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User Capacity Scaling Laws for Fading

Multiple-Access Channels

Hengameh Keshavarz, Liang-Liang Xie, and Ravi R. Mazumdar

Department of Electrical & Computer Engineering,

University of Waterloo, Waterloo N2L 3G1, Canada

e-mail: hkeshavarz@ieee.org; {llxie, mazum }@ecemail.uwaterloo.ca

Abstract

In this paper, the maximum number of simultaneous active transmitters (i.e. user capacity) is

obtained in in the many user case for fading multiple-access channels in which a minimum rate must be

maintained for all active users. The results are presented in the form of scaling laws as the numbers of

transmitters increases and three commonly used fading models, namely, Rayleigh, Rician and Nakagami

are considered. It is shown that in all three cases the capacity scales double logarithmically in the number

of users and differ only by constants that depend on the distributions. We also show that a scheduling

policy that maximizes the number of simultaneous active transmitters can be implemented in a distributed

fashion.

I. INTRODUCTION

In a wireless environment channel states vary dynamically and users experience different

fading conditions. As a result, some users have high channel gains while other users experience

poor channel conditions. Delay-sensitive applications such as video and voice need users to

maintain a minimum rates. Due to limited transmission power, it is not always possible for all

users to maintain a minimum rate. A reasonable strategy is to allow users with good channel

conditions to be active while others remain silent during each time slot. This is often referred

to as opportunistic scheduling. Hence it is desirable to have an opportunistic scheduling policy

that maximizes the number of active users which can be meet the minimum-rate constraint.

In broadcast channels, the transmitter can allocates its total transmit power to different re-

ceivers according to their channel states. There is thus a fundamental trade-off between the total

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throughput and the minimum rate achievable for all the receivers. The basic idea is to adapt

power allocation to the variations of the channel states. The transmission rate for a receiver is

increased when its channel state becomes better, thus higher rates can be achieved at the expense

of less power. This raises the issue of the trade-off between ergodic capacity and outage capacity,

for which, extensive studies have been given in [1], [2], [3] in the context of broadcast channels.

In [4], a power allocation scheme is considered to maximize the number of active receivers

maintaining a minimum rate while allocating no power to the other inactive receivers. Under

the assumption of independent Rayleigh fading channels for different receivers with the channel

gain distribution hi∼ CN(0,1), it is shown that the maximum number of active receivers is

asymptotically with probability converging to 1 given by ln(P lnn)/Rmin, where P is the total

transmit power and Rmin denotes the pre-set minimum rate (in nats), as the total number of

receivers n goes to infinity.

User capacity of CDMA cellular networks with different assumptions have been investigated

in the literature. In [5], downlink user capacity in a single-cell with successive interference

cancellation is asymptotically derived using order statistics. In [5], user capacity is defined as

the expected number of simultaneous receivers. They present a complicated expression in terms

of the normal distribution function for the downlink user capacity. In [6], capacity and asymptotic

user capacity (the limit when spreading gains tend to infinity with the constraint of finite system

load) is considered for a single-cell CDMA uplink with successive interference cancellation.

In multiple-access channels, the total throughput increases by the number of active transmitters.

As in the downlink case, although delay-sensitive applications need transmitters to maintain a

minimum rate, it is not always possible for all users to keep this minimum rate due to limited

transmission power. In this paper, user capacity of fading multiple-access channels in which a

minimum rate must be maintained for all active transmitters is analyzed in the case of many

users. The joint decoding scheme is used at the receiver since it is well known that this decoding

scheme maximizes the total transmitted rate. In this case, messages sent by all transmitters are

simultaneously decoded at the receiver. While the number of active transmitters in each slot

depends on the specific channel states, the asymptotic behavior is can be precisely characterized

when the total number of transmitters n is large. Three fading distributions, Rayleigh, Rician, and

Nakagami are considered. These cover the commonly used models for wireless communication

channels. For example, if there are multiple indirect paths between transmitter and receiver,

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with no distinct dominant path, Rayleigh fading is appropriate from the central limit theorem.

If there is a dominant component, say line-of-sight (LOS), in addition to indirect paths, the

Rician distribution is appropriate. Nakagami fading occurs in the case of relatively large delay-

time spreads, with different clusters of reflected waves. Within any one cluster, the delay times

are approximately equal for all waves, and as a result the envelope of each cumulated cluster

signal is Rayleigh distributed. Since the average time delay differs significantly between clusters,

Nakagami fading follows from a sum of multiple Rayleigh-faded signals [8].

Under the assumption of independent Rayleigh fading channels for all transmitters, it is shown

that the maximum number of active transmitters, Mn, is of the order lnlnn. Moreover, we show

that there is a sharp concentration effect in the probability distribution of Mn; in other words, the

lower bound of Mndiffers from the upper bound by ǫ which can be made arbitrarily small. The

results obtained for Rayeigh fading are also generalized to Rician and Nakagami fading. It is

shown that user capacity of these fading channels also scales double logarithmically with the total

number of transmitters in the system. The difference between user capacity of Rayleigh fading

channels and Rician or Nakagami fading channels is just a constant shift. Note that because of

joint decoding at the receiver and individual power constraints at each transmitter, the duality

(see [7]) of broadcast and multiple-access channels can not be simply used to extend the results

in [4] to the uplink, although the asymptotic results follow the same scaling law.

This paper is organized as follows: In section II, the system model is introduced. In section

III, the power allocation scheme maximizing the number of active transmitters in the system is

proposed. In section IV, the asymptotic analysis of user capacity is presented in Rayleigh fading

channels and these results are extended to Rician and Nakagami fading channels in section V.

Finally, simulation results are shown in section VI.

II. SYSTEM MODEL

Consider a multiple-access channel with one receiver and n transmitters. The receiver and

all transmitters are each equipped with single antenna and are assumed know channel state

information (CSI) perfectly. The received signal is represented by

Y (t) = hi(t)Xi(t) + Z(t) ;i = 1,··· ,n

(1)

where Xi(t) denotes the ith user transmitted signal, Y (t) refers to the received signal, Z(t) ∼

CN(0,σ2), and hi(t) denotes the time-varying channel gain of the path from transmitter i to

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) t ( Y

2

) t (

X

. . .

) t (

Ζ

1

t (X)

n

t ()X

1

) t (

h

n

) t (h

2

) t (

h

. . .

Fig. 1.Fading multiple-access channel

the receiver and t is the time index. hi, i = 1,...,n are assumed to be constant during each

time block. Without loss of generality, assume that |h1| ≤ |h2| ≤ ··· ≤ |hn|. It is assumed that

joint decoding is exploited at the receiver. This decoding scheme maximizes the sum rate and

achieves a set of rates satisfying the following conditions.

Ri ≤ ln

?

?

1 +Pi|hi|2

σ2

?

;i = 1,...,n

(2)

Ri+ Rj

≤ ln

...

1 +Pi|hi|2+ Pj|hj|2

σ2

?

;i,j = 1,...,n

(3)

n

?

i=1

Ri ≤ ln

?

1 +

?n

i=1Pi|hi|2

σ2

?

(4)

where Riand Pidenote the ith user’s achievable rate (in nats) and transmitted power respectively.

III. SCHEDULING POLICY

As in the case of broadcast channels, to decrease delay in a multiple-access channel, a

minimum rate constraint could be considered for all active transmitters. That is, each transmitter

maintains a minimum rate or remains silent during each time slot. Due to time-varying channel

states and limited transmitted power, it is not always possible for all transmitters to keep

Rmin. Hence, the following scheduling policy is proposed to maximize the number of active

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transmitters.

max{m}

Ri≥ Rmin,

Pi= P,

(5)

i = n − m + 1,...,n

i = n − m + 1,...,n

(6)

(7)

That is, users with high channel gains are allowed to transmit data while other transmitters are

inactive. As messages sent by all active transmitters are decoded simultaneously, each user’s

signal is not affected by interference from other active transmitters. Hence, all users are allowed

to transmit data with maximum power. For simplicity, it is assumed that all transmitters have

the same power constraint; however, different individual power constraints can be considered

without much difficulty.

With fixed P and Rmin, the maximum number of active transmitters completely depends

on the channel gains hi, i = 1,...,n. In general these are not known a priori since they are

random. However when the distribution of channel gains is known, the asymptotic behavior of

the maximum Mncan be obtained when the total number of transmitters is large enough. This

asymptotic behavior is determined even without knowledge of the exact channel gains for each

transmitter, although channel gains are required for scheduling active transmitters. The practical

implementation issues and a decentralized scheduling policy for systems with a large number of

transmitters will be discussed in Section VI.

IV. ASYMPTOTIC ANALYSIS

We use the following standard notation in the sequel o(·), O(·), and ∼ to denote the follow-

ing: for any positive infinite sequences f(n) and g(n), n = 1,2,..., f(n) = o(g(n)) means

f(n)

g(n)= 0; f(n) = O(g(n)) means limsup

n→∞

We begin with the case of Rayleigh fading. Let Mndenote the maximum number of simulta-

lim

n→∞

f(n)

g(n)< ∞; f(n) ∼ g(n) means lim

n→∞

f(n)

g(n)= 1.

neous active transmitters (out of n transmitters) that can be supported with a rate greater than or

equal to Rmin. Note, Mnis random which depends on the channel gains. Consider independent

Rayleigh fading channels for different transmitters, i.e., the channel gains hi, i = 1,...,n are

independent realizations of the complex Gaussian distribution; as a result, |hi|2, i = 1,...,n are

independent realizations of the exponential distribution.

Theorem 4.1: Under the assumption of independent Rayleigh fading channels for different

transmitters with channel gains hi ∼ CN(0,1), i = 1,...,n , for any ǫ > 0, the maximum

March 20, 2008 DRAFT

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number of active transmitters, Mnsatisfies:

P{⌊ν(n) − ǫ⌋ ≤ Mn≤ ν(n) + ǫ} → 1,

as n → ∞,

(8)

where ⌊x⌋ denotes the floor of x, n is the total number of transmitters, and

ν(n) =

1

Rminln

?Pν(n)

σ2

lnn

?

.

(9)

Remark: In theorem 4.1, the channel gain variance equals one; however, this result can be easily

generalized to any arbitrary variance by normalization. That is, dividing the channel gain by its

variance results in a channel gain with the normal distribution.

Proof: Consider the multiple-access channel (1) with independent channel gains hi ∼

CN(0,1), for i = 1,...,n; as a result, |hi|2∼ Exponential(1). For any fixed h0 > 0, we

can characterize the number of “good” channels with |hi|2greater than h0as the following.

Let p0= 1−P(|hi|2≤ h0) = e−h0. That is, with probability p0, a channel is good (i.e. a user

can be activated). Consider a Bernoulli sequence:

xi=

1,

with probability p0

0,

with probability 1 − p0

(10)

for i = 1,2,...,n. Then, the number of transmitters having good channels has the same

distribution as Mn=?n

For any large integer m, if minn−m+1≤i≤n|hi|2= |hn−m+1|2≥ σ2emRmin/(mP),

?

?

...

?

Clearly, based on (2)-(4), the minimum-rate constraint is satisfied for all m active transmitters.

i=1xi, which satisfies the Binomial distribution B(n,p0).

ln1 +P|hi|2

σ2

?

?

≥ ln

?

?

1 +emRmin

m

?

≥ Rmin ;

?

i = 1,··· ,m

ln1 +P|hi|2+ P|hj|2

σ2

≥ ln1 + 2emRmin

m

≥ 2Rmin ;i,j = 1,··· ,m

ln1 +P?m

i=1|hi|2

σ2

?

≥ ln?1 + emRmin?≥ mRmin.

(11)

Next, we show that if m ≤ ν(n)−ǫ (see (9) for ν(n) definition), |hn−m+1|2≥ σ2emRmin/(mP)

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holds with probability approaching one. Let h0= σ2emRmin/(mP). Then,

p0= 1 − P(|hi|2≤ h0) = exp(−h0) = exp

?

?

?

−σ2emRmin

mP

−σ2eRmin(ν(n)−ǫ)

P(ν(n) − ǫ)

−λln(n)

?

∗

≥ exp

?

= exp

ν(n)

ν(n) − ǫ

?

∼ n−λ

(12)

where λ = e−ǫRmin< 1. Inequality ∗ holds for any sufficiently large m and n. Since np0≥ n1−λ

and m ≤ ν(n) − ǫ, it can be readily seen that as n → ∞,

1

2p0

n2np0

As m − 1 ≤ np0, using the Chernoff bound on the sum of independent Poisson trials [9, page

70],

(np0− m + 1)2

∼n2p2

0

=np0

2

≥n1−λ

2

→ ∞.

(13)

P(Mn≤ m − 1) ≤ exp

?

−1

2p0

(np0− m + 1)2

n

?

,

(14)

or

P(Mn≥ m) ≥ 1 − exp

?

−1

2p0

(np0− m + 1)2

n

?

.

(15)

Then, from (13), the probability in (15) approaches one for any m ≤ ν(n)−ǫ. Thus, we proved

that as n → ∞, with probability approaching 1, there are at least Mn= ⌊ν(n)−ǫ⌋ transmitters

with |hi|2≥ σ2emRmin/(mP), for which the minimum rate constraint is satisfied and can be

activated.

Next, we prove Mn≤ ν(n) + ǫ holds with probability approaching one. First, we show that

for any δ > 0, the best transmitter should have the channel gain |hn|2≥´h0= σ2emRmin/(mPδ),

with Pδ= P + δ. Otherwise, if maxn−m+1≤i≤n|hi|2<´h0,

?

ln1 +

?n

i=n−m+1P|hi|2

σ2

?

< ln

?

?

?PemRmin

1 +mP´h0

σ2

?

(16)

= ln 1 +PemRmin

Pδ

?

∼ ln

Pδ

?

< mRmin

which violates (4). Hence, to show that

P(Mn≤ ν(n) + ǫ) → 1,

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or

P(Mn> ν(n) + ǫ) → 0,

we only need to show that

P

?

|hn|2≥σ2e(ν(n)+ǫ)Rmin

Pδ(ν(n) + ǫ)

?

→ 0.

Define p1 = 1 − P(|hi|2≤´h0) = e−´ h0with´h0 = σ2e(ν(n)+ǫ)Rmin/(Pδ(ν(n) + ǫ)). The

probability that all the transmitters have channel gains less than´h0equals (1 − p1)n. Hence,

?

which tends to 0 if and only if

P

|hn|2≥σ2e(ν(n)+ǫ)Rmin

Pδ(ν(n) + ǫ)

?

= 1 − (1 − p1)n,

(17)

?

1 − exp(−´h0)

?n

=

?

1 − exp

?

−σ2e(ν(n)+ǫ)Rmin

Pδ(ν(n) + ǫ)

??n

→ 1.

(18)

Since

?

1 − exp

?

−σ2e(ν(n)+ǫ)Rmin

Pδ(ν(n) + ǫ)

??exp

?

σ2e(ν(n)+ǫ)Rmin

Pδ(ν(n)+ǫ)

?

→ e−1,

(18) holds if

n · exp

?

−σ2e(ν(n)+ǫ)Rmin

Pδ(ν(n) + ǫ)

?

= n · exp

?

−P ln(n)ν(n)

λPδ(ν(n) + ǫ)

?

∼ n1−

P

λPδ → 0

which holds by choosing δ < (eǫRmin− 1)P. As ν(n) is given by a nonlinear fixed-point

equation, finding a closed-form expression for this function is complicated. However, ν(n) can

be computed by iterative fixed-point algorithms.

Remark 4.1: In [4], the maximum number of active receivers in broadcast channels with

Z(t) ∼ CN(0,σ2) was shown to be given by

1

Rminln

?P

σ2lnn

?

(19)

where P denotes total transmitted power. In multiple-access case with each transmitter having

power P, Pν(n) is total transmitted power of the system. Thus the result has the the same

interpretation as for broadcast channels, since (9) can be also written as

1

Rminln

?Ptotal

σ2

lnn

?

(20)

where Ptotaldenotes total transmitted power in the system.

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We can also use the above to characterize the convergence rate that is of importance in

determining how large n should be in order for the estimates to hold.

Corollary 4.1: The probabilities in theorem 4.1 converge to 1 at the following rates:

P(Mn< ⌊ν(n) − ǫ⌋) = o

?

exp

?

−n1−λ

2 + ˜ σ

??

(21)

and

P(Mn> ν(n) + ǫ) = o

?

n1−

1

λ(1+˜ σ)?

(22)

where λ = e−ǫRmin< 1, and ˜ σ > 0 can be arbitrarily small.

Proof: To prove (21), we only need to show that for m = ⌊ν(n) − ǫ⌋,

1

2p0

As m = o(n), for sufficiently large n,

(np0− m + 1)2

n

≥n1−λ

2 + ˜ σ

(23)

(np0)2

(2 + ˜ σ)np0

≤

1

2p0

(np0− m + 1)2

n

≤(np0)2

2np0

Hence, with the following modification, (23) is proved.

(np0)2

(2 + ˜ σ)np0

=n1−λ

2 + ˜ σ

To prove (22), noting (17),

P(Mn> ν(n) + ǫ) ≤ P

?

|hn|2≥σ2(e(ν(n)+ǫ)Rmin)

Pδ(ν(n) + ǫ)

?

?

?

?

?

= 1 −

1 − exp

?

−σ2(e(ν(n)+ǫ)Rmin)

Pδ(ν(n) + ǫ)

−P ln(n)ν(n)

λPδ(ν(n) + ǫ)

??n

= On · exp

?

?

??

= On1−

P

λPδ

= on1−

1

λ(1+˜ σ)?

.

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V. OTHER FADING CHANNELS

We now show that subject to constant shifts, theorem 4.1 is also true for Rician and Nakagami

fading models. Note that all these distributions have exponentially decaying tails and thus it is

reasonable to believe that the double logarithmic scaling law should hold for all distributions

with exponential tails.

A. Rician Fading Channels

Theorem 5.1: Under the assumption of independent Rician fading channels for different trans-

mitters with channel gains hi∼ CN(¯ µ,2), i = 1,...,n and for any ǫ > 0, the maximum number

of active transmitters, Mnis bounded as

P(⌊ν1(n) − ǫ⌋ ≤ Mn≤ ν1(n) + ǫ) → 1,

as n → ∞,

(24)

where n is the total number of transmitters, and

ν1(n) =

1

Rminln

?2Pν1(n)

σ2

lnn

?

.

(25)

Remark: In theorem 5.1, the channel gain variance equals two because the resulting distribution

(i.e. the non-central Chi-square distribution with two degrees of freedom) is easy to work with;

however, this theorem can be easily generalized to any arbitrary variance by normalization.

Proof: Consider multiple-access channel (1) with independent gains hi ∼ CN(¯ µ,2), for

i = 1,··· ,n; as a result, |hi| ∼ Rice(1, ¯ µ) and |hi|2∼ NCχ2

distribution with two degrees of freedom) with the cumulative distribution function

2(¯ µ2) (i.e. non-central Chi-square

FNCχ2

2(x;2, ¯ µ2) =

∞

?

j=0

e−¯ µ2/2(¯ µ2/2)j

j!

γ(j + 1,x/2)

Γ(j + 1)

where γ(a,x) and Γ(a) denote respectively the lower incomplete gamma and the gamma func-

tions and are defined as

Γ(a) =

?∞

?x

0

ta−1e−tdt

γ(a,x) =

0

ta−1e−tdt

(26)

If Γ(a,x) denotes the upper incomplete gamma function,

lim

x→∞γ(a,x) = lim

x→∞[Γ(a) − Γ(a,x)] = Γ(a).

(27)

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Furthermore, if a is an integer,

Γ(a,x) = (a − 1)!e−x

a−1

?

k=0

xk

k!

(28)

Defining Bernoulli random variable (10), we show that for any ǫ > 0, if m ≤ ν1(n) −

ǫ, minn−m+1≤i≤n|hi|2≥ σ2emRmin/(mP) holds with probability approaching one. Let h0 =

σ2emRmin/(mP). Then,

p0= 1 − P(|hi|2≤ h0) = 1 − FNCχ2

2(h0;2, ¯ µ2)

= 1 − FNCχ2

2(σ2emRmin

mP

2(σ2eRmin(ν1(n)−ǫ)

P(ν1(n) − ǫ);2, ¯ µ2)

∞

?

∞

?

;2, ¯ µ2)

⋄

≥ 1 − FNCχ2

= 1 − e−¯ µ2/2

j=0

(¯ µ2/2)j

j!Γ(j + 1)γ

?

j + 1,λlnn

ν1(n)

ν1(n) − ǫ

?

∼ 1 − e−¯ µ2/2

j=0

(¯ µ2/2)j

j!Γ(j + 1)γ (j + 1,λlnn).

(29)

Inequality ⋄ holds for any sufficiently large m and n. Using (27) and (28), it is clear that as

n → ∞,

p0 ≥ 1 − e−¯ µ2/2

j=0

∞

?

∞

?

(¯ µ2/2)j

j!Γ(j + 1)[Γ(j + 1) − Γ(j + 1,λlnn)]

= n−λe−¯ µ2/2

j=0

(¯ µ2/2)j

j!

j

?

k=0

(λlnn)k

and

1

2p0

(np0− m + 1)2

n

≈np0

2

≥n1−λ

2

e−¯ µ2/2

∞

?

j=0

(¯ µ2/2)j

j!

j

?

k=0

(λlnn)k→ ∞.

(30)

Hence, using (15), we prove that as n → ∞, with probability approaching 1, there are at least

Mn = ⌊ν1(n) − ǫ⌋ transmitters with |hi|2≥ σ2emRmin/(mP), for which the minimum rate

constraint is satisfied and can be activated.

Next, we prove the upper bound, i.e., Mn≤ ν1(n) + ǫ holds with probability approaching 1.

Similar to the proof of theorem 4.1, we only need to show that

P

?

|hn|2≥σ2(e(ν1(n)+ǫ)Rmin)

Pδ(ν1(n) + ǫ)

?

→ 0

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12

which tends to 0 if and only if

ng(n) = n(1 − FNCχ2

2(σ2(e(ν1(n)+ǫ)Rmin)

Pδ(ν1(n) + ǫ)

(¯ µ2/2)j

j!Γ(j + 1)Γ

;2, ¯ µ2))

= ne−¯ µ2/2

∞

?

∞

?

j=0

?

?

j + 1,P lnn

λPδ

ν1(n)

ν1(n) − ǫ

?

?

∼ ne−¯ µ2/2

j=0

(¯ µ2/2)j

j!Γ(j + 1)Γ

j + 1,P lnn

λPδ

= ne−¯ µ2/2

?clnn

j=0

?

(¯ µ2/2)j

j!Γ(j + 1)Γ

(¯ µ2/2)j

j!Γ(j + 1)Γ

?

j + 1,P lnn

λPδ

?

+

∞

?

j=clnn

?

j + 1,P lnn

λPδ

??

→ 0.

(31)

Assume c <

P

λPδis selected such that the following expansion of the incomplete gamma function

for large x can be applied to the first summation [10, page 263].

Γ(a,x) ∼ e−xxa−1

?

1 +a − 1

x

+(a − 1)(a − 2)

x2

+ ···

?

(32)

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Hence, using j! ≥ (j/2)j/2and defining j0= 2

?

(¯ µ2/2)j

j!Γ(j + 1)Γ

¯ µ2P

P−λPδ

?2,

ng1(n) = ne−¯ µ2/2

clnn

?

j=0

?

j + 1,P lnn

λPδ

?

∼ n1−

P

λPδe−¯ µ2/2

clnn

?

j=0

(¯ µ2P lnn

2λPδ)j

j!j!

≤ n1−

P

λPδe−¯ µ2/2

∞

?

j=0

?

??

?

??

∞

?

¯ µ2P lnn

2λPδ

?j

j!

j

2

?j

≤ n1−

P

λPδe−¯ µ2/2

j0

?

j=0

¯ µ2P lnn

2λPδ

?j

j!

j

2

?j

?

??

+n1−

P

λPδe−¯ µ2/2

j=j0

¯ µ2P lnn

2λPδ

?j

j!

j0

2

?j

∼ n1−

P

λPδe−¯ µ2/2

∞

?

P

λPδ)→ 0

j=0

?

(P−λPδ)lnn

2λPδ

j!

?j

= e−¯ µ2/2n

1

2(1−

(33)

March 20, 2008DRAFT

Page 14

14

which holds by choosing δ < (eǫRmin− 1)P. Using Stirling’s approximation (in asymptotic

equality ⊲) for sufficiently large clnn, the second summation in (31),

nh2(n) = ne−¯ µ2/2

∞

?

j=clnn

(¯ µ2/2)j

j!Γ(j + 1)Γ

?

j + 1,P lnn

λPδ

?

= n1−

P

λPδe−¯ µ2/2

∞

?

(¯ µ2/2)j

j!

j=clnn

(¯ µ2/2)j

j!

j

?

k=0

?

P lnn

λPδ

k!

?k

≤ ne−¯ µ2/2

∞

?

∞

?

∞

?

?∞

j=clnn

⊲∼ ne−¯ µ2/2

j=clnn

(e ¯ µ2/2)j

jj√2πj

∼ ne−¯ µ2/2

j=clnn

1

jj

≤ ne−¯ µ2/2

clnn

dx

xx

1

≤ ne−¯ µ2/2

(clnn)(clnn−2)

clnn

(clnn)(clnn)

?∞

clnn

dx

x2

= ne−¯ µ2/2

(34)

which tends to zero since

lnn + lnlnn

(clnn)ln(clnn)∼

lnn

(clnn)ln(clnn)→ 0.

(35)

Hence, according to (33) and (34), (31) holds by choosing δ < (eǫRmin− 1)P.

B. Nakagami Fading Channels

Theorem 5.2: Under the assumption of independent Nakagami fading channels for different

transmitters with channel gains |hi| ∼ Nakagami(µ,ω), i = 1,...,n and for any ǫ > 0, the

maximum number of active transmitters Mnis bounded as

P(⌊ν2(n) − ǫ⌋ ≤ Mn≤ ν2(n) + ǫ) → 1,

as n → ∞,

(36)

where n is the total number of transmitters, and

ν2(n) =

1

Rminln

?ωPν2(n)

µσ2

lnn

?

.

(37)

March 20, 2008 DRAFT

Page 15

15

Proof: In Nakagami fading channels, the cumulative distribution function of |hi|2is given

by

F(x;µ,ω) =γ(µ,µ

ωx)

Γ(µ)

where µ denotes the shape parameter and ω controls distribution spread. Defining Bernoulli

random variable (10), we show that for any ǫ > 0, if m ≤ ν2(n) − ǫ, minn−m+1≤i≤n|hi|2≥

σ2emRmin/(mP) holds with probability approaching one. Let h0= σ2emRmin/(mP). Then,

p0= 1 − P(|hi|2≤ h0) ≥ 1 − F

?σ2eRmin(ν2(n)−ǫ)

γ(µ,λlnn

P(ν2(n) − ǫ);µ,ω

ν2(n)

ν2(n)−ǫ)

Γ(µ)

?

= 1 −

∼ 1 −γ(µ,λlnn)

Γ(µ,λlnn)

Γ(µ)

n−λ(λlnn)µ−1

Γ(µ)

Γ(µ)

=

∼

.

(38)

Using the fact lnn = o(nǫ) for any ǫ > 0, it is clear that as n → ∞,

1

2p0

n

Hence, by (15), we prove that as n → ∞, with probability approaching 1, there are at least

Mn = ⌊ν2(n) − ǫ⌋ transmitters with |hi|2≥ σ2emRmin/(mP), for which the minimum rate

constraint is satisfied.

(np0− m + 1)2

≈np0

2

≥n1−λ(λlnn)µ−1

2Γ(µ)

→ ∞.

(39)

Next, we prove the upper bound, i.e., Mn≤ ν2(n) + ǫ holds with probability approaching 1.

Similar to the proof of theorem 4.1, we only need to show that

P

?

|hn|2≥σ2(e(ν2(n)+ǫ)Rmin)

Pδ(ν2(n) + ǫ)

?

→ 0.

March 20, 2008DRAFT