On the Delay-Throughput Tradeoff in Distributed Wireless Networks
aa r X i v : . [ c s . I T ] O c t SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 On the Delay-Throughput Tradeoff in DistributedWireless Networks { jabouei, alireza, khandani } @cst.uwaterloo.ca Abstract
This paper deals with the delay-throughput analysis of a single-hop wireless network with n transmitter/receiverpairs. All channels are assumed to be block Rayleigh fading with shadowing, described by parameters ( α, ̟ ) , where α denotes the probability of shadowing and ̟ represents the average cross-link gains. The analysis relies on thedistributed on-off power allocation strategy (i.e., links with a direct channel gain above a certain threshold transmitat full power and the rest remain silent) for the deterministic and stochastic packet arrival processes. It is alsoassumed that each transmitter has a buffer size of one packet and dropping occurs once a packet arrives in thebuffer while the previous packet has not been served. In the first part of the paper, we define a new notion ofperformance in the network, called effective throughput , which captures the effect of arrival process in the networkthroughput, and maximize it for different cases of packet arrival process. It is proved that the effective throughputof the network asymptotically scales as log n ˆ α , with ˆ α , α̟ , regardless of the packet arrival process. In the secondpart of the paper, we present the delay characteristics of the underlying network in terms of the packet droppingprobability. We derive the sufficient conditions in the asymptotic case of n → ∞ such that the packet droppingprobability tend to zero, while achieving the maximum effective throughput of the network. Finally, we studythe trade-off between the effective throughput, delay, and packet dropping probability of the network for differentpacket arrival processes. In particular, we determine how much degradation will be enforced in the throughput byintroducing the aforementioned constraints. Index Terms
Throughput maximization, delay-throughput tradeoff, dropping probability, Poisson arrival process. ∗ This work is financially supported by Nortel Networks and the corresponding matching funds by the Natural Sciences and EngineeringResearch Council of Canada (NSERC), and Ontario Centers of Excellence (OCE). ∗ The material in this paper was presented in part at the IEEE International Symposium on Information Theory (ISIT), Nice, France, June24-29, 2007 [1].
UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 I. I
NTRODUCTION
As the demand for higher data rates increases, effective resource allocation emerges as the primaryissue in wireless networks in order to satisfy Quality of Service (QoS) requirements. Central to the studyof resource allocation schemes, the distributed power control algorithms for maximizing the networkthroughput have attracted significant research attention [2]–[7]. Moreover, achieving a low transmissiondelay is an important QoS requirement in wireless networks [8]. In particular, for buffer-limited userswith real-time services (e.g., interactive games, live sport videos, etc), too much delay results in droppingsome packets. Therefore, the main challenge in wireless networks with real-time services is to utilize anefficient power allocation scheme such that the delay is minimized, while achieving a high throughput.The throughput maximization problem in cellular and multihop wireless networks has been extensivelystudied in [9]–[13]. In these works, delay analysis is not considered. However, it is shown that the highthroughput is achieved at the cost of a large delay [14]. This problem has motivated the researchers tostudy the relation between the delay characteristics and the throughput in wireless networks [15]–[18].In particular, in most recent literature [14], [19]–[26], the tradeoffs between delay and throughput havebeen investigated as a key measure of the network’s performance. The first studies on achieving a highthroughput along with a low-delay in ad hoc wireless networks are framed in [17] and [18]. This line ofwork is further expanded in [14], [20] and [21] by using different mobility models. El Gamal et al. [14]analyze the optimal delay-throughput scaling for some wireless network topologies. For a static randomnetwork with n nodes, they prove that the optimal tradeoff between throughput T n and delay D n is givenby D n = Θ( nT n ) . Reference [14] also shows that the same result is achieved in random mobile networks,when T n = O (1 / √ n log n ) . Neely and Modiano [21] consider the delay-throughput tradeoff for mobile adhoc networks under the assumption of redundant packet transmission through multiple paths. Sharif andHassibi [22] analyze the delay characteristics and the throughput in a broadcast channel. They propose analgorithm to reduce the delay without too much degradation in the throughput. This line of work is furtherextended in [23] by demonstrating that it is possible to achieve the maximum throughput and short-termfairness simultaneously in a large-scale broadcast network.In [27], we addressed the throughput maximization of a distributed single-hop wireless network with K links, where the links are partitioned into a fixed number ( M ) of clusters each operating in a subchannelwith bandwidth WM . We proposed a distributed and non-iterative power allocation strategy, where theobjective for each user is to maximize its best estimate (based on its local information, i.e., direct channelgain) of the average sum-rate of the network. Under the block Rayleigh fading channel model withshadowing effect, it is established that the average sum-rate in the network scales at most as Θ(log K ) inthe asymptotic case of K → ∞ . This order is achievable by the distributed threshold-based on-off scheme UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 (i.e., links with a direct channel gain above certain threshold τ n transmit at full power and the rest remainsilent). In addition, in the strong interference scenario, the on-off power allocation scheme is shown to bethe optimal strategy. Moreover, the optimum threshold level that achieves the maximum average sum-rateof the network is obtained as τ n = log n − n + O (1) , where n = KM is the number of links ineach cluster. We also optimized the average network’s throughput in terms of the number of the clusters, M . It is proved that the maximum average sum-rate of the network, assuming on-off power allocationscheme, is achieved at M = 1 . However, [27] only focuses on the network throughput and other issues(like delay and packet dropping probability) were not addressed in this work.In this paper, we follow the distributed single-hop wireless network model proposed in [27] with M = 1 (which is the case with the maximum throughput) and address the delay-throughput tradeoff ofthe network. The channels are assumed to be block Rayleigh fading with shadowing (the same model asin [27]), where the transmission block is assumed to be equal to the fading block (which is assumed to beequal for all links). Moreover, the links are assumed to be synchronous. The assumption of block Rayleighfading with synchronous users is used in many works in the literature (like [28] for the point-to-pointscenario, [29] for the multiple-access channel, and [22] and [23] for the broadcast scenario). We considera buffer-limited network, in which the users have a buffer size of one packet. This assumption introduces dropping event in the network, which is defined as the event when a packet is arrived in the buffer whilethe previous packet has not been served yet. Although the assumption of one packet buffer size is harshfor many practical applications, it simplifies the analysis while giving a good insight about the worstcase performance in the network. Noting the optimality of on-off power allocation scheme in terms ofachieving the maximum order of the sum-rate throughput [27], we use it in this work. Therefore, for anylink, if the direct channel is above a pre-determined threshold and there is any packet in the buffer, thetransmitter sends that packet during a transmission block with full power and if not, remains silent.In the first part, we define a new notion of throughput, called effective throughput , which describes the actual amount of data transmitted through each links. This notion captures the effect of arrival processby taking into account the full buffer probability . We compute the optimum threshold level τ n , and thecorresponding maximum effective throughput of the network, for each packet arrival process. It is provedthat the effective throughput of the network scales as log n ˆ α , with ˆ α , α̟ , regarding the packet arrivalprocess. This throughput scaling is exactly the same as what we had derived in [27], i.e., the case ofbacklogged users. Moreover, we show that the maximum throughput is achieved in the strong interferencescenario , in which the interference term dominates the noise. As an interesting consequence, the resultsof this section are valid even without the assumption of synchronization between the users or equality oftheir fading coherence time (fading blocks). UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 In the second part, we present the delay characteristics of the underlying network in terms of the packetdropping probability for deterministic and stochastic packet arrival processes. We derive the sufficientconditions in the asymptotic case of n → ∞ such that the packet dropping probability of the links tendsto zero, while achieving the maximum effective throughput of the network, asymptotically. The importanceof this result is showing the fact that the loss in the network performance due to the limited buffer size canbe made negligible in the asymptotic regime of n → ∞ . In the subsequent section, we study the tradeoffbetween the effective throughput of the network and other performance measures, i.e., packet droppingprobability and delay for different arrival processes. In particular, we determine how much degradationwill be enforced in the throughput by introducing the aforementioned constraints, and how much thisdegradation depends on the arrival process. The setup in this paper is quite different from that of with theon-off Bernoulli scheme in [30]. In fact, we utilize a distributed approach using local information, i.e.,direct channel gains, while [30] relies on a central controller which studies the channel conditions of allthe links and decides accordingly. Furthermore, we consider a homogeneous network model without pathloss. This differs from the geometric models considered in [14], [20] and [21], which are based on thedistance between the source and the destination (i.e., power decay-versus-distance law).The rest of the paper is organized as follows. In Section II, the network model and objectives aredescribed. The throughput maximization of the underlying network is presented in Section III. The delaycharacteristics in terms of the packet dropping probability are analyzed in Section IV. Section V establishesthe tradeoff between the throughput, delay, and packet dropping probability in the underlying network.Simulation results are presented in in section VI. Finally, in Section VII, an overview of the results andconclusions are presented. Notations:
For any functions f ( n ) and g ( n ) [31]: • f ( n ) = O ( g ( n )) means that lim n →∞ (cid:12)(cid:12)(cid:12) f ( n ) g ( n ) (cid:12)(cid:12)(cid:12) < ∞ . • f ( n ) = o ( g ( n )) means that lim n →∞ (cid:12)(cid:12)(cid:12) f ( n ) g ( n ) (cid:12)(cid:12)(cid:12) = 0 . • f ( n ) = ω ( g ( n )) means that lim n →∞ f ( n ) g ( n ) = ∞ . • f ( n ) = Ω( g ( n )) means that lim n →∞ f ( n ) g ( n ) > . • f ( n ) = Θ( g ( n )) means that lim n →∞ f ( n ) g ( n ) = c , where < c < ∞ . • f ( n ) ∼ g ( n ) means that lim n →∞ f ( n ) g ( n ) = 1 . • f ( n ) ≈ g ( n ) means that f ( n ) is approximately equal to g ( n ) , i.e., if we replace f ( n ) by g ( n ) in theequations, the results still hold.Throughout the paper, we use log( . ) as the natural logarithm function and N n for representing the set { , , · · · , n } . Also, E [ . ] represents the expectation operator, and P { . } denotes the probability of the givenevent. UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 T T T T R R R R Fig. 1. A distributed single-hop wireless network with n = 4 . II. N
ETWORK M ODEL AND P ROBLEM D ESCRIPTION
A. Network Model
In this work, we consider a distributed single-hop wireless network, in which n pairs of nodes , indexedby { , ..., n } , are located within the network area (Fig. 1). We assume the number of links, n , is knowninformation for the users. All the nodes in the network are assumed to have a single antenna. Also, itis assumed that all the transmissions occur over the same bandwidth. In addition, we assume that eachreceiver knows its direct channel gain with the corresponding transmitter, as well as the interference powerimposed by other users. However, each transmitter is assumed to be only aware of the direct channel gainto its corresponding receiver. The power of Additive White Gaussian Noise (AWGN) at each receiver isassumed to be N .We assume that the time axis is divided into slots with the duration of one transmission block, which isdefined as the unit of time. The channel model is assumed to be Rayleigh flat-fading with the shadowingeffect. The channel gain between transmitter j and receiver i at time slot t is represented by the randomvariable L ( t ) ji . For j = i , the direct channel gain is defined as L ( t ) ji , h ( t ) ii , where h ( t ) ii is exponentiallydistributed with unit mean (and unit variance). For j = i , the cross channel gains are defined based on a The term “pair” is used to describe the transmitter and the related receiver, while the term “user” is used only for the transmitter. In this paper, channel gain is defined as the square magnitude of the channel coefficient . In the sequel, we use the superscript ( t ) for some events to show that the events occur in time slot t . UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 shadowing model as follows : L ( t ) ji , β ( t ) ji h ( t ) ji , with probability α , with probability − α, (1)where h ( t ) ji s have the same distribution as h ( t ) ii s, ≤ α ≤ is a fixed parameter, and the random variable β ( t ) ji , referred to as the shadowing factor , is independent of h ( t ) ji and satisfies the following conditions: • β min ≤ β ( t ) ji ≤ β max , where β min > and β max is finite, • E (cid:2) β ( t ) ji (cid:3) , ̟ ≤ .All the channels in the network are assumed to be quasi-static block fading, i.e., the channel gainsremain constant during one block and change independently from block to block. In other words, L ( t ) ji isindependent of L ( t ′ ) ji for t = t ′ . Moreover, the fading block of all channels are assumed to be equal toeach other and this value is equal to the duration of the transmission block for all users. This model isalso used in [22] and [23]. Also, users are assumed to be synchronous to each other. However, as we willsee later, the results of the paper are still valid even in the cases that the users are not synchronous orthe fading block (coherence time) of the channels are not equal. B. On-Off Power Allocation Strategy
In [27], we have shown that a distributed scheme, called threshold-based on-off scheme , achieves themaximum order of the sum-rate throughput in a single-hop wireless network with n links, under the blockRayleigh fading channel model possibly with shadowing effect, in the asymptotic regime of n → ∞ .Moreover, in the strong interference scenario, the on-off power allocation scheme is the optimal strategy,in terms of the sum-rate throughput, assuming the availability of direct channel gains at the transmitters.Motivated by the results of [27], we assume that all the links utilize the threshold-based on-off powerallocation strategy proposed in [27] . Unlike most of the works in the literature that assume backloggedusers, here we assume a practical model for the packet arrivals in which the buffer of each link is notnecessarily full (of packet) all the time. Based on this observation, we adopt the on-off power allocationscheme during each time slot t as follows:1- Based on the direct channel gain, the transmission policy is p ( t ) i = , if h ( t ) ii > τ n and the buffer of link i is full at time slot t , Otherwise , (2) For more details, the reader is referred to [32] and [33] and references therein. We consider a homogeneous network in the sense that all the links have the same configuration and use the same protocol. Thus, thetransmission strategy for all users are agreed in advance. In fact, if there is no packet in the buffer, it does not make sense for the user to be active, even if its channel is good.
UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 where p ( t ) i denotes the transmission power of user i at time slot t and τ n is a pre-specified threshold levelthat is a function of n and also depends on the channel model and packet arrival process.2- Knowing its corresponding direct channel gain, each active user i transmits a Gaussian signal withfull power and the rate equal to: R ( t ) i = E h ( t ) ii ,I ( t ) i " log h ( t ) ii p ( t ) i I ( t ) i + N ! nats/channel use , (3)where I ( t ) i = P nj =1 j = i L ( t ) ji p ( t ) j is the power of the interference term seen by receiver i ∈ N n at time slot t . The above rate is achievable by encoding and decoding over arbitrarily large number ( M ) of blocks.More precisely, assuming the number of channel uses per each transmission block to be N , the i th transmitter maps the message m ∈ { m , m , · · · , m L } , where L = 2 MNR ( t ) i , to a Gaussian codeword ofsize M N , C m ∈ {C , C , · · · , C L } . In the k th block, if p ( t ) i = 1 , the transmitter sends the k th portion of C m ,denoted by C m ( k ) . At the receiver side, the decoder considers only the blocks in which the transmitter wastransmitting with full power, denoted by { a , · · · , a l } , and is able to decode the message m , if L ≤ NlR ,where R , E h ( t ) ii ,I ( t ) i (cid:20) log (cid:18) h ( t ) ii I ( t ) i + N (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p ( t ) i = 1 (cid:21) . Noting that as M → ∞ , l ≈ M P { p ( t ) i = 1 } , and R ( t ) i = P { p ( t ) i = 1 } R , it is concluded that the rate R ( t ) i is achievable. As we will see later, in the optimalperformance regime, which is the strong interference regime, encoding and decoding over single blocksis sufficient to achieve (3). C. Packet Arrival Process
One of the most important parameters in the network analysis is the model for the packet arrivalprocess. The packet arrival process is a random process which is described by either the arrival time ofthe packets or the interarrival time between the subsequent packets. These quantities may be modeled bythe deterministic or stochastic processes (Fig. 2). In this paper, we consider the following packet arrivalprocesses: • Poisson Arrival Process (PAP):
In this process, the number of arrived packets in any interval of unitlength is assumed to have a Poisson distribution with the parameter λ . This process is a commonlyused model for random and mutually independent packet arrivals in queueing theory [34]. • Bernoulli Arrival Process (BAP):
In this process, at any given time slot, the probability that a packetarrives is ρ , λ . Moreover, the arrival of the packets in different slots occurs independently. Thismodel has been used in many works in the literature such as [21] and [35]. • Constant Arrival Process (CAP):
In this process, packets arrive continuously with a constant rate of λ packets per unit length (Fig. 2-b) [36]. We choose the parameter ρ as λ to be consistent with other packet arrival processes. UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 k x + k x + k x + k x TransmitterBuffer k A t k A t + k A t + k A t + k A t + (a) TransmitterBuffer lllll k A t k A t + k A t + k A t + k A t + k A t + (b) Fig. 2.
A schematic figure for a) stochastic packet arrival process, b) constant packet arrival process.
It is assumed that the packet arrival process for all links is the same. Let us denote t ( i ) A k as the time instantof the k th packet arrival into the buffer of link i . It is observed from Fig. 2-a that t ( i ) A k = P k − j =1 x ( i ) j + t ( i )0 where t ( i )0 is the starting time for link i , and the random variable x ( i ) j is the interarrival time defined as x ( i ) j , t ( i ) A j +1 − t ( i ) A j , (4)with E [ x ( i ) j ] = λ . For the CAP, x ( i ) j = λ and t ( i ) A k = ( k − λ + t ( i )0 , while for the PAP, x ( i ) j ’s are independentsamples of an exponential random variable x with the probability density function (pdf) f X ( x ) = 1 λ e − λ x , x > . (5)Also for the BAP, x ( i ) j ’s are independent samples of a geometric random variable X with the probabilitymass function (pmf) p X ( m ) , P { X = m } = (1 − ρ ) m − ρ, m = 1 , , ..., (6)with ρ , λ .We represent t ( i ) D k as the time instant at which either the k th arriving packet departs the buffer of link i for the transmission or drops from the buffer. In such configuration, we have the following definition: Definition 1 (Delay):
The random variable D ( i ) k , t ( i ) D k − t ( i ) A k for each link i is defined as the delaybetween the departure and the arrival time of each packet k , expressed in terms of the number of timeslots. In this work, we assume that the buffer size for each transmitter is one packet. Due to the this limitationon the buffer size and the on-off power allocation strategy, the existing buffered packet may be dropped if For analysis simplicity, we assume that λ is an integer number. UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 it is not served before the arrival of the next packet. Mathematically speaking, the event that the droppingof packet k occurs in link i ∈ N n is defined as B i ≡ n D ( i ) k ≥ t ( i ) A k +1 − t ( i ) A k o (7) ≡ n D ( i ) k ≥ x ( i ) k o . (8)Therefore, the packet dropping probability in each link i ∈ N n , denoted by P { B i } , can be obtained as P { B i } = P n D ( i ) k ≥ x ( i ) k o (9) = Z ∞ P n D ( i ) k ≥ x ( i ) k (cid:12)(cid:12)(cid:12) x ( i ) k = x o f X ( x ) dx, for PAP , (10) = ∞ X m =1 P n D ( i ) k ≥ x ( i ) k (cid:12)(cid:12)(cid:12) x ( i ) k = m o p X ( m ) , for BAP , (11) = P n D ( i ) k ≥ λ o , for CAP . (12)where f X ( x ) and p X ( m ) are defined as (5) and (6), respectively. In Section IV, we will obtain P { B i } for different packet arrival processes in terms of λ and τ n . D. Objectives
Part I: Throughput Maximization:
The main objective of the first part of this paper is to maximize thethroughput of the underlying network. To address this problem, we first define a new notion of throughput,called effective throughput , which denotes the actual amount of data transmitted through the links. In orderto derive the effective throughput, we obtain the full buffer probability of a link for the deterministic andstochastic packet arrival processes. Then, we compute the optimum threshold level τ n , and the maximumeffective throughput of the network, for each packet arrival process. Part II: Delay Characteristics:
The main objective of the second part is to formulate the packetdropping probability of each link in the underlying network based on the aforementioned packet arrivalprocesses in terms of the number of links ( n ), λ , and the parameter of the on-off power allocation scheme( τ n ). This analysis enables us to derive the sufficient conditions in the asymptotic case of n → ∞ suchthat the packet dropping probabilities tend to zero, while achieving the maximum effective throughput ofthe network. Part III: Delay-Throughput-Dropping Probability Tradeoff:
The main goal of the third part is tostudy the tradeoff between the effective throughput of the network and other performance measures, i.e.,the dropping probability and the delay-bound ( λ ) for different packet arrival processes. In particular, weare interested to determine how much degradation will be enforced in the throughput by introducing theother constraints, and how much this degradation depends on the packet arrival process. UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 III. T
HROUGHPUT M AXIMIZATION
A. Effective Throughput
In this section, we aim to derive the maximum throughput of the network with a large number ( n )of links, based on using the distributed on-off power allocation strategy. We present a new performancemetric in the network, called effective throughput , which is a function of the threshold level τ n and λ .Let us start with the following definition. Definition 2 (Effective Throughput):
Under the on-off power allocation strategy, the effective throughputof each link i , i ∈ N n , is defined (on a per-block basis) as T i , lim L →∞ L L X t =1 R ( t ) i I ( t ) i , (13) where R ( t ) i is defined as (3) and I ( t ) i is an indicator variable which is equal to , if user i transmits attime slot t , and otherwise. Furthermore, the effective throughput of the network is defined as T eff , n X i =1 T i . (14)The quantity T i represents the average amount of information conveyed through link i in a long periodof time. This metric is suitable for real-time applications, where the packets have a certain amount ofinformation and certain arrival rates. It should be noted that I ( t ) i = 1 is equivalent to the case in whichthe buffer is full and the channel gain h ( t ) ii is greater than the threshold level τ n at time slot t . Definingthe full buffer event as follows C ( t ) i ≡ { Buffer of link i is full at time slot t } , (15)we have P n I ( t ) i = 1 o = P n h ( t ) ii > τ n , C ( t ) i o (16) ( a ) = P n h ( t ) ii > τ n o P n C ( t ) i o (17) = q n ∆ n , (18)where q n , P n h ( t ) ii > τ n o , and ∆ n , P n C ( t ) i o is the full buffer probability . In the above equations, ( a ) follows from the fact that the full buffer event depends on the packet arrival process as well as the directchannel gains h ( t ′ ) ii , for t ′ < t , which is independent of the channel gain h ( t ) ii (due to the block fadingchannel model). Thus, I ( t ) i = , with probability q n ∆ n , , with probability − q n ∆ n . (19) UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 It is observed that I ( t ) i is a Bernoulli random variable with parameter q n ∆ n . In fact, q n ∆ n is the probabilityof the link activation which is a function of n . In the sequel, we derive ∆ n for the aforementioned packetarrival processes. B. Full Buffer Probability
Let us denote t ( i ) a as the time instant the last packet has arrived in the buffer of link i before or at thesame time t . The event C ( t ) i implicitly indicates that during X ( t ) i , t − t ( i ) a time slots, the channel gainof link i is less than the threshold level τ n . Clearly, X ( t ) i is a random variable which varies from zeroto infinity for the stochastic packet arrival processes and is finite for the CAP . Under the on-off powerallocation scheme and using the block fading model property, the full buffer probability can be obtainedas ∆ n = E h (1 − q n ) X ( t ) i i , (20)where the expectation is computed with respect to X ( t ) i , and q n , P n h ( t ) ii > τ n o = e − τ n . Lemma 1
Let us denote the full buffer probability of an arbitrary link i ∈ N n , for the Poisson, Bernoulliand constant arrival processes as ∆ P APn , ∆ BAPn and ∆ CAPn , respectively. Then, ∆ P APn = 11 + λ log(1 − q n ) − , (21) ∆ BAPn = 11 + ( λ − q n , (22) ∆ CAPn = 1 − (1 − q n ) λ λq n . (23) Proof:
For the PAP, since X ( t ) i is an exponential random variable, (20) can be simplified as ∆ P APn = Z ∞ λ (1 − q n ) x e − λ x dx (24) = 11 + λ log(1 − q n ) − . (25)Also for the BAP, X ( t ) i is a geometric random variable with parameter ρ = λ . Thus, (20) can besimplified as ∆ BAPn = ∞ X m =0 (1 − q n ) m ρ (1 − ρ ) m (26) ( a ) = 11 + ( λ − q n , (27) Note that, here we assume that if a packet arrives at time t and the channel gain is greater than τ n at this time, the packet will betransmitted. As we will show in Lemma 1, ∆ n is independent of index i . UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 where ( a ) follows from the following geometric series: ∞ X m =0 x m = 11 − x , | x | < . (28)For the CAP, the full buffer probability in (20) can be written as ∆ CAPn ( a ) = λ − X m =0 (1 − q n ) m P { X ( t ) i = m } (29) ( b ) = λ − X m =0 (1 − q n ) m λ (30) ( c ) = 1 − (1 − q n ) λ λq n , (31)where ( a ) follows from Fig. 2-b, in which X ( t ) i varies from zero to λ − and ( b ) follows from the factthat for the deterministic process, X ( t ) i has a uniform distribution. In other words, for every value of m ∈ [0 , λ − , P { X ( t ) i = m } = λ . Also, ( c ) comes from the following geometric series: s X m =0 x m = 1 − x s +1 − x . (32)Having derived the full buffer probability, we obtain the effective throughput of the network in thefollowing section. C. Effective Throughput of the Network
Rewriting (13), the effective throughput of link i can be obtained as T i = lim L →∞ L L X t =1 R ( t ) i I ( t ) i (33) ( a ) = E h R ( t ) i I ( t ) i i (34) = E h R ( t ) i I ( t ) i (cid:12)(cid:12)(cid:12) I ( t ) i = 1 i P n I ( t ) i = 1 o + E h R ( t ) i I ( t ) i (cid:12)(cid:12)(cid:12) I ( t ) i = 0 i P n I ( t ) i = 0 o (35) ( b ) = q n ∆ n E h R ( t ) i (cid:12)(cid:12)(cid:12) h ( t ) ii > τ n , C ( t ) i i (36) ( c ) = q n ∆ n E " log h ( t ) ii I ( t ) i + N !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ( t ) ii > τ n , (37)where the expectation is computed with respect to h ( t ) ii and the interference term I ( t ) i . In the above equations, ( a ) follows from the ergodicity of the channels (due to the block fading model), which implies that theaverage over time is equal to average over realization. ( b ) results from (16)-(18) and E (cid:2) R ( t ) i I ( t ) i (cid:12)(cid:12) I ( t ) i =0 (cid:3) = 0 . Finally, ( c ) results from the fact that the term log (cid:18) h ( t ) ii I ( t ) i + N (cid:19) is independent of C ( t ) i . UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 In order to derive the effective throughput, we need to find the statistical behavior of I ( t ) i which isperformed in the following lemmas: Lemma 2
Under the on-off power scheme, we have E h I ( t ) i i = ( n −
1) ˆ αq n ∆ n , (38) Var h I ( t ) i i ≤ ( n − ακq n ∆ n ) , (39) where ˆ α , α̟ and κ , E (cid:20)(cid:16) β ( t ) ji (cid:17) (cid:21) .Proof: See Appendix I.
Lemma 3
The maximum effective throughput is achieved at λ = o ( n ) and the strong interference regimewhich is defined as E [ I ( t ) i ] = ω (1) , i ∈ N n .Proof: Suppose that λ = o ( n ) which implies that λ = Ω( n ) . Using (37), we have T i ≤ q n ∆ n E " log h ( t ) ii N !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ( t ) ii > τ n (40) ( a ) ≤ q n ∆ n log E h h ( t ) ii (cid:12)(cid:12)(cid:12) h ( t ) ii > τ n i N (41) = q n ∆ n log (cid:18) τ n + 1 N (cid:19) , (42)where ( a ) comes from the concavity of log( . ) function and Jensen’s inequality , E [log x ] ≤ log( E [ x ]) , x > . Following (21) - (23), it is revealed that ∆ n ≤ min (cid:16) , λq n (cid:17) for all packet arrival processes.Substituting in (42), we have T i ≤ λ log (cid:18) λ + 1 N (cid:19) ∼ log log λλ , (43)which follows from the fact that the maximum value of q n ∆ n log (cid:16) τ n +1 N (cid:17) with the condition of ∆ n ≤ min (cid:16) , λq n (cid:17) is attained at q n = λ . Noting that λ = Ω( n ) , we have T i ≤ Θ (cid:0) log log nn (cid:1) .Now, suppose that λ = o ( n ) but E [ I ( t ) i ] = ω (1) , or equivalently, E [ I ( t ) i ] = O (1) for some i . Since E [ I ( t ) i ] = ( n −
1) ˆ αq n ∆ n , the condition E [ I ( t ) i ] = O (1) implies that there exists a constant c such that q n ∆ n ≤ cn . Noting (21) - (23), it follows that either ∆ n ∼ λq n or ∆ n = Θ(1) . In the first case, the UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 condition q n ∆ n ≤ cn implies that n ≤ cλ which cannot hold due to the assumption of λ = o ( n ) .Therefore, we must have q n ≤ c ′ n , for some constant c ′ . Substituting in (42) yields T i ≤ c ′ n log (cid:18) τ n + 1 N (cid:19) ( a ) ≤ c ′ n log (cid:18) n/c ′ ) + 1 N (cid:19) = Θ (cid:18) log log nn (cid:19) , (44)where ( a ) results from the fact that q n log (cid:16) τ n +1 N (cid:17) is an increasing function of q n and reaches itsmaximum at the boundary which is c ′ n .In the sequel, we present a lower-bound on the effective throughput of link i in the region λ = o ( n ) and E [ I ( t ) i ] = ω (1) and show that this lower-bound beats the upper-bounds derived in the other regions,proving the desired result. For this purpose, using (37), we write T i ( a ) ≥ q n ∆ n log τ n E h I ( t ) i (cid:12)(cid:12)(cid:12) h ( t ) ii > τ n i + N o ( b ) = q n ∆ n log (cid:18) τ n ( n −
1) ˆ αq n ∆ n + N o (cid:19) ( c ) ≈ q n ∆ n log (cid:18) τ n ( n −
1) ˆ αq n ∆ n (cid:19) , (45)where ( a ) follows from the convexity of the function log(1+ bx + a ) with respect to x and Jensen’s inequality, ( b ) results from the independence of I ( t ) i from h ( t ) ii , and ( c ) follows from neglecting the term N withrespect to ( n −
1) ˆ αq n ∆ n due to the strong interference assumption. Setting q n = log nn and λ = n log n ,it is easy to check that τ n ( n − αq n ∆ n = o (1) and hence, log (cid:16) τ n ( n − αq n ∆ n (cid:17) ≈ τ n ( n − αq n ∆ n which givesthe effective throughput as τ n ( n − α = Θ (cid:0) log nn (cid:1) which is greater than the throughput obtained in the otherregimes.Due to the result of Lemma 3, we restrict ourselves to the case of λ = o ( n ) and the strong interferenceregime in the rest of the paper. Lemma 4
Let us assume < α ≤ is fixed and we are in the strong interference regime (i.e., E h I ( t ) i i = ω (1) ). Then with probability one (w. p. 1), we have I ( t ) i ∼ ( n −
1) ˆ αq n ∆ n , (46) as n → ∞ . More precisely, substituting I ( t ) i by ( n −
1) ˆ αq n ∆ n does not change the asymptotic effectivethroughput of the network.Proof: See Appendix II.
UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 Lemma 5
The effective throughput of the network for large values of n can be obtained as T eff ≈ nq n ∆ n log (cid:18) τ n n ˆ αq n ∆ n (cid:19) . (47) Proof:
Using (37), the effective throughput of the network in the asymptotic case of n → ∞ isobtained as T eff = n X i =1 T i (48) ( a ) ≈ nq n ∆ n E " log h ( t ) ii ( n −
1) ˆ αq n ∆ n + N !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ( t ) ii > τ n (49) ( b ) ≈ nq n ∆ n E " log h ( t ) ii n ˆ αq n ∆ n !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ( t ) ii > τ n , (50)where ( a ) results from the strong interference assumption and Lemma 4, and ( b ) follows from approxi-mating ( n −
1) ˆ αq n ∆ n + N by n ˆ αq n ∆ n due to the strong interference assumption and large values of n .A lower-bound on (50) can be written as T l eff = nq n ∆ n log (cid:18) τ n n ˆ αq n ∆ n (cid:19) . (51)Furthermore, due to the concavity of log( . ) function and Jensen’s inequality, an upper-bound on T eff canbe given as T u eff = nq n ∆ n log E h h ( t ) ii (cid:12)(cid:12)(cid:12) h ( t ) ii > τ n i n ˆ αq n ∆ n = nq n ∆ n log (cid:18) τ n + 1 n ˆ αq n ∆ n (cid:19) . (52)In order to prove that the above upper and lower bounds have the same scaling, it is sufficient to show thatthe optimum threshold value ( τ n ) is much larger than one. For this purpose, we note that if τ n = O (1) ,then the effective throughput of the network will be upper-bounded by T eff ( a ) ≤ τ n + 1ˆ α (53) = O (1) , (54)where ( a ) follows from log(1 + x ) ≤ x . In other words, the effective throughput of the network doesnot scale with n , while the throughput of Θ(log n ) , as will be shown later, is achievable. This suggeststhat the optimum threshold value must grow with n , and hence, the bounds given in (51) and (52) areasymptotically equal to nq n ∆ n log (cid:16) τ n n ˆ αq n ∆ n (cid:17) and this completes the proof of the lemma. Lemma 6
The maximum effective throughput of the network is obtained in the region that τ n = o ( n ˆ αq n ∆ n ) . UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 Proof:
Rewriting the expression of the effective throughput of the network from (47) and noting thefact that log(1 + x ) ≤ x , for x ≥ , we have T eff ≈ nq n ∆ n log (cid:18) τ n n ˆ αq n ∆ n (cid:19) ≤ τ n ˆ α . (55)It can be shown that if the condition τ n = o ( n ˆ αq n ∆ n ) is not satisfied, the ratio log ( τnn ˆ αqn ∆ n ) τnn ˆ αqn ∆ n is strictlyless than one. Having τ n = o ( n ˆ αq n ∆ n ) results in log (cid:16) τ n n ˆ αq n ∆ n (cid:17) ≈ τ n n ˆ αq n ∆ n yielding the upper-bound τ n ˆ α . This means that to achieve the maximum throughput, the interference should not only be strong butalso be much larger than τ n . Observation -
An interesting observation of Lemmas 3-6 is that there is no need to have synchronizationbetween the users or equality of the fading blocks (coherence time) of the channels to obtain theseresults. This is due to the fact that during a transmission block (which is equal to the fading blockof the corresponding direct channel), the receiver observes different samples of interference I i (due toasynchronousy between the users). However, as the interference is strong, from the result of Lemma4, all samples of interference asymptotically almost surely scale as n ˆ αq n ∆ n , and hence, the receiver isstill capable of decoding the message correctly if the transmission rate is below q n ∆ n log (cid:16) τ n n ˆ αq n ∆ n (cid:17) .Moreover, the encoding and decoding do not need to be performed over large number of blocks. In fact,in the blocks where h ( t ) ii > τ n , the transmitter sends data with the rate log (cid:16) τ n n ˆ αq n ∆ n (cid:17) nats/channel useand the decoder will be able to decode the packet information correctly.Having the expression for the effective throughput of the network in (47), in the next theorem, we findthe optimum value of q n (or equivalently τ n ) in terms of n and λ for the aforementioned packet arrivalprocesses, i.e.: ˆ q n = arg max q n T eff . (56)As shown in the proof of Lemma 5, since the optimum threshold value is much larger than one, theoptimizer ˆ q n is sufficiently small, i.e., ˆ q n = o (1) . Theorem 1
Assuming the Poisson packet arrival process and large values of n , the optimum solution for(56) is obtained as q P APn = δ log nn (57) for some constant δ . Furthermore, the maximum effective throughput of the network asymptotically scalesas log n ˆ α , for λ = o (cid:16) n log n (cid:17) .Proof: See Appendix III.
UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 Theorem 2
Assuming the Bernoulli packet arrival process and large values of n , the optimum solutionfor (56) is obtained as q BAPn = δ log nn (58) for some constant δ . Furthermore, the maximum effective throughput of the network asymptotically scalesas log n ˆ α , for λ = o (cid:16) n log n (cid:17) .Proof: See Appendix IV.
Theorem 3
Assuming a deterministic packet arrival process, the optimum solution of (56) and thecorresponding maximum effective throughput of the network are asymptotically obtained asi) q CAPn = δ log nn and T eff ≈ log n ˆ α , for λ = o (cid:16) n log n (cid:17) ,ii) q CAPn = δ ′ log nn and T eff ≈ log n ˆ α , for λ = Θ (cid:16) n log n (cid:17) ,iii) q CAPn = log „ λ log2 λn ˆ α « λ and T eff ≈ log n ˆ α , for λ = ω (cid:16) n log n (cid:17) and λ = o (cid:16) n log n (cid:17) ,for some constants δ and δ ′ .Proof: See Appendix V.The above theorems imply that the effective throughput of the network scales as log n ˆ α , regardless of thepacket arrival process. Note that this value is the same as the sum-rate scaling of the same network withbacklogged users [27], which is an upper-bound on the effective throughput of the current setup. In otherwords, the effect of the real-time traffic in the throughput (which is captured in the full buffer probability)is asymptotically negligible. However, we did not consider the effect of dropping on the calculations.In the subsequent section, we include the dropping probability in the analysis and find the maximumeffective throughput of the network such that the dropping probability approaches zero.IV. D ELAY A NALYSIS
In this section, we first formulate the packet dropping probability in the underlying network in termsof the number of links ( n ) and λ for the aforementioned packet arrival processes. Then, we derive thesufficient conditions for the delay-bound ( λ ) in the asymptotic case of n → ∞ such that the packetdropping probabilities tend to zero, while achieving the maximum effective throughput of the network. Lemma 7
Let us denote the packet dropping probability of a link i , i ∈ N n , for the Poisson, Bernoulliand constant arrival processes as P (cid:8) B P APi (cid:9) , P (cid:8) B BAPi (cid:9) and P (cid:8) B CAPi (cid:9) , respectively. Then, P (cid:8) B P APi (cid:9) = 11 + λ log(1 − q n ) − , (59) P (cid:8) B BAPi (cid:9) = (1 − q n )( λq n ) − − q n )( λq n ) − , (60) P (cid:8) B CAPi (cid:9) = (1 − q n ) λ . (61) UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 Proof:
Recalling t ( i ) A k as the time instant of the k th packet arrival into the buffer of link i , eachuser i is active at time slot t ≥ t ( i ) A k only when h ( t ) ii > τ n . In other words, assuming the buffer is full, notransmission (or no service) occurs in each slot with probability − q n . From (4) and (7)-(11), since thetime duration between subsequent packet arrivals is x ( i ) k , the packet dropping probability for a link i isobtained as P { B i } = E h (1 − q n ) x ( i ) k i , (62)where the expectation is computed with respect to x ( i ) k . For the PAP, since x ( i ) k is an exponential randomvariable, (62) can be simplified as P (cid:8) B P APi (cid:9) = Z ∞ λ (1 − q n ) x e − λ x dx (63) = 11 + λ log(1 − q n ) − . (64)Also for the BAP, x ( i ) k is a geometric random variable with parameter ρ = 1 λ . Thus, (62) can besimplified as P (cid:8) B BAPi (cid:9) = ∞ X m =1 (1 − q n ) m ρ (1 − ρ ) m − (65) = ρ − ρ ∞ X m =1 [(1 − q n )(1 − ρ )] m (66) ( a ) = (1 − q n )( λq n ) − − q n )( λq n ) − , (67)where ( a ) comes from the following geometric series: ∞ X m =1 x m = x − x , | x | < . (68)According to Fig. 2-a, x ( i ) k for the CAP is a deterministic quantity and is equal to λ . Thus, we have P (cid:8) B CAPi (cid:9) = (1 − q n ) λ . (69)It should be noted that (64), (67) and (69) are valid for every value of q n ∈ [0 , . In particular, in theextreme case of q n = 1 , P (cid:8) B CAPi (cid:9) = P (cid:8) B P APi (cid:9) = P (cid:8) B BAPi (cid:9) = 0 .We are now ready to prove the main result of this section. In the next theorem, we derive the sufficientconditions on λ , such that the corresponding packet dropping probabilities tend to zero, while achievingthe maximum effective throughput of the network. Theorem 4
For the optimum q n obtained in Theorems 1-3 resulting in the maximum effective throughputof the network,i) lim n →∞ P (cid:8) B P APi (cid:9) = 0 , if λ P AP = ω (cid:16) n log n (cid:17) and λ P AP = o (cid:16) n log n (cid:17) , UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 ii) lim n →∞ P (cid:8) B BAPi (cid:9) = 0 , if λ BAP = ω (cid:16) n log n (cid:17) and λ BAP = o (cid:16) n log n (cid:17) ,iii) lim n →∞ P (cid:8) B CAPi (cid:9) = 0 , if λ CAP = ω (cid:16) n log n (cid:17) and λ CAP = o (cid:16) n log n (cid:17) .Proof: i) From (59), we have P (cid:8) B P APi (cid:9) = 11 − λ P AP log(1 − q P APn ) . (70)It follows from (70) that achieving P (cid:8) B P APi (cid:9) = ǫ results in λ P APǫ = 1 − ǫ − log(1 − q P APn ) ( a ) ≈ ǫ − − q P APn , (71)where ( a ) comes from q P APn = o (1) and the approximation log(1 − z ) ≈ − z, | z | ≪ . Noting the fact thatthe optimum value of q P APn scales as Θ (cid:16) log nn (cid:17) , having λ P AP = ω (cid:16) n log n (cid:17) results in lim n →∞ P (cid:8) B P APi (cid:9) =0 . On the other hand, from Theorem 1, the condition λ P AP = o (cid:16) n log n (cid:17) is required to achieve the maximum T eff , and this completes the proof of the first part of the Theorem.ii) It is realized from (60) that achieving P (cid:8) B BAPi (cid:9) = ǫ results in λ BAPǫ = 1 q BAPn (cid:2) (1 − q BAPn ) ǫ − − (1 − q BAPn ) (cid:3) ≈ ǫ − q BAPn , (72)for small enough ǫ . Noting the fact that the optimum value of q BAPn scales as Θ (cid:16) log nn (cid:17) , having λ BAP = ω (cid:16) n log n (cid:17) results in lim n →∞ P (cid:8) B BAPi (cid:9) = 0 . On the other hand, from Theorem 2, λ BAP = o (cid:16) n log n (cid:17) guarantees achieving the maximum effective throughput of the network.iii) From (61), we have P (cid:8) B CAPi (cid:9) = e λ CAP log(1 − q CAPn ) (73) ( a ) ≈ e − q CAPn λ CAP (74)where ( a ) follows from log(1 − z ) ≈ − z, | z | ≪ for q CAPn = o (1) . To achieve P (cid:8) B CAPi (cid:9) = ǫ , wemust have λ CAPǫ = 1 q CAPn log ǫ − . (75)It follows from (74) that setting q CAPn λ CAP = ω (1) makes e − q CAPn λ CAP → . Using part ( iii ) in Theorem 3,it turns out that choosing λ CAP = ω (cid:16) n log n (cid:17) satisfies q CAPn λ CAP = ω (1) which yields lim n →∞ P (cid:8) B CAPi (cid:9) =0 . We also need the condition λ CAP = o (cid:16) n log n (cid:17) to ensure achieving the maximum effective throughputof the network. UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 Remark 1-
It is worth mentioning that the delay-bound ( λ ) in each link for the CAP scales the sameas that of for the PAP and BAP. However, P (cid:8) B CAPi (cid:9) decays faster than P (cid:8) B P APi (cid:9) and P (cid:8) B BAPi (cid:9) interms of λ , when n tends to infinity (exponentially versus linearly).An interesting conclusion of Theorem 4 is the possibility of achieving the maximum effective throughputof the network while making the dropping probability approach zero. More precisely, there exists some ǫ ≪ such that P { B i } ≤ ǫ , ∀ i ∈ N n , while achieving the maximum T eff of log n ˆ α . This is true for allaforementioned arrival processes. However, for arbitrary values of ǫ , there is a tradeoff between increasingthe throughput, and decreasing the dropping probability and the delay-bound ( λ ). This tradeoff is studiedin the next section. V. T HROUGHPUT -D ELAY -D ROPPING P ROBABILITY T RADEOFF
In this section, we study the tradeoff between the effective throughput of the network and otherperformance measures, i.e., the dropping probability and the delay-bound ( λ ) for different packet arrivalprocesses. In particular, we would like to know how much degradation will be enforced in the throughputby introducing the other constraints, and how much this degradation depends on the packet arrival process. A. Tradeoff Between Throughput and Dropping Probability
In this section, we assume that a constraint P { B i } ≤ ǫ must be satisfied for the dropping probability.It can be easily shown that the constraint P { B i } ≤ ǫ is equivalent to P { B i } = ǫ . The aim is tocharacterize the degradation on the effective throughput of the network in terms of ǫ for different packetarrival processes. First, we consider PAP.Looking at the equations (21) and (59), it turns out that P (cid:8) B P APi (cid:9) = ∆
P APn . Hence, the condition P (cid:8) B P APi (cid:9) = ǫ is translated to ∆ P APn = ǫ . Therefore, using (47), the effective throughput of the networkcan be written as T eff ≈ nq n ǫ log (cid:18) τ n n ˆ αq n ǫ (cid:19) . (76)From the above equation, it can be realized that the effective throughput of the network is equal to theaverage sum-rate of the network with nǫ users in the case of backlogged users, which is given in [27] as log( nǫ )ˆ α for the case of nǫ ≫ or ǫ = ω ( n ) . Also, the optimum value of q n is shown to scale as δ log ( nǫ ) nǫ for some constant δ and hence, the optimum value of λ is given as ǫ − q n = nδ log ( nǫ ) . Let us denote ∆ T eff as the degradation in the effective throughput of the network, which is defined as the difference betweenthe maximum effective throughput in the case of no constraint on P { B i } (Theorem 1-3) and the case UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 with constraint on P { B i } . Using Theorem 1, ∆ T eff for the PAP can be written as ∆ T eff ≈ log n ˆ α − log( nǫ )ˆ α = log( ǫ − )ˆ α , (77)for ǫ = ω (cid:0) n (cid:1) . Moreover, for values of ǫ such that log( ǫ − ) = o (log n ) , it can be shown that the scalingof the effective throughput of the network is not changed, i.e., T eff ∼ log n ˆ α .For the BAP, and using (22) and (60), we have P (cid:8) B BAPi (cid:9) = 1 − q n λ − q n ( a ) ≈
11 + ( λ − q n = ∆ BAPn , (78)where ( a ) follows from the fact that q n = o (1) . Therefore, similar to the case of the PAP, we have P (cid:8) B BAPi (cid:9) ≈ ∆ BAPn = ǫ and as a result, the rest of the arguments hold. In particular, ∆ T eff ≈ log( ǫ − )ˆ α . (79)For the CAP, and using (23) and (61), we have (1 − q n ) λ = ǫ = ⇒ λq n ≈ log( ǫ − ) , (80)which gives ∆ CAPn = 1 − (1 − q n ) λ λq n (81) ≈ ǫ − ) . (82)Hence, using (47), the effective throughput of the network can be written as T eff ≈ n log( ǫ − ) q n log τ nn log( ǫ − ) ˆ αq n ! , (83)which is equal to the average sum-rate of a network with n log( ǫ − ) backlogged users and is asymptoticallyequal to log “ n log( ǫ − ” ˆ α , for values of ǫ satisfying log( ǫ − ) = o ( n ) . Therefore, the degradation in the effectivethroughput of the network for the CAP can be expressed as ∆ T eff ≈ log n ˆ α − log (cid:16) n log( ǫ − ) (cid:17) ˆ α = log log( ǫ − )ˆ α . (84) In the case of ǫ = O ( n ) , it is easy to see that the effective throughput of the network does not scale with n . UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 Comparing the expressions of ∆ T eff for the Poisson, Bernoulli and constant packet arrival processes, itfollows that the degradation in the effective throughput of the network in the cases of PAP and BAP bothgrow logarithmically with ǫ − , while in the case of CAP it grows double logarithmically. In other words,the degradation in the throughput in the cases of the PAP and BAP is much more substantial comparedto the CAP. This fact is also observed in the simulation results in the next section. B. Tradeoff Between Throughput and Delay
In this section, we aim to find the tradeoff between the effective throughput of the network and thedelay-bound ( λ ), for a given constraint on the dropping probability, i.e., P { B i } ≤ ǫ .
1) PAP:
Using (21) and (59), it follows that for a given λ and ǫ ≪ , we have q n ≈ ǫ − λ , = ⇒ τ n ≈ log( λǫ ) , (85)and q n ∆ n ≈ λ . (86)Substituting q n ∆ n and τ n from the above equations in (47) yields T eff ≈ nλ log (cid:18) λ log( λǫ ) n ˆ α (cid:19) . (87)It can be verified that T eff has a global maximum at λ P APopt ≈ n ˆ α log ( n ˆ αǫ − ) . In other words, for λ < λ P APopt ,there is a tradeoff between the throughput and delay, meaning that increasing λ results in increasing boththe throughput and delay. However, the increase in the throughput is logarithmic while the delay increaseslinearly with λ . It should be noted that the region λ > λ P APopt is not of interest, since increasing λ from λ P APopt results in decreasing the throughput and increasing the delay which is not desired.
2) BAP:
Due to the similarity between the values of P { B i } and ∆ n for the PAP and the BAP, theresults obtained for the PAP are also valid for the BAP.
3) CAP:
Using (23) and (61), it follows that for a given λ and ǫ ≪ , we have q n ≈ log( ǫ − ) λ , = ⇒ τ n ≈ log (cid:18) λ log( ǫ − ) (cid:19) , (88)and q n ∆ n ≈ λ . (89)As can be observed, all the results for the cases of PAP and BAP are extendable to the case of CAPby substituting ǫ − with log( ǫ − ) . In particular, the optimum value for λ can be written as λ CAPopt ≈ UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 n ˆ α log ( n ˆ α log( ǫ − )) , and for λ < λ CAPopt , the effective throughput of the network can be given as T eff ≈ α log (cid:16) λ log( ǫ − ) (cid:17) . This means that in the region λ < λ CAPopt , which is the region of interest, there is atradeoff between the throughput and delay such that by increasing λ , T eff increases logarithmically, whilethe delay increases linearly with λ . Furthermore, comparing the value of λ opt for the PAP and BAP withthe CAP, it is realized that λ CAPopt > λ
P APopt and λ CAPopt > λ
BAPopt . This fact is also observed in the simulations.VI. N
UMERICAL R ESULTS
In this section, we present some numerical results to evaluate the tradeoff between the effectivethroughput of the network and other performance measures, i.e., dropping probability and the delay-bound ( λ ) for different packet arrival processes. For this purpose, we assume that all users in the networkfollow the threshold-based on-off power allocation policy. In addition, the shadowing effect is assumedto be lognormal distributed with mean ̟ = 0 . , variance and α = 0 . . Furthermore, we assume that n = 500 and N = 1 .Figures 3 and 4 show the effective throughput of the network versus λ ǫ for the PAP, BAP and CAPand different values of ǫ . It is observed from these figures that for a given constraint on the droppingprobability (e.g., ǫ = 0 . ), and for λ < λ opt , increasing λ results in increasing both the throughput anddelay. However, the increase in the throughput is logarithmic while the delay increases linearly with λ as expected. Also, increasing λ from λ opt results in decreasing the throughput and increasing the delaywhich is not desired. Furthermore, comparing the value of λ opt for the PAP and BAP with the CAP, it isrealized that λ CAPopt > λ
P APopt and λ CAPopt > λ
BAPopt , as expected.To evaluate the degradation in the effective throughput of the network in terms of dropping probability,we plot T eff versus log ǫ − for different packet arrival processes in Fig. 5. It can be seen that the degradationin the throughput in the cases of the PAP and BAP is much more substantial compared to the CAP, asexpected. Hence, the performance of the underlying network with the CAP is better than that of the PAPand BAP from the delay-throughput and delay-dropping probability tradeoff points of view.VII. C ONCLUSION
In this paper, the delay-throughput of a single-hop wireless network with n links was studied. Weconsidered a block Rayleigh fading model with shadowing, described by parameters ( α, ̟ ) , for thechannels in the network. The analysis in the paper relied on the distributed on-off power allocation strategy for the deterministic and stochastic packet arrival processes. It was also assumed that each transmitter hasa buffer size of one packet and dropping occurs once a packet arrives in the buffer while the previouspacket has not been served. In the first part of the paper, we defined a new notion of performance in thenetwork, called effective throughput , which captures the effect of arrival process in the network throughput, UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 l e N e t w o r k ’ s E ff ec t i ve Th r oughpu t e =0.1 e =0.05 e =0.02 (a) l e N e t w o r k ’ s E ff ec t i ve Th r oughpu t e =0.1 e =0.05 e =0.02 (b) Fig. 3.
Effective throughput of the network versus λ ǫ for N = 1 , n = 500 , α = 0 . , and different values of ǫ a) PAP and b)BAP.UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 l e N e t w o r k ’ s E ff ec t i ve Th r oughpu t e =0.1 e =0.05 e =0.02 Fig. 4. Effective throughput of the network versus λ ǫ for the CAP and N = 1 , n = 500 , α = 0 . , and different values of ǫ . and maximize it for different cases of arrival process. It was proved that the effective throughput of thenetwork asymptotically scales as log n ˆ α , with ˆ α , α̟ , regardless of the packet arrival process. In thesecond part of the paper, we presented the delay characteristics of the underlying network in terms of thepacket dropping probability. We derived the sufficient conditions in the asymptotic case of n → ∞ suchthat the packet dropping probability tend to zero, while achieving the maximum effective throughput ofthe network. Finally, we studied the trade-off between the effective throughput, delay, and packet droppingprobability of the network for different packet arrival processes. It was shown from the numerical resultsthat the performance of the deterministic packet arrival process is better than that of the Poisson and theBernoulli packet arrival processes, from the delay-throughput and throughput-dropping probability tradeoffpoints of view. A PPENDIX IP ROOF OF L EMMA χ ( t ) j , L ( t ) ji p ( t ) j , where L ( t ) ji is independent of p ( t ) j , for j = i . Note that P n p ( t ) j = 1 o = P n h ( t ) jj > τ n , C ( t ) j o (A-1) ( a ) = q n ∆ n , (A-2) UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 N e t w o r k ’ s E ff ec t i ve Th r oughpu t log e −1 CAPBAPPAP
Fig. 5. Effective throughput of the network versus log ǫ − for different packet arrival processes and N = 1 , n = 500 , α = 0 . . where ( a ) follows from (18). Thus for the on-off power scheme, we have E h p ( t ) j i = q n ∆ n . (A-3)Under a quasi-static Rayleigh fading channel model, it is concluded that χ ( t ) j ’s are independent andidentically distributed (i.i.d.) random variables with E h χ ( t ) j i = E h L ( t ) ji p ( t ) j i = ˆ αq n ∆ n , (A-4)Var h χ ( t ) j i = E (cid:20)(cid:16) χ ( t ) j (cid:17) (cid:21) − E h χ ( t ) j i (A-5) ( a ) ≤ ακq n ∆ n − ( ˆ αq n ∆ n ) , (A-6)where E (cid:20)(cid:16) h ( t ) ji (cid:17) (cid:21) = 2 , E (cid:20)(cid:16) β ( t ) ji (cid:17) (cid:21) , κ and ˆ α , α̟ . Also, ( a ) follows from the fact that (cid:16) p ( t ) j (cid:17) ≤ p ( t ) j .Thus, E (cid:20)(cid:16) p ( t ) j (cid:17) (cid:21) ≤ E h p ( t ) j i = q n ∆ n . The interference I ( t ) i = P nj =1 j = i χ ( t ) j is a random variable with mean µ n and variance ϑ n , where µ n , E h I ( t ) i i = ( n −
1) ˆ αq n ∆ n , (A-7) ϑ n , Var h I ( t ) i i ≤ ( n − ακq n ∆ n − ( ˆ αq n ∆ n ) ) ≤ ( n − ακq n ∆ n ) . (A-8) UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 A PPENDIX
IIP
ROOF OF L EMMA
Central Limit Theorem [37, p. 183], we obtain P n | I ( t ) i − µ n | < ψ n o ≈ − Q (cid:18) ψ n ϑ n (cid:19) (B-1) ( a ) ≥ − e − ψ n ϑ n , (B-2)for all ψ n > such that ψ n = o (cid:16) n ϑ n (cid:17) . In the above equations, the Q ( . ) function is defined as Q ( x ) , √ π R ∞ x e − u / du , and ( a ) follows from the fact that Q ( x ) ≤ e − x , ∀ x > . Selecting ψ n =( nq n ∆ n ) √ ϑ n , we obtain P {| I ( t ) i − µ n | < ψ n } ≥ − e − ( nq n ∆ n ) . (B-3)Therefore, defining ε , ψ n µ n , noting that as ϑ n = O ( nq n ∆ n ) (from (A-8) in Appendix I) and µ n =Θ( nq n ∆ n ) , we have ε = O (cid:16) ( nq n ∆ n ) − (cid:17) , it reveals that P { µ n (1 − ε ) ≤ I ( t ) i ≤ µ n (1 + ε ) } ≥ − e − ( nq n ∆ n ) . (B-4)Noting that nq n ∆ n → ∞ , it follows that I ( t ) i ∼ µ n , with probability one.A PPENDIX
IIIP
ROOF OF T HEOREM τ n yields ∂ T eff ∂τ n ( a ) = nq n (cid:20) ∂ ∆ n ∂τ n − ∆ n (cid:21) log (cid:18) τ n n ˆ αq n ∆ n (cid:19) + nq n (1 + τ n )∆ n − τ n ∂ ∆ n ∂τ n n ˆ αq n ∆ n + τ n (C-1) ( b ) ≈ nq n (cid:20) ∂ ∆ n ∂τ n − ∆ n (cid:21) τ n n ˆ αq n ∆ n + nq n (1 + τ n )∆ n − τ n ∂ ∆ n ∂τ n n ˆ αq n ∆ n + τ n , (C-2)where ( a ) comes from q n = e − τ n and ∂q n ∂τ n = − q n . Also, ( b ) follows from Lemma 6 and using theapproximation log(1 + x ) ≈ x , for x ≪ . Setting (C-2) equal to zero yields n ˆ αq n ∆ n = (cid:18) ∆ n − ∂ ∆ n ∂τ n (cid:19) τ n . (C-3)It should be noted that (C-3) is valid for every packet arrival process. Recalling from (21), the full bufferprobability for the PAP is given by ∆ P APn = 11 + λ log(1 − q n ) − (C-4) ( a ) ≈
11 + λq n , (C-5) UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 where ( a ) follows from the fact that for q n = o (1) , log(1 − q n ) − ≈ q n . In this case, ∂ ∆ PAPn ∂τ n = ∂ ∆ PAPn ∂q n ∂q n ∂τ n = λq n (1+ λq n ) , which results in ∆ P APn − ∂ ∆ P APn ∂τ n ≈ λq n ) = (cid:0) ∆ P APn (cid:1) . (C-6)Thus for the Poisson arrival process, (C-3) can be simplified as n ˆ αq n = τ n . (C-7)It can be verified that the solution for (C-7) is τ P APn = log n − n + O (1) . (C-8)Using q n = e − τ n , we conclude that q P APn = δ log nn , (C-9)for some constant δ .To satisfy the condition of lemma 6, we should have τ n n ˆ αq n ∆ P APn ≪ , (C-10)Using (C-5), (C-8), and (C-9), it yields λ P AP = o (cid:18) n log n (cid:19) . (C-11)Thus, the maximum effective throughput of the network obtained in (47) can be written as T eff ≈ τ n ˆ α . (C-12)A PPENDIX
IVP
ROOF OF T HEOREM ∂ ∆ BAPn ∂τ n = ∂ ∆ BAPn ∂q n ∂q n ∂τ n = − q n ∂ ∆ BAPn ∂q n = q n ( λ − λ − q n ) . In this case, ∆ BAPn − ∂ ∆ BAPn ∂τ n = 1(1 + ( λ − q n ) = (cid:0) ∆ BAPn (cid:1) . (D-1)Thus for the Bernoulli arrival process, (C-3) can be simplified as n ˆ αq n = τ n . (D-2)It can be observed that (D-2) is exactly equal to (C-7) and hence, its solution can be written as τ BAPn = log n − n + O (1) , (D-3)and q BAPn = δ log nn , (D-4) UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 for some constants δ . Similarly, the maximum effective throughput of the network for the BAP is obtainedas T eff ≈ τ n ˆ α , (D-5)which is achieved under the condition λ BAP = o (cid:18) n log n (cid:19) . (D-6)A PPENDIX VP ROOF OF T HEOREM ∂ ∆ CAPn ∂τ n = ∂ ∆ CAPn ∂q n ∂q n ∂τ n (E-1) = − q n ∂ ∆ CAPn ∂q n (E-2) = 1 − (1 − q n ) λ λq n − (1 − q n ) λ − (E-3) = ∆ CAPn − (1 − q n ) λ − . (E-4)Hence, ∆ CAPn − ∂ ∆ CAPn ∂τ n = (1 − q n ) λ − . In this case, (C-3) can be simplifies as n ˆ αq n (cid:2) − (1 − q n ) λ (cid:3) ( λq n ) = (1 − q n ) λ − τ n . (E-5)or n ˆ α = τ n λ q n (1 − q n ) λ − [1 − (1 − q n ) λ ] . (E-6)Since q n = o (1) , we have (1 − q n ) λ − = e ( λ −
1) log(1 − q n ) ( a ) ≈ e − λq n , and − (1 − q n ) λ ( b ) ≈ − e − λq n . Itshould be noted that ( a ) and ( b ) are valid under the condition λq n = o (1) . Thus, (E-6) can be simplifiedas n ˆ α = τ n λ q n e − λq n [1 − e − λq n ] , (E-7)or ν log ν − (1 − ν ) = Ψ , (E-8)where ν , e − λq n and Ψ , n ˆ ατ n λ . For this setup, we have the following cases: Case 1: Ψ ≫ As we will show the condition λq n = o (1) is satisfied for the optimum q n and the corresponding λ . UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 It is realized from (E-8) that for Ψ ≫ , ν = 1 − ǫ , where ǫ = o (1) . Thus, (E-8) can be simplified as Ψ ≈ log(1 − ǫ ) − ǫ (E-9) ( a ) ≈ ǫǫ (E-10) = 1 ǫ , (E-11)where ( a ) follows from the Taylor series expansion log(1 − z ) = − P ∞ k =1 z k k ≈ − z, | z | ≪ . Since ν , e − λq n and ν = 1 − ǫ , we have e − λq n = 1 − , (E-12) = ⇒ q n ( a ) ≈ λ = τ n n ˆ α , (E-13)where ( a ) follows from the fact that as λq n = o (1) , we have e − λq n ≈ − λq n . It can be verified that thesolution for (E-13) is τ CAPn = log n − n + O (1) . (E-14)Using q n = e − τ n , we conclude that q CAPn = δ log nn , (E-15)for some constant δ .The above results are valid for Ψ , n ˆ ατ n λ ≫ or λ = o (cid:16) n log n (cid:17) . Also, it can be verified that λq n = o (1) ,and therefore the approximations (1 − q n ) λ − ≈ e − λq n and − (1 − q n ) λ ≈ − e − λq n are valid in thisregion.To satisfy the condition of Lemma 6, we must have τ n n ˆ αq CAPn ∆ CAPn ≪ . (E-16)From (23), (E-14) and noting that as λ = o (cid:16) n log n (cid:17) , (cid:2) − (1 − q n ) λ (cid:3) ≈ − e − λq n ≈ λq n , we can write τ n n ˆ αq CAPn ∆ CAPn ≈ λ log nn ˆ α [1 − (1 − q n ) λ ] (E-17) ≈ log nn ˆ αq n = O (cid:18) n (cid:19) , (E-18)which means that the condition of Lemma 6 is automatically satisfied in this region. Thus, the maximumeffective throughput of the network obtained in (47) can be simplified as T eff ≈ τ n ˆ α ≈ log n ˆ α . (E-19) UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 Case 2:
Ψ = Θ(1)
From (E-8) which gives ν log ν − (1 − ν ) = Ψ = Θ(1) , we conclude that ν , e − λq n = Θ(1) . Thus, q n = c λ (E-20) ( a ) = c τ n n ˆ α (E-21)where c and c are constants and ( a ) follows from Ψ , n ˆ ατ n λ = Θ(1) . It can be verified that the solutionfor (E-21) is τ CAPn = log n − n + O (1) . (E-22) q CAPn = δ ′ log nn , (E-23)for some constant δ ′ .The above results are valid for Ψ , n ˆ ατ n λ = Θ(1) or λ = Θ (cid:16) n log n (cid:17) . Also, it can be verified that λq n = o (1) , and therefore, the approximations (1 − q n ) λ − ≈ e − λq n and − (1 − q n ) λ ≈ − e − λq n arevalid in this region.Similar to the argument in Case 1, the condition of Lemma 6 is satisfied, and therefore, the maximumeffective throughput of the network is obtained as T eff ≈ τ n ˆ α ≈ log n ˆ α . (E-24) Case 3: Ψ ≪ It is concluded from (E-8) that ν log ν − (1 − ν ) = Ψ , where Ψ = o (1) . In this case, ν = o (1) , and therefore, ν log ν − ≈ Ψ . The solution for this equation is ν ≈ Ψlog(Ψ) − . In other words, e − λq n ≈ n ˆ αλτ n log (cid:16) λτ n n ˆ α (cid:17) . (E-25)Thus, λq n ≈ log (cid:18) λτ n n ˆ α (cid:19) + log log (cid:18) λτ n n ˆ α (cid:19) (E-26) ( a ) ≈ log (cid:18) λτ n n ˆ α (cid:19) , (E-27)where ( a ) follows from λq n = ω (1) which comes from ν = o (1) . The solution for the above equationcan be written as τ n = log λ − f ( λ ) or q n = e f ( λ ) λ = o (1) , where we assume f ( λ ) = o (log λ ) . Substitutingin (E-27), we obtain e f ( λ ) = log (cid:18) λ (log λ − f ( λ )) n ˆ α (cid:19) (E-28) = log (cid:18) λ log λn ˆ α (cid:19) + 2 log (cid:18) − f ( λ )log λ (cid:19) (E-29) ( a ) ≈ log (cid:18) λ log λn ˆ α (cid:19) , (E-30) UBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, OCTOBER 2009 where ( a ) follows from the fact f ( λ ) = o (log λ ) . Thus, using τ n = log λ − f ( λ ) , it yields τ CAPn = log λ − log log (cid:18) λ log λn ˆ α (cid:19) . (E-31)It should be noted that (E-31) is derived from (E-25) for Ψ , n ˆ ατ n λ ≪ . This translates the condition n ˆ ατ n λ ≪ to n ˆ αλ log λ ≪ , which incurs that λ = ω (cid:16) n log n (cid:17) .Also, in the following we show that the condition λq n = o (1) is satisfied. It follows from (E-27) that λq n = log (cid:16) λτ n n ˆ α (cid:17) λ (E-32) ( a ) ≤ log (cid:16) λ log λn ˆ α (cid:17) λ (E-33) ( b ) = o (1) , (E-34)where ( a ) follows from (E-31) and ( b ) comes from λ = ω (cid:16) n log n (cid:17) .To satisfy the condition of Lemma 6, we must have τ n n ˆ αq CAPn ∆ CAPn ≪ . (E-35)From (23) and (E-31), we can write τ n n ˆ αq CAPn ∆ CAPn ≈ λ log λn ˆ α [1 − e − λq n ] (E-36) ( a ) ≈ λ log λn ˆ α , (E-37)where ( a ) follows from e − λq n = o (1) . In order to have λ log λn ˆ α = o (1) , one must have λ = o (cid:16) n log n (cid:17) . In thiscase, the maximum effective throughput of the network can be simplified as T eff ≈ τ n ˆ α ≈ log λ ˆ α . (E-38)Noting that λ satisfies λ = ω (cid:16) n log n (cid:17) and λ = o (cid:16) n log n (cid:17) , it follows that log λ ∼ log n . In other words, T eff ≈ log n ˆ α . A CKNOWLEDGMENT
The authors would like to thank V. Pourahmadi of CST Lab. for the helpful discussions.R
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