Delay Analysis for Fair Power Allocation Strategyin Multi Antenna Broadcast Channels

Alireza Borhani and Soroush AkhlaghiFaculty of Engineering, Shahed University, Tehran, Iran

Emails: {aborhani, akhlaghi}@shahed.ac.ir

Abstract— This paper concerns delay analysis for FairPower Allocation Strategy (FPAS) in multi antenna BroadcastChannel (BC). It is assumed the transmitter aims at sendingindividual information to a large number of single-antenna users(𝑛) through a common wireless channel. Assuming there is atotal transmit power constraint, the FPAS attempts to select themaximum number of active users in which each can receive theminimum rate constraint of 𝑅𝑚𝑖𝑛. Noting this and assuming thesystem delay is defined as the minimum number of channel usessuch that each user with probability approaching one is selectedat least once, we show that for the case of single-antenna BC,the expected delay of FPAS scales as 𝑅𝑚𝑖𝑛

𝑛 log𝑛loglog𝑛

+ 𝜔(

𝑛loglog𝑛

).

Then, using our earlier proposed FPAS for multi-antenna BC,it is shown the expected delay of this strategy behaves like𝑅𝑚𝑖𝑛

𝑛 log𝑛𝑀 loglog𝑛−𝜃(1)

+ 𝜔(

𝑛loglog𝑛

), where 𝑀 denotes the number

of transmit antennas. Moreover, it is shown for sufficientlylarge (𝑘) number of channel uses, the average number ofservices received by a randomly selected user for single-antenna

and multi-antenna BC scale as 𝑘 loglog𝑛𝑛𝑅𝑚𝑖𝑛

(1 + 𝑂(

√log𝑘𝑘

))

and𝑘𝑀 loglog𝑛

𝑛𝑅𝑚𝑖𝑛

(1 +𝑂(

√log𝑘𝑘

)), respectively.

Index terms — Broadcast Channel, Power Allocation, DelayAnalysis, Minimum Rate Constraint.

I. INTRODUCTION

Quality of Servise (QoS) is mainly involved with someimportant features like achievable rate, delay, and fairness.Indeed, QoS and its problems have been recognized as oneof the main concerns in many communications systems, morespecifically in broadcast systems. In many applications, theusers are required to get a service with a constant rate and fora certain period of time. In this regard, [1] proposes an efficientway to support QoS in wireless channels based on maximizingthe number of users that can be supported with the desiredQoS. Moreover, there have been some attempts to make abalance between the system delay and the achieved throughputin wireless networks [2], [3]. The problem of throughput-delay trade off in multicast channels for different schedulingalgorithms is presented in [4]. Similarly, some other worksare devoted to the case of delay limited systems under fadingconstraint [5], [6]. Subsequently, the fairness maximizationfor multi antenna BCs in terms of maximizing the numberof active users is addressed in [7]. Also, [8] explores at theexpected delay of a fair resource allocation method in singleantenna BCs under the minimum rate constraint.

This paper at first concerns a variant issue in a Rayleighbroadcast channel when there is a base station (is equipped

with single transmit antenna) aims at sending individual in-formation to many users. Accordingly, in [9] it is shown theoptimum power allocation strategy in terms of maximizing thesum-rate is to choosing the best user, meaning to allocate thewhole power to the corresponding codebook of the selecteduser. From now on, we call this strategy as the Best UserSelection Strategy (BUSS). Although, the BUSS is able toachieve the sum-rate capacity of BC channel, it renders anexcess delay to happen [10], as only one user is being servicedat a given time. This may not be a wise strategy, specifically,for the case of large users. Motivated by this, in [11] anelegant power allocation method is proposed which is ableto simultaneously achieves the asymptotic sum-rate capacity 1

and to maximize the number of active users, assuming eachuser is subject to a minimum rate constraint. In what follows,we call this strategy as the Fair Power Allocation Strategy(FPAS).

As noting earlier, the main advantage of the aforementionedstrategy is approaching the maximum number of active usersat the same time [11] which dramatically decreases the systemdelay. This motivated us to explore the expected delay ofthe FPAS in this paper. The system delay is defined as theminimum number of channel uses which guarantees each usersuccessfully receives at least 𝑝 packets. In [10], it is provedthat in a homogenous single antenna BC and for the case of𝑝=1, the expected delay of the BUSS, scales as 𝐷=𝑛 log(𝑛).

Similarly, we define the delay as the minimum numberof channel uses in which each user successfully receives atleast one packet. Accordingly, we will show later that for thecase of single antenna BC with 𝑛 users, the expected delaybehaves like 𝑅𝑚𝑖𝑛

𝑛 log𝑛loglog𝑛 + 𝜔

(𝑛

loglog𝑛

), which lies between

two external points; the result of round-robin scheduling with𝐷1=𝑛, which serves as a lower bound and 𝐷2=𝑛 log(𝑛) asthe upper bound which refers to the case that merely one useris being serviced at a given time [10].

Then, using our proposed FPAS for multi-antenna BC [7], itis demonstrated that the expected delay of this strategy behaveslike 𝑅𝑚𝑖𝑛

𝑛 log𝑛𝑀 loglog𝑛−𝜃(1) + 𝜔

(𝑛

loglog𝑛

)2.

The remainder of this paper is organized as follows, Sec-tion II presents the system model, formalizes the problemand finally discusses on the applying Fair Power AllocationStrategy in single and multiple antenna BC, separately. Next,

1The optimality of this scheme is limited to the case of large users.2Note that, applying FPAS to the MIMO BC introduces a marginal gap

between the achievable sum-rate and the sum-rate capacity.

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for two aforementioned cases, the expected delay of thisstrategy is analyzed in Section III, in details. Section IVshows the simulation results, and finally Section V summarizesfindings.

Throughout the paper, vectors and scalers are bold facelower case and simple lower case, respectively. Also theHermitian operation is denoted by (.)𝐻 . Moreover, accordingto the Kunth’s notation [12], for any function 𝑓(𝑛) and𝑔(𝑛), 𝑓(𝑛) = O(𝑔(𝑛)), 𝑓(𝑛) = 𝜃(𝑔(𝑛)), 𝑓(𝑛) = o(𝑔(𝑛))and 𝑓(𝑛) = 𝜔(𝑔(𝑛)) are equivalent to lim𝑛→∞∣ 𝑓(𝑛)𝑔(𝑛) ∣ < ∞,

lim𝑛→∞∣ 𝑓(𝑛)𝑔(𝑛) ∣ = 𝑐, lim𝑛→∞∣ 𝑓(𝑛)𝑔(𝑛) ∣= 0 and lim𝑛→∞∣ 𝑓(𝑛)𝑔(𝑛) ∣=∞, respectively.

II. PROBLEM FORMULATION

This section first provides the problem formulation for thesingle-antenna BC and discusses the corresponding FPAS insuch channel [11], then presents the problem formulation ofmulti-antenna BC, and briefly discusses findings in [7].

A. Single Transmit Antenna Model

We consider a single antenna BC with 𝑛 users, each ofsingle receive antenna. Assuming the transmitted informationat time instant 𝑡 is s(𝑡), thus the received signal at the 𝑖’threceiver can be represented as follows

𝑦𝑖(𝑡) = ℎ𝑖s(𝑡) + 𝑤𝑖(𝑡) 𝑖 = 1, 2, ..., 𝑛 , (1)

where ℎ𝑖 denotes the channel gain of the 𝑖’th user and isassumed to be available at both the transmitter and the receiversides. Moreover, the channel gains are supposed to be constantacross the coding block, and varies for each block. 𝑤𝑖(𝑡)represents a scaler additive white gaussian noise with variance𝜎2, i.e., 𝑤𝑖(𝑡)∼𝒞𝒩 (0,𝜎2). Further more, it is assumed theaverage transmit power is unit, i.e., 𝐸{∣s(𝑡)∣2} = 1.

It is demonstrated that the single antenna gaussian broadcastchannel is stochastically degraded [9]. As a result, by usingsuperposition coding at the transmitter and applying successiveinterference cancelation at the receivers, one can increasethe received Signal to Interference plus Noise Ratio (SINR)of the 𝑖’th user by canceling those interfering signals forwhich the corresponding channel strength outperforms that ofthe 𝑖’th user in the ordered list. In other words, assumingusers depending on their channel strength are indexed in adescending order, i.e., ∣ℎ1∣2>∣ℎ2∣2>...>∣ℎ𝑛∣2, the interferingsignals arising from users 𝑘 = 1, . . . , 𝑖 − 1 can be readilycanceled at the 𝑖’th receiver. As a result, the sum-rate capacityunder gaussian code book assumption becomes

∑𝑛𝑖=1𝑅𝑖,

where 𝑅𝑖 is represented as follows

𝑅𝑖 < log(1 +𝑝𝑖

1/∣ℎ𝑖∣2 +∑𝑖−1

𝑘=1 𝑝𝑘) for 𝑖 = 1, . . . , 𝑛. (2)

where 𝑝𝑖 for 𝑖 = 1, . . . , 𝑛 denotes the power allocated to the𝑖’th user, and it is assumed the received noise power is one.Accordingly, noting (2), the equivalent noise of the 𝑖’th usercan be thought as 𝑁𝑖=

1∣ℎ𝑖∣2 .

It should be noted that if one could find a power allocationstrategy such that the corresponding SINR of each active

user exceeds 𝛼 − 1, then 𝑅𝑚𝑖𝑛 = log(1 + SINR) = log(𝛼)would be achievable by all active users. Noting above, in [11]for the case of Rayleigh channels it is proved that 𝑅𝑚𝑖𝑛=log(𝛼) for 𝑚 best users is achievable if 𝑚𝑎𝑥1≤𝑖≤𝑚𝑁𝑖<

1𝛼𝑚 .

Accordingly, noting users are labeled in a descending order,i.e., 𝑁1 < 𝑁2 < . . . < 𝑁𝑚, thus 𝑚 is defined such that𝑁𝑚 < 1

𝛼𝑚 . Moreover, the assigned power to the 𝑖’th useris computed as 𝑝𝑖 =

𝐶𝛼𝑚−𝑖 for 𝑖 = 1, . . . ,𝑚, where 𝐶 is

a constant value and is defined such that the total powerconstraint is satisfied, thus 𝐶 = (1 − 1/𝛼). In this case, itis shown that the maximum number of active users scales as𝑚= log(log(𝑛))

𝑅𝑚𝑖𝑛(for more details refer to [11]). Clearly, as the

minimum rate of each user is 𝑅𝑚𝑖𝑛, the achievable sum-rate ofsuch channel asymptotically approaches the sum-rate capacityof such channel, i.e., log(log(𝑛)).

B. Multiple Transmit Antenna Model

A MIMO broadcast channel with 𝑀 transmit antennas and𝑛 single-antenna users is considered. Assuming the transmittedinformation vector at time instant 𝑡 is s(𝑡), the received signalat the 𝑖’th receiver can be represented as follows

𝑦𝑖(𝑡) = h𝑇𝑖 s(𝑡) + 𝑤𝑖(𝑡) 𝑖 = 1, 2, ..., 𝑛 , (3)

where h𝑖 is the 𝑀 × 1 channel gain vector of the 𝑖’th usercomposed of 𝑀 independent, identically distributed (i.i.d.) el-ements which are drawn from a circularly symmetric gaussiandistribution, i.e., 𝒞𝒩 (0, 1

𝑀 ), and are assumed to be perfectlyavailable at the affiliated receiver. Moreover, it is assumedthe channel gains are constant across the coding block, andvaries for each block. 𝑤𝑖(𝑡) represents a scaler additive whitegaussian noise with variance 𝜎2, i.e., 𝑤𝑖(𝑡) ∼ 𝒞𝒩 (0,𝜎2).Further more, it is assumed the average transmit power perantenna is unit, i.e., 𝐸{s𝐻(𝑡)s(𝑡)} =𝑀 .

Assuming the transmitter makes use of random beamform-ing and referring to [7], one can decompose the originalMIMO BC to 𝑀 parallel single antenna BCs, each hav-ing 𝑛

𝑀 ( 𝜁𝑀−1 )

𝑀 (1+O( log(𝑛)

𝑛

)) users, where 𝜁 is chosen such

that 𝜁≪𝜎2. In this case, it is demonstrated that the maximumnumber of active users over all beamforming directions scalesas 𝑚= 𝑀 log(log(𝑛))

𝑅𝑚𝑖𝑛− 𝜃(1) [7]. Accordingly, it is shown the

achievable sum-rate behaves like 𝑀 log log(𝑛)− 𝜃(1). Notingthe maximum sum-rate of the original MIMO BC scales as𝑀 log(log(𝑛)) [13], it turns out that the proposed methodachieves this throughput within a constant gap.

III. DELAY ANALYSIS

There are some applications for which there is a stringentdelay constraint and they can not tolerate an excess delay.One of the main advantages of our proposed user selectionalgorithm is approaching the maximum number of active userswhich dramatically decreases the system delay. The systemdelay is defined as the minimum number of channel uses whichguarantees each of 𝑛 users successfully receives 𝑝 packets.Accordingly, in [10] it is proved that in a homogenous singleantenna BC and for the case of 𝑝 = 1, the expected delay of thebest strategy in terms of throughput maximization, sending to

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the best user in terms of the corresponding channel gain, scalesas 𝑛 log(𝑛). Furthermore, for multiple antenna BC with 𝑀transmit antennas, the expected delay is shown asymptoticallybecomes 𝑛 log(𝑛)

𝑀+𝑂(𝑀2/𝑛) [10].Similarly, we define the delay as the minimum number of

channel uses for which each user successfully receive at leastone packet. In what follows, we derive the expected delay ofthe FPAS for single antenna BC and for the case of 𝑝 = 1, andthen extend the results to the multi-antenna BC when randombeamforming is employed at the transmitter [7].

We consider a single transmit antenna BC with large (𝑛)number of users. The problem is to find the minimum num-ber of channel uses, dubbed 𝐷𝑚𝑖𝑛, for which all users arebeing serviced successfully with probability approaching one(w.p.1). In this regard, we define a successful service S asall users receive at least once. To this end, we propose usinga probabilistic viewpoint to seek for conditions which leads to𝑃𝑟{S } → 1. Let S1, ...,S𝑛 denote, respectively, the totalnumber of times for which the users indexed 1 to 𝑛 arebeing serviced. As is noted earlier, the successful event Scorresponds to the event that each user is being serviced atleast once, i.e., S1 ∕= 0, ...,S𝑛 ∕= 0. The following theoremcomputes 𝐷𝑚𝑖𝑛 which guarantees the successful event tohappen w.p.1 .

Theorem 1: For large number (𝑛) of users, assuming 𝑚users are being serviced simultaneously at a given time, thenecessary and sufficient condition to have 𝑃𝑟{S } → 1 is

𝐷𝑚𝑖𝑛∼= 𝑛 log(𝑛)

𝑚 + 𝜔( 𝑛𝑚 ).

Proof: Refer to [8]Theorem 1 states that in a broadcast channel with 𝑛 users,

for 𝑚 concurrent transmissions, the expected delay behaveslike 𝑛 log(𝑛)

𝑚 . Referring to the minimum delay Round-Robinscheduling with 𝐷1 = 𝑛, the corresponding 𝐷𝑚𝑖𝑛 of theaforementioned scheduling could theoretically approaches 𝐷1if 𝑚 = log(𝑛), which is not feasible as the maximumnumber of active users is at most log(log(𝑛))

𝑅𝑚𝑖𝑛. Also, this is

interesting to note that having one active user at a giventime corresponds to the case of 𝑚 = 1, thus the expecteddelay becomes 𝑛 log(𝑛) which is in accordance to the resultreported in [10]. Finally, referring to theorem 1 and notingthe maximum possible value of 𝑚 is at most log(log(𝑛))

𝑅𝑚𝑖𝑛, the

expected delay of a single antenna BC [8], 𝐷𝑆𝐴𝑚𝑖𝑛, becomes

𝐷𝑆𝐴𝑚𝑖𝑛 = 𝑅𝑚𝑖𝑛

𝑛 log(𝑛)

log(log(𝑛))+ 𝜔

( 𝑛

log(log(𝑛))

). (4)

As is noted earlier, using random beamforming strategytogether with FPAS, the MIMO BC can be thought as 𝑀parallel channels each having 𝑚 = log(log(𝑛))

𝑅𝑚𝑖𝑛− 𝜃(1) active

users. As a result, noting 𝑚𝑀 users are being serviced ata given time, the probability that a given user out of 𝑛existing users is selected is simply computed as 𝑚𝑀

𝑛 . Thus,following the same approach as is used in theorem 1, it can beconcluded that 𝐷𝑚𝑖𝑛

∼= 𝑛 log(𝑛)𝑚𝑀 +𝜔( 𝑛

𝑚𝑀 ). Finally, after somemanipulations, the expected delay of the proposed strategy for

the case of multi antenna BCs, 𝐷𝑀𝐴𝑚𝑖𝑛, behaves like

𝐷𝑀𝐴𝑚𝑖𝑛

∼= 𝑅𝑚𝑖𝑛

𝑀

( 𝑛 log(𝑛)

log(log(𝑛))− 𝜃(1)+ 𝜔

( 𝑛

log(log(𝑛))

)). (5)

The following theorem indicates the average number ofservices received by a randomly selected user after 𝑘 trans-missions.

Theorem 2: Assume there are 𝑛 users in a single antennaBC. The average number of services received by a randomlyselected user for large (𝑘) number of channel uses behaves

like 𝑘𝑚𝑛

(1 +𝑂(

√log(𝑘)

𝑘 ))

w.p.1.Proof: Recall that for each channel use, the probabil-

ity that a specific user is being serviced is 𝑚𝑛 . Under the

assumption of 𝑘 channel uses, the average number (S ) ofgetting services by this user follows a Binomial distribution,that is B(𝑘, 𝑚𝑛 ) 3. Using gaussian approximation for a binomialdistribution when 𝑘 is large enough, we have

𝑃𝑟{𝑘𝑚𝑛

(1− 𝛿) < S <𝑘𝑚

𝑛(1 + 𝛿)}

≈ 1− 2𝑄( 𝑘𝑚

𝑛 𝛿√𝑘𝑚𝑛 (1− 𝑚

𝑛 )

)(6)

It can be verified that by setting 𝛿=√

2( 𝑛𝑚−1) log(𝑘)

𝑘 and noting

the approximation 𝑄(𝑥)= 1√2𝜋𝑥

𝑒−𝑥2

2 for 𝑥≫1, it follows

𝑃𝑟{𝑘𝑚𝑛

(1− O(

√log(𝑘)

𝑘))<S<

𝑘𝑚

𝑛

(1 + O(

√log(𝑘)

𝑘))}

∼ 1− o(1

𝑘) (7)

It means for sufficient large number of channel uses, theaverage number of services to a randomly selected user ina single antenna BC, S

𝑆𝐴, scales as

S𝑆𝐴

=𝑘𝑚

𝑛

(1 +𝑂(

√log(𝑘)

𝑘)), (8)

with probability approaching one.Under the assumption of employing the FPAS at the transmitterand noting the number of active users in the single antennascenario, 𝑚 = log(𝑙𝑜𝑔(𝑛))

𝑅𝑚𝑖𝑛, the average number of services to

a randomly selected user in a single antenna BC, S𝑆𝐴

𝐹𝑃𝐴𝑆 ,behaves like

S𝑆𝐴

𝐹𝑃𝐴𝑆 =𝑘 loglog𝑛

𝑛𝑅𝑚𝑖𝑛

(1 +𝑂(

√log𝑘

𝑘)). (9)

Accordingly, for the case of MIMO BC with 𝑛 singleantenna users, the average number of services to a ran-domly selected user is a Binomial distribution with param-eters 𝐵(𝑘, 𝑚𝑀

𝑛 ). Thus, noting 𝑚 = log(log(𝑛))𝑅𝑚𝑖𝑛

− 𝜃(1) andfollowing the same approach as is used in Theorem 2, the av-erage number of services achieved by employing the proposed

3B(𝑘, 𝑝) denotes a Binomial distribution with parameters 𝑘 and 𝑝, respec-tively, as the number of trials and the probability of success.

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500 550 600 650 700 750 800 850 900 950 10001000

2000

3000

4000

5000

6000

7000

8000

Total number of users

Del

ay (

The

num

ber

of p

acke

ts)

BUSS, Numerical resultFPAS, Numerical resultFPAS, Analytical resultBUSS, Analytical result

Fig. 1. The expected delay of the BUSS as compared to that of the FPAS,where 𝑀 = 1 and 𝑅𝑚𝑖𝑛 is set to one.

method, S𝑀𝐴

𝐹𝑃𝐴𝑆 , becomes (w.p.1.)

S𝑀𝐴

𝐹𝑃𝐴𝑆 =𝑘𝑀 log(log(𝑛))

𝑛𝑅𝑚𝑖𝑛

(1 +𝑂(

√log(𝑘)

𝑘)). (10)

IV. SIMULATION RESULTS

This section aims at comparing the expected delay of thefollowing scheduling methods in a Rayleigh fading single andmultiple transmit antenna broadcast channel: (i) the BUSSand (ii)the FPAS. Figs. 1 and 2 compare the system delay ofthe aforementioned methods for single and multiple transmitantenna BCs, respectively. Accordingly, for different numberof users ranging from 500 to 1000, the expected delay ofthese scheduling methods are simulated and compared with thetheoretical results, showing there is a close agreement betweenthe computed delays derived by the analytical results to thatof the simulations. Referring to Figs. 1 and 2, there is a majordifference between the expected delay of these two methods.Moreover, the small gap between the theoretical results to thatof the simulations is due to the fact that merely the main termsof the analytical results are considered and the ordering partsare ignored.

Figs. 3 and 4 compare the average number of gettingservices for a randomly selected user for both the BUSS andFPAS methods and for different values of 𝑅𝑚𝑖𝑛

4, where thetotal number of channel uses 𝑘 is set to 𝐷𝐵𝑈𝑆𝑆 = 𝑛 log(𝑛).Noting 𝑘=𝑛 log(𝑛), and referring to theorem 2, the averagenumber of services to a randomly selected user in a MIMOBC for the BUSS with 𝑚 = 𝑀 5 and the FPAS with 𝑚 =𝑀 log(log(𝑛))

𝑅𝑚𝑖𝑛respectively scale as

S𝑀𝐴

𝐵𝑈𝑆𝑆 =𝑀 log(𝑛)(1 +𝑂(𝑛−

12 )), (11)

4Regarding the value of 𝑅𝑚𝑖𝑛, one can consider a system with channelbandwidth of 200Khz. Consequently, the transmission rate of 0.1 bits persample is equivalent to 0.1× 2𝐵𝑊 = 40𝐾 bits per second.

5𝑚 is regarded as the number of active users in a concurrent transmission.

500 550 600 650 700 750 800 850 900 950 1000400

600

800

1000

1200

1400

1600

1800

2000

Total number of users

Del

ay (

The

num

ber

of p

acke

ts)

BUSS, NumericalFPAS, NumericalFPAS, AnalyticalBUSS, Numerical

Fig. 2. The expected delay of the BUSS as compared to that of the FPAS,where 𝑀 = 4 and 𝑅𝑚𝑖𝑛 is set to one.

and

S𝑀𝐴

𝐹𝑃𝐴𝑆 =𝑀 log(𝑛) log(log(𝑛))

𝑅𝑚𝑖𝑛

(1 +𝑂(𝑛−

12 )). (12)

Moreover, Figs. 3 and 4 indicate that there is a major gapbetween two power allocation strategies when relying on theaverage number of getting services for a randomly selecteduser.

V. CONCLUSION

A multi antenna broadcast channel with sufficiently largenumber of single antenna users (𝑛) is considered. In thispaper, the expected delay of a Fair Power Allocation Strategies(FPAS) has been studied both analytically and on the basis ofsimulation results. In analytical study, the results showed thatthe system delay is decreased through a double logarithmicscale when the transmitter uses the FPAS instead of the BestUser Selection Strategy (BUSS).

In the simulation study, the results demonstrated that there isa close agreement between the analytic and numerical results.In fact, the numerical experiment showed that the expecteddelay of the FPAS is substantially less than the delay of theBUSS. Moreover, illustrating the experiment results as well,when the FPAS is employed at the transmitter, the averagenumber of services to a randomly selected user is considerablygreater than the case of using the BUSS.

REFERENCES

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[2] N. Bansal and Z. Liu, “Capacity, delay and mobility in wireless ad-hocnetworks,” in Proc. IEEE INFOCOM, vol. 52, no. 6, pp. 1553–1563,Apr. 2003.

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Fig. 3. The average number of services for a randomly selected user for twodifferent power allocation strategies, 𝑅𝑚𝑖𝑛 = 0.1.

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Fig. 4. The average number of services for a randomly selected user for twodifferent power allocation strategies, it is assumed that 𝑀 = 4.

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with partial side information,” IEEE Trans. Inf. Theory, vol. 51, no. 2,pp. 506–522, Feb 2005.

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