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A Method for Cross-layer Analysis of TransmitBuffer Delays in Message Index Domain

Sami Akin and Markus Fidler

Abstract—In data transmission systems, quality-of-service con-straints are commonly defined in the form of buffer overflowprobability or delay violation probability at a transmitter buffer.Some of the studies that employ the large-deviation principlehave taken buffer overflow probability as the quality-of-serviceconstraint and performed the associated analyses in the timedomain. The delay violation probability has been investigatedthrough buffer overflow probability given that there exists aconstant data arrival rate at or a constant service rate froma buffer. These studies cultivated the concepts of effective band-width and effective capacity. Different from the existing studies,we investigate the performance of a transmitter buffer in themessage index domain rather than the time domain by taking thewaiting time (buffering delay) as the primary quality-of-serviceconstraint. We provide two new concepts: effective inter-arrivaltime and effective service time, which are the duals of effectivebandwidth and effective capacity, respectively, in the messageindex domain. The effective inter-arrival time of a data arrivalprocess determines the maximum constant service time for amessage that can sustain the arrival process under a stochasticwaiting time constraint, and the effective service time of a dataservice process determines the minimum constant inter-arrivaltime between successive messages arriving at a buffer that theservice process can sustain. We show that the effective capacity ofa service process or the effective bandwidth of an arrival processcan be obtained through the effective service time or effectiveinter-arrival time of the corresponding process, respectively.We analyze a typical data dissemination and collection task invehicular networks using a broadcast downlink and a slottedAloha uplink transmission.

Index Terms—Cross-layer analysis, large-deviation principle,quality-of-service, effective capacity, effective bandwidth, effec-tive service time, effective inter-arrival time, message indexdomain.

I. INTRODUCTION

With communication devices progressively taking part inour daily lives, the need for faster and more reliable datatransmission techniques is growing. Therefore, investigatingdelay-intolerant transmission models is fundamental. In theview of this motivation, cross-layer performance measures thatcan form a bridge between the limits in the physical layerand the buffer overflow or delay violation performance inthe data-link layer are important. Herein, some of the initialattempts have focused on the large-deviation principle andproposed effective bandwidth. We can define the effectivebandwidth of a stochastic data arrival process at a transmitterbuffer as the minimum necessary constant service rate fromthe buffer that can support the arrival process under a buffer

S. Akin and M. Fidler are with the Institute of Communications Technology,Leibniz Universitat Hannover, 30167 Hanover, Germany. E-mails: {sami.akinand markus.fidler}@ikt.uni-hannover.de.

This work was supported by the European Research Council under StartingGrant–306644.

overflow probability constraint [1]–[5]. The effective band-width of relevant data arrival processes has been investigatedin various studies. For instance, Elwalid et al. showed that itis possible to assign a notional effective bandwidth to eachof the general Markovian data traffic sources at a buffer,which is an explicitly identified and simply computed quantitywith provably correct properties in the natural asymptoticregime of small loss probabilities [6]. Effective bandwidthwas studied in bandwidth sharing mechanisms as well. Maoet al. studied a multi-queue generalized processor sharingsystem with arrivals that are Markov modulated fluid processesand derived the effective bandwidth values of these arrivals[7]. Likewise, the authors in [8] investigated variable bitrate video sources with real-time constraints and presented aconsistent estimator for the effective bandwidth of an arrivalprocess with its confidence interval. Effective bandwidth alsotook attention from other research perspectives as well. Liet al. established a link between effective bandwidth andnetwork calculus and expressed effective bandwidth withinthe framework of a probabilistic version of network calculus,where both data arrival and service processes are specified interms of probabilistic bounds [9].

The effective bandwidth concept fits well into transmissionscenarios where data service rates from a buffer are almostconstant (or constant for a long period of time) but data arrivalprocesses at the same buffer show a stochastic nature. Inwireless communications, on the other hand, the stochastic na-ture of wireless service channels comes into sight more whencompared to data arrival processes. At this juncture, effectivecapacity, the dual of effective bandwidth with respect to thebuffer, emerges as a measure to demonstrate the substantialperformance levels in wireless channels. The concept, whichcan be interpreted as the maximum constant data arrival rateat a buffer that a random service process can support undercertain quality-of-service constraints, was initially provided inthe work of Chang [3, Example 2.5], and leveraged in wirelesscommunications under the term effective capacity by Wu andNegi in [10], [11]. Following the advent of effective capacity,the wireless communications research community welcomedthe notion with a significant recognition. In [12], consideringan ON-OFF wireless channel model, which is independentand identically distributed across time and users, the authorsstudied a multi-user formulation of effective capacity, andshowed that the effective capacity in the channel decreaseswith the increasing burstiness in the channel. Furthermore, Liuet al. analyzed the effective capacity of a class of multiple-antenna systems and evaluated the system performance inlow signal-to-noise ratio regimes [13]. Similarly, focusing onmultiple-antenna systems, the authors in [14] maximized the

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effective capacity of multiple-input multiple-output systemswith channel covariance feedback and showed that the largerthe delay requirement is, the more the spatial degrees of free-dom are used to avoid low instantaneous transmission rates.Likewise, Femenias et al. combined adaptive modulation andcoding in the physical layer with an automatic-repeat-requestprotocol in the data-link layer and obtained the effective ca-pacity of a system by employing a multidimensional physicallayer Markov model [15]. Moreover, effective capacity wasimplemented in cognitive radio research as well. Particularly,the authors obtained the effective capacity of a network, whichis composed of secondary users, and determined the powerallocation policies that maximize the effective capacity of thesecondary users’ relay channels [16]. In addition, the authorsin [17] studied the effective capacity of cognitive radios wherethe secondary users perform transmission in a multi-bandenvironment. For further relevant channel models, we referthe interested readers to [18]–[22], in which effective capacitywas examined.

In the aforementioned studies, the research was based ona quality-of-service constraint in the form of buffer overflowprobability in the steady-state. Specifically, letting q(t) be thenumber of messages (bits or packets) in a data buffer at timeinstant t, the research of effective bandwidth and effectivecapacity provided an approximation for the steady-state bufferoverflow probability Pr{q(t → ∞) ≥ qth}, where qth is thedesired buffer threshold. In other words, effective bandwidthwas addressed as the minimum service rate that can sustainthe desired buffer overflow probability constraint for a givenlarge buffer threshold when the data arrival process has astochastic nature. Likewise, effective capacity was introducedas the maximum data arrival rate that can guarantee the desiredbuffer overflow probability constraint for a given large bufferthreshold when the service process has randomly varyingtransmission rates. Basically, employing effective bandwidthor effective capacity, we can determine certain settings fora service process or an arrival process, respectively, witha desired buffer overflow probability and a defined bufferthreshold.

Meanwhile, based on the research established around thebuffer overflow probability constraint, the delay violationprobability of first-come first-served systems was consideredunder the assumption of either a constant data arrival rate ora constant service rate. For instance, considering the relationa · d(n) = q(t), where a is the constant data arrival rate andd(n) is the delay experienced by message n that leaves thebuffer at time instant t [23], the steady-state delay violationprobability of a message in the buffer can be expressed as afunction of the steady-state buffer overflow probability [10],[13], [19]. Similarly, when there exists a stochastic data arrivalprocess that is served from a buffer at a constant service ratec, we know that the messages q(t) that are in the buffer att are served from the buffer in q(t)

c time units so that thedelay experienced by message n that enters the buffer at t isd(n) = q(t)

c . To the best of our knowledge, an approach thatprecisely treats the delay violation probability constraint whenboth arrival and service processes show a stochastic natureis not available in effective bandwidth and capacity studies,

because there is not a straightforward linear relation betweenthe message backlog and the delay experienced by messagesin a buffer. Furthermore, in order to calculate the effectivebandwidth or the effective capacity of a given arrival or serviceprocess, respectively, Markov processes were exploited toproject the arrival or service process onto analytical schemes.While Markov processes and certain analytical results thatalready exist help ease the effective bandwidth and effectivecapacity depictions, it may be difficult to obtain closed-formsolutions for certain data arrival and service processes. Forexample, the effective capacity of certain hybrid-automatic-repeat-request protocols were provided in [24]–[26], wherethe solutions require certain numerical techniques to be eval-uated or assumptions that facilitate the analysis. Herein, theintroduction of an approach that simplifies the performancemeasure expressions in certain transmission scenarios becomesnecessary.

In this paper, different from the approach in the effectivebandwidth and capacity calculations, we perform analysisin the message index domain rather than the time domain.Specifically, while the number of messages arriving at orthe number of messages leaving a data buffer in a certaintime frame matters in the effective bandwidth or capacityanalysis, respectively, the inter-arrival time between successivemessages arriving at a buffer and the service time that isspent for each message are employed in our analysis. Weinitially characterize the waiting time1 (buffering delay) foreach message in a buffer, and then we show that the progress ofthe waiting time in the message index domain takes the sameform as the progress of the backlog in the same buffer in thetime domain. We define the waiting time for one message asthe time spent in the buffer, i.e., the time between the momentthe message enters the buffer and the moment it is servedfrom the buffer. Therefore, the waiting time of a message in abuffer can be considered as the buffering delay the message isexposed to. More specifically, the contributions of the paperare the following:

1) Characterizing the waiting time for a message in a trans-mitter buffer, we perform the steady-state waiting timeanalysis and obtain an approximation for the waitingtime violation probability as an exponential functionof the waiting time threshold and a quality-of-serviceparameter.

2) Given a stochastic service process, we characterize theminimum constant inter-arrival time between successivemessages arriving at the transmitter buffer, which allowsto sustain the given waiting time violation probabilitywith a given waiting time threshold. We define it as theeffective service time of the stochastic service process.

3) Given a stochastic data arrival process, we administer themaximum constant service time for one message, belowwhich the service process can guarantee the waiting timeviolation probability with a given waiting time threshold.We further define it as the effective inter-arrival time ofthe stochastic data arrival process, which can be regarded

1We use the term waiting time and buffering delay interchangeably.

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Fig. 1. Transmitter buffer and service channel.

as the dual of effective service time with respect to thebuffer.

4) We identify that the effective capacity of a stochasticservice process or the effective bandwidth of a stochasticarrival process can be obtained through the effectiveservice time of the corresponding service process or theeffective inter-arrival time of the corresponding arrivalprocess, respectively.

5) We apply the waiting time analysis to a practical settingof message dissemination and collection schemes forvehicular ad hoc networks in both downlink and uplinkscenarios.

The rest of the paper is as follows: In Section II, weprovide the state-of-the-art regarding the effective bandwidthand capacity notions, and their relation to the buffer overflowprobability constraint. We perform the buffer analysis inthe message index domain in Section III. In particular, wecharacterize the waiting time for messages in a buffer andachieve the steady-state waiting time analysis that providesus the effective inter-arrival time of an arrival process at thebuffer and the effective service time of a service process fromthe buffer. We exemplify our analytical results with an ON-OFF channel model and an ON-OFF data arrival process. Wedemonstrate the fitness of our results in message disseminationand collection schemes in vehicular ad hoc networks in SectionIV and conclude the paper in Section V. We provide a list offrequently used notations in Table I.

II. STATE-OF-THE-ART

One critical concern of providing quality-of-service guaran-tees in communications is to keep the buffer overflow probabil-ity or delay violation probability at a transmitter below certainvalues. One may choose to control the data arrival process at orthe data service process from the transmitter buffer, or both,in order to limit the aforementioned probabilities. However,this becomes a challenging task when the arrival and serviceprocesses vary randomly. Particularly as seen in Figure 1, letaq(t) be the number of messages arriving at the buffer andsq(t) be the number of messages that can be served2 from thebuffer at time instant t. Subsequently, assuming that the buffersize is infinite and the service process is work-conserving, we

2The number of messages leaving the buffer is equal to sq(t) if the numberof messages in the buffer is greater than or equal to sq(t). On the other hand,the number of messages leaving the buffer is equal to the number of messagesin the buffer if the number of messages in the buffer is smaller than sq(t).

TABLE INOTATIONS

Symbol Descriptiont Time instant or time frame index

aq(t) Number of messages arriving at a bufferat time instant t

Aq(t) Cumulative number of messages arrivingat a buffer until time instant t

sq(t) Available service in units of messagesfrom a buffer at time instant t

Sq(t) Available cumulative service in units ofmessages from a buffer until time instant t

D(t) Cumulative number of messages departingfrom a buffer until time instant t

q(t) Number of messages in a bufferat time instant t

qth Buffer overflow thresholdθq Decay rate of the tail distribution of q(t)

ΛAq (θq) Log-moment generating function of Aq(t)ΛSq (θq) Log-moment generating function of Sq(t)aq(θq) Effective bandwidth of Aq(t) (minimum

constant service rate from a buffer)sq(−θq) Effective capacity of Sq(t) (maximum

constant arrival rate at a buffern Message index

Mn Message nad(n) Inter-arrival time between messages Mn−1

and Mn

Ad(n) Time when message Mn arrives at a buffersd(n) Service time of message Mn−1

Sd(n) Cumulative service time of messages M1

to Mn−1

U(n) Time when message Mn starts servicefrom a buffer

d(n) Waiting time of message Mn in a bufferdth Waiting time thresholdθd Decay rate of the tail distribution of d(n)

ΛAd(θd) Log-moment generating function of Ad(n)ΛSd(θd) Log-moment generating function of Sd(n)ad(−θd) Effective inter-arrival time of Ad(n)

(maximum constant service time)sd(θd) Effective service time of Sd(n)

(minimum constant inter-arrival time)

can express the backlog (queue length) at time instant t, i.e.,the number of messages in the buffer, as follows:

q(t) = [q(t− 1) + aq(t)− sq(t)]+, (1)

where [·]+ = max{·, 0}. The time-line representation of thebacklog with the data arrival and service processes is shownin Fig. 2.

The aforementioned backlog characterization in (1) canreflect the buffer dynamics intuitively. Nevertheless, we canalso justify (1) by invoking cumulative arrival and serviceprocesses. We note that we will use this method as an approachin waiting time characterization as well. While we can perform

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Fig. 2. Time frame representation of the backlog at the transmitter buffer.

the following analysis with an initial backlog (i.e., q(0) 6= 0),we consider q(0) = 0 and set aq(t) = 0 and sq(t) = 0for t ≤ 0 for the sake of brevity. Basically, we let the totalnumber of messages arriving at the buffer in t time instants beAq(t) =

∑tτ=1 aq(τ) and the available total service provided

in t time instants be Sq(t) =∑tτ=1 sq(τ). Subsequently, the

total number of messages that depart from the buffer in thefirst t time instants is given as

D(t) = min{Aq(t), D(t− 1) + sq(t)}.

The cumulative departure process, D(t), and the backlog, q(t),which is the difference between Aq(t) and D(t), are displayedin Fig. 3. Now, using the relation between the cumulativedeparture process and the backlog in the buffer, we have

q(t) = Aq(t)−D(t)

= Aq(t)−min{Aq(t), D(t− 1) + sq(t)}= max{0, Aq(t)−D(t− 1)− sq(t)}= [Aq(t− 1)−D(t− 1) + aq(t)− sq(t)]+

= [q(t− 1) + aq(t)− sq(t)]+,

which confirms (1). Additionally, d(t) in Fig. 3 indicates theduration that a message, which arrives at the buffer at timeinstant t, spends in the buffer until it is served from the bufferin the first-come first-served order.

It is further known that q(∞) will converge in distributionto a finite random variable when both aq(t) and sq(t) arestationary and ergodic, and when E{aq(t)} < E{sq(t)} [3].Now, given that aq(t) and sq(t) are independent of each other,we have a unique θ?q such that

ΛAq (θ?q ) + ΛSq (−θ?q ) = 0 (2)

andθ?q = − lim

qth→∞

log Pr{q(∞) ≥ qth}qth

, (3)

where qth is the threshold and θ?q is the decay rate ofthe tail distribution of the backlog, q(t) [3, Theorem 2.1].Above, ΛAq (θq) = limt→∞

1t logE

{eθqAq(t)

}and ΛSq (θq) =

limt→∞1t logE

{eθqSq(t)

}are the asymptotic log-moment

generating functions of the arrival and service processes,respectively, that are the Gartner-Ellis limits and differentiablefor all θq ∈ R. As for the physical meaning of θ?q , the steady-state backlog is expressed with an exponential function of θ?qand qth [4], i.e.,

Pr{q(∞) ≥ qth} ≈ e−θ?q qth .

Specifically, larger θ?q implies stricter quality-of-service con-straints, while smaller θ?q means looser quality-of-serviceconstraints.

Fig. 3. Graphical representation of arrival and service processes in time frame.Aq(t), Sq(t) and Dq(t) are the cumulative data arrival, service and departureprocesses in the time domain, respectively.

Now, setting a constant service rate, i.e., sq(t) = sq for allt > 0, we have ΛSq (θq) = θqsq . Then, using the relation (2),we obtain

sq =ΛAq (θ

?q )

θ?q= limt→∞

1

tθ?qlogE

{eθ?qAq(t)

}= aq(θ

?q ) (4)

for θ?q > 0. Specifically, given an arrival process, Aq(t),the term aq(θq) denotes the effective bandwidth of the sameprocess for any given θq > 0, where θq is the quality-of-service metric. Similarly, setting a constant data arrival rate,i.e., aq(t) = aq for all t > 0, we have ΛAq (θq) = θqaq . Then,using again the relation (2), we obtain

aq =ΛSq (−θ?q )

−θ?q= limt→∞

−1

tθ?qlogE

{e−θ

?qSq(t)

}= sq(−θ?q )

(5)

for θ?q > 0. Here, sq(−θq) denotes the effective capacity ofthe given service process, Sq(t), for any given θq > 0.

It is worth mentioning that another approximation for thebuffer overflow probability for given θq > 0 when qth is smallis provided in [10] as

Pr{q(∞) ≥ qth} ≈ εe−θqqth ,

where ε is the probability that the buffer is not empty. Further,for the special case of a constant arrival rate, the delay bound,i.e., Pr{d(∞) ≥ dth}, for a given delay threshold dth isexpressed as [10]

Pr{d(∞) ≥ dth} ≈ εe−θq sq(−θq)dth .

The delay probability approximation comes from the fact thatthe data arrival rate at the buffer is constant and set to theeffective capacity (the maximum sustainable constant dataarrival rate). Particularly, dth = qth

sq(−θq) and d(∞) = q(∞)sq(−θq) .

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Fig. 4. Graphical representation of arrival and service processes in message.Ad(n) is the time instant at which message n arrives at the buffer, Sd(n) isthe time instant at which the service is available for message n, and Ud(n)is the time instant at which message n is served from the buffer.

Notice that by employing the queue balance equation in (2),we can obtain the buffer overflow probability as an exponentialfunction of the buffer threshold, qth, and the decay rate,θ?q , when either the arrival process or the service process isstochastic, or when both processes are stochastic. However, itis difficult to obtain a closed-form solution for the delay viola-tion probability when both the arrival and service processes arestochastic. In most practical systems, both the data arrival andservice rates vary over time. Therefore, it is necessary to havea methodology that provides the delay violation probabilityin a buffer when both the arrival and service processes arestochastic. Further notice that in order to obtain the effectivebandwidth (4) and the effective capacity (5) in some commu-nications models, we resort to either numerical techniques orparticular assumptions. For instance, the effective capacity ofa general class of hybrid-automatic-repeat-request protocolscan be obtained by invoking numerical search techniques[24]. Or, a closed-form solution for the effective capacityof the aforementioned system can be obtained under less-strict quality-of-service constraints only [25]. However, if weperform analysis in the domain spanned by message indicesrather than time instants, we can characterize the waitingtime violation probability when both the arrival and serviceprocesses are stochastic, and we can formulate the systemperformance with closed-form solutions in systems where itis difficult to establish straightforward analytical expressionsfor effective bandwidth and capacity. In the next section, weshow the details of our approach and methodology in whichwe obtain a waiting time violation probability in the formof an exponential function by performing our analysis in themessage index domain.

Fig. 5. Transmitter buffer with delay representation.

III. ANALYSIS IN MESSAGE INDEX DOMAIN

As seen in Fig. 3, we analyze the effective bandwidth of agiven data arrival process and the effective capacity of a givenservice process by positioning the time and the number ofmessages on the x-axis and y-axis, respectively, and performanalysis in the time domain. However, we can resort to adifferent approach by invoking the pseudo-inverse functionsof the cumulative arrival and service processes. This approachwas used, e.g., in deterministic data traffic regulation [27, Sec.6.2.3]. Intuitively, the pseudo-inverse corresponds to rotatingand flipping the graph in Fig. 3. Particularly, the numberof messages and the number of time instants are positionedon the x-axis and y-axis, respectively, as shown in Fig. 4.Suitably, considering that the messages arrive at the bufferfrom a source one by one, let each message be denoted by Mn

for n ∈ {1, 2, · · · }, where n is the message index. As seenin Fig. 5, message Mn+1 arrives at the buffer in ad(n + 1)time units after message Mn arrives at the buffer. Similarly,message Mn+2 arrives at the buffer in ad(n + 2) time unitsafter message Mn+1 arrives at the buffer. Therefore, ad(n+1)and ad(n + 2) can be referred to as the inter-arrival timebetween messages Mn and Mn+1 and the inter-arrival timebetween messages Mn+1 and Mn+2, respectively.

We define the waiting time of a message as the timedifference between the arrival of the message at the buffer andthe start of the service of that message. If the message arrivesat an idle system (the buffer is empty and there is no othermessage in service), it is immediately served from the bufferand the waiting time of the message is zero. On the otherhand, if the system is busy, the message waits for previousmessages (in service and possibly in the buffer) to completeservice in the first-come first-served order. We assume that theservice becomes available for message Mn in sd(n) time unitsafter message Mn−1 starts service from the buffer. Likewise,the service becomes available for message Mn−1 in sd(n−1)time units after message Mn−2 starts service. Since we assumethat the service process has a work-conserving policy, i.e.,the system does not idle when there are messages in thesystem, we can regard sd(n) and sd(n − 1) as the servicetime of message Mn−1 and the service time of message Mn−2,respectively.

Hence, following the aforementioned system description inthe sequel, we initially characterize the waiting time a messageexperiences in the buffer, and then perform the steady-statewaiting time analysis and provide the effective service andinter-arrival time expressions for given data service and arrivalprocesses, respectively.

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A. Waiting Time Characterization

Although the following analysis can be performed withan initial backlog in a buffer or an initial waiting time fora message, we assume that there are no messages in thebuffer before message M1 arrives at the buffer. Therefore, thewaiting time of M1 is equal to zero, i.e., d(1) = 0 whered(n) represents the waiting time of message Mn. MessageM1 starts service immediately after arriving at the buffer andfinishes service after sd(2) time units so that service becomesavailable for message M2, which comes to the buffer in ad(2)time units after message M1. If the service becomes availablefor message M2 before message M2 comes to the buffer, i.e.,sd(2) ≤ ad(2), message M2 does not wait in the buffer tobe served. Basically, the waiting time for M2 is zero. Onthe other hand, if M2 is in the buffer before the service isavailable for M2, i.e., sd(2) > ad(2), message M2 waits inthe buffer for sd(2) − ad(2) time units. Hence, the waitingtime for M2 is expressed as d(2) = [sd(2) − ad(2)]+. Asfor message M3, we know that M3 arrives at the bufferin ad(3) time units after M2 comes to the buffer and thatthe service is ready for M3 in sd(3) time units after M2

starts service from the buffer. Accordingly, if M2 does notwait in the buffer because sd(2) ≤ ad(2), the waiting timefor M3 is d(3) = [sd(3) − ad(3)]+. On the other hand, ifM2 waits in the buffer for sd(2) − ad(2) time units becausesd(2) > ad(2), the service becomes available for M3 insd(2)−ad(2)+sd(3) time units. As a result, the waiting timefor M3 becomes [sd(2)−ad(2)+sd(3)−ad(3)]+. Combiningthe two cases, the waiting time for M3 can be expressed as[[sd(2)−ad(2)]+ +sd(3)−ad(3)]+ = [d(2)+sd(3)−ad(3)]+.Generalizing for message Mn, we can characterize the waitingtime as follows:

d(n) = [d(n− 1) + sd(n)− ad(n)]+ for n ≥ 2, (6)

where d(n) = 0, sd(n) = 0 and ad(n) = 0 for n < 2. Clearly,the waiting time increases with the service time and decreaseswith the inter-arrival time. The waiting time, d(n), and thebacklog, q(n), are shown graphically in Fig. 4. The backlog,q(n), provides the number of messages in the buffer whenmessage Mn is being served.

Above, we developed the waiting time characterization in(6) inductively. Considering a formal proof, let U(n) be thetime instant at which message Mn starts service, and it isexpressed as

U(n) = max {Ad(n), U(n− 1) + sd(n)} , (7)

where Ad(n) =∑ni=1 ad(i) is the time instant at which

message Mn arrives at the buffer. Therefore, the waiting timefor Mn is given as

d(n) = U(n)−Ad(n)

= max {Ad(n), U(n− 1) + sd(n)} −Ad(n)

= max {0, U(n− 1)−Ad(n) + sd(n)}= [U(n− 1)−Ad(n− 1)− ad(n) + sd(n)]+

= [d(n− 1) + sd(n)− ad(n)]+,

which confirms the characterization in (6).

B. Steady-State Waiting Time AnalysisAnalogous to the time domain model in (1) that defines

the backlog in data units, the message index domain model(6) evaluates the backlog in time units. This is a consequenceof the first-come first-served assumption that implies that theentire backlog q(Ad(n)) that exists at the time of arrival ofmessage Mn is cleared during the waiting time d(n) of thatmessage. Corresponding to the backlog analysis [3], we canconclude from (6) that the steady-state waiting time d(∞)converges in distribution to a finite random variable whenboth sd(n) and ad(n) are stationary and ergodic, and whenE{sd(n)} < E{ad(n)}. Recalling that the service process iswork-conserving and that the arrival and service processes areindependent of each other, we have a unique θ?d such that

ΛSd(θ?d) + ΛAd(−θ?d) = 0 (8)

andθ?d = − lim

dth→∞

log Pr{d(∞) ≥ dth}dth

,

where dth is the waiting time threshold and θ?d is thedecay rate of the tail distribution of the waiting time,d(n) [3, Theorem 2.1]. The Gartner-Ellis limits that aredifferentiable for all θd ∈ R are given as follows:ΛSd(θd) = limn→∞

1n logE

{eθdSd(n)

}and ΛAd(θd) =

limn→∞1n logE

{eθdAd(n)

}, where Sd(n) =

∑nm=1 sd(m)

and Ad(n) is given in (7). Hence, for a large dth, we have thefollowing approximation for the waiting time in the steady-state for one message:

Pr{d(∞) ≥ dth} ≈ e−θ?ddth . (9)

Now, considering a constant inter-arrival time betweenconsecutive messages, i.e., ad(n) = ad, and then benefitingfrom the relation in (8) along with ΛAd(θd) = θdad, we obtain

ad =ΛSd(θ?d)

θ?d= limn→∞

1

nθ?dlogE

{eθ?dSd(n)

},

which is the minimum constant inter-arrival time betweenconsecutive messages that the defined service process cansustain for θ?d > 0. We note that θ?d is different from thedecay rate, θ?q , of the tail distribution of the queue length, q(t).In particular, θ?d expresses the strictness of the waiting timeviolation probability constraint, while θ?q shows the strictnessof the buffer overflow probability constraint.

Definition 1 (Effective Service Time): Given a serviceprocess Sd(n) that is work-conserving and stationary withrandomly distributed service time for a message, the minimumconstant inter-arrival time between consecutive messages ar-riving at the buffer that the service process can sustain underthe quality-of-service constraints defined by the decay rate ofthe tail distribution of the waiting time, θd > 0, is calledeffective service time, and it is expressed as

sd(θd) =ΛSd(θd)

θd= limn→∞

1

θdnlogE

{eθdSd(n)

}. (10)

In (10), if the service time samples are independent andidentically distributed, the effective service time of the givendata service process becomes

sd(θd) =1

θdlogE

{eθdsd(n)

}for any n ≥ 2. (11)

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Notice that when θd goes to zero in (11), i.e., when thereare no quality-of-service constraints, the effective service timeapproaches the average service time in the channel. Basically,

limθd→0

sd(θd) = limθd→0

ΛSd(θd)

θd= E {sd(n)} .

On the other hand, when θd goes to infinity, i.e., when there arethe strictest quality-of-service constraints, the effective servicetime becomes the maximum service time, i.e.,

limθd→∞

sd(θd) = limθd→∞

ΛSd(θd)

θd= max {sd(n)} .

Similarly, considering a constant service time for eachmessage, i.e., sd(n) = sd, and benefiting from the relationin (8) along with ΛSd(θd) = θdsd, we obtain

sd =ΛAd(−θ?d)

−θ?d= limn→∞

−1

nθ?dlogE

{e−θ

?dAd(n)

},

which is the maximum allowable constant service time for amessage that sustains the defined arrival process for θ?d > 0.

Definition 2 (Effective Inter-arrival Time): Given an arrivalprocess Ad(n) that is stationary with randomly distributedinter-arrival time between consecutive messages, the maximumallowable constant service time for a message being servedfrom the buffer that supports the arrival process under thequality-of-service constraints defined by the decay rate of thetail distribution of the waiting time, θd > 0, is called effectiveinter-arrival time, and it is expressed as

ad(−θd) =ΛAd(−θd)−θd

= limn→∞

−1

θdnlogE

{e−θdAd(n)

}.

(12)In (12), if the inter-arrival time samples are independent

and identically distributed, the effective inter-arrival time ofthe given data arrival process becomes

ad(−θd) =−1

θdlogE

{e−θdad(n)

}for any n ≥ 2. (13)

Notice again that when θd in (13) goes to zero, i.e., when thereare no quality-of-service constraints, the effective inter-arrivaltime approaches the average inter-arrival time of the process,i.e.,

limθd→0

ad(−θd) = limθd→0

−1

θdlogE

{e−θdad(n)

}= E{ad(n)}.

On the other hand, when θd goes to infinity, i.e., when there arethe strictest quality-of-service constraints, the effective inter-arrival time approaches the minimum inter-arrival time of theprocess, i.e.,

limθd→∞

ad(−θd) = limθd→∞

−1

θdlogE

{e−θdad(n)

}= min{ad(n)}.

To illustrate the concepts of effective service time andeffective inter-arrival time, we consider the basic scenario ofON-OFF processes. ON-OFF processes are a relevant modelto characterize random outages, e.g., of wireless block fadingchannels. Further, the ON-OFF model has the convenientproperty that it can be defined equivalently in the time domainas well as in the message domain permitting us to drawimportant conclusions on the connection of the two domains.

Fig. 6. Effective service time, sd(θd), vs. waiting time decay rate, θd(dB) =10 log10(θd).

1) ON-OFF Service Channel Model: Let us consider adiscrete-time ON-OFF channel model with a frame dura-tion of T time instants as the service process. The channelis ON with probability ρ and it is OFF with probability(1− ρ). The channel state changes from one frame to anotherindependently. In each frame, if the channel is ON, onemessage is successfully served from the buffer. Otherwise,the transmission of the message fails, and then the messageis re-transmitted in the next frame. Therefore, the probabilitydistribution of the service time of message Mn is given asPr{sd(n) = iT} = ρ(1−ρ)i−1 for i ∈ {1, 2, · · · }. As a result,the effective service time of the process (the minimum constantinter-arrival time that can be sustained) is characterized asfollows:

sd(θd) =1

θdlogE

{esd(n)θd

}=

1

θdlog

{ ∞∑i=1

ρ(1− ρ)i−1eiTθd

}

=1

θdlog

{ρeTθd

1− (1− ρ)eTθd

}(14)

when 0 < θd < − log{1−ρ}T and sd(θd) = ∞ when

− log{1−ρ}T ≤ θd. Notice that the effective service time ap-

proaches Tρ when θd goes to zero, which is the average service

time and hence the reciprocal of the average transmission rateof the channel.

In Fig. 6, we set the transmission frame to 0.1 milliseconds,i.e., T = 0.1, and plot the effective service time as a functionof the decay rate parameter, θd. The effective service timeapproaches the average service time, which is 0.1

ρ , as θddecreases to zero, whereas the effective service time goes toinfinity with increasing θd. Herein, for a better understandingof the results in Fig. 6, let us consider that we are given awaiting time violation probability constraint, Pr{d(n) > dth}for large n, and a waiting time threshold, dth. Then, we canuse (9) to calculate the decay rate of the tail distribution ofthe waiting time, θd, as follows: θd ≈ − log Pr{d(n)>dth}

dth. Now,

having θd, we can easily determine the effective service time

8

of the defined ON-OFF service process through (14), which isthe minimum inter-arrival time between consecutive messagesthat arrive at the transmitter buffer such that the waiting timeviolation probability constraint for the desired waiting timethreshold is guaranteed.

Connection to the queue decay rate, θq: Let us againconsider the aforementioned scenario and assume that theconstant data arrival rate is a messages per time instant, andhence, the constant inter-arrival time between two consecutivemessages is 1

a time instants. Noting the relations in (8) and(14), we have a unique waiting time decay rate such that

1

a=

1

θ?dlog

{ρeTθ

?d

1− (1− ρ)eTθ?d

}. (15)

Moreover, noting the relation given in (2) and (5), and thatthe service rate is sq(t) = 1 message per one frame of T timeinstants if the service channel is ON with probability ρ andsq(t) = 0 messages per time frame if the service channel isOFF with probability (1− ρ), we have a unique queue decayrate such that

a = limt→∞

−1

tTθ?qlogE

{e−θ

?qSq(t)

}(16)

= − 1

Tθ?qlogE

{e−θ

?qsq(t)

}(17)

= − 1

Tθ?qlog{ρe−θ

?q + 1− ρ

}. (18)

Notice that t is the time frame number and T is the frameduration in (16). Because a is given in messages per timeinstant, we divide the expression in (16) by T . Moreover, (17)follows from the fact that the channel state changes from oneframe to another independently. Hence, noting that the momentgenerating function of the service is E{eθsq(t)} = ρeθq+1−ρ,we obtain the result in (18). Above, the right-hand-side of (18)for given θq provides us the effective capacity of the serviceprocess. Then, solving (15) and (18), we obtain

0 =1− ρ+ ρe−θ?da − e−θ

?dT

and

0 =1− ρ+ ρe−θ?q − e−aθ

?qT ,

respectively. Since the aforementioned balance expressions areobtained when the buffer is in steady-state, we verify a linearrelation between the decay rate of the waiting time and thedecay rate of the backlog such that θ?d = aθ?q .

In the following theorem, given any stationary and ergodicdata service process or data arrival process, we generalize theaforementioned result and provide the relation between theeffective service time and capacity of a given service process,and the relation between the effective inter-arrival time andbandwidth of a given arrival process.

Theorem 1: Given that the effective service time of a dataservice process is sd(θd) for θd > 0 and the effective capacityof the same service process is sq(−θq) for θq > 0, there existsa reciprocal relation between sd(θd) and sq(−θq) such that

sd(θd)sq(−θq) = 1 and θq = sd(θd)θd. (19)

Likewise, given that the effective inter-arrival time of a dataarrival process is ad(−θd) for θd > 0 and the effectivebandwidth of the same arrival process is aq(θq), there existsa reciprocal relation between ad(−θd) and aq(θq) such that

ad(−θd)aq(θq) = 1 and θq = ad(−θd)θd. (20)

Proof: See Appendix A. �Connection of message index and time domain models:

Theorem 1 lays ground for a method for construction of timedomain models from message index domain models and viceversa. First, notice that the decay rate of the tail distributionof the queue length in the steady-state, θq , in (19) is thelog-moment generating function of the service process in themessage index domain. In particular, we have with (10) that

θq = sd(θd)θd = ΛSd(θd). (21)

Now, define f(θd) = ΛSd(θd) to be a function of the decayrate of the tail distribution of the waiting time in the steady-state, θd. Then, we can solve θq = f(θd) for θd to express θdas a function of θq as follows: θd = f−1(θq), where f−1(θq)is the inverse function. In general, when we are given theeffective service time of a service process in the message indexdomain, we can use that sd(θd)sq(−θq) = 1 from (19) to reachthe effective capacity of the same service process in the timedomain by taking the reciprocal of the effective service timeand substituting f−1(θq) for θd, i.e.,

sq(−θq) =1

sd(f−1(θq)).

For instance, noting the log-moment generating functionof the ON-OFF service channel model in (14) as f(θd) =

log{

ρeTθd

1−(1−ρ)eTθd

}, and letting θq = f(θd) we can solve for

θd = f−1(θq) to show that

θd =1

Tlog

{eθq

ρ+ (1− ρ)eθq

}=− 1

Tlog{ρe−θq + 1− ρ

}. (22)

Then, plugging (22) into the effective service time (14) andtaking the reciprocal of (14), we express the effective capacityof the ON-OFF service channel model as

sq(−θq) =1

sd(θd)= − 1

Tθqlog{ρe−θq + 1− ρ

}, (23)

which also confirms (18).Overall, we know that achieving closed-form solutions for

the effective capacity or bandwidth of certain processes inthe time domain may be intractable. On the other hand, wecan carry out an analysis in the message index domain, andthen we can arrive at solutions for the effective capacity orbandwidth by invoking Theorem 1. Similarly, if results in themessage index domain cannot be achieved, we can acquiresolutions through an analysis in the time domain as well.

2) ON-OFF Data Arrival Process: In addition to the ON-OFF channel model described in Section III-B1, let us alsoconsider an ON-OFF arrival process3 with an ON probability

3Our aims are twofold in this analysis. First, in practical systems, boththe arrival and service processes are generally stochastic. Second, ON-OFFarrival and service process models provide a clear insight into the analyticalfindings from a practical perspective to understand the buffer dynamics.

9

λ. In particular, a message arrives at the buffer with probabilityλ in a frame of T time instants. Noting that the arrivalprocess has two states, i.e., ON and OFF states, we furtherassume that the arrival process changes its state from oneframe to another independently. Therefore, the probabilitydistribution for the inter-arrival time between message Mn−1

and Mn is given as Pr{ad(n) = iT} = λ(1 − λ)i−1 fori ∈ {1, 2, · · · }. Hence, regarding independent and identicallydistributed inter-arrival time samples, we formulate the asymp-totic log-moment generating function of the arrival processas follows: ΛAd(θd) = log

{λeTθd

1−(1−λ)eTθd

}for 0 < θd <

− log{1−λ}T and ΛAd(θd) =∞ for − log{1−λ}

T ≤ θd. Hence, weinvoke the relation (8) and have the following characterization:

log

{ρeTθ

?d

1− (1− ρ)eTθ?d

}+ log

{λe−Tθ

?d

1− (1− λ)e−Tθ?d

}= 0,

which provides the decay rate of the tail distribution of thewaiting time as

θ?d =1

Tlog

{1− λ1− ρ

}(24)

where λ < ρ. Notice that the decay rate increases with theincreasing ON probability in the service channel while itdecreases with the increasing ON probability in the arrivalprocess.

Non-linear relation between θd and θq: Let us againconsider the communications scenario in Section III-B2. Weknow that the decay rate of the tail distribution of the waitingtime is given in (24). As for the decay rate of the taildistribution of the queue, we start with defining the servicerate and the arrival rate in one time frame. In particular, theservice rate is sq(t) = 1 message per time frame if the servicechannel is ON with probability ρ and sq(t) = 0 messagesper time frame if the service channel is OFF with probability(1 − ρ). Similarly, the arrival rate is aq(t) = 1 message pertime frame if the data arrives at the buffer with probability λand aq(t) = 0 messages per time frame if the data does notarrive at the buffer with probability (1 − λ). Noting that thestate transitions occur independently in both the service andarrival process, and invoking the relation in (2), we can showthat

1

Tlog{λeθ

?q + 1− λ

}+

1

Tlog{ρe−θ

?q + 1− ρ

}= 0.

Given that 0 < θ?q , we can obtain the decay rate of the taildistribution of the queue length as

θ?q = log{ρλ

}+ log

{1− λ1− ρ

}(25)

= log{ρλ

}+ Tθ?d. (26)

Specifically, (26) shows that there is not always a linearrelationship between θ?d and θ?q when both the arrival andservice processes are stochastic. Following this result, we cansee that by taking the buffer overflow probability constraint asthe primary quality-of-service metric, we may not easily reacha conclusion on the buffering delay performance because ofa possible non-linear relation between θ?d and θ?q . Therefore,

the results in this paper provide a straightforward method toobtain a performance measure regarding the waiting time in atransmitter buffer.

IV. VEHICULAR AD HOC NETWORKS

In this section, we substantiate our analytical results inpractical settings. We consider first a reliable broadcast in adownlink scenario, where a transmitter employs a fountainencoder and receivers send feedback to the transmitter onlywhen they stop receiving the transmitted data packets, e.g.,when they have successfully decoded the message. Then, weemploy the slotted Aloha transmission protocol as a relevantmulti-access protocol in an uplink scenario. These two scenar-ios can be embedded in message dissemination and collectionschemes for vehicular ad hoc networks. For instance, in orderto minimize packet arrival time and bandwidth occupancy,the authors studied fountain codes-based data disseminationtechniques in vehicular ad hoc networks and showed the ef-fectiveness of their techniques when compared with traditionaldata dissemination approaches [28]. Furthermore, the authorsin [29] studied mobile slotted Aloha protocols in vehicular adhoc networks and validated their solutions through simulationsthat confirm a dramatic improvement in the scalability of thesolution and in the overall protocol performance. Besides,we consider a time-constrained4 communications scenario.Particularly, in the downlink scenario, a base station broad-casts a time-constrained message to the vehicles that are incertain proximity to the base station. In the uplink scenario,the vehicles in close proximity pass their time-constrainedmessages to the base station in a multi-access manner. We alsonote that while we analyze these two protocols for numericalpresentations, we emphasize that our analytical methods arefor a general class of communications scenarios.

A. Downlink Broadcast Transmission

We consider a downlink scenario in which a transmitter(base station) sends a message to L receivers as seen in Fig. 7.The transmitter partitions the message into K encoding (data)packets and composes N encoded packets by randomly form-ing a combination of the encoding packets, where K < N .Subsequently, the transmitter broadcasts the encoded packetsone-by-one in frames of T time units. During the transmissionof each message, when a receiver is able to collect K? of theencoded packets successfully, where K ≤ K? < N , it candecode the message. On the other hand, when the receiverrealizes in any time frame that it is not possible to recoverthe message even if it receives the future encoded packets,it stops the reception of the encoded packets. In both cases,the receiver sends an acknowledgment to the transmitter toinform about its status. Subsequently, the transmitter stopsthe transmission when it collects acknowledgments from allthe receivers or when it completes the transmission of allthe N encoded packets regardless of the number of received

4We assume messages have a limited lifetime as, e.g., in the case of periodicupdates where each message supersedes the previous ones. The lifetime isreflected by a delivery deadline. Hence, messages are time-constrained in thesense that delivery is important before the deadline but not afterwards.

10

Fig. 7. System model and communication scenario.

acknowledgments. The transmitter in this scenario can beassociated with a fountain encoder [30] or a Luby transform(LT) encoder [31], [32], where the only difference comes fromthe acknowledgments that the receivers send when they aredone with the reception. On the other hand, the transmitterscenario is different than the conventional automatic-repeat-request transmission protocols in such a way that the receiversdo not confirm every successful packet reception but theysend acknowledgments only once when they either decode thewhole message or fail to decode the whole message.

Accordingly, let ρl denote the probability of successfulreception of an encoded packet by the lth receiver for l ∈{1, · · · , L}. Principally, we consider an ON-OFF channelmodel between the transmitter and the lth receiver, i.e., thechannel is ON with probability ρl and OFF with probability1−ρl. Let us initially assume 2K? ≤ N+1. Then, noting thatone receiver needs at least K? encoded packets, we realize thata receiver may declare a success in decoding and then send anacknowledgment to the transmitter in one of the time framesstarting from the (K?)th time frame. Therefore, the probabilityof successful decoding in the jth time frame is(

j − 1

K? − 1

)ρK

?

l (1− ρl)j−K?

for j ∈ {K?, · · · , N}. On the other hand, one receiver maydeclare a failure after the (N −K?)th time frame because thereceiver may realize that it is not possible to obtain at least K?

encoded packets at the end of the N th frame. For instance, letus assume that a receiver has collected zero encoded packetat the end of the (N −K?)th frame. Hence, it missed (N −K?) encoded packets due to transmission errors. Basically, thereceiver has to obtain the encoded packets in the followingK? time frames correctly so that it can recover the message.If the receiver fails to receive one encoded packet in the lastK? time frames, it declares a failure and sends a negativeacknowledgment to the transmitter. Hence, the probability of

decoding failure is expressed as(j − 1

j − 1−N +K?

)ρj−1−N+K?

l (1− ρl)N−K?+1

for j ∈ {N −K? + 1, · · · , N}. Consequently, we formulate(27) that provides us the probability of sending an acknowledg-ment to the transmitter for the lth receiver in the jth time frame.Furthermore, when we have N + 1 < 2K?, the probability ofsending the acknowledgment is given in (28).

Recall that the transmitter stops the transmission of amessage when either it receives acknowledgments from all thereceivers or it reaches the transmission deadline, which is setto the total transmission duration of the N encoded packets.Now, let pj be the probability that the transmitter stops thetransmission of the message that is in transition at the endof the jth time frame. Herein the probability that some ofthe receivers send their acknowledgments before the jth timeframe and the last receiver sends its acknowledgment or thelast receivers send their acknowledgments in the jth time frameis expressed as

p1 =

L∏l=1

p1l and pj =

L∏l=1

j∑i=1

pli −L∏l=1

j−1∑i=1

pli,

where 2 ≤ j ≤ N−1. Moreover, because the transmitter stopsthe transmission at the end of the N th transmission frame dueto the transmission deadline, we have

pN = 1−N−1∑j=1

pj .

If the transmitter stops the transmission of a message andremoves it from the buffer after the jth time frame, we candeclare that the service time of the message is jT time units.Herein, because the ON-OFF process in each channel betweenthe transmitter and the receivers is composed of independentstate transitions, the effective service time becomes

sd(θd) =1

θdlog

N∑j=1

pjejTθd

(29)

for 0 < θd. Notice that when θd goes to zero, the effectiveservice time approaches the average service time, which isT∑Nj=1 jpj . On the other hand, when θd goes to infinity,

the effective service time approaches the maximum serviceduration, which is NT time units.

For numerical presentations that are shown in Fig. 8 andFig. 9, we assume that the probability of successful receptionof an encoded packet by one receiver is uniformly distributedbetween ρmin and 1, i.e., ρmin ≤ ρl ≤ 1 and

fρl(ρl) =

{ 11−ρmin

, ρmin ≤ ρl ≤ 1,

0, otherwise.(30)

We set the number of receivers to 100 and 1000, i.e., L = 100and L = 1000, in Fig. 8 and Fig. 9, respectively, and plotthe effective service time as a function of the waiting timedecay rate, θd. Given that the transmission time frame is 1milliseconds, i.e., T = 1, we observe that the effective servicetime goes to the average service time in the channel with

11

pjl =

0, 1 ≤ j ≤ K? − 1(j−1K?−1

)ρK

?

l (1− ρl)j−K?

, K? ≤ j ≤ N −K?(j−1

j−1−N+K?

)ρj−1−N+K?

l (1− ρl)N−K?+1 +

(j−1K?−1

)ρK

?

l (1− ρl)j−K?

, N −K? + 1 ≤ j ≤ N(27)

pjl =

0, 1 ≤ j ≤ N − 1(j−1

N−K?

)ρj−1−N+K?

l (1− ρl)N−K?+1, N −K? + 1 ≤ j ≤ K? − 1(

j−1N−K?

)ρj−1−N+K?

l (1− ρl)N−K?+1 +

(j−1K?−1

)ρK

?

l (1− ρl)j−K?

, K? ≤ j ≤ N(28)

Fig. 8. Effective service time, sd(θd), vs. waiting time decay rate, θd (dB).

decreasing θd, while it approaches the maximum service timefor one message in the channel with increasing θd, whichis NT milliseconds. We further observe that the number ofreceivers does not matter with increasing θd. Specifically,larger θd means that a message arriving at the buffer is to betransmitted almost without waiting in the buffer. Therefore,the inter-arrival time (effective service time) increases to themaximum, which is NT seconds, and the value of L does notmatter anymore. Notice that when the inter-arrival time is NT ,the data arrival rate decreases but each message arriving at thebuffer is served without waiting in the buffer. On the otherhand, with decreasing θd, the waiting time constraint becomesloose and the inter-arrival time decreases to the average servicetime in the channel. Since the average service time is afunction of L because the state transition probabilities dependon L, the effective service time decreases with decreasing Las seen in Fig. 8 and Fig. 9, where the average service timeis higher when L is higher.

B. Slotted Aloha

We consider a multi-access scenario with L transmitterssending messages to a receiver in frames of T time units asseen in Fig. 7. Each transmitter enters the channel and sendsits message with probability ρ, and each message is receivedsuccessfully by the receiver when there is no collision. Follow-ing a successful reception of a message, the receiver sends anacknowledgment to the respective transmitter. Subsequently,

Fig. 9. Effective service time, sd(θd), vs. waiting time decay rate, θd (dB).

the transmitter removes the message from its queue. On theother hand, if the receiver fails receiving any message, it doesnot send any acknowledgment. Meanwhile, the transmittersthat do not get any acknowledgment enter the channel againwith probability ρ and repeat the transmission of the messagesthat are not acknowledged. In particular, the probability ofremoving a message from a transmitter buffer as a result ofits successful reception by the receiver in any time frame isp = ρ(1 − ρ)L−1. Hence, p becomes the probability of thechannel being ON for the transmitter. Now, noting that thechannel is ON with probability p and OFF with probability(1 − p) for one transmitter and that the transmitters optentering the channel randomly, we consider that the channelstates change independently from one time frame to another.Therefore, we can refer to Section III-B1 and express theeffective service time of the channel for one transmitter asfollows:

sd(θd) =1

θdlog

{peTθd

1− [1− p]eTθd

}(31)

for 0 < θd < − log{1−p}T , and sd(θd) = ∞ for − log{1−p}

T ≤θd. Let us also assume that there exists a transmission deadlineover the transmission duration of a message from a transmitter.Noting that a transmitter can attempt to send a message for

12

Fig. 10. Effective service time, sd(θd), vs. channel access probability, ρ.

maximum N times, the effective service time becomes

sd(θd) =1

θdlog

{(1− p)N−1eNTθd

+ peTθd1− (1− p)N−1e(N−1)Tθd

1− (1− p)eTθd

}. (32)

The effective service time expressions in (31) and (32)provide us the minimum constant inter-arrival time betweenthe consecutive messages arriving at one transmitter buffersuch that the quality-of-service constraint in the form ofsteady-state waiting time violation probability is sustained bythe defined service process. Moreover, notice in (31) that therange of probability of entering the channel under the quality-of-service constraint is bounded as

1− e−Tθd < p ≤ maxρ

{ρ(1− ρ)L−1

}. (33)

Basically, given a decay rate constraint, θd, the maximumnumber of transmitters is limited as follows:

Lmax = max

{L : max

ρ

{ρ(1− ρ)L−1

}> 1− e−Tθd

}.

(34)Thus, we can conclude that when the number of transmittersis greater than Lmax for given θd, the quality-of-serviceconstraint in the form of waiting time violation probabilityis generally not attainable. Additionally, because we needto minimize the effective service time such that the numberof messages arriving at one transmitter buffer increases, wehave to maximize the ON probability, p. We can show thisby taking the first derivative of the expression in (31) withrespect to p, which is always negative. Therefore, we haveto find the channel access probability, ρ, that maximizes theON probability, p. By taking the derivative of p with respectto ρ and setting it to zero, we find that the optimal channelaccess probability that minimizes the effective service time isρ? = 1

L . The probability ρ? = 1L is also the probability that

maximizes the average service rate in the channel.In Fig. 10 and Fig. 11, we set the transmission frame to

1 milliseconds, i.e., T = 1, and plot the effective service

Fig. 11. Effective service time, sd(θd), vs. waiting time decay rate, θd (dB).

time as a function of the channel access probability for givenwaiting time decay rate values in Fig. 10 and as a function ofthe waiting time decay rate parameter given that the optimalchannel access probability is employed in Fig. 11. Regardlessof θd, the effective service time is minimized when the channelaccess probability is set to one over the number of transmitters,i.e., ρ = 1

L , as seen in Fig. 10. Another observation isthe increase of the effective service time with increasing θd.Moreover, the effective service time goes to infinity withθd increasing beyond − log{1−p}

T , which is −11.27, −10.29,−9.01 and −7.20 dB for L = 50, 40, 30 and 20, respectively,as seen in Fig. 11. In addition, the effective service time goesto the average service time in the channel with decreasing θdin all cases.

Remark 1: As seen in (29), a closed-form expression existsfor the effective service time of the aforementioned downlinkscenario. On the other hand, a simple closed form solutionfor the effective capacity of the same scenario may not beobtained. Noting that the hybrid-automatic-repeat-request withincremental redundancy protocol is same with the aforemen-tioned downlink scenario when the number of users is set toL = 1, we refer to [24, Theorem 1] for the effective capacity ofthe downlink scenario with one user, which requires extensivenumerical calculations. Basically, we observe that an analyticalclosed-form solution can be obtained for the effective servicetime of a system while it is difficult to have a simple expressionfor the effective capacity of the same system. Similarly, as forthe effective capacity of the uplink scenario, because thereexists an analogy between the aforementioned uplink scenarioand the hybrid-automatic-repeat-request type-I protocol, weagain refer to [24, Theorem 1]. We can see that a simpleclosed-form solution for the effective capacity of one serviceprocess may not be possible when there exists a transmissiondeadline, whereas the effective service time of the same serviceprocess can be obtained easily as seen in (32). Therefore,invoking Theorem 1, we can obtain the effective capacity ofa service process through the effective service time of thesame service process and the effective bandwidth of an arrivalprocess through the effective inter-arrival time of the same

13

arrival process. Similarly, when it becomes difficult to obtainthe effective service time of a service process or the effectiveinter-arrival time of an arrival process, we can again resort toTheorem 1 to obtain them through the effective capacity ofthe corresponding service process and the effective bandwidthof the corresponding arrival process, respectively.

V. CONCLUSION

In this paper, we have approached the data queueing prob-lem by employing the steady-state waiting time violationprobability rather than the buffer overflow probability asthe primary quality-of-service constraint. Different than theexisting studies, we have not operated in the time domainbut the message index domain. We have characterized thewaiting time for one message in a buffer. Subsequently, wehave formulated the steady-state analysis between the dataarrival and service processes, and then we have identified theeffective service time and effective inter-arrival time. Underthe waiting time violation probability constraint, the effectiveservice time of a service process yields the minimum constantinter-arrival time between consecutive messages arriving at abuffer and the effective inter-arrival time of an arrival processyields the maximum constant service time for a message thatis in transition in the service channel. In particular, we haveshown that we can sustain a desired waiting time violationprobability by controlling the inter-arrival time between themessages arriving at a buffer and the service time spent for amessage during its transition in a service channel. Moreover,we have identified the reciprocal relation between the effectivecapacity and service time of a service process and the recipro-cal relation between the effective bandwidth and inter-arrivaltime of an arrival process. Finally, we have substantiated ouranalytical results in numerical demonstrations where we haveemployed a message dissemination and collection scenarioas it is common, e.g., in vehicular ad hoc networks with adownlink broadcast channel and an uplink multi-access slottedAloha protocol.

APPENDIX

A. Proof of Theorem 1

Let sd(θd) be the effective service time of the serviceprocess from a data buffer for θd > 0 and the inter-arrivaltime between messages arriving at the buffer be constant.Now, recall that the unique θ?d that is the decay rate of thetail distribution of the waiting time in (8) is

θ?d = − limdth→∞

Pr{d(∞) ≥ dth}dth

, (35)

where sd(θ?d) equals the constant inter-arrival time between

messages arriving at the buffer. We re-write (35) as

θ?d = − limdth→∞

Pr{d(∞)sd(θ?d) ≥

dthsd(θ?d)

}sd(θ?d) dth

sd(θ?d)

. (36)

Moreover, let us consider that message Mn leaves the bufferat time instant U(n) and the waiting time of message Mn isd(n). Hence, the backlog in the buffer at time instant U(n)

is q(U(n)) = d(n)sd(θ?d) when message Mn leaves the buffer.

Subsequently, in the steady-state, i.e., when n goes to infinity,we have q(∞) = d(∞)

sd(θ?d) . Then, we re-organize (36) as

sd(θ?d)θ?d = − lim

qth→∞

Pr {q(∞) ≥ qth}qth

, (37)

where qth = dthsd(θ?d) is the buffer overflow threshold. With

(37) the decay rate of the tail distribution of the backlog (3)follows as θ?q = sd(θ

?d)θ?d. Further, the constant inter-arrival

time between consecutive messages sd(θ?d) implies a constantarrival rate at the buffer of 1

sd(θ?d) . At the same time, theconstant arrival rate is equal to the effective capacity, so thatwe have sq(−θ?q ) = 1

sd(θ?d) which completes the proof of (19).Now, let ad(−θd) be the effective inter-arrival time of the

arrival process at a data buffer for θd > 0 and the servicetime for one message from the buffer be constant. Then, were-write (35) as

θ?d = − limdth→∞

Pr{

d(∞)ad(−θ?d) ≥

dthad(−θ?d)

}ad(−θ?d) dth

ad(−θ?d)

, (38)

where ad(−θ?d) equals the constant service time of messages.Note that when message Mn enters the buffer at time instantAd(n), the backlog is q(Ad(n)). Because the service time forone message, i.e., ad(−θ?d), is constant, the waiting time thatmessage Mn experiences is d(n) = q(Ad(n))ad(−θ?d). Then,we have d(∞) = q(∞)ad(−θ?d) in the steady-state and re-organize (38) as

ad(−θ?d)θ?d = − limqth→∞

Pr {q(∞) ≥ qth}qth

, (39)

where qth = dthad(−θ?d) . With (39), the decay rate of the tail

distribution of the backlog is θ?q = ad(−θ?d)θ?d. Further, sincethe service time for each message from the buffer is constant,the effective bandwidth (constant service rate) equals aq(θ?q ) =

1ad(−θ?d) , which confirms the result in (20).

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PLACEPHOTOHERE

Sami Akin received the B.S. degree in electrical andelectronics engineering from Bogazici University,Istanbul, Turkey in 2005, and the Ph.D. degreein electrical engineering from the University ofNebraska-Lincoln, Lincoln, NE, US in 2011. SinceDecember 2011, he has been with the Institute ofCommunications Technology at Leibniz UniversitatHannover, Hanover, Germany as a post-doctoralresearch fellow. He was the technical group leaderof the Cognitive Radio for Audio Systems (CoRAS)project funded by Lower Saxony Ministry of Science

and Culture. Currently, he is a research member of the Towards a UnifiedInformation and Queueing Theory (UnIQue) project funded by EuropeanResearch Council (ERC) Starting Grant. His research interests are in thegeneral areas of wireless communications, signal processing, informationtheory, queueing theory, and network calculus with a focus on wirelesscommunications and networks under quality of service (QoS) constraints.

PLACEPHOTOHERE

Markus Fidler (M04–SM08) received the Doctoraldegree in computer engineering from RWTH AachenUniversity, Aachen, Germany, in 2004. He was aPostdoctoral Fellow with the Norwegian Universityof Science and Technology, Trondheim, Norway, in2005 and with the University of Toronto, Toronto,ON, Canada, in 2006. During 2007 and 2008, he wasan Emmy Noether Research Group Leader at Tech-nische Universitat Darmstadt, Darmstadt, Germany.Since 2009, he has been a Professor of communi-cations networks at Leibniz Universitat Hannover,

Hanover, Germany. He was awarded a Starting Grant of the EuropeanResearch Council in 2012.