Random Access With Multi-Packet Reception SUBMITTED TO THE IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS

Aditya Dua Department of Electrical Engineering Stanford University 350 Serra Mall, Stanford CA 94305 Email: [email protected]

Abstract The queuing performance of a finite-user slotted-Aloha type random access protocol with multipacket reception capability at the base station is considered. The system evolution under this protocol can be described by a finite dimensional Markov chain, which is not amenable to analysis due to complex interactions between user queues. A “user-centric” analysis approach is proposed, which focuses on the evolution of the one-dimensional Markov chain associated with the queue of a typical user in the system, assuming that all other users are in “steady-state”. While the approach is quite general, this paper addresses the special case of identical users with Bernoulli arrivals in order to obtain insightful analytical expressions for various quantities of interest (throughput, average delay etc.). Both static and Rayleigh faded wireless channels are studied, under suitable assumptions on the underlying power control mechanisms. Sufficient stability conditions as well as asymptotic performance measures in the infinite power and infinite user regime are derived for both cases. The fundamental throughput versus power trade-off inherent in the protocol is established. Index Terms Random access, power control, slotted-Aloha, multi-packet reception, wireless networks.

I. I NTRODUCTION Random access protocols offer a simple, decentralized way of regulating access to a shared wireless channel, for instance on the uplink of a cellular communication system. The performance of random access is controlled by varying two sets of parameters — the power with which each user transmits and the probability with which each user attempts to access the shared channel. Choosing these control parameters to enhance system performance is an important and interesting problem, both from an engineering and theoretical perspective. Classical analysis of randomized multiple access schemes like slotted-Aloha [1] has focused on the the so called collision model, where at most one user can successfully communicate with a centralized base station (BS) or access point (AP) in a time-slot. Under this model, if multiple users attempt transmissions in a time-slot, then all of them are unsuccessful. Another extensively studied paradigm is the capture model, wherein at most one user can successfully communicate with the BS even in the presence of interfering transmissions, provided the user’s signal is sufficiently strong vis-`a-vis the interfering signals. However, in code division multiple access (CDMA) systems, where the BS can decode multiple simultaneous interfering transmissions (say, using a Rake receiver), it is possible for more than one user to concurrently communicate with the BS in a time-slot. Random access with multi-packet reception (MPR) has received considerable attention in the literature in recent times. A generalized MPR model was first introduced by Ghez et. al. in [2]. They modeled the number of successful transmissions in a time-slot as a random variable which is a function of the number of attempted transmissions only, and provided stability conditions for the case of indistinguishable nodes. del Angel e. al. [3] considered slotted-Aloha from an information theoretic perspective under the assumptions of infinite users and no fading. They studied dynamic control of transmission probability and randomized power control to improve the throughput and spectral efficiency of the system. Naware et. al. examined the stability and delay properties of finite-user slotted-Aloha with MPR capability in [4]. They extensively characterized the performance of a two-user system (N = 2), and also provided sufficient stability conditions for the N > 2 case for fixed transmission probabilities. Sant et. al. [5] provided sufficient stability

conditions for finite-user slotted-Aloha with capture under Markov modulated wireless channel fading. In other related work, Ghez et. al. [6], Tsbyakov [7], and del Angel et. al. [8] studied dynamic control policies for stabilizing slotted-Aloha with MPR and optimizing its performance under a variety of modeling assumptions. Queuing performance of slotted-Aloha with capture over wireless channels exhibiting Rayleigh fading and shadowing was considered by van der Plas et. al. in [9]. Their focus was on the assessment of receiver capture probabilities under a general wireless propagation model. Yu et. al. studied the stability properties of slotted-Aloha with capture under Rayleigh fading in the infinite-user regime in [10]. Peh et. al. [11] proposed a multi-bit feedback algorithm for optimizing the performance of slotted-Aloha in a Rayleigh faded environment. Dua [12] studied random access over Rayleigh faded channels with power and transmission probability control in a convex optimization framework. In this paper, we focus on queuing aspects of a slotted-Aloha type random access protocol with MPR capability at the BS, under both static and fading channel conditions. We study a multiple capture model where a user’s transmission is successful if her received signal to interference plus noise ratio (SINR) is above a certain threshold γ ? , independent of everything else. We examine two scenarios — (a) a static environment where the wireless channel from each user to the BS is time-invariant, and (b) a mobile environment where the wireless channel from each user to the BS exhibits Rayleigh fading. The evolution of a system with N users can be described by an N-dimensional Markov chain (under Markovian arrival and channel processes), which is extremely hard to analyze for steadystate behavior owing to the complex interactions between queues of different users. To simplify the problem, we abandon the holistic approach for a user-centric approach, wherein we analyze a one-dimensional Markov chain which captures the evolution of the queue of a typical user in isolation, given that this user perceives the rest of the system in steady-state. While this method is applicable in great generality, to obtain insightful analytical formulae we restrict attention to the symmetric case, where packets arrive to queues of different users according to independent Bernoulli processes with the same parameter p. We study the static channel case under the

assumption of perfect power control, i.e., transmissions of all users are received with the same power P at the BS. For the fading case, we assume perfect slow power control [11], i.e., the transmissions of all users are received with the same power P at the BS on an average. The key quantity which captures the coupling between a user’s queue and the rest of the system is η — the effective probability of transmission by a typical user in steady-state, which is less than the actual probability ∆ with which a user with a non-empty queue attempts transmission. The probability of successful transmission by a user and all other quantities of interest (average backlog etc.) are then expressed as a function of η. We provide sufficient conditions (choice of ∆) under which η is unique, i.e., the system converges to a unique steady-state. We examine the maximum throughput the system can sustain as a function of P . We also study the behavior of the system in two asymptotic regimes — infinite user (N → ∞) and infinite power (P → ∞). The user-centric approach also allows us to obtain closed form expressions for average packet delay perceived by a typical user as a function of η. A. Paper outline The system model for random access with MPR is described in Section II, and the static channel case is treated in great detail. A corresponding treatment of random access with MPR over Rayleigh faded wireless channels is provided in Section III. The delay performance for both scenarios is examined in Section IV. The theoretical results developed in each section are illustrated via numerical examples. Finally, concluding remarks are given in Section V. B. Notation Some notations used throughout the paper are enumerated here for convenience. N denotes the set of natural numbers, Z+ denotes the set of non-negative integers, and R denotes the set of real numbers. For any set A, |A| denotes the cardinality of A and A\a denotes the set difference between A and {a}. For any x ∈ R, bxc denotes the greatest integer smaller than x, while dxe denotes the smallest integer greater than x. All vectors are denoted in boldface. The abbreviation w.p. is used to represent “with probability”. Finally, the “big-oh” notation f (N) = O (g(N)) is used to indicate that ∃ c > 0 such that f (N) ≤ cg(N) for large enough N.

II. S TATIC C HANNEL A. System model Consider the uplink of a time-slotted wireless communication system with N users communicating with a centralized base-station (BS) or access-point (AP). A time-slot is divided into two logical sub-slots. In the first sub-slot of every time-slot, a packet arrives to the transmission queue of each user w.p. p, independent of all past and future arrivals. The arrival processes are assumed to be independent across users. In the second sub-slot, every user with a non-empty queue transmits a packet w.p. ∆, independent of the decision of all other users. The power at which a user’s packet is received at the BS is determined by the transmitted power and the power attenuation from the user to the BS. In this section, we examine a “static channel” scenario where the power attenuation from transmitters to the BS is constant. This would be the case for a wireless system with fixed/static users. In particular, if the ith user transmits a packet with power Pi , it is received at the BS with power Pi Gi , for some fixed Gi ∈ (0, 1]. Under this model, the SINR for the ith user in the tth time-slot (if she transmits) is given by γit = X

P i Gi P j Gj + σ 2

,

(1)

j∈S t \i

where σ 2 denotes the ambient noise power at the BS and S t denotes the set of users who transmit in the tth time-slot (active set). The ith user’s transmission in the tth time-slot is successful if the received SINR is above a threshold γ ? , i.e., γit ≥ γ ? . All users who transmit in a time-slot receive instantaneous and error-free one bit feedback (ACK/NAK) regarding the outcome of their transmission at the end of the time-slot by the BS. No feedback is given to inactive users. We assume perfect power control, i.e., packets of all users are received with equal power at the BS. In particular, each user adjusts her transmit power so that Pi Gi = P ∀ i. This helps combat the so called near-far effect, wherein transmissions of “strong” users (near the BS) can potentially overwhelm transmissions of “weak” users (far from the BS). With perfect power control, all users have an equal opportunity to transmit packets successfully (regardless of their

distance from the BS), which is a reasonable requirement if they all observe the same packet arrival rate p. The received SINR for the ith user in the tth time-slot can be rewritten as γit =

(|S t |

1 P , = t 2 − 1)P + σ |S | − 1 + γ0−1

(2)

P is the received signal-to-noise ratio (SNR) for a user if she alone transmits in a σ2 time-slot. Note that it is essential to have γ0 ≥ γ ? , else communication is not feasible even in

where γ0 ,

the absence of interference. The necessary and sufficient condition for the ith user’s transmission in the tth time-slot to be successful is γit ≥ γ ? , which translates to (see also [3]) 1 1 |S | ≤ 1 + ? − , k ? (γ0 ). γ γ0 t

(3)

The condition in (3) is interpreted as follows — given the received SNR γ0 and transmission probability ∆ for all users, all transmissions are successful in a time-slot if k ? (γ0 ) or fewer users transmit, and all transmissions are unsuccessful else. Note that k ? (γ0 ) is a non-decreasing 1 ? ? ? ? function of γ0 . Also, observe that k (γ ) = 1 and kmax , lim k (γ0 ) = . In fact, γ0 →∞ γ? γ0 ≥ 1 if γ1? ∈ N ? ? (4) k (γ0 ) = kmax ∀ γ ≥ 1 if 1 ∈ / N. 0

1/γ ? −b1/γ ? c

"

γ?

! 1 ? There are disjoint intervals I1 , I2 , . . . , Ikmax with Ik = 1 , such that at most 1 − k ? ? γ γ 1 k users can concurrently transmit successfully when γ0 ∈ Ik . Consequently, 1 is the −k+1 γ? minimum γ0 needed for k users to be able to simultaneously communicate with the BS. 1 , −k+1

Remarks: Only one user can successfully transmit in a time-slot when γ0 = γ ? . However, ? no more than kmax users can simultaneously transmit in a time-slot even when γ0 = ∞ (infinite

power). Intuitively, the overall throughput increases with k ? (γ0 ), which in turn increases with γ0 . This suggests a fundamental power versus throughput trade-off for random access with MPR, which is explored in greater detail later in this section.

B. A user-centric approach Let Qti ∈ Z+ denote the number of backlogged packets in the ith user’s queue in the tth timeslot. The system evolution is fully captured by a discrete-time Markov chain whose state in the tth time-slot is the N-tuple Qt , (Qt1 , . . . , QtN ). While the system dynamics are straightforward to describe, analyzing the Markov chain to characterize steady-state behavior is rather cumbersome. We therefore do not pursue this holistic approach further. Instead, we adopt a user-centric perspective, where we focus on the queue of a typical user, assuming that all other users are in steady-state. By symmetry, all users have the same steady-state backlog probability distribution {πn , n ∈ Z+ }, where πn , lim P(Qti = n), provided the limit exists. The queue of each user can t→∞

be modeled as a discrete-time birth-death Markov chain [13]. An “upward” transition or birth occurs when a packet arrives to the queue and either an attempted transmission is unsuccessful or no transmission is attempted. A “downward” transition or death occurs when no packet arrives to the queue and a transmission attempt is successful. Let α and β respectively denote the birth and death probability for a typical user’s Markov chain. Then, α = p(1 − ∆s),

β = (1 − p)∆s,

(5)

where s is the probability of successful transmission for a typical user in steady-state. The user index i has been suppressed due to symmetry considerations. Further, the dependence of α, β, and s on system parameters N, p, ∆, and γ0 has been suppressed for brevity. We will make the dependence explicit whenever deemed necessary. For fixed γ0 , a transmission by a user is successful if at most k ? (γ0 ) − 1 other users transmit simultaneously. The (random) number of users who interfere with a transmission follow a binomial distribution with parameters N − 1 and η, where η is the effective probability of transmission by a typical user in steady-state. Note that η ≤ ∆, because a user with an empty queue cannot transmit even if she wants to. A user can transmit in (the second sub-slot of) a time-slot if she either has a non-empty queue (w.p. 1 − π0 ), or she has an empty queue but a

packet arrives in the first sub-slot of the time-slot (w.p. π0 p). Consequently, η = ∆ (1 − (1 − p)π0 ) .

(6)

The probability of successful transmission is then given by s=

k ? (γ0 )−1

X j=0

N −1 j η (1 − η)N −1−j . j

(7)

The steady-state distribution of each user’s Markov chain is determined by β π1 , π0 = α

β πn = πn−1 + πn+1 ∀ n ∈ N. α+β X πn = 1 to obtain The above equations are solved subject to the constraint α α+β

(8)

n∈Z+

n α α πn = 1 − , β β

n ∈ Z+ .

(9)

∆s − p . Now, substituting this expression Substituting for α and β from (5) in (9) gives π0 = (1 − p)∆s p for π0 in (6) yields η = . Finally, combining with (7) we have ψN (η; γ0 ) = p, where s ψN (η; γ0) ,

k ? (γ0 )−1

X j=0

N − 1 j+1 η (1 − η)N −1−j . j

(10)

Thus, for fixed γ0 and p, η must necessarily satisfy ψN (η; γ0) = p in steady-state. Moreover, ∆ must be chosen to ensure that a steady-state exists (equivalently, the system is stable). Remarks: For a steady-state to exist, ψN (η; γ0 ) = p must have at least one solution. It is not clear a priori if multiple solutions exists, and if they correspond to different steady-states. We now study sufficient conditions on ∆ for the existence of a unique steady-state. C. A sufficient stability condition Since ψN (0; γ0) = ψN (1; γ0) = 0, Rolle’s theorem implies that ψN (η; γ0) has at least one local maximum on the interval (0, 1). The following lemma establishes the uniqueness of this extreme point.

? Lemma 1: For fixed N and γ0 , the function ψN (η; γ0 ) has a unique maximizer ηN (γ0 ) on the ? 1 k (γ0 ) 1 ? ? interval (0, 1). Further, ηN (γ0 ) ∈ and ψN (ηN (γ0 ); γ0 ) = O . , N N N

Proof: See Section VI-A. Lemma 1 implies that ψN (η; γ0) = p has two distinct solutions (0 < η1 < η2 < 1) for any ? p < p?N (γ0 ) , ψN (ηN (γ0 ); γ0 ) and no solution for p > p?N (γ0 ). For the latter case, no steady-

state exists for the system, regardless of the choice of ∆. In other words, for given N and γ0 , p?N (γ0) is the maximum permissible arrival rate (per user) for which the system is stable. Lemma 1 tells us that this quantity decreases inversely with the number of users in the system, irrespective of the amount of power the users are allowed to transmit. For a user’s queue to be stable, her Markov chain should have a well-defined steady-state distribution. This happens p only when α < β, or equivalently p < ∆s from (5). Since η = in steady-state, a necessary s condition for stability is η < ∆. The next result says that choosing ∆ to ensure ψN (∆; γ0 ) > p is sufficient to ensure stability of the system. Theorem 1: For fixed γ0 and any ∆ such that ψN (∆; γ0 ) > p, lim sup E[Qti ] < ∞ ∀ i, t→∞

provided p < p?N (γ0 ). Proof: See Section VI-B. Remarks: Lemma 1 implies that ψN (x; γ0 ) > p for x ∈ (η1 , η2 ) and ψN (x; γ0 ) ≤ p else. It therefore follows from Theorem 1 that choosing ∆ ∈ (η1 , η2 ) is sufficient to ensure stability of the system. Further, for this choice of ∆, the system has a unique steady-state, and η1 is the effective transmission probability of a typical user in this unique steady-state.

D. Asymptotic regimes Consider the quantity TN? (γ0 ) , Np?N (γ0 ), which is the maximum cumulative load for a which a symmetric N-user system is stable. From the definition of ψN (η; γ0) and p?N (γ0 ) TN? (γ0 )

=

k ? (γ0 )−1

X j=0

N −1 ? ? (ηN (γ0 ))j+1(1 − ηN (γ0 ))N −1−j . j

(11)

1 k ? (γ0 ) , for , Now consider the infinite-user regime, namely N → ∞. Since ∈ N N ? every γ0 , ηN (γ0 ) = µN (γ0 )/N for some µN (γ0) ∈ [1, k ? (γ0 )]. Denote µ∞ (γ0 ) , lim µN (γ0 ). ? ηN (γ0 )

N →∞

?

Substituting in (11), taking the limit N → ∞, and using finiteness of k (γ0 ) k ? (γ0 )−1 ? T∞ (γ0 )

, lim

N →∞

TN? (γ0 )

=

X

e−µ∞ (γ0 )

j=0

µ∞ (γ0 )j+1 . j!

(12)

1 ? For γ0 ∈ I1 , µN (γ0 ) = 1 ∀ N, implying µ∞ (γ0 ) = 1 and T∞ (γ0 ) = . For any γ0 ∈ / I1 , µ∞ (γ0 ) e is computed numerically. ? Next, consider the infinite power regime, namely γ0 → ∞. From (4), k ? (γ0 ) = kmax for all γ0

big enough. It follows that µN (γ0 ) , µN (∞) (independent of γ0 ) for all γ0 big enough. Thus, the maximum permissible cumulative load in the infinite power regime is TN? (∞)

, lim

γ0 →∞

TN? (γ0 )

=

d1/γ ? e−1

X j=0

N −1 (µN (∞))j+1 (1 − µN (∞))N −1−j . j

(13)

Finally, the maximum permissible cumulative load in the infinite power and user regime is d1/γ ? e−1 ? T ? , lim TN? (∞) , lim T∞ (γ0 ) = N →∞

γ0 →∞

X j=0

?

e−µ

(µ? )j+1 , j!

(14)

where µ? = lim µN (∞) = lim µ∞ (γ0 ). We conclude this section with an illustrative numerical N →∞

γ0 →∞

example. Numerical Example 1: Let γ ? = 0.1, implying lim k ? (γ0 ) = 10. The interval Ik is of the γ0 →∞ 1 1 for k = 1, 2, . . . , 10. Fig. 1 depicts ψN (η; γ0) as a function of η , form Ik = 11 − k 10 − k for different γ0 for N = 20. The innermost curve corresponds to γ0 ∈ I1 and the outermost curve corresponds to γ0 ∈ I10 . Fig. 2 shows µN (γ0 ) as a function of γ0 for four different values of N. Each curve is a step function with k th jump at the boundary between the intervals Ik and Ik+1 . Note that for fixed γ0 , µN (γ0 ) is a decreasing sequence. Fig. 3 depicts the maximum permissible cumulative load TN (γ0 ) as a function of γ0 for four different values of N. Once again, each curve is a step function with jumps at interval boundaries. Further, TN (γ0 ) is a decreasing sequence for fixed γ0 . This plot is an instance of the fundamental throughput versus

power curve for random access with MPR under static channel conditions. Note that the trade-off curve is concave, as intuition would suggest. For this example, the minimum γ0 needed to sustain a cumulative load of 1 is 0.1228 ∈ I2 . In contrast, pure TDMA (with centralized scheduling) can support a cumulative load of 1 with γ0 = γ ? = 0.1. Thus, random access requires ∼ 22.8% more power and a multi-packet reception capability of 2 to match the performance of TDMA in static channel conditions. In other words, decentralized control with contentions for channel access comes at a cost. However, a cumulative load of much more than 1 can be sustained by random access with MPR by suitable choice of γ0 . In particular, for this example we see that T ? ≈ 5.97, whereas T ? = 1 for TDMA. III. FADING C HANNEL A. System model and sufficient stability conditions We now examine the case were the channel of each user undergoes fading according to an independent and identically distributed (i.i.d.) stochastic process. The fading realizations for different users are assumed to be independent of each other. We focus on a specific fading process, namely Rayleigh fading, which is widely used in the literature to model wireless communication channels for mobile users [14]. It is known that the (random) received power under Rayleigh fading follows an exponential distribution. We assume perfect slow power control [11], i.e., the average received power for each user at the BS is the same for every transmission (say P ). Thus, under i.i.d. Rayleigh fading the received power for the ith user in the tth time-slot (if she transmits) can be expressed as Pit = P Xit, where Xit is an exponentially distributed random variable with unit mean. As noted earlier, Xit and Xiτ are independent if t 6= τ , and Xit and Xjτ are independent ∀ t, τ if i 6= j. The received SINR for the ith user in the tth time-slot (if she transmits) is given by γit = X

j∈S t \i

P Xit P Xjt + σ 2

= X

Xit Xjt + γ0−1

.

(15)

j∈S t \i

As before, the ith user’s transmission is successful if γit ≥ γ ? . However, unlike the static channel case, the probability of successful transmission for the ith user depends not only on the size of

the active set S t , but also on the instantaneous fading realizations for the members of the set. Thus, the probability of successful transmission for the ith user conditioned on S t is t X 1 X si (S t ) = P Xjt − ?i ≤ − . γ γ0 t

(16)

j∈S \i

By symmetry, the above probability depends on S t only through its size and is the same for all

users, i.e., si (S t ) ≡ s(|S t |) ∀ i. Lemma 2: The probability of successful transmission for a typical user under i.i.d. Rayleigh |S t |−1 1 ? fading conditioned on the active set S t is s(|S t |) = e−γ /γ0 ? 1+γ Proof: See Section VI-C. Once again, let η denote the effective probability of transmission by a typical user in steadystate. The random variable |S t |−1, which is the number of users that interfere with a transmission, is binomially distributed with parameters N − 1 and η. Therefore, by averaging with respect to the binomial distribution the probability of successful transmission is computed as s=

N −1 X j=0

N −1 γ?η N −1 j N −1−j −γ ? /γ0 η (1 − η) s(j) = 1 − e . j 1 + γ?

(17)

Similar to the static channel case, it can be shown that the effective probability of transmission for each user in steady-state must satisfy ηs = p, or equivalently ξN (η; γ0 ) = p where N −1 γ?η ? e−γ /γ0 . ξN (η; γ0) = η 1 − ? 1+γ

(18)

γ? , implying ξN (η; γ0) = a(γ0 )η(1−bη)N −1 . 1 + γ? 1 ? . Interestingly, Simple calculus shows that ξN (η; γ0 ) achieves a unique maximum at ηN = Nb ? ηN is independent of γ0 , unlike the static channel scenario. The corresponding maximum value N −1 a(γ0 ) 1 ? ? is given by pN (γ0 ) , ξN (ηN ; γ0) = 1− , which is the maximum permissible Nb N arrival rate (per user) for which an N-user system is stable. For any stable p, ξN (η; γ0) = p has For brevity we denote a(γ0 ) = e−γ

? /γ 0

and b =

at most two solutions. In particular, if p < ξN (1; γ0 ) then ξN (η; γ0) = p has a unique solution η1 . If p > ξN (1; γ0 ), then ξN (η; γ0 ) = p has two distinct solution η1 < η2 . Along the lines of our

discussion following Theorem 1, it follows that stability can be ensured by choosing ∆ ∈ (η1 , 1] when p < ξN (1; γ0) and by choosing ∆ ∈ (η1 , η2 ) when p > ξN (1; γ0 ). Theorem 2: For fixed N, γ0 and any ∆ such that ξN (∆; γ0 ) > p, lim sup E[Qti ] < ∞ ∀ i, t→∞

provided p < p?N (γ0 ). Proof: The proof is based on a Lyapunov technique and is similar to the proof for Theorem 1. We omit the details.

B. Asymptotic regimes Analogous to the static channel case, consider TN? (γ0 ) = Np?N (γ0 ), the maximum permissible cumulative load for which the system is stable. In the infinite-user regime, ? T∞ (γ0 )

, lim

N →∞

TN? (γ0 )

−(1+γ ? /γ0 )

=e

1 1+ ? γ

(19)

In the infinite power regime, TN? (∞)

, lim

γ0 →∞

TN? (γ0 )

=

1 1− N

N −1 1 1+ ? . γ

(20)

Finally, in the infinite-user and power regime,

1 1+ ? . (21) T , lim , lim γ0 →∞ N →∞ γ 1 1 ? ? Remarks: Note that for the fading scenario T∞ (γ ) = 2 1 + ? , whereas for the static e γ 1 1 ? ? ? ≈ 0.582, the maximum permissible cumulative channel scenario T∞ (γ ) = . For γ < e e−1 load in the infinite-user regime under i.i.d. Rayleigh fading is greater than that under static ?

TN? (∞)

? T∞ (γ0 )

1 = e

channel conditions when γ0 = γ ? . In fact, this assertion is true for any finite value of N. In other words, there is a “statistical gain” in throughput under fading conditions when γ ? is small enough. This can be explained as follows: Setting γ0 = γ ? dictates that under static channel conditions at most one user can successfully transmit in a time-slot. However, under fading conditions, more than one user can potentially succeed in a time-slot, depending upon the fading realizations. The likelihood of such an event however decreases as γ ? increases.

To further illustrate this point, consider N = 2 with γ0 = γ ? and fading realizations X1 , X2 in a particular time-slot. Suppose that both users transmit in this time-slot. The received SINRs X2 X1 . It is easily verified that for the two-users are given by γ1 = −1 and γ2 = X2 + γ 0 X1 + γ0−1 packets of both users are successfully received at the BS if X1 , X2 satisfy the linear constraints X2 ≤ (X1 − 1)/γ ? and X2 ≥ γ ? X1 + 1. The constraints define a conical region on the (X1 , X2 ) 2 1 − γ? plane (Fig. 7). It can be shown that the constraints are satisfied w.p. e− 1−γ? , which ? 1+γ ? is a monotonically decreasing function of γ . Remarks: Strictly non-zero throughput is achieved in the fading scenario even when γ0 < γ ? , contrary to the static channel scenario where necessarily γ0 ≥ γ ? for non-zero throughput. This is attributed to the fact that the power attenuation for a user can sometimes be greater than 1 due to fading, and hence her transmission can be successful even if her received SNR is lower than γ ? on an average. As expected, TN (γ0 ) is a decreasing function of γ0 , with lim TN (γ0 ) = 0 ∀ N. γ0 →0

We conclude this section with an illustrative numerical example. Numerical Example 2: Let the target SINR γ ? = 0.1. Fig. 4 shows ξN (η; γ0 ) as a function of η for different γ0 , for fixed N = 20. Fig. 5 depicts TN? (γ0 ) for different N as a function of γ0 . As remarked, TN? (γ0 ) > 0 even for γ0 < γ ? . A visual inspection suggests that TN? (γ0 ) is convex for γ0 < γ ? and concave for γ0 > γ ? , which is consistent with intuition. Fig. 6 depicts TN? (γ0 ) as a function of γ0 for N = 20 for the static channel case and fading case. Since γ ? < 0.582, the maximum permissible load is greater in the fading case for γ0 = γ ? , as discussed previously. However, for big enough γ0 the maximum permissible load for the static channel case is greater. In the infinite user and power regime, T ? ≈ 5.97 for the static channel case and T ? ≈ 4.05 for the fading case. Interestingly, T ? = 1 for pure TDMA under i.i.d. Rayleigh fading, which is the same as the T ? under static channel conditions with γ0 = γ ? . Thus, in terms of throughput, TDMA in fading conditions behaves like random access system without MPR (pure slotted Aloha) in static conditions.

IV. D ELAY P ERFORMANCE In this section we briefly examine the average delay performance of random access with MPR. Recall that the evolution of each user’s queue is described by a birth-death Markov chain with birth probability α, death probability β, and steady-state distribution {π(n), n ∈ Z} given by (9). The average backlog in a typical user’s queue in steady-state is computed as Qave =

∞ X

nπn =

n=0

α/β . 1 − α/β

(22)

Substituting for α and β from (5) and using Little’s law [15], the average delay per packet is given by D ave =

Qave η − ∆p = , p p(∆ − η)

(23)

Note that (23) applies to both static and fading scenarios. For the former case, η is the smallest solution to ψN (η; γ0) = p, assuming ∆ is chosen such that ψN (∆; γ0 ) > p. For the latter case, η is the smallest solution to ξN (η; γ0) = p, assuming ∆ is chosen such that ξN (∆; γ0 ) > p. We conclude this section with two numerical examples. Numerical Example 3: Fix N = 20 and γ ? = 0.1. Fig. 8 depicts D ave for the static channel case as a function of ∆ for different γ0 . The k th curve in the upper-half of the plot corresponds to γ0 ∈ Ik . For each curve, p = 0.9p?N (γ0 ). For fixed relative load (as a fraction of the maximum), the lowest achievable average delay decreases as γ0 increases. The lower-half of the plot shows D ave versus ∆ for fixed p for three different choices of k ? (γ0 ). There is a visible saturation effect as γ0 increases, implying a convex power versus delay trade-off curve. Numerical Example 4: Fix N = 20 and γ ? = 0.1. Fig. 9 depicts D ave for the i.i.d. Rayleigh fading case as a function of ∆ for different γ0 . The upper-half of the figure shows D ave versus ∆ for fixed relative load p = 0.9p?N (γ0 ). Each curve corresponds to a different choice of γ0 . The lower-half of the plot shows D ave versus ∆ for fixed p. The saturation effect is evident as γ0 increases. A 10dB increase in γ0 from γ ? to 10γ ? produces a massive improvement in average delay performance, while a 10dB increase from 10γ ? to 100γ ? has minimal impact.

Remarks: While the focus of this paper was on the case of symmetric users with Bernoulli packet arrivals, the proposed user-centric approach is in principle applicable to more general scenarios. For instance, packet arrival and channel processes modulated by finite-state Markov chains (to model bursty arrivals and channels with memory) are easily incorporated into this framework. Matrix geometric techniques can be used to solve for the steady-state, however, it may not always be possible to obtain “neat” closed-form solutions. The case of asymmetric users can also be treated within this framework. In particular, if there are K > 1 classes of users (with pk as a the arrival rate for class k users), the steady-state is identified by the Ktuple η = (η1 , . . . , ηK ), where ηk is the effective transmission probability of a class k user in steady-state. The probabilities η1 , . . . , ηK are solutions to a set of K non-linear equations. The case K = 1 was treated in this paper. V. C ONCLUSIONS This paper examined queuing aspects of a finite-user slotted Aloha type random access protocol with multi-packet reception capability at the base station. Queuing theoretic analysis of random access protocols is complicated due to complex interactions between queues of different users. A “user-centric” approach is proposed in this paper, which makes steady-state analysis more tractable. The approach focuses on the queue of a typical user in the system, assuming that this user observes all other users in steady-state. While the method is applicable quite generally, this paper treated the special case of identical users with Bernoulli arrivals in order to obtain analytical expressions for quantities of interest like throughput and average delay. Both static wireless channels (for fixed users) and Rayleigh faded channels (for mobile users) were considered. Sufficient conditions for system stability were also derived in the paper. Obtaining conditions which are both necessary and sufficient for stability of a random access protocol with MPR is still an unsolved problem. R EFERENCES [1] D.P. Bertsekas and R.G. Gallagher, Data Networks, Prentice Hall, 1991.

[2] S. Ghez, S. Verd´u, and S.C. Schwartz, “Stability properties of slotted-Aloha with multi-packet reception capability”, IEEE Transactions on Automatic Control, vol. 33, no. 7, pp. 640-649, Jul. 1988. [3] G. del Angel and T.L. Fine, “Information capacity and power control for slotted-Aloha random-access systems”, IEEE Transactions on Information Theory, vol. 51, no. 12, pp. 4074-4090, Dec. 2005. [4] V. Naware, G. Mergen, and L. Tong, “Stability and delay of finite-user slotted-Aloha with multi-packet reception”, IEEE Transactions on Information Theory, vol. 51, no. 7, pp. 2636-2656, Jul. 2005. [5] J. Sant and V. Sharma, “Performance analysis of a slotted-Aloha protocol on a capture channel with fading”, Queueing Systems, vol. 34, pp. 1-35, 2000. [6] S. Ghez, S. Verd´u, and S.C. Schwartz, “Optimal decentralized control in the random access multi-packet channel”, IEEE Transactions on Automatic Control, vol. 34, no. 11, pp. 1153-1163, Nov. 1989. [7] B. Tsbyakov, “Packet multiple access for channel with binary feedback, capture, and multiple reception”, IEEE Transactions on Information Theory, vol. 50, no. 6, pp. 1073-1085, Jun. 2004. [8] G. del Angel and T.L. Fine, “Optimal power and retransmission control policies for random access systems”, IEEE/ACM Transactions on Networking, vol. 12, no. 6, pp. 1156-1166, Dec. 2004. [9] C. van der Plas and J.M.G. Linnartz, “Stability of mobile slotted ALOHA network with Rayleigh fading, shadowing, and near-far effect”, IEEE Transactions on Vehicular Technology, vol. 39, no. 4, pp. 359-366, Nov. 1990. [10] Y. Yu, X. Cai, and G.B. Giannakis, “On the instability of slotted-Aloha with capture”, Proceedings of IEEE WCNC, vol. 2, pp. 728-732, Atlanta, GA, 2004. [11] M. Peh, S. Hanly, and P. Whiting, “Random-access over fading channels”, Proceedings of IEEE Globecom, vol. 2, pp. 888-892, San Francisco, CA, 2003. [12] A. Dua, “Power controlled random access”, Proceedings of IEEE ICC, vol. 6, pp. 3514-3518, Paris, France, Apr. 2004. [13] S. Karlin and H.M. Taylor, A First Course in Stochastic Processes, Academic Press, 1975. [14] D. Tse and P. Viswanath, Fundamentals of Wireless Communications, Cambridge University Press, May 2005. [15] J. Walrand, An Introduction to Queueing Networks, Prentice Hall, 1988. [16] R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press, Aug. 1995. [17] P.R. Kumar and S.P. Meyn, “Stability of queuing networks and scheduling policies”, IEEE Trans. on Automatic Control, vol. 40, no. 2, pp. 251-260, Feb. 1995.

VI. A PPENDIX A. Proof of Lemma 1 Proof: The proof is based on inductive arguments. Denoting k ? (γ0 )−1

η)

N −1−j

, it follows from definition ψN (η; γ0) =

X j=0

j ωN (η)

N − 1 j+1 η (1 − = j

j ωN (η).

0 Base case: Suppose γ0 ∈ I1 , so that k ? (γ0 ) = 1. In this case, ψN (η; γ0 ) = ωN (η) = 1 ? . Further, η(1 − η)N −1 . Simple calculus shows that the unique maximizer is ηN (γ0 ) = N

N −1 N 1 1 1 1 1− . Using the inequality 1 − < [16], ψN (1/N; γ0 ) < ψN (1/N; γ0) = N N N e 1 1 , as desired. , implying ψN (1/N; γ0 ) = O (N − 1)e N ? Inductive step: Assume that ψN (η; γ0) has a unique maximizer ηN (γ0 ) ∈ [1/N, k/N] for

γ0 ∈ Ik (i.e., k ? (γ0 ) = k). Now, suppose that the received SNR is γ00 ∈ Ik+1 , i.e., k ? (γ00 ) = k +1. j k Since ωN (·)’s are functions of η alone, ψN (η; γ00 ) = ψN (η; γ0 ) + ωN (η). The first two derivatives j of ωN (η) with respect to η are given by

j + 1 − Nη = η(1 − η) N(N − 1)η 2 − 2(N − 1)(j + 1)η + j(j + 1) j 00 j (ωN ) (η) = ωN (η) . η 2 (1 − η)2 j 0 (ωN ) (η)

j ωN (η)

(24)

j j Since ωN (η) > 0 ∀ η ∈ (0, 1), it follows that ωN (η) is a monotonically increasing function

for η ∈ [0, (j + 1)/N] and a monotonically decreasing function for η ∈ [(j + 1)/N, 1]. This observation immediately implies that ψN (η; γ00 ) is monotonically increasing for η ∈ [0, 1/N] and monotonically decreasing for η ∈ [(k + 1)/N, 1]. Thus, ψN (η; γ00 ) must necessarily achieve its maximum on the interval [1/N, (k + 1)/N]. r j+1 (j + 1)(N − j + 1) Γ j+1 Γ j Further, ωN (η) is concave for η ∈ , Γ = − , + , N N N N N −1 since the second derivative is negative on that interval. Since j < N, we get Γ > 1, implying k that ωN (η) is concave for η ∈ [k/N, (k + 1)/N]. Inductive arguments can be used to show that

ψN (η; γ0) is also concave for η ∈ [k/N, (k + 1)/N] (details omitted due to space constraints). Consequently, their sum ψN (η, γ00 ) is concave for η ∈ [k/N, (k + 1)/N]. From the inductive assumption, ψN (η; γ0 ) has a unique maximizer on [1/N, k/N]. Further, k ωN (η) is monotonically increasing on [1/N, k/N]. Thus, ψN (η; γ00 ), which is the sum of the two,

has at most one maximizer on [1/N, k/N]. The two possible cases are treated separately: 0 k 0 1) ψN (η; γ00 ) has a maximizer on [1/N, k/N]: In this case, ψN (k/N; γ00 ) < 0. Also, (ωN ) ((k+ 0 0 1)/N) = 0 and ψN ((k + 1)/N, γ0) < 0, implying ψN ((k + 1)/N; γ00 ) < 0. The concavity 0 of ψN (η; γ00 ) on [k/N, (k + 1)/N] implies that necessarily ψN (η, γ00 ) < 0 on that interval. ? Thus, there is a unique maximizer ηN (γ00 ) ∈ [1/N, (k + 1)/N].

2) ψN (η; γ00 ) does not have a maximizer on [1/N, k/N]: In this case, ψN (η; γ00 ) is monotoni-

0 0 cally increasing on [1/N, k/N], implying ψN (k/N; γ0 ) > 0. Also, ψN ((k + 1)/N; γ00 ) < 0, ? 0 as in case 1. Thus, ∃ ηN (γ00 ) ∈ [k/N, (k + 1)/N] such that ψN (η ? (γ00 ); γ00 ) = 0. The ? concavity of ψN (η; γ00 ) on the interval [k/N, (k + 1)/N] ensures uniqueness of ηN (γ00 ). ? Thus, there is a unique maximizer ηN (γ00 ) ∈ [1/N, (k + 1)/N].

In summary, subject to the inductive assumption that ψN (η; γ0) has a unique maximizer on [1/N, k/N] when γ0 ∈ Ik , ψN (η; γ0) has a unique maximizer on [1/N, (k + 1)/N] when γ0 ∈ Ik+1 . The desired result follows from the principle of mathematical induction. j Recall that ωN (η) achieves its maximum at η = (j + 1)/N. The maximum value is given by

j+1 N −1−j j+1 N −1 j+1 j+1 = 1− . j N N N Nk x N < < e−x [16] and 1 − Using the inequalities k! N k j ωN

j ωN

j+1 N

1 < N

N −1 N −j−1 {z | O(1)

j ωN (η)

j+1

N e−(j+1) (j + 1)j+1 , · N {z − 1} j! | {z } | } ·

O(1)

(25)

(26)

independent of N

k ? (γ0 )−1 X 1 j ? ωN (η), it follows . Since k (γ0 ) is bounded and ψN (η; γ0 ) = N j=0

=O 1 , as desired. ψN (η; γ0) = O N implying

B. Proof of Theorem 1 Proof: Consider the time and state dependent quadratic Lyapunov function V t (Qt ) , hQt , Qt i, where hx, yi denotes the inner-product between vectors x and y. Define the set C t , {k : Qtk > 0}. If k ∈ C t , i.e., Qtk > 0, the evolution of the queue of the k th user is described by

Qt+1 = k

Qt + 1 ; w.p. p(1 − ∆st ) k

Qtk

; w.p. (1 − p)(1 − ∆st ) + p∆st

Qtk − 1 ; w.p. (1 − p)∆st ,

(27)

where st is the probability of successful transmission for a queue transmitting in the tth time-slot. If k ∈ / C t , i.e., Qtk = 0, the evolution of the queue of the k th user is described by Qt+1 k

Qt + 1 ; w.p. p(1 − ∆st ) k = Qt ; w.p. 1 − p + p∆st . k

(28)

2 t 2 t t t t t It follows that E (Qt+1 k ) − (Qk ) |Qk ≤ 2(p − ∆s )Qk + B , with equality when k ∈ C , where B t = (1 − p)∆st + p(1 − ∆st ). Now, the k th user transmits a packet w.p. ∆ if k ∈ C t , and

w.p. p∆ if k ∈ / C t . The “worst-case” arises when all users have a packet to transmit, and each of them attempts a transmission w.p. ∆. The probability of successful transmission is therefore lower-bounded by t

s ≥

k ? (γ0 )−1

X j=0

N −1 ∆j (1 − ∆)N −1−j , smin . j

(29)

The expected conditional one-step drift of V can now be upper-bounded as ψN (∆)

z }| { E V t+1 (Qt+1 ) − V t (Qt )|Qt ≤ −2(∆smin −p)hQt , 1i + NB, | {z }

(30)

where 1 is an N-length vector with all unit entries. Since p, ∆, st ∈ [0, 1], B t < C ∀ t for some p constant C > 0. Noting that hQt , 1i ≥ V t (Qt ), for any > 0 p E V t+1 (Qt+1 ) − V t (Qt )|Qt ≤ −2 V t (Qt ) + NC,

(31)

The desired result follows from Foster’s stability criteria for Markov chains [17].

C. Proof of Lemma 2 Proof: Let X be a random variable with probability density function (p.d.f.) fX (x). The Z ∞ Laplace transform associated with the density of X is given by LX (y) = e−yx fX (x)dx. −∞

The following facts will come in handy: • •

1 . 1+y If X1 and X2 are two independent random variables with well defined p.d.f.s associated

If X is exponentially distributed with mean 1, then LX (y) =

with Laplace transforms LX1 (y) and LX2 (y) respectively, the Laplace transform associated

with the p.d.f. of the random variable X = X1 + X2 is LX (y) = LX1 (y)LX2 (y). If X is exponentially distributed with mean 1, the Laplace transform associated with the 1 . p.d.f. of X 0 = −aX for fixed a > 0 is given by LX 0 (y) = 1 − ay X Xt Denoting Z = Xjt − ?i for convenience, and using the above facts it follows that LZ (y) = γ j∈S t \i |S t |−1 γ? 1 . Using a partial fraction expansion technique and applying the inverse 1+y γ? − y |S t |−1 1 ? ? eγ z for z ≤ 0. Consequently, Laplace transform, fZ (z) = γ ? 1+γ •

Z −1 |S t |−1 γ0 1 1 −γ ? /γ0 s(|S |) = P Z ≤ − fZ (z)dz = = e . γ0 1 + γ? −∞ t

0.35

0.3

k*(γ0) = { 1,2,...,10 }

ψN(η; γ0)

0.25

0.2

0.15

0.1

0.05

0

Fig. 1.

0

0.1

0.2

0.3

0.4

0.5

η

0.6

0.7

0.8

0.9

1

ψN (η; γ0 ) as a function of η for different γ0 with fixed N = 20 and γ ? = 0.1.

(32)

11

10

9

8

µN(γ0)

7

6

5

4

3

N=10 N=20 N=40 N=80

2

1

Fig. 2.

0

0.2

0.4

0.6

0.8

γ0

1

1.2

1.4

1.6

µN (γ0 ) as a function of γ0 for four different values of N with fixed γ ? = 0.1.

11 10 9 8

T*N(γ0)

7 6 5 4 3

N=10 N=20 N=40 N=80

2 1 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

γ0

Fig. 3.

TN? (γ0 ) (static channel) as a function of γ0 for four different values of N with fixed γ ? = 0.1.

0.25

γ0=γ*/2 γ =γ* 0 γ0=2γ* γ =10γ* 0 γ0=100γ* γ0=1000γ*

0.2

ξN(η; γ0)

0.15

0.1

0.05

0

Fig. 4.

0

0.1

0.2

0.3

0.4

0.5

η

0.6

0.7

0.8

0.9

1

ξN (η; γ0 ) as a function of η for different values of γ0 with fixed N = 20 and γ ? = 0.1.

5

4.5

4

3.5

T*N(γ0)

3

2.5

2

1.5

N=5 N=10 N=20 N=40 N=80

1

γ*

0.5

0

−1

10

0

10

1

10

2

10

γ0 (log scale)

Fig. 5.

TN? (γ0 ) (Rayleigh fading) as a function of γ0 for five different values of N with fixed γ ? = 0.1.

7

Static channel Rayleigh fading 6

5

T*N(γ0)

4

3

2

1

0 −2 10

−1

0

10

1

10

10

γ0 (log scale)

Fig. 6. A comparison of TN? (γ0 ) for the static channel case and Rayleigh fading case as a function of γ0 with fixed N = 20 and γ ? = 0.1.

X2

1 1 − γ?

1

1

1 1 − γ?

X1

Fig. 7. A two-user example to illustrate that more than one user can simultaneously succeed in the fading case even when γ0 = γ ? . The shaded conical region corresponds to the fading realizations for which both users can transmit simultaneously and succeed.

60 50

k*(γ ) = {1,2,...,10} 0

Dave

40 30 20 10 0 0.05

0.1

0.15

0.2

0.25

∆

0.3

0.35

0.4

0.45

0.5

60

k*(γ )=3 0 k*(γ )=4 0 k*(γ )=10

50

Dave

40

0

30 20 10 0

0

0.1

0.2

0.3

∆

0.4

0.5

0.6

0.7

Fig. 8. Dave as a function of ∆ for the static channel case for different values of γ0 ; The upper-half corresponds to a fixed relative load (90% of maximum) while the lower-half corresponds to a fixed load. .

40

γ0=γ* γ0=2γ* γ0=10γ* γ0=100γ*

Dave

30

20

10

0

0.4

0.5

0.6

∆

0.7

0.8

0.9

1

40

γ =γ* 0 γ0=2γ* γ0=10γ* γ =100γ*

Dave

30

0

20

10

0

0

0.1

0.2

0.3

0.4

0.5

∆

0.6

0.7

0.8

0.9

1

Fig. 9. Dave as a function of ∆ for the Rayleigh fading case for different values of γ0 ; The upper-half corresponds to a fixed relative load (90% of maximum) while the lower-half corresponds to a fixed load. .

Aditya Dua Department of Electrical Engineering Stanford University 350 Serra Mall, Stanford CA 94305 Email: [email protected]

Abstract The queuing performance of a finite-user slotted-Aloha type random access protocol with multipacket reception capability at the base station is considered. The system evolution under this protocol can be described by a finite dimensional Markov chain, which is not amenable to analysis due to complex interactions between user queues. A “user-centric” analysis approach is proposed, which focuses on the evolution of the one-dimensional Markov chain associated with the queue of a typical user in the system, assuming that all other users are in “steady-state”. While the approach is quite general, this paper addresses the special case of identical users with Bernoulli arrivals in order to obtain insightful analytical expressions for various quantities of interest (throughput, average delay etc.). Both static and Rayleigh faded wireless channels are studied, under suitable assumptions on the underlying power control mechanisms. Sufficient stability conditions as well as asymptotic performance measures in the infinite power and infinite user regime are derived for both cases. The fundamental throughput versus power trade-off inherent in the protocol is established. Index Terms Random access, power control, slotted-Aloha, multi-packet reception, wireless networks.

I. I NTRODUCTION Random access protocols offer a simple, decentralized way of regulating access to a shared wireless channel, for instance on the uplink of a cellular communication system. The performance of random access is controlled by varying two sets of parameters — the power with which each user transmits and the probability with which each user attempts to access the shared channel. Choosing these control parameters to enhance system performance is an important and interesting problem, both from an engineering and theoretical perspective. Classical analysis of randomized multiple access schemes like slotted-Aloha [1] has focused on the the so called collision model, where at most one user can successfully communicate with a centralized base station (BS) or access point (AP) in a time-slot. Under this model, if multiple users attempt transmissions in a time-slot, then all of them are unsuccessful. Another extensively studied paradigm is the capture model, wherein at most one user can successfully communicate with the BS even in the presence of interfering transmissions, provided the user’s signal is sufficiently strong vis-`a-vis the interfering signals. However, in code division multiple access (CDMA) systems, where the BS can decode multiple simultaneous interfering transmissions (say, using a Rake receiver), it is possible for more than one user to concurrently communicate with the BS in a time-slot. Random access with multi-packet reception (MPR) has received considerable attention in the literature in recent times. A generalized MPR model was first introduced by Ghez et. al. in [2]. They modeled the number of successful transmissions in a time-slot as a random variable which is a function of the number of attempted transmissions only, and provided stability conditions for the case of indistinguishable nodes. del Angel e. al. [3] considered slotted-Aloha from an information theoretic perspective under the assumptions of infinite users and no fading. They studied dynamic control of transmission probability and randomized power control to improve the throughput and spectral efficiency of the system. Naware et. al. examined the stability and delay properties of finite-user slotted-Aloha with MPR capability in [4]. They extensively characterized the performance of a two-user system (N = 2), and also provided sufficient stability conditions for the N > 2 case for fixed transmission probabilities. Sant et. al. [5] provided sufficient stability

conditions for finite-user slotted-Aloha with capture under Markov modulated wireless channel fading. In other related work, Ghez et. al. [6], Tsbyakov [7], and del Angel et. al. [8] studied dynamic control policies for stabilizing slotted-Aloha with MPR and optimizing its performance under a variety of modeling assumptions. Queuing performance of slotted-Aloha with capture over wireless channels exhibiting Rayleigh fading and shadowing was considered by van der Plas et. al. in [9]. Their focus was on the assessment of receiver capture probabilities under a general wireless propagation model. Yu et. al. studied the stability properties of slotted-Aloha with capture under Rayleigh fading in the infinite-user regime in [10]. Peh et. al. [11] proposed a multi-bit feedback algorithm for optimizing the performance of slotted-Aloha in a Rayleigh faded environment. Dua [12] studied random access over Rayleigh faded channels with power and transmission probability control in a convex optimization framework. In this paper, we focus on queuing aspects of a slotted-Aloha type random access protocol with MPR capability at the BS, under both static and fading channel conditions. We study a multiple capture model where a user’s transmission is successful if her received signal to interference plus noise ratio (SINR) is above a certain threshold γ ? , independent of everything else. We examine two scenarios — (a) a static environment where the wireless channel from each user to the BS is time-invariant, and (b) a mobile environment where the wireless channel from each user to the BS exhibits Rayleigh fading. The evolution of a system with N users can be described by an N-dimensional Markov chain (under Markovian arrival and channel processes), which is extremely hard to analyze for steadystate behavior owing to the complex interactions between queues of different users. To simplify the problem, we abandon the holistic approach for a user-centric approach, wherein we analyze a one-dimensional Markov chain which captures the evolution of the queue of a typical user in isolation, given that this user perceives the rest of the system in steady-state. While this method is applicable in great generality, to obtain insightful analytical formulae we restrict attention to the symmetric case, where packets arrive to queues of different users according to independent Bernoulli processes with the same parameter p. We study the static channel case under the

assumption of perfect power control, i.e., transmissions of all users are received with the same power P at the BS. For the fading case, we assume perfect slow power control [11], i.e., the transmissions of all users are received with the same power P at the BS on an average. The key quantity which captures the coupling between a user’s queue and the rest of the system is η — the effective probability of transmission by a typical user in steady-state, which is less than the actual probability ∆ with which a user with a non-empty queue attempts transmission. The probability of successful transmission by a user and all other quantities of interest (average backlog etc.) are then expressed as a function of η. We provide sufficient conditions (choice of ∆) under which η is unique, i.e., the system converges to a unique steady-state. We examine the maximum throughput the system can sustain as a function of P . We also study the behavior of the system in two asymptotic regimes — infinite user (N → ∞) and infinite power (P → ∞). The user-centric approach also allows us to obtain closed form expressions for average packet delay perceived by a typical user as a function of η. A. Paper outline The system model for random access with MPR is described in Section II, and the static channel case is treated in great detail. A corresponding treatment of random access with MPR over Rayleigh faded wireless channels is provided in Section III. The delay performance for both scenarios is examined in Section IV. The theoretical results developed in each section are illustrated via numerical examples. Finally, concluding remarks are given in Section V. B. Notation Some notations used throughout the paper are enumerated here for convenience. N denotes the set of natural numbers, Z+ denotes the set of non-negative integers, and R denotes the set of real numbers. For any set A, |A| denotes the cardinality of A and A\a denotes the set difference between A and {a}. For any x ∈ R, bxc denotes the greatest integer smaller than x, while dxe denotes the smallest integer greater than x. All vectors are denoted in boldface. The abbreviation w.p. is used to represent “with probability”. Finally, the “big-oh” notation f (N) = O (g(N)) is used to indicate that ∃ c > 0 such that f (N) ≤ cg(N) for large enough N.

II. S TATIC C HANNEL A. System model Consider the uplink of a time-slotted wireless communication system with N users communicating with a centralized base-station (BS) or access-point (AP). A time-slot is divided into two logical sub-slots. In the first sub-slot of every time-slot, a packet arrives to the transmission queue of each user w.p. p, independent of all past and future arrivals. The arrival processes are assumed to be independent across users. In the second sub-slot, every user with a non-empty queue transmits a packet w.p. ∆, independent of the decision of all other users. The power at which a user’s packet is received at the BS is determined by the transmitted power and the power attenuation from the user to the BS. In this section, we examine a “static channel” scenario where the power attenuation from transmitters to the BS is constant. This would be the case for a wireless system with fixed/static users. In particular, if the ith user transmits a packet with power Pi , it is received at the BS with power Pi Gi , for some fixed Gi ∈ (0, 1]. Under this model, the SINR for the ith user in the tth time-slot (if she transmits) is given by γit = X

P i Gi P j Gj + σ 2

,

(1)

j∈S t \i

where σ 2 denotes the ambient noise power at the BS and S t denotes the set of users who transmit in the tth time-slot (active set). The ith user’s transmission in the tth time-slot is successful if the received SINR is above a threshold γ ? , i.e., γit ≥ γ ? . All users who transmit in a time-slot receive instantaneous and error-free one bit feedback (ACK/NAK) regarding the outcome of their transmission at the end of the time-slot by the BS. No feedback is given to inactive users. We assume perfect power control, i.e., packets of all users are received with equal power at the BS. In particular, each user adjusts her transmit power so that Pi Gi = P ∀ i. This helps combat the so called near-far effect, wherein transmissions of “strong” users (near the BS) can potentially overwhelm transmissions of “weak” users (far from the BS). With perfect power control, all users have an equal opportunity to transmit packets successfully (regardless of their

distance from the BS), which is a reasonable requirement if they all observe the same packet arrival rate p. The received SINR for the ith user in the tth time-slot can be rewritten as γit =

(|S t |

1 P , = t 2 − 1)P + σ |S | − 1 + γ0−1

(2)

P is the received signal-to-noise ratio (SNR) for a user if she alone transmits in a σ2 time-slot. Note that it is essential to have γ0 ≥ γ ? , else communication is not feasible even in

where γ0 ,

the absence of interference. The necessary and sufficient condition for the ith user’s transmission in the tth time-slot to be successful is γit ≥ γ ? , which translates to (see also [3]) 1 1 |S | ≤ 1 + ? − , k ? (γ0 ). γ γ0 t

(3)

The condition in (3) is interpreted as follows — given the received SNR γ0 and transmission probability ∆ for all users, all transmissions are successful in a time-slot if k ? (γ0 ) or fewer users transmit, and all transmissions are unsuccessful else. Note that k ? (γ0 ) is a non-decreasing 1 ? ? ? ? function of γ0 . Also, observe that k (γ ) = 1 and kmax , lim k (γ0 ) = . In fact, γ0 →∞ γ? γ0 ≥ 1 if γ1? ∈ N ? ? (4) k (γ0 ) = kmax ∀ γ ≥ 1 if 1 ∈ / N. 0

1/γ ? −b1/γ ? c

"

γ?

! 1 ? There are disjoint intervals I1 , I2 , . . . , Ikmax with Ik = 1 , such that at most 1 − k ? ? γ γ 1 k users can concurrently transmit successfully when γ0 ∈ Ik . Consequently, 1 is the −k+1 γ? minimum γ0 needed for k users to be able to simultaneously communicate with the BS. 1 , −k+1

Remarks: Only one user can successfully transmit in a time-slot when γ0 = γ ? . However, ? no more than kmax users can simultaneously transmit in a time-slot even when γ0 = ∞ (infinite

power). Intuitively, the overall throughput increases with k ? (γ0 ), which in turn increases with γ0 . This suggests a fundamental power versus throughput trade-off for random access with MPR, which is explored in greater detail later in this section.

B. A user-centric approach Let Qti ∈ Z+ denote the number of backlogged packets in the ith user’s queue in the tth timeslot. The system evolution is fully captured by a discrete-time Markov chain whose state in the tth time-slot is the N-tuple Qt , (Qt1 , . . . , QtN ). While the system dynamics are straightforward to describe, analyzing the Markov chain to characterize steady-state behavior is rather cumbersome. We therefore do not pursue this holistic approach further. Instead, we adopt a user-centric perspective, where we focus on the queue of a typical user, assuming that all other users are in steady-state. By symmetry, all users have the same steady-state backlog probability distribution {πn , n ∈ Z+ }, where πn , lim P(Qti = n), provided the limit exists. The queue of each user can t→∞

be modeled as a discrete-time birth-death Markov chain [13]. An “upward” transition or birth occurs when a packet arrives to the queue and either an attempted transmission is unsuccessful or no transmission is attempted. A “downward” transition or death occurs when no packet arrives to the queue and a transmission attempt is successful. Let α and β respectively denote the birth and death probability for a typical user’s Markov chain. Then, α = p(1 − ∆s),

β = (1 − p)∆s,

(5)

where s is the probability of successful transmission for a typical user in steady-state. The user index i has been suppressed due to symmetry considerations. Further, the dependence of α, β, and s on system parameters N, p, ∆, and γ0 has been suppressed for brevity. We will make the dependence explicit whenever deemed necessary. For fixed γ0 , a transmission by a user is successful if at most k ? (γ0 ) − 1 other users transmit simultaneously. The (random) number of users who interfere with a transmission follow a binomial distribution with parameters N − 1 and η, where η is the effective probability of transmission by a typical user in steady-state. Note that η ≤ ∆, because a user with an empty queue cannot transmit even if she wants to. A user can transmit in (the second sub-slot of) a time-slot if she either has a non-empty queue (w.p. 1 − π0 ), or she has an empty queue but a

packet arrives in the first sub-slot of the time-slot (w.p. π0 p). Consequently, η = ∆ (1 − (1 − p)π0 ) .

(6)

The probability of successful transmission is then given by s=

k ? (γ0 )−1

X j=0

N −1 j η (1 − η)N −1−j . j

(7)

The steady-state distribution of each user’s Markov chain is determined by β π1 , π0 = α

β πn = πn−1 + πn+1 ∀ n ∈ N. α+β X πn = 1 to obtain The above equations are solved subject to the constraint α α+β

(8)

n∈Z+

n α α πn = 1 − , β β

n ∈ Z+ .

(9)

∆s − p . Now, substituting this expression Substituting for α and β from (5) in (9) gives π0 = (1 − p)∆s p for π0 in (6) yields η = . Finally, combining with (7) we have ψN (η; γ0 ) = p, where s ψN (η; γ0) ,

k ? (γ0 )−1

X j=0

N − 1 j+1 η (1 − η)N −1−j . j

(10)

Thus, for fixed γ0 and p, η must necessarily satisfy ψN (η; γ0) = p in steady-state. Moreover, ∆ must be chosen to ensure that a steady-state exists (equivalently, the system is stable). Remarks: For a steady-state to exist, ψN (η; γ0 ) = p must have at least one solution. It is not clear a priori if multiple solutions exists, and if they correspond to different steady-states. We now study sufficient conditions on ∆ for the existence of a unique steady-state. C. A sufficient stability condition Since ψN (0; γ0) = ψN (1; γ0) = 0, Rolle’s theorem implies that ψN (η; γ0) has at least one local maximum on the interval (0, 1). The following lemma establishes the uniqueness of this extreme point.

? Lemma 1: For fixed N and γ0 , the function ψN (η; γ0 ) has a unique maximizer ηN (γ0 ) on the ? 1 k (γ0 ) 1 ? ? interval (0, 1). Further, ηN (γ0 ) ∈ and ψN (ηN (γ0 ); γ0 ) = O . , N N N

Proof: See Section VI-A. Lemma 1 implies that ψN (η; γ0) = p has two distinct solutions (0 < η1 < η2 < 1) for any ? p < p?N (γ0 ) , ψN (ηN (γ0 ); γ0 ) and no solution for p > p?N (γ0 ). For the latter case, no steady-

state exists for the system, regardless of the choice of ∆. In other words, for given N and γ0 , p?N (γ0) is the maximum permissible arrival rate (per user) for which the system is stable. Lemma 1 tells us that this quantity decreases inversely with the number of users in the system, irrespective of the amount of power the users are allowed to transmit. For a user’s queue to be stable, her Markov chain should have a well-defined steady-state distribution. This happens p only when α < β, or equivalently p < ∆s from (5). Since η = in steady-state, a necessary s condition for stability is η < ∆. The next result says that choosing ∆ to ensure ψN (∆; γ0 ) > p is sufficient to ensure stability of the system. Theorem 1: For fixed γ0 and any ∆ such that ψN (∆; γ0 ) > p, lim sup E[Qti ] < ∞ ∀ i, t→∞

provided p < p?N (γ0 ). Proof: See Section VI-B. Remarks: Lemma 1 implies that ψN (x; γ0 ) > p for x ∈ (η1 , η2 ) and ψN (x; γ0 ) ≤ p else. It therefore follows from Theorem 1 that choosing ∆ ∈ (η1 , η2 ) is sufficient to ensure stability of the system. Further, for this choice of ∆, the system has a unique steady-state, and η1 is the effective transmission probability of a typical user in this unique steady-state.

D. Asymptotic regimes Consider the quantity TN? (γ0 ) , Np?N (γ0 ), which is the maximum cumulative load for a which a symmetric N-user system is stable. From the definition of ψN (η; γ0) and p?N (γ0 ) TN? (γ0 )

=

k ? (γ0 )−1

X j=0

N −1 ? ? (ηN (γ0 ))j+1(1 − ηN (γ0 ))N −1−j . j

(11)

1 k ? (γ0 ) , for , Now consider the infinite-user regime, namely N → ∞. Since ∈ N N ? every γ0 , ηN (γ0 ) = µN (γ0 )/N for some µN (γ0) ∈ [1, k ? (γ0 )]. Denote µ∞ (γ0 ) , lim µN (γ0 ). ? ηN (γ0 )

N →∞

?

Substituting in (11), taking the limit N → ∞, and using finiteness of k (γ0 ) k ? (γ0 )−1 ? T∞ (γ0 )

, lim

N →∞

TN? (γ0 )

=

X

e−µ∞ (γ0 )

j=0

µ∞ (γ0 )j+1 . j!

(12)

1 ? For γ0 ∈ I1 , µN (γ0 ) = 1 ∀ N, implying µ∞ (γ0 ) = 1 and T∞ (γ0 ) = . For any γ0 ∈ / I1 , µ∞ (γ0 ) e is computed numerically. ? Next, consider the infinite power regime, namely γ0 → ∞. From (4), k ? (γ0 ) = kmax for all γ0

big enough. It follows that µN (γ0 ) , µN (∞) (independent of γ0 ) for all γ0 big enough. Thus, the maximum permissible cumulative load in the infinite power regime is TN? (∞)

, lim

γ0 →∞

TN? (γ0 )

=

d1/γ ? e−1

X j=0

N −1 (µN (∞))j+1 (1 − µN (∞))N −1−j . j

(13)

Finally, the maximum permissible cumulative load in the infinite power and user regime is d1/γ ? e−1 ? T ? , lim TN? (∞) , lim T∞ (γ0 ) = N →∞

γ0 →∞

X j=0

?

e−µ

(µ? )j+1 , j!

(14)

where µ? = lim µN (∞) = lim µ∞ (γ0 ). We conclude this section with an illustrative numerical N →∞

γ0 →∞

example. Numerical Example 1: Let γ ? = 0.1, implying lim k ? (γ0 ) = 10. The interval Ik is of the γ0 →∞ 1 1 for k = 1, 2, . . . , 10. Fig. 1 depicts ψN (η; γ0) as a function of η , form Ik = 11 − k 10 − k for different γ0 for N = 20. The innermost curve corresponds to γ0 ∈ I1 and the outermost curve corresponds to γ0 ∈ I10 . Fig. 2 shows µN (γ0 ) as a function of γ0 for four different values of N. Each curve is a step function with k th jump at the boundary between the intervals Ik and Ik+1 . Note that for fixed γ0 , µN (γ0 ) is a decreasing sequence. Fig. 3 depicts the maximum permissible cumulative load TN (γ0 ) as a function of γ0 for four different values of N. Once again, each curve is a step function with jumps at interval boundaries. Further, TN (γ0 ) is a decreasing sequence for fixed γ0 . This plot is an instance of the fundamental throughput versus

power curve for random access with MPR under static channel conditions. Note that the trade-off curve is concave, as intuition would suggest. For this example, the minimum γ0 needed to sustain a cumulative load of 1 is 0.1228 ∈ I2 . In contrast, pure TDMA (with centralized scheduling) can support a cumulative load of 1 with γ0 = γ ? = 0.1. Thus, random access requires ∼ 22.8% more power and a multi-packet reception capability of 2 to match the performance of TDMA in static channel conditions. In other words, decentralized control with contentions for channel access comes at a cost. However, a cumulative load of much more than 1 can be sustained by random access with MPR by suitable choice of γ0 . In particular, for this example we see that T ? ≈ 5.97, whereas T ? = 1 for TDMA. III. FADING C HANNEL A. System model and sufficient stability conditions We now examine the case were the channel of each user undergoes fading according to an independent and identically distributed (i.i.d.) stochastic process. The fading realizations for different users are assumed to be independent of each other. We focus on a specific fading process, namely Rayleigh fading, which is widely used in the literature to model wireless communication channels for mobile users [14]. It is known that the (random) received power under Rayleigh fading follows an exponential distribution. We assume perfect slow power control [11], i.e., the average received power for each user at the BS is the same for every transmission (say P ). Thus, under i.i.d. Rayleigh fading the received power for the ith user in the tth time-slot (if she transmits) can be expressed as Pit = P Xit, where Xit is an exponentially distributed random variable with unit mean. As noted earlier, Xit and Xiτ are independent if t 6= τ , and Xit and Xjτ are independent ∀ t, τ if i 6= j. The received SINR for the ith user in the tth time-slot (if she transmits) is given by γit = X

j∈S t \i

P Xit P Xjt + σ 2

= X

Xit Xjt + γ0−1

.

(15)

j∈S t \i

As before, the ith user’s transmission is successful if γit ≥ γ ? . However, unlike the static channel case, the probability of successful transmission for the ith user depends not only on the size of

the active set S t , but also on the instantaneous fading realizations for the members of the set. Thus, the probability of successful transmission for the ith user conditioned on S t is t X 1 X si (S t ) = P Xjt − ?i ≤ − . γ γ0 t

(16)

j∈S \i

By symmetry, the above probability depends on S t only through its size and is the same for all

users, i.e., si (S t ) ≡ s(|S t |) ∀ i. Lemma 2: The probability of successful transmission for a typical user under i.i.d. Rayleigh |S t |−1 1 ? fading conditioned on the active set S t is s(|S t |) = e−γ /γ0 ? 1+γ Proof: See Section VI-C. Once again, let η denote the effective probability of transmission by a typical user in steadystate. The random variable |S t |−1, which is the number of users that interfere with a transmission, is binomially distributed with parameters N − 1 and η. Therefore, by averaging with respect to the binomial distribution the probability of successful transmission is computed as s=

N −1 X j=0

N −1 γ?η N −1 j N −1−j −γ ? /γ0 η (1 − η) s(j) = 1 − e . j 1 + γ?

(17)

Similar to the static channel case, it can be shown that the effective probability of transmission for each user in steady-state must satisfy ηs = p, or equivalently ξN (η; γ0 ) = p where N −1 γ?η ? e−γ /γ0 . ξN (η; γ0) = η 1 − ? 1+γ

(18)

γ? , implying ξN (η; γ0) = a(γ0 )η(1−bη)N −1 . 1 + γ? 1 ? . Interestingly, Simple calculus shows that ξN (η; γ0 ) achieves a unique maximum at ηN = Nb ? ηN is independent of γ0 , unlike the static channel scenario. The corresponding maximum value N −1 a(γ0 ) 1 ? ? is given by pN (γ0 ) , ξN (ηN ; γ0) = 1− , which is the maximum permissible Nb N arrival rate (per user) for which an N-user system is stable. For any stable p, ξN (η; γ0) = p has For brevity we denote a(γ0 ) = e−γ

? /γ 0

and b =

at most two solutions. In particular, if p < ξN (1; γ0 ) then ξN (η; γ0) = p has a unique solution η1 . If p > ξN (1; γ0 ), then ξN (η; γ0 ) = p has two distinct solution η1 < η2 . Along the lines of our

discussion following Theorem 1, it follows that stability can be ensured by choosing ∆ ∈ (η1 , 1] when p < ξN (1; γ0) and by choosing ∆ ∈ (η1 , η2 ) when p > ξN (1; γ0 ). Theorem 2: For fixed N, γ0 and any ∆ such that ξN (∆; γ0 ) > p, lim sup E[Qti ] < ∞ ∀ i, t→∞

provided p < p?N (γ0 ). Proof: The proof is based on a Lyapunov technique and is similar to the proof for Theorem 1. We omit the details.

B. Asymptotic regimes Analogous to the static channel case, consider TN? (γ0 ) = Np?N (γ0 ), the maximum permissible cumulative load for which the system is stable. In the infinite-user regime, ? T∞ (γ0 )

, lim

N →∞

TN? (γ0 )

−(1+γ ? /γ0 )

=e

1 1+ ? γ

(19)

In the infinite power regime, TN? (∞)

, lim

γ0 →∞

TN? (γ0 )

=

1 1− N

N −1 1 1+ ? . γ

(20)

Finally, in the infinite-user and power regime,

1 1+ ? . (21) T , lim , lim γ0 →∞ N →∞ γ 1 1 ? ? Remarks: Note that for the fading scenario T∞ (γ ) = 2 1 + ? , whereas for the static e γ 1 1 ? ? ? ≈ 0.582, the maximum permissible cumulative channel scenario T∞ (γ ) = . For γ < e e−1 load in the infinite-user regime under i.i.d. Rayleigh fading is greater than that under static ?

TN? (∞)

? T∞ (γ0 )

1 = e

channel conditions when γ0 = γ ? . In fact, this assertion is true for any finite value of N. In other words, there is a “statistical gain” in throughput under fading conditions when γ ? is small enough. This can be explained as follows: Setting γ0 = γ ? dictates that under static channel conditions at most one user can successfully transmit in a time-slot. However, under fading conditions, more than one user can potentially succeed in a time-slot, depending upon the fading realizations. The likelihood of such an event however decreases as γ ? increases.

To further illustrate this point, consider N = 2 with γ0 = γ ? and fading realizations X1 , X2 in a particular time-slot. Suppose that both users transmit in this time-slot. The received SINRs X2 X1 . It is easily verified that for the two-users are given by γ1 = −1 and γ2 = X2 + γ 0 X1 + γ0−1 packets of both users are successfully received at the BS if X1 , X2 satisfy the linear constraints X2 ≤ (X1 − 1)/γ ? and X2 ≥ γ ? X1 + 1. The constraints define a conical region on the (X1 , X2 ) 2 1 − γ? plane (Fig. 7). It can be shown that the constraints are satisfied w.p. e− 1−γ? , which ? 1+γ ? is a monotonically decreasing function of γ . Remarks: Strictly non-zero throughput is achieved in the fading scenario even when γ0 < γ ? , contrary to the static channel scenario where necessarily γ0 ≥ γ ? for non-zero throughput. This is attributed to the fact that the power attenuation for a user can sometimes be greater than 1 due to fading, and hence her transmission can be successful even if her received SNR is lower than γ ? on an average. As expected, TN (γ0 ) is a decreasing function of γ0 , with lim TN (γ0 ) = 0 ∀ N. γ0 →0

We conclude this section with an illustrative numerical example. Numerical Example 2: Let the target SINR γ ? = 0.1. Fig. 4 shows ξN (η; γ0 ) as a function of η for different γ0 , for fixed N = 20. Fig. 5 depicts TN? (γ0 ) for different N as a function of γ0 . As remarked, TN? (γ0 ) > 0 even for γ0 < γ ? . A visual inspection suggests that TN? (γ0 ) is convex for γ0 < γ ? and concave for γ0 > γ ? , which is consistent with intuition. Fig. 6 depicts TN? (γ0 ) as a function of γ0 for N = 20 for the static channel case and fading case. Since γ ? < 0.582, the maximum permissible load is greater in the fading case for γ0 = γ ? , as discussed previously. However, for big enough γ0 the maximum permissible load for the static channel case is greater. In the infinite user and power regime, T ? ≈ 5.97 for the static channel case and T ? ≈ 4.05 for the fading case. Interestingly, T ? = 1 for pure TDMA under i.i.d. Rayleigh fading, which is the same as the T ? under static channel conditions with γ0 = γ ? . Thus, in terms of throughput, TDMA in fading conditions behaves like random access system without MPR (pure slotted Aloha) in static conditions.

IV. D ELAY P ERFORMANCE In this section we briefly examine the average delay performance of random access with MPR. Recall that the evolution of each user’s queue is described by a birth-death Markov chain with birth probability α, death probability β, and steady-state distribution {π(n), n ∈ Z} given by (9). The average backlog in a typical user’s queue in steady-state is computed as Qave =

∞ X

nπn =

n=0

α/β . 1 − α/β

(22)

Substituting for α and β from (5) and using Little’s law [15], the average delay per packet is given by D ave =

Qave η − ∆p = , p p(∆ − η)

(23)

Note that (23) applies to both static and fading scenarios. For the former case, η is the smallest solution to ψN (η; γ0) = p, assuming ∆ is chosen such that ψN (∆; γ0 ) > p. For the latter case, η is the smallest solution to ξN (η; γ0) = p, assuming ∆ is chosen such that ξN (∆; γ0 ) > p. We conclude this section with two numerical examples. Numerical Example 3: Fix N = 20 and γ ? = 0.1. Fig. 8 depicts D ave for the static channel case as a function of ∆ for different γ0 . The k th curve in the upper-half of the plot corresponds to γ0 ∈ Ik . For each curve, p = 0.9p?N (γ0 ). For fixed relative load (as a fraction of the maximum), the lowest achievable average delay decreases as γ0 increases. The lower-half of the plot shows D ave versus ∆ for fixed p for three different choices of k ? (γ0 ). There is a visible saturation effect as γ0 increases, implying a convex power versus delay trade-off curve. Numerical Example 4: Fix N = 20 and γ ? = 0.1. Fig. 9 depicts D ave for the i.i.d. Rayleigh fading case as a function of ∆ for different γ0 . The upper-half of the figure shows D ave versus ∆ for fixed relative load p = 0.9p?N (γ0 ). Each curve corresponds to a different choice of γ0 . The lower-half of the plot shows D ave versus ∆ for fixed p. The saturation effect is evident as γ0 increases. A 10dB increase in γ0 from γ ? to 10γ ? produces a massive improvement in average delay performance, while a 10dB increase from 10γ ? to 100γ ? has minimal impact.

Remarks: While the focus of this paper was on the case of symmetric users with Bernoulli packet arrivals, the proposed user-centric approach is in principle applicable to more general scenarios. For instance, packet arrival and channel processes modulated by finite-state Markov chains (to model bursty arrivals and channels with memory) are easily incorporated into this framework. Matrix geometric techniques can be used to solve for the steady-state, however, it may not always be possible to obtain “neat” closed-form solutions. The case of asymmetric users can also be treated within this framework. In particular, if there are K > 1 classes of users (with pk as a the arrival rate for class k users), the steady-state is identified by the Ktuple η = (η1 , . . . , ηK ), where ηk is the effective transmission probability of a class k user in steady-state. The probabilities η1 , . . . , ηK are solutions to a set of K non-linear equations. The case K = 1 was treated in this paper. V. C ONCLUSIONS This paper examined queuing aspects of a finite-user slotted Aloha type random access protocol with multi-packet reception capability at the base station. Queuing theoretic analysis of random access protocols is complicated due to complex interactions between queues of different users. A “user-centric” approach is proposed in this paper, which makes steady-state analysis more tractable. The approach focuses on the queue of a typical user in the system, assuming that this user observes all other users in steady-state. While the method is applicable quite generally, this paper treated the special case of identical users with Bernoulli arrivals in order to obtain analytical expressions for quantities of interest like throughput and average delay. Both static wireless channels (for fixed users) and Rayleigh faded channels (for mobile users) were considered. Sufficient conditions for system stability were also derived in the paper. Obtaining conditions which are both necessary and sufficient for stability of a random access protocol with MPR is still an unsolved problem. R EFERENCES [1] D.P. Bertsekas and R.G. Gallagher, Data Networks, Prentice Hall, 1991.

[2] S. Ghez, S. Verd´u, and S.C. Schwartz, “Stability properties of slotted-Aloha with multi-packet reception capability”, IEEE Transactions on Automatic Control, vol. 33, no. 7, pp. 640-649, Jul. 1988. [3] G. del Angel and T.L. Fine, “Information capacity and power control for slotted-Aloha random-access systems”, IEEE Transactions on Information Theory, vol. 51, no. 12, pp. 4074-4090, Dec. 2005. [4] V. Naware, G. Mergen, and L. Tong, “Stability and delay of finite-user slotted-Aloha with multi-packet reception”, IEEE Transactions on Information Theory, vol. 51, no. 7, pp. 2636-2656, Jul. 2005. [5] J. Sant and V. Sharma, “Performance analysis of a slotted-Aloha protocol on a capture channel with fading”, Queueing Systems, vol. 34, pp. 1-35, 2000. [6] S. Ghez, S. Verd´u, and S.C. Schwartz, “Optimal decentralized control in the random access multi-packet channel”, IEEE Transactions on Automatic Control, vol. 34, no. 11, pp. 1153-1163, Nov. 1989. [7] B. Tsbyakov, “Packet multiple access for channel with binary feedback, capture, and multiple reception”, IEEE Transactions on Information Theory, vol. 50, no. 6, pp. 1073-1085, Jun. 2004. [8] G. del Angel and T.L. Fine, “Optimal power and retransmission control policies for random access systems”, IEEE/ACM Transactions on Networking, vol. 12, no. 6, pp. 1156-1166, Dec. 2004. [9] C. van der Plas and J.M.G. Linnartz, “Stability of mobile slotted ALOHA network with Rayleigh fading, shadowing, and near-far effect”, IEEE Transactions on Vehicular Technology, vol. 39, no. 4, pp. 359-366, Nov. 1990. [10] Y. Yu, X. Cai, and G.B. Giannakis, “On the instability of slotted-Aloha with capture”, Proceedings of IEEE WCNC, vol. 2, pp. 728-732, Atlanta, GA, 2004. [11] M. Peh, S. Hanly, and P. Whiting, “Random-access over fading channels”, Proceedings of IEEE Globecom, vol. 2, pp. 888-892, San Francisco, CA, 2003. [12] A. Dua, “Power controlled random access”, Proceedings of IEEE ICC, vol. 6, pp. 3514-3518, Paris, France, Apr. 2004. [13] S. Karlin and H.M. Taylor, A First Course in Stochastic Processes, Academic Press, 1975. [14] D. Tse and P. Viswanath, Fundamentals of Wireless Communications, Cambridge University Press, May 2005. [15] J. Walrand, An Introduction to Queueing Networks, Prentice Hall, 1988. [16] R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press, Aug. 1995. [17] P.R. Kumar and S.P. Meyn, “Stability of queuing networks and scheduling policies”, IEEE Trans. on Automatic Control, vol. 40, no. 2, pp. 251-260, Feb. 1995.

VI. A PPENDIX A. Proof of Lemma 1 Proof: The proof is based on inductive arguments. Denoting k ? (γ0 )−1

η)

N −1−j

, it follows from definition ψN (η; γ0) =

X j=0

j ωN (η)

N − 1 j+1 η (1 − = j

j ωN (η).

0 Base case: Suppose γ0 ∈ I1 , so that k ? (γ0 ) = 1. In this case, ψN (η; γ0 ) = ωN (η) = 1 ? . Further, η(1 − η)N −1 . Simple calculus shows that the unique maximizer is ηN (γ0 ) = N

N −1 N 1 1 1 1 1− . Using the inequality 1 − < [16], ψN (1/N; γ0 ) < ψN (1/N; γ0) = N N N e 1 1 , as desired. , implying ψN (1/N; γ0 ) = O (N − 1)e N ? Inductive step: Assume that ψN (η; γ0) has a unique maximizer ηN (γ0 ) ∈ [1/N, k/N] for

γ0 ∈ Ik (i.e., k ? (γ0 ) = k). Now, suppose that the received SNR is γ00 ∈ Ik+1 , i.e., k ? (γ00 ) = k +1. j k Since ωN (·)’s are functions of η alone, ψN (η; γ00 ) = ψN (η; γ0 ) + ωN (η). The first two derivatives j of ωN (η) with respect to η are given by

j + 1 − Nη = η(1 − η) N(N − 1)η 2 − 2(N − 1)(j + 1)η + j(j + 1) j 00 j (ωN ) (η) = ωN (η) . η 2 (1 − η)2 j 0 (ωN ) (η)

j ωN (η)

(24)

j j Since ωN (η) > 0 ∀ η ∈ (0, 1), it follows that ωN (η) is a monotonically increasing function

for η ∈ [0, (j + 1)/N] and a monotonically decreasing function for η ∈ [(j + 1)/N, 1]. This observation immediately implies that ψN (η; γ00 ) is monotonically increasing for η ∈ [0, 1/N] and monotonically decreasing for η ∈ [(k + 1)/N, 1]. Thus, ψN (η; γ00 ) must necessarily achieve its maximum on the interval [1/N, (k + 1)/N]. r j+1 (j + 1)(N − j + 1) Γ j+1 Γ j Further, ωN (η) is concave for η ∈ , Γ = − , + , N N N N N −1 since the second derivative is negative on that interval. Since j < N, we get Γ > 1, implying k that ωN (η) is concave for η ∈ [k/N, (k + 1)/N]. Inductive arguments can be used to show that

ψN (η; γ0) is also concave for η ∈ [k/N, (k + 1)/N] (details omitted due to space constraints). Consequently, their sum ψN (η, γ00 ) is concave for η ∈ [k/N, (k + 1)/N]. From the inductive assumption, ψN (η; γ0 ) has a unique maximizer on [1/N, k/N]. Further, k ωN (η) is monotonically increasing on [1/N, k/N]. Thus, ψN (η; γ00 ), which is the sum of the two,

has at most one maximizer on [1/N, k/N]. The two possible cases are treated separately: 0 k 0 1) ψN (η; γ00 ) has a maximizer on [1/N, k/N]: In this case, ψN (k/N; γ00 ) < 0. Also, (ωN ) ((k+ 0 0 1)/N) = 0 and ψN ((k + 1)/N, γ0) < 0, implying ψN ((k + 1)/N; γ00 ) < 0. The concavity 0 of ψN (η; γ00 ) on [k/N, (k + 1)/N] implies that necessarily ψN (η, γ00 ) < 0 on that interval. ? Thus, there is a unique maximizer ηN (γ00 ) ∈ [1/N, (k + 1)/N].

2) ψN (η; γ00 ) does not have a maximizer on [1/N, k/N]: In this case, ψN (η; γ00 ) is monotoni-

0 0 cally increasing on [1/N, k/N], implying ψN (k/N; γ0 ) > 0. Also, ψN ((k + 1)/N; γ00 ) < 0, ? 0 as in case 1. Thus, ∃ ηN (γ00 ) ∈ [k/N, (k + 1)/N] such that ψN (η ? (γ00 ); γ00 ) = 0. The ? concavity of ψN (η; γ00 ) on the interval [k/N, (k + 1)/N] ensures uniqueness of ηN (γ00 ). ? Thus, there is a unique maximizer ηN (γ00 ) ∈ [1/N, (k + 1)/N].

In summary, subject to the inductive assumption that ψN (η; γ0) has a unique maximizer on [1/N, k/N] when γ0 ∈ Ik , ψN (η; γ0) has a unique maximizer on [1/N, (k + 1)/N] when γ0 ∈ Ik+1 . The desired result follows from the principle of mathematical induction. j Recall that ωN (η) achieves its maximum at η = (j + 1)/N. The maximum value is given by

j+1 N −1−j j+1 N −1 j+1 j+1 = 1− . j N N N Nk x N < < e−x [16] and 1 − Using the inequalities k! N k j ωN

j ωN

j+1 N

1 < N

N −1 N −j−1 {z | O(1)

j ωN (η)

j+1

N e−(j+1) (j + 1)j+1 , · N {z − 1} j! | {z } | } ·

O(1)

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independent of N

k ? (γ0 )−1 X 1 j ? ωN (η), it follows . Since k (γ0 ) is bounded and ψN (η; γ0 ) = N j=0

=O 1 , as desired. ψN (η; γ0) = O N implying

B. Proof of Theorem 1 Proof: Consider the time and state dependent quadratic Lyapunov function V t (Qt ) , hQt , Qt i, where hx, yi denotes the inner-product between vectors x and y. Define the set C t , {k : Qtk > 0}. If k ∈ C t , i.e., Qtk > 0, the evolution of the queue of the k th user is described by

Qt+1 = k

Qt + 1 ; w.p. p(1 − ∆st ) k

Qtk

; w.p. (1 − p)(1 − ∆st ) + p∆st

Qtk − 1 ; w.p. (1 − p)∆st ,

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where st is the probability of successful transmission for a queue transmitting in the tth time-slot. If k ∈ / C t , i.e., Qtk = 0, the evolution of the queue of the k th user is described by Qt+1 k

Qt + 1 ; w.p. p(1 − ∆st ) k = Qt ; w.p. 1 − p + p∆st . k

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2 t 2 t t t t t It follows that E (Qt+1 k ) − (Qk ) |Qk ≤ 2(p − ∆s )Qk + B , with equality when k ∈ C , where B t = (1 − p)∆st + p(1 − ∆st ). Now, the k th user transmits a packet w.p. ∆ if k ∈ C t , and

w.p. p∆ if k ∈ / C t . The “worst-case” arises when all users have a packet to transmit, and each of them attempts a transmission w.p. ∆. The probability of successful transmission is therefore lower-bounded by t

s ≥

k ? (γ0 )−1

X j=0

N −1 ∆j (1 − ∆)N −1−j , smin . j

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The expected conditional one-step drift of V can now be upper-bounded as ψN (∆)

z }| { E V t+1 (Qt+1 ) − V t (Qt )|Qt ≤ −2(∆smin −p)hQt , 1i + NB, | {z }

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where 1 is an N-length vector with all unit entries. Since p, ∆, st ∈ [0, 1], B t < C ∀ t for some p constant C > 0. Noting that hQt , 1i ≥ V t (Qt ), for any > 0 p E V t+1 (Qt+1 ) − V t (Qt )|Qt ≤ −2 V t (Qt ) + NC,

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The desired result follows from Foster’s stability criteria for Markov chains [17].

C. Proof of Lemma 2 Proof: Let X be a random variable with probability density function (p.d.f.) fX (x). The Z ∞ Laplace transform associated with the density of X is given by LX (y) = e−yx fX (x)dx. −∞

The following facts will come in handy: • •

1 . 1+y If X1 and X2 are two independent random variables with well defined p.d.f.s associated

If X is exponentially distributed with mean 1, then LX (y) =

with Laplace transforms LX1 (y) and LX2 (y) respectively, the Laplace transform associated

with the p.d.f. of the random variable X = X1 + X2 is LX (y) = LX1 (y)LX2 (y). If X is exponentially distributed with mean 1, the Laplace transform associated with the 1 . p.d.f. of X 0 = −aX for fixed a > 0 is given by LX 0 (y) = 1 − ay X Xt Denoting Z = Xjt − ?i for convenience, and using the above facts it follows that LZ (y) = γ j∈S t \i |S t |−1 γ? 1 . Using a partial fraction expansion technique and applying the inverse 1+y γ? − y |S t |−1 1 ? ? eγ z for z ≤ 0. Consequently, Laplace transform, fZ (z) = γ ? 1+γ •

Z −1 |S t |−1 γ0 1 1 −γ ? /γ0 s(|S |) = P Z ≤ − fZ (z)dz = = e . γ0 1 + γ? −∞ t

0.35

0.3

k*(γ0) = { 1,2,...,10 }

ψN(η; γ0)

0.25

0.2

0.15

0.1

0.05

0

Fig. 1.

0

0.1

0.2

0.3

0.4

0.5

η

0.6

0.7

0.8

0.9

1

ψN (η; γ0 ) as a function of η for different γ0 with fixed N = 20 and γ ? = 0.1.

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11

10

9

8

µN(γ0)

7

6

5

4

3

N=10 N=20 N=40 N=80

2

1

Fig. 2.

0

0.2

0.4

0.6

0.8

γ0

1

1.2

1.4

1.6

µN (γ0 ) as a function of γ0 for four different values of N with fixed γ ? = 0.1.

11 10 9 8

T*N(γ0)

7 6 5 4 3

N=10 N=20 N=40 N=80

2 1 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

γ0

Fig. 3.

TN? (γ0 ) (static channel) as a function of γ0 for four different values of N with fixed γ ? = 0.1.

0.25

γ0=γ*/2 γ =γ* 0 γ0=2γ* γ =10γ* 0 γ0=100γ* γ0=1000γ*

0.2

ξN(η; γ0)

0.15

0.1

0.05

0

Fig. 4.

0

0.1

0.2

0.3

0.4

0.5

η

0.6

0.7

0.8

0.9

1

ξN (η; γ0 ) as a function of η for different values of γ0 with fixed N = 20 and γ ? = 0.1.

5

4.5

4

3.5

T*N(γ0)

3

2.5

2

1.5

N=5 N=10 N=20 N=40 N=80

1

γ*

0.5

0

−1

10

0

10

1

10

2

10

γ0 (log scale)

Fig. 5.

TN? (γ0 ) (Rayleigh fading) as a function of γ0 for five different values of N with fixed γ ? = 0.1.

7

Static channel Rayleigh fading 6

5

T*N(γ0)

4

3

2

1

0 −2 10

−1

0

10

1

10

10

γ0 (log scale)

Fig. 6. A comparison of TN? (γ0 ) for the static channel case and Rayleigh fading case as a function of γ0 with fixed N = 20 and γ ? = 0.1.

X2

1 1 − γ?

1

1

1 1 − γ?

X1

Fig. 7. A two-user example to illustrate that more than one user can simultaneously succeed in the fading case even when γ0 = γ ? . The shaded conical region corresponds to the fading realizations for which both users can transmit simultaneously and succeed.

60 50

k*(γ ) = {1,2,...,10} 0

Dave

40 30 20 10 0 0.05

0.1

0.15

0.2

0.25

∆

0.3

0.35

0.4

0.45

0.5

60

k*(γ )=3 0 k*(γ )=4 0 k*(γ )=10

50

Dave

40

0

30 20 10 0

0

0.1

0.2

0.3

∆

0.4

0.5

0.6

0.7

Fig. 8. Dave as a function of ∆ for the static channel case for different values of γ0 ; The upper-half corresponds to a fixed relative load (90% of maximum) while the lower-half corresponds to a fixed load. .

40

γ0=γ* γ0=2γ* γ0=10γ* γ0=100γ*

Dave

30

20

10

0

0.4

0.5

0.6

∆

0.7

0.8

0.9

1

40

γ =γ* 0 γ0=2γ* γ0=10γ* γ =100γ*

Dave

30

0

20

10

0

0

0.1

0.2

0.3

0.4

0.5

∆

0.6

0.7

0.8

0.9

1

Fig. 9. Dave as a function of ∆ for the Rayleigh fading case for different values of γ0 ; The upper-half corresponds to a fixed relative load (90% of maximum) while the lower-half corresponds to a fixed load. .