Energy-saving Predictive Resource Planning and...

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Energy-saving Predictive Resource Planning andAllocation

Chuting Yao, Chenyang Yang and Zixiang Xiong

Abstract—Predictive resource allocation is an emerging ap-proach to improve the performance of mobile systems as humanbehavior is reported predictable by leveraging big data analyt-ics. Yet what information can be predicted by big data, whatinformation need to be predicted for wireless access optimization,how to translate the information, and how to exploit the syntheticknowledge for allocating radio resources are not well-understoodand largely explored. In this paper, we are concerned with thelater two issues. In particular, we devise energy-saving resourceplanning and allocation policy for multiple base stations (BS) toserve mobile users with non-real-time (NRT) traffic by exploitingthe user, network, and application levels of context information,where real-time traffic may occupy partial resources of each BS.Inspired by the solution from an energy minimization problemwith future instantaneous information, a low complexity multi-timescale predictive policy is proposed. Upon the arrival of eachNRT user request, the resource planning is made with the user andnetwork level context information, defined as the average channelgains of the NRT users and the statistics of residual bandwidthafter serving real-time traffic, with which the scheduling, power al-location and BS sleeping can be accomplished after instantaneouschannel information and residual network resource are availableat each BS in each time slot. Simulation results show that theproposed policy can dramatically reduce the energy consumed bythe BSs for serving the NRT traffic.

Index Terms—Predictive resource allocation, big data, contextinformation, energy saving, user mobility

I. INTRODUCTION

Inspired by the recent finding that human behavior is highlypredictable [1] and as the big data analytics flourishes, im-proving the performance of wireless systems by exploiting thepredicted information has started to draw attention, which isreferred to as the predictive, anticipatory, or context-awareresource allocation (or wireless access, networking) in theliterature [2–6].

The human behavior related information is a kind of contextinformation, and context-awareness is not a new concept in thedomain of computer science. In [7], the context is defined asany information that can be used to characterize the situation ofan entity, where the entity can be a person, place, or object thatis considered relevant to the interaction between a user and anapplication. In the domain of wireless communications, contextinformation is classified into application level (e.g., quality of

This work was supported by National Natural Science Foundation of China(NSFC) under Grant 61120106002, National Basic Research Program of Chinaunder Grant 2012CB316003, and NSFC under Grant 61429101.

C. Yao and C. Yang are with the School of Electronics and InformationEngineering, Beihang University, Beijing 100191, China. (E-mail: {ctyao,cyyang}@buaa.edu.cn), and Z. Xiong is with the Department of Electrical andComputer Engineering, Texas A&M University, College Station, TX 77843,USA. (E-mail: [email protected])

service (QoS)), network level (e.g., congestion status), userlevel (e.g., location or mobility pattern), and device level [8].

To support predictive wireless access, the context informa-tion needs to be first acquired, often by translating from the pre-dicted and/or measured information. Then, predictive resourceallocation can be optimized to achieve a specific objective byexploiting the context information.

While predictive resource allocation has been investigatedin the past years either for phone calls [4, 9, 10] or for videostreaming [2, 3, 5, 11–14] , the following questions still largelyremain open. What specific information can be predicted bybig data analytics? What information needs to be predicted tofacilitate wireless resource management? How to translate theoriginally predictable information? How to exploit the syntheticknowledge for allocating resources to different kinds of trafficin a predictive manner?

A. Related Works on Predicting or Measuring Information

According to the reports in the literature, the relevant in-formation that can be predicted or measured includes mobilitypattern, radio map, and traffic map.

Mobility pattern prediction: The analysis on human travelpatterns shows that people move along particular routes withhigh predictability [1, 15]. User positioning is the basis ofmobility pattern prediction, which can be reported from asmart phone assisted by Global Positioning Systems (GPS),or estimated with the measurements from sensors [16], or byusing the signals from WiFi or cellular systems [17]. Basedon the GPS measurements along a vehicle’s past trips, thealgorithms for predicting the end-to-end route of the vehiclewere developed in [18]. In [19], not only the future location auser will visit but also the arrival time and how long it will stay,i.e., the trajectory that has both time and location information,were predicted. In [20], the trajectory of a mobile user waspredicted based on its current position and direction, and thehistory of its trajectories. In [21], the destination, mobility path,and subsequent transitions of road segments along the trajectoryof a user within future half an hour were predicted. A morecomprehensive survey can be found in [6].

Radio map construction: With the predicted trajectory ofa mobile user, the average channel gains can be predictedwith the help of a radio map (or path-loss map, coveragemap). The radio map can be constructed based on drive testmeasurements [22–24], which however is of high-cost. Anattractive alternative is to use mobile terminals equipped withGPS to complement the measurements [25]. Considering thatthe samples measured by drive tests or reported by mobile usersover discrete geographical coordinates are far from complete,

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the radio map was reconstructed from random measurements bymatrix completion in [26], and was constructed by two kernel-based adaptive online algorithms in [27].

Traffic map prediction: To predict the congestion status of anetwork, one possible way is to predict the traffic load variationover time and location. Many recent works have reported thatthe dynamics of traffic load exhibit periodical characteristicamong days and even hours, see [28] and references therein.This implies that the traffic load, i.e., the average request arrivalrate, is highly predicable. In [29], the authors predict the next-day traffic load with time granularity of one hour by usingcompressive sensing. In [30], the authors predict the trafficvolumes in a day with a resolution of five minutes by leveragingthe spatiotemporal correlation of nominal traffic and the sparsenature of anomalies.

B. Related Works on Predictive Resource Allocation

The techniques exploiting the predicted information are verydifferent for real-time (RT) traffic (e.g., phone calls and videoconference) and non-real-time (NRT) traffic (e.g., video stream-ing and file downloading), both in the underlying mechanismand in the required information.

For the RT traffic, predictive wireless access has been in-vestigated to improve the connection-level quality of service(QoS), say reduce the call dropping rates. Considering thatthe information bits are generated randomly by each user andthe RT service is with high priority, the major mechanismis to preserve resource for the RT traffic. In fact, mobilityprediction has long been used for mobility management toassist handover, where the location prediction granularity isusually rough, say one cell. In [9], with the predicted next-cellconnection and hand-off time, dynamical resource preservationand call admission control method was proposed to improve theQoS. In [10], by using future user location together with roadtopology information, an efficient resource preservation schemewas proposed to achieve a target handoff dropping probabilityby blocking fewer new calls. In [4], by exploiting the predictedhandoff time and available bandwidth under the assumption ofknowing future locations and all incoming/outgoing handoffs,a resource preservation and call admission control scheme wasproposed to reduce the call dropping rate.

For the NRT traffic, not only the QoS of each user but also theperformance of a network can be improved by exploring futureinformation. One simple reason is that for typical NRT trafficsuch as video steaming, the videos to be transmitted can bestored meanwhile the delay requirement is not stringent, suchthat the users can be served when they are in good channelcondition. By assuming perfect instantaneous data rate predic-tion, the transmission time and total power consumed at thebase station (BS) were respectively minimized under the QoSconstraint in [2, 3], the energy consumed at the BS was savedby closing antennas while guaranteeing the QoS in [12], andthe QoS was improved in [14], all for stored video steaming.Noticing that the rate prediction is inevitably inaccurate, predic-tion errors were introduced into future instantaneous data ratein [13], and then a robust rate allocation was designed to controlthe QoS of video streaming. Since instantaneous data rate

depends on instantaneous channel information, which is onlypredictable within the channel coherence time, a more realisticassumption is knowing the statistics of the future data rates.By assuming perfect average data rate prediction, the storedvideo was delivered efficiently without compromising delay byusing the method proposed in [11]. By exploiting predicteddata rate distribution derived from the location and number ofactive users, the transmission time was minimized meanwhilethe playback interruptions of video steaming was eliminated in[31]. In [32], the average rates at different locations measuredin the past days were stored and used as the future average rateprediction with the help of user trajectory, with which the QoSof video streaming was improved. In these research efforts, theknowledge of the future achievable rates is a cornerstone for thepredictive wireless access, which however was not explicitlyconnected with the predictable information. In [33], by usingthe perfect trajectory prediction and a radio map obtained fromreal measurement to predict the future average data rate, thethroughput of the system with proportional fair scheduling wasimproved. In [5], by assuming perfect average channel gainprediction, the energy efficiency (EE) of a multi-carrier systemswas maximized under the QoS constraint of video streaming.In all existing predictive resource allocation polices except [5],the major mechanism to exploit future information is to pre-allocate future data rates for mobile users at the time instancewhen the predicted information is obtained. As a result, thepolicies are in one timescale, despite that the user trajectory,the traffic map, and the small scale channel information arepredictable in different timescales, which naturally calls formultiple timescale resource allocation. Furthermore, all previ-ous works except [5, 13, 31] do not consider the impact of time-varying small scale fading channels on the prediction errorsof the achievable rate, which however is essential for mobilesystem design.

C. Motivation and Contribution of This Work

Both RT and NRT traffic exist in real-world networks. In thispaper, we consider a network with both classes of traffic, anddesign predictive resource allocation for NRT users by treatingRT users as background traffic.

In such practical network with hybrid traffic, the inaccuracyof rate prediction for a NRT user comes not only from the chan-nel uncertainty (e.g., small scale channel fading) but also fromthe available resources uncertainty (e.g., resources occupied byrandomly arrived RT traffic and other NRT users). Moreover,the predictable context information, say user trajectories andtraffic map, are usually predicted in a large timescale (sayin seconds, minutes or even hours) since their variation isrelatively in large timescale. Yet the radio resource in wirelesssystems is often allocated in a much small timescale, since it de-pends on fast channel fading that varies in the timescale of mil-liseconds. How to exploit the information predictable/availablein different timescales together for predictive resource alloca-tion is still open.

Different from previous studies using predicted future ratesfor resource allocation, we propose an alternative approach,which is able to exploit the predictable information from bigdata explicitly. As a result, the proposed policy is able to be

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implemented in multiple timescales. Specifically, we exploitcontext information from three levels, i.e., application, networkand user levels, and design energy-minimizing resource alloca-tion for the users with NRT service when RT service occupiespartial transmission resource.

In the following of this work, the user and network levelcontext information are defined as the future average channelgains of NRT users and the statistics of residual bandwidthafter serving RT traffic, respectively, which can be translatedfrom the predictable trajectory and traffic map and are “largescale” in nature. The NRT traffic is often modeled as best efforttraffic and is served right after the requests arrive if resourceis available. As a result, the spectral efficient or energy efficientresource allocation for this kind of traffic is usually optimized tomaximize the capacity or the instantaneous EE of the network.Nonetheless, if we know the expected deadline to accomplishthe service either by user subscription or by analyzing the userapplication [34] or even the user behavior, it is possible tosave energy dramatically with long term planning for resourceallocation by waiting for better channel condition and networkstatus. To demonstrate how such an application level contextinformation can be exploited to save energy, we artificiallymodel the NRT service as transmitting a given amount ofdata before a deadline after the request arrives, which cancharacterize content delivery traffic, or even some emergingmobile video traffic based on subscription or proactive pushingbased on user preference prediction [35].

As [7] mentioned, “realizing the need for context is onlythe first step toward using it effectively”. To find the wayto exploit such “large scale” information, we first formulatea total energy minimization problem for a single NRT userwith future instantaneous information. The obtained optimalsolution not only provides insight on how we should use contextinformation to make a resource allocation plan but also servesas a performance upper bound. By finding the key parametersfor resource planning by synthesizing user and network levelcontext information, a resource allocation policy includingBS sleeping, time slot scheduling, and power allocation isproposed. Then, we extend the single-user predictive resourceplanning and allocation to the practical scenario of multiplerandomly arrived NRT users. To complete the transmission forevery NRT user before its “deadline”, we need to cope withthe uncertainty in predicting network level context informationcaused by the newly arrived NRT user request. To this end,we update the network level context information to reflect theresource occupied by both the RT traffic and other users withNRT traffic at each BS, and make a transmission plan for eachNRT user to assist user scheduling.

The major contributions of this paper are summarized asfollows.• We propose a multi-timescale predictive resource allo-

cation for NRT traffic to exploit three levels of contextinformation, in contrast to existing policies that are allin one timescale. Our policy includes resource planningwhen the user initiates a request and plan updating whennew requests arrive randomly, both only depending onstatistical information, and resource allocation accordingto the plan depending on instantaneous information.

• We find two key parameters translated from the predictableinformation in making a plan for the BSs to decide when,where and with how much radio resource to serve the NRTuser, in contrast to existing policies that allocate futuredata rate.

• The proposed policy is of low-complexity and viable forpractice use. Simulation results show that for single NRTuser case, the performance of the policy approaches theupper bound obtained with all the future instantaneousinformation, which implies that only future statistical in-formation is necessary. For both single and multiple NRTcases, the proposed policy provides substantial energy-saving gain over those not exploiting the context infor-mation, even when the predicted statistical information isimperfect.

The rest of the paper is organized as follows. System modelis introduced in Section II. In Section III, we formulate the op-timization problem with perfect future information and providethe solution for single NRT user. In Section IV, we propose thepredictive resource planning and allocation policy for singleuser case, and extend it into multi-user case in section V.Simulation results are provided in Section VI and the paper isconcluded in Section VII.

II. SYSTEM MODEL

Consider a multicell downlink system with M BSs, whereeach BS is equipped with Nt antennas and operates in a time-slotted fashion over a frequency band with bandwidth Wmax.The maximal transmit power of each BS is pmax. Each BSserves randomly arrived requests of RT traffic, and also servesthe users demanding NRT traffic (called NRT users) who enterthe coverage of the BS. The BSs are connected with a controlunit (CU), who gathers the context information from the NRTusers and configures the resource for these users.

To capture the essence of the problem and simplify thenotations, we first consider the scenario of one user with NRTtraffic, and then extend the results to multi-user scenario. As-sume that the NRT user is with a single antenna, who may moveacross the cells during transmission.

A. Traffic Model

The RT traffic such as voice and video conference has strictQoS provision in terms of delay, and needs to be served im-mediately after the requests arrive at a BS. To ensure the QoSof the RT traffic, a given fraction of the resources should bereserved for each request [36]. In practice, since the requestsarrive randomly, the resource occupied by these requests istime-varying. The transmit power and bandwidth allocated tothe RT traffic at the ith BS in the tth time slot are denoted aspti,RT and W t

i,RT, respectively.The NRT traffic such as file downloading and advertisements

is usually served with best effort. In the forthcoming analysis,we show that the energy consumed by the BSs can be reducedby introducing an “expected deadline” for the NRT traffic.Specifically, a “QoS” is introduced to a NRT user, which ismodelled as conveying a given number of B bits within aduration of T time slots, which is referred to as the application

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α1Large-scale variation α2 · · · αj

T time slots are divided into Tf frame

· · · αTf−1 αTf

h(j−1)Ts+1Small-scale variation · · ·one frame contains Ts time slots

hjTs

Fig. 1. Illustration of channel model.

The transmission of the NRT traffic can be started in any time after the instant request arrives

(set as the 1st time slot) and before the deadline. Since such traffic is not urgent, the NRT user

can be served with the residual transmit power and bandwidth of a BS.

For simplicity, we assume that the user is only accessed to the closest BS. Denote mi =

[m1i , . . . ,m

Ti ]H as the indicator of scheduling status during the T time slots for the NRT user at

the ith BS, where mti ∈ {0, 1}. When mt

i = 1, the user is scheduled in the tth time slot by the

ith BS. When mti = 0, the user is not scheduled in the tth time slot by the ith BS, which will

be turned into sleeping mode if there is no RT traffic in the ith cell.

B. Channel Model

We divide the expected transmission duration T into Tf frames, where each frame contains

Ts , T/Tf time slots. Suppose that the large scale channel gains including path-loss and

shadowing may vary among frames along the trajectory of the user, and the small scale fading

gains stay constant in each time slot and change independently among time slots as shown in

Fig. 1.

For the NRT user, the received signal in the tth time slot can be expressed as

yt = mti

√αd

tTse(ht)Hwt

√ptxt + nt, (1)

where xt is the transmit symbol for the user with E{|xt|2} = 1 and E{·} denotes expectation, pt is

the transmit power, wt ∈ CNt×1 is the beamforming vector, αdtTse is the large scale channel gain

between the user and the closest BS in the tth time slot, d·e is the ceiling function, ht ∈ CNt×1 is

the independent and identically distributed (i.i.d.) small scale fading channel vector, (·)H denotes

the conjugate transpose and nt is the noise with variance σ2. Since the NRT user is scheduled

∆

Fig. 1. Illustration of channel model. ∆ is the duration of each time slot.

level context information.1 The value of T can be set accordingto the prediction on the user behavior, e.g., set as the minimalof the duration within which the mobility of the user can bepredicted and the duration that the user anticipates the down-loading can be completed. It is reasonable to assume that thevalue of T is much longer than the transmission time, say, onthe order of seconds or even minutes. Since such a “deadline”is artificially introduced to save energy, it is not a real harddeadline required by the user. The remaining bits not conveyedbefore the deadline can be transmitted in the subsequent Tl

time slots (referred to as flexible time). If the value of T is setappropriately and T1 � T , violating the “deadline” will notdegrade the user experience.

The transmission of the NRT traffic can be started in anytime instance after the request arrives (set as the 1st time slot)and before the deadline. Since such traffic is not urgent, theNRT user can be served with the residual transmit power andbandwidth of a BS.

For simplicity, we assume that the user is only accessed tothe closest BS. Denote mi = [m1

i , . . . ,mTi ]H as the indicator

of scheduling status during the T time slots for the NRT userat the ith BS, where mt

i ∈ {0, 1}. When mti = 1, the user is

scheduled in the tth time slot by the ith BS. When mti = 0, the

user is not scheduled in the tth time slot by the ith BS, whichwill be turned into sleeping mode if there is no RT traffic in theith cell.

B. Channel Model

We divide the expected transmission duration T into Tfframes, where each frame contains Ts , T/Tf time slotsand each time slot is with duration ∆. The large scale channelgains including path-loss and shadowing may vary amongframes along the trajectory, and the small scale fading gainsare assumed staying constant in each time slot and changingindependently among time slots (reflecting the worst case interm of predictability), as shown in Fig. 1.

Then, in the tth time slot, the achievable rate of the NRT userin nats is

Rt = mtiW

ti ln(1 + gtpt), (1)

where W ti , Wmax − W t

i,RT is the remaining bandwidththat can be used for the NRT user in the tth time slot,pt is the transmit power, gt , αd

tTse‖ht‖2/(Gσ2) =

αdtTse‖ht‖2/(GN0W

ti ) is the equivalent channel gain, αd

tTse

1For example, a user initiates a request to download a movie but plans towatch the movie when he is available, say when he is on the way to home. Thecontent requested by the user can be analyzed from the application on the user’sterminal after the user initiates the request [34] and is reported to the CU. Then,the CU informs the closest BS to fetch the required file via backhual and corenetworks from the server where the file is stored.

is the large scale channel gain between the user and the closestBS in the tth time slot, ht ∈ CNt×1 is the independent andidentically distributed (i.i.d.) small scale fading channel vector,N0 is the noise power spectrum density, and G is the signal-to-noise ratio gap between the capacity-achieving and practicalmodulation and coding selection policy [37]. d·e, (·)H and‖ ·‖ denote ceiling function, conjugate transpose and Euclideannorm, respectively.

C. Power Model

Because we strive to save energy by long term resourceallocation for the NRT traffic, the RT traffic is regarded asbackground traffic. In the tth time slot, the power consumed bythe M BSs for the background traffic (refer to as basic powerin the sequel) can be modeled as [38]

ptB =∑Mi=1

(1ξpti,RT + 1(pti,RT > 0)(pact− psle) + psle

), (2)

where ξ is the power amplifier efficiency, pact and psle are thecircuit power consumed at a BS in active and sleeping mode,respectively, and 1(x) = 1 when x is true, 1(x) = 0 otherwise.

When there are both RT and NRT traffic in the network, thetotal power consumed by both traffic at the BSs in the tth timeslot can be derived as,

pttot =∑Mi=1

1ξ (pti,RT +mt

ipt)+

∑Mi=1

(1(pti,RT +mt

ipt > 0)(pact − psle) + psle

). (3)

III. RESOURCE ALLOCATION WITH FUTUREINSTANTANEOUS INFORMATION

To provide a performance upper bound and obtain insights onhow to design a viable predictive resource allocation exploitingboth average and instantaneous information, in this section weassume that in the 1st time slot all the instantaneous informationduring T time slots are known, including the equivalent channelgain gt of the NRT user, the available bandwidth for the NRTuser W t

i and transmit power pti,RT occupied by the RT traffic atthe ith BS for t = 1, . . . , T .

The scheduling and power allocation for the NRT user tominimize the overall energy consumed by the BSs in the T timeslots under the constraint of transmittingB bits, i.e.,B ln 2 nats,within the T time slots can be formulated as follows,minp,m

∑Tt=1 p

ttot∆ (4a)

s.t.∑Tt=1m

tiW

ti ln(1 + gtpt) = B ln 2

∆ , (4b)pt ≥ 0,mt

ipt + pti,RT ≤ pmax, t = 1, . . . , T, i = 1, . . . ,M,

(4c)

where p = [p1, . . . , pT ]H is the power allocated to the NRTuser during all the T time slots, and m = [m1, . . . ,mM ] is thescheduling status matrix of all the M BSs during all time slots.(4b) is the “QoS” constraint of the NRT user, and (4c) is thepower constraint of every BS.

Since the NRT user is only accessed to the closest BS ineach time slot, we can omit subscript i for notational simplicity.Then, mt, ptRT, and W t respectively represent the indicator ofwhether the NRT user is scheduled by its closest BS, the power

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allocated to the RT traffic, and the bandwidth available for theNRT user at its closest BS in the tth time slot.

According to whether the closest BS is occupied by thebackground traffic, we divide the T time slots into busy timeand idle time. Denote Toc = {t|ptRT > 0} with cardinality Toc

as the index set of the busy time slots, and Tid = {t|ptRT = 0}with cardinality Tid = T − Toc as the index set of the idle timeslots. Then, Tid is the complementary set of Toc.

To exploit the residual transmit power and bandwidth at theBS, the NRT user may be served in both kinds of time slots.Then, from (2) and (3), the overall power consumed by the BSsin the T time slots can be rewritten as follows,∑Tt=1 p

ttot=

∑Tt=1p

tB︸︷︷︸

Basic Power

+∑Tt=1

1ξm

tpt

︸︷︷︸Transmit Power

+∑t∈Tid

mt(pact − psle)︸︷︷︸

Circuit Power

,

which includes the transmit and circuit powers consumed forbackground traffic, transmit power for NRT traffic, and extracircuit power for the NRT traffic in idle time slots. Sincethe basic power is not affected by the resource allocation forthe NRT traffic, minimizing the overall power consumption isequivalent to minimizing the sum of the second and third terms.

Denote N = {t|mt = 1, t ∈ Tid} as the index set of sched-uled idle time slots for the NRT user. Then, N ,

∑t∈Tid

mt

is the number of the scheduled idle time slots. During the non-scheduled idle time slots, the transmit power allocated to theNRT user is zero, i.e., pt = 0,mt = 0, t ∈ Tid−N , and the BSis turned into sleeping mode. Then, problem (4) is equivalentlyto the problem of minimizing the total power consumed by theBSs for the NRT user, which is

minp,N

∑t∈Toc∪N

1ξpt +N(pact − psle) (5a)

s.t.∑t∈Toc∪N W

t ln(1 + gtpt) = B ln 2∆ , (5b)

pt ≥ 0, pt + ptRT ≤ pmax, t ∈ Toc ∪N ,N ⊆ Tid. (5c)

To solve problem (5), we first optimize pt with given N andthen search the optimal value ofN from Tid to 1 that minimizesthe total power consumption.

When N is given, the circuit power consumption is given.Then, we only need to minimize the total transmit power, i.e.,

minp

∑t∈Toc∪N

1ξpt, (6)

s.t. (5b), (5c).

whose optimal solution can be found by standard water-filling,i.e.,

pt =(

W t

Wmaxν − 1

gt

)pmax−ptRT

0, t ∈ Toc ∪N , (7)

where (·)pmax−ptRT0 represents 0 ≤ pt ≤ pmax − ptRT, and ν is

the water-filling level with expression

ν=Wmaxexp(B ln 2/∆−

∑t∈T W

t ln(1+gtpt)−∑t∈T W

t ln(W tgt)∑t∈T W

t

),

(8)where T = {t|pt = pmax − ptRT} and T = {t|pt ∈ (0, pmax −ptRT)} are the index set of the time slots when the NRT user isserved with all residual transmit power and when pt < pmax −ptRT.

With (7), (5a) becomes a function of N . Then, the optimalnumber of the scheduled idle time slots can be found from

N∗ = arg minN∑t∈Toc∪N

1ξpt + N(pact − psle) by bisec-

tion searching since the transmit power increases and circuitpower decreases as N decreases. With N∗, the optimal powerallocation during all the T time slots can be obtained as

pt∗ =

{(W t

Wmaxν∗ − 1

gt

)pmax−ptRT

0, t ∈ Toc ∪N ∗

0, t ∈ Tid −N ∗,(9)

where ν∗ is the optimal water-filling level obtained from prob-lem (6) with N∗, and N ∗ is the index set of the scheduledN∗ idle time slots. From the water-filling structure of powerallocation in (9), we know that N ∗ contains the N∗ idle timeslots with highest equivalent channel gains. Hence, N ∗ ={t|gt ≥ g∗th, t ∈ Tid}, where g∗th is the optimal threshold toselect the N∗ idle time slots. Since pt∗ > 0 for t ∈ N ∗, for anygt ≥ g∗th, from (9) we have

ν∗ − 1g∗th≥ 0. (10)

Then, the optimal scheduling indicator during the idle timeslots can be obtained as mt∗ = 1(gt ≥ g∗th), t ∈ Tid, and theindicator during the busy time slots is obtained from mt∗ =1(pt∗ > 0), t ∈ Toc. When pt∗ = 0, t ∈ Tid − N , the BS isturned into sleeping mode.

Observation: The power allocated in the tth time slot de-pends on the instantaneous information in this time slot includ-ing the equivalent channel gain gt and available bandwidth W t

for the NRT user and the transmit power ptRT for RT traffic, aswell as the information in other time slots implicitly includedin the water-filling level ν∗ and threshold g∗th. It is noteworthythat the power allocation among T time slots shares the samewater-filling level ν∗ and threshold g∗th.

IV. RESOURCE ALLOCATION WITH CONTEXTINFORMATION

The predictive policy in previous section is not viable inpractice, because when optimizing the resource allocation forthe NRT user in the 1st time slot, the future information gt,W t, pt in the time slot t > 1 is hard to predict if not impossible,owing to the small scale channel fading.

Fortunately, only the water-filling level ν∗ and threshold g∗thdepend on the instantaneous information in future time slots,which do not change across the T time slots, and hence canserve as the resource planning parameters after estimated inthe 1st time slot. Inspired by this observation, in the sequel weestimate these two parameters from problem (5) with the helpof the user level and network level context information.

A. Context Information

In the following, we first introduce the context information tobe exploited, which can be obtained by the CU by using relevantprediction methods in literature.• Network level context information With big data analyt-

ics, the average arrival rate is predictable within a duration,say one day [29, 30]. With the predicted arrival rate of theRT traffic arrival, the probability that

(1− l

L

)· 100% of

the bandwidth is occupied by RT traffic in each frame,i.e., Pjl , Pr(W t = l

LWmax), can be obtained to reflectthe average resource utilization status of a BS, wherej = d tTs e, L is the maximal number of users with RT

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traffic that the BS is able to serve in one time slot si-multaneously. For mathematical tractability and notationalsimplicity, the transmit power reserved for the RT trafficis assumed in proportion to the occupied bandwidth,2 i.e.,ptRT = (1− W t

Wmax)pmax. We assume that the resource uti-

lization probabilities Pjl , j = 1, . . . , Tf , l = 1, . . . , L areknown at the CU as the network level context information.

• User level context information Recent studies have re-ported that the trajectory of a mobile user is predictablewithin a duration, say half an hour [20, 21]. Then, with thepredicted future location and the help of a radio map [22],the large scale fading gains can be obtained. We assumethat the large scale channel gains of the NRT user duringthe Tf frames (i.e., totally T time slots) α1, . . . , αTf areknown at the CU as the user level context information.Denote gt , αd

tTse‖ht‖2/(GN0Wmax), then the equiv-

alent channel gain can be rewritten as gt = Wmax

W t gt. For

Rayleigh fading, gt in the j = d tTs eth frame followsGamma distribution, whose probability density function(pdf) is f j(g). With αj , the pdf is estimated as

f j(g)= 1Γ(Nt)g

(GN0Wmax

αj g)Nt

exp(−GN0Wmax

αj g). (11)

Note that the two levels of context information are in “largescale”, which do not depend on instantaneous channel gains andinstantaneous excess resources available for the NRT user.

B. Estimating Water-filling Level and Threshold

To estimate the water-filling level ν∗ and threshold g∗thfrom problem (5) by exploiting the context information, wetransform the objective function and constraint. To help under-standing, we first rewrite problem (5) via dividing the objectivefunction and “QoS” constraint by T as

minp,N

1T

∑t∈Toc∪N

1ξpt + N

T (pact − psle) (12a)

s.t. 1T

∑t∈Toc∪N W

t ln(1 + gtpt) = B ln 2T∆ , (12b)

pt ≥ 0, pt + ptRT ≤ pmax, t ∈ Toc ∪N ,N ⊆ Tid, (12c)

whose solution is the same as problem (5).In what follows, we transform problem (12) into a problem

with optimization variables ν∗ and g∗th by using the relation ofpt∗ with ν∗ and g∗th in (9).

Proposition 1: When the small scale fading and the availablebandwidth for the NRT user are ergodic in each frame,3 theobjective function in (12a) becomes (13), and the constraint in(12b) becomes (14).

Proof: See Appendix A.

2Simulation results show that this assumption has minor impact on theperformance of the proposed policy, which are not shown for space limitation.By assuming that each RT request occupies fixed amount of resources in eachtime slot, the serving time for RT request depends on its channel condition andthe amount of data required to be transmitted, which affects Pj

l .3The small scale channels are ergodic because they are assumed i.i.d. in each

frame and they change in the timescale of ms but the average channel gainschange in the timescale of second. The random arrival of RT traffic is oftenmodeled as Poisson or interrupted Poisson process in literature, which are allstationary. Since a fixed amount of resource is reserved in each time slots foreach RT request [36], the available resources in the time slots within each frameare stationary. Simulation results show that this assumption has minor impacton the performance of the proposed policy, which are not shown due to spacelimitation.

The maximal and minimal power constraints in (12c) areguaranteed by the function (·)a0 in (9). Then, the parametersν∗ and g∗th can be obtained from the following optimizationproblem with (13) as the objective function and (14) as theconstraint,

P1 : minν,gth

Φp(ν, gth; Pjl , fj(g))

s.t. ΨR(ν, gth; Pjl , fj(g)) = B ln 2

T∆ ,

which is equivalent to problem (12) when the channel andavailable bandwidth are ergodic.

Since the optimal water-filling level and threshold satisfy(10), adding ν − 1

gth≥ 0 as a constraint shrinks the feasible

region but dose not change the optimal solution of problem P1.Besides, only the statistical information are necessary to solveproblem P1, i.e., f j(g) and Pjl . By using the user level andnetwork level context information, the water-filling level andthreshold can be estimated from the following problem,

P2 : minν,gth

Φp(ν, gth; Pjl , fj(g)) (16a)

s.t. ΨR(ν, gth; Pjl , fj(g)) = B ln 2

T∆ , (16b)ν − 1

gth≥ 0. (16c)

The objective function and constraints of problem P2 aredifferentiable with respect to the variables ν and gth since theyare the integral of continuous functions with expressions in(13) and (14). Hence, the optimal solution, ν∗ and g∗th, shouldsatisfy the Karush-Kuhn-Tucker (KKT) conditions [39]. Sinceconstraint (16b) is one of KKT conditions, ν∗ and g∗th shouldsatisfy (16b). Further considering the other KKT conditions, wecan obtain the following proposition.

Proposition 2: ν∗ and g∗th satisfy the following equation,

(ν∗− 1g∗th

)+ξ(pact−psle)−ν∗ ln(ν∗g∗th) = 0, ν∗g∗th > 1. (17)

Proof: See Appendix B.Due to the maximal transmit power constraint, the solution

may not exist, i.e., problem P2 is infeasible. If problem P2 isfeasible, we have following proposition.

Proposition 3: ν∗ and g∗th are the unique solution of the twoequations in (16b) and (17) if problem P2 is feasible, and bothν∗ and ΨR(ν∗, g∗th; Pjl , f

j(g)) decrease with g∗th.Proof: See Appendix C.

This suggests that we can use two-tier bisection searching tofind the optimal solution of problem P2. In the inner tier, wefind ν∗ with given g∗th by bisection searching from (17). In theouter tier, we find g∗th by bisection searching from (16b). Thedetails are given in Algorithm 1.

C. Proposed Predictive Resource Allocation Policy with Con-text Information

The BS sleeping, scheduling and power allocation policy isimplemented in two timescale.

1) Resource planning: When the request of the NRT userarrives at the 1st time slot, the CU estimates the two parametersν∗ and g∗th with Algorithm 1 by using context information.Then, the CU sends the two parameters to the BSs who mayserve the NRT user, which serve as a “ruler” for the BSs todecide when, where and with how much resource to serve theuser.

7

Φp(ν, gth; Pjl , fj(g))

, 1ξ

1Tf

∑Tfj=1

∑L−1l=1 Pjl

lL

∫ 1(ν−pmax)∞01ν

(ν − 1

g

)f j(g)dg + 1

ξ1Tf

∑Tfj=1 PjL

∫ 1(ν−pmax)∞0gth

(ν − 1

g

)f j(g)dg

+ 1ξ

1Tf

∑Tfj=1

∑Ll=1 Pjl

lL

∫∞1

(ν−pmax)∞0

pmaxfj(g)dg + pact−psle

Tf

∑Tfj=1 PjL

∫∞gthf j(g)dg. (13)

ΨR(ν, gth; Pjl , fj(g))

, 1Tf

∑Tfj=1

∑L−1l=1 Pjl

lLWmax


ln(νg)f j(g)dg + 1Tf

∑Tfj=1 PjLWmax


ln(νg)f j(g)dg

+ 1Tf

∑Tfj=1

∑Ll=1 Pjl

lLWmax

∫∞1

(ν−pmax)∞0

ln(1 + pmaxg)f j(g)dg = B ln 2T∆ . (14)

Algorithm 1 Finding water-filling level ν∗ and threshold g∗thInput: estimated large scale channel information in the future

Tf frames, α1, . . . , αTf , the resource utilization probabilityP1l , . . . ,P

Tfl , l = 1, . . . , L, and precision ε0.

Output: estimated water water-filling level ν∗ and estimatedthreshold g∗th .

1: Initialize by finding feasible ν(0) and gth(0) satisfying (17)and making ΨR(ν(0), gth(0); Pjl , f

j(g)) larger than B ln 2T∆ .

If no initial values for ν(0) and gth(0) can be found, thenν∗ =∞, g∗th = 0. Otherwise, go to next step.

2: Iteration i = i+ 1, update gth(i) = 12 (gH + gL).

3: Compute ν(i) from equation (17) with given gth(i) bybisection searching.

4: Update the range of gth according to the relationof ΨR(ν(i), gth(i); Pjl , f

j(g)) and B ln 2T∆ . If

ΨR(ν(i), gth(i); Pjl , fj(g)) > B ln 2

T∆ , then gL = gth(i).Otherwise, gH = gth(i).

5: If |gH − gL| > ε0, return to step 2. Otherwise, go to nextstep.

6: return g∗th = gth(i) and ν∗ = ν(i).

2) Resource allocation: When the values of gt, W t andptRT are available at the BS closest to the NRT user in the tthtime slot, the BS computes optimal power allocation with (9).With pt∗, the optimal scheduling indicator and BS sleeping areobtained.

V. EXTENSION TO MULTIPLE NRT USERS SCENARIO

To minimize the total energy consumed by multiple NRTusers, user scheduling also needs to be optimized. This makesthe problem a combinational optimization problem, whosesolution is hard to find even with the perfect future informa-tion with complexity exponentially increasing with the numberof arrived NRT users and the duration of their deadline, asanalyzed later. When only context information is available,the problem becomes even more challenging in the scenarioof randomly arrived NRT users. This motivates us to find analternative way to extend the predictive resource allocation forsingle NRT user into multi-user case.

First, the user level context information cannot be exploitedusing the same way as in the single NRT user scenario, if eachBS serves multiple NRT users with spatial division multipleaccess (SDMA). This is because with SDMA the equivalentchannel gain g of a NRT user not only depends on the channelof this user but also on the small scale channel gains of otherco-scheduled NRT users. Before user scheduling, g and hencef j(g) cannot be obtained, but without f j(g) for each user, theresource planning can not be made for multiple NRT users.To circumvent this difficulty, time division multiple access(TDMA) or frequency division multiple access (FDMA) canbe employed to serve multiple NRT users in different timeslots or frequency. Then, each BS can employ maximum ratiotransmission to serve each NRT user in each time slot, andhence f j(g) can still be obtained as in (11). For easy exposition,we consider TDMA.4

Second, since there may exist multiple NRT users in eachcell who compete for the time slots, the network level contextinformation (i.e., the resource utilization status) for each usershould also reflect the resource occupied by other arrived NRTrequests except the background traffic.

In the sequel, the subscript k will be added to the notationsto distinguish different NRT users, say, the kth NRT user needsto convey Bk bits before Tk time slots after its request arrives.

A. Resource Planning Using Context Information

Upon each request arrival, the CU needs to make the resourceallocation plan for each of the NRT users. Except finding thewater-filling level and threshold to assist time slot scheduling,power allocation, and BS sleeping, the CU also needs to deter-mine the number of bits ought to be accumulatively conveyed inthe end of each frame to assist user scheduling in each time slot,by using the user and network levels of context information.

1) Updating network level context information: With theuser level context information from the NRT users who havesent their requests, the CU can obtain the number of NRT users

4FDMA techniques such an orthogonal frequency division multiple accesscan also be employed, but the network information update will be morecomplicated, where the CU needs to plan how many subcarriers should beassigned to multiple users.

8

in the coverage of each BS. Denote the NRT user set of the ithBS in the jth frame as Kji .

With the user level and the original network level contextinformation (which only consider the resource utilization statusof the background traffic), the CU can make a rough estimatefirst for the water-filling level and threshold for a NRT user inKji without considering other arrived NRT users using Algo-rithm 1, and then for the power allocated to the NRT user with(9). According to such estimation, multiple NRT users in Kjimay be allocated with non-zero power at the same time slot,which reflects the conflict in preference. Without resolving theconflict, a large portion of the data of some NRT users maynot be transmitted before their expected “deadline” due to themutual interference. To avoid this, in what follows we firstderive the probability that the conflict happens for each NRTuser, with which the network level context information can beupdated considering the resources occupied by both the RT andNRT traffic.

Since the NRT users in Kji have different thresholds andwater-filling levels, the probability that non-zero power is al-located to each NRT user in the time slots of the jth frame,is different. It means that some users may prefer to transmitin the jth frame and others may not. When the ith BS isbusy with background traffic (i.e., l = 1, . . . , L − 1), theprobability that non-zero power will be allocated to the kthNRT user in Kji , denoted as Pjtr,l,k, can be derived from (9) asPjtr,l,k = Pr(ptk > 0, j = d tTs e) = Pr( W t

Wmaxν∗k − W t

Wmax

1gtk>

0, j) = Pr(gtk >1ν∗k, j) by considering gtk = Wmax

W t gtk. When

the BS does not serve the RT traffic (i.e., l = L), the probabilitycan also be derived from (9) as Pjtr,l,k = Pr(gtk > g∗th, j). Then,by using the user level context information of the kth NRT user,the probability that the user prefers to transmit in the jth frameis

Pjtr,l,k =

∫∞1ν∗k

f jk(g)dg, l = 1, . . . , L− 1,∫∞g∗th,k

f jk(g)dg, L.(18)

Since the resource allocation of multiple NRT users is inde-pendent after the resource plans are made for them, the conflictprobability for the kth user in Kji can be derived as

Pjcon,l,k=Pjtr,l,k(1−∏q∈Kji ,q 6=k

(1−Pjtr,l,q)), l=1, . . . , L, (19)

where 1−∏q∈Kji ,q 6=k(1− Pjtr,l,q) is the probability that other

users in Kji prefer to transmit.To handle the conflict, we can assign different time slots to

multiple NRT users. Because Pjtr,l,k ≤ 1, i.e., the NRT usersmay not transmit in all the assigned time slots, the time slotsassigned to different users are not necessarily to be all different.On the other hand, since a NRT user can transmit only in theassigned time slots, which may not be with high channel gains,the energy consumed for the user may increase. To balancethe interference caused by the conflict and the energy increasecaused by not using the best channels, the NRT users whohave high conflict probabilities are only allowed to use sometime slots in the jth frame, and those who have low conflictprobabilities can employ all the time slots. Here we introducea time resource use ratio ζjl,k ∈ [0, 1], which is the percentage

of the time slots assigned to the kth NRT user in the jth framewhen (l/L)×100% of bandwidth is available for the NRT usersin the frame.

To control the conflict probability not exceeding a smallvalue of θ, we divide the NRT users in Kji into two types.For the first type of NRT users, denoted as Kji , the conflictprobability is lower than θ, and their time resource use ratiosare ζjl,k = 1, k ∈ Kji . For the second type, i.e., the NRT users inKji − Kji , the conflict probability is higher than θ, and the timeresource use ratio is ζjl,k , ζjl,k + sjl , where ζjl,k is the fractionof the time slots orthogonally allocated to different NRT usersand sjl is the shared time slots among the NRT users, as shownin Fig. 2.

Proposition 4: When (l/L) × 100% of bandwidth resourcesis available at the ith BS for the NRT users in the jth frame, thetime resource use ratio for the kth user in Kji − Kji to controlthe conflict probability as θ can be approximated as

ζjl,k ≈Pjcon,l,k

θ+∑k∈Kj

i−Kj

i(Pjcon,l,k−θ)

. (20)

Proof: See Appendix D.When θ = 0, all the NRT users in Kji are assigned withdifferent time slots to avoid conflicts with others. This reflectsa conservative estimation on the network resources, since NRTusers may not prefer to transmit during all the assigned timeslots. For example, if there are two NRT users with conflictprobability Pjcon,l,1 = Pjcon,l,2, then each of them will beassigned half time slots in the jth frame, i.e., ζjl,k = 0.5.However, since ζjl,kPjtr,l,k ≤ 0.5, the percentage of the timeslots truly transmitted in the jth frame by each user is less than0.5.

With the introduced time resource use ratio for each NRTuser, the network level context information for the kth user canbe updated as follows,

Pjl,k = ζjl,kPjl,k. (21)

2) Making resource allocation plan: With the user levelcontext information and the updated network level contextinformation, the CU can refine the estimate of the water-fillinglevel and threshold for the kth NRT user using Algorithm 1,denoting as ν∗k and g∗th,k.

In order to help multi-user scheduling for ensuring the “QoS”of all arrived NRT users, the CU needs to make a plan foreach NRT user of how many bits should be transmitted ineach frame with ν∗k and g∗th,k. Since such a decision is madewithout knowing future instantaneous channel information, itis more reasonable to determine the data amount ought tobe accumulatively conveyed at the end of each frame, whichreflects a transmission progress plan for each NRT user. Thisis because less number of bits can be conveyed in a framewith deep fading. For the kth NRT user, the accumulativelytransmitted data (in nats) until the last time slot of the J th framecan be derived as follows,

ΛB(k, J) =∑JTst=1 R

tk∆ =

∑JTst=1 W

tk ln(1 + gtkp

tk)∆. (22)

B. Resource Allocation with Instantaneous InformationWhen the small scale channel fading and the available band-

width for the NRT user are ergodic in each frame, by using

9

16

ζjl,k1= 1

Assigned time resource use ratio in the jth frame

ζjl,k2= 1

ζjl,k3sjl

ζjl,k4sjl

ζjl,k5sjl

NRT users k1, k2 ∈ Kji

NRT users k3, k4, k5 ∈ Kji − Kji

Fig. 1. Example of the time resource use ratio, ζjl,k is the percentage of time slots assigned to the kth NRT user in the jth

frame when lL× 100% of bandwidth at the ith BS is available for NRT users.

With the introduced time resource use ratio for each NRT user, the network level context

information for the kth user can be computed as follows,

Pjl,k = ζjl,kP

jl,k. (22)

2) Making Resource Allocation Plan: With the user level context information and the updated

network level context information, the CU can refine the estimation of the water-filling level and

threshold for the kth NRT user using Algorithm 1, denoting as ν∗k and g∗th,k.

In order to help multi-user scheduling for ensuring the QoS of all arrived NRT users, the CU

needs to make a plan for each NRT user of how many bits should be transmitted in each frame

with ν∗k and g∗th,k. Since such a decision is made without knowing future instantaneous channel

information, it is more reasonable to determine the data amount ought to be accumulatively

conveyed at the end of each frame, which reflects the planned transmission progress for each

NRT user. This is because less number of bits can be conveyed in a frame with deep fading.

For the kth NRT user, the accumulatively transmitted data (in nats) until the last time slot of

the J th frame can be derived as follows,

ΛB(k, J) =∑JTs

t=1 Rtk∆t =

∑JTst=1 W

tk ln(1 + gtkp

tk)∆t. (23)

By using the similar way to obtain ΨR in (15), (23) can be further simplified when the small

scale channel fading in each frame and the available bandwidth for the NRT user are ergodic.

Then, upon substituting the user level context information and updated network level context

information, (23) can be derived as follows,

Fig. 2. Example of the time resource use ratio, ζjl,k is the percentage of time slots assigned to the kth NRT user in the jth frame when (l/L)×100% of bandwidthat the ith BS is available for NRT users.

the similar way to derive (14) we can show that (22) hassimilar expression as in (14), which is a function of ν∗k , g∗th,k,Pjl,k, and f jk(g). In fact, we can show that ΛB(k, Tf,k) =

Ts∆ΨR(ν∗k , g∗th,k; Pjl,k, f

jk(g)) = Bk ln 2.

It indicates that if each BS can serve the scheduled NRT usersaccording to the planned transmission progress, all the NRTusers can complete the transmission of their required amountof data before the expected “deadlines”. This suggests that thetask of user scheduling is to catch up the transmission progressΛB(k, J) in the J th frame for the kth NRT user.

In the tth time slot, the set of NRT users that enter thecoverage of the ith BS is Kd

tTse

i . To maximize the probabilitythat all the NRT users can convey the required data before theirdeadlines, the ith BS should schedule the NRT users from theset of users who have not caught up with the progress (denotedas Kd

tTse

i,pro , {k|ΛB(k, d tTs e) −∑tτ=1R

τk∆ > 0, k ∈ Kd

tTse

i },where

∑tτ=1R

τk∆ is the actually transmitted amount of data

depending on the resource allocation), and transmit as manybits as possible until a given number of bits are conveyed ineach frame. Therefore, the BS should select the NRT user withthe maximal data rate from the set Kd

tTse

i,pro as,

k∗ = arg max

{Rtk, k ∈ K

d tTs ei,pro

}. (23)

C. Predictive Resource Allocation Policy with Context Infor-mation

The proposed policy is implemented in multiple timescale asfollows.

1) Resource planning: When the request of the kth NRTuser arrives, the CU makes the resource allocation plan accord-ing to the context information of the kth user and other arrivedNRT users. Then, the CU sends the parameters ν∗k , g∗th,k andΛB(k, J), J = 1, · · · , Tf,k to the BSs who may serve the kthNRT user. When a new NRT user arrives, the water-filling level,threshold and the transmission progress for each NRT user areupdated.

2) Resource allocation: When the values of gtk,W t and ptRT

are available at the BS closest to the kth NRT user in the tth timeslot, the BS computes the power allocation for the kth NRT userwith (9) and also the corresponding transmit rate Rtk from (1).Then, the BS selects one user to serve in the tth time slot with(23), and the power allocated to the user is pt∗k∗ , with which theBS sleeping and time slot scheduling indicator can be obtained.

Complexity Analysis: With future instantaneous informa-tion, the resource for multiple users arrived in T time slots canbe allocated, while with context information only the resourceallocation plans can be made and need to be re-made when anew NRT user arrives. To simplify the comparison and get arule of thumb on the complexity, we consider the case whereK NRT users arrive in the 1st time slots and the deadlines areall T = TfTs time slots. With instantaneous information, themulti-user resource allocation needs to find the binary variablesmtik (indicating whether the kth user is scheduled by the ith BS

in the tth time slot) and the continuous variables ptk in each timeslot. The searching space ofmt

ik is 2KT . For a given schedulingstatus of multiple users, the solutions ofK transmit power min-imization problems are with water-filling structure and can beobtained with complexity in an order of O(KT log2 T ). Then,the complexity is in an order of O(2KTKT log2 T ). For theproposed predictive resource allocation policy, most computa-tion operations come from making the resource plan. Since thecomplexity to compute Tf numerical integrations for each userscales with T , the complexity for updating the network levelcontext information is in an order of O(KT log2

21ε0

), whereε0 is the bisection searching precision. Then, the complexityfor making the resource plan for K users is in an order ofO(K2T 2 log4

21ε0

). It is shown that the proposed policy has amuch lower complexity.

VI. SIMULATION RESULTS

In this section, we evaluate the proposed predictive resourceallocation by simulations.

Consider a network with multiple hexagonal macro cellseach with radius D = 250 m, where each BS is equipped withNt = 4 antennas. Each frame contains 100 time slots (i.e.,Ts = 100) and the interval of each time slot is ∆ = 10 ms,hence the interval of each frame is one second. The maximaltransmit power and bandwidth of each BS are pmax = 40 W andWmax = 10 MHz. The path loss model is 35.3 + 37.6 log10(d),where d is the distance between the BS and user in meter[40]. The noise variance is σ2 = −95 dBm for 10 MHz. Thecircuit power consumption of a BS in active mode is pact =233.2 W [38], while the circuit power consumption in sleepingmode is set to psle = 10 W as an optimistic estimate [41]and psle = 150 W as a conservative estimate [38]. The poweramplifier efficiency is ξ = 21.3% [38]. The BSs serve multiplemobile NRT users who randomly arrive at different locations

10

and times, during which each BS also serves randomly arrivedbackground traffic, as shown in Fig. 3.

The request arrival of the background traffic is modeled asPoisson process with average rate λi for the ith BS. Eachrequest of background traffic is assumed to occupy the samebandwidth of 2 MHz (i.e., 20% of 10 MHz) with transmitpower 8 W (i.e., 20% of 40 W). To reflect the difference inrequests and channel conditions of different RT users, theirservice time follows exponential distribution with mean V = 2time slots [36]. Then, each BS can serve at most L = 5 requestsof background traffic in one time slot, and the newly arrivedrequest will not be admitted if the BS has been fully occupied.In the simulation, the background traffic load will not changeduring the Tf frames but may differ for different BSs. Twotypical average arrival rates are considered, where λi = 20and λi = 50 requests/s respectively reflect a light and a heavyload.5 The corresponding network level context informationobtained from simulation is listed in Table I and illustrated inFig. 4, where Pjl = Pr(W t = l

LWmax) is the probability that(l/L)×100% of bandwidth is available for the NRT users in thejth frame (e.g., when λi = 20 requests/s and V = 2 time slots,Pj5 = 0.6018, which indicates that the time slots with availablebandwidthWmax account for 60.18% of all the time slots in thejth frame).

The requests of the NRT users also arrive following Poissonprocess but with average rate λNRT. After initiating the requestat random locations in the roads, the NRT users move acrossmultiple cells along the roads with speeds shown in Fig. 3. Therequested file of each NRT user is set with the same size of Bbits and the expected “deadline” is set as Tf = 100 frames, i.e.,100s, after it arrives. Shannon Capacity is considered in (1), i.e.,G = 0 dB. The above setup will be be used in the simulationunless otherwise specified.

In the sequel, we first simulate the single NRT user case withperfect context information to show the impact of backgroundtraffic load, circuit power, and file size on the energy saving.Then, we show the impact of imperfect context informationconsidering the prediction errors on traffic load, route, velocityand positioning. Finally, we simulate the multiple NRT userscase.

A. Single NRT User Case: Perfect Context Information

In the simulation, the NRT user only moves across two cellsin a line during the 100s after it sends the request. The nearestdistance between the user and the BSs is 100 meters. When thedeadline is set longer, the NRT user can move across more cells,while the results are similar. The results are obtained from 100Monte Carlo trails, where the trajectory stays the same but thesmall scale fading channel is subject to i.i.d. Rayleigh blockfading and the resource occupied by background traffic variesaccording to the Poisson process with given average requestarrival rate. Since context information is exploited to saveenergy for the NRT traffic, we only show the energy consumed

5In practice, traffic load may not be stationary, and the predicted trafficvolume is often with rough resolution (say five minutes [30]). We also simulatea scenario where λi in different frames are randomly selected, and we use thetime-averaged average rate in all frames to compute Pj

l in each frame. Theresults are very close to those obtained with stationary traffic load in the sequel,and hence are not shown for conciseness.

for this kind of traffic in the single user case, which is thetotal energy consumption minus the basic energy consumption,∑Tt=1(pttot − ptB)∆.To show the impact of different levels of context information

on saving energy, the following approaches are simulated. Withall approaches, the BS is turned into sleeping mode whenthere is no power allocated to the NRT user and there is nobackground traffic.

1) SE-maximizing without context information (Legend“SE-No context”): Without considering context informa-tion, the BS closest to the NRT user serves the user asbest effort service that maximizes the achievable rate ineach time slot, i.e., pt∗ = pmax − ptRT.

2) EE-maximizing with application level context informa-tion (Legend “EE-App Context”): With the applicationlevel context information, the BS can obtain the remain-ing data to be transmitted for the NRT user and theremaining time before the deadline in each time slot, andhence can compute the expected transmit rate in the re-maining duration as Rt0. Then, according to the principleof EE-optimal resource allocation (i.e., maximizing theinstantaneous EE in each time slot under the constraintof the QoS required by the service), the BS closest to theNRT user serves the user with optimal transmit power aspt∗ = arg max Rt

pttot−ptB, s.t. Rt ≥ Rt0, pmax − ptRT ≥

pt ≥ 0.3) Resource allocation with network, user and application

level context information (Legend “All Context ”): Thisis the proposed predictive resource allocation.

4) Resource allocation with all future information (Legend“Upper Bound”): This is the optimal solution in sectionIII with perfect future instantaneous information.

In Fig. 5 and Fig. 6, we show the transmission behavior ofthese approaches and the corresponding energy consumptionunder different arrival rates of background traffic, circuit pow-ers, and file sizes. We can see that the proposed predictiveresource allocation policy performs almost the same as theupper bound, and its energy saving gain is dramatic whenthe file size is relatively small. Comparing Fig. 5(a) and 5(b),we can see that the proposed policy transmits more when theBS is heavyly loaded, since there are more chances to exploitthe excess resources without extra circuit energy consumptionin this simulation setting. Comparing Fig. 5(c) and 5(d), wecan observe that when the circuit power gap between activeand sleeping modes (i.e., pact − psle) is larger, more bits aretransmitted when the NRT user is in the center of the cell.

B. Single NRT User Case: Imperfect Context Information

In the following, we show the impact of the imperfect pre-diction of the network level context information reflected bythe average arrival rate of RT traffic and the user level contextinformation due to prediction uncertainty/errors of route, veloc-ities and positioning, respectively.

In the simulation, the user’s real route is the same as in lastsubsection, the real speed is 10 m/s, λ1 = λ2 = 20 requests/s,psle = 150 W, B = 5 Gbits (the same as in Fig. 5(c)). In [18],the route prediction precision is defined as the probability that

11

10m

/s

10m/s10m/s

10m/s10m/s

10m/s

10m/s

50m

50m

10m

/s10m

/s

10m

/s

10m

/s10m

/s

10m

/s10m

/s

10m

/s10m

/s

5m

/s5m

/s

5m/s

5m/s

5m/s

5m/s 5m/s

5m/s5m/s

5m/s

5m/s

5m/s

150m

Trajectory 1

Trajectory 2

150m

Real Trajectory Estimated Trajectory

500m

150m

Road 1

Road 2

150m

NRT user 1

150m

150m

RT users Real Trajectory of NRT user NRT users Estimated Trajectory of NRT user

10m

/s

5m/s

5m/s5m/s

5m/s

10m/s

5m

/s

5m

/s

10m

/s

100m

Real Trajectory Estimated Trajectory

10m/s

10m/s

10

0m

Background Traffic

NRT Users

Fig. 3. Illustration of the multi-cell network. The speed marked on each road is the speed for the NRT users on it.

TABLE INETWORK LEVEL INFORMATION FOR THE iTH BS IN THE jTH FRAME WITH ARRIVAL RATE λi

Pj5 Pj

4 Pj3 Pj

2 Pj1 Pj

0λi = 20 requests/s, V = 2 time slots 0.6018 0.3056 0.0776 0.0131 0.0017 0.0002λi = 50 requests/s, V = 2 time slots 0.2812 0.3569 0.2262 0.0961 0.0303 0.0093

Occupied by RT Traffic

Available for NRT Traffic with Extra Circuit Energy Consumption

Ban

dwid

thB

andw

idth

Time slots in Framejth20, 2i V

Time slots in Framejth50, 2i V Available for NRT Traffic without Extra Circuit Energy Consumption

Fig. 4. Illustration of network level context information, where the time slots is re-ordered according to the bandwidth status.

the route can be predicted correctly where several routes arepredicted at the beginning of the trip, denoted as η. Accordingto the prediction results in [18], η ranges from 40% to 97%.For easy exposition, we consider that two routes are predictedby the CU where route 1 is real, as shown in Fig. 7(a) (thenthe probability that the prediction result is route 1 will be η,and the probability that the prediction result is route 2 willbe 1 − η). In the 100 monte carlo trails, the route predictionresult is randomly generated according to the probability ofη. In practice, if the route prediction is incorrect, the CU canreplan the resource allocation. For example, the CU can checkthe predicted position with the position reported from GPSin each frame, and if the error exceeds a threshold, it willupdate the water-filling level and threshold according to theleft bits for the remaining route. To reflect velocity predictionerrors in the considered horizon, we employ the results in [42]that predict the location within the duration around 100∼200seconds. Specifically, the velocity prediction error is a zeromean Gaussian random variable with standard deviation σv =1∼5 m/s. Considering that the positioning error is within 10meters with GPS [17], and the predicted trajectory may deviatefrom the real trajectory (e.g., in different lanes of the road), weconsider two deviations Ad = 5 m and Ad = 10 m.

In Fig. 7(b), we provide the average energy consumption forserving NRT user when only the network level information pre-

diction is inaccurate and the corresponding water-filling leveland threshold are computed with the predicted average arrivalrate. The actual average arrival rate is λi = 20 requests/s, andthe predicted average arrival rate is a constant within 16 ∼ 24requests/s considering that the maximal prediction error of thetraffic volume in every five minutes is within 20% as reported in[30]. It can be observed that the proposed policy is robust withprediction errors on the network level information.

In Fig. 7(c), we show the average energy consumption forserving NRT user versus the prediction precision of route η,meanwhile the velocity prediction is with prediction errors ofdifferent standard deviations. It can be observed that with there-planning, the proposed policy is robust to the predictionuncertainty of the route. With prediction errors on the velocity,the energy consumed by the proposed policy slightly increases,even when the correct route prediction probability is low, sayη = 40%.

In Fig. 7(d), we show the impact of the deadline settingfor the NRT traffic and the prediction errors of positioning onenergy saving where the speeds are accurate. The predictedpositions are always far from BSs with deviation Ad. Whenthey are near, the flexible time will be longer (from one frameshown in Fig. 5(a) and 5(b) to five frames) and little moreenergy is consumed, which is not shown in the figure. It canbe seen that EE-maximizing approach consumes more energy

12

0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8x 10

7

Frame

Ave

rage

Tra

nsm

it R

ate

(bps

)

SE−No ContextEE−App ContextAll ContextUpper Bound

(a) λ1 = λ2 = 20, psle = 150 W, B = 500 Mbits

0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8x 10

7

Frame

Ave

rage

Tra

nsm

it R

ate

(bps

)


(b) λ1 = 20, λ2 = 50, psle = 150 W, B = 500 Mbits

0 20 40 60 80 100 1200

2

4

6

8

10

12x 10

7

Frame

Ave

rage

Tra

nsm

it R

ate

(bps

)


(c) λ1 = λ2 = 20, psle = 150 W, B = 5 Gbits

0 20 40 60 80 100 1200

2

4

6

8

10

12x 10

7

Frame

Ave

rage

Tra

nsm

it R

ate

(bps

)


(d) λ1 = 20, λ2 = 20, psle = 10 W, B = 5 Gbits

Fig. 5. Average transmit rate in each frame, Tf = 100. The small pike indicates the average rate in the “flexible time”.

(a) (b) (c) (d)0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Ave

rage

Ene

rgy

Con

sum

ptio

n of

NR

T T

raffi

c (J

)


100% 100%

100%

100%

60% 60%1.2%0.7% 1%0.5%

46%38%38%

75%

53%53%

Fig. 6. Average energy consumed at the BS for serving the NRT user. Thevalue on the top of each bar is the percentage of the energy consumed by thecorresponding approach compared to “SE-No Context”.

than the proposed predictive resource allocation, and the energyconsumption of the proposed policy decreases as the deadlineincreases. With accurate large scale channel information, theproposed policy almost performs the same as the upper boundfor different deadlines. Inaccurate positioning leads to moreenergy consumption, which increases with the deadline slightly.

In Table II, we provide the completion ratio achieved by theproposed policy, which is defined as the ratio of the numberof bits transmitted before the deadline to B. With accuratelarge scale channel information, most of the bits are transmittedbefore the “deadline”, which makes the flexible time very short.With inaccurate large scale channel information due to position-ing, all the bits are transmitted within the “deadline”, becausethe channel condition obtained from the predicted trajectoryis worse than the real channel condition in the consideredsimulation setting.

C. Multiple NRT Users Case: Perfect Context Information

To show the contribution of the energy consumed by theNRT users, here we evaluate the average total energy consumedby the BSs for both NRT users and background traffic in a

13

100m

100m

10 m/s

Real RoutePredicted Route 1 Predicted Route 2

5 m/s

1 20 2

3

20

50

(a)

16 17 18 19 20 21 22 23 24Predicted Average Arrival Rate of RT Traffic (request/s)

5000

5200

5400

5600

5800

6000

6200

Ave

rage

Ene

rgy

Con

sum

ptio

n of

NR

T T

raffi

c (J

) EE-App ContextAll ContextUpper Bound

Actual average arrival rate of RT traffic

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Prediction Precision of Route 2

5000

5200

5400

5600

5800

6000

6200

Ave

rage

Ene

rgy

Con

sum

ptio

n of

NR

T T

raffi

c (J

)

EE-App ContextAll Context without Replan, <v=0m/s

All Context with Replan, <v=5m/s



Upper Bound

(c)

100 150 200 250Deadline Tf (seconds)

1000

2000

3000

4000

5000

6000

7000

8000

Ave

rage

Ene

rgy

Con

sum

ptio

n of

NR

T T

raffi

c (J

)

EE-App ContextAll Context, Ad=10

All Context, Ad=5

All Conext, Ad=0

Upper Bound

(d)

Fig. 7. (a) Real and predicted routes. (b) Impact of imperfect network level context information. (c) Impact of imperfect route and velocity. (d) Impact ofpositioning errors and longer deadline. The energy consumed by traditional transmission (i.e., “SE-No Context”) is 1.338 × 104 J, which is not shown in thefigure. When all prediction errors are taken into account, e.g., the predicted average arrival rate is 24 requests/s, η = 40%, σv = 5 m/s, and Ad = 10 m, theenergy consumption is 5459 J.

TABLE IIAVERAGE COMPLETION RATIO WITH PREDICTION ERRORS VS. DEADLINE

Deadline 100 s 125 s 150 s 175 s 200 s 225 s 250 sCompletion Ratio, Ad = 0 99.88% 99.66% 99.68% 99.73% 99.82% 99.53% 99.49%

Completion Ratio, Ad = 5 and 10 100% 100% 100% 100% 100% 100% 100%

duration of 200 frames. The NRT users randomly arrive thecells following Poisson process in a duration of 100 frames.In the subsequent 100 frames, the background traffic continuesto arrive randomly but no requests of NRT users arrive. EachNRT user requests a file with size B = 500 Mbits and thedeadline is set as Tf = 100 frames. The results are obtainedfrom 100 Monte Carlo trails, where the request arrival rates andthe trajectories of multiple NRT users change randomly in everytrail. In Fig. 8, we show the average total energy consumptionof three values of λNRT. It can be seen that although theenergy consumption of NRT traffic increases with the arrivalrate of NRT requests (i.e., with the number of NRT users), thepredictive policy we proposed reduces the energy consumed by

the NRT traffic remarkably.

VII. CONCLUSION

In this paper, we proposed a predictive resource planningand allocation policy for non-real-time (NRT) traffic by ex-ploring different information available/predictable in differenttimescales, when partial resources of the BS is occupied bybackground traffic. Three levels of context information wereexploited to save the energy consumed at the BSs. We firstderived a policy for a single NRT user, where the resourceplanning is made by using two parameters translated from thepredictable information as a “ruler” to decide when, where andwith how much resources to serve the user. We then extended

14

0.1 0.3 0.50

1

2

3

4

5

6

7

8

9

10x 10

4

λNRT=

Tot

al E

nerg

y C

onsu

mpt

ion

of B

ackg

ound

Tra

ffic

and

NR

T T

raffi

c (J

)

NRT, SE−No ContextNRT, All ContextRT

λNRT=λNRT=

Average Arrival Rate of NRT Users per Second λNRTFig. 8. Average total energy consumption of the system vs. average arrivalrate of NRT users. The average arrival rate of background traffic is λi = 20requests/s, psle = 150 W, B = 500 Mbits, and Tf = 100 frames (i.e., 100 s).

the policy to the practical scenario with multiple randomlyarrived NRT users except the background traffic, where theplanning further decides how many bits ought to be transmittedto each NRT user in the end of each frame. Simulation resultsshowed that by exploiting the future average channels gainsand network resource usage probability translated from thepredictable information with big data and by setting a long“deadline”, the proposed policy can save energy remarkably,even when the prediction errors are taken into account.

APPENDIX APROOF OF PROPOSITION 1

According to (9), gt = Wmax

W t gt and considering that N ∗

is the set of scheduled idle time slots with cardinality N∗ =∑Tid

1(gt ≥ g∗th), the objective function with variables pt

and N in (12a) can be re-written as an objective function withvariables ν and gth, which is

1T

∑t∈Toc∪N

1ξ

(W t

Wmaxν − W t

Wmax

1gt

)pmax−ptRT

0+∑

Tid1(gt≥gth)

T (pact − psle). (A.1)

When the small scale fading and the available bandwidth areergodic in each frame, the time average can be replaced byensemble average. By considering N = {t|gt ≥ gth, t ∈ Tid}and pmax − ptRT = W tpmax/Wmax, the first term in (A.1)can be derived as in (A.2), where (a) is obtained by taking theexpectation over W t, (b) is obtained by considering the PDF ofgt in (11), and (c) comes from the fact that the maximum andminimum values of (ν − 1

g )pmax

0 are positive value pmax and 0,respectively.

Again, considering that the small scale channel and availablebandwidth are ergodic, the second term in (A.1) can be furtherderived as,

∑Tid

1(gt≥gth)

TfTs(pact − psle)

=pact−psle

Tf

∑Tfj=1PjLEg{1(g ≥ gth)}

=pact−psle

Tf

∑Tfj=1PjL

∫∞gthf j(g)dg. (A.3)

Together with (A.2), we obtain (13).By substituting (9) into (12b), considering thatN = {t|gt ≥

gth, t ∈ Tid}, pmax − ptRT = W tpmax/Wmax, and gt =Wmax

W t gt, constraint (12b) can be derived as

B ln 2T∆ = 1

T

∑t∈Toc∪N W

t ln(1 + gtpt)

= 1TsTf

∑t∈Toc∪Tid

W t ln(1 + pmaxgt)1(gt > 1

(ν−pmax)∞0)+

1TsTf

∑t∈Toc

W t ln(νgt)1( 1(ν−pmax)∞0

≥ gt ≥ 1ν )+

1TsTf

∑t∈Tid

Wmax ln(νgt)1( 1(ν−pmax)∞0

≥ gt ≥ gth).

When gt and W t are ergodic, the time average can be replacedby ensemble average, then

B ln 2T∆

(a)= 1Tf

Tf∑j=1

L∑l=1

PjllLWmax

∫∞1

(ν−pmax)∞0

ln(1 + pmaxg)f j(g)dg+

1Tf

Tf∑j=1

L−1∑l=1

PjllLWmax


ln(νg)f j(g)dg+

1Tf

Tf∑j=1

PjLWmax


ln(νg)f j(g)dg, (A.4)

where (a) is obtained by taking expectation over W t and gt

similar to (A.2). Therefore, proposition 1 is proved.

APPENDIX BPROOF OF PROPOSITION 2

From constraint (16c), ν∗ and g∗th satisfy ν∗ ≥ 1g∗th≥ 0, i.e.,

ν∗g∗th ≥ 1. The KKT conditions of problem P2 can be writtenas

∂L∂ν

∣∣ν=ν∗,gth=g∗th

= 0, ∂L∂gth


= 0, ν∗g∗th ≥ 1,

ρ(ν∗ − 1g∗th

) = 0, ρ ≥ 0, (16b),

where L is the Lagrange function, which is (B.1).From ∂L

∂ν


= 0 and ∂L∂gth


= 0, wehave (B.2) and (B.3).

When ρ > 0, since ρ(ν∗ − 1g∗th

) = 0 in KKT conditions,ν∗ − 1

g∗th= 0. However, by substituting ν∗ − 1

g∗th= 0

and considering pact − psle > 0, (B.3) becomes −(pact −psle)f(g∗th) 1

Tf

∑Tfj=1 PjL−ρ 1

(g∗th)2 < 0, i.e., one KKT conditionwill not be satisfied. This indicates that ρ = 0, i.e., ν∗g∗th > 1is proved.

When ρ = 0, since 1Tf

∑Tfj=1

(∑L−1l=1 Pjl

lL

∫ 1(ν∗−pmax)∞01ν∗

f j(g)dg + PjL∫ 1

(ν∗−pmax)∞0g∗th

f j(g)dg)

is the summation of in-tegrals of pdf function and is always positive, (B.2) becomes,

1ξ + Wmaxω

ν∗ = 0. (B.4)

Since f(g∗th) 1Tf

∑Tfj=1 PjL > 0 always holds, (B.3) can be

further simplified as follows,1ξ

(1g∗th− ν∗

)− (pact − psle)− ωWmax ln(g∗thν

∗) = 0. (B.5)Substituting ω = − ν∗

ξWmaxobtained from (B.4) into (B.5),

we can obtain (17).

15

1T

∑t∈Toc∪N

1ξ

(W t

Wmaxν − W t

Wmax

1gt

) Wt

Wmaxpmax

0= 1TfTs

∑t∈Toc

1ξ

W t

Wmax

(ν − 1

gt

)pmax

0+ 1TfTs

∑t∈Tid

1ξ

(ν − 1

gt

)pmax

01(gt ≥ gth)

(a)= 1ξ

1Tf

∑Tfj=1

∑L−1l=1 Pjl

lLEg

{(ν − 1

g

)pmax

0

}+ 1

ξ1Tf

∑Tfj=1 PjLEg

{(ν − 1

g

)pmax

01(g ≥ gth)

}

(b)= 1ξ

1Tf

∑Tfj=1

∑L−1l=1 Pjl

lL

∫∞0

(ν − 1

g

)pmax

0f j(g)dg + 1

ξ1Tf

∑Tfj=1 PjL

∫∞gth

(ν − 1

g

)pmax

0f j(g)dg

(c)= 1ξ

1Tf

∑Tfj=1

∑L−1l=1 Pjl

lL


(ν − 1

g

)f j(g)dg + 1

ξ1Tf

∑Tfj=1 PjL


(ν − 1

g

)f j(g)dg

+ 1ξ

1Tf

∑Tfj=1

∑Ll=1 Pjl

lL

∫∞1

(ν−pmax)∞0

pmaxfj(g)dg, (A.2)

L = 1ξ

1Tf

Tf∑j=1

L−1∑l=1

PjllL


(ν − 1

g

)f j(g)dg + 1

ξ1Tf

Tf∑j=1

PjL∫ 1

(ν−pmax)∞0gth

(ν − 1

g

)f j(g)dg

+ 1ξ

1Tf

Tf∑j=1

L∑l=1

PjllL

∫∞1

(ν−pmax)∞0

pmaxfj(g)dg + pact−psle

Tf

Tf∑j=1

PjL∫∞gthf j(g)dg

+ω(

1Tf

Tf∑j=1

L∑l=1

PjllLWmax

∫∞1

(ν−pmax)∞0

ln(1+pmaxg)f j(g)dg+ 1Tf

Tf∑j=1

L−1∑l=1

PjllLWmax


ln(νg)f j(g)dg

+ 1Tf

Tf∑j=1

PjLWmax


ln(νg)f j(g)dg − BTf

)− ρ(ν − 1

gth

). (B.1)

(1ξ + ωWmax

ν∗

)1Tf

∑Tfj=1

(∑L−1l=1 Pjl

lL

∫ 1(ν∗−pmax)∞01ν∗

f j(g)dg+PjL∫ 1

(ν∗−pmax)∞0g∗th

f j(g)dg)−ρ = 0, (B.2)

(1ξ

(1g∗th− ν∗

)− (pact − psle)− ωWmax ln(g∗thν

∗))f(g∗th) 1

Tf

∑Tfj=1 PjL − ρ 1

(g∗th)2 = 0. (B.3)

APPENDIX CPROOF OF PROPOSITION 3

By taking derivation on both sides of (17), we have∂ν∗

∂g∗th+ 1

(g∗th)2 − ∂ν∗

∂g∗thln(ν∗g∗th)− ν∗( 1

ν∗∂ν∗

∂g∗th+ 1g∗th

) = 0. (C.1)

Since ν∗g∗th > 1 from proposition 1, i.e., ln(ν∗g∗th) > 0,(C.1) can be re-written as

∂ν∗

∂g∗th= 1

g∗th

(1g∗th− ν∗

)/ln(ν∗g∗th). (C.2)

Since ν∗g∗th > 1 and g∗th > 0, we know that 1g∗th− ν∗ < 0.

Therefore, ∂ν∗

∂g∗th< 0, i.e., ν∗ is a decreasing function of g∗th.

In the sequel, we prove that ΨR(ν∗, g∗th; Pjl , fj(g)) = I1+I2

decreases with g∗th, where

I1, 1Tf

∑Tfj=1

∑L−1l=1 Pjl

lLWmax

∫ 1(ν∗−pmax)∞01ν∗

ln(ν∗g)f j(g)dg+

1Tf

∑Tfj=1

L−1∑l=1

PjllLWmax

∫∞1

(ν∗−pmax)∞0

ln(1 + pmaxg)f j(g)dg,

I2 , 1Tf

∑Tfj=1 PjLWmax

∫ 1(ν∗−pmax)∞0g∗th

ln(ν∗g)f j(g)dg+

1Tf

Tf∑j=1

PjLWmax

∫∞1

(ν∗−pmax)∞0

ln(1 + pmaxg)f j(g)dg.

By taking the derivation of I1 over g∗th, we have

∂I1∂g∗th

= ∂I1∂ν∗

∂ν∗

∂g∗th

= 1Tf

Tf∑j=1

L−1∑l=1

PjlWmaxl

Lν∗

∫ 1(ν∗−pmax)∞01ν∗

f j(g)dg · ∂ν∗∂g∗th

<0, (C.3)

because 1Tf

∑Tfj=1

∑L−1l=1 PjlWmax

lLν∗

∫ 1(ν∗−pmax)∞01ν∗

f j(g)dg

is a summation of integrals of pdf functions, which is a positivevalue, and ∂ν∗

∂g∗th< 0 as shown in (C.2).

By taking derivation of I2 over g∗th, we have∂I2∂g∗th

= 1Tf

Tf∑j=1

PjLWmax

(1ν∗

∂ν∗

∂g∗th

1(ν∗−pmax)∞0∫

g∗th

f j(g)dg−f j(g∗th) ln(ν∗g∗th))

<0, (C.4)

because ν∗g∗th > 1 and ∂ν∗

∂g∗th< 0, ∂I2

∂g∗th< 0.

Therefore, we know that ΨR(ν∗, g∗th; Pjl , fj(g)) is a de-

creasing function of g∗th. This means that if there exists g∗thsatisfying (16b), it will be the unique solution of (16b). Furtherconsidering that ν∗ is a monotonic function of g∗th from (17),ν∗ and g∗th is the unique solution of the two equations (16b)and (17) if the solution exists.

16

APPENDIX DPROOF OF PROPOSITION 4

From the definition of ζjl,k and sjl as illustrated in Fig. 2, sjlpercentage of time resources are employed by all the NRT usersarrived in the network, and the conflict probability Pjcon,j,k canbe obtained with (19). Then, the conflict probability for the kthNRT user after considering the time resource use ratio of twotypes of NRT users can be obtained as,

(ζjl,kPjpcon,j,k + sjlPjcon,j,k)(ζjl,k + sjl )

−1 = θ, k ∈ Kji − Kji ,(D.1)

where Pjpcon,j,k is the conflict probability of the kth second typeNRT user with the first type NRT users when the user employsζjl,k fraction of the time slots. Since the conflict probability ofthe first type NRT users is low, Pjpcon,j,k is small. Then, (D.1)can be approximated as

sjlPjcon,j,k(ζjl,k + sjl )

−1 ≈ θ, k ∈ Kji − Kji . (D.2)

Further considering that∑k∈Kji−K

jiζjl,k + sjl = 1, sjl and ζjl,k

can be obtained when θ > 0 as,

sjl ≈(1 +

∑k∈Kji−K

ji

Pjcon,l,k−θθ

)−1

and

ζjl,k ≈Pjcon,l,k−θ

θ

(1 +

∑k∈Kji−K

ji

Pjcon,l,k−θθ

)−1

Then, we have (20) when θ > 0.When θ = 0, any orthogonal time slot assignment among

all the NRT users can satisfy the conflict constraint. To unifythe expression of the time resource use ratio, we can chooseζjl,k = Pjcon,l,k/

∑k∈Kji−K

ji

Pjcon,l,k as the time resource userratio. Considering the two cases of θ > 0 and θ = 0,proposition 4 is proved.

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