Delay Analysis in Static Poisson Network

Yi Zhong∗, Wenyi Zhang∗, and Martin Haenggi† ∗Key Laboratory of Wireless-Optical Communications, CAS; University of Science and Technology of China †University of Notre Dame, Notre Dame, IN, USA Emails: [email protected], [email protected], [email protected]

Abstract—We investigate the delay of the discrete-time slotted Previous analyses have yielded only bounds of arrival rate for ALOHA network where the sources are distributed as a Poisson which the system is stable [4]–[6]. In [7], the stability and delay . Each of the sources is paired with a destination at of high-mobility networks are analyzed by combining queueing a given distance and a buffer of infinite capacity. The network is assumed to be static, i.e., the sources and the destinations are theory and stochastic geometry. In high-mobility networks, the generated at first and remain static during all the time slots. sizes of queues and the serving rates are decoupled, which, Employing tools from as well as point process however, does not hold in static networks. In [8], we derived theory, we obtain upper bounds and lower bounds for the cumulative the sufficient conditions and necessary conditions for stability of distribution function (cdf) of the conditional mean delay. Numerical static Poisson networks, and in this paper, we extend that work results show that the gap between the upper and lower bounds is small, and the results also reveal how these bounds vary with system and analyze the delay of such networks. parameters. II. SYSTEM MODEL Index Terms—ALOHA, delay, dominant system, interacting queues, Poisson bipolar model, queueing theory, static network. We consider a discrete-time slotted ALOHA network with transmitters and receivers distributed as a Poisson bipolar process I. INTRODUCTION [3, Def. 5.8], i.e., we model the locations of the transmitters {x }⊂Rd λ In the literature, the protocol of slotted ALOHA is studied as a PPP Φ= i of intensity . Each transmitter is r extensively. However, most works concentrate on the capacity attached with a receiver at a fixed distance 0 and a random x ∈ analysis and assume that terminals are backlogged. To make the orientation. In the analysis, we condition on 0 Φ at which a r |x | model more practically relevant, each terminal should provide typical transmitter under consideration is located, where 0 = 0 a buffer for queueing. This problem is complex because it is the distance between this point and the origin at which the involves interacting queues, i.e., the serving rate of each queue corresponding receiver is located (see Fig. 1). The time is divided depends on the sizes of queues, the analysis of which should into discrete slots with equal duration, and each transmission combine queueing theory and multi-access information theory and attempt occupies exactly one time slot. The network is assumed is notoriously difficult to cope with. to be static, i.e., the locations of the nodes are generated once at Previous analyses of interacting queues are mostly based on an the beginning and then kept unchanged during all the time slots. oversimplified physical layer and consider a discrete-time slotted ALOHA system. Each terminal attempts to transmit the head-of- line packet in each time slot with a certain if its buffer is not empty. If two or more terminals transmit in the same time slot, a collision occurs. Only the stability region has been studied [1] for this simplified ALOHA system. In practical networks, concurrent transmissions lead to interference, which cannot be accurately modeled as collisions. Moreover, the randomness in the deployment makes accurate analysis complicated. In this work, we model a large-scale network by point process theory, which is widely used to analyze the performance of wireless networks [2], [3]. A common and meaningful model is the (PPP), in which each transmitter is modeled as one point of the PPP. We combine queueing theory Fig. 1. A snapshot of the bipolar model with ALOHA. and stochastic geometry to analyze the delay in a static ALOHA network, i.e., the transmitters and the receivers are generated at We assume the Rayleigh block fading model, in which the first and remain static during all the time slots. Although ALOHA power fading coefficients remain constant over each time slot is a very simple MAC, the stability and delay analysis is still an and are spatially and temporally independent with exponential open problem, and studying over it helps us understand more distribution of mean 1. Let α be the path loss exponent and hk,x complex protocols. If each transmitter maintains a buffer to store be the fading coefficient between transmitter x and the considered the generated packets, the analysis becomes complex since the receiver located at origin o in time slot k. All transmitters are serving rate of each queue depends on the status of other queues, assumed to transmit at unit power. The power spectral density the channel status and the ALOHA protocol. We derive upper of the thermal noise is set as N0 and the bandwidth is W .We bounds and lower bounds for the cdf of the delay, and by slightly assume that if its SINR is above a threshold θ, a link can be relaxing the results, we obtain the closed-form results. successfully used for information transmission. Each transmitter has a buffer of infinite capacity to store the on the realization of the PPP as the conditional mean delay.1 generated packets. The packets are generated at each transmitter In the following discussions, we analyze the of the according to a with arrival rate λa (0 ≤ λa ≤ 1) conditional mean delay. Specifically, we derive upper bounds packets per time slot. The arrival processes at different transmit- and lower bounds for the cdf of the conditional mean delay, ters are independent, and each transmitter attempts to send its which is quite suitable to characterize the delay performance. head-of-line packet with probability p if its buffer is not empty. Unless otherwise specified, we assume that the delay mentioned We assume that the feedback of each transmission attempt, either in this paper is evaluated in terms of time slots. Let DΦ be the successful or failed, is instantaneous so that each transmitter conditional mean delay conditioned on the realization of the PPP; is aware of the outcome. If the transmission attempt fails, the thus it is a random variable uniquely defined by the PPP and transmitter attempts to retransmit the packet at the next time slot formally defined as follows. with probability p; otherwise, it deletes the packet from the buffer. + Definition 1. Let A(t) be the number of packets that arrived For any time slot k ∈ N , let Φk be the set of transmitters that at the transmitter x0 in [0,t]. Let Ti,Φ be the time in number are transmitting in time slot k. The interference at the typical of time slots between the arrival of the ith packet and the receiver located at the origin o in time slot k is moment of successfully transmitting it to the receiver at the origin  −α conditioned on the realization of the PPP Φ. Then, the conditional Ik = hk,x|x| 1(x ∈ Φk). (1) mean delay is defined as x∈Φ\{x0} A(t) i=1 Ti,Φ When the typical transmitter is allowed to transmit, the SINR of DΦ  lim . (3) t→∞ A t the typical receiver in time slot k is ( ) A. Lower Bound h r−α  k,x0 0 . SINRk = −α (2) In order to derive a lower bound for the cdf of the mean delays WN0 + hk,x|x| 1(x ∈ Φk) x∈Φ\{x0} in the overall network, we consider a dominant system. Assume The relevant probability measure of the point process in this that in the dominant system the typical transmitter behaves exactly paper is the Palm probability Px0 . Correspondingly, the expec- the same as that in the original system. However, for the other tation Ex0 is taken with respect to the measure Px0 . Whether transmissions in the dominant system, we assume that when the the transmission of the typical transmitter x0 is successful or queues at the transmitters become empty, the transmitters continue p not is uncertain because of the four sources of randomness, i.e., to transmit “dummy” packets with the ALOHA probability , thus the bursty arrival of traffic, the fading, the ALOHA protocol continuing to cause interference to the other transmissions no k matter whether their queues are empty or not. The queue size at and the realization of PPP. Let CΦ be the event that the typical transmission succeeds conditioned on the realization of the PPP each transmitter in the dominant system will never be smaller than k that in the original system, resulting in larger delay. Therefore, the Φ and the status of each queue in time slot k, i.e., CΦ consists of two events: that the transmission is scheduled by the ALOHA and cdf we obtain under these assumptions will be a lower bound for k the cdf of the mean delay in the original system. The following also successful in time slot k. The success event CΦ relies on the realization of the PPP since the locations of serving transmitter theorem gives the lower bound for the cdf of the conditional mean and the interferers are different for different realizations of the delay in the overall network. Ck PPP. Meanwhile, the success event Φ relies on the index of the Theorem 1. Given a slotted ALOHA system with the transmitters time slot k because the statuses of the queues at the interferers distributed as a PPP and with a packet arrival being Bernoulli change over the time slot. When the queue at an interfering processes, a lower bound of the cdf of the conditional mean delay transmitter is empty, no interference will be caused by the said in the overall network is transmitter. Even if the realization of the PPP Φ and the time slot  ∞   x0 1 1 1 jω α index k are given, whether the typical link is successful or not is P (DΦ ≤ T ) ≥ + Im p exp − jωθr WN0 2 π ω 0 still uncertain because of the effect of ALOHA and fading. Let  0  x k 2 P 0 (C ) be the success probability conditioned on the PPP Φ in (2 + 2λaT − λa) − 8Tλa − λa +2 λa Φ −jω ln + k T time slot .  4  2 ∞ p jω − πλ − − p r r ω. 2 1 α −α +1 d d (4) III. DELAY STATISTICS 0 1+θr0 r For the typical link in the network, the delay considered in Proof: The success probability for the typical link given Φ x this paper includes two parts, one is the queueing delay, which in the dominant system, denoted by P 0 (CΦ), is the same for measures the delay between the time when a packet arrives at the each time slot, which is given by (7). Given the realization of the queue and the time when it starts to be served, the other is the point process Φ, the queueing system at the typical transmitter is service time, which is the time to transmit said packet. In the static equivalent to a Geo/G/1 queue, or a discrete-time single server network, given the locations of the transmitters and receivers, the retrial queue [9], [10]. In the equivalent Geo/G/1 queue, the time success are different for different transmissions; thus, is slotted and the packets arrive according to a Bernoulli process the serving rates are also different, which results in different mean delays for different transmissions. Therefore, the mean delay has a 1Strictly speaking, there is no typical transmitter in a realization of the PPP. Since there is no commonly accepted terminology, we still call the link from x0 statistical distribution for the overall network. To avoid confusion, to o the typical link, even when conditioning on Φ (and on x0 ∈ Φ). When taking we denote the mean delay of the typical transmitter conditioning the expectation w.r.t. Φ later, the notion of typicality has its usual meaning. with intensity λa packets per time slot. The arrival process is processes, for all t>0, a lower bound of the cdf of the also called geometric arrival process since the probability that a conditional mean delay in the overall network is λ  packet arrives in a time slot is a, and the number of time slots x0 −t α P (DΦ ≤ T ) ≥ max 0, 1 − p exp tθr0 WN0 between two adjacent arrivals is a geometric random variable.  x ∞ −t The success probability is P 0 (CΦ), thus the service times of p −2πλ 1 − +1− p rdr θrαr−α packets are independent and identically distributed with geometric 0 1+ 0 DΦ 2  distribution. From [9], we get the conditional mean delay as (2 + 2λaT − λa) − 8Tλa − λa +2 λa t . λ −λ Px0 (C ) + ln + (12) 1 a a Φ Px0 C >λ 4T 2 Px0 (C ) + 2(Px0 (C )−λ )Px0 (C ) if ( Φ) a DΦ = Φ Φ a Φ (5) x0 When t is a positive integer, denoted by n, we get a closed-form ∞ if P (CΦ) ≤ λa. lower bound x0 Noticing that the queue is stable only when P (CΦ) >λa,we x0 lb P (DΦ ≤ T ) ≥ p (T,n) (13) get the cdf of the conditional mean delay as

where   Px0 D ≤ T Px0 Px0 C ≥ ( Φ )= ( Φ) lb −n α  p (T,n) = max 0, 1 − p exp nθr0 WN0 2 (2 + 2λaT − λa) − 8Tλa − λa +2 λa + . (6) n 4T 2 δ δ 2 −i Γ(i − δ)Γ(n − i + δ) +πλδn(1 − p) θ r0 ((1 − p) − 1) The success probability for the typical link conditioned on Φ Γ(i +1)Γ(n − i +1)  i=1 2 in the dominant system is evaluated as (2 + 2λaT − λa) − 8Tλa − λa +2 λa +n ln + . x0 x0 (14) P (CΦ)=pP (SINR >θ| Φ) 4T 2  pPx0 h r−α >θ WN I | n T = k,x0 0 ( 0 + k) Φ When is chosen as the optimal value for given , i.e., lb nmax(T ) = argmax + p (T,n) x0 α n∈N , we get a lower bound that = pE exp − θr0 WN0 n  is superior to other lower bounds corresponding to other as α −α x0 lb − θr0 hk,x|x| 1(x ∈ Φk) | Φ P (DΦ ≤ T ) ≥ p (T,nmax(T )), (15) x∈Φ\{x0} t> Px0 C   Proof: For all 0, the cdf of ( Φ) is α p   p − θr WN − p . x0 = exp 0 0 α −α +1 (7) Px0 {Px0 C <λ } Px0 e−t ln(P (CΦ)) >e−t ln(λa) . 1+θr0 |x| ( Φ) a = (16) x∈Φ\{x0} By applying the Markov inequality, we obtain Δ x Y P 0 CΦ The moment generating function of =ln( ( )) is x0 x0 x0 1 −t ln(P (CΦ)) P {P (CΦ) <λa} < E e s α e−t ln(λa) MY (s)=p exp − sθr0 WN0  p−t t λ tθrαWN ∞ p s = exp ln( a)+ 0 0 − πλ − − p r r .  ∞ 2 1 α −α +1 d (8) p −t 0 1+θr0 r − πλ − − p r r . 2 1 α −α +1 d (17) 0 1+θr0 r The cdf of Y , denoted by FY (y)=P (Y ≤ y), follows from Therefore, the cdf of the conditional mean delay is the Gil-Pelaez Theorem [11] as   x −t α ∞ −jωy P 0 D ≤ T ≥ , − p tθr WN 1 1 Im{e MY (jω)} ( Φ ) max 0 1 exp 0 0 FY (y)= − dω. (9)  ∞ 2 π ω p −t 0 − πλ − − p r r 2 1 α −α +1 d x0 1+θr r The cdf of P (CΦ) is evaluated as 0 0 λ T − λ 2 − Tλ − λ λ  x0 x0 (2 + 2 a a) 8 a a +2 a P {P (CΦ) ≤ λa} = FY (ln(λa)) +t ln + . T  ∞ −jω ln(λ ) 4 2 1 1 Im{e a MY (jω)} + = − dω. By setting t = n ∈ N , we get π ω (10)  2 0 ∞ p −n − − p r r Therefore, we have 1 α −α +1 d 0 1+θr0 r  ∞    n ∞ α −α i x0 1 1 1 jω α (a) (θr r ) r P DΦ ≤ T p − jωθr WN0 i i 0 ( )= + Im exp 0 = − Cn(1 − (1 − p) ) dr 2 π 0 ω − p θrαr−α n   i=0 0 (1 + (1 ) 0 ) 2 (2 + 2λaT − λa) − 8Tλa − λa +2 λa n −jω ln + (b) n δ δ 2 −i Γ(i − δ)Γ(n − i + δ) 4T 2 = − δ(1 − p) θ r0 ((1 − p) − 1) .   2 Γ(i +1)Γ(n − i +1) ∞ p jω i=1 − πλ − − p r r ω. 2 1 α −α +1 d d (11) Ci n / i n − i n / i n − i 0 1+θr0 r where n = ! ( !( )!)=Γ( +1) (Γ( +1)Γ( +1)) is the binomial coefficient and δ =2/α.(a) holds from the Thus, for the original system, we get the result in Theorem 1. binomial expansion and the exchange of summation and integral. In order to simplify the lower bound, we derive a lower bound (b) follows from the relationship between the beta function by using the Markov inequality. B x, y 1 tx−1 − t y−1 t ( )= 0 (1 ) d and the gamma function and Corollary 1. Given a slotted ALOHA system with the transmitters from the fact that the term for i =0equals to zero. Thus, we get distributed as a PPP and with a packet arrival being Bernoulli the results in Corollary 1. B. Upper Bound Therefore, the cdf of the conditional mean delay for the To derive an upper bound of the cdf of the conditional mean modified system is evaluated as  ∞   delay, we consider a modified system as follows: if a packet in an x0 1 1 1 jω α P (DΦ ≤ T )= + Im p exp − jωθr WN0 interfering transmitter is not scheduled by the ALOHA or failed 2 π ω 0  0  for the transmission, it will be dropped rather than retransmitted. 2 (2 + 2λaT − λa) − 8Tλa − λa +2 λa −jω ln + In this way, since the interference in the modified system is T  4  2 always smaller than that in the original system and the packets ∞ jω λap will not accumulate at the interfering transmitters, the cdf of the −2πλ 1 − +1− λap rdr dω. θrαr−α (24) conditional mean delay for the modified system will be an upper 0 1+ 0 bound of that for the original system. Since the cdf of the conditional mean delay of the modified system is only an upper bound of the original system, we get the Theorem 2. An upper bound of the cdf of the conditional result in Theorem 2. mean delay of the slotted ALOHA system with the transmitters In the following, we use the Markov inequality to derive an distributed as a PPP and with a packet arrival being Bernoulli upper bound that is slightly easier to evaluate. processes is    ∞ Corollary 2. An upper bound of the cdf of the conditional x0 1 1 1 jω α P (DΦ ≤ T ) ≤ + Im p exp − jωθr0 WN0 mean delay in the slotted ALOHA system with the transmitters 2 π ω  0  distributed as a PPP and with a Bernoulli packet arrival is 2 (2 + 2λaT − λa) − 8Tλa − λa +2 λa  −jω ln + x0 t α T P (DΦ ≤ T ) ≤ p exp − tθr WN0  4  2 0 ∞ λ p jω  a ∞ λ p t −2πλ 1 − α +1− λap rdr dω. (18) a 1+θr r−α −2πλ 1 − +1− λap rdr 0 0 1+θrαr−α 0 0  Proof: In the modified system, the interfering transmitter 2 (2 + 2λaT − λa) − 8Tλa − λa +2 λa is active if there is a packet arriving at the queue and the −t ln + , (25) 4T 2 transmission is scheduled by ALOHA, thus the probability is λap. t> t n From (6), we get the cdf of the conditional mean delay as for all 0. When is a positive integer, denoted by ,weget a lower bound in closed form as x0 x0 x0  P (DΦ ≤ T )=P P (CΦ) ≥  Px0 D ≤ T ≤ pn − nθrαWN 2 ( Φ ) exp 0 0 (2 + 2λaT − λa) − 8Tλa − λa +2 λa + . (19) n 4T 2 δ 2 i Γ(i − δ)Γ(n − i + δ) −πλnδθ r (1 − (1 − λap) ) By introducing the modified system, packets at the interfering 0 Γ(i +1)Γ(n − i +1)  i=1  transmitters are dropped if the transmitters are silenced due to 2 (2 + 2λaT − λa) − 8Tλa − λa +2 λa ALOHA or if the transmission fails due to the SINR condition, −n ln + . T (26) thus an interfering transmitter is active with probability λap. 4 2 Similar to the derivation of (7), we get the success probability When the optimum value is chosen for n, we have   for the typical link conditioned on Φ in the modified system as x0 n α  P (DΦ ≤ T ) ≤ min p exp − nθr0 WN0 x0 α n∈N+ P (CΦ)=p exp − θr0 WN0  n λap δ 2 i Γ(i − δ)Γ(n − i + δ) +1− λap . −πλnδθ r (1 − (1 − λap) ) θrα|x|−α (20) 0 i n − i 1+ 0 i=1 Γ( +1)Γ( +1) x∈Φ\{x0}   λ T − λ 2 − Tλ − λ λ Δ x (2 + 2 a a) 8 a a +2 a The moment generating function of Y =ln(P 0 (CΦ)) is −n ln + . (27) 4T 2 s α MY (s)=p exp − sθr0 WN0 Proof: For all t>0, by applying the Markov inequality, we  ∞ λ p s obtain the following inequality − πλ − a − λ p r r . 2 1 α +1 a d (21) x0 x0 θr r−α P {P (CΦ) <λa} 0 1+ 0   x0 x0 t t = P (P (CΦ)) <λ The cdf of Y can be derived as follows by applying the Gil- a t Pelaez Theorem given by (9). −t x0 > 1 − λa E (P (CΦ))  ∞ −jωy 1 1 Im{e MY (jω)} t α FY (y)= − dω. (22) =1− p exp − t ln(λa) − tθr0 WN0 2 π 0 ω  ∞ λ p t − πλ − a − λ p r r . Therefore, we have 2 1 α −α +1 a d (28) 0 1+θr0 r Px0 {Px0 C ≥ x}  ( Φ)  =  ∞ Therefore, 1 1 1 jω α + Im p exp − jωθr WN0 − jω ln (x) x0 x0 t α π ω 0 P {P (CΦ) ≥ x}≤p exp − t ln(x) − tθr0 WN0 2  0   ∞ λ p jω ∞ λ p t − πλ − a − λ p r r ω. − πλ − a − λ p r r . 2 1 α −α +1 a d d (23) 2 1 α −α +1 a d (29) 0 1+θr0 r 0 1+θr0 r From (6), we get an upper bound for the cdf of the conditional 1 mean delay for the modified system as follows Lower bound in Theorem 1  0.5 Lower bound in closed form in Corollary 1 Upper bound in Theorem 2 x0 t α Probability P (DΦ ≤ T ) ≤ p exp − tθr0 WN0 Upper bound in closed form in Corollary 2 0  2 4 6 8 10 12 14 ∞ t Conditional Mean Delay (λ=0.01) λap −2πλ 1 − +1− λap rdr 1 1+θrαr−α 0 0  2 (2 + 2λaT − λa) − 8Tλa − λa +2 λa 0.5

−t ln + . (30) Probability 4T 2 0 x 2 4 6 8 10 12 14 Since P 0 (DΦ ≤ T ) ≤ 1, we give an upper bound in Corollary 2 Conditional Mean Delay (λ=0.03) for the cdf of the conditional mean delay for the original system. 1 By the same derivations as Corollary 1, we get the results. 0.5

C. Comparison of Lower and Upper Bounds Probability 0 2 4 6 8 10 12 14 In this section, we compare the lower and upper bounds for the Conditional Mean Delay (λ=0.05) cdf of conditional mean delay based on their numerical evaluation. Fig. 3. Comparison of lower bound and upper bound for the cdf of conditional 1 mean delay for different transmitter intensities λ. The parameters are set as θ = 10dB, r0 =1, N0 = −173dBm, W =20MHz, α =4and λa =0.1. Lower bound in Theorem 1 0.5 Lower bound in closed form in Corollary 1 Upper bound in Theorem 2 Probability Upper bound in closed form in Corollary 2 0 2 4 6 8 10 12 14 show that the gap between the non-closed form upper and lower Conditional Mean Delay (λ =0.05) a bounds is small. As for the closed form upper and lower bounds, 1 the gap is small for small arrival rate λa and small intensity of transmitters λ. 0.5

Probability ACKNOWLEDGEMENT 0 2 4 6 8 10 12 14 Conditional Mean Delay (λ =0.1) The research of Y. Zhong and W. Zhang has been supported a 1 by the National Basic Research Program of China (973 Program) through grant 2012CB316004, and National Natural Science 0.5 Foundation of China through grant 61379003. Probability

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