Effective Capacity for Renewal Service Processes with Applications to HARQ Systems
Zheng Shi, Theodoros Tsiftsis, Weiqiang Tan, Guanghua Yang, Shaodan Ma, Mohamed-Slim Alouini
aa r X i v : . [ c s . I T ] D ec Effective Capacity for Renewal Service Processeswith Applications to HARQ Systems
Zheng Shi, Theodoros Tsiftsis, Weiqiang Tan, Guanghua Yang, Shaodan Ma, and Mohamed-Slim Alouini
Abstract —Considering the widespread use of effective capacityin cross-layer design and the extensive existence of renewal ser-vice processes in communication networks, this paper thoroughlyinvestigates the effective capacity for renewal processes. Basedon Z-transform, we derive exact analytical expressions for theeffective capacity at a given quality of service (QoS) exponent forboth the renewal processes with constant reward and with vari-able rewards. Unlike prior literature that the effective capacity isapproximated with no many insightful discussions, our expressionis simple and reveals further meaningful results, such as themonotonicity and bounds of effective capacity. The analyticalresults are then applied to evaluate the cross-layer throughputfor diverse hybrid automatic repeat request (HARQ) systems,including fixed-rate HARQ (FR-HARQ, e.g., Type I HARQ,HARQ with chase combining (HARQ-CC) and HARQ withincremental redundancy (HARQ-IR)), variable-rate HARQ (VR-HARQ) and cross-packet HARQ (XP-HARQ). Numerical resultscorroborate the analytical ones and prove the superiority of ourproposed approach. Furthermore, targeting at maximizing theeffective capacity via the optimal rate selection, it is revealed thatVR-HARQ and XP-HARQ attain almost the same performance,and both of them perform better than FR-HARQ.
Index Terms —Effective capacity, hybrid automatic repeat re-quest, QoS exponent, renewal process.
I. I
NTRODUCTION G has been envisioned to offer extremely high data rate (onthe order of Gbps), ultra-reliability (higher than 99.999%)and very low latency (sub-1ms) [1]. To fulfill these demandingrequirements, plenty of existing works prefer to evaluateand devise communication systems from physical-layer per-spective. However, the constraints of link-layer quality ofservice (QoS) were rarely considered in the literature, suchas queue length limitation and maximum allowable delay[2]. Unfortunately, physical-layer models can not capture thecharacteristics of these QoS requirements, which depends onthe queueing behavior of the connection model. Ignoring theseQoS limitations grossly overestimates the network perfor-mance [3], and many QoS-constrained applications cannot besupported [4]. Therefore, it is imperative to come up with a
Zheng Shi, Theodoros Tsiftsis and Guanghua Yang are with the Schoolof Electrical and Information Engineering, Jinan University, Zhuhai 519070,China (e-mails: [email protected], [email protected],[email protected]).Weiqiang Tan is with the School of Computer Science and Educa-tional Software, Guangzhou University, Guangzhou 510006, China (e-mail:[email protected]).Shaodan Ma is with the Department of Electrical and Computer En-gineering, University of Macau, Macao S.A.R., China (e-mail: [email protected]).Mohamed-Slim Alouini is with CEMSE Division, King Abdullah Uni-versity of Science and Technology, Thuwal 23955-6900, Saudi Arabia (e-mail:[email protected]). cross-layer performance metric that combines physical-layerparameters and QoS requirements together.To address the aforementioned issue, the concept of effec-tive capacity was developed in [2] initially by considering thefinite buffer size in practice. This concept has been extensivelyemployed to evaluate the maximum supportable arrival rategiven a QoS exponent, where the QoS exponent affects thestatistical QoS constraints, including buffer overflow probabil-ity and delay-violation probability. Furthermore, the effectivecapacity enables us to optimally design cross-layer parame-ters for various wireless systems subject to statistical QoSconstraints [5], [6]. However, most of the relevant literatureassumed perfect knowledge of channel state information (CSI)at the transmitter, which is an impractical assumption due tothe unpredictable noise, quantization errors, etc. Particularly,in the absence of perfect CSI, the retransmission techniqueof hybrid automatic repeat request (HARQ) is frequentlyutilized to enhance the system reliability. Nonetheless, theintroduction of HARQ would make the queueing behaviour ofthe connection vastly involved, and consequently impedes theanalysis of effective capacity given diverse QoS requirements.Instead, in [7], large deviation was adopted to convert thephysical-layer throughput of HARQ-IR into effective capacityapproximately. Moreover, the concept of effective throughputwas developed to bypass the complex effective capacity in[3]. In [8], an accurate approximation of effective capacityunder small QoS exponent was obtained on the basis of thecumulants of renewal processes. Whereas, the concept of theeffective capacity in [8] represents the maximum arrival ratethat can be supported by HARQ systems. No matter whetherthe conveyed message can be recovered by receiver or not,every HARQ cycle will be counted as a success. Obviously,it will overestimate the link-layer throughput particularly forhigh probability of the decoding failures. Therefore, only thegoodput of HARQ systems was considered into the formula-tion of the effective capacity in [9], and the effective capacitywas obtained by using the recurrence relation approach. Thesimilar results were further extended to examine the outageeffective capacity of the buffer-aided diamond relay systemsin [10]. However in both [9], [10], the analytical results areonly applicable to fixed-rate HARQ (FR-HARQ) schemes,wherein the transmission rates remain constant during allHARQ rounds.Unfortunately so far, the effective capacities of moreadvanced and complex HARQ schemes, e.g., variable-rateHARQ (VR-HARQ) [11] and cross-packet HARQ (XP-HARQ) [12], have never been investigated due to the challengeof analyzing more complicated service process. Moreover, even if the closed-form expressions have been derived forthe effective capacity of the conventional HARQ schemes[9], [10], the complex expressions of the effective capacityprovided little insights and it is also difficult to extend theanalytical results to the general case. Hence, they will impedethe effective cross-layer design of HARQ systems to furtherenhance the system performance. To combat this issue andgeneralize the analytical results, we notice that the HARQtransmission model can actually be described by a renewalreward process [13]. Specifically, the event that the trans-mitter halts HARQ transmissions for the current message isrecognized to be a renewal, and the number of the transmittedinformation bits reflects the reward received from the renewal.In this paper, we first derive a simple exact expression forthe effective capacity of the network services that followsconstant reward renewal process. The results are further ex-tended to the general renewal process with variable rewards.The simple analytical expressions not only offer accurateapproximation for the effective capacity under small QoS ex-ponent, but also facilitate the extraction of further meaningfulinsights. In particular, the effective capacity decreases withthe QoS exponent and is bounded. The analytical results arethen applied to calculate link-layer throughputs for differentHARQ systems, including the conventional FR-HARQ (e.g.,Type I HARQ, HARQ with chase combining (HARQ-CC),HARQ with incremental redundancy (HARQ-IR)), VR-HARQand XP-HARQ. Numerical examples are finally presented toconfirm the proposed approach compared with the alreadyexisting ones. Furthermore, aiming to maximize the effectivecapacity through the optimal rate selection, the numericalresults reveal that XP-HARQ and VR-HARQ reach almost thesame performance in terms of the optimal effective capacity.The remainder of this paper is structured as follows. SectionII presents preliminaries on effective capacity and HARQschemes. In Section III, the effective capacity for the renewalprocess with constant reward is derived by means of Laplacetransform and Z-transform, respectively. Section IV then ex-tends the results to the general renewal reward process. Theanalytical results are further applied to evaluate the link-layerthroughput of various HARQ systems in Section V. Numer-ical results are presented for verifications and discussions inSection VI. Section VII finally concludes this paper.II. P
RELIMINARIES
A. Effective Capacity
The cross-layer model of the typical HARQ system withlimited buffer is shown in Fig. 1. By assuming a constantarrival rate µ , buffer overflows will happen if the queuelength Q exceeds the buffer threshold τ , and it can notbe avoided under limited buffer size. Based on the theoryof large deviations, the buffer overflow probability can beapproximated as Pr ( Q ≥ τ ) ≈ e − θτ for large values of τ [14], where Q and τ stand for the stationary queue lengthand buffer’s threshold, respectively. It is worth noting that theQoS exponent θ plays a critical role in connecting the physicaland link layers. The QoS exponent θ implies how fast thebuffer overflow probability Pr ( Q ≥ τ ) decays exponentially with τ . Accordingly, once a constraint is imposed on the bufferoverflow probability, the QoS exponent θ can be determinedas θ ≈ − ln (Pr ( Q ≥ τ )) /τ . To meet the requirement ofthe QoS exponent, the constant arrival rate µ should beproperly chosen. To this end, the concept of the effectivecapacity was developed to specify the maximum supportablearrival rate, i.e., C e = max { µ } [2]. Assume that the serviceprocess satisfies the G¨artner-Ellis theorem [14], given the QoSexponent θ , the effective capacity is explicitly given by thelimit [2] C e = − lim t →∞ θt ln E (cid:8) e − θS t (cid:9) , (1)where E {·} is the expectation operator and S t is the time-accumulated service process representing the total amount ofreward received until time t . Moreover, the definition of theeffective capacity was further expanded to the case with finitetime t in [9], i.e., C e,t = − θt ln E (cid:8) e − θS t (cid:9) . B. HARQ Schemes
This paper offers a through analysis of the effective ca-pacity for various HARQ systems. As a widely used reliabletransmission technique, HARQ leverages both the forwarderror correction coding and automatic repeat request suchthat the currently received packet could be combined withthe erroneously received packets to diminish the probabilityof decoding failures. As shown in Fig. 1, according to theHARQ mechanism, the delivered message is first encodedinto a long codeword, and the generated codeword is thenbroken into multiple subcodewords. These subcodewords willbe sent sequentially in different HARQ rounds upon thereception of negative acknowledgment (NACK) messages,while the feedback of the positive acknowledgment (ACK)messages notifies the transmitter about the initiation of anew HARQ process for the next information message. Onthe basis of whether all the subcodeword lengths are fixedor not, the HARQ scheme can be classified into FR-HARQand VR-HARQ. According to different encoding/decodingoperations performed at the transceiver, the conventional FR-HARQ can be further categorized into three types, i.e., Type IHARQ, HARQ-CC and HARQ-IR [13]. To be specific, TypeI HARQ does not require the aid of the buffer at the receiverbecause the failed packets are directly discarded. Whereas,both HARQ-CC and HARQ-IR have the dedicated bufferto store the erroneously received packets, and their majordifference lies in that diversity combining and code combiningare employed for the decodings of HARQ-CC and HARQ-IR, respectively. In contrast with the conventional FR-HARQ,VR-HARQ assumes the variable lengths of the subcodewordsfor further throughput enhancement [11]. It is worth notingthat both FR- and VR-HARQ do not include new informationbits into retransmissions, they may yield the waste of mutualinformation. To overcome this issue, VR-HARQ is proposedto add new information bits to retransmissions for possibleredundant mutual information [12].Note that the cross-layer HARQ system can be modelledby using the renewal reward process [13]. Specifically, theevent that HARQ retransmissions stop for the current mes-sage is treated as a renewal, and each time a renewal takes NA C K NA C K A C K A C K NA C K ……0110010…… Fig. 1: Cross-layer model for the buffer-limited HARQ System.place we receive a reward, which is the total amount ofthe delivered information bits. Most of the network servicesobey renewal reward processes, which incorporate the HARQchannel service as a special case. Therefore, to ease extension,the effective capacity of the renewal service process withconstant reward is derived first in Section III. A unified closed-form expression of the effective capacity is then obtained forthe complex renewal service process with variable rewardsin Section IV. The analytical results are then employed toobtain the effective capacity of various HARQ schemes. Forthe simplification of the analysis, the receptions of the HARQfeedback messages are assumed to be error- and delay-free.
C. Two Useful Theorems
To start with the analysis, two important theorems will berepeatedly used in the sequel. The first theorem is utilized toderive the effective capacity in a matrix form.
Theorem 1. If z , · · · , z K are distinct, the zero points ofthe function φ ( z ) does not coincide with z , · · · , z K , and φ ( z ) has no poles. The inverse Z-transform of ψ ( z ) , φ ( z ) / Q Kk =1 ( z − z k ) can be written in a compact form as Z − { ψ ( z ) } ( t ) = 12 π i I C ψ ( z ) z t − dz = det F det Z , (2) where i = √− , Z − ( · ) denotes the inverse Z-transform, C is the contour path of the integration which encir-cles all of the poles z , · · · , z K , F = h(cid:0) z ij − (cid:1) i,j i is a Vandermonde matrix of size K × K and Z = h(cid:0) z ij − (cid:1) i, ≤ j ≤ K − , (cid:0) z it − φ ( z i ) (cid:1) i,K i with size K × K and det · refers to the determinant operation.Proof. Please see Appendix A.The second theorem is invoked to prove the monotonicityof the effective capacity with respect to the QoS exponent θ . Theorem 2. If f ( x ) is twice differentiable and f (0) = 0 , η ( x ) = f ( x ) /x is increasing whenever f ′′ ( x ) ≥ anddecreasing otherwise.Proof. Please see Appendix B. III. E
FFECTIVE C APACITY OF THE R ENEWAL P ROCESSWITH C ONSTANT R EWARD
If each time a renewal occurs with a constant reward R , thetotal amount of the accumulated reward is S t = RN t , where N t denotes the renewal counting process. More precisely, thecounting process N t is defined as N t = max { n : S n ≤ t } ,where S n = P ni =1 X i and { X i , i ∈ N } is an independent andidentically distributed (i.i.d.) sequence of interarrival times.Accordingly, the effective capacity of the renewal serviceprocess reduces to [8] C e = − lim t →∞ θt ln E (cid:8) e − θRN t (cid:9) . (3)We denote by f X ( x ) the probability density function (PDF)of each X i . Moreover, E (cid:8) e − θRN t (cid:9) is the expectation takenover N t , such that E (cid:8) e − θRN t (cid:9) = ∞ X n =0 e − θRn Pr ( N t = n ) , (4)and Pr ( N t = n ) is given by Pr ( N t = n ) = Pr {S n ≤ t } − Pr {S n +1 ≤ t } = F n ( t ) − F n +1 ( t ) , (5)where F n ( t ) represents the cumulative distribution function(CDF) of S n , i.e., F n ( t ) = Pr {S n ≤ t } = Pr ( n X i =1 X i ≤ t ) . (6) A. Laplace Transform-Based Analysis
The independence among X i ’s motivates us to calculate(6) through Laplace transform. Denote by F ( s ) the Laplacetransform of X , i.e., F ( s ) = E ( e − sX ) = R ∞ e − sx f X ( x ) dx .By applying the convolutional property of Laplace transformto (6), it follows that F n ( t ) = L − { ( F ( s )) n /s } ( t ) , where L − stands for the operator of the inverse Laplace transform.By substituting this result into (5), we get Pr ( N t = n ) = L − { ( F ( s )) n (1 − F ( s )) /s } ( t ) . Putting this result into (4) together with the formula of the sum of a geometric series,we reach E (cid:8) e − θRN t (cid:9) = L − (cid:26) − F ( s ) s (1 − e − θR F ( s )) (cid:27) ( t ) , (7)where (cid:12)(cid:12) e − θR F ( s ) (cid:12)(cid:12) < . By substituting (7) into (3), theevaluation of the effective capacity can thus be enabled.By taking the Poisson service process as an example, N t follows a Poisson distribution with mean λt , E (cid:8) e − θRN t (cid:9) = P ∞ k =0 e − θRk e − λt ( λt ) k /k ! = exp (cid:0) − λt + λte − θR (cid:1) . On theother hand, the interarrival time is found to be exponen-tially distributed with mean / ( λt ) . Thus, it follows that F ( s ) = λt/ ( s + λt ) . By using (7), E (cid:8) e − θRN t (cid:9) = L − (cid:8) / (cid:0) s + λt − λte − θR (cid:1)(cid:9) ( t ) = exp (cid:0) − λt + λte − θR (cid:1) ,which consequently justifies the analytical result.Unfortunately, the application of Laplace transform toHARQ systems ( F ( s ) is written in the form of P q k e − sk )usually results in an infinite number of poles in (7), andmost of the time there is no closed-form expression for theeffective capacity. Moreover, the computation of (7) relyingon numerical methods entails higher complexity and accuracydue to the calculation of exp( − sk ) for extremely high s . Thesefacts hinder the thorough analysis of the effective capacity.However, if the sequence { X i , i ∈ N } is a discrete renewalprocess (e.g., HARQ channel service), Z-transform can beapplied to investigate the effective capacity for more insights. B. Z-Transform-Based Analysis If X , X , · · · , X i , · · · are i.i.d. discrete and non-negativeinteger random variables with the probability mass function(pmf) Pr( X i = k ) = q k , k ∈ [0 , K ] , Z-transform of thedistribution of X i is expressed as X ( z ) = P Kk =0 q k z − k .Similarly, by applying the convolutional property of Z-transform to (6), we have F n ( t ) = Z − { ( X ( z )) n z/ ( z − } .Combining the latter with (5) leads to Pr ( N t = n ) = Z − n zz − ( X ( z )) n (1 − X ( z )) o . From (4), E (cid:8) e − θRN t (cid:9) canbe rewritten as E (cid:8) e − θRN t (cid:9) = 12 π i I C z (1 − X ( z ))( z −
1) (1 − e − θR X ( z )) z t − dz. (8)where (cid:12)(cid:12) e − θR X ( z ) (cid:12)(cid:12) < . By using the definition of X ( z ) , E (cid:8) e − θRN t (cid:9) can be further expressed as E (cid:8) e − θRN t (cid:9) = 1 − e θR − q e − θR π i I C z t + K (1 −X ( z )) z − Q i ∈ Φ ( z − z i ) dz, (9)where Φ , (cid:8) z i : z iK (cid:0) − e − θR X ( z i ) (cid:1) = 0 (cid:9) . Clearly, z i = 0 if q K = 0 , Φ is therefore equivalent to (cid:8) z i : X ( z i ) = e θR (cid:9) .Furthermore, < | z i | < can be proved by applyingtriangle inequality to X ( z i ) = e θR . We assume that z i ’s aredistinct for analytical tractability. Thus the cardinality of Φ is K , z K (cid:0) − e − θR X ( z ) (cid:1) = (1 − q e − θR ) Q Kk =1 ( z − z k ) thenfollows. The integrand in (9) has K distinct poles z i ∈ Φ , andit is worthwhile to mention that z = 1 is not a pole because lim z → (1 − X ( z )) / ( z −
1) = constant . Accordingly, by usingTheorem 1, E (cid:8) e − θRN t (cid:9) is obtained as E (cid:8) e − θRN t (cid:9) = 1 − e θR − q e − θR det B det A , (10) where A = h(cid:0) z ij − (cid:1) i,j i is a Vandermonde matrix of size K × K and B = h(cid:0) z ij − (cid:1) i, ≤ j ≤ K − , (cid:0) z it + K / ( z i − (cid:1) i,K i with size K × K . Plugging (10) into (3) leads to C e = − lim t →∞ ln ( − e θR ) det B (1 − q e − θR )det A θt = − lim t →∞ θt ln det B . (11)Without loss of generality, let z be the largest in absolutevalue in Φ , i.e., | z | = max {| z i | , i ∈ [1 , K ] } , we have C e = − lim t →∞ ln z t + K θt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) z ij − (cid:1) i, ≤ j ≤ K − , (cid:16) z i z (cid:17) t + K z i − i,K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ln z − θ = ln ζθ , (12)where ζ , z − . In the following, some discussions areexpanded on the basis of (12).
1) Calculation of ζ : It is clear from (12) that ζ ≥ , and ζ satisfies K X k =0 q k ζ k = e θR . (13)By applying Jensen’s inequality to (13), we have K X k =0 q k ζ k ≥ ζ K P k =0 kq k = ζ E ( X ) . (14)Thus, ζ ≤ e θR E ( X ) follows. Note ζ is larger than or equal to 1, ζ belongs to [1 , e θR E ( X ) ] . The latter eases the calculation of ζ .More specifically, let us define g ( x ) , P Kk =0 q k x k − e θR .Since g ( x ) is an increasing function of x if x > , ζ isdefinitely the unique zero point of g ( x ) within the range x ∈ [1 , e θR E ( X ) ] . Accordingly, the bisection method can beadopted to calculate ζ .
2) Approximation of C e : If θ is sufficiently small, theeffective capacity can be approximated by C e ≈ R/ E ( X ) − θR Cov ( X ) / (cid:16) E ( X ) (cid:17) from [8]. This can be further con-firmed by using Taylor expansion of (12). To proceed, wedefine u ( θ ) = ln ζ and h ( u ) = P Kk =0 q k e ku such that h ( u ( θ )) = e θR . (15)By using Taylor series expansion, u ( θ ) can be expanded as u ( θ ) = u (0) + u ′ (0) θ + u ′′ (0)2 θ + o (cid:0) θ (cid:1) , (16)where o ( · ) refers to the little-O notation, u ′ and u ′′ denotethe first and second derivatives of u with respect to (w.r.t.) θ , respectively. u (0) and its derivatives can be determined asfollows.It is readily found from (15) that u (0) = 1 . Taking thefirst derivative w.r.t. θ at the both sides of (15) leads to dhdu u ′ ( θ ) = Re θR . With the definition of h ( u ) , u ′ ( θ ) canthen be obtained as u ′ ( θ ) = Re θR / P Kk =0 q k ke ku ( θ ) . Hence, u ′ (0) = R/ E ( X ) . Similarly, taking the first derivative of u ′ ( θ ) w.r.t. θ gives u ′′ ( θ ) as u ′′ ( θ ) = R e θR K X k =0 q k ke ku ( θ ) ! − × K X k =0 q k ke ku ( θ ) ! − K X k =0 q k e ku ( θ ) K X k =0 q k k e ku ( θ ) . (17)Thus, u ′′ (0) = − R Cov ( X ) / ( E ( X )) , where Cov( · ) de-notes the covariance operator. Substituting u (0) , u ′ (0) and u ′′ (0) into (16) together with the definition of u ( θ ) arrivesat ζ ≈ exp (cid:16) θR/ E ( X ) − θ R Cov ( X ) / (cid:16) E ( X ) (cid:17)(cid:17) . Con-sequently, putting ζ into (12) leads to the approximation of C e , which coincides with the result in [8].
3) Properties of Effective Capacity:
By applying theCauchy-Schwarz inequality to (17), we have u ′′ ( θ ) ≤ . WithTheorem 2, the effective capacity is found to be a decreasingfunction of QoS exponent θ . Moreover, as proved in AppendixC, C e is bounded as RK ≤ C e ≤ min (cid:18) RK − ln q K Kθ , R E ( X ) (cid:19) , (18)and lim θ →∞ C e = R/K if q K > .
4) Extension to the General Case:
The result of (12) canbe extended to a general case when { X i } is a non-negativecontinuous renewal process. Specifically, the horizontal axisof the distribution of X i can be partitioned into a numberof equal intervals, each with length ∆ x . This discretizationleads to a new discrete random variable ˜ X i with pmf given by Pr( ˜ X i = k ) = R ( k +1)∆ xk ∆ x f X ( x ) dx , q k . Therefore, a similarapproach in Subsection III-B can be adopted to approximatethe effective capacity, and the approximation can become moreaccurate by using a higher-resolution discretization. As ∆ x → , it is readily proved that the exact expression of the effectivecapacity is C e = ln ζθ , (19)where ζ is the solution to the following equation E (cid:0) ζ X (cid:1) = e θR , ζ ≥ . (20) ζ ∈ [1 , e θR E ( X ) ] , and the proof can be found in Appendix D.Furthermore, it is readily proved that the same properties asshown in Subsections III-B1-III-B3 also apply to the generalcase except for the bounds of the effective capacity in (18).Nevertheless, (18) is applicable to the discrete renewal serviceprocess with non-integer interarrival time, and K stands forthe maximum interarrival time herein.IV. E FFECTIVE C APACITY OF THE G ENERAL R ENEWAL R EWARD P ROCESS
In general, the reward earned each time could be variableapart from constant [15]. In communication systems, thereward of the i -th renewal commonly varies with the lengthof the renewal interval X i and the channel/termination state S i of each renewal . We denote by ( X i , S i ) the i -th renewalevent state, and denote by R ( X i , S i ) the reward received fromthe i -th renewal. Likewise, the effective capacity of discreterenewal service processes is derived first, and the analyticalresults are further extended to that of continuous ones.By favor of the definition of the renewal counting process N t , S t is given by S t = N t X i =1 R ( X i , S i ) . (21)Note that the renewal depends on the interarrival time andthe termination state, we define R k,s , R ( X i = k, S i = s ) for notational convenience, where k ∈ [1 , K ] and s ∈ [1 , v k ] .Moreover, denote by q k,s the probability that the renewal eventis ended with interval length k and termination state s , i.e., q k,s , Pr ( X i = k, S i = s ) . Thus, the probability that theinterarrival time is k can be obtained as Pr ( X i = k ) = v i X j =1 Pr ( X i = k, S i = j ) = v i X j =1 q k,j . (22)Adding up the probabilities q k,s of all the renewal statesequals to one, i.e., X Kk =1 X v k s =1 q k,s = 1 . (23)To facilitate the analysis, we define a vector of re-newal counting processes that count the numbers of R k,s ’sachieved by the network service up to time t as n t = (cid:16) ( n t,k, , · · · , n t,k,v k ) Kk =1 (cid:17) , where n t,k,s represents the num-ber of the reward R k,s ’s earned up until time t , and thesubscript t is omitted in the sequel for simplicity. Moreover,since the maximum interval between two consecutive renewalsis K , n t should satisfy the following constraint as [ t − K + 1] + ≤ X Kk =1 X v k s =1 kn k,s ≤ t, (24)where [ x ] + = max { , x } is the projection onto the nonneg-ative orthant. With the above definition, the total amount ofreward given in (21) can be rewritten as S t = X Kk =1 X v k s =1 n k,s R k,s . (25)Therefore, the corresponding effective capacity can be ob-tained as C e = − lim t →∞ θt ln E n e − θ P Kk =1 P vks =1 n k,s R k,s o| {z } ϕ ( t ) , (26)where ϕ ( t ) is given by (27), shown at the top of the nextpage. Different from Section III, it is impossible to directlyapply Z-transform to derive the moment generating function(mgf) of S t , ϕ ( t ) , due to the complex vector form of n t . Toaddress this challenge, the effective capacity with finite timeis obtained by using the multinomial distribution of n t . For example, S i represents the outcome whether the receiver succeeds todecode the message after the termination of HARQ. Moreover, S i could alsobe the channel gains if perfect CSI is known at the transmitter, and adaptive& modulation scheme is adopted. ϕ ( t ) = X K P k =1 vk P s =1 kn k,s ∈ [ [ t − K +1] + ,t ] e − θ K P k =1 vk P s =1 n k,s R k,s Pr (cid:16) n t = (cid:16) ( n k, , · · · , n k,v k ) Kk =1 (cid:17)(cid:17) . (27) A. Effective Capacity with Finite Time
To proceed, Pr (cid:16) n t = (cid:16) ( n k, , · · · , n k,v k ) Kk =1 (cid:17)(cid:17) is derivedfirst. To do so, we define τ = t − P Kk =1 P v k s =1 kn k,s to showthe time of the last renewal passed before time t . Clearly,it follows from (24) that τ is bounded as ≤ τ ≤ K ,and the interval of the last renewal should be larger than τ ,i.e., X N t +1 > τ . According to the law of total probability, Pr (cid:16) n t = (cid:16) ( n k, , · · · , n k,v k ) Kk =1 (cid:17)(cid:17) is given by (28), shownat the top of the page, where step (a) holds because of theindependence among renewals, and step (b) holds by using(22). More specifically, the vector n t − τ follows a multinomialdistribution with the pmf given by (29), shown at the top of thispage, where (cid:0) nn , ··· ,n Q (cid:1) = n ! / ( n ! · · · n Q !) is the multinomialcoefficient and n = P Qq =1 n q .By substituting (28) and (29) into (27) together with somealgebraic manipulations, ϕ ( t ) is finally rewritten as (30) atthe top of the following page. Specifically, if ≤ t < K , (30)becomes ϕ ( t ) = X t P k =1 vk P s =1 kn k,s ∈ [0 ,t ] e − θ t P k =1 vk P s =1 R k,s n k,s × t P k =1 v k P s =1 n k,s ( n k, , · · · , n k,v k ) tk =1 t Y k =1 v k Y s =1 q k,sn k,s × K X i = t − t P k =1 vk P s =1 kn k,s +1 v i X j =1 q i,j , ≤ t < K, (31)and ϕ (0) = 1 . However, for large t , the computation of ef-fective capacity with (30) causes higher complexity overhead.To address this issue, we turn to the Z-transform to derive theeffective capacity in a closed-form. B. Z-Transform-Based Analysis
By virtue of the recurrence relation of multinomial coeffi-cient [16, eq.26.4.10], (30) can be rewritten as (32), shown atthe top of the next page. By assuming t ≥ K and switching theorder of summations along with some rearrangements, ϕ ( t ) is represented by a homogeneous difference equation as ϕ ( t ) − K X κ =1 a κ ϕ ( t − κ ) = 0 , t ≥ K, (33)where a κ = P v κ j =1 q κ,j e − θR κ,j . As proved in Appendix E, byapplying Z-transform to (33), ϕ ( t ) can be expressed as ϕ ( t ) = K X l =1 ϕ ( K − l ) det ˜ B l det ˜ A , t ≥ K, (34) where ˜ A = (cid:20)(cid:16) ˜ z j − i (cid:17) i,j (cid:21) , ˜ B l = (cid:20)(cid:16) ˜ z j − i (cid:17) i, ≤ j ≤ K − , (cid:16)P Kκ = l a κ ˜ z t + l − κ − i (cid:17) i,K (cid:21) , and ˜ z i ’s are K distinct roots of the polynomial z K − P Kκ =1 a κ z K − κ = 0 . Without loss of generality,we define | ˜ z | = max {| ˜ z i | , i ∈ [1 , K ] } . Plugging (34) into(26) leads to C e = − lim t →∞ θt ln ˜ z t K X l =1 ϕ ( K − l ) ˜ z l − κ − × det "(cid:16) ˜ z j − i (cid:17) i, ≤ j ≤ K − , (cid:18) K P κ = l a κ (cid:16) ˜ z i ˜ z (cid:17) t + l − κ − (cid:19) i,K det ˜ A = ln ˜ z − θ = ln ˜ ζθ , (35)where ˜ ζ = ˜ z − . In analogous to Subsection III-B, somediscussions are carried out as follows based on (35).
1) Calculation of ζ : It is obvious from (35) that ˜ ζ is a realnumber and ˜ ζ ≥ . Similarly, we define ˜ g ( x ) = P Kκ =1 a κ x κ − . Hence, ˜ ζ is the zero point of ˜ g ( x ) and uniquely exists owingto its increasing monotonicity within x ≥ . The monotonicityenables us to find the zero point ˜ ζ with bisection method, and ˜ ζ lies in [1 , exp ( θ E ( R ( X, S )) / E ( X ))] , where the upper boundis proved in Subsection IV-B3.
2) Approximation of C e : By defining ˜ u ( θ ) = ln ˜ ζ and ˜ h (˜ u ) = P Kκ =1 P v κ j =1 q κ,j e − θR κ,j e κ ˜ u , we have ˜ h (˜ u ) = 1 . Inanalogous to Subsection III-B2, ˜ u ( θ ) can be approximated forsmall θ as ˜ u ( θ ) ≈ ˜ u (0) + ˜ u ′ (0) θ + ˜ u ′′ (0)2 θ , (36)where ˜ u ′ and ˜ u ′′ denote the first and second derivatives of ˜ u w.r.t. θ , respectively. Clearly, ˜ u (0) = 0 . ˜ u ′ ( θ ) and ˜ u ′′ ( θ ) aregiven by ˜ u ′ ( θ ) = P Kκ =1 P v κ j =1 q κ,j R κ,j e − R κ,j θ e κ ˜ u P Kκ =1 P v κ j =1 κq κ,j e − R κ,j θ e κ ˜ u , (37)and (38) at the top of the next page, respectively. Hence, ˜ u ′ (0) and ˜ u ′′ (0) can be respectively obtained as ˜ u ′ (0) = P Kκ =1 P v κ j =1 q κ,j R κ,j P Kκ =1 κ P v κ j =1 q κ,j = E ( R ( X, S )) E ( X ) , (39) ˜ u ′′ (0) = − E ( R ( X, S ) E ( X ) − E ( R ( X, S )) X ) ( E ( X )) . (40)(39) implies that ˜ u ′ (0) refers to the average reward over time.Particularly for the HARQ schemes, ˜ u ′ (0) is the so-called thelong term average throughput (LTAT) [13]. Pr (cid:16) n t = (cid:16) ( n k, , · · · , n k,v k ) Kk =1 (cid:17)(cid:17) = K X i = τ +1 Pr (cid:16) n t = (cid:16) ( n k, , · · · , n k,v k ) Kk =1 (cid:17)(cid:12)(cid:12)(cid:12) X N t +1 = i (cid:17) Pr ( X N t +1 = i ) ( a ) = Pr (cid:16) n t − τ = (cid:16) ( n k, , · · · , n k,v k ) Kk =1 (cid:17)(cid:17) K X i = τ +1 Pr ( X N t +1 = i ) ( b ) = Pr (cid:16) n t − τ = (cid:16) ( n k, , · · · , n k,v k ) Kk =1 (cid:17)(cid:17) K X i = τ +1 v i X j =1 q i,j , (28) Pr (cid:16) n t − τ = (cid:16) ( n k, , · · · , n k,v k ) Kk =1 (cid:17)(cid:17) = K P k =1 v k P s =1 n k,s ( n k, , · · · , n k,v k ) Kk =1 K Y k =1 v k Y s =1 q k,sn k,s . (29) ϕ ( t ) = X K P k =1 vk P s =1 kn k,s ∈ [ [ t − K +1] + ,t ] e − θ K P k =1 vk P s =1 R k,s n k,s K P k =1 v k P s =1 n k,s ( n k, , · · · , n k,v k ) Kk =1 K Y k =1 v k Y s =1 q k,sn k,s K X i = t − K P k =1 vk P s =1 kn k,s +1 v i X j =1 q i,j . (30) ϕ ( t ) = X K P k =1 vk P s =1 kn k,s ∈ [ [ t − K +1] + ,t ] e − θ K P k =1 vk P s =1 R k,s n k,s K Y k =1 v k Y s =1 q k,sn k,s K X i = t − K P k =1 vk P s =1 kn k,s +1 v i X j =1 q i,j × K X κ =1 v κ X ν =1 K P k =1 v k P s =1 n k,s − n , , · · · , n ,v ) , · · · , ( n κ, , · · · , n κ,ν − , · · · , n κ,v κ ) , · · · , ( n K, , · · · , n K,v K ) . (32) ˜ u ′′ ( θ ) = − K P κ =1 v κ P j =1 q κ,j R κ,j e − R κ,j θ e κ ˜ u + ˜ u ′ K P κ =1 v κ P j =1 κq κ,j R κ,j e − R κ,j θ e κ ˜ u ! K P κ =1 v κ P j =1 κq κ,j e − R κ,j θ e κ ˜ u + K P κ =1 v κ P j =1 κq κ,j R κ,j e − R κ,j θ e κ ˜ u − ˜ u ′ K P κ =1 v κ P j =1 κ q κ,j e − R κ,j θ e κ ˜ u ! K P κ =1 v κ P j =1 q κ,j R κ,j e − R κ,j θ e κ ˜ u K P κ =1 v κ P j =1 κq κ,j e − R κ,j θ e κ ˜ u ! , (38)Substituting (39) and (40) into (36) along with (35), theeffective capacity can be approximated as C e ≈ E ( R ( X, S )) E ( X ) − θ E n ( R ( X, S ) E ( X ) − E ( R ( X, S )) X ) o E ( X )) . (41)
3) Properties of Effective Capacity:
It is proved in Ap-pendix F that the effective capacity is a decreasing functionof θ . Moreover, C e is bounded as R ˆ κ, ˆ j K ≤ C e ≤ min (cid:26) R ˆ κ, ˆ j ˆ κ − ln q ˆ κ, ˆ j ˆ κθ , E ( R ( X, S )) E ( X ) (cid:27) , (42) where (ˆ κ, ˆ j ) = arg ( κ,j ) min q κ,j > R κ,j . Accordingly, as θ approaches to infinity, C e is bounded as R ˆ κ, ˆ j K ≤ lim θ →∞ C e ≤ R ˆ κ, ˆ j ˆ κ . (43)
4) Extension to the Continuous Ones:
Similarly to Subsec-tion III-B4, the result of (35) can also be extended to thecase that { ( X i , S i ) } is a non-negative continuous renewalprocess and the reward is variable. Specifically, the horizon-tal and vertical axis of the distribution of ( X i , S i ) can bepartitioned into a number of equal rectangles, each with area ∆ x × ∆ s . This discretization leads to two new discrete randomvariables ( ˜ X i , ˜ S i ) with pmf given by Pr( ˜ X i = k, ˜ S i = l ) = R k ∆ x ( k − x R l ∆ s ( l − s f X, S ( x, s ) dxds , q k,l . Therefore, the similar approach in Subsection IV-B can be employed toapproximate the effective capacity. As ∆ x, ∆ s → , we getthe exact expression of the effective capacity that is the sameas (19), where ζ is the solution to the following equation E X, S (cid:16) e − θ R ( X, S ) ζ X (cid:17) = 1 , ζ ≥ . (44)Similarly, the effective capacity of the continuous case alsofollows same properties as shown in Subsections IV-B1-IV-B3except for the bounds of the effective capacity in (42) and(43). Nevertheless, (42) and (43) are applicable to the discreterenewal service process with non-integer interarrival time,where K is the maximum interarrival time.V. A PPLICATIONS TO
HARQ S
YSTEMS
A. Fixed-Rate HARQ Systems
Since HARQ transmissions can be modelled by the renewal-reward process [13], the proceeding analysis of the effectivecapacity can be used to characterize the cross-layer throughputof HARQ [8]. To this end, a renewal of HARQ transmissionsis defined as an event that the receiver successfully receivesthe message or the maximum number of transmissions isreached. It is assumed that the number of transmissions foreach message is allowed up to K due to congestion avoidance.In addition, N t is the number of renewals that occur up untiltime t , and X k is the random time between two consecutiverenewals.We assume that each delivered packet contains b informationbits, and each one is first encoded into a long codeword.Following the FR-HARQ scheme, the generated codeword isthen partitioned into K subcodewords, and each subcodewordconsists of L symbols. Suppose that each symbol duration isnormalized to unit. More specifically, X i is the time intervaldemanded in the successful delivery of the i -th message, i.e., X k = kL and k ≤ K . The pmf of X k , i.e., q k is given by[13] Pr( X = kL ) , q k = p k − − p k ( k = K ) , k ∈ [1 , K ] , (45)where p k is the outage probability after k HARQ roundsand the indicator function ( A ) is one whenever A is trueand zero otherwise. By applying capacity-achieving codes,the outage expressions for three different HARQ schemes aregiven respectively by [13] p k = Pr (log (1 + max ( γ , · · · , γ k )) ≤ R ) , Type IPr (cid:18) log (cid:18) k P l =1 γ l (cid:19) ≤ R (cid:19) , CCPr (cid:18) k P l =1 log (1 + γ l ) ≤ R (cid:19) , IR . (46)where R = b/L is the transmission rate (normalized rewardper renewal), γ l = γ T α l represents the signal-to-noise ratio(SNR) of the l -th transmission, γ T and α l correspond tothe transmit SNR and the l -th channel gain, respectively and p = 1 by convention. The outage performance of HARQschemes under various fading channels has been extensively investigated. In particular, the outage probabilities of Type IHARQ and HARQ-CC have been derived in closed-form byconsidering general fading channels in [17, eqs. (8), (11)].Moreover, the outage expression of HARQ-IR has been givenin [18, eq. (17)]. Nonetheless, the specific expressions areomitted here due to space limitation.
1) Maximum Arrival Rate:
In [8], the effective capacity isused to assess the maximum arrival rate of HARQ schemes. Todo so, by putting (46) into (45) and then combining with (12)and (13), the effective capacities of the three HARQ schemescan be calculated as C e = ln ζ/θ , where P K k =1 q k ζ kL = e θb .By introducing ¯ ζ = ζ L and ¯ θ = Lθ , the effective capacity canbe obtained as C e = ln ¯ ζ ¯ θ , (47)where P K k =1 q k ¯ ζ k = e ¯ θ R . From (47), the effective capacity ofthe HARQ scheme is equivalent to that of a new renewal ser-vice process with the interarrival time ranged from 1 to K andthe constant reward R . Moreover, C e can be approximated forsmall θ by using the Taylor series expansion as illustrated inSubsection III-B2. With (18), the effective capacity is boundedas R/K ≤ C e ≤ R / P K− k =0 p k and lim θ →∞ C e = R /K .Since both the decoding successes and failures are countedas rewards, the effective capacity obtained in (47) indicatesthe maximum service rate supported by HARQ systems. Inaddition, the maximum arrival rate can be used to evaluate thethroughput of the lossless HARQ scheme, which is allowedto have an infinite number of transmissions, i.e., K = ∞ , toguarantee no decoding failures.
2) Outage Effective Capacity:
As pointed out in SubsectionV-A1, the maximum arrival rate does not imply that all theincoming data will be successfully delivered to the receiverbecause of the presence of decoding failures. In particular,the outage events usually take place for the truncated HARQschemes, i.e. K < ∞ . To address this issue, the outageeffective capacity is proposed in [9], [10]. Unlike the max-imum arrival rate studied in Subsection V-A1, the reward ofeach renewal is a two-value function for the truncated HARQschemes. More specifically, the reward is b if the message issuccessfully decoded and zero otherwise. Therefore, the ana-lytical results in Section IV can be used herein to evaluate theoutage effective capacity. For truncated HARQ schemes, X i and S i represent the number of HARQ transmissions involvedinto the delivery of the i -th message, and the termination statewhether the receiver successfully recovers the message or not,respectively. Hence, the joint pmf q k,s of truncated HARQschemes is given by Pr( X = kL, S = s ) = q k,s = (cid:26) p k − − p k , k ≤ K & s = S p K , k = K & s = F , (48)where the notations S and F denote the success and the failureof the decoding, respectively. The reward function R ( X, S ) is explicitly given by R ( X = kL, S = s ) = (cid:26) b, k ≤ K & s = S0 , k = K & s = F . (49) By using (35), the outage effective capacity can be obtainedas C oute = ln ζ/θ , where ζ satisfies P K k =1 q k, S e bθ ζ kL + q K , F ζ K L = 1 . Similarly, with the same definitions of ¯ ζ = ζ L and ¯ θ = Lθ , the outage effective capacity can be rewritten as C oute = ln ¯ ζ/ ¯ θ , where ¯ ζ satisfies K X k =1 q k, S e R ¯ θ ¯ ζ k + q K , F ¯ ζ K = 1 , ¯ ζ ≥ . (50)Accordingly, (50) implies that C oute is equivalent to the outageeffective capacity of a simplified HARQ service process withthe interarrival time ranged from 1 to K , the QoS expo-nent ¯ θ and the constant reward R . This result is consistentwith [9]. By using (41), the outage effective capacity canbe approximated for small θ . Moreover, from (42), since ( K L, F) = arg ( κ,j ) min q κ,j > R ( X, S ) , the outage effectivecapacity is bounded as ≤ C oute ≤ min (cid:26) − ln q K , F K ¯ θ , E ( R ( X, S )) E ( X ) (cid:27) , (51)where the LTAT E ( R ( X, S )) / E ( X ) is given by [19] E ( R ( X, S )) E ( X ) = R (1 − p K ) P K− k =0 p k . (52)As expected, the outage effective capacity is less than or equalto the LTAT due to the constraint of the limited buffer length.In addition, (43) indicates that lim θ →∞ C oute = 0 . (53)This is different from the maximal arrival rate because erro-neously received messages are counted as null rewards. As θ increases, the buffer overflow probability becomes muchmore stringent, which consequently leads to the continuousdeclination of the outage effective capacity. B. Variable-Rate HARQ-IR Systems
Unlike the conventional FR-HARQ schemes, the transmis-sion rate of the HARQ-IR scheme could be changeable fromone transmission to another. More specifically, we assumethat the length of the k -th subcodeword is L k . By applyingcapacity achieving channel coding, an outage event happenswhen the accumulated mutual information is below b . Theoutage probability after k HARQ rounds is thus written as[11] ˆ p k = Pr k X l =1 R l log (1 + γ l ) < ! , (54)where R k = b/L k for k ∈ [1 , K ] . From (54), the outageprobability of VR-HARQ scheme can be derived in closed-form if the fading channels are independently Nakagami-m distributed among different HARQ rounds. The channelgains α , · · · , α K are independent Gamma random variablesunder the circumstance, and we assume α l ∼ G ( m l , Ω l /m l ) ,where m l and Ω l stand for the fading order and averagechannel power gain, respectively. As proved in Appendix G,by using Mellin transform, ˆ p k can be derived in terms of thegeneralized Fox’s H function as (55), shown at the top of the next page, where the explicit definition of the generalizedFox’s H function Y m,np,q ( · ) is omitted here to conserve space(see [20]).The joint pmf of the interarrival time ˆ X and the terminationstate ˆ S , ˆ q k,s , is given by Pr (cid:18) ˆ X = X kl =1 L l , ˆ S = s (cid:19) = ˆ q k,s = (cid:26) ˆ p k − − ˆ p k , k ≤ K & s = Sˆ p K , k = K & s = F . (56)Moreover, the corresponding reward function R ( ˆ X, ˆ S ) isexpressed as R ( ˆ X = X kl =1 L l , ˆ S = s ) = (cid:26) b, k ≤ K & s = S0 , k = K & s = F . (57)Therefore, the effective capacity is obtained as C oute =ln ζ/θ , where ζ is given by P K k =1 ˆ q k, S e bθ ζ P kl =1 L l +ˆ q K , F ζ P K l =1 L l = 1 . By defining ˆ θ = bθ and ˆ ζ = ζ b , theeffective capacity can be rewritten as C e = ln ˆ ζ/ ˆ θ , where ˆ ζ satisfies K X k =1 ˆ q k, S e ˆ θ ˆ ζ k P l =1 R l − + ˆ q K , F ˆ ζ K P l =1 R l − = 1 . (58)With (41), the effective capacity can be approximated. Inanalogous to (51), the effective capacity is bounded as ≤ C e ≤ min − ln ˆ q K , F ˆ θ P K l =1 R l − , E (cid:16) R ( ˆ X, ˆ S ) (cid:17) E (cid:16) ˆ X (cid:17) , (59)where the LTAT E (cid:16) R ( ˆ X, ˆ S ) (cid:17) / E (cid:16) ˆ X (cid:17) is given by [11] E (cid:16) R ( ˆ X, ˆ S ) (cid:17) E (cid:16) ˆ X (cid:17) = 1 − ˆ p KK− P k =0 R k +1 − ˆ p k . (60)Furthermore, the result lim θ →∞ C oute = 0 similar to (53) canbe obtained. C. Cross-Packet HARQ Systems
In the retransmissions of both the FR- and VR-HARQschemes, the retransmitted subcodewords do not involve newinformation bits. Particularly for FR-HARQ, the provisioningof the throughput close to ergodic capacity is prevented fromexploiting the possible redundant mutual information. Instead,the XP-HARQ scheme was devised to avoid the waste ofmutual information to substantially improve the throughput[12]. More specifically, the XP-HARQ scheme introduces newinformation bits in retransmissions besides the redundancybits. We assume the number of the new introduced informationbits in the k -th HARQ round is b k , and the number of theinformation bits in the initial transmission is b . During the k -th HARQ round, the currently introduced information bitsis first concatenated with all the previously introduced ones toconstruct a long message. The resultant message thus contains P kl =1 b l information bits, and is then encoded into the k -thcodeword of length L symbols. According to the decoding ˆ p k = Y k, ,k +1 " (1 , , , (cid:16) , R , m γ T Ω , m (cid:17) , · · · , (cid:16) , R k , m k γ T Ω k , m k (cid:17) , (0 , , , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k Y l =1 (cid:18) m l γ T Ω l (cid:19) R l , (55)conditions of XP-HARQ studied in [12], the outage event takesplace after k HARQ rounds if and only if the accumulatedmutual information is below the number of the deliveredinformation bits in the current and previous HARQ rounds.Therefore, the outage probability of the XP-HARQ schemecan be obtained as [12] ˘ p k = Pr ( k [ κ =1 κ X l =1 L log (1 + γ l ) < κ X l =1 b l !) = Pr k [ κ =1 κ X l =1 log (1 + γ l ) | {z } I κ < κ X l =1 ˘ R l , (61)where ˘ R l = b l /L and S ( · ) represents the union of events.Unfortunately, due to the presence of the correlation amongthe accumulated mutual informations per symbol I , · · · , I K ,it is intractable to derive an exact expression for (61) and thereis still no readily available result in the literature. In this paper, ˘ p k is computed by conducting Monte Carlo simulations.In order to obtain the effective capacity of the XP-HARQscheme, the pmf of the interarrival time ˘ X and the terminationstate ˘ S , ˘ q k,s , is derived as Pr (cid:16) ˘ X = kL, ˘ S = s (cid:17) = ˘ q k,s = (cid:26) ˘ p k − − ˘ p k , k ≤ K & s = S˘ p K , k = K & s = F . (62)And the reward function R ( ˘ X, ˘ S ) is given by R ( ˘ X = kL, ˘ S = s ) = (cid:26) P kl =1 b l , k ≤ K & s = S0 , k = K & s = F . (63)Accordingly, the effective capacity of XP-HARQ is cal-culated as C oute = ln ζ/θ , where ζ is determined by P K k =1 ˘ q k, S e θ P kl =1 b l ζ kL + ˘ q K , F ζ kL = 1 . By defining ˘ θ = Lθ and ˘ ζ = ζ L , we have C e = ln ˘ ζ/ ˘ θ , where ˘ θ is given by K X k =1 ˘ q k, S e ˘ θ k P l =1 ˘ R l ˘ ζ k + ˘ q K , F ˘ ζ k = 1 . (64)With (41), the effective capacity can be approximated. Inanalogous to (51), the effective capacity is bounded as ≤ C e ≤ min − ln ˘ q K , F K ˘ θ , E (cid:16) R ( ˘ X, ˘ S ) (cid:17) E (cid:16) ˘ X (cid:17) , (65)where the LTAT of XP-HARQ, E (cid:16) R ( ˘ X, ˘ S ) (cid:17) / E (cid:16) ˘ X (cid:17) , isgiven by [12] E (cid:16) R ( ˘ X, ˘ S ) (cid:17) E (cid:16) ˘ X (cid:17) = P K k =1 ˘ R k (˘ p k − − ˘ p K ) P K− k =0 ˘ p k . (66) Furthermore, the similar result to (53) can be demonstrated,i.e., lim θ →∞ C e = 0 .Finally, the calculations of the effective capacity for variousHARQ schemes are briefly summarized in Table I.VI. V ERIFICATIONS AND D ISCUSSIONS
In this section, numerical examples are presented for verifi-cations and discussions. Unless otherwise specified, we set γ T = 20 dB, R = 4 bps/Hz, K = 5 and b = 1080 bits[21]. In addition, we assume that all HARQ links experienceindependent Rayleigh fading with unit average power, i.e., E ( α l ) = Ω l = 1 and m l = 1 for l ∈ [1 , K ] . A. FR-HARQ Scheme
To start with, the maximum arrival rate against the QoSexponent θ for the three conventional FR-HARQ schemes isdepicted in Fig. 2, and the approximate results obtained in [8]are provided for comparison. It is observed that C e decreaseswith θ , which is consistent with our analysis. This is becausethat higher θ represents stricter queuing constraints imposedon buffer overflow probability, which limits the increase ofservice arrival rate, and consequently results in the decreaseof the capacity. As shown in Fig. 2, the gap between theapproximate and the exact results grows with θ , which justifiesthe significance of the exact analysis, especially, for large θ .As θ approaches to infinity, the maximum arrival rate tendsto a lower bound that is given by (18), i.e., lim θ →∞ C e = R/K = 0 . bps/Hz. Moreover, Fig. 2 further substantiates thatHARQ-IR is superior to other HARQ schemes in terms of themaximum arrival rate.In Fig. 3, the outage effective capacity C e is plotted versusthe QoS exponent. It can be seen that the tendency of C e forthe three conventional HARQ schemes is the same as Fig. 2,and their approximations given by (41) are also shown forcomparison. It is readily found that the approximate resultscoincide with the exact ones under small QoS exponent.Unlike the maximum arrival rate in Fig. 2, the outage effec-tive capacity approaches to zero as θ tends to infinity. Thisdifference is due to the fact that the amount of unsuccessfullydelivered data is not counted as reward while computing theoutage effective capacity. The smaller θ means the stricterQoS constraint, which needs smaller arrival rate µ to ensurea tighter constraint on buffer overflow probability.Fig. 4 illustrates the impact of the transmit SNR γ T on theeffective capacity, and also shows the comparison between themaximum arrival rate and the outage effective capacity. It isnot out of expectation that both the maximum arrival rate andthe effective capacity increase with γ T . Nonetheless, both ofthem are found to be bounded no matter how high γ T is,because the bounds of the maximum arrival rate given in (18)manifest that . bps/Hz ≤ C e ≤ R / E ( X ) < R = 4 bps/Hz.Whereas, the bounds of the outage effective capacity given by TABLE I: The effective capacity for various HARQ schemes.
HARQ Metrics C e θ ζ E ( R ( X, S )) E ( X ) FR-HARQ-MaximumArrival Rate C e = ln ¯ ζ/ ¯ θ ¯ θ = Lθ K P k =1 q k ¯ ζ k = e ¯ θ R R K− P k =0 p k FR-HARQ-OutageEffective Capacity K P k =1 q k, S e R ¯ θ ¯ ζ k + q K , F ¯ ζ K = 1 R (1 − p K ) K− P k =0 p k VR-HARQ C e = ln ˆ ζ/ ˆ θ ˆ θ = bθ K P k =1 ˆ q k, S e ˆ θ ˆ ζ k P l =1 R l − + ˆ q K , F ˆ ζ K P l =1 R l − = 1 − ˆ p KK− P k =0 R k +1 − ˆ p k XP-HARQ C e = ln ˘ ζ/ ˘ θ ˘ θ = Lθ K P k =1 ˘ q k, S e ˘ θ k P l =1 ˘ R l ˘ ζ k + ˘ q K , F ˘ ζ k = 1 K P k =1 ˘ R k (˘ p k − − ˘ p K ) K− P k =0 ˘ p k -4 -3 -2 -1 M a x i m u m a rr i v a l r a t e C e [ bp s / H z ] QoS exponent Type I-Exact Type I-App.[8] CC-Exact CC-App.[8] IR-Exact IR-App.[8]
Fig. 2: The maximum arrival rate versus the QoS exponent forFR-HARQ schemes. -4 -3 -2 -1 O u t a g e e ff ec ti v e ca p ac it y C e [ bp s / H z ] QoS exponent Type I-Exact Type I-App.-eq.(41) CC-Exact CC-App.-eq.(41) IR-Exact IR-App.-eq.(41)
Fig. 3: The outage effective capacity versus the QoS exponentfor FR-HARQ schemes.(42) show that bps/Hz ≤ C e ≤ R (1 − p K ) / E ( X ) < R =4 bps/Hz. B. VR-HARQ Scheme
Fig. 5 shows the effective capacity of the VR-HARQschemes against the effect of the QoS exponent, and thenotation r herein is defined as r = [ R , · · · , R K ] . It isobserved in Fig. 5 that the exact results perfectly agree withthe approximate ones for small θ . Similarly to Fig. 3, theeffective capacity decreases to zero as θ increases to infinityin Fig. 5. Unsurprisingly, the increase of the transmit SNRimproves the effective capacity. For example, for fixed values Type I HARQ C e [ bp s / H z ] Transmit SNR [d ] Maximum arrival rate Outage effective capacity
Fig. 4: The comparison between the maximum arrival rate andthe outage effective capacity with parameter θ = 10 − . -4 -3 -2 -1 E ff ec ti v e ca p ac it y C e [ bp s / H z ] QoS exponent Exact App.-eq.(41)=[4,3,3,2,2], dB=[2,2,3,3,4], dB=[4,3,3,2,2], dB
Fig. 5: The effective capacity versus the QoS exponent for theVR-HARQ schemes.of r = [4 , , , , bps/Hz and θ = 10 − , the effective capac-ity increases by 1.5bps/Hz if the transmit SNR is increasedfrom dB to dB. Furthermore, it is seen from this figurethat the rate selection has a significant impact on the effectivecapacity. C. XP-HARQ Scheme
The effective capacity for XP-HARQ scheme is plottedagainst the QoS exponent in Fig. 6, wherein ˘ r = [ ˘ R , · · · , ˘ R K ] .It can be observed that there is a perfect agreement betweenthe exact results and the approximate ones under a small θ .Similarly, the effective capacity of the XP-HARQ scheme is -5 -4 -3 -2 -1 =[4,1,1,1,1], dB=[3,1,1,1,1], dB=[4,1,1,1,1], dB E ff ec ti v e ca p ac it y C e [ bp s / H z ] QoS exponent Exact App.-eq.(41)
Fig. 6: The effective capacity versus the QoS exponent for theXP-HARQ scheme. O p ti m a l e ff ec ti v e ca p ac it y C e [ bp s / H z ] Transmit SNR [d ] FR-HARQ-IR- =10 -3 VR-HARQ- =10 -3 XP-HARQ- =10 -3 FR-HARQ-IR- =10 -5 VR-HARQ- =10 -5 XP-HARQ- =10 -5 Fig. 7: Optimal effective capacities of different HARQ-IRschemes with parameter K = 2 .a decreasing function of the QoS exponent as well as thetransmit SNR. In addition, it is seen from Fig. 6 that thetransmission rates of the XP-HARQ considerably influencesthe effective capacity.Note that the selection of transmission rates has a criticalimpact on the effective capacity, the effective capacity can bemaximized via optimizing transmission rates of the HARQ-IRscheme. To compare the three different HARQ-IR schemes,including FR-HARQ-IR, VR-HARQ and XP-HARQ, Fig. 7exhibits their optimal effective capacities versus the transmitSNR. Nevertheless, the optimal design of transmission ratesfor HARQ schemes is out of the scope of this paper dueto the non-convexity of the objective function. Similarly to[12], we conduct an exhaustive search over the space offollowing available transmission rates: FR-HARQ-IR and VR-HARQ adopt R , R k ∈ { . , . , , · · · , . } bps/Hz, whileXP-HARQ adopts ˘ R ∈ { . , . , , · · · , . } bps/Hz forthe initial round and ˘ R k ∈ { , . , . , · · · , . } bps/Hz forthe others. Fig. 7 reveals that VR-HARQ and XP-HARQcan achieve almost the same optimal effective capacity, whileFR-HARQ performs the worst. This is because that VR-HARQ turns to the variable-length coding to fully exploitthe statistical information of fading channels. Whereas, XP-HARQ attempts to incorporate new information bits intoretransmission to make the utmost of the possible redundantmutual information. VII. C ONCLUSIONS
This paper has derived a unified exact formula for theeffective capacity of the renewal network service process at a given QoS exponent, with which the cross-layer throughputfor various HARQ systems has been accurately evaluated,including FR-HARQ (e.g., Type I HARQ, HARQ-CC andHARQ-IR), VR-HARQ and XP-HARQ. The formula not onlyhas gained many insightful results, but also has paved the wayfor the exact cross-layer design by integrating the channelmodel of physical-layer to QoS requirements of link-layer.Specifically, the effective capacity has been found to decreasewith QoS exponent as well as be bounded. Furthermore, ifthe transmission rates are optimally chosen to maximize theeffective capacity, it has been shown that VR-HARQ and XP-HARQ achieve almost the same performance, and both ofthem surpass FR-HARQ.A
PPENDIX AP ROOF OF T HEOREM ψ ( z ) can be expressed as Z − { ψ ( z ) } ( t ) = K X i =1 Res z t − φ ( z ) K Q k =1 ( z − z k ) , z = z i = K X i =1 z it − φ ( z i ) K Q j =1 ,j = i ( z i − z j ) , (67)where the notation of Res( f ( x ) , u ) stands for the residueof f ( x ) at pole x = u . Identifying (67) with the Laplaceexpansion of Vandermonde determinant yields Z − { ψ ( z ) } ( t )= K P i =1 ( − K + i z it − φ ( z i ) K Q ≤ m = i
A. Decreasing Monotonicity of C e w.r.t. θ By defining ℓ t ( θ ) = − ln E (cid:8) e − θS t (cid:9) /t , the effective ca-pacity can be expressed as C e = lim t →∞ ℓ t ( θ ) /θ . Morespecifically, ℓ t (0) = 0 and the second derivative of ℓ t ( θ ) w.r.t. θ is less than or equal to zero because ℓ ′′ t ( θ ) = − E (cid:8) S t e − θS t (cid:9) E (cid:8) e − θS t (cid:9) − (cid:0) E (cid:8) S t e − θS t (cid:9)(cid:1) ( E { e − θS t } ) t ≤ , (80)where the last step holds by using Cauchy-Schwarz inequality.Hence, we get lim t →∞ ℓ t (0) = 0 and lim t →∞ ℓ t ′′ ( θ ) ≥ . Byusing Theorem 2, the decreasing monotonicity of C e w.r.t. θ is thus proved. B. Bounds of the Effective Capacity
Owing to the decreasing monotonicity of the effectivecapacity, the effective capacity is upper bounded as C e ≤ lim θ → C e , where the upper bound is given by lim θ → C e = lim θ → ˜ u ( θ ) θ = ˜ u ′ (0) = E ( R ( X, S )) E ( X ) . (81)On the other hand, we first define the lowest reward as R ˆ κ, ˆ j ,where (ˆ κ, ˆ j ) = arg ( κ,j ) min q κ,j > R κ,j . Hence, the effectivecapacity satisfies K X κ =1 v κ X j =1 q κ,j e − θ ( R κ,j − R ˆ κ, ˆ j ) e κθC e = e θR ˆ κ, ˆ j . (82) ϕ ( t ) = det "(cid:16) ˜ z j − i (cid:17) i, ≤ j ≤ K − , (cid:18) K P κ =1 κ P l =1 a κ ϕ ( K − l ) ˜ z t + l − κ − i (cid:19) i,K det ˜ A = K X l =1 ϕ ( K − l ) det ˜ B l z }| {(cid:16) ˜ z j − i (cid:17) i, ≤ j ≤ K − , K X κ = l a κ ˜ z t + l − κ − i ! i,K det ˜ A . (79)From (82), it follows that q ˆ κ, ˆ j e ˆ κθC e ≤ K X κ =1 v κ X j =1 q κ,j e − θ ( R κ,j − R ˆ κ, ˆ j ) e κθC e ≤ e KθC e . (83)Thus C e is bounded as R ˆ κ, ˆ j K ≤ C e ≤ R ˆ κ, ˆ j ˆ κ − ln q ˆ κ, ˆ j ˆ κθ . (84)Combining (81) and (84) leads to (42). Consequently, wecomplete the proof. A PPENDIX GP ROOF OF (55)From (54), ˆ p k can be rewritten as ˆ p k = Pr Y kl =1 (1 + γ l ) R l | {z } G k < = F G k (2) , (85)where F G k ( x ) refers to the CDF of G k . By using the Mellintransform, the Mellin transform w.r.t. F G k ( x ) can be obtainedby using [22, eq.8.3.15] as {M F G k } ( s ) = − s {M f G k } ( s + 1) , (86)where f G k ( x ) denotes the PDF of G k . The Mellin transformw.r.t. f G k ( x ) can be expressed as {M f G k } ( s ) = E (cid:8) G ks − (cid:9) ( a ) = k Y l =1 E n (1 + γ l ) s − R l o = k Y l =1 Z ∞ (1 + x ) s − R l f γ l ( x ) dx. (87)where step (a) holds because of the independence amongfading channels. Note γ l = γ T α l and α l ∼ G ( m l , Ω l /m l ) ,the PDF of γ l is given by f γ l ( x ) = (cid:18) m l γ T Ω l (cid:19) m l x m l − e − mlγT Ω l x Γ ( m l ) . (88) By substituting (88) into (87) yields {M f G k } ( s ) = k Y l =1 (cid:18) m l γ T Ω l (cid:19) m l ×
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