TRANSACTIONS IEEE ON INFORMATION THEORY, VOL., NO., FEBRUARY 2009
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Stability of Finite Population ALOHA with Variable Packets
Predrag R. Jelenkovi and Jian Tan c
AbstractALOHA is one of the most basic Medium Access Control (MAC) protocols and represents a foundation for other more sophisticated distributed and asynchronous MAC protocols, e.g., CSMA. In this paper, unlike in the traditional work that focused on mean value analysis, we study the distributional properties of packet transmission delays over an ALOHA channel. We discover a new phenomenon showing that a basic nite population ALOHA model with variable size (exponential) packets is characterized by power law transmission delays, possibly even resulting in zero throughput. These results are in contrast to the classical work that shows exponential delays and positive throughput for nite population ALOHA with xed packets. Furthermore, we characterize a new stability condition that is entirely derived from the tail behavior of the packet and backoff distributions that may not be determined by mean values. The power law effects and the possible instability might be diminished, or perhaps eliminated, by reducing the variability of packets. However, we show that even a slotted (synchronized) ALOHA with packets of constant size can exhibit power law delays when the number of active users is random. From an engineering perspective, our results imply that the variability of packet sizes and number of active users need to be taken into consideration when designing robust MAC protocols, especially for ad-hoc/sensor networks where other factors, such as link failures and mobility, might further compound the problem. Index TermsALOHA, medium access control, power laws, heavy-tailed distributions, light-tailed distributions, adhoc/sensor networks.
I. I NTRODUCTION ALOHA represents one of the rst and most basic distributed Medium Access Control (MAC) protocols [1]. It is easy to implement since it does not require any user coordination or complicated controls and, thus, represents a basis for many modern MAC protocols, e.g., Carrier Sense Multiple Access (CSMA). Basically, ALOHA enables multiple users to share a common communication medium (channel) in a completely uncoordinated manner. Namely, a user attempts to send a packet over the common channel and, if there are no other user (packet) transmissions during the same time, the packet is considered successfully transmitted. Otherwise, if the transmissions of more than one packet (user) overlap, we say that there is a collision and the colliding packets need to be retransmitted. Each user retransmits a packet after waiting for an independent (usually exponential/geometric) period of time, making ALOHA entirely decentralized and asynchronous. The desirable properties of ALOHA, including its low complexity
Predrag R. Jelenkovi and Jian Tan are with the Department of Elecc trical Engineering, Columbia University, New York, 10027 USA e-mail: [e-mail], [e-mail] The preliminary version of this paper has appeared in [8].
and distributed/asynchronous nature, make it especially benecial for wireless sensor networks with limited resources as well as for wireless ad hoc networks that have difculty in carrier sensing due to hidden terminal problems and mobility. Furthermore, because of these properties ALOHA represents a basis for many more sophisticated MAC protocols, e.g., CSMA. Traditionally, the performance evaluation of ALOHA has focused on mean value (throughput) analysis, the examples of which can be found in every standard textbook on networking, e.g., see [3], [11], [10]; for more recent references see [9] and the references therein (due to space limitations, we do not provide comprehensive literature review on ALOHA in this paper). However, it appears that there are no explicit and general studies (more than two users) of the distributional properties of ALOHA, e.g., delay distributions. In this regard, in Subsection II-A, we consider a standard nite population ALOHA model with variable length packets [4], [2] that have an asymptotically exponential tail. Surprisingly, we discover a new phenomenon that the distribution of the number of retransmissions (collisions) and time between two successful transmissions follow power law distributions, as stated in Proposition II.1 of Subsection II-B, Theorem IV.1 of Subsection IV-A on starting behavior as well as Theorem IV.2 of Subsection IV-B on steady state behavior. Based on this observation, we derive new stability conditions for nite population ALOHA with variable packets in Theorem III.1 of Section III. Informally, our theorem shows that when the exponential decay rate of the packet distribution is smaller than the parameter of the exponential backoff distribution and the arrival rate, even the nite population ALOHA may have zero throughput. This is contrary to the common belief that the nite population ALOHA system always has a positive, albeit possibly small, throughput. Furthermore, even when the long term throughput is positive, the high variability of power laws (innite variance when the power law exponent is less than 2) may cause unstable buffer content (queue sizes), implying periods of very high congestion, long delays, and low throughput. It also may appear counterintuitive that the system is characterized by power laws even though the distributions of all the variables (arrivals, backoffs and packets) of the system are of exponential type. However, this is in line with the results in [5], [12], [6], which show that job completion times in systems with failures where jobs restart from the very beginning exhibit similar power law behavior. Our study in [6] was done in the communication context where job completion times are represented by document/packet transmission delays, e.g., ARQ protocol. It may also be worth noting that [6] reveals
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the existence of power law delays regardless of how light or heavy the packet/document and link failure distributions may be (e.g., Gaussian), as long as they have proportional hazard functions. Furthermore, from a mathematical perspective, Proposition III.1, Theorems IV.1 and IV.2 analyze a more complex setting than the one in [6], [12] and, thus, require a novel proof. Hence, when compared with [6], [12], this paper both discovers a new related phenomenon in a communication MAC layer application area and provides a novel analysis of it. As already stated in the abstract, the preceding power law phenomenon is a result of combined effects of packet variability and collisions. Hence, one can see easily that the power law delays can be eliminated by reducing the variability of packets. Indeed, for slotted ALOHA with constant size packets the delays are geometrically distributed. However, we show in Section V that, when the number of users sharing the channel is geometrically distributed, the slotted ALOHA exhibits power law delays as well. In Section VI, we illustrate our results with simulation experiments, which show that the asymptotic power law regime is valid even for relatively small delays and reasonably large probability values. Furthermore, the distribution of packets/number of users in practice might have a bounded support. To this end, we show by a simulation experiment that this situation results in distributions that have power law main body with an exponentiated (stretched) support in relation to the support of the packet size/number of active users. Hence, although exponentially bounded, the delays may be prohibitively long. In practical applications, we may have combined effects of both variable packets and a random number of users, implying that the delay and congestion is likely to be even worse than predicted by our results. Thus, from an engineering perspective, one has to pay special attention to the packet variability and the number of users when designing robust MAC protocols, especially for ad-hoc/sensor networks where link failures [6], mobility and many other factors might further worsen the performance. In summary, the rest of the paper is organized as follows. In Section II, we provide the description and the preliminary power law bounds. Then, we present our new stability conditions that are based on packet distribution decay rates in Section III. Further distributional properties for the number of retransmissions and delays are investigated in Section IV. Section V contains the results on power laws in slotted ALOHA with random number of users. Experimental validation of our results can be found in Section VI. The paper is concluded in Section VII. Finally, some of the more technical proofs are postponed to Section VIII. II. P OWER L AWS IN THE F INITE P OPULATION ALOHA WITH VARIABLE S IZE PACKETS In this section we show that the variability of packet sizes, when coupled with the contention nature of ALOHA, is a cause of power law delays. This study is motivated by the well-known fact that packets in todays Internet have variable
sizes. To further emphasize that packet variability is a sole cause of power laws, we assume a nite population ALOHA model where each user can hold (queue) up to one packet at a time since the increased queueing only further exacerbates the problem. In addition, in Section V we show that the user variability in an innite population model may be a cause of power law delays as well. In the remainder of this section, we describe the model and introduce the necessary notation in Subsection II-A and present the preliminary results in Subsection II-B. A. Model Description Consider M 2 users sharing a common communication link (channel) of unit capacity. Each user can hold at most one packet in its queue and, when the queue is empty, a new packet is generated after an independent (from all other variables) exponential time with mean 1/. Each packet has an independent length that is equal in distribution to a generic random variable L. A user with a newly generated packet attempts its transmission immediately and, if there are no other users transmitting during the same time, the packet is considered successfully transmitted. Otherwise, if the transmissions of more than one packet overlap, we say that there is a collision and the colliding packets need to be retransmitted; for a visual representation of the system see Figure 1. After a collision, each participating user waits (backoffs) for an independent exponential period of time with mean 1/ and then attempts to retransmit its packet. Each such user continues this procedure until its packet is successfully transmitted and then it generates a new packet after an independent exponential time of mean 1/. Let {U (t)}t0 denote the } number of users that are in { (t) backoff state at time t and Li denote the packet
1iU (t)
sizes of all the U (t) number of active users at time t.
1
111 000 111 000 1111 0000 1111 0000
11 00 11 00 11 11111 00 00000 11 11111 00 00000 111 000
succeed
T
2
. . .
M
111111 000000 111111 000000
retransmit collide
11 00 11 00
11 00
Finite population ALOHA model with variable packet sizes.
Fig. 1.
From the perspective of the receiver, let {Ci }i1 be an increasing sequence of positive time points when either a collision or successful transmission occurs with C0 = 0. Let {Dm }m1 be the sequence of time points when the receiver successfully receives the mth packet and dene Tm = Dm Dm1 to be the transmission time for the mth successfully received packet with a convention D0 = 0. Correspondingly, we can dene Nm to be the number of (re)transmissions in the interval (Dm1 , Dm ] for the mth successful transmission. { Now, from the perspective of user i, , 1 i M , dene } (i) Dm to be the sequence of time points when user i sucm1
cessfully sends the mth packet and dene Tm = Dm Dm1
(i)
(i)
(i)
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to be the transmission time for the mth successfully transmit(i) ted packet with a convention D0 = 0. By the same fashion, (i) we can dene Nm to be the number of (re)transmissions in (i) (i) the interval (Dm1 , Dm ] for the mth successful transmission. We will study the stability of this model as well as the asymptotic properties of the distributions of Nm , Tm and (i) (i) Nm , Tm . B. Power Law Bounds In the rest of this subsection, we present preliminary results for the nite ALOHA with variable packets, described in the preceding subsection. Let x y = min(x, y), x y = max(x, y), and , , = denote inequalities and equality in distribution, respectively. Basically, ALOHA model can be viewed as a state dependent channel with failures where the failure rate depends on the number of backoffed users and the sizes of the packets present in the system. Hence, this model can be viewed as a generalization of the problem stated in [6], [7]. The following proposition shows that the distributions of the number of retransmissions and the delays in our ALOHA model are always sandwiched between two power laws, which is obtained by uniformly bounding the variable collision (failure) rates independently of the state of the channel. Proposition II.1 Assume that , for > 0,
x d d d
that only measures the delay caused by the collisions of the (i) (i) second type, then Tm T m . Now, consider the system at the moment when user i has successfully initiated its transmission. At that moment a number of users ( M 1) can be in the backoffed state (exponential with rate for each user) and the remaining ones are waiting for the new packets to arrive (exponential with rate for each user). Hence, the time until another user attempts to access the channel is upper bounded by an exponential time of rate (M 1)( ). Therefore, given L, the probability that there is a collision of the second type is lower bounded by 1 eL(M 1)() , implying that [ ] (i) (i) P Nm > n P[N m > n] [( )n ] E 1 eL(M 1)() = P[N > n], (6) since the repetitions of exponential times of rate (M 1)() are independent due to the memoryless property. Condition (1) implies that, for any > 0, there exists x such that P[L > x] e(+ )x for all x x . Then, if we dene random variable L with P[L > x] = e(+ )x , x 0, we obtain d L L 1(L > x ), resulting in [( [( 1 eL
1(L >x )(M 1)() (M 1)()
lim
log P[L > x] = , x
P [N > n] E (1) E
)n ]
1 eL
)n
] 1(L > x ) . (7)
and let N and N be two random variables with distributions [( )n ] P [N > n] = E 1 eL(M 1)() (2) and [ ] P N >n =E [( 1 eL(M 1)() M ( )
d d
)n ] .
Then, uniformly for all m and i,
(i) N Nm N
(3) (4) (5)
Noticing that for any > 0, there exists 0 < x < 1 such that 1 x e(1+)x for all 0 x x , we can choose x large enough, such that [ ] L (M 1)() P [N > n] E e(1+ )ne 1(L > x ) [ ] L (M 1)() = E e(1+ )ne [ ] L (M 1)() E e(1+ )ne 1(L x ) [ ] L (M 1)() E e(1+ )ne n, (8)
) where = e(1+ )e < 1. Now, for any 0 < x < 1, [ ] [ ] log x (+ )L P e <x =P L > = x, +
x (M 1)(
and
log P [N > n] = , n log n (M 1)( ) ] [ log P N > n lim = . n log n (M 1)( ) lim
Similarly, there exist T and T such that (3), (4) and (5) are (i) satised for the corresponding expressions for Tm , T and T (replacing N by T ). Proof: We begin with studying Nm . First, we prove the lower bound. Note that a collision for user i may occur for two different reasons. Either, when user i attempts to access the channel it collides with the already existing transmission, or after user i successfully starts its transmission it is interrupted later by some other user that tries to access the channel. Now, (i) if N m only counts the collisions due to the second reason, (i) (i) (i) then clearly Nm N m . Similarly, if T m is the total time
(i)
implying that e(+ )L = U , where U is a uniform random variable between 0 and 1. Thus, we can derive from (8) ] [ (M 1)()/(+ ) n. P[N > n] E en(1+ )U Now, since E[eU
1/
d
] ( + 1)/
(9)
as , one can easily obtain log P [N > n] + , log n (M 1)( ) n lim which, by passing 0, proves the lower bound. (10)
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Next, we prove the upper bound. We observe that a successful transmission has two steps. First, the user has to initiate the transmission successfully (grab the channel). Second, after accessing the channel it has to complete the transmission without interruptions from other users. We will bound these events by independent ones as described below. After a successful transmission or a collision, user i will attempt to access the channel after an exponential time of rate no smaller than ; each other user will compete to access the channel after an exponential time of rate no larger than . Once user i grabs the channel, its transmission will be successful if the rst channel access time of all the other users is larger than L. Note that the rst access time of the other users is exponential with rate upper bounded by (M 1)(). Therefore, given L, the probability of a collision (failure) is upper bounded by 1 eL(M 1)() . M ( ) Furthermore, due to the memoryless property of exponential distribution the probability of n successive collisions, given L, can be upper bounded by independent events with probabilities given by the preceding expression. Therefore, after unconditioning, we obtain [( )n ] [ ] (i) P Nm > n E 1 eL(M 1)() M ( ) ] [ (11) =P N >n . Next, using 1x ex and dening /(M ()), we derive, for x > 0, [ ] [ ] L(M 1)() P N > n E ene 1 (L > x ) ( )n + 1 ex (M 1)() [ ] L1(L>x )(M 1)() E ene + n , (12) where 1 ex (M 1)() < 1. Now, condition (1) implies that, for any 0 < < , we can choose x such that P[L > x] e( )x for all x x . Thus, by dening an exponential random variable L with P[L > x] = e( )x , x 0, we obtain L1(L > x ) L . Therefore, (12) implies [ ] [ ] L (M 1)() P N > n E ene + n . (13) Similarly as in the proof of the lower bound, we know d e( )L = U is a uniform random variable between 0 and 1. Thus, (13) implies ] [ [ ] (M 1)()/( ) P N > n E enU + n . By (9), we obtain ] [ log P N > n , lim n log n (M 1)( )
d
First, user i initiates an attempt to grab the channel; for the jth attempt, denote by {Xj }j1 the idle period where user i either is waiting for a new packet to arrive or is in its backoff state for the mth packet. Hence, X1 is exponential with rate and Xj , j > 1 are exponential with rate . Second, after user i makes an attempt to access the channel, it either collides with other users that are transmitting packets or starts transmitting its own packet; for the jth attempt, denote by {Yj }j1 the period during which there are no transmissions from other users after user i starts sending the mth packet. Note that if user i fails to grab the channel for the jth attempt, then Yj = 0; if user i successfully grabs the channel for this attempt, then it spends time Yj transmitting the mth packet without interference from other users. Thus, we have
(i) Nm
(i) Tm
=
j=1
(i) Nm 1
Xj +
j=1
Yj + L.
(14)
Since {Xj } is a sequence of exponential random variables with rate equal to either or , we can always nd two i.i.d. exponential sequences, {X, X j }j1 and {X, X j }j1 , such that X j Xj X j . (15) Additionally, observe that when user i successfully grabs the channel, Yj is stochastically upper bounded by an exponential random variable with rate (M 1)( ), and thus, we can construct a sequence of i.i.d. exponential random variables {Y , Y j } such that Yj Y j , (16) where {Y j } is independent of {X j }. First, we prove the upper bound. Using the union bound, (i) Nm ] [ (i) (Xj + Yj ) + L > 2t P Tm > 2t P P
j=1
(i) Nm
j=1
(i) (Xj + Yj ) > t, Nm
[ ] t (i) + P Nm > + P [L > t] 2E[X + Y ] t/(2E[X+Y ]) (Xj + Yj ) > t P [ (i) + P Nm >
j=1
t 2E[X + Y ]
] t + P [L > t] . 2E[X + Y ]
By (3), (15) and (16), we obtain t/(2E[X+Y ]) [ ] (i) (X j + Y j ) > t P Tm > 2t P [ +P N > ] t + P [L > t] 2E[X + Y ] I1 + I2 + I3 , (17)
j=1
which, by passing 0, nishes the proof of the upper bound. (i) Now, we prove the result for Tm . Observe that each attempt for user i to transmit the mth packet consists of two steps.
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which, by dening random variable T with the following distribution [ ] P T > 2t min{I1 + I2 + I3 , 1}, [ ] ] [ (i) implies P Tm > 2t P T > 2t , i.e.,
(i) Tm T . d
which, combined with the estimates for I2 (t), implies that
t
lim
log P [T > t] = . log t (M 1)( )
(20)
(18)
For (17), applying Chernoff bound, we derive I1 = O(en ) for some > 0. Condition (1) implies I3 = O(en ) for some other > 0. To compute I2 , using (5), we obtain, [ ] t log P N > 2E[X+Y ] lim = , t log t (M 1)( ) which, combined with the estimates for I1 and I3 , implies that [ ] log P T > t lim = . (19) t log t (M 1)( ) Next, we prove the lower bound. It is easy to obtain (i) Nm 1 ] [ (i) (Xj + Yj ) + L > t P Tm > t P P P
j=1
(i) Nm 1
Combining (19) and (20) completes the proof. The following lemma studies the distribution of the number of retransmissions that occur from a point when there is a departure until the system becomes full. For the two sequences {Ci } and {Dm } dened in Subsection II-A, noting that {Dm } is a subsequence of {Ci }, we can dene the position of Dm in {Ci } by hm min{i 0 : Ci = Dm }. Let f Nm , m 0 be the total number of both collisions and departures until the system becomes full and all the users are backlogged (a collision occurs) for the rst time after Dm , f i.e., Nm min{l hm : U (Cl +) = M, l hm }, where U (Cl +) represents the right hand limit of U (t) at time Cl . { } (t) Recall that Li represents the packet sizes of all
1iU (t)
the U (t) number of active users at time t. Lemma II.1 For any nite values {Li m }1iU (Dm ) at time Dm , uniformly for all m > 0, we have ( ) [ f ] P Nm > n = O e n (21) where the constant (D ) {Li m }1iU (Dm ) . > 0 does not depend on
(D )
j=1
Xj > t
(i) Xj > t, Nm >
(i) Nm 1
j=1
[ ] 2t (i) P Nm > +1 E[X] (i) Nm 1 (i) Xj t, Nm P
j=1
2t + 1 E[X]
Remark 1 We believe that it is possible to prove a tighter ex[ f ] ponential bound P Nm > n = O (en ), but the preceding Weibull bound sufces for our proofs. The proof of Lemma II.1 is presented in Section VIII. III. S TABILITY In this Section, we derive the stability condition of nite population ALOHA with variable packets. Corollaries III.1 and III.2 are based on Proposition II.1; Proposition III.1 studies the distributional properties of the upper bound for the number of (re)transmissions and transmission delay for each successfully received packet observed at the receiver. Using these results, we derive the stability condition in Theorem III.1. We use lim to denote both lim and lim, i.e., lim means that the corresponding two statements with respect to lim and lim are true. From Proposition II.1, we can easily obtain the following two corollaries. Note that in Corollary III.1 we use lim with respect to m since the existence of the stationary (i) (i) region for Nm and Tm is not established. At this point of our analysis, we could not nd an easy argument for resolving this, maybe minor, technical issue. Corollary III.1 If = > 0, then, as n , [ ] (i) log P Nm > n lim . log n (M 1) m Corollary III.2 If 0 < and > (M 1), then the system has a positive throughput. If > 0 and < (M 1), then the system has a zero throughput.
2t + 1 , E[X]
which, by recalling Xj X j and using (3), yields [ ] 2t/E[X] [ ] 2t (i) (i) P Tm > t P Nm > Xj t + 1 P E[X] j=1 [ ] 2t/E[X] 2t X j t P N > + 1 P E[X] j=1 I1 (t) I2 (t). Now, dene a random variable T with P [T > t] implying
(i) Tm T . d
max{I1 (t) I2 (t), 0},
Next, by Churnoff bound, we obtain I2 (t) O (et ) for some > 0. Using (5), we derive [ ] 2t log P N > E[X] + 1 lim = , t log t (M 1)( )
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j Proof: Let N (t) min{j : m=1 Tm t} be the counting process for the number of successfully transmitted packets observed at the receiver from time 0 until time t. By the same fashion, we can dene the counting process j (i) N (i) (t) min{j : m=1 Tm t} for user i, 1 i M , which represents the number of successfully transmitted packets observed at user i from time 0 until time t. Clearly, we have N (t) =
M i=1
N (i) (t)
(22)
where N (t), N (i) (t) all go to innity almost surely as t . Recalling the proof corresponding to T and T in Proposition { } can always construct on the same probability II.1, we { } space T j j1 and T j j1 , two sequences of i.i.d. copies of {T } and {T }, such that T j Tm T j . Dene j (i) (i) N (t) min{j : m=1 T m t} and N (i) (t) min{j : j (i) m=1 T m t} for user i, 1 i M . By the preceding denitions, we can easily obtain N (i) (t) N (i) (t) N
(i) (i)
and using the union bound, we obtain [ ] f P[Nm > n] = P Nm > n, Nm1 < Nm [ ] f + P Nm > n, Nm1 Nm [ ] f f f P Nm Nm1 + Nm1 > n, Nm1 < Nm [ ] f + P Nm1 > n [ ] n f f P Nm Nm1 > , Nm1 < Nm 2 [ [ n] n] f f + P Nm1 > + P Nm1 > . (27) 2 2 By Lemma (II.1), we know that for some 0 < < 1, [ [ [ n] n] n] f f f P Nm1 > + P Nm1 > 2P Nm1 > 2 2 2 n 2 . (28) 2
f Observe that Nm1 < Nm implies that there exists time < Dm at which each user has a packet and is in backoffed status. Thus, by recalling the notation dened for Lemma II.1, we can denote the packet sizes held by M active users at () time by Li , 1 i M . In addition, we know that right after time Dm1 , one user has just successfully transmitted a packet. Thus, at time (when the system is full) there is at least one new packet with size equal in distribution to L in the system. Therefore, we obtain [ ] n f f P Nm Nm1 > , Nm1 < Nm 2 ( (M )) n 2 (Dm ) 1 (M 1) eLi E 1 M i=1 [( ) n ] 1 L(M 1) 2 e E 1 , M
(t).
(23)
Thus, if and > (M 1), then N (i) (t) N (i) (t) 1 lim = [ ] > 0, t t E T t t lim (24)
since ]T has a power law tail with index greater than one [ (E T < ) by Proposition II.1. If and < (M 1), then N (i) (t) N (t) 1 lim lim = = 0, t t t t E [T ]
(i)
(25)
since ]T has a power law tail with index smaller than one [ (E T = ) by Proposition II.1. Combining (22), (24) and (25), we nish the proof.
which, in conjunction with (27) and (28), implies, uniformly for all m, [( ) n ] n 1 L(M 1) 2 P[Nm > n] E 1 e + 2 2 . M Now, we can dene a random variable N which satises, for integer n, P[N > n] is equal to { [( } ) n ] n 1 L(M 1) 2 min 1, E 1 e + 2 2 , M implying Nm N . By using the same approach as in calculating (11), we obtain log P[N > n] = , n log n (M 1) lim
d
Proposition III.1 For an ALOHA system with nite size packets at t = 0 and under condition (1) on asymptotically exponential packet sizes, there exist N and T such that the number of transmissions Nm and the transmission time Tm satisfy Nm N , with P[N > n] P[T > t] lim = lim = . n t log n log t (M 1) (26)
d
Tm T
d
f Proof: Recalling the denition of Nm before Lemma II.1
which nishes the proof of the result on N in equation (26). follows similar arguments as in proving the The proof for T (i) result on Tm in Proposition II-B. Combining Theorem III.1 and Corollary III.2, we obtain the following theorem. Observe that this theorem is slightly more
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general than Corollary III.2 since it shows that > (M 1) is enough for positive throughput, i.e., the additional condition in Corollary III.2 is not needed. Theorem III.1 Under condition (1), if > (M 1), the ALOHA system has a positive throughput. Conversely, if > 0 and < (M 1), then, the system has a zero throughput. Remark 2 For the critical case = (M 1), if L has an exact exponential tail, i.e., P[L > x] cex and , (i) (i) then, the limiting distributions of Nm and Tm would have exact power law tails of index 1, and therefore, have innite means. Remark 3 The condition and < (M 1) yields a zero throughput. However, it appears that one could obtain a positive throughput by decreasing for xed and in this case. Specically, we conjecture that the throughput of the system is positive when is small enough and M > (M 1) > . Proof: The second statement of this theorem is the same as the second statement of Corollary III.2. Given Proposition III.1, the rst statement can be easily derived using basically the same arguments as in the proof of Corollary III.2, and thus we omit the details. IV. A PPROXIMATION OF THE D ISTRIBUTIONS OF Nm Tm A. Starting Behavior In this subsection, we study the number of retransmissions Nm and the transmission delay Tm for the mth successfully transmitted packet observed at the receiver when the system starts from an empty state. This result characterizes the starting behavior of our ALOHA model for small (nite) m. Furthermore, since ALOHA tends to accumulate with time longer packets, it would make sense to dene a modied ALOHA which, after a nite (possibly large) number of successful transmissions, refreshes itself by discarding all the packets currently present in the system. Hence, for this modied ALOHA, the following theorem describes the steady state behavior as well. Theorem IV.1 Under condition (1), assume that at time t = 0 the system is empty U (0) = 0, then, for any xed m M , the number of transmissions Nm and the transmission time Tm satisfy
n
Remark 6 This theorem indicates that the distribution tails of Nm and Tm are essentially power laws when the packet distribution is approximately exponential ( ex ). Thus, the nite population ALOHA may exhibit high variations, e.g., the system has innite average transmission time when 0 < M /(M 1) < 1; and when 1 < M /(M 1) < 2, the transmission time has nite mean but innite variance. It might be worth noting that this may even occur when the expected packet length is much smaller than the expected backoff time EL 1/. Proof of Theorem IV.1: We rst prove the logarithmic asymptotics for Nm , based on which a similar result can be proved for Tm . First, we begin with proving the lower bound for Nm . We construct a special event with a positive probability that guides system the from time 0 up to time Dm1 . Denote by E1 the event that only one of the users has packets to send and all the other M 1 users are empty from time 0 through time Dm1 ; additionally, we require that the sizes of these arriving packets be less than a constant k 1 with P[L k 1] > 0 and that each new arrival be within a unit interval after the previous departure. This construction implies Dm1 (m 1)k, and therefore, by time Dm1 the probability that the system evolves according to E1 is lower bounded by ( )m1 (M 1)(m1)k P[E1 ] (1 e )P[L k 1] e > 0. (30) Next, immediately after time Dm1 , observe that the whole system becomes empty according to our construction. Then, we build another special event E2 that leads M users to have i.i.d. packets with sizes that are larger than 1 in their buffers after time Dm1 . To this point, we require that each of the M users have a packet with size larger than 1 arriving to the system after Dm1 and that their arriving points be within [Dm1 , Dm1 + 1]. This event happens with probability (1e )M P[L > 1]M . Notice that, immediately after the M th packet arrives, there are either M 1 or M users in the backoff status, depending on whether the M th arrived packet collides with others upon arrival or not. If the M th packet does not collide with others upon arrival, we require that a retransmission occur within one unit of time after it arrives, which happens with a probability greater than 1 e(M 1) . These requirements can guarantee that there exists a time [Dm1 , Dm1 + 2) with = min{Cn | U (Cn +) = M, Cn > Dm1 }, at which each user in the system has a packet and is in the backoff status. The probability that the event E2 happens is lower bounded by P[E2 ] (1 e )M P[L > 1]M (1 e(M 1) ) > 0.
Now, given E1 and E2 , we can denote by Nm the number of retransmissions between (, Dm ], implying Nm Nm . Then, recalling the notation dened before Lemma II.1 and dening { } ( ) ( ) ( ) Lo min L1 , L2 , , LM , we obtain [( (M ))n ] 1 L( ) (M 1) P[Nm > n | E1 , E2 ] = E 1 e i M i=1 )n ] [( . (31) E 1 eLo (M 1)
AND
lim
P[Nm > n] P[Tm > t] M = lim = . (29) t log n log t (M 1)
Remark 4 A special case of this theorem when U (Cm +) = M with all the packets in the system being i.i.d. and equal in distribution to L was proved in Theorem 1 of [8]. Remark 5 Note that this result still holds even if we allow m to be a slowly growing function of n for Nm , e.g., m = o(log n) (or m = o(log t) for Tm ).
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It is easy to check that the complementary cumulative distribution function P[Lo x] satises log P[Lo x] = M , x which, by using the same technique as in estimating (6), yields
x
lim
log P[Nm > n | E1 , E2 ] M . log n (M 1) n
lim
a packet through the channel, it immediately generates a new packet in its buffer and goes into the backoff state, i.e., we can interpret that the arrival and departure happen at the same time. Therefore, the system evolves as if it always had M packets available and all users remained in the backoff state over the entire operation. Let L(Dm ) be the minimum of the packet sizes of the other M 1 users except the one departing at time Dm . Lemma IV.1 Assume that = > /(M 1) and lim sup 1 log yx ( P[L > x] P[L > y] ) (35)
Finally, using P[Nm > n] P[E1 , E2 ]]P[Nm > n | E1 , E2 ] completes the proof of the lower bound for Nm . Next, we proceed with the proof of the upper bound for Nm . Using the same approach as in evaluating (27), we obtain [ ] n n f f P[Nm > n] P Nm Nm1 > , Nm1 < Nm + 2 2 . 2 (32) f Now, we observe that Nm < Nm implies that there exists time < Dm at which each user has a packet at hand and is in the backoff status. Thus, we denote the packet sizes held () by M users at time by Li , 1 i M . In addition, we know that at time the total number of packets, including those still present in the system and those already successfully transmitted, is less than m + M since the system has only M users. Denote the sizes of the rst m + M packets arriving to the system by {L1 , L2 , , Lm+M } and its order statistics by L(1) L(2) L(m+M ) , and we obtain f f P[Nm Nm1 > n, Nm1 < Nm ] (M [( ))n ] 1 L() (M 1) e i E 1 M i=1 [( )n ] 1 L(M ) (M 1) E 1 e . (33) M
y y<x<y
for 0 < < 1. Then, there exists p > 0 such that for any xed y,
m
lim P [L(Dm ) > y] > p.
(36)
Remark 7 We believe that a stronger result limm P [L(Dm ) > y] = 1 for all y is also true, but the preceding lemma sufces for our proofs. Furthermore, a careful examination of our proof shows that the result is also true for min{, } > /(M 1), but we avoid this generalization due to considerable notational complications. Remark 8 It is easy to see that condition (35) holds for a broad range of distributions from exponential family, e.g., Gamma distribution, ex ex with 0 < < 1, etc. The proof of Lemma IV.1 is presented in Section VIII. By using this lemma, we can derive the following theorem that characterizes the limiting steady state behavior of our ALOHA model. Theorem IV.2 Under condition (35), if = > /(M 1), we obtain lim lim P[Nm > n] P[Tm > t] = lim lim t m log n log t = . (M 1)
Since L(M ) is the M th largest value among Li , 1 i m + M , we know P[L(M ) > x] = M . x x Then, by (32), (33) and using the same approach as in estimating (11), one derives lim log P[Nm > n] M , n log n (M 1) lim (34)
n m
which completes the proof of the upper bound. The proof for the logarithmic asymptotics of Tm is based (i) on similar arguments as in proving Tm in Proposition II.1 and, thus, we omit the details.
(37)
B. Limiting Steady State Behavior When the system keeps running for a long period of time, we can show that the preceding upper bound, presented in Theorem III.1, is attainable when = > /(M 1). In order to study this situation, rst we establish the following lemma that characterizes the growth of the packet sizes in the system immediately after a departure at time Dm . Noting that = , we can assume that once a user successfully transmits
Proof: First, we prove the result for Nm . The upper bound is implied by Proposition III.1 and thus, we only need to prove the lower bound. Recalling the denition of L(Dm ) in the paragraph before Lemma IV.1 and using Lemma IV.1, we obtain that there exist p > 0 and m0 > 0 such that for all m > m0 , [ ] log n P L(Dm1 ) > > p. (38) (M 1) Since there is a new packet with size equal in distribution to L arriving to the system at time Dm1 (see the discussion before Lemma IV.1), and the packet sizes of the other M 1
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users are lower bounded by L(Dm1 ), we obtain [ ] log n P[Nm > n] P Nm > n, L(Dm1 ) > (M 1) [( (M ))n 1 L(Dm1 ) (M 1) E 1 e i M i=1 ( )] log n 1 L(Dm1 ) > (M 1) ( [( 1 eL(M 1) E 1 M ))n M 1 L(Dm1 )(M 1) e +
variable since sensors may switch between sleep and active modes, as shown in Figure 2; similarly in ad hoc wireless networks the variability of users may arise due to mobility, new users joining the network, etc.
Active Sleep
11 00 11 00 11 00 11 00 11 00 11 00
Sink
Fig. 2.
Random number of active neighbors in a sensor network.
More formally, consider a slotted ALOHA model (e.g., see Section 4.2.2 of [3]) with packets/slots of unit size and a i=1 random number of users M 1 that are xed over time. ( )] log n This model can be viewed as a rst order approximation of 1 L(Dm1 ) > (M 1) a real system where the number of users change very slowly. [ ] Similarly as in Section II, each user holds at most one packet log n P L(Dm1 ) > at a time and after a successful transmission a new packet is (M 1) )n ] generated according to an independent Bernoulli process with [( 1 L(M 1) M 1 1 e , success probability 1e , > 0. In case of a collision, each E 1 M n M colliding user backs off according to an independent geometric (39) random variable with parameter e , > 0. Denote the where we use the independence between the new packet size number of slots where transmissions are attempted but failed and the total time between two successful packet transmissions and L(Dm1 ) at time Dm1 in the last inequality. Combining (38) and (39) yields, for n large enough, as N and T , respectively. P[Nm > n] is lower bounded by ( )n Theorem V.1 If = and there exists > 0, such that M 1 1 p 1 log P[M > x] M n lim = , [( )n ] x x 1 L(M 1) E 1 e then, we have M (1 (M 1)/(M n)) ( )n [( )n ] log P[N > n] log P[T > t] M 1 1 lim = lim = . (40) p 1 E 1 eL(M 1) , n t log n log t M n which, by noting that ( )n M 1 1 lim 1 = e(M 1)/M > 0 n M n and using the same approach as in calculating (6), completes the proof the lower bound. The result on Tm can be proved by using the same approach as in proving the result on Tm in Proposition II.1. V. P OWER L AWS IN S LOTTED ALOHA WITH R ANDOM N UMBER OF U SERS Remark 9 Similarly as in Theorem IV.1, this result shows that the distributions of N and T are essentially power laws, i.e., P[T > t] t/ and, clearly, if < , then EN = ET = . Proof: Since = , we can consider a situation where all the users are backlogged, i.e., have a packet to send. In this case the total number of collisions between two successful transmissions is geometrically distributed given M , ( )n M e(M 1) (1 e ) P[N > n | M ] = 1 , n N, 1 eM since, given M , 1 eM is the conditional probability that there is an attempt to transmit a packet, and 1 eM M e(M 1) (1 e ) is the conditional probability that there is a collision. Therefore, [( )n ] M e(M 1) (1 e ) P[N > n] = E 1 . (41) 1 eM On the other hand, we have [( )t ] P[T > t] = E 1 M e(M 1) (1 e ) , t N. (42)
It is clear from the preceding section that the power law delays arise due to the combination of collisions and packet variability. Hence, it is reasonable to expect an improved performance when this variability is reduced. Indeed, it is easy to see that the delays are geometrically bounded in a slotted ALOHA with constant size packets and a nite number of users. However, in this section we will show that, when the number of users sharing the channel has asymptotically an exponential distribution, the slotted ALOHA exhibits power law delays as well. Situations with random number of users are essentially predominant in practice, e.g., in sensor networks, the number of active sensors in a neighborhood is a random
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Now, following the same arguments as in the proof of Proposition II.1, we can prove (40). Actually, using part i) of Theorem 2.1 in [7], we can compute the exact asymptotics of T under more restrictive conditions. Theorem V.2 If = and F (x) P[M > x] satises ( ) 1 F (x) ex (e x x)1 , where () is regularly varying with index > 0, then, as t , P[T > t] ( + 1) . (t)
VI. S IMULATION E XAMPLES In this section, we illustrate our theoretical results with simulation experiments. In particular, we emphasize the characteristics of the studied ALOHA protocol that may not be immediately apparent from our theorems. For example, in practice, the distributions of packets and number of random users might have bounded supports. We show that this situation may result in truncated power law distributions for the transmission delays. To this end, it is also important to note that the delay distribution has a power law main body with a stretched support in relation to the support of L and M and, thus, may result in very long, although, exponentially bounded delays. Example 1 (Finite population model) For the nite population model described in Subsection II-A, we compare the starting and steady state behavior in this experiment.
0
with mean 1 and the arrival intervals and backoffs follow an exponential distribution with mean 2/3. The simulation experiments that each repeatedly measure 105 samples are shown in Figure 3, which indicates a power law transmission delay. We can see from the gure that, as M gets large (M = 10, 20), the slopes of the distributions that represent the power law exponents on the log / log plot are essentially the same, as predicted by our Theorem IV.1. Next, we compare the starting behavior with the steady state behavior predicted by Theorem IV.2. In this setting, we set M = 3 and choose i.i.d. packet sizes that follow an exponential distribution with mean 1. In addition, we assume that arrival intervals and backoffs are exponential with mean 1.5. The starting behavior is represented by repeatedly measuring 105 number of the transmission times for the rst packet (m=1) in a system that is initially empty and the steady state distribution is obtained by continuously measuring the transmission times of the packets with indexes from m = 105 to m = 107 . The plot in Figure 4 shows that the transmission time distribution of the rst packet for the starting behavior has a slope M /((M 1)) = 2.25, and the steady state transmission time distribution has a slope /((M 1)) = 0.75, as predicted by equations (29) and (37) in the log-log scale, respectively.
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Fig. 4. Comparing starting behavior and steady state behavior for nite population ALOHA with variable size packets.
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Fig. 3. Starting behavior: transmission time distribution for the rst successfully transmitted packet for nite population ALOHA with variable size packets.
First, we verify Theorem IV.1 on the starting behavior by plotting the empirical distribution of time T1 for the rst successful transmission in a system that is initially empty. In this regard, we conduct four experiments for M = 2, 4, 10, 20 users, respectively. The packets are assumed i.i.d. exponential
Example 2 (Random number of users) As stated in Section V, the situation when the number of users M is random may cause heavy-tailed transmission delays even for slotted ALOHA. However, in many practical applications the number of active users M may be bounded, i.e., the distribution P[M > x] has a bounded support. Thus, from equation (42) it is easy to see that the distribution of T is exponentially bounded. However, this exponential behavior may happen for very small probabilities, while the delays of interest can fall inside the region of the distribution (main body) that behaves as the power law. This example is aimed to illustrate this important phenomenon. Assume that initially M 1
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Fig. 5. Illustration of the stretched support of the power law main body when the number of users is min(M, K), where M is geometrically distributed.
users have unit size packets ready to send and M follows geometric distribution with mean 3. The backoff times of the colliding users and the arrival intervals of the new packets are independent and geometrically distributed with mean 2. We take the number of users to have nite support [1, K] and show how this results in a truncated power law distribution for T in the main body, even though the tails are exponentially bounded. This example is parameterized by K where K ranges from 6 to 14 and for each K we set the number of users to be equal to MK = min(M, K). We plot the distribution of P[T > t], parameterized by K, in Figure 5. From the gure we can see that, when we increase the support of the distributions from K = 6 to K = 14, the main (power law) body of the distribution of T increases from less than 5 to almost 700. This effect is what we call the stretched support of the main body of P[T > t] in relation to the support K of M . In fact, it can be rigorously shown that the support of the main body of P[T > t] grows exponentially fast. Furthermore, it is important to note that, if K = 14 and the probabilities of interest for P[T > t] are bigger than 1/500, then the result of this experiment is basically the same as for K = ; see Figure 5. VII. C ONCLUDING R EMARKS AND F URTHER E XTENSIONS In this paper, we show that a basic nite population ALOHA model with exponential packets is characterized by power law transmission delays, possibly even resulting in zero throughput. Based on these results, we establish a new stability condition that is entirely derived from the tail behavior of the packet and backoff distributions. Note that at any moment of time the nite population ALOHA model from Subsection II-A( can be described ) as (t) (t) (t) a Markov process for the state vector L1 , L2 , , LM , where Li is the packet size of user i at time t. However, this Markov process is not easy to analyze in the sense that it has innitely, possibly uncountably, many states with complicated transitions, where long packets tend to accumulate in the
(t)
system since the short ones are easier to pass. Hence we conjecture, based on our initial simulation experiments, that in the steady state the system may have multiple functional forms for the power law exponent for different values of , and . The complete characterization of the stability of this Markov process and the full understanding of the spatial interactions and temporal correlations of packet sizes in the system remain a challenging problem. In this paper, we provide a partial picture of the system behavior. Furthermore, from an engineering perspective, it is important to study more sophisticated MAC protocols, including CSMA and RTS/CTS scheme, since ALOHA represents the basis for these more practical MAC protocols. This power law effect and the possible instability for our ALOHA model might be diminished, or perhaps eliminated, by reducing the variability of packets. However, we show that even a slotted (synchronized) ALOHA with packets of constant size can exhibit power law...