IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 36, NO. 5 , SEPTEMBER

REFERENCES [ I ] A. J. Rainal, “Origin of Rice’s formula,” IEEE Trans. Inform. Theory, vol.

[2] S . 0. Rice, “Distribution of the duration of fades in radio transmission,”

[3] -, “Mathematical analysis of random noise,” Bell Syst. Tech. J . , vol.

[4] A. J. Rainal, “First and second passage times of sine wave plus noise,”

[51 D. Slepian, “On the zeros of Gaussian noise,” in Time Series Analysis, M.

[6] M. Kac and D. Slepian, “Large excursions of Gaussian processes,” Ann.

IT-34, pp. 1383-1387, Nov. 1988.

Bell Syst. Tech. J . , vol. 37, pp. 581-635, May 1958.

24, pp. 46-156, Jan. 1945.

Bell Syst. Tech. J . , vol. 47, pp. 2239-2258, Dec. 1968.

Rosenblatt, Ed.

Math. Statist., vol. 30, pp. 1215-1228, Dec. 1959.

New York: Wiley, 1963, ch. 6, pp. 104-1 15.

On the Cutoff Point for Pairwise Enabling for Infinite Users in a Multiaccess System

3’. c. YAO AND F. K. HWANG

Abstract-Consider a multiaccess channel shared by an infinite set of users each of which, independently, has a message to transmit with probability p . Pairwise enabling is the scheduling algorithm which enables the users to transmit by pairs, and if a collision occurs, lets the two users transmit separately in the next two time slots. Molle showed that pairwise enabling is optimal for 0.5 s p I 1/@ and is not optimal for p s 0.430. Let p’ denote the cutoff point for pairwise enabling, i.e., pairwise enabling is optimal for p ’ ~ p i I / f i hut not optimal for p < p’. In this correspondence Molle’s results are improved by showing that 0.4300 < p’ < 0.4745. In addition, we show that 0.4400 < p o holds when we assume that the 1-feedback also reveals the identity of the transmitter.

I. INTRODUCTION

Consider an infinite set of users (labeled by the set of natural numbers) who share a multiaccess channel for communication. Each user is assumed to be active, i.e., having a message to transmit, independently with probability p . Such a set of users is called an infinite Bernoulli sequence by Molle. The channel is a slotted ternary feedback channe7, i.e., at any given time slot the channel can be in one of the three states, 0,1, e. It is in state 0 if no user transmits. It is in state 1 if exactly one user transmits (the transmission is considered successful). It is in state e if more than one user transmit simultaneously, in which case the transmissions collide and have to be rescheduled. Define throughput as the proportion of time the channel is in state 1. The problem is to schedule the users for transmissions (we do not know in advance who are the active users) to maximize the throughput. The maximum throughput is called the channel capacity and denoted by C,.

Define individual enabling as the scheduling algorithm which enables users to transmit one by one. Define painvise enabling as the algorithm that enables users to transmit by pairs, and if a collision occurs (when both users are active), then lets the two users transmit separately in the next two time slots. Molle [5] proved that individual enabling is optimal for p 2 1/fi and not optimal for p < l/&. In a recent interesting paper [6] he further proved that painvise enabling is optimal for 1/2 I p I 1/ fi and not optimal for p I 0.430. Therefore the cutoff point p’ for painvise enabling lies between 0.430 and 0.5. In this corre-

Manuscript received July 21, 1988; revised January 16, 1990. Y. C. Yao is with the Department of Statistics, Colorado State University,

F. K. Hwang is with the AT&T Bell Laboratories, 2C379, Murray Hill, NJ

IEEE Log Number 9036191.

Fort Collins, C O 80523.

07974.

1990 1183

spondence we reduce the upper range to 0.4745 by sharpening Molle’s methods. By assuming that the 1-feedback also reveals the identity of the transmitter, as assumed in [1]-[3] but not by Molle, we increase the lower range to 0.4400.

11. THE GENERAL APPROACH

Call a user satisfied if the user is included in an enabled set which contains at most one active user. Molle [6], using an idea of Mikhailov and Tsybakov [4] first proposed for the Poisson model, scores an enabling by giving 1 point to each inactive user satisfied and 1 + q points for each active user satisfied. Let H ( n , q ) denote an upper bound (to be specified later) for the expected score when a set of n users is enabled. Consider a set of M Bernoulli users each of which has probability p of being active. Then there is an expected number of M(1+ p q ) points to be scored when all users in that set are satisfied. Thus for any q 2 0

w 1 + P 4 )

max H ( n , 9) E, =

n

is a lower bound on the expected number of enablings to satisfy a Bernoulli set of M users, while

Mp mnaxH(nd?) _- - E, l / p + q

is an upper bound on the capacity C,. This upper bound appears in Theorem 1 of Molle [6], which also holds for M + m.

iMolle derived

Define H,(n, 4) = n(1- p)”-’[l- p + ( n + q)p]. He proved the following.

Lemma 1: For 0 I q I 1, p 2 q / ( l + q) , and n 2 2, H ( n , q ) = H J n , q) .

Lemma 2: For all n 2 1 , p 2 1 / 4 and O r q r 1, H,(n ,q) at- tains its maximum at the least n for which H,(n ,q) > H,(n + 1,q).

Using these two results and setting

1-2p2 q = 1 - 2 p ( l - p )

he showed that H ( n , q ) r 1 + q and C, I 2p/(l +2p2) for 0.5 - < p I l / f i . Since 2p / ( l +2p2) is the throughput for painvise enabling, it must be optimal for p in this range.

Note that the existence of a high scoring enabling results in a weak lower bound for the expected number of enablings, and consequently, a weak upper bound on the capacity. In particu- lar, enabling a pair of Bernoulli items is such a high scoring enabling and essentially determines Molle’s upper bound 0.5 of p’. Our idea is to mix each high scoring enabling with some low scoring enablings and thus bring down the expected score per enabling when the enablings are analyzed as a block. We also need to reset the q value to bring down the score of enabling a single user known to be active.

In the next section, we need the following stronger version of Lemma 2, the proof of which is implicitly given in [6].

Lemma 2’: For p 2 1/4, 0 5 q I 1, (1 - p)’-“{H,(n + 1,q)- H J n , q) } is monotonically decreasing in n.

We also need the following lemma (see Lemma 1 of [6]). Lemma 3: If, at some algorithmic step, r users are selected

for transmission, none of whom were selected at a previous step

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1184 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 36, NO. 5, SEPTEMBER 1990

that resulted in either an idle slot or a successful transmission, then, no matter what history led up to the selection of those r users,

P{none of the r users are active} = Z, I (1 - p)‘

v(1- P)‘- 1 - ( 1 - P )

P(exact1y one of the r users is active} I r . ( l - Z r ) .

111. THE OFTIMALITY OF PAIRWISE ENABLING FOR 0.4745 I p I 0.5

Without loss of generality, we consider only algorithms in which no enabling includes some already satisfied users, and no enabling is performed when it is known to contain at least two active users.

Any given algorithm can be represented as a rooted tree in which each node is associated with an enabling and the outlinks from a node correspond to the possible outcomes of the associ- ated enabling. Each enabling is assigned a natural number in such a way that the first enabling (i.e., the root of the tree) is assigned the number 1, and the number of an enabling is less than that of another whenever the former is performed before the latter.

We are now ready to define “blocks” of enablings recursively. Let enabling i be a highest level enabling which has not been included in any block. If the size of enabling i is not 2, then let this enabling itself form a block. If the size is 2, then the new block will contain enabling i and at least two of its descendents. To define the new block rigorously, we note that the outcome of enabling i could be either “at most one active user” or “two active users.” In the case of “at most one active user,” both users are satisfied so that neither will appear in any subsequent enablings. In the case of “two active users,” there must be at least two further enablings, each intersecting with enabling i in one user, on every path of the rooted tree starting from enabling i . We will call the first such enabling a first intersecting enabling (with enabling i ) and the second a second intersecting enabling. Denote by 1,; . ., 1, all the first intersecting enablings with enabling i, and by m l ; * . , m k all the second intersecting en- ablings with enabling i . We now build a new block consisting of enablings i , l , ; . . , l j , m , ; . ‘, m k .

Note that there may be more than one first intersecting enabling. As we commented before, in the rooted tree, below the outlink “two active users” from the node “enabling i,” there must be at least one first intersecting enabling and one second intersecting enabling, so that a block led by an enabling of size 2 has at least three enablings.

Furthermore, note that none of the enablings 1,; . , I j , m , ; . ., m k can be included in a previous block. Suppose to the contrary that one of these enablings, say, 1 is already included in a block starting with an enabling j of two active users. Then j must be an ancestor of i and 1 must intersect j. If i also intersects j , then i, hence I , cannot be a first or second inter- secting enabling with j since i is not in the block of j. If i does not intersect j , then I contains two known active users, one from i and one from j. This contradicts our assumption that no enabling is performed when it is known to contain at least two active users. Therefore, each enabling belongs to one and only one block. Let Q = p/( l - p ) so that 0 I q I 1 for 0 I p I 0.5.

Lemma 4: For 0.4145 I p I 0.5 and for each block, the ex- pected score per enabling is bounded by

2( 1 + P4) 2( 1 - P + P 2 ) -=

1 + 2 p 2 ( 1 - p ) ( 1 + 2 p 2 ) .

Pro08 If the block consists of an enabling of size 1, then the expected score for this enabling equals

I , + ( l - I , ) ( l + q ) = 1+ q - qz,

< l + q I - 2(1 + w, since p I 0.5, - 1 + 2 p 2 ’

where I , is the (conditional) probability that the user is inactive. If the block consists of an enabling of size r 2 3 and I , is the (conditional) probability that all the r users are inactive, then by Lemma 3 the expected score is bounded by

I H ( r , q ) , s i n c e O I Z r I ( 1 - p ) ‘

= H , ( r , q ) , by Lemma 1

2(1+ P4) I H,(3 ,q ) I ~

1 + 2p2

Here, the last inequality follows from p 2 0.4745. To prove the second to last inequality, note that H,(3, q ) > H1(4, q ) for 0.4745 I p 5 0.5 by direct verification. By Lemma 2’, for n 2 3,

( l - p ) ’ - “ { H , ( n + l , q ) - H , ( n , q ) }

- < (1 - p ) - * { H I ( 4 , 4 ) - Hl(3,q)) < 0.

So, H J r , q ) I H1(3, q ) for r 2 3. Finally, consider a block led by an enabling i of size 2 and followed by enablings 1,; . .,I,, m , ; . . , m k . Let Z,, I;,Zi) be the (conditional) probabilities that the number of active users in enabling i equals 0, 1, and 2, respectively. We have, by Lemma 3

z, + I; + I $ = 1 ,

2 P ( l - P ) 1 - ( 1 - p)2

z2 I ( 1 - p)2, z; I ( 1 - Z,)

Given that enabling i is reached, the expected number of enablings in the block to be performed equals 1 + 2Zi), since exactly one of enablings 1,; . ., l j and exactly one of enablings m , , . . . , m k will be performed when enabling i consists of two active users. Consider the case that enabling i consists of two active users. Then any subsequent enabling of size r 2 1 that intersects enabling i in one user and also contains r - 1 other users not in enabling i has the expected score

( r + q ) (conditional probability

that the r - 1 users are all inactive)

- < ( r + q ) ( l - p)‘-’ I I + q ,

since (2+ qX1- p ) I 1 + q , and

( r + I + q ) ( l - p)‘

( r + q ) ( l - P ) ‘ ~ ’ r + q

( r + 1 + q ) ( l - P I - -

( r + 1)(1-0.4) < I , f o r r l 2 . - <

r

So, the expected total score of enablings in the block is bounded by

21, + ( 2 + q ) Z ; + 1;(2(1+ 4)).

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 36, NO. 5, SEPTEMBER 1990 1185

Hence, the expected score per enabling for the block is bounded by

21, + ( 2 + q ) I i + 2( 1 + q ) I i ’ 1+2Ii‘

2 I2 + (2 + q ) 1; - ( 1 + q ) 1+21i’

= l + q +

1 - q = l + q + -

1 +21$

1 - q 2 ( 1 - p + p 2 ) I l + q + - - - - - - -

1 + 2.p’ (1 - p)( 1 + 2p2)

where the last inequality follows since

2 - 2 p

2 - p 2 - p 2 - P 2 - P - 1 - p2 . 5 -( 1 - p ) 2 + - -

2 - 2 P P P = -1‘ + -

This proves Lemma 4. 0

Theorem 1: Pairwise enabling is optimal for 0.4745 I p I 0.5.

Proof: From Lemma 4 and the discussion in Section 11, the capacity is upper bounded by

2(1+ p q ) / ( l + 2 P 2 ) 2P =-

l / P + 4 1+2p2

which is the throughput of pairwise enabling. Theorem 1 follows immediately.

1 Fig. 1. Algorithm T .

feedback is 1, we also know who the transmitter is. To establish the nonoptimality of painvise enabling for small p , it seems natural to compare it with triplewise enabling algorithms. Among the many different triplewise enabling algorithms, the algorithm T defined in Fig. 1 compares favorably with painvise enabling when p I 0.4400. This algorithm takes advantage of the infor- mation about the transmitter in the case of 1-feedback.

In Fig. 1, the first four users are denoted by A , B , C, and D. Each circled node is labeled by the set enabled at that stage. Outlinks from a node are labeled by the various outcomes of the enabling. An unlabeled outlink means that it applies to all outcomes. T means that we apply the same algorithm to the remaining unsatisfied users. It is straightforward to verify that the throughput of T is

4p - (1 - ~ ) ~ p - 3( 1 - p ) 2 p 2

1. [ ( 1 - p)3+3(1- p ) * p ] + 4 . ( 1 - p ) 2 p 2 + 3 . 2 ( 1 - p y p 2 +5.(1- p ) p 3 + 4 . ( 1 - p ) p 3 + 5 . 2 ( 1 - p ) p 3 + 7 . P 4

Note: The value 0.4745 is determined by solving H,(3, q) I 2(1+ pq)/( l+ 2p2) to obtain which is greater than 2 p / ( l + 2 p 2 ) if and only if

p 3(1- p)2(1 + 2 p ) ( 1 +2p2) - 2

p ( 2 - 3( 1 - p)’( 1 +2P2)] 4 = - 2

1 - P

i.e.,

A ( p ) = 2( p2 + 1 - p) 2 3(1- p)’( 1 +2p2)( 1 + p - p 2 )

= B( p ) .

It can be shown that A ( p ) < B ( p ) at p = 0.4744 and A ( p ) > B ( p ) at p = 0.4745. Also A ’ ( p ) > B ’ ( p ) for 0.474 < p < 0.5. (Actually, A‘(p ) > - 0.104 > - 2.5 > B‘(p ) for 0.474 < p < 0.5).

0

IV. THE NONOPTIMALITY OF PAIRWISE ENABLING FOR p I 0.4400 WHEN THE FEEDBACK REVEALS THE

IDENTITY OF THE TRANSMITTER

We now assume that a transmission always includes the transmitter’s identity as assumed in [1]-[3]. Thus when the

1 - 5 p 2 -4p3 + lop4 -4p5 > 0.

Computations show that this inequality holds for p I 0.4400.

V. SOME CONCLUDING REMARKS

The high-scoring blocks that essentially determine our upper bound 0.4745 for p o are those enablings on triples. Thus it seems that we might be able to combine such an enabling with some subsequent enablings into a block to lower the expected score per enabling just as we did for a pair. While the principle is sound, there is a difficulty which does not show in the pair case. Namely, we cannot rule out the possibility that a subse- quent enabling may intersect with more than one triple with feedback e. Then we are faced with the problem of “to which block such an enabling should be assigned.” However, we be- lieve that the true value of p o is closer to its lower range than its upper range.

1186 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 36, NO. 5, SEPTEMBER 1990

Incidentally, it may be tempting to establish similar optimality results for the nonhomogeneous (nonidentical p’s) case. How- ever, painvise enabling is no longer unique, depending on how the different p’s are paired. This nonuniqueness makes the nonhomogeneous case more difficult to analyze. Nevertheless, it seems reasonable to conjecture that the optimal algorithm is a pairwise enabling algorithm whenever all the p’s are between p” and l/&.

and let X ( t ) (-m < t <a) be a wide sense stationary process with covariance function R(t - s) of the form (0 < 6 < T )

~ ( t - s> = jT-’ e i ( ‘+S)cdF( t ) . - 7 r + 6

Then if D < i, with probability one we have

REFERENCES R. W. Chen and F. K. Hwang, “k-definite group testing and its applica- tion to polling in computer networks,” Congressus Numerantium, vol. 47, pp. 145-159, Dec. 1985. F. K. Hwang and X. M. Chang, “Cutoff points for roll call protocols in multiple access systems,” IEEE Trans. Inform. Theory, vol. IT-33, pp. 577-581, July 1987. n = l 1. Kessler and M. Sidi, “Mixing collision resolution algorithms exploiting information of successful messages,” IEEE Trans. Inform. Theory, to appear. V. A. Mikhailov and B. S . Tsybakov, “An upper bound to capacity of a random multiple access system,” Probl. Peredach. Inform., vol. 17, pp. 90-95, Jan.-Mar. 1981. M. L. Molle, “On the capacity of infinite population multiple access protocols,” IEEE Trans. Inform. Theory, vol. IT-28, no. 2, pp. 396-401, May 1982. M. L. Molle, “On the cutoff point for pairwise enabling in multiple access systems,” preprint. n = - N

where

For any D < (1) converges uniformly in mean-square on com- pact subsets of the t-axis.

Proof: For any positive integer N define the process

G ( t ) X(tn) Gr(tn)(t -

N

x N ( t ) = .

We then have

A Note on Sampling of Bandlimited Stochastic Processes

KRISTIAN SEIP

Abstract --It is pointed out that an irregular sampling theorem of the author can be used to generalize the results of a paper by Piranashvili on interpolation of stochastic processes.

For any real 5 f,(‘) = cif' belongs to the Banach space g,[, of [3] with I l f s l l = 1. Thus from the concluding estimate in the proof

The paper [2] of Piranashvili illustrates how the Shannon sampling theorem by simple arguments yields a sampling expan- sion for a very general class of bandlimited processes (the fact that Piranashvili’s processes are indeed bandlimited is estab- lished by Lee in [l]). The purpose of this note is to point out that all the interpolation results of [2] can be generalized by using the sampling theorems of [3].

We shall confine ourselves to discuss only the important case of wide sense stationary processes. The general case is slightly more technical, but i t is covered in the same simple way as shown next, that is by substituting an estimate for the cardinal series by an estimate from [3]. The interested reader may consult [2] and [3] for details and [l] for the interpretation of Piranashvili’s processes as bandlimited.

The generalization of Corollary 3 of [2] reads as the following

of Theorem 2 of [3] we have

E [ I X ( t > - XN( t> I?]

for N > It1 with C some absolute constant. This inequality en- sures the mean-square convergence. Since it implies

c E [ I X ( t ) - XN(t)I’] Crn N > It/

when D < i, we find that the proof is complete. 0

theorem. Remark: Alternatives to (I) with faster convergence can be obtained from Theorem 3 of [3], see the remark following Theorem 1: Let ( t n ) f l E Z be a sequence of real numbers such

that Corollary 3 of [2].

REFERENCES [ I ] A. J. Lee, “On band-limited stochastic processes,” SIAM J. Appl. Math.,

vol. 30, no. 2, pp. 269-277, 1976. [2]

[3]

Z. A. Piranashvili, “On the problem of interpolation of random pro- cesses,” Theory Prob. Appl., vol. 12, pp. 647-657, 1967. K. Seip, “An irregular sampling theorem for functions bandlimited in a generalized sense,” SIAM J. Appl. Math., vol. 47, no. 5, pp. 11 12-1 116,

Manuscript received October 15, 1989; revised February 12, 1990. The author is with the Division of Mathematical Sciences, University of

IEEE Log Number 9036389. 1987. Trondheim, N-7034 Trondheim-NTH, Norway.

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