UMMARY OF NOTATIONS K number of users M number of sensors T c channel coherence time T f duration of one frame D max required delay bound in radio access network D q max queueing delay bound φ duration for data transmission in each frame N t number of antennas at the BS ε qk queueing delay violation probability of the k th user ε qc transmission error probability of the k th user ε hc proactive packet dropping probability of the k th user ε D overall packet loss probability N sc k number of subchannels allocated to the k th user N c k number of subcarriers allocated to the k th user W c bandwidth of each subchannel B bandwidth of each subcarrier n sk blocklength of channel coding of the k th user W k total bandwidth allocated to the k th user s k ( n ) achievable rate with finite blocklength of the k th user in the n th frame s ∞ k ( n ) capacity of the k th user in the n th frame h k channel vector of the k th user µ k average channel gain of the k th user g k normalized instantaneous channel power gain of the k th user P k ( n ) transmit power allocated to the k th user in the n th frame N single-sided noise spectral density u number of bits in one packet f − ( x ) inverse of Q-function f g ( x ) probability density function of normalized instantaneous channel gain A k a set consists of the indices of the nodes that lie in the area of interestw.r.t. the k th user a i ( n ) the number of packets uploaded to the BS from the i th node b k ( n ) number of packets departed from the k th queue in the n th frame Q k ( n ) queue length of the k th user in the n th frame E Bk ( θ k ) effective bandwidth of the arrival process to the k th user θ k the QoS exponent of the k th user P UB Dk upper bound of queueing delay violation probability of the k th queue π l probability that there are l packets in the queue λ k average packet rate of the k th Poisson process λ on k average packet rate in the “ON” state of the k th IPP α − average duration of “OFF” state of IPP β − average duration of “ON” state of IPP α − average duration of the first state of SPP α − average duration of the second state of SPP λ I k average packet rate in the first state of the k th SPP λ II k average packet rate in the second state of the k th SPP ξ k ratio of average arrival rate to service rate of the k th queue γ k required SNR of the k th user η k buffer non-empty probability of the k th queue P th k maximal transmit power that can be allocated to the k th user D q max (cid:44) D max − T f . If the queueing delay bound is notsatisfied, then a packet will become useless and has to bedropped. Denote the reactive packet dropping probability dueto queueing delay violation as ε qk . As detailed later, to satisfythe requirement imposed on the queueing delay for each packet ( D q max , ε qk ) and ε ck to the k th user, the required transmitpower may become unbounded in deep fading. To guaranteeQoS with finite transmit power, we proactively drop severalpackets in the queue under deep fading and control the overallreliability. Denote the proactive packet dropping probabilityfor the k th user as ε hk .Then, the overall reliability for the k th user can be charac-terized by the overall packet loss probability, which is − (1 − ε ck )(1 − ε qk )(1 − ε hk ) ≈ ε ck + ε qk + ε hk ≤ ε D , (1)where the approximation is accurate since ε ck , ε qk , and ε hk areextremely small. B. Channel Model
We consider block fading, where the channel remains con-stant within a coherence interval and varies independentlyamong intervals. Denote the channel coherence time as T c .Since the required delay bound D max is very short, it isreasonable to assume that T c > D max > D q max , as shownin Fig. 2. In the following, we consider such a representative For instance, for users with velocities less than km/h in a vehiclecommunication system operating in carrier frequency of GHz, the channelcoherence time is larger than ms, which exceeds the delay bound of eachpacket. For other applications like smart factory, the velocities of sensors areslow or even zero, and hence T c (cid:29) ms. scenario for typical applications of URLLC, which is morechallenging than the other case with T c ≤ D q max . Since T f should be less than D max and the channel coding is performedwithin φ of each frame, such a channel (i.e., T f < T c ) isreferred to as quasi-static fading channel as in [27]. ...Duration of one frame (i.e. TTI) Coherence time of channelRequired delay bound max q D max D UL delay... f T c T DL delayData transmissionControl signaling
Fig. 2. Relation of the required delay bound, channel coherence time, frameduration and TTI. The UL transmission delay is equal to T f , and the same tothe DL transmission delay. Denote the average channel gain of the k th user as µ k ,and the corresponding channel vector in a certain coherenceinterval as h k ∼ CN (0 , ∈ C N t × with independent andidentically distributed (i.i.d.) zero mean and unit varianceGaussian elements. Denote the size of each packet as u bits.According to the Shannon capacity formula with infiniteblocklength coding, when µ k and h k are perfectly known atthe BS, the maximal number of packets that can be transmittedto the k th user in the n th frame can be expressed as s ∞ k ( n ) = φBN c k u ln 2 ln (cid:20) µ k P k ( n ) g k N BN c k (cid:21) (packets) , (2)where P k ( n ) is the transmit power allocated to the k th usern the n th frame, g k = h Hk h k , N is the single-sided noisespectral density, B is the separation among subcarriers, N c k is number of subcarriers allocated to the k th user, and [ · ] H denotes the conjugate transpose. When the bandwidth allo-cated to the k th user, W k = BN c k , is smaller than coherencebandwidth, the channel is flat fading and the channel gainsover N c k subcarriers are approximately identical. We firstconsider flat fading channel, which is applicable for manyscenarios of tactile internet and utra-reliable MTC where thenumber of users is large. We then discuss how to applythe proposed framework to frequency-selective channels inSection V.The number of symbols transmitted in one frame (alsoreferred to as the blocklength of channel coding) for the k thuser, n sk , is determined by the bandwidth and duration, i.e. n sk = φW k . To ensure the ultra-low latency, the transmissionduration φ is very short. Considering that the bandwidth foreach user is limited, n sk is far from infinite, and hence s ∞ k ( n ) is not achievable. The maximal achievable rate with finiteblocklength coding is with very complicated expression [27].By using the normal approximation in [27], the maximalnumber of packets that can be transmitted to the k th userin the n th frame can be accurately approximated as s k ( n ) ≈ φBN c k u ln 2 (cid:40) ln (cid:20) µ k P k ( n ) g k N BN c k (cid:21) − (cid:115) V k φBN c k f − ( ε ck ) (cid:41) (packets) , (3)where f − ( x ) is the inverse of Q-function, and V k is givenby [27] V k = 1 − (cid:104) µ k P k ( n ) g k N BN c k (cid:105) . (4)(3) is obtained for interference-free systems, which is valid forthe considered OFDMA (and also for time division multipleaccess or space division multiple access with zero-forcingbeamforming). To consider other multiple access techniqueswhere interference cannot be completely avoided, the achiev-able rate with finite blocklength in interference channelsshould be used, which however is not available in the literatureuntil now.As shown in [23], if (2) is used to design resource allocationwith finite blocklength coding, then the queueing delay and thequeueing delay violation probability will be underestimated.As a result, the allocated resource is insufficient for ensuringthe queueing performance. This indicates that to guaranteeultra-low latency and ultra-high reliability, (3) should beapplied. C. Queueing Model
In the n th frame, the k th user requests the packets uploadedfrom its nearby nodes. The indices of the nodes that lie in thearea of interest w.r.t. the k th user constitute a set A k withcardinality |A k | . As illustrated in Fig. 3, the index set of thenearby nodes of the k th user is A k = { k +1 , ..., k + m } . Then, the number of packets waited in the queue for the k th user atthe beginning of the ( n + 1) th frame can be expressed as Q k ( n + 1) = max { Q k ( n ) − s k ( n ) , } + (cid:88) i ∈A k a i ( n ) , (5)where a i ( n ) , i ∈ A k is the number of packets uploaded tothe BS from the i th nearby node of the k th user.We consider the scenario that the inter-arrival time betweenpackets could be shorter than D q max (otherwise the queueingdelay is zero), which happens when the packets for a targetuser are randomly uploaded from multiple nearby nodes, i.e. |A k | > . At the first glance, such a scenario seems tooccur with a low probability. However, to ensure the ultra-high reliability of ε D = 0 . % ∼ . %, the scenario ofnon-zero queueing delay is not negligible. Denote the numberof packets departed from the k th queue in the n th frameas b k ( n ) . If all the packets in the queue can be completelytransmitted in the n th frame, then b k ( n ) = Q k ( n ) . Otherwise, b k ( n ) = s k ( n ) . Hence, we have b k ( n ) = min { Q k ( n ) , s k ( n ) } . (6) User k Node k+m
Buffers in BS k s n k Q n
Node k+1 ... k m a n k a n DLUL User 1 ... ... Q n ......
Node k k a n ...... s n ... Fig. 3. Queueing model at the BS.
Using (5) and (6), the evolution of the queue length can bedescribed as follows, Q k ( n + 1) − Q k ( n ) = (cid:88) i ∈A k a i ( n ) − b k ( n ) . (7)III. E NSURING THE Q UEUEING D ELAY R EQUIREMENT
In this section we employ effective bandwidth to representthe queueing delay requirement. We validate that effectivebandwidth can be applied in the short delay regime for Poissonarrival process, and then extend the discussion to IPP and SPP.
A. Representing Queueing Delay Constraint with EffectiveBandwidth
For stationary packets arrival process { (cid:80) i ∈A k a i ( n ) , n =1 , , ... } , the effective bandwidth is defined as [19] E Bk ( θ k ) = lim N →∞ N T f θ k ln (cid:40) E (cid:34) exp (cid:32) θ k N (cid:88) n =1 (cid:88) i ∈A k a i ( n ) (cid:33)(cid:35)(cid:41) (packets/s) , (8)here θ k is the QoS exponent for the k th user. A larger valueof θ k indicates a smaller queueing delay bound with givenqueueing delay violation probability. Remark 1:
When the queueing delay bound is not longerthan the channel coherence time, the service process is con-stant within the delay bound with given resources such astransmit power and bandwidth, and the power allocation overfading channel is channel inversion in order to guaranteequeueing delay [33]. This is also true when achievable ratein (3) is applied, as explained in what follows. To satisfythe queueing delay requirement of the k th user ( D q max , ε qk ) in fading channels, the constant service rate should be no lessthan the effective bandwidth of the arrival process of the user.By setting s k ( n ) in (3) equal to E Bk ( θ k ) , P k ( n ) g k is constant,i.e., the power allocation is channel inversion, which is notalways feasible in practical fading channels. We will showhow to handle this issue in the next section.When the k th user is served with a constant rate equal to E Bk ( θ k ) , the steady state queueing delay violation probabilitycan be approximated as [20] Pr { D k ( ∞ ) > D q max } ≈ η k exp {− θ k E Bk ( θ k ) D q max } , (9)where η k is the buffer non-empty probability and the approx-imation is accurate when D q max → ∞ (i.e. queue length islarge enough) [19]. Since η k ≤ , we have Pr { D k ( ∞ ) > D q max } ≤ exp {− θ k E Bk ( θ k ) D q max } (cid:44) P UB D k . (10)If the upper bound in (10) satisfies P UB D k = exp {− θ k E Bk ( θ k ) D q max } = ε qk , (11)then the queueing delay requirement ( D q max , ε qk ) can be satis-fied. In other words, if the number of packets transmitted inevery frame to the k th user is a constant that satisfies s k ( n ) = T f E Bk ( θ k ) ( packets ) , (12)then ( D q max , ε qk ) can be ensured [19]. When the k th queue isserved by the constant service process { s k ( n ) , n = 1 , , ... } that satisfies (12), the departure process in (6) becomes b k ( n ) = min { Q k ( n ) , T f E Bk ( θ k ) } ( packets ) . (13)If the departure process { b k ( n ) , n = 1 , , ... } satisfies (13),then ( D q max , ε qk ) can be guaranteed. Satisfying (13) does notrequire constant service process. For example, when Q k ( n ) =0 , the buffer is empty, then no service is needed. B. Validating the Upper Bound P UB D k in (10) with Represen-tative Arrival Processes1) Representative arrival processes: The aggregation ofpackets that are independently generated by |A k | nodes lie inthe concerned area w.r.t the k th user (i.e. (cid:80) i ∈A k a i ( n ) in (5)) canbe modeled as a Poisson process in vehicle communication andother MTC applications [34, 35]. Denote the average packetrate of the k th Poisson process as λ k . Since the features of traffic, say burstiness and autocorrela-tion, have large impact on the delay performance of queueingsystems [29, 36], and the effective bandwidth for real-worldarrival processes is hard to obtain, we also consider anothertwo representative traffic models.As shown in [37], the event-driven packet arrivals in ve-hicular communication networks can be modelled as a burstyprocess, IPP. When no event happens, no sensor sends packetsto the BS. When an event happens (e.g., a sudden brake) anddetected by nearby sensors, the sensors send the packets tothe BS. IPP has two states. In the “OFF” state, no packetarrives. In the “ON” state, packets arrive at the buffer of theBS according to a Poisson process with average packet rate λ on k packets/frame. The durations that the process stays in“OFF” and “ON” states are exponential distributed with meanvalues of α − and β − frames, respectively.Both Poisson process and IPP are renewal processes, whichcannot characterize the autocorrelation of a traffic. In [37], SPPis used to model the aggregation of event-driven packets andperiodic packets in vehicle communication networks. Similarto IPP, SPP has two states, where the durations that a SPPstays in the first state and the second state are exponentialdistributed with mean values of α − and α − frames, respec-tively. In the two states, packets arrive at the buffer of theBS according to Poisson processes with average packet rates λ I k and λ II k packets/frame, respectively. Therefore, a SPP isdetermined by parameters ( λ I k , λ II k , α I , α II ) .The effective bandwidths of Poisson process, IPP and SPPare provided in Appendix A.
2) Validating the upper bound:
The approximation in (9)is accurate when the delay bound is sufficiently large and ε qk is very small [19, 28]. However, it is unclear how large D q max needs to be for an accurate approximation. One possible reasonis that it is very difficult to obtain an accurate distribution ofthe queueing delay.In fact, what really concerned here is whether the upperbound in (10) is applicable to our problem. If P UB D k is indeed anupper bound of Pr { D k ( ∞ ) > D q max } , then a transmit policyoptimized under the constraint in (12) or (13) can satisfy thequeueing delay requirement. In what follows, we derive thequeueing delay distribution for Poisson process, which canbe used to validate the upper bound in short D q max regimenumerically. For arrival processes that are more bursty thanPoisson process, the upper bound in (10) is applicable [38].When a Poisson arrival process is served by a constantservice process { s k ( n ) , n = 1 , , ... } , the well-known M/D/1queueing model can be applied [36]. For a discrete stateM/D/1 queue, the CCDF of the steady state queue lengthcan be expressed as Pr { Q k ( ∞ ) > L } = 1 − L (cid:80) l =1 π l , where π l = Pr { Q k ( ∞ ) = l } is the probability that there are l packetsin the queue, i.e., π = 1 − ξ k , π = (1 − ξ k )( e ξ k − ,π l = (1 − ξ k ) × e lξ k + l − (cid:88) j =1 e jξ k ( − l − j (cid:34) ( jξ k ) l − j ( l − j )! + ( jξ k ) l − j − ( l − j − (cid:35) , ( l ≥ , (14)with ξ k = λ k /s k ( n ) [36]. For a Poisson arrival process servedby a constant service rate T f s k ( n ) = E Bk ( θ k ) , Pr { D k ( ∞ ) > D q max } = Pr { Q k ( ∞ ) > E Bk ( θ k ) D q max } . (15)Then, from (14), the CCDF of the queueing delay can bederived as Pr { D k ( ∞ ) > T f L/s k ( n ) } = Pr { Q k ( ∞ ) > L } = 1 − L (cid:88) l =0 π l , (16)which is too complicated to obtain a closed-form constraint onqueueing delay due to expressions of π l in (14). Nonetheless,(16) can be used to validate the upper bound P UB D k in (10)numerically.IV. A F RAMEWORK FOR C ROSS - LAYER T RANSMISSION O PTIMIZATION
In this section, we first show that the required transmitpower to guarantee the queueing delay and transmissionerror probability requirement for some packets may becomeunbounded for any given bandwidth and N t , owing to D q max We consider the case where Q k ( n ) ≥ T f E Bk ( θ k ) , then b k ( n ) = T f E Bk ( θ k ) . If a transmit power can guarantee sucha departure rate, then for the other case where Q k ( n ) With the proactive packet dropping mechanism, the totaltransmit power is bounded by K (cid:80) k =1 P th k . To find the minimalresources required to ensure the QoS, we optimize the cross-layer transmission strategy, which includes a transmit powerallocation policy P k ( n ) and a proactive packet dropping policy b dk ( n ) for single user scenario and also includes a bandwidthallocation policy for multi-user scenario, to minimize K (cid:80) k =1 P th k with given total bandwidth of the system.According to (18), P k ( n ) depends on the values of γ k and P th k . Given the values of γ k and ε hk , the minimal value of P th k can be obtained from (22) by letting the equality hold.Moreover, the required SNR γ k is determined by ε ck and ε qk according to (17). Therefore, the power allocation policy andthe minimal P th k are uniquely determined by the values of ε ck , ε qk and ε hk .According to (20), the number of packets to be dropped b dk ( n ) depends on s th k , which can be obtained from (19) after P th k and ε ck are obtained.This indicates that to optimize the power allocation policyand packet dropping policy that minimize K (cid:80) k =1 P th k , we only need to control ε qk , ε ck , and ε hk .For easy exposition, we first consider single user case, andthen extend to multi-user scenario. 1) Single-user Scenario: When K = 1 , the index k canbe omitted for notational simplicity. We consider the case that Q ( n ) > . For Q ( n ) = 0 , no power is allocated, i.e., P ( n ) =0 . The values of ε c , ε q , and ε h that minimize P th can beobtained from the following problem, min ε q ,ε c ,ε h P th (23)s.t. ε h = (cid:90) N BN c γαP th − ln (cid:16) µP th gN BN c (cid:17) ln (1 + γ ) f g ( g ) dg, (23a) ln (1 + γ ) = T f u ln 2 φBN c E B ( θ ) + (cid:115) VφBN c f − ( ε c ) , (23b) ε c + ε q + ε h ≤ ε D and ε c , ε q , ε h ∈ R + , (23c)where constraint (23a) and constraint (23b) are the single-usercase of (17) and (22), respectively, E B ( θ ) depends on thesource as well as ( D q max , ε q ) , and R + represents the positivereal number. In the following, we propose a two-step method to find theoptimal solution of problem (23).In the first step, ε h ∈ (0 , ε D ) is fixed. Given ε h , P th in theright hand side of (23a) increases with γ . Hence, minimizing P th is equivalent to minimizing γ .For Poisson process, the optimal values of ε c and ε q thatminimize the required γ can be obtained by solving thefollowing problem, min ε q ,ε c T f u ln 2 ln (1 /ε q ) φBN c D q max ln (cid:104) T f ln(1 /ε q ) D q max λ (cid:105) + (cid:115) VφBN c f − ( ε c ) (24)s.t. ε c + ε q ≤ ε D − ε h , (24a)where the effective bandwidth in (A.2) is used to derive theobjective function. As proved in Appendix C, the objectivefunction in (24) is strictly convex in ε c and ε q , and hence theproblem is convex. To ensure the stringent QoS requirement,the required SNR γ is high, in this case V ≈ as shown in (4).Then, there is a unique solution of ε c and ε q that minimizes γ .Denote the minimal SNR obtained from problem (24) as γ ∗ .Since the right hand side of (23a) decreases with P th , for given ε h and γ ∗ , the value of P th can be obtained numerically viabinary searching [42] as a function of ε h , denoted as P th ( ε h ) .In the second step, we find the optimal ε h ∈ (0 , ε D ) thatminimizes P th ( ε h ) . Since there is no closed-form expressionof P th ( ε h ) , exhaustive searching is needed to obtain the The distribution of channel gain f g ( g ) depends on the number of antennas N t . Therefore, the optimal solution of problem (23) will depend on N t . Wewill illustrate the impact of N t via numerical results in the next section. ptimal ε h in general. However, numerical results indicatethat P th ( ε h ) first decreases and then increases with ε h . Withthis property, we can find the optimal solution of ε h andthe required transmit power to ensure ε D via the exact linearsearch method [42].As proved in Appendix D, the solution obtained from thetwo-step method is the global optimal solution of problem (23)if the solutions of both steps are global optimal. Impact of traffic feature: To show the impact of burstinesson the cross-layer optimization, we consider IPP with fixedaverage packet rate in two asymptotic cases, i.e. C → and C → ∞ , where C is the variance coefficient that can beused to characterize burstiness [29]. To show the impact ofburstiness, we keep the average packet rate of IPP, αα + β λ on ,as a constant. Then, the average packet rate can be expressedas λ on δ , and C = 1 + δλ on (1+ δ ) α [29], where δ = β/α .When α → ∞ , C → , the effective bandwidth of theIPP can be expressed as E B ( θ ) = λ on T f θ (1+ δ ) (cid:0) e θ − (cid:1) , whichis the same as the effective bandwidth of a Poisson processwith average packet rate λ on δ . When α → , C → ∞ , theeffective bandwidth of the IPP can be expressed as E B ( θ ) = λ on T f θ (cid:0) e θ − (cid:1) , which is the same as the effective bandwidth ofa Poisson process with average packet rate λ on .To show the impact of autocorrelation, we consider aSPP with 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 ofthe effective bandwidth of it can be obtained by substituting λ = λ on into (A.1). Therefore, the effective bandwidth of SPPis less than that of a Poisson process with average packet rate max { λ I , λ II } . Remark 2: For IPP, when C increases from to ∞ ,the effective bandwidth (i.e. the required constant servicerate) increases δ times. For SPP, the required constantservice rate does not exceed the upper bound, which equalsto the effective bandwidth of a Poisson process with averagepacket rate max { λ I , λ II } . This indicates that the service raterequirement is still finite for IPP with C → ∞ or for SPPwith any values of α I and α II . Therefore, the burstiness andautocorrelation will not change the proposed framework. 2) Multi-user Scenario: In this case, we jointly optimize N c k , ε ck , ε qk , and ε hk , with which we can obtain the optimalcross-layer strategy including bandwidth allocation, powerallocation and packet dropping policies. The optimizationproblem in the multi-user scenario is formulated as min N c k ,ε qk ,ε ck ,ε hk k =1 , ,...,K P tot (cid:44) K (cid:88) k =1 P th k (25)s.t. ε hk = (cid:90) N BN c kγkµkP th k − ln (cid:16) µ k P th k gN BN c k (cid:17) ln (1 + γ k ) f g ( g ) dg, (25a) ln (1 + γ k ) = T f u ln 2 φBN c k E Bk ( θ k ) + (cid:115) V k φBN c k f − ( ε ck ) , (25b) ε ck + ε qk + ε hk ≤ ε D and ε ck , ε qk , ε hk ∈ R + , (25c) K (cid:88) k N c k ≤ N cmax , N c k ∈ Z + , k = 1 , ..., K, (25d)where N cmax is the maximal number of subcarriers for DLtransmission. Since N c k is integer, this is a mixed-integerprogramming problem.Given the values of N c k , k = 1 , ..., K , the problem canbe decomposed into K single-user problems similar to (23),which can be solved by the two-step method. Then, the powerallocation policy among subsequent TTIs and the packet drop-ping policy can be obtained similarly to those in the single-user scenario, i.e., (18) and (20). We refer to the K single-user problems as subproblem I . Since binary search and exactlinear search methods are applied in solving subproblem I, thecomplexity of the two-step method is O (log ( ε D ∆ h ) log ( ε D ∆ c )) . The complexity of problem (25) is determined by theinteger programming that optimizes N c k , k = 1 , ..., K withgiven ε ck , ε qk , ε hk to minimize the objective function in (25).We refer this integer programming as subproblem II . Since N c k ≥ , the remaining number of subcarriers is N cmax − K .To solve problem (25), we need to allocate the remainingsubcarriers to K users. Thus, subproblem II includes around K N cmax − K feasible solutions. To reduce complexity, a heuristicalgorithm is proposed, as listed in Table II. The basic idea issimilar to the steepest descent method [42]. The subcarrierallocation algorithm includes N cmax − K steps. In each step,one subcarrier is allocated to one of the K users that leadsto the steepest total transmit power descent. The proposedalgorithm only needs to solve subproblem I for K ( N cmax − K ) times, and hence the complexity is O ( K ( N cmax − K )) . Fur-ther considering the complexity of the two-step method forsolving subproblem I, the overall complexity of the proposedalgorithm is O (cid:0) K ( N cmax − K ) log ( ε D ∆ h ) log ( ε D ∆ c ) (cid:1) .V. A PPLYING THE F RAMEWORK TO F REQUENCY - SELECTIVE C HANNEL If the number of users is not very large, the bandwidthallocated to a user (say W k = BN c k in problem (25)) could belarger than the coherence bandwidth. In this section, we showhow to apply the framework to frequency-selective channel.We divide the bandwidth allocated to the k th user into N sc k subchannels, where each subchannel consists of multiple sub-carriers. The bandwidth of each subchannel is W c that is lessthan the coherence bandwidth. Then, the subcarriers withineach subchannel subject to flat fading, while the subchannelssubject to frequency-selective fading. To study the delay andreliability performance, we first need to find the achievablerate with finite blocklength. As shown in Appendix E, the By solving problem (25), the bandwidth (i.e., the number of subcarriers)allocation is obtained. With constraint (25d), the total number of subcarriersallocated to all the K users is less than the maximal number of subcarriersof the system. Therefore, we can always find a subcarrier allocation policy,with which each subcarrier is only allocated to one user. The complexity of a searching algorithm depends on the stopping criterion.Here, the iterations stop if | ε hk ( i ) − ε hk ( i + 1) | < ∆ h or | ε ck ( i ) − ε ck ( i +1) | < ∆ c is satisfied, where ε hk ( i ) and ε ck ( i ) are the results obtained after i iterations.ABLE IIS UBCARRIER A LLOCATION A LGORITHM Input: Number of users K , total number of subcarrers N cmax ,duration for data transmission in each DL frame φ , packet size u , noise spectral density N , number of transmit antennas N t ,average channel gains of users µ k , k = 1 , ..., K . Output: Subcarrier allocation N c ∗ k , k = 1 , ..., K . Set N c k (0) := 1 , k = 1 , ..., K . Set l := 1 . Solve subproblem I with N c k (0) = 1 , and obtain the total transmitpower P tot (0) . while l ≤ N cmax − K do Set ˆ k := 1 while ˆ k ≤ K do N cˆ k ( l ) := N cˆ k ( l − 1) + 1 ; N c k ( l ) := N c k ( l − , k (cid:54) = ˆ k . Solve subproblem I with N c k ( l ) , and obtain ˆ P totˆ k ( l ) . ˆ k := ˆ k + 1 . end while k ∗ := arg min ˆ k ˆ P totˆ k ( l ) . N c k ∗ ( l ) := N c k ∗ ( l − 1) + 1 ; N c k ( l ) := N c k ( l − , k (cid:54) = k ∗ . l := l + 1 . end while return N c ∗ k = N c k ( l − , k = 1 , ..., K . number of packet that can be transmitted in one frame can beobtained as, s fs k ≈ φW c u ln 2 N sc k (cid:88) j =1 ln (cid:20) µ k P kj ( n ) g kj N W c (cid:21) − (cid:115) V k φW c f − ( ε ck ) (packets) , (26)where P kj ( n ) is the transmit power allocated to the j thsubchannel of the k th user in the n th frame, g kj is theinstantaneous channel gain on the j th subchannel of the k thuser, and V k = N sc k − N sc k (cid:80) j =1 1 (cid:104) µkPkj ( n ) gkjN W c (cid:105) . Since the channelgains could be arbitrarily close to zero, the required transmitpower to guarantee queueing delay is also unbounded.The packet rate in (26) can be achieved if all the packets ina frame are coded in one block with length W c N sc k φ (calledthe optimal coding scheme), as illustrated in Fig. 4(a). Bysubstituting (26) into (12), we cannot obtain the required SNRto ensure ( D q max , ε qk ) and ε ck as that in (17). This is becauseeach channel coding block consists of packets transmittedover multiple subchannels with different instantaneous channelgains. As a result, it is very challenging to derive and optimizethe proactive packet dropping probability that guarantees theQoS.To overcome this difficulty, we consider a suboptimal cod-ing scheme that the packets to be transmitted on differentsubchannels are coded independently. As illustrated in Fig.4(b), the blocklength of the suboptimal coding scheme is W c φ .With shorter blocklength, the suboptimal coding scheme cansupport lower packet rate for a given ε ck , thus the requiredresources with the suboptimal channel coding scheme arehigher than that with the optimal scheme in order to achieve the same QoS [43]. Nonetheless, with the optimal scheme,if a block is not decoded without error, then all the packetstransmitted in one frame will be lost. By contrast, with thesuboptimal scheme, if the packets in one block is not decodedsuccessfully, the packets in other blocks can still be decodedcorrectly. This suggests that the packet transmission errorswith the suboptimal scheme is less busty than those with theoptimal scheme. Frequency Time c W ( a ) Optimal scheme4 packets Frequency Time ( b ) Suboptimal scheme2 packets2 packets1 block with length c W c W c W c W Fig. 4. Illustration of two channel coding schemes, where four packets needto be transmitted in a frame and W k = 2 W c . When the number of packets transmitted over each sub-channel is E Bk ( θ k ) /N sc k , the constraints on proactive packetdropping probability, queueing delay violation probability andtransmission error probability can be obtained by replac-ing BN c k and E Bk ( θ k ) in (25a) and (25b) with W c and E Bk ( θ k ) /N sc k , respectively. In this way, the proposed frame-work can be applied over frequency-selective channel.In what follows, we analyze the rate loss. With the subop-timal scheme, the number of packets that can be transmittedover the N sc k subchannels can be expressed as follows, ˜ s fs k ≈ φW c u ln 2 N sc k (cid:88) j =1 ln (cid:20) µ k P kj ( n ) g kj N W c (cid:21) − (cid:115) ˜ V fs kj φW c f − ( ε ck ) (packets) , (27)where the number of packets transmitted in each subchannelis obtained by replacing bandwith BN c k in (3) with W c , andhence ˜ V fs kj = 1 − (cid:104) µkPkj ( n ) gkjN W c (cid:105) . From (26) and (27), wecan derive the gap between s fs k and ˜ s fs k as, s fs k − ˜ s fs k ≈ √ φW c u ln 2 N sc k (cid:88) j =1 (cid:113) ˜ V fs kj − (cid:112) V k f − ( ε ck ) , which shows that s fs k − ˜ s fs k ∼ O ( N sc k − (cid:112) N sc k ) , and thusthe gap between s fs k and ˜ s fs k increases with N sc k . From (27),we have ˜ s fs k ∼ O ( N sc k ) , hence ( s fs k − ˜ s fs k ) / ˜ s fs k ∼ O (1) . Thismeans that the normalized rate loss ( s fs k − ˜ s fs k ) / ˜ s fs k approachesto a constant when N sc k is large. Some applications like safe messages transmission in vehicle networksmay prefer such suboptimal scheme, which is also applicable for flat fadingchannels. Here y ( N sc k ) ∼ O (cid:0) x ( N sc k ) (cid:1) means y ( N sc k ) /x ( N sc k ) approaches to aconstant when N sc k is large. I. S IMULATION AND N UMERICAL R ESULTS In this section, we first validate that the effective bandwidthcan be used as a tool to optimize resource allocation inshort delay regime for Poisson process, IPP and SPP. Then,we show the optimal values of ε qk , ε ck and ε hk , and therequired maximal transmit power for both Poisson process andIPP. Next, we compare the required transmit power of theproposed algorithm with the global optimal policy obtained byexhaustive searching.A single-BS scenario is considered in the sequel. The usersare uniformly distributed with distances from the BS as m ∼ m. The arrival process of each user is modeled asPoisson process, IPP, or SPP with average rate packets/s,i.e., each user requests the safety messages from nearbysensors, and each sensor uploads packets to the BS withaverage rate packets/s [37]. Other parameters are listed inTable III, unless otherwise specified. TABLE IIIP ARAMETERS [6, 37] Overall reliability requirement ε D − . E2E delay requirement D max msQueueing delay requirement D q max . msDuration of each frame (equals to TTI) . msDuration of data transmission in oneframe φ . msSingle-sided noise spectral density N − dBm/HzPacket size u bytesPath loss model 10 lg( µ k ) 35 . . d k ) Average duration of “OFF” or “ON”state α − or β − s (i.e. frames) The CCDFs of queue length and queueing delay for thepackets to the k th user are shown in Fig. 5, where (15) isused to translate the CCDF of the queueing delay into theCCDF of queue length. To obtain the upper bound in (10), Pr { D k ( ∞ ) > D th } ≤ exp {− θ k E Bk ( θ k ) D th } is computed bychanging D th from to D q max . The CCDFs of queueing delayare obtained via Monte Carlo simulation by generating arrivalprocess and service process during frames. Numericalresults in Fig. 5(a) indicate that for Poisson process, theupper bound derived by effective bandwidth works when themaximal queue length is short. Simulation results in Fig. 5(b)show that the upper bound also works for IPP and SPP. In fact,it has been observed in [44] that effective bandwidth can beused for resource allocation under statistical queueing delayrequirement when D q max is small, if the TTI is much shorterthan the delay bound.The optimal solution of problem (23) and the requiredmaximal transmit power for both Poisson and IPP are shownin Fig. 6. The results in Fig. 6(a) show that ε ck , ε qk and ε hk are in the same order of magnitude with different valuesof N t . In fact, similar to ε hk , when either ε ck or ε qk is set The optimal values of ε qk , ε ck and ε hk and the required transmit power forSPP are similar to that for IPP, and hence the results for SPP are omitted forconciseness. −8 −6 −4 −2 Queue length L (packets) CCD F o f queue l eng t h P r { Q k ( ∞ ) > L } Upper bound, Q max = 10M/D/1, Q max = 10Upper bound, Q max = 5M/D/1, Q max = 5Upper bound, Q max = 3M/D/1, Q max = 3Q max ε kq (a) Poisson arrivals, where ε qk = 10 − . −8 −6 −4 −2 Queueing delay D th (ms) P r { D k ( ∞ ) > D t h } exp[− θ k E Bk ( θ k )D th ]Poisson, λ = 0.1 pakcet/frameIPP λ on =2 λ , α −1 = β −1 =10 framesSPP (0.5 λ ,1.5 λ , α , β )D qmax ε qk (b) Poisson arrival, IPP and SPP, where C = 1001 for the IPP.Fig. 5. Validating the upper bound in (10). as zero, the required transmit power will become infinite,because E Bk ( θ k ) → ∞ when ε qk = 0 (as can be clearlyseem from (A.2)) and f − ( x ) → ∞ (and hence s k ( n ) in(3) approaches infinity) when ε ck = 0 . This implies that theoptimal probabilities will also be in the same order when othersystem parameters change. On the other hand, Fig. 6(b) showsthat compared with ε ck = ε qk = ε hk , the required maximaltransmit power only reduces ∼ % with the optimized ε ck , ε qk and ε hk when N t ≥ . This implies that dividing the requiredpacket loss probability equally to the three probabilities willcause minor performance loss.Moreover, the optimal queueing delay violation probabilityfor IPP is higher than that for Poisson process. This indicatesthat bursty arrival processes lead to higher queueing delayviolation probability. Furthermore, P th decreases extremelyfast as N t increases. This agrees with the intuition: increasingthe number of transmit antennas is an efficient way to reducethe required maximal transmit power thanks to the spatialdiversity. −8 Number of transmit antennas N t P r obab ili t y PoissonIPP ε h ε q ε c (a) Optimal values of ε c , ε q and ε h that minimize the required transmitpower. −2 Number of transmit antennas N t R equ i r ed m a x i m a l t r an s m i t po w e r P m a x ( W ) IPP, ε c = ε q = ε h = ε D /3IPP, optimized ε c , ε q , ε h Poisson, ε c = ε q = ε h = ε D /3Poisson, optimized ε c , ε q , ε h 10% power savingwith optimization2~5% power savingwith optimization25% powersaving with optimization 40W(46dBm) (b) Required maximal transmit power.Fig. 6. Single-user scenario, where user-BS distance is m, N c = 4 , B = 0 . MHz, and α = β . TABLE IVR EQUIRED T RANSMIT P OWER , N cmax = 16 , B = 0 . MH Z , AND N t = 8 Number of users K . W . W . WExhaustive Searching . W . W . W The required K (cid:80) k =1 P th k obtained by the proposed algorithmand the global optimal solution with exhaustive searching areprovided in Table IV. The results illustrate that the proposedalgorithm is near-optimal. Because the complexity of exhaus-tive search method is extremely high when N cmax and K arelarge, we only provide results with small values of N cmax and K .The number of dropped packets is determined by thedistribution of channel gain, which depends on the propagationenvironments and N t as well. In Fig. 7, we provide the numberof dropped packets over Nakagami- m fading channel withdifferent values of m and N t . We consider the worst case that all the users are located at the edge of the cell (i.e., user-BS distance is m). Since the average channel gains of allthe users are the same, the total bandwidth and transmit powerare equally allocated to all the users. Then, N c k = N cmax /K and P th k = P max /K . We set ε c = ε q = ε D / . ε hk is calculatedfrom (22), where f g ( g ) = ( mg ) m − ( m − exp ( − mg ) when N t = 1 and m > [40] and f g ( g ) = N t − g N t − e − g when N t > and m = 1 [39]. All the results in the figure are obtainedunder constraint ε hk ≤ ε D / . The results when N t = 1 and m = 1 are not shown, because constraint ε hk ≤ ε D / cannotbe satisfied under the transmit power constraint. The numberof dropped packets in transmitting packets with proactivepacket dropping policy is ε hk . To show the performancegain of proactive packet dropping, we also provide the resultsfor an intuitive packet dropping policy, which simply dropsall the packets to the k th user when g k < N BN c k γ k µ k P th k . We cansee that proactive packet dropping policy can help reduce thenumber of dropped packets. N u m be r o f d r opped pa ck e t s i n t r an s m i ss i on s Intuitive, m=3, Nt=1Proactive, m=3, Nt=1Intuitive, m=5, Nt=1Proactive, m=5, Nt=1Intuitive, m=1, Nt=4Proactive, m=1, Nt=4 Fig. 7. Number of dropped packets over Nakagami- m fading channel, where N cmax = 1024 and P max = 46 dBm. VII. C ONCLUSIONS In this paper, we studied how to optimize resource allocationto guarantee ultra-low latency and ultra-high reliability forradio access networks in typical application scenarios wherethe required delay is shorter than channel coherence time.Both queuing delay and transmission delay were consideredin the latency, and the transmission error probability, queueingdelay violation probability, and packet dropping probabilitywere taken into account in the reliability. We first showed thatthe required transmit power to ensure the QoS is unboundedwhen queueing delay bound is shorter than channel coherencetime. To satisfy the QoS requirement with finite transmitpower, a proactive packet dropping mechanism was proposed.A framework for optimizing resource allocation to ensurethe stringent QoS was established, where a queue state andchannel state information dependent transmit power allocationand packet dropping policies were optimized for single usercase, and bandwidth allocation was further optimized formulti-user scenario, to minimize the required maximal transmitpower of the BS. How to apply the proposed framework torequency-selective channel was also addressed. Simulationresults validated that effective bandwidth can be used tooptimize resource allocation for Poisson process, IPP andSPP, which are representative traffic models to characterizingperformance of a system with queueing. Numerical resultsshowed that the transmission error probability, queueing delayviolation probability, and packet dropping probability are inthe same order of magnitude, and setting the three packet lossprobabilities equal will cause minor power loss.A PPENDIX AE FFECTIVE BANDWIDTH OF S EVERAL R ELEVANT A RRIVAL P ROCESSES Poisson arrival process: The effective bandwidth of Pois-son process is given by E Bk ( θ k ) = λ k T f θ k (cid:0) e θ k − (cid:1) (packets/s) . (A.1)Substituting (A.1) into (11), we can obtain the required QoSexponent θ k = ln (cid:104) T f ln(1 /ε qk ) λ k D q max + 1 (cid:105) . Then, (A.1) can be re-expressed as a function of ( D q max , ε qk ) as E Bk ( θ k ) = ln(1 /ε qk ) D q max ln (cid:104) T f ln(1 /ε qk ) λ k D q max + 1 (cid:105) (packets/s) . (A.2) IPP: The effective bandwidth of the IPP can be expressedas [45] E Bk ( θ k ) = Ω2 θ k T f (packet/s) , (A.3)where Ω (cid:44) [ (cid:0) e θ k − (cid:1) λ on k − ( α + β )] + (cid:113) [( e θ k − λ on k − ( α + β )] + 4 α ( e θ k − λ on k .Substituting (A.3) into (11), the QoS exponent θ k canbe obtained from Ω = − T f ln ε qk D q max numerically. SPP: Deriving the effective bandwidth of autocorrelatedprocesses is much harder than that of renewal processes. Toovercome this difficulty, we provide an upper bound of theeffective bandwidth of SPP. Without loss of generality, weassume λ I k ≤ λ II k .Consider a Poisson process with average arrival rate λ II k ,the arrival rate in the first state of SPP is less than that of thePoisson process. Thus, the effective bandwidth of the SPP isless than that of the Poisson process, which can be obtainedby substituting λ k = λ II k into (A.1).A PPENDIX BU PPER BOUND OF THE PACKET DROPPING PROBABILITY Proof. To derive ε hk , we introduce an upper bound of b dk ( n ) as follows, b Uk ( n ) = (cid:26) max (cid:0) T f E Bk ( θ k ) − s th k , (cid:1) , if Q k ( n ) > , , if Q k ( n ) = 0 , considering that b Uk ( n ) = b dk ( n ) when Q k ( n ) ≥ T f E Bk ( θ ) or Q k ( n ) = 0 , and b Uk ( n ) > b dk ( n ) when < Q k ( n ) < T f E Bk ( θ k ) . Then, we can derive an upper bound of E [ b dk ( n )] as E [ b Uk ( n )] = η k (cid:90) N BN c kγkµkP th k ( T f E Bk ( θ k ) − s th k ) f g ( g ) dg. Substituting E [ b Uk ( n )] into (21), we obtain an upper bound ofthe packet dropping probability as ε hk ≤ (cid:90) N BN c kγkµkP th k (cid:20) − s th k T f E Bk ( θ k ) (cid:21) f g ( g ) dg, (B.1)where η k = Pr { Q k ( n ) > } = E { (cid:80) i ∈A k a i ( n ) } / E [ s k ( n )] = E { (cid:80) i ∈A k a i ( n ) } / [ T f E Bk ( θ k )] is applied.By substituting s th k in (19) and considering (17), we have s th k T f E Bk ( θ k ) ≈ ln (cid:16) µ k P th k g k N BN c k (cid:17) − (cid:113) V k φBN c k f − ( ε ck )ln (1 + γ k ) − (cid:113) V k φBN c k f − ( ε ck ) . (B.2)Because a packet is dropped only if it will be transmitted indeep fading, i.e. g k → , V k in (4) approaches , and then(B.2) can be further accurately approximated by s th k T f E Bk ( θ k ) ≈ ln (cid:16) µ k P th k g k N BN c k (cid:17) ln (1 + γ k ) . (B.3)Substituting (B.3) into (B.1), we obtain the approximationin (22). A PPENDIX CP ROOF OF THE CONVEXITY OF THE OBJECTIVE FUNCTIONIN (24) Proof. For the Q-function f Q ( x ) = √ π (cid:82) ∞ x exp (cid:16) − τ (cid:17) dτ ,we have f (cid:48) Q ( x ) ∆ = − √ π e − x / < , and f (cid:48)(cid:48) Q ( x ) = x √ π e − x / > when x > . Thus, f Q ( x ) is an decreasingand strictly convex function when x > , i.e. f Q ( x ) < . .Since the inverse function of a decreasing and strictly convexfunction is also strictly convex [42], f − ( ε c ) is strictly convexwhen ε c < . (which is true for any application). Hence, thesecond term of (24) is strictly convex.To prove that the first term of (24) is strictly convex, we firstderive its second order derivative. Denote y = − ln ( ε q ) and z = T f D q max λ > . After removing the non-relevant constants,the first term of (24) can be expressed as f ( y ) = y ln(1+ zy ) ,and its second order derivative is derived as d fd ( ε q ) = (cid:18) d fdy (cid:19) (cid:18) dydε q (cid:19) + (cid:18) dfdy (cid:19) (cid:32) d yd ( ε q ) (cid:33) . (C.1)After some regular derivations, we can obtain that dydε q = − ε q , d yd ( ε q ) = (cid:18) ε q (cid:19) , (C.2) dfdy = (1 + zy ) ln (1 + zy ) − zy [ln (1 + zy )] (1 + zy ) , (C.3) fdy = 2 z y − (cid:0) z + z y (cid:1) ln (1 + zy )[ln (1 + zy )] (1 + zy ) . (C.4)After substituting (C.2), (C.3) and (C.4) into (C.1), we canfinally obtain that d fd ( ε q ) = (cid:110) (1 + zy ) [ln (1 + zy )] − (cid:0) z + zy + z y + z y (cid:1) ln (1 + zy ) + 2 z y (cid:111) × (cid:110) [ln (1 + zy )] (1 + zy ) ( ε q ) (cid:111) − . (C.5)Since the denominator is positive, we only need to showthe numerator is positive. Denote the numerator of (C.5) as f mun ( x, z ) , where x = yz . Then, we have f mun ( x, z ) =(1 + x ) [ln(1 + x )] − ( x + x ) ln(1 + x ) − [(2 + x ) ln(1 + x ) − x ] z. (C.6)For ε q < − , which is true for applications with ultra-highreliability requirement, y > − ln (cid:0) − (cid:1) > , and then x > z . Moreover, (2 + x ) ln(1 + x ) − x > , ∀ x > . Then, wecan obtain a lower bound of f mun ( x, z ) as follows, f LB ( x ) =(1 + x ) [ln(1 + x )] − ( x + x ) ln(1 + x ) − [(2 + x ) ln(1 + x ) − x ] x/ . (C.7)When x = 0 , f LB ( x ) = 0 . To prove f LB ( x ) > , ∀ x > , wesubstitute ν = x +1 into (C.7) and prove f (cid:48) LB ( ν ) > , ∀ ν > .It is not hard to derive that f (cid:48) LB ( ν ) = 20 ν (ln ν ) + (10 ν − ν ) ln ν + (3 ν − ν − ν . (C.8)Denote the numerator of (C.8) as f LBnum ( ν ) , which equalszero when ν = 0 . Besides, f (cid:48) LBnum ( ν ) = 40 ν (ln ν ) + (10 + 36 ν ) ln ν + 4( ν − > , ∀ ν > . As a result, f (cid:48) LB ( ν ) > , and hence f LB ( x ) increases with x .Therefore, we have f LB ( x ) > , ∀ x > . This completes theproof. A PPENDIX DP ROOF 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 c , ˜ ε h ) and ˜ P max ,respectively. Given ˜ ε h , we can obtain the global minimaltransmit power P max (˜ ε h ) ≤ ˜ P max by solving problem (24),which is for Poisson arrival process. In the second step, theglobal optimal ε h ∗ is obtained such that P max ∗ ≤ P max (˜ ε h ) .Therefore, P max ∗ ≤ ˜ P max .A PPENDIX EA CHIEVABLE RATE OVER FREQUENCY - SELECTIVECHANNEL Denote the channel vector on the j th subchannel ofthe k th user as h kj ∈ C N t × . Then, the channel matrix over frequency-selective channel is equivalent toa N t N sc k × N sc k MIMO channel with bandwidth W c ,i.e., H k = diag (cid:0) h k , h k , ..., h kN sc k (cid:1) and H Hk H k = diag (cid:0) g k , g k , ..., g kN sc k (cid:1) , where g kj = h Hkj h kj is the channelgain on the j th subchannel allocated to the k th user and alsoone of the eigenvalues of H Hk H k . Then, by substituting theeigenvalues into (96) and (97) in [27], the number of packetsthat can be transmitted in one frame can be expressed as (26).R EFERENCES[1] C. She, C. Yang, and T. Quek, “Cross-layer transmission design fortactile internet,” in Proc. IEEE Globecom , 2016.[2] 3GPP, Study on Scenarios and Requirements for Next Generation AccessTechnologies . Technical Specification Group Radio Access Network,Technical Report 38.913, Release 14, Oct. 2016.[3] G. P. Fettweis, “The tactile internet: Applications & challenges,” IEEEVehic. Tech. Mag. , vol. 9, no. 1, pp. 64–70, Mar. 2014.[4] P. Popovski, et al. , “Deliverable d6.3 intermediate system evaluationresults.” ICT-317669-METIS/D6.3, 2014.[5] 3GPP, Further Advancements for E-UTRA Physical Layer Aspects .Technical Specification Group Radio Access Network, Technical Report36.814, Release 9, Mar. 2010.[6] A. Osseiran, F. Boccardi and V. Braun, et al. , “Scenarios for 5G mobileand wireless communications: The vision of the METIS project,” IEEECommun. Mag , vol. 52, no. 5, pp. 26–35, May. 2014.[7] S.-Y. Lien, S.-C. Hung, K.-C. Chen, and Y.-C. Liang, “Ultra-low-latencyubiquitous connections in heterogeneous cloud radio access networks,” IEEE Wireless Commun. , vol. 22, no. 3, pp. 22–31, Jun. 2015.[8] F. Capozzi, G. Piro, L. Grieco, G. Boggia, and P. Camarda, “Downlinkpacket scheduling in LTE cellular networks: Key design issues and asurvey,” IEEE Commun. Surveys Tuts. , vol. 15, no. 2, pp. 678–700,2013.[9] M. Simsek, A. Aijaz, M. Dohler, J. Sachs, and G. Fettweis, “5G-enabledtactile internet,” IEEE J. Select. Areas Commun. , vol. 34, no. 3, pp. 460–473, Mar. 2016.[10] S. A. Ashraf, F. Lindqvist, R. Baldemair, and B. Lindoff, “Control chan-nel design trade-offs for ultra-reliable and low-latency communicationsystem,” in IEEE Globecom Workshops , 2015.[11] P. Kela and J. Turkka, et al. , “A novel radio frame structure for 5G denseoutdoor radio access networks,” in Proc. IEEE VTC Spring , 2015.[12] Y. Polyanskiy, H. V. Poor, and S. Verd´u, “Channel coding rate in thefinite blocklength regime,” IEEE Trans. Inf. Theory , vol. 56, no. 5, pp.2307–2359, May 2010.[13] K. Niu, K. Chen, J. Lin, and Q. T. Zhang, “Polar codes: Primary conceptsand practical decoding algorithms,” IEEE Commun. Mag , vol. 52, no. 7,pp. 192–203, Jul. 2014.[14] D. Ohmann, M. Simsek, and G. P. Fettweis, “Achieving high availabilityin wireless networks by an optimal number of Rayleigh-fading links,”in IEEE Globecom Workshops , 2014.[15] F. Kirsten, D. Ohmann, M. Simsek, and G. P. Fettweis, “On the utilityof macro- and microdiversity for achieving high availability in wirelessnetworks,” in Proc. IEEE PIMRC , 2015.[16] G. Pocovi, B. Soret, M. Lauridsen, K. I. Pedersen, and P. Mogensen,“Signal quality outage analysis for ultra-reliable communications incellular networks,” in IEEE Globecom Workshops , 2015.[17] O. N. C. Yilmaz, Y.-P. E. Wang, N. A. Johansson, N. Brahmi, S. A.Ashraf, and J. Sachs, “Analysis of ultra-reliable and low-latency 5Gcommunication for a factory automation use case,” in IEEE ICCWorkshops , 2015.[18] N. A. Johansson, Y.-P. E. Wang, E. Eriksson, and M. Hessler, “Radioaccess for ultra-reliable and low-latency 5G communications,” in IEEEICC Workshops , 2015.[19] C. Chang and J. A. Thomas, “Effective bandwidth in high-speed digitalnetworks,” IEEE J. Sel. Areas Commun. , vol. 13, no. 6, pp. 1091–1100,Aug. 1995.[20] D. Wu and R. Negi, “Effective capacity: A wireless link model forsupport of quality of service,” IEEE Trans. Wireless Commun. , vol. 2,no. 4, pp. 630–643, Jul. 2003.[21] B. Soret, P. Mogensen, K. I. Pedersen, and M. C. Aguayo-Torres,“Fundamental tradeoffs among reliability, latency and throughput incellular networks,” in IEEE Globecom Workshops , Dec. 2014.22] A. Aijaz, “Towards 5G-enabled tactile internet: Radio resource alloca-tion for haptic communications,” in Proc. IEEE WCNC , 2016.[23] S. Schiessl, J. Gross, and H. Al-Zubaidy, “Delay analysis for wirelessfading channels with finite blocklength channel coding,” in Proc. ACMMSWiM , 2015.[24] M. C. Gursoy, “Throughput analysis of buffer-constrained wirelesssystems in the finite blocklength regime,” in Proc. IEEE ICC , 2011.[25] S. Xu, T.-H. Chang, S.-C. Lin, C. Shen, and G. Zhu, “On the convexityof energy-efficient packet scheduling problem with finite blocklengthcodes,” in IEEE Globecom Workshops , 2015.[26] R. A. Berry, “Optimal power-delay tradeoffs in fading channels—small-delay asymptotics,” IEEE Trans. Inf. Theory , vol. 59, no. 6, pp. 3939–3952, Jun. 2013.[27] W. Yang, G. Durisi, T. Koch, and Y. Polyanskiy, “Quasi-static multiple-antenna fading channels at finite blocklength,” IEEE Trans. Inf. Theory ,vol. 60, no. 7, pp. 4232–4264, Jul. 2014.[28] W. Whitt, “Tail probabilities with statistical multiplexing and effectivebandwidths in multi-class queues,” Telecommunication Systems , vol. 2,no. 1, pp. 71–107, 1993.[29] J. Wu, Y. Bao, G. Miao, S. Zhou, and Z. Niu, “Base station sleepingcontrol and power matching for energy-delay tradeoffs with burstytraffic,” IEEE Trans. Veh. Technol. , vol. 65, no. 5, pp. 3657–3675, May2016.[30] G. Zhang, T. Q. S. Quek, M. Kountouris, A. Huang, and H. Shan,“Fundamentals of heterogeneous backhaul design—analysis and opti-mization,” IEEE Trans. Commun. , vol. 64, no. 2, pp. 876–889, Feb.2016.[31] D. Feng, L. Lu, Y. Yuan-Wu, G. Y. Li, S. Li, and G. Feng, “Device-to-device communications in cellular networks,” IEEE Commun. Mag. ,vol. 52, no. 4, pp. 49–55, Apr. 2014.[32] C. She, C. Yang, and T. Q. S. Quek, “Uplink transmission design withmassive machine type devices in tactile internet,” in IEEE GlobecomWorkshops , 2016.[33] J. Tang and X. Zhang, “Quality-of-service driven power and rateadaptation over wireless links,” IEEE Trans. Wireless Commun. , vol. 6,no. 8, pp. 3058–3068, Aug. 2007.[34] M. Khabazian, S. Aissa, and M. Mehmet-Ali, “Performance modeling ofsafety messages broadcast in vehicular ad hoc networks,” IEEE Trans.Intell. Transp. Syst. , vol. 14, no. 1, pp. 380–387, Mar. 2013.[35] G. R1-120056, “Analysis on traffic model and characteristics for MTCand text proposal.” Technical Report, TSG-RAN Meeting WG1 Fundamentals of Queueing Theory . Wiley,1985.[37] H. A. Omar, W. Zhuang, A. Abdrabou, and L. Li, “Performance evalu-ation of VeMAC supporting safety applications in vehicular networks,” IEEE Trans. Emerg. Topics Comput. , vol. 1, no. 1, pp. 69–83, Aug.2013.[38] G. L. Choudhury, D. M. Lucantoni, and W. Whitt, “Squeezing the mostout of ATM,” IEEE Trans. Commun. , vol. 44, no. 2, pp. 203–217, Feb.1996.[39] I. E. Telatar, Capacity of multi-antenna Gaussian channels , 1995.[40] A. Goldsmith, Wireless Communications . Cambridge University Press,2005.[41] I.-H. Hou, V. Borkar, and P. R. Kumar, “A theory of QoS for wireless,”in Proc. IEEE INFOCOM , 2009.[42] S. Boyd and L. Vandanberghe, Convex Optimization . Cambridge Univ.Press, 2004.[43] G. Durisi, T. Koch, and P. Popovski, “Toward massive, ultrareliable, andlow-latency wireless communication with short packets,” Proc. IEEE ,vol. 104, no. 9, pp. 1711–1726, Aug. 2016.[44] B. Soret, M. C. Aguayo-Torres, and J. T. Entrambasaguas, “Capacitywith explicit delay guarantees for generic sources over correlatedRayleigh channel,” IEEE Trans. Wireless Commun. , vol. 9, no. 6, pp.1901–1911, Jun. 2010.[45] M. Ozmen and M. C. Gursoy, “Wireless throughput and energy effi-ciency with random arrivals and statistical queuing constraints,”