Backlog and Delay Reasoning in HARQ Systems

Transcription

1 Backlog ad Delay Reasoig i HARQ Systems Sami Aki ad Markus Fidler Istitute of Commuicatios echology Leibiz Uiversität Haover {sami.aki ad markus.fidler}@ikt.ui-haover.de Abstract Recetly, hybrid-automatic-repeat-request (HARQ) systems have bee favored i particular state-of-the-art commuicatios systems sice they provide the practicality of error detectios ad correctios aliged with repeat-requests whe eeded at receivers. he queueig characteristics of these systems have take cosiderable focus sice the curret techology demads data trasmissios with a miimum delay provisioig. I this paper, we ivestigate the effects of physical layer characteristics o data lik layer performace i a geeral class of HARQ systems. Costructig a state trasitio model that combies queue activity at a trasmitter ad decodig efficiecy at a receiver, we idetify the probability of clearig the queue at the trasmitter ad the packet-loss probability at the receiver. We determie the effective capacity that yields the maximum feasible data arrival rate at the queue uder quality-ofservice costraits. I additio, we put forward o-asymptotic backlog ad delay bouds. Fially, regardig three differet HARQ protocols, amely ype-i HARQ, HARQ-chase combiig (HARQ-CC) ad HARQ-icremetal redudacy (HARQ-IR), we show the superiority of HARQ-IR i delay robustess over the others. However, we further observe that the performace gap betwee HARQ-CC ad HARQ-IR is quite egligible i certai cases. he ovelty of our paper is a geeral cross-layer aalysis of these systems, cosiderig ecodig/decodig i the physical layer ad delay aspects i the data-lik layer. I. INRODUCION Due to the dyamic ature of wireless media, trasmissio errors easily occur, especially whe chael fadig is too strog. Hece, i order to esure a error-free (or errormiimized) trasmissio, error detectio ad correctio techiques such as automatic-repeat-request (ARQ) ad forwarderror-correctio (FEC) are egaged i certai classes of commuicatios systems. Specifically i ARQ-furished systems, the receiver feeds the trasmitter with either a positive ackowledgmet (ACK) or a egative ACK (NACK) after each data packet is received correctly or icorrectly so that the trasmitter ca either sed the ext packet or resed the same packet, respectively. O the other had, i FEC each packet is ecoded by itroducig redudat data bits for the receiver to detect ad correct errors. While FEC provides a fixed trasmissio rate, it does ot adjust error correctio capacity with respect to chael variatios. However, ARQ ca adapt to the chael variatios by resedig the trasmitted packets. I that respect i ARQ systems, the trasmissio of a packet ca be completed i a shorter period if the chael atteuatio is weak, whereas it ca be completed i a loger period if his work was supported by the Europea Research Coucil uder Startig Grat the chael atteuatio is strog. hus, i order to provide better error correctio performace, FEC ad ARQ have bee merged to form hybrid-arq (HARQ). Several HARQ protocols have bee suggested ad aalyzed. For istace, origially proposed by Wozecraft ad Horstei [], ype-i HARQ (HARQ-) relies o a bleded process of error detectios ad error correctios while the retrasmissio of oe packet is iitiated if this process fails. Subsequetly, ulike HARQ- i which the past erroeously acquired sigals are discarded, ype-ii HARQ was itroduced i []. Additioal iformatio is trasmitted to the receiver whe the receiver fails decodig the acquired sigal. O that accout, oe of the leadig studies itroduced a adaptivefeedback codig strategy that performs ype-ii HARQ with icremetal redudacy (HARQ-IR) [3]. Likewise, a widelykow method of packet combiig techique, amed ype- II HARQ chase combiig (HARQ-CC), has bee aalyzed i [4]. I HARQ-CC, the repeated trasmissios of a packet, before beig decoded, are liearly added i order to obtai a power gai. From the o, ype-i ad ype-ii HARQ protocols have bee aalyzed from various perspectives such as codig performace [] [7], outage probability [8], ad power optimizatio [9], []. Furthermore, a lik-level performace compariso of these two techiques has bee ivestigated i [], ad it was show that HARQ-IR ca be sigificatly better tha HARQ-CC, especially for high chael-codig rates ad high modulatio orders. Meawhile, sice HARQ systems may require the retrasmissios of packets, delay-sesitive traffic cocers have become a research focus as well. For istace, cosiderig adaptive modulatio techiques, Villa et al. ivestigated latecycostraied etworks []. I aother lie of research, the author i [3] formulated the desig of a HARQ system as a stochastic cotrol problem where the trasmitter cotrols the code rate of each packet i order to achieve a miimum overall average delay. Similarly i [4], effective capacity [], a dual of effective badwidth [6], has bee characterized as the performace metric. A two-dimesioal cotiuoustime Markov chael model has bee employed i order to partitio the istataeous data rate received at the destiatio ito a fiite umber of states each of which is represetig a mode of operatio of the HARQ scheme. I additio, Huag et al. coducted the aalysis of queueig performace of HARQ systems by modelig the wireless chael as a twostate cotiuous Markov process ad derived the queueig

2 For istace, the adoptio of liear codig ad modulatio schemes i HARQ protocols ad their delay performace ca be easily aalyzed usig this coduct. he orgaizatio of the rest of the paper is as follows. I Sectio II, we describe the system model. he, we discuss i Sectio III the system performace regardig packet-loss probability, effective capacity, ad o-asymptotic performace bouds. Fially, we provide the coclusio i Sectio IV, ad the relegated proofs i Appedices A ad B. Fig.. Chael model. he trasmitter iitially stores the data packets i its buffer, subsequetly ecodes, modulates ad forwards each data packet to the receiver through the wireless chael. he, the receiver feeds the trasmitter with ACK/NACK depedig o its decodig performace. delay from geeratig fuctios of the fiite-buffer-capacity system [7]. Usig the framework of the statistical etwork calculus, the authors ivestigated the delay performace of block codes with more or less redudacy [8]. Fially, we ote that takig the first order expasio of the effective capacity uder loose quality-of-service (QoS) costraits ito accout, the authors i [9] studied the impact of deadlie costraits, outage probability, ad QoS costraits o the eergy efficiecy i HARQ systems, ad compared HARQ- CC ad HARQ-IR protocols. Differet tha the above studies, we evaluate the queueig performace of a broad rage of HARQ wireless systems by modelig the trasmitter queue activity aliged with the decodig performace at the receiver as a discrete M-state Markov process with a trasmissio deadlie strategy for each data packet. o the best of our kowledge, a cross-layer examiatio of a geeral class of HARQ systems, regardig the effect of wireless chael atteuatio o decodig performace i physical layer together with queueig performace i data-lik layer, has ot bee performed. Sigly, we ca express our cotributios with the followig: We idetify a approach ad costruct a state-trasitio model for ay HARQ system to ivestigate how advaced trasmissio schemes at the physical layer affect the queueig performace at the data lik layer. Usig the above model, we aalyze the iterplay betwee the packet-loss probability at the receiver due to trasmissio deadlies, ad the steady-state probability of clearig the trasmitter queue. We obtai a closed-form expressio for the effective capacity of HARQ systems, which provides the maximum feasible arrival rate at the trasmitter queue uder QoS costraits such as the asymptotic probabilities of buffer overflow ad delay violatio. Cosiderig the stability coditio of the queue, we aalyze the o-asymptotic backlog ad delay bouds of these systems. I this paper, we provide a geeral framework to ivestigate the iteractios betwee the QoS cocers, ad the codig ad modulatio techiques embedded i HARQ protocols. II. SYSEM MODEL I this sectio, we preset i detail the trasmissio ad chael model, a geeral view of HARQ systems, ad the state trasitio model of these systems. A. rasmissio ad Chael Model We cosider a poit-to-poit trasmissio system i which oe trasmitter ad oe receiver commuicate over a wireless fadig chael as see i Figure. I this model, it is assumed that the trasmitter iitially divides the available data ito packets ofbits ad stores them i its buffer. he, it performs the ecodig, modulatio, ad trasmissio of each packet i time-slots of secods with the first-come first-served policy. It is further assumed that each packet should be trasmitted i maximum M secods due to a trasmissio deadlie for each packet where M is a iteger. Specifically, if a packet is ot received correctly by the receiver at the ed of the M th slot, it will be discarded from the trasmitter queue. Durig the trasfer of each packet, if the packet is successfully decoded at the ed of ay slot, the receiver feeds the trasmitter with a ACK, ad the trasmitter removes the packet from its queue. Otherwise, the receiver seds a NACK, ad the trasmitter, depedig o the trasmissio protocol, either repeats the same ecoded packet or seds the ext partitio of a larger codeword i the ext slot. We ote that a removal of ay data packet from the buffer, either as a result of a successful decodig by the receiver or due to the trasmissio deadlie, is cosidered to be a packet service from the queue. All alog, the discrete-time chael iput-output relatio i the t th symbol istat is give as y(t) = h(t)x(t)+w(t) for t =,,, where x(t) is the complex chael iput ad y(t) is the complex chael output. Above, {w(t)} forms a idepedet ad idetically distributed (i.i.d.) sequece of additive zeromea circularly symmetric, complex Gaussia radom oise variables with a variace E{ w(t) } = σw. Furthermore, h(t) deotes the fadig coefficiet betwee the trasmitter ad the receiver with a arbitrary margial distributio ad a fiite variace E{ h(t) } = E{z(t)} = σh <. Note that, here ad throughout the paper, z(t) = h(t) deotes the magitude-square of the fadig coefficiets. We further assume that chael side iformatio (CSI) is available at the receiver, i.e., the receiver kows the actual value of h(t), while the trasmitter is oly aware of chael statistics. Besides, due to a limited power budget, the chael iput is subject to the

3 Fig.. State-trasitio model with M states. he umber of bits removed from the trasmitter queue is i State, while it is i other states. followig average power costrait: E{ x(t) } P/B where B is the available chael badwidth. Sice we assume that B complex symbols per secod are trasmitted, the average power of the system is costraied by P. Fially, we cosider a block-fadig chael model ad assume that the fadig coefficiet stays costat for a slot duratio of secods ad chages idepedetly from oe slot to aother, i.e., h(lb) = h(lb + ) = = h((l + )B ) = h l i the l th slot, ad h l = z l. his chael model has bee widely used to represet slowly-varyig, flat fadig chaels [, ad refereces therei]. I the above model, the istataeous ergodic chael capacity i the l th slot is give by C l = Blog {+γz l } bits/sec, for l =,,, () where the average trasmitted sigal-to-oise ratio (SNR) is γ = P Bσ. I (), the capacity is achievable whe x(t) is zeromea Gaussia-distributed, i.e.,x(t) CN(, P/B) [, Ch. 9.]. Sice a block-fadig chael is cosidered, we deote the chael fadig power ad the istataeous chael capacity i the l th slot by z l ad C l, respectively. Additioally, we remark that if the trasmitter would kow the istataeous chael fadig coefficiet h l, it could sed the data with a rate equal to the istataeous chael capacity. Sice i practice h l is geerally ot available at the trasmitter, we cosider a more realistic sceario, where a wireless lik is equipped with a HARQ protocol, which ca be modeled as a discrete-time fiite state Markov chai. B. HARQ I geeral, we assume that before beig delivered ito the chael, a packet of bits is ecoded ad modulated ito a codeword of legth either B or M B complex symbols. We ca easily deduce that sice oe fadig slot Codewords ca be composed of symbols from ay modulatio techique with the iput distributio Q(x). he, the capacity i () is replaced with the achievable rate expressed as the mutual iformatio betwee x ad y, i.e., I(x;y). he physical layer framework of this study ad the aalysis i the rest of the paper ca be easily adopted to ay practical liear modulatio techique by usig Q(x) ad I(x;y). is secods, ad the available badwidth is B Hz, a total of B complex symbols ca be trasmitted i oe slot. herefore, a codeword of B complex symbols is assumed to be repeatedly trasmitted i each slot util either it is decoded by the receiver or the trasmissio deadlie is reached. HARQ- ad HARQ-CC are two of these protocols that perform iterative trasmissios of data packets. O the other had, a codeword of MB complex symbols is abstracted ito M sub-codewords that are trasmitted cosecutively i each slot agai util either it is decoded or the trasmissio deadlie is reached, e.g., HARQ-IR. Particularly, the first partitio (codeword or sub-codeword of legth B complex symbols) is set to the receiver i the first attempt. If a successful decodig is performed, the receiver seds a ACK to the trasmitter, ad the the packet is removed from the queue. O the other had, if a decodig failure occurs, the receiver seds a NACK to the trasmitter ad requests the secod partitio. he, the trasmitter seds the secod partitio i the secod attempt. Equivaletly, followig a successful decodig i ay attempt, the packet is removed from the queue; otherwise with a decodig failure, the correspodig ext partitio is trasmitted i the ext attempt. However, at the ed of the M th attempt, the packet is removed from the queue without waitig for ay ACK or NACK sice the trasmissio deadlie for a packet is reached. We fially ote that due to a decodig failure at the ed of the M th attempt, if a packet is removed from the queue ad ot trasmitted agai, we deem it as a lost packet at the receiver i order to differetiate it from the other packet removals. So, the ratio of the lost packets to the total umber of packets served by the trasmitter is a very critical QoS measure sice it etails the average reliable trasmissio rate to the receiver. C. State rasitio Model For a aalytical presetatio, we model the queue activity at the ed of each slot as a discrete-time Markov process. As see i Fig., followig a decodig success at the ed of I order to differetiate ay slot from the slot i which a packet is trasmitted for the first time, we use attempt rather tha slot, uless otherwise eeded for clarity. 3

4 the m th attempt, the system eters State with probability - p m, whereas if a decodig failure occurs the system eters State m with probability p m where m {,,M }. O the other had, regardless of the decodig result at the ed of M th attempt, the system goes to State with probability sice the trasmitted packet is removed from the queue due to the trasmissio deadlie. We ote that a packet trasmitted i the M th attempt is decoded either correctly with probability -p M or icorrectly with probability p M. Accordigly, we have the followig state trasitio matrix P : p p p p M p p P = p p M Now, otig that the aalysis i the paper is for a geeral HARQ protocol, we cosider the followig three differet HARQ protocols i a Rayleigh 3 chael fadig eviromet for a profoud examiatio. ) ype-i HARQ (HARQ-): I this protocol, the trasmitter ecodes ad modulates its data packets, of bits, ito oe codeword by usig a chael code of a codebook C C B of legth B over the complex umbers. he codeword is trasmitted over the chael i oe slot. We assume that if the trasmissio rate is lower tha or equal to the istataeous chael capacity i (), i.e., C l, the receiver is able to decode the codeword reliably. O the other had, if the trasmissio rate is greater tha the capacity, i.e., > C l, the receiver caot recover the packet correctly. Featurig a receiver that does ot perform accumulatio of data, i.e., that uses oly the curret received sigal i ay slot i order to retrieve the packet, the decodig failure probability i ay slot is give by } p = Pr{ > C l = Pr{κ > z l } = e κ/σ h where the probability desity fuctio of expoetiallydistributed z l (i.e., Rayleigh distributed h l ) is f zl (z l ) = e z l/σ h ad κ = ( B )/γ. Sice ay state trasitio does ot deped o the past trasitios, we have p = p = = p M. ) ype-ii HARQ-CC: I this protocol, we agai assume that the trasmitter ecodes ad modulates its data packets, of bits each, by usig a chael code of a codebook C C B of legth B over the complex umbers. However, differet tha the receiver structure i Sectio II-C, we devise a receiver that makes use of the received sigals i the earlier slots by implemetig maximum ratio combiig. Hece, the average accumulated mutual iformatio at the receiver at the 3 he Rayleigh fadig chael assumptio fits well with HARQ schemes i practical mobile wireless systems []. However, other chael models ca be adopted easily ito this study. ed of the m th slot for m M is expressed as [] { } C l+m l = B m m log +γ z l+i bits/sec. i= Above, while Cl l is the chael capacity i the first slot, C l+k l ca be cosidered as the average chael capacity i the first k slots. Notig that the receiver ca decode a packet at the ed of m th slot whe m Cl+m l, we have the followig state trasitio probabilities: { } p m = Pr m > Cl+m l (m ) > Cl+m l = Pr{ m > } Cl+m l { } Pr (m ) > Cl+m l { Pr κ > } m i= z l+i γ ( ) m,κ/σ = { Pr κ > h } = m i= z (m )γ(m,κ/σ l+i h ) where m {,,M}, ad p = e κ/σ h. Above, γ(a,b) is the lower icomplete gamma fuctio where γ(a, b) = b ta e t dt, ad κ is as defied i Sectio II-C. Note that the sum of expoetially-distributed radom variables is gamma-distributed. 3) ype-ii HARQ-IR: I this protocol, differet tha the other two protocols, we assume that the trasmitter ecodes ad modulates its data packets, of bits each, by usig a chael code of a codebook C C MB of legth MB over the complex umbers. he, the trasmitter divides the codewords ito M partitios of the same legth with B complex symbols. I each slot, oe partitio is set to the receiver, ad the receiver decodes the message usig the curret partitio combied with the previously received partitios related to the ecoded data packet. Hece, followig the result i [], we ca see that the receiver ca decode the trasmitted data packet at the ed of the m th slot, if m i= Blog (+γz l+i ). Subsequetly, we ca express the state trasitio probabilities for m {,,M} as follows: { } Pr p m = Pr γκ+ > m i= (+γz l+i) { γκ+ > m i= (+γz l+i) where p = e κ/σ h, ad κ is as defied i Sectio II-C. III. SYSEM PERFORMANCE I this sectio, we cocetrate o the performace measures of the HARQ systems with the above defied state-trasitio model. We iitially obtai the packet-loss probability at the receiver, ad the determie the effective capacity which provides the maximum sustaiable arrival rate at the buffer uder certai QoS costraits. We fially provide the oasymptotic performace measures i the form of backlog ad delay bouds. } 4

5 π ( p lost )/, Reliable hroughput (Mbits/sec) HARQ 3 4 6, Number of Data Bits Ecoded ad Modulated ito Oe ime Slot π ( p lost ) Fig. 3. Reliable throughput, bits/sec., v.s. umber of bits ecoded ad modulated ito B symbols i oe slot,, whe γ = db. π, Rate of Clearig the Queue ( db) ( db) HARQ ( db) ( db) ( db) HARQ ( db) M, Maximum Number of rasmissios Fig. 4. Steady state probability of State, π, as a fuctio of trasmissios deadlie, M, for three differet protocols whe γ = ad db. A. Packet-Loss Probability Let π = [π,,π M ] be the steady-state probability vector of the above state trasitio model i Fig. where M j= π j = ad π = Pπ. hus, we ca obtai the steadystate probabilities as follows: i π i = π p j ad π = j= + M i= i j= p j where i {,,M }. Recallig that at the ed of the M th slot, data packets are removed from the queue due to a decodig failure with probability p M ad due to a decodig success with probability p M, we ca easily express the packet-loss probability, i.e., the ratio of the lost packets to the total umber of packets that are served by the trasmitter, with p lost = p M π M π = p M M j= p j = M j= Note that while π is the steady-state probability of removig a packet from the queue i oe slot, p lost is the probability of a packet ot reachig the receiver. We employ the aforemetioed HARQ protocols, ad plot the average reliable throughput that arrives at the receiver, i.e., π ( p lost) bits/sec, as a fuctio of, i.e., the umber of bits that are ecoded ad modulated ito B symbols i oe slot, whe γ = db i Fig. 3. We cosider the followig settigs: he average chael fadig power is σh =, the slot duratio is = µsec, ad the trasmissio badwidth is B = MHz. We observe that while HARQ-IR sigificatly outperforms the other two protocols, HARQ- is the least achievig protocol for all. Fixig a costat umber of bits served i all protocols 4, we further compare the queue clearig performaces of all the protocols, π, as a fuctio of the trasmissio deadlie, M, whe γ = ad db i Fig. 4. We ca see for ay γ that the probability π decreases geerally with icreasig M, the it coverges after certai M 4 Uless stated otherwise, here ad throughout the paper, we cosider the give trasmissio settigs, ad choose that maximizes the reliable throughput at the receiver whe HARQ- is employed. p j. p lost, Packet Loss Probability 4 6 ( db) ( db) 8 HARQ ( db) ( db) ( db) HARQ ( db) M, Maximum Number of rasmissios Fig.. Packet-loss probability, p lost, as a fuctio of trasmissios deadlie, M, for three differet protocols whe γ = ad db. i all HARQ protocols. All alog, HARQ-IR coverges very quickly, while HARQ-CC performs very close to HARQ-IR. O the other had, HARQ- accomplishes cosiderably less i geeral compared to the ype-ii HARQ protocols for all γ values. Fially, we plot the packet-loss probability as a fuctio of M i Fig.. Ulike π, p lost is cotiuously decreasig with icreasigm. HARQ-IR has a sigificatly lower packetloss probability, ad the performace gap betwee HARQ-IR ad the others icreases dramatically with agai icreasig M. B. Effective Capacity Eve though it seems to be advatageous to icrease M i order to decrease p lost, we ca promptly ifer that the average delay of a packet i the buffer will icrease with icreasigm. herefore, we propose the effective capacity that characterizes the asymptotic decay rate of buffer occupacy. It idetifies the maximum costat arrival rate that a give service process ca support i order to guaratee a desired statistical QoS specified with the QoS expoet θ []. Defiig Q(t) as the statioary queue legth at time t, ad θ as the decay rate of the tail distributio of the queue legth Q(t), we ca express the fol- logpr(q(t) q) lowig: lim q q = θ. herefore, we have the followig approximatio for larger q: Pr(Q(t) q) e θq, which meas that larger θ refers to strict buffer costraits,

6 . ρ S (θ), Effective Capacity (Mbits/sec) HARQ Miimum Service Rate (//M) Average Service Rate (π /) ρ S (θ), Effective Capacity (Mbits/sec) HARQ Miimum Service Rate (//M) Average Service Rate (π /). 4 3 θ, Decay Rate of ail Distributio (db) θ, Decay Rate of ail Distributio (db) Fig. 6. Effective capacity, ρ S (θ), as a fuctio of decay rate of the tail distributio, θ, for three differet protocols whe γ = db. Fig. 7. Effective capacity, ρ S (θ), as a fuctio of decay rate of the tail distributio, θ, for three differet protocols whe γ = db. ad smaller θ implies looser costraits. Furthermore, it is show i [3] that Pr(D(t) d) c Pr(Q(t) q) for costat arrival rates, where D(t) deotes the steady-state delay experieced i the buffer, ad c is a positive costat. I the above formulatio, we have q = ad. herefore, effective capacity provides us with the maximum arrival rate whe the system is subject to the statistical queue legth or delay costraits i the forms of Pr(Q(t) q) e θq or Pr(D(t) d) ce θad/, respectively. For a give QoS expoet θ >, effective capacity, ρ S (θ), is give by ρ S (θ) = Λ( θ) θ = lim t θt log ee{e θs(,t) } () wheres(τ,t) = t k=τ+r(k) is the time-accumulated service process, ad r(k) for k =,,... is the discrete-time, statioary ad ergodic stochastic service process. We ote that Λ(θ) is the asymptotic log-momet geeratig fuctio of S(, t), ad is give by Λ(θ) = lim t t loge{ e θs(,t)}. Heceforth i the ext result, we provide the effective capacity for ay give HARQ protocol. heorem : For the HARQ system with the state-trasitio model give i Sectio II-C, the effective capacity for a give QoS expoet θ is give by ρ S (θ) = θ log ( e p e θ y ) bits/sec (3) where y is the oly uique real positive root of f(y) where f(y) =y M p p y M p M p p M M i= e (M )θ ( p i )p i p p i e iθ y M i. Proof: See Appedix A. he real positive root of f(y) ca be foud by usig umerical techiques. For istace, bisectio method ca be efficietly used to fid the solutio sice it has oly oe real positive root. Employig, agai, the above HARQ protocols, we plot the effective capacity, ρ S (θ), as a fuctio of the decay rate, θ, whe γ = db ad γ = db i Fig. 6 ad Fig. (4) 7, respectively. Above, we set the trasmissio deadlie to M = 4. I both figures, the superiority of HARQ-IR is clearly see, ad HARQ-CC performs close to HARQ-IR. We further see that while the effective capacity of each protocol goes to the average service rate π of the same protocol with decreasig θ, all of them go to the same miimum service rate with icreasig θ. M C. No-asymptotic Performace Bouds So far, we have cosidered the effective capacity that deals with asymptotic characteristics. However, o-asymptotic performace bouds regardig the statistical characterizatios of backlog ad delay are also of iterest for system desigers. For a o-asymptotic aalysis we use the framework of the stochastic etwork calculus [4] [7]. Followig the model i [4, Defiitio 7..], we defie a statistical affie boud for the above chael model for ay θ as follows: { E e θs(τ,t)} e θ[ρs(θ)(t τ) σs(θ)] () where ρ S (θ) is the effective capacity ad σ S (θ) is a slack term that defies a iitial service delay. Although the expressio i () seems to be a upper boud, due to θ, i fact () is a lower boud o the expected amout of service. Now, usig Cheroff s lower boud Pr{X x} e θx E { e θx} for θ, the so-called expoetially bouded fluctuatio model described i [8] with parameters ρ S (θ) >, b, ad Pr{S(τ,t) < ρ S (θ)(t τ) b} ε(b) follows, where ε(b) = e θσs(θ) e θb specifies a expoetially decayig deficit profile of the service. A sample path guaratee follows by the use of the uio boud as: where Pr{ τ [,t] : S(τ,t) < ρ S(θ)(t τ) b} ε (b) ε (b) = eθσs(θ) e θδe θb (6) ad ρ S (θ) = ρ S(θ) δ with free parameter < δ ρ S (θ) a for a arrival rate a. For a detailed derivatio, we refer the iterested reader to [7]. All i all, perceive that the backlog Q(t) = max τ [,t] {a(t τ) S(τ,t)} has a statistical boud 6

7 d, Delay Boud (msec) HARQ d, Delay Boud (msec) HARQ ε, Violatio Probability γ, Sigal to Noise Ratio (db) Fig. 8. Delay boud, d, as a fuctio of the violatio probability, ε, for three differet protocols whe γ = ad a =.4 Mbits/sec with = 8. Fig.. Delay boud, d, as a fuctio of sigal-to-oise ratio, γ, whe M = 4 ad a =.6 Mbits/sec. d, Delay Boud (msec) ε, Violatio Probability HARQ d, Delay Boud (msec) HARQ γ, Sigal to Noise Ratio (db) Fig. 9. Delay boud, d, as a fuctio of the violatio probability, ε, for three differet protocols whe γ = ad a =.8 Mbits/sec ad =. Fig.. Delay boud, d, as a fuctio of sigal-to-oise ratio, γ, whe M = 3 ad a =.8 Mbits/sec. q = max τ [,t] {a(t τ) [ρ S (θ)(t τ) b] +} that may fail with probability Pr{Q(t) > q} ε (b), where [x] + = if x <, ad [x] + = x otherwise accouts for the fact that S(τ,t), geerally. If a ρ S (θ) for stability, b q = a ρ S (θ) δ follows for all t. Accordigly, the delay boud Pr{D(t) > d} ε (b) ca be expressed with d = q/a. I (7), b/(ρ S (θ) δ) provides us with the iitial latecy caused by the variability of the service. Fially, by iversio of (6), b ca be easily obtaied for ay give ε by b = σ S (θ) ( log(ε )+log ( e θδ)). (8) θ As for the existece of the slack term σ S (θ) i (8), cosiderig the result [4, 7..6.ii], we provide the followig lemma: Lemma : If S(τ,t) has a evelope rate ρ S (θ) <, for every ǫ >, there exists σ S (θ) < such that S(τ,t) is (σ S (θ),ρ S (θ) ǫ)-upper costraied (). Proof: See Appedix B. As see i Fig. 8 ad Fig. 9, we plot the delay boud threshold, d, as a fuctio of the violatio probability, ε, whe (7) γ = db ad γ = db whe the arrival rates are a =.4 Mbits/sec ad a =.8 Mbits/sec while = 8 bits/slot ad = bits/slot are ecoded ad modulated ito symbols, respectively. We observe that HARQ- has a very poor performace whe compared with the other ype-ii HARQ protocols. For istace, d is aroud 78 msec whe HARQ- is utilized, whereas it is aroud msec whe the ype-ii HARQ protocols are egaged for ε = 9 ad γ = db. Similarly, whe γ = db, d is reduced from aroud msec to aroud 4 msec. We remark that the arrival rates are differet ad are close to the average service rates i both plots. We agai observe that HARQ-CC ad HARQ-IR achieve delay bouds very close to each other. Furthermore, settig a fixed arrival rate a =.6 Mbits/sec ad a =.8 Mbits/sec, we plot the delay boud threshold, d, as a fuctio of SNR, γ, i Fig. ad Fig. whe M = 4 ad 3, respectively, while ε = 6 ad = 36 bits/slot. As expected, the delay boud decreases with icreasig γ. However, the decrease i d coverges i all HARQ protocols. his is due to the fixed trasmissio slot legth,. It is worth metioig that whe the delay is the mai cocer, expedig trasmissio eergy above some value is useless, ad that whe there is eough eergy provided for data trasmissio, employig a simplified 7

8 d, Delay Boud (msec) 3 3 HARQ π / (Mbits/sec) (HARQ ) π / (Mbits/sec) () π / (Mbits/sec) () d, Delay Boud (msec) 3 3 HARQ π / (Mbits/sec) (HARQ ) π / (Mbits/sec) () π / (Mbits/sec) () a, Data Arrival Rate (Mbits/sec) a, Data Arrival Rate (Mbits/sec) Fig.. Delay boud, d, as a fuctio of data arrival rate, a, whe γ = db. Fig. 3. Delay boud, d, as a fuctio of data arrival rate, a, whe γ = db. techology is more profitable for system desigers. As for the delay performace regardig the variatios i data arrival rates at the buffer, we plot d as a fuctio of a i Fig. ad Fig. 3 whe γ = db ad db while = bits/slot ad bits/slot, respectively. We agai ote ε = 6. I all protocols ad cases, whe the arrival rate is greater tha the average service rate, the delay threshold goes to ifiity sice the system is ot stable ay more. he performace of HARQ- is oticeably degraded with icreasig a whe compared to other HARQ protocols. However, a system furished with HARQ- is more appreciated whe arrival rates are lower. IV. CONCLUSION I this paper, we have cosidered HARQ systems uder QoS costraits such as asymptotic ad o-asymptotic delay ad backlog bouds. We have addressed a practical settig i which data packet trasmissio uder a trasmissio deadlie is performed. We have iitially established a state-trasitio model to defie the queueig characteristics of the system, the we have idetified the steady-state probability of clearig the queue at the trasmitter ad the packet-loss probability at the receiver. We have further derived the effective capacity that features the arrival rate at the trasmitter queue uder QoS costraits. We have the furished the aalysis with the oasymptotic delay ad backlog bouds with respect to a give violatio probability. We have fially preseted the umerical results comparig three differet HARQ protocols. A. Proof of heorem APPENDIX I [4, Chap. 7, Example 7..7], it is show for Markov modulated processes that Λ(θ) = θ θ log e sp{pφ(θ)} = θ log e sp{υ} where sp{υ} is the spectral radius of the matrix Υ, P is the trasitio matrix of the uderlyig Markov process, ad φ(θ) = diag{φ (θ),...,φ M (θ)} is a diagoal matrix, compoets of which are the momet geeratig fuctios of the processes i M states. he rates (umber of bits leavig the queue) supported by the above chael model with the state trasitio model described i the previous sectio ca be see as a Markov modulated process ad hece the setup cosidered i [4] ca be applied immediately ito our settig. We have bits served from the queue i State while we have bit served i other states. herefore, we have φ(θ) = diag{e θ,,...,}. he, we have ( p )e θ p p M p e θ Υ = p. (9) p M Note that Υ i (9) is a o-egative matrix, i.e., Υ, with each elemet greater or equal to zero, i.e., υ ij, where Υ = [υ ij ]. [9, Chap. 8, Corollary 8.3.3] states that if Υ, x = [x,,x M ], ad x the = max x x sp(υ) =max x x [ { mi mi M i M x i j= υ ij x j ( p )e θ + x M + M i= ( p i)x i x, p e θx x,p x x,,p M x M x M }]. () Now, let us assume that the maximum of the miimum i () is obtaied wheever p e θx x = p x x = = p M x M x M = ( p )e θ + x M + M i= ( p i)x i x. () Note that wheever x i for i =,,M chages, the miimum i () will decrease, but it will ot be the maximum of the miimum values. herefore, sp(υ) will be obtaied wheever () is provided. Holdig the equality i (), we 8

9 have ( ) i p i p i p x x i = x p i e ( i)θ x for i= M-. () he, defiig y = x x, ad isertig y ad () ito the equality i (), we have sp(υ) = p e θ y ad y = p M ( p i )p i p + p p i i= e iθ y i + p M p p M e (M )θ y M. (3) Reorderig (3), we obtai the expressio f(y) i (4). I order to aalyze the roots of f(y), we ivoke the followig theorem: heorem (Cauchy s heorem): Let g(x) = x b x b, where all the umbers b i are o-egative ad at least oe of them is o-zero. he polyomial g(x) has a uique positive root x ad the absolute values of the other roots do ot exceed x [3]. Note that f(y) has coefficiets that are o-egative ad at least oe of them is o-zero. herefore, there is oly oe uique real positive root of f(y), deoted by y, which gives the spectral radius of Υ. B. Proof of Lemma We ca ote from () that for every ǫ >, there exists t < such that θ sup logee θs(τ,τ+t) t( ρ S (θ)+ǫ) τ t ( ρ S (θ)+ǫ) (4) whe t > t. 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