Analytical average throughput and delay estimations for LTE

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Analytical average throughput and delay estimations for LTEuplink cell edge users q

Spiros Louvros ⇑, Michael ParaskevasComputer & Informatics Engineering Department – CIED, Technological Educational Institute (TEI) of Western Greece, Greece

a r t i c l e i n f o

Article history:Received 30 March 2013Received in revised form 13 March 2014Accepted 18 March 2014Available online 10 May 2014

a b s t r a c t

Estimating average throughput and packet transmission delay for worst case scenario (celledge users) is crucial for LTE cell planners in order to preserve strict QoS for delay sensitiveapplications. Cell planning techniques emphasize mostly on cell range (coverage) andthroughput predictions but not on delay. Cell edge users mostly suffer from throughputreduction due to bad coverage and consequently unexpected uplink transmission delays.To estimate cell edge throughput a common practice on international literature is theuse of simulation results. However simulations are never accurate since MAC scheduleris a vendor specific software implementation and not 3GPP explicitly specified. This paperskips simulations and proposes an IP transmission delay and average throughput analyticalestimation using mathematical modeling based on probability delay analysis, thus offeringto cell planners a useful tool for analytical estimation of uplink average IP transmission.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Nowadays IP based multi-service wireless cellular networks mobile handsets are requesting reliable data transmissionfrom QoS perspective point of view [1–4]. In 3GPP standards four negotiated QoS profiles are defined based on four existingQoS classes [3]. These QoS classes define specific attributes related to traffic integrity which QoS profiles should include,which among others are mean and peak throughputs, precedence, delivery delay and Service Data Units (SDU) error ratio[3]. A new generation of wireless cellular network since 2010, called Enhanced UTRAN (E-UTRAN) or Long Term Evolution(LTE) workgroup of 3GPP, has been evolved providing advantages to services and users [4,5]. LTE requirements, compared toprevious mobile broadband networks (HSPA, 3G), pose strong demands on throughput and latency, requesting new multipleaccess techniques over air interface and simplified network architecture [6,7]. Using OFDM/SC-FDMA technology a minimumgroup of 12 sub-carriers of total 180 kHz bandwidth is known as Resource Block (RB). In a frequency-time domain resourcegrid a Schedule Block (SB), a unit of resource allocated by MAC scheduler, is defined as a resource unit of total 180 kHz band-width (12 sub-carriers of 15 kHz each) in the frequency domain and 1ms sub-frame duration (known also as TransmissionTime Interval (TTI)) in time domain.

From cell planning perspective uplink is always the weakest link in the power-link budget and throughput analysis, forboth outdoor and indoor to outdoor coverage. MAC scheduler, residing in eNodeB, is responsible for dynamically allocatinguplink/downlink resources [8]. The primary goal of uplink scheduler is the ability to allocate an appropriate amount ofconsecutive resources in the SC-FDMA with the appropriate transport format, modulation to appropriately map symbols

http://dx.doi.org/10.1016/j.compeleceng.2014.03.0080045-7906/� 2014 Elsevier Ltd. All rights reserved.

q Reviews processed and approved for publication by Editor-in-Chief Dr. M. Malek.⇑ Corresponding author. Tel.: +30 2631058484.

E-mail addresses: [email protected] (S. Louvros), [email protected] (M. Paraskevas).

Computers and Electrical Engineering 40 (2014) 1552–1563

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to bits and coding to protect data and transmitted power per TTI. The secondary goal of scheduler functionality is to appro-priately manage the transmission of uplink SB among neighbor cells to suppress as much as possible the inter-cell interfer-ence (ICI). Mobile operators face quite often QoS problems in case of bad coverage (coverage limited environment) orinterference (Interference limited environment), due to low scheduling decisions of the uplink scheduler. A lot of researchhas been performed on international literature regarding ICI and scheduling decisions focusing on throughput estimationsand coverage cell range probabilities. In [9] authors performed a survey of the Inter Cell Interference Cancellation (ICIC)3GPP feature [10] for interference coordination on LTE MAC scheduler. Interference coordination has also been proposedon [11] where network planning issues have been considered together with remote radio head by the authors. Resource allo-cation on LTE uplink has been also extensively studied on international literature so far in conjunction with throughput per-formance and expected delay of service. To analyse allocation of resources is not easy since MAC scheduler functionality isnot standardized by 3GPP; it is rather left on vendor (Ericsson, Nokia, HUAWEI, etc.) implementations trying to make moreefficient use of available resources for good coverage users. 3GPP describes only the general procedures for scheduling func-tionality and standardizes three functional blocks to be implemented, Scheduler block, Signal to Interference and Noise Ratio(SINR) estimation block and Link Adaptation block. Uplink Scheduler block and SINR block exist in eNodeb; however foruplink transmission Link Adaptation block is implemented on user equipment (UE). In order to depict the MAC functionalityfrom vendor specific solutions, system simulations or drive tests are extensively used on papers in international literature.Indeed authors in [12] proposed a new resource allocation method well-suited for the uplink scenario of LTE allocating fre-quency spectrum among cell users with the goal of maximizing the system’s overall throughput. In [13] authors used powerand packet delay as two important metrics to propose an innovative resource allocation technique for LTE uplink. Authors in[14] proposed a new resource allocation scheme based on the knowledge of buffer statuses and channel conditions to reducethe waste of system resources and improve the aggregate throughput. Although all these research papers have been consid-ering MAC functionality, their proposals are validated based on general or public simulators which do not depict reality sincethe vendor specific MAC software implementation is not public released.

A major metric, not considered so far on international literature, is the evaluation of overall IP packet transmission delayas a function of scheduler resource allocation decisions and channel conditions. Prediction evaluation is considered to besplit into three distinct delay contributions:

� N, number of allocated SB from uplink scheduler: The number of allocated SB is directly related to throughput or in otherwords to packet delay. This delay is also affected by the selected spatial multiplexing mode (MIMO or Transmission diver-sity), number of expected retransmissions, size of IP service packets and the selected MAC packet size. Many researchpapers exist in international literature using either theoretical simulations or analytical probabilistic models trying tocombine packet delay and resource allocation principles. In [15] a semi-analytical macroscopic probabilistic model hasbeen proposed trying to capture channel conditions and MAC resource allocations for different cell load conditions. In[16] authors try to analytically model expected interference and expected channel conditions and combine it withMAC scheduler decisions and throughput. End-to-end QoS performance of Bandwidth and QoS Aware (BQA) schedulerfor LTE uplink, together with delay sensitive traffic thresholds, is evaluated in heterogeneous traffic environment in[17]. A very good approach has been proposed on [18] where packet delays may be deduced from buffer status reports(BSR) from UE’s in LTE uplink. However these delays have not been directly correlated to the expected throughput con-ditions neither the MAC scheduler IP buffering. Although all aforementioned papers have studied the expected number ofresources allocated from MAC decisions they do not consider the reality since allocation of resources from MAC scheduleris vendor specific and only vendor official simulators [19] or drive tests could depict the reality; consequently there is notmuch work on such a topic on international literature. One important such drive test reference is on [20] which will beused later on the mathematical analysis.� n, Scheduler decision: Second expected transmission delay contribution relies on the fact that MAC scheduler never

schedules each UE every TTI = 1 ms due to capacity reasons, QoS service priority issues and finally due to Channel QualityIndex (CQI) reports per UE radio channel conditions; hence an inherent delay has to be considered in the total delay cal-culation. Again this is vendor specific and any analytical estimation has to rely either on public simulators or analyticalmathematical modeling. Few papers exist on international literature. One very good research paper is [21] where authorshave derived a mathematical model for delay estimations. An oldest approach [22] indicates also an innovative algorithmto consider end-to-end delay constraints on MAC scheduler decisions.� P0, UE transmission buffer delay: Third expected transmission delay contribution is the buffer delay on UE transmission

buffer due to QoS class identifier (QCI) scheduling core network priorities. This is a topic considered in seldom in otherpapers in international literature; however its contribution to transmission delay calculations is vital.

All aforementioned research papers never combine predicted delays with cell planning principles and constraints andmost of predicted results are generated from public LTE simulators not following vendor specific solutions; thus estimationsare not accurate for specific network equipments. This paper proposes an analytical mathematical model to predict bufferdelay as an integral part of overall packet transmission delay estimation; uplink delay is considered as a cell planning con-straint, according to 3GPP QoS restrictions, realizing a very interesting metric for operators to understand how the cell plan-ning and coverage conditions affect the uplink packet transmission delays [15]. Moreover average transmission uplinkthroughput is predicted to be considered as analytical tool for cell planning algorithm.

S. Louvros, M. Paraskevas / Computers and Electrical Engineering 40 (2014) 1552–1563 1553


Rest of the paper is organized as follows. On Section 2 an analytical mathematical model, using one Lemma and oneimportant Theorem, is proposed calculating the probability of n packets existing in the system either in scheduled blocksor in the transmission buffer. On Section 3 an explicit calculation for non-delay probability on UE buffer is proposed anda mathematical Theorem is also stated. On Section 4 an overall uplink average IP throughput formula, considering uplinkair interface transmission delay as input, is proposed for cell planning analytical predictions. Applications on cell planningand parameter justifications are analytically presented on Section 5 and final conclusions on Section 6. Finally on AppendicesA and B formal mathematical proofs on delay probabilities for Lemma and Theorems of Sections 2 and 3 are explicitlyprovided.

2. IP packet probability modeling

LTE services are based solely on IP technology. IP service packets are going to be segmented through RLC/MAC layer intoMAC segments and then properly scheduled over SBs on air interface resources [23]. Each MAC packet is supposed to betransmitted completely over the air interface before starting transmission of next MAC packet in a duration of TTI = 1 ms.A number of uplink MAC packets will be buffered on UE transmitter before being scheduled and mapped into SBs; upon arri-val to the eNodeB receiver will be acknowledged on the PDSCH downlink channel. In our mathematical model analysis we doconsider IP segmented packets arriving from upper layers to MAC layer where a single server, known in our case as MACscheduler unit, schedules packets to several resources. Our resources SB in our mathematical model are called channels; con-sequently we do consider in general m parallel channels.

IP packets, before scheduling, are buffered into a queue with finite length. Queue is considered to be empty if there are narrived packets in the system and the occupied resources are less than maximum m channels (SB) available in the radiointerface, otherwise queue contains IP packets. IP packets arrival process is considered to be Poisson with k packet rate ofarrival. Service time lo is considered to be constant for all parallel channels and the reasoning behind constant service timeis the small deviations in transmission delays due mostly on processor load fluctuations. It has to be clear that transit timeeffects are neglected on this analysis since there are no transit effects when scheduler operates as a continuous schedulingprocess. Fig. 1 presents the mathematical model in block diagram format. Considering queue equilibrium, mathematicalanalysis considers always m > k. Define pn the probability of existing specifically n packets in both queue and service at agiven time s and pn the probability that no more than n packets exists in the model at given time s. Since service time isconsidered to be constant a good assumption might be to consider typical unit of time to be the service time lo. FollowingLemma 1 and Theorem 1 provide the probability that specifically n packets exist in the system at the unit of time. Proofs areanalytically provided in Appendix A.

Fig. 1. Scheduler block diagram considering buffering.

1554 S. Louvros, M. Paraskevas / Computers and Electrical Engineering 40 (2014) 1552–1563


Lemma 1. Overall probability pn that specifically n packets exist in the system at the unit of time equals:

pn ¼ pm �kn

n!e�k þ

Xn

k¼0

pmþk �kn�k

ðn� kÞ! e�k � pm �kn

n!e�k; ð1Þ

where pm defines the probability that no more than zero packets exist in the queue as long as m packets exist in the server atthe beginning of unit of time (corresponding proof in Appendix A).

Theorem 1. The analytical solution of overall probability pn, using Laurent series expansion, equals (corresponding proof inAppendix A):

PðzÞ ¼X1n¼0

pnzn ¼ ðk�mÞðz� z1Þðz� z2Þ . . . ðz� zm�1Þðz� 1Þð1� z1Þð1� z2Þ . . . ð1� zm�1Þ½1� zmekð1�zÞ� ; ð2Þ

3. Non-delay probability estimation

To proceed with maximum throughput analysis the non-delay probability P0 in the scheduler system has to be estimated.no delay means non-existent IP packets in the buffer or better that there are n < m occupied channels over the air interface,non-delay probability could be explicitly calculated as:

P0 ¼ pm�1 ¼Xm�1

n¼0

pn; ð3Þ

To calculate analytically pn from (2) and substitute into (3) it is not easy; in order to facilitate the calculation of non-delayprobability we should skip the analytical calculations of pn and proceed to another method on Appendix B.

Theorem 2. The non-delay probability is calculated to be (corresponding proof in Appendix B):

P0 ¼ 10�X1k¼1

1k 1�

Xm�1

l¼0

ðkkÞll! e�kk

" #; ð4Þ

4. LTE air interface total delay analysis

IP packets, arriving on MAC scheduler, are segmented into MAC packet segments (SDU) completely transmitted over airinterface before transmission of next IP packet taking place. Scheduling decisions are mostly decided based on several attri-butes like QoS profile, radio link quality reports and UE uplink buffer sizes (signaled uplink to the eNodeB MAC layer usingthe uplink packet physical channel PUCCH) [24–27]. In order to proceed further with our analytical model a TCP/UDP IPpacket of MI variable bits and average hMIi bits per packet is considered to be segmented into total hMI/Mmaci number ofMAC packets of variable length Mmac (bits per packet), containing a fixed number of Mover header bits per packet [15]. Totalaverage number of transmitted bits will be hMIi + hd MI/MmaceiMover where factor hdMI/MmaceiMover indicates the MAC over-head. Average transmission delay is expected to be increased due to existing retransmissions over Hybrid Automatic RepeatRequest (HARQ) [26–28]. Indeed real radio channel conditions with dispersive channel characteristics introduce ISI and thusBit Error Rate (BER) on the receiver especially in low SNR cellular areas [29–32]. In this scenario we also consider corruptedpackets to be uncorrelated between each other; thus if one MAC packet is corrupted and retransmission is requested, nextMAC packet of the TCP/IP original packet could be also corrupted or not, without any previous memory of the previouspacket condition. Assuming that the average number of MAC retransmissions is nmac, average TCP/IP packet transmissiondelay time could be estimated as:

Tretrdelay

D E¼ ð1þ nmacÞhMIi þ ð1þ nmacÞhdMI=Mmacei �Mover

M � N � nTTITs þ nTs þ ð1�P0ÞTs; ð5Þ

where nTTI is the number of transmitted bits per SB depending on Link Adaptation and Modulation Scheme of eNodeB firm-ware. N is the average allocated number of 180 kHz radio block units of bandwidth per TTI, considering also the constraintthat 0 < 0.18N 6 BW where BW is the allocated radio bandwidth in MHz , ranging from 1.4 to maximum 20 MHz, and M is thenumber of antenna ports (in case of MIMO implementation). Factor (1 �Po) is the delay probability in the UE transmissionbuffer for a MAC packet. Finally n is an integer indicating the number of TTIs one MAC packet is not scheduled by schedulerin a total scheduling period and Ts is TTI duration of 1 ms; depends mainly on service QCI, on CQI reports, on UE transmittermean packet waiting time on the buffer and on cell load.

Finally IP average transmission data rate hRdatai in the worst scenario is then estimated as:

hRdatai ¼hMIiTretr

delay

D E ¼ hMIið1þnmacÞhMIiþð1þnmacÞhdMI=Mmacei�Mover

M�N�nTTIþ nþ ð1�P0Þ

� �Ts

; ð6Þ



5. Results and discussion

Average number of retransmissions nmac depends explicitly on the maximum number of attempts v and on the size of theMAC packet Mmac., considering also LTE MAC Scheduler priority rules estimated to be [15]:

nmac ¼1� ð1� pÞv

p; ð7Þ

Assuming that each MAC packet could be retransmitted maximum v times (operator determined parameter cell planning;in Ericsson technology defined by parameter transmissionTargetError, range [1, . . . , 200]), what is left to be further estimatedis parameter v which influences scheduling and delay over air interface. 3GPP standards do not provide any strict restrictionon maximum number of retransmissions, leaving it on vendor specific firmware implementation. According to cell planningconsiderations maximum number of retransmissions could be estimated indirectly by considering 3GPP specifications onQoS restrictions. Indeed following 3GPP standards there is always a strict delay restriction on LTE services regarding themaximum cell range with a restricted delay time smax > TTI ms depending on service [15,23]. Hence due to HARQ functionone MAC packet will be retransmitted a maximum number of v times as long as delay budget never exceeds smax:

smax ¼ vTs þ nTs ) v ¼ smax � nTs

Ts; ð8Þ

Substituting (8) into (7) we have the estimated average number of retransmissions [15]:

nmac ¼1� ð1� ð1� pbÞ

Mmac Þv

ð1� pbÞMmac

¼ 1� ð1� ð1� pbÞMmac Þ

ð1� pbÞMmac

smax�nTsTs

; ð9Þ

where pb is defined as the average bit error probability of MAC packet bits. Average bit error probability could be estimatedby real drive tests or LTE radio simulations, as evaluated on [15]; it depends explicitly on SINR in the cell planning area and isaffected from maximum cell range for cell edge users.

Average number of TCP/UDP IP bits per packet, hMIi, is considered for most applications to be 1500 bytes. Relying on 3GPPMAClayer uplink mapping, hdMI/Mmacei could be estimated considering also that MAC payload carried in one subframe of anuplink RB will vary depending on the coding and modulation scheme selected from Link Adaptation algorithm. 3GPP defineprecisely the corresponding data rate at MAC Layer [24]. As an example Fig. 2 illustrates three modulation schemes in worstchannel conditions (cell edge users).

Considering the worst scenario for uplink user on cell edge, Link Adaptation Block will decide on QPSK modulationscheme with Transmission Diversity spatial mode. Following Fig. 2 Mmac = 96 bits per TTI; thus MI/Mmac = (1500 � 8)/96 = 125 and hdMI/Mmacei= 125 MAC packet segments per IP packet. Moreover due to Transmission Diversity spatial modeM = 1. Mover is the estimated overhead due to RLC/MAC packet formation. RLC/MAC overhead on LTE, based on 3GPP MACstandards [24] is considered to be Mover = 20 bytes = 160 bits.

What is left to be estimated is the number of MAC allocated SB, N per service. Since MAC scheduling decisions rely onvendor specific software, average number of allocated SB in all possible cell ranges of LTE coverage could be only estimatedeither by drive tests or simulations. However, specifically from cell planning principles for worst scenario of cell edge users,estimation could be based on a planning target SINR ratio (also known on international literature as c0,target). The number of

Fig. 2. Uplink channel mapping per modulation scheme, 3GPP standards.



allocated resource blocks N, considering uniform power distribution of nominal UE power PUE over all transmitted resourceblocks, is estimated as:

c0;target ¼PUE=ðLpath � NÞðNRB þ IRBÞ

) N ¼ PUE

Lpath � ðNRB þ IRBÞ � c0;t arg et; ð10Þ

Expected worst scenario pathloss Lpath is calculated based on existing certain defined pathloss models for LTE in interna-tional literature. A well defined formula for 2.5 GHz LTE microcell outdoor to outdoor coverage is [15]:

Fig. 3. LTE physical user plane resources on uplink.

Fig. 4. Cell bandwidth vs. available radio resources (channels).

Fig. 5. Average throughput estimation vs. IP packet arrival rate on UE uplink buffer.



Lpath½dB� ¼39þ 20log10 d½m�ð Þ; &10 m < d 6 45 m�39þ 67log10 d½m�ð Þ; &d > 45 m

� �; ð11Þ

NRB per resource block is considered to be the background wideband noise, calculated as �111.44 dB [32]. At worst cell con-ditions we do suppose maximum uplink UE uplink power of PUE = 31.76 dBm = 1.5 W. Interference could be estimated eitherby drive tests or by simulations. A good approximation for cell edge scenario might be in the range of [�90, . . . , �70] dB [32].Number of transmitted bits per SB, nTTI, could be easily calculated for worst case cell edge UEs. From Fig. 2, Link Adaptationblock will allocate QPSK modulation which implies 2 bits per symbol together with TX diversity. One SB on a sub-frame of1 ms contains 14 � 12 = 168 resource elements (RE) and two OFMD symbols (24 RE) of the subframe are allocated for sound-ing reference signals, according to Fig. 3, [26]. Thus the available user plane resource elements are calculated to be:nTTI = (168 � 24) � 2 = 288 bits/ms.

Number of channels m in (4) depends on available allocated bandwidth on cell. Fig. 4 defines the number of availableradio resources (channels) per allocated cell bandwidth, based on 3GPP [26]. Finally, considering the overall transmissiondelay in (5), the number of TTIs one MAC packet is not scheduled by scheduler n has to be estimated. This is indeed hiddeninside the algorithm of vendor specific MAC Scheduler functionality; thus direct calculation is impossible. Following thenRef. [21] simulations, average scheduling delay for normal load (number of available users) conditions is considered to bein the range of n 2 [1, . . . , 5].

Fig. 5 presents the curve expected average throughput vs. IP packet arrival rate for cell edge users in case of LTE frequencyband of 2.6 GHz, hMii = 1500 bytes = (1500 � 8) bits, pb = 0.1, worst case 3GPP specs [24] provide Mmac = 96 bits and N = 1 forco,target = �5 dB and Mover = 24 bits, smax = 0.1 s (conversational voice or live video streaming), n = 5, M = 1 (SISO scenariowithout diversity), nTTI = 288 bits, cell range d = 500 m, NRB + IRB = �80 dB, PUE = 0.75 W, m = 6 (cell bandwidth 1.4 MHz).Average uplink throughput is estimated to be 1.085 kbps. This result is compliant with international literature simulationestimations; indeed following [33] on Fig. 5 for SISO and 1.4 MHz bandwidth the estimated throughput is less than 10 kbps.The small deviation between the simulation result and our analytical estimation is due to imperfections in the analyticalMAC number of retransmissions and the allocation of N resource blocks. However it provides indeed a good estimationfor cell planning initial calculations.

6. Conclusions

Cell coverage affects the scheduler decisions and thus the user throughput due to degraded CQI reports in bad channelcondition areas. Scheduler is vendor specific implementation and it is difficult to use analytical models in order to estimateaverage uplink transmission rate. Cell planners are very much interested in predicting MAC scheduler decisions in order totune properly cell ranges and expected delays. In this paper an analytical mathematical method, based on delay probabilitiesand 3GPP QoS standards, has been demonstrated to facilitate the estimation of average uplink throughput. Model is based onIP transmission delays taking into account three different factors that influence the IP data packet transmission delay. Pro-posed analysis has been applied specifically for cell edge users, giving a good prediction tool for cell planning worst serviceconditions. However and without loss of generality this analysis could be applied for any cell distance inside the cell cover-age. For future improvements, a more analytical and accurate model for HARQ number of retransmissions nmac should beimplemented; moreover a detailed calculation of number of allocated resource blocks N vs. SINR, BER or cell distance shouldbe simulated to perform scheduler functionality. Finally allocation of resource blocks on scheduler is affected from inter-cell

Fig. 6. Contour areas for non-delay complex analysis calculations.



interference. An analytical mobility model of neighbor cell edge users is needed to contribute to analytical SINR predictionsand thus more accurate estimations of allocated resource blocks N.

Appendix A

Proof of Lemma 1. The probability, in the unit of time, that specifically zero packets exists in the queue and m packets inservice po could be calculated as the intersection of (the probability pm that no more than zero packets exist in the queue aslong as m packets exist in the server at the beginning of unit of time) and (the probability (Poisson distribution) of zeroarrivals during the considered time interval), that is:

p0 ¼ pm \ e�k ¼ pm � e�k; ðA:1Þ

Using same reasoning the probability that specifically one packet exists in the queue p1 at the unit of time could be cal-culated as:

p1 ¼ ðpm \ ke�kÞ [ ðpmþ1 \ e�kÞ ¼ pm � ke�k þ pmþ1 � e�k; ðA:2Þ

Considering the general case, the overall probability pn that specifically n packets exists in the system at the unit of timeequals:

pn ¼ pm �kn

n!e�k þ pmþ1 �

kn�1

ðn� 1Þ! e�k þ :::þ pnþme�k ¼ pm �kn

n!e�k þ

Xn

k¼0

pmþk �kn�k

ðn� kÞ! e�k � pm �kn

n!e�k; ðA:3Þ

Proof of Theorem 1. Expanding into Laurent series P(z):

PðzÞ ¼X1n¼0

pnzn ¼ pme�kX1n¼0

ðkzÞn

n!þ e�k

X1n¼0

knXn

k¼0

pmþkzn�kzk

kkðn� kÞ!

!� pme�k

X1n¼0

ðkzÞn

n!)

PðzÞ ¼ ðpm � pmÞekð1�zÞ þ e�kX1n¼0

knXn

k¼0

pmþkzn�kzk

kkðn� kÞ!

!;

By definition of pn and pm obviously pm ¼Pm

n¼0pn, hence:

PðzÞ ¼Xm

n¼0

pn � pm

!ekð1�zÞ þ e�k

X1n¼0

knXn

k¼0

pmþkzn�kzk

kkðn� kÞ!

!) PðzÞ ¼ pm�1 � ekð1�zÞ þ e�k �

X1n¼0

knXn

k¼0

pmþkzn�kzk

kkðn� kÞ!

!; ðA:4Þ

Following the summations and after appropriate mathematical calculations, considering also PmðzÞ ¼Pm

n¼0pnzn as the def-inition of finite Laurent series, Eq. (4) is then simplified into:

PðzÞ ¼ PmðzÞ � pmzm

1� zmekð1�zÞ ; ðA:5Þ

Since 0 6 pn 6 1, P(z) is a regular function bounded into the unit circle on the complex space jzj 6 1. Numerator of (5) con-sists of two polynomials of mth order. Both Pm(z) and pmzm are analytical functions inside the simple curve jzj 6 1 and alsobounded into the unit circle on the complex space jzj 6 1. Since jpmzmj 6 jPm(z)j on jzj 6 1 then both have same number ofzeroes inside jzj 6 1 and since they are polynomials of mth order they have m zeroes inside jzj 6 1, denoted as z1, z2, . . . , zm

respectively, leading into a closed form function of P(z):

PðzÞ ¼ Aðz� z1Þðz� z2Þ . . . ðz� zmÞ1� zmekð1�zÞ ; ðA:6Þ

Considering (3) and the nominator of (A.6) it could be shown that z = 1 is a root; indeed:

limz!1 PmðzÞ � pmzmð Þ ¼ limz!1

Xm

n¼0

pnzn � pmzm

!¼Xm

n¼0

pn � pm ¼ 0; ðA:7Þ

consequently (A.6) could be rewritten as

PðzÞ ¼ Aðz� z1Þðz� z2Þ . . . ðz� zm�1Þ � ðz� 1Þ1� zmekð1�zÞ ; ðA:8Þ

Total probability condition for P(z) holds:

limz!1PðzÞ ¼ limz!1

X1n¼0

pnzn ¼X1n¼0

pn ¼ 1) A ¼ k�mð1� z1Þð1� z2Þ . . . ð1� zm�1Þ

;



Finally using the Laurent series:

PðzÞ ¼X1n¼0

pnzn ¼ ðk�mÞðz� z1Þðz� z2Þ . . . ðz� zm�1Þðz� 1Þð1� z1Þð1� z2Þ . . . ð1� zm�1Þ 1� zmekð1�zÞ½ � ; ðA:9Þ

Appendix B

Proof of Theorem 2. Indeed we could use complex analysis, starting from (3) and the following observation:

pm ¼ pm � pm�1 ¼Xm

n¼0

pn �Xm�1

n¼0

pn; ðB:1Þ

Considering Laurent series expansion function H(z):

HðzÞ ¼X1k¼0

pkzk; ðB:2Þ

Combining (3) and (B.1) and taking into account that p�1 is meaningless in our analysis:

pm¼ pm�pm�1)X1n¼0

pnzn¼X1n¼0

pnzn�X1n¼0

pn�1zn)PðzÞ¼HðzÞ�X1l¼�1

plzlþ1)PðzÞ¼HðzÞ�z

X1l¼0

plzl)HðzÞ¼ PðzÞ

1�zð Þ ;

ðB:3Þ

Substituting (A.9) into (B.1) then:

HðzÞ ¼X1k¼0

pkzk ¼ ðk�mÞðz� z1Þðz� z2Þ . . . ðz� zm�1Þð1� z1Þð1� z2Þ . . . ð1� zm�1Þ½1� zmekð1�zÞ� ; ðB:4Þ

Differentiating (m � 1) times with respect to z, dividing by factor (m � 1)! and setting z = 0, non-delay probability couldbe calculated as:

P0 ¼ pm�1 ¼ðk�mÞXm�1

l¼1

ð1� zlÞ; ðB:5Þ

To calculate roots z1, z2, . . . , zm�1, we have to rely into complex analysis and the generalized argument theorem fromcomplex calculus [34]. We shall select function f(z) as f(z) = log(z � 1) and we do select an analytical function inside a contourC in the z-plane which should have number of poles and zeroes inside the contour. We do select an exponential functionwhich has m multiple z = 0 poles inside the contour C and z1, z2, . . . , zm�1 zeroes:

hðzÞ ¼ 1� ekz

ekzm ¼ 1�

X1n¼0

ðkzÞnn!

ekzm ; ðB:6Þ

Following the generalized argument theorem we integrate over the contour area C:

12pi

ZC

f ðzÞh0ðzÞ=hðzÞdz ¼ 12pi

ZC

logðz� 1Þh0ðzÞ=hðzÞdz ¼ �piþXm�1

l¼1

logð1� zlÞ; ðB:7Þ

Taking logarithmic function of P0 on (B.5), substituting to (B.7) and integrating by parts:

12pi

ZC

logðz� 1Þh0ðzÞ=hðzÞdz ¼ �piþ logðm� kÞ � logðP0Þ ¼1

2pi½logðz� 1Þ log h�

��C

� 12pi

ZC

log hz� 1

� dz; ðB:8Þ

What is left is to calculate the left part on (B.8) and solve for non-delay probability. Singularity point z = 1 should defi-nitely be avoided splitting contour C into two contour parts, C1 with radius R and center at z = 1 and C2 with radius r alsoat center at z = 1, as described in Fig. 6. We then have to calculate the integral over C1, C2 and remaining line paths amongthese circles. Starting with the integrals over contour area C1 the extreme points of calculation have to be defined as a circlewith extreme polar coordinate points (R, h = 0) and (R, h = 2p). Then considering Fig. 6, expressing the circle in complex polarcoordinates: z � 1 = Reih) log(z � 1) = log R + ih and for function h(z) from (B.6):

log hðzÞ ¼ log 1� ekðz�1Þ

zm

� �¼ log 1� ekReih

ð1þ ReihÞm

!; ðB:9Þ

From (B.8) and considering the contour C1 in extreme points:




��2p

0� 1

2pi

ZC

log hz� 1

� dz ¼ �piþ logðm� kÞ � logðP0Þ; ðB:10Þ

Substituting polar coordinates into (B.10) and expanding complex exponential with Euler formula:


��2p

0¼ 1

2piðlog Rþ ihÞ log 1� ekRfcos hþi sin hg

ð1þ Rfcos hþ i sin hgÞm� � ��

2p

0

¼ log 1� ekR

ð1þ RÞm� �

; ðB:11Þ

Hence considering (B.10) and (B.11) the contribution of contour area C1 will be:

12pi

ZC1

logðz� 1Þ h0

hdz ¼ 1


��C1

� 12pi

ZC1

log hz� 1

� dz) 12pi

ZC1

logðz� 1Þh0

hdz

¼ log 1� ekR

ð1þ RÞm� �

� 12pi

ZC1

log hz� 1

� dz; ðB:12Þ

To proceed we do have to calculate the contribution of remaining paths on Fig. 6 to the closed path integral on (B.8). Wedo start our analysis from the general form of generalized argument theorem:

12pi

Zlogðz� 1Þh0ðzÞ=hðzÞdz ¼ 1

2pi

Z z¼1þR

z¼1þrlogðz� 1Þh0ðzÞ=hðzÞdz

��h¼0þZ z¼1þr

z¼1þRlogðz� 1Þh0ðzÞ=hðzÞdz

��h¼2p

ðB:13Þ

¼ 12pi½logðz� 1Þ log h�

��h¼0� 1

2pi

Z z¼1þR

z¼1þr

log hz� 1

dzþ 12pi½logðz� 1Þ log h�

��h¼2p� 1

2pi

Z z¼1þr

z¼1þR

log hz� 1

dz; ðB:14Þ

Substituting polar coordinates:

12pi

Zlogðz� 1Þh0ðzÞ=hðzÞdz ¼ 1

2pilogðRþ ihÞ � log 1� ekReih

ð1þ ReihÞm

!" #��h¼0

� 12pi

Z z¼1þR

z¼1þr

log 1� ekReih

ð1þReihÞm

� �Reih

dz

þ 12pi

logðr þ ihÞ � log 1� ekreih

ð1þ reihÞm

!" #��h¼2p

þ 12pi

Z z¼1þR

z¼1þr

log 1� ekReih

ð1þReihÞm

� �Reih

dz;

ðB:15Þ

Eliminating same factors and using Euler expansion in (B.15) the contribution of remainings into the integral over contourC is:

12pi

Zlogðz� 1Þh0ðzÞ=hðzÞdz ¼ � log 1� ekR

ð1þ RÞm� �

þ log 1� ekr

ð1þ rÞm� �

; ðB:16Þ

Final contribution will be the other contour area C2. To proceed we consider again (B.8) and taking into account polarcoordinates for internal circle, z = 1 + reih, finally we get:

12pi

ZC2

logðz� 1Þ h0

hdz ¼ 1

2pi

ZC2

logðreihÞh0ð1þ reihÞ

hð1þ reihÞ dð1þ reihÞ ¼ 12p

Z h¼0

h¼2p½log r þ ih�reih h0ð1þ reihÞ

hð1þ reihÞ dh; ðB:17Þ

Considering function h(z) from (B.6) it is obvious that, taking Laurent series expansion around z = 1, it behaves as(m � k)(reih) + O(r); consequently from (B.17):

limz!1h0ðzÞ=hðzÞ � 1=ðz� 1Þ ¼ 1reih) 1

2p

Z h¼0

h¼2p½log r þ ih�reih h0ð1þ reihÞ

hð1þ reihÞ dh ¼ 12p

Z h¼0

h¼2p½log r þ ih�dh

¼ �pi� log r; ðB:18Þ

and

limz!1hðzÞ � ðz� 1Þh0ðz ¼ 1Þ ¼ ðm� kÞðz� 1Þ ) logðhð1þ reihÞÞ ¼ logðm� kÞ þ logðreihÞ¼ logðm� kÞ þ log r þ ih) log r ¼ limz!1 logðhð1þ reihÞÞ � logðm� kÞ � ih; ðB:19Þ

Substituting (B.19) to (B.18) then:

12p

Z h¼0

h¼2p½log r þ ih�dh ¼ �pi� limz!1 logðhð1þ reihÞÞ þ logðm� kÞ � ih

� ��h ¼ 0z! 1

¼ �pi� logðhð1þ reihÞÞ þ logðm� kÞ;

ðB:20Þ



Combining (B.8), (B.12), (B.16) and (B.20) we finally get:

12pi

ZC

logðz� 1Þh0ðzÞ=hðzÞdz ¼ log 1� ekR

ð1þ RÞm� �

� 12pi

ZC1

log hz� 1

� dz� log 1� ekR

ð1þ RÞm� �

þ log 1� ekr

ð1þ rÞm� �

� pi� limz!1 logðhð1þ reihÞÞ þ logðm� kÞ � ih

¼ �piþ logðm� kÞ � logðP0Þ; ðB:21Þ

Hence combining (B.21) with (B.9):

12pi

ZC1

log hz� 1

� dz ¼ logðP0Þ ¼1

2pi

ZC1

1z� 1

� log 1� ekðz�1Þ

zm

� �dz; ðB:22Þ

Using Laurent series expansion around z = 1 with convergence inside the circle ekðz�1Þ

zm

�� < 1:

logðP0Þ ¼ �1

2pi

ZC1

1z� 1

�X1k¼1

ekkðz�1Þ

kzm

!dz ¼ �

X1k¼1

12pki

ZC1

1z� 1

� ekkðz�1Þ

kzm

� �dz

; ðB:23Þ

To calculate above integral we do use residues theorem, hence:

12pi

ZC1

1z� 1

� ekkðz�1Þ

zm

� �dz ¼ 1�

Xm�1

l¼0

ðkkÞl

l!e�kk; ðB:24Þ

And finally the non-delay probability equals:

logðP0Þ ¼ �X1k¼1

1k

1�Xm�1

l¼0

ðkkÞl

l!e�kk

" #) P0 ¼ 10

�X1k¼1

1k 1�

Xm�1

l¼0

kkð Þll! e�kk

" #; ðB:25Þ

References

[1] 3GPP TS 23.060 V.8.5.1 service description; 2009.[2] ETSI. GSM Specification Service description, Stage 1, 1999 (02.60); Service description, Stage 2, 1999 (03.60).[3] 3GPP TS 23.107. Quality of Service (QoS) concept and architecture. WCDMA.[4] 3GPP TS 29.212 v8.8.0. Policy and Charging Control over Gx Reference Point. Technical Specification Group Core Network & Terminals.[5] 3GPP TR 25.913. Feasibility study of evolved UTRA and UTRAN.[6] Dahlman, Parkvall, Skold, Beming. 3G evolution: HSPA and LTE for mobile broadband. Oxford, UK: Academic Press; 2007.[7] 3GPP TS 25.104. Base station (BS) radio transmission and reception (FDD).[8] Pokhariyal A, Kolding TE, Mogensen PE. Performance of downlink frequency domain packet scheduling for the UTRAN long term evolution. In: Proc of

IEEE international symposium on personal, indoor and mobile radio communications. Helsinki; September 11–14, 2006.[9] Lindbom L, Love R, Krishnamurthy S, Yao C, Miki N, Chandrasekhar V. Enhanced inter-cell interference coordination for heterogeneous networks in

LTE-advanced: a survey. arXiv:1112.1344v2 [cs.IT] Cornell University library; 2011.[10] 3GPP TS 36.302 v8.1.0. Service provided by the physical layer; 2009.[11] Kin Jaewon, Lee Donghyun, Sung Wonjin. Interference coordination of heterogeneous LTE systems using remote radio heads. EURASIP J Adv Signal

Process 2013. http://asp.eurasipjournals.com/content/2013/1/90.[12] Jar M, Fettweiss G. Throughput maximization for LTE uplink via resource allocation. In: IEEE international symposium of wireless communication

systems (ISWCS 2012). Paris; August 28–31, 2012. p. 146–50.[13] Reyhani A, Song Shaowen, Primak SL, Shami A. Heterogeneous delay-power resource allocation in uplink LTE. In: IEEE wireless and mobile networking

conference (WMNC 2013). Dubai; April 23–25, 2013. p. 1–6.[14] Chiapin Wang, Xingrong Li. A buffer-aware resource allocation scheme for 4G LTE systems. In: IEEE 7th international symposium on consumer

electronics (ISCE 2013). Hsinchu; June 3–6, 2013. p. 157–8.[15] Louvros S, Iossifides AC, Aggelis K, Baltagiannis A, Economou G. A semi-analytical macroscopic MAC layer model for LTE uplink. In: Proc Of 5th IFIP

international conference on new technologies, mobility and security (NTMS 2012). Turkey Instanbul; May 2012.[16] Novlan Thomas D, Dhillon Harpreet S, Andrews Jeffrey G. Analytical modelling of uplink cellular networks. arXiv:1203.1304v3. [cs.IT] Cornell

University library.[17] Marwat SNK, Zaki Y, Goerg C, Weerawardane T. Design and performance analysis of bandwidth and QoS aware LTE uplink scheduler in heterogeneous

traffic environment. In: IEEE 8th international wireless communications and mobile computing conference (IWCMC 2012). Limassol; August 27–31,2012. p. 499–504.

[18] Baid A, Madan Ritesh, Sampath Ashwin. Delay estimation and fast iterative scheduling policies for LTE uplink. In: IEEE 10th international symposiumon modeling and optimization in mobile, ad-hoc and wireless networks (WiOpt 2012). Paderborn Germany; May 14–18, 2012. p. 89–96.

[19] http://www.rapidio.org/education/applications/Ericsson2_lte_web.pdf.[20] http://www.ericsson.com/ericsson/corpinfo/publications/review/2008_03/files/LTE.pdf.[21] Asheralieva Alia, Mahata Kaushik, Khan Jamil Y. Delay and loss due to uplink scheduling in LTE network. Wireless access flexibility. Berlin: Springer-

Verlang; 2013. pp. 1–12.[22] Delgado O, Jaumard B. Scheduling and resource allocation in LTE uplink with a delay requirement. In: IEEE 8th annual communication networks and

services research conference (CNSR 2010). Montreal QC, Canada; May 11–14, 2010. p. 268–75.[23] 3GPP TS 23.203. Policing and charging control architecture. Rel-11, V11.4.0; 2011.[24] 3GPP TS 36.321. Evolved universal terrestrial radio access (E-UTRA); medium access control (MAC) protocol specification (Release 8). V8.1.0; 2008.[25] Szczescny D, Showk A, Hessel S, Bilgic A, Hildebrand U, Frascolla V. Performance analysis of LTE protocol processing on an ARM based mobile plattform.

In: International symposium on system-on-chip (SOC 2009). Tampere; October 5–7, 2009. p. 56–63.[26] 3GPP TS 36.211. Evolved Universal Terrestrial Radio Access (E-UTRA); Physical Channels and Modulation. Rel-10, V10.4.0; 2011.



[27] 3GPP TS 36.300. Evolved Universal Terrestrial Radio Access (E-UTRA) and Evolved Universal Terrestrial Radio Access Network (E-UTRAN); OverallDescription. Stage 2, Rel-11, V11.0.0; 2011.

[28] Kolehmainen N, Puttonen J, Kela P, Ristaniemi T, Henttonen T, Moisio M. Channel quality indication reporting schemes for UTRAN long term evolutiondownlink. In: IEEE vehicular technology conference (VTC Spring 2008). Singapore; May 11–14, 2008. p. 2522–6.

[29] Pokhariyal A, Kolding TE, Mongensen PE. Performance of downlink frequency domain packet scheduling for the UTRAN long term evolution. In: IEEE17th international symposium on personal indoor and mobile radio communications. Helsinki; September 11–14, 2006. p. 1–5.

[30] Louvros S, Angelis K, Baltagiannis A. LTE cell coverage planning algorithm optimizing user cell throughput. In: Proceedings of 11th IEEE internationalconference on telecommunications (ConTEL 2011). Graz; June 15–17, 2011. p. 51–8.

[31] Liu H, Li G. OFDM-based broadband wireless networks; design and optimisation. John Wiley & Sons; 2005.[32] Monghal G, Pedersen KI, Kovacs IZ, Mongensen PE. QoS oriented time and frequency domain packet schedulers for the UTRAN long term evolution.

IEEE vehicular technology conference (VTC Spring 2008). Singapore; May 11–14, 2008. p. 2532–6.[33] Blumenstein J, Ikuno JC, Prokopec J, Rupp M. Simulating the long term evolution uplink physical layer. In: Proceedings of international symposium

ELMAR 2011. Zadar; September 14–16, 2011. p. 141–4.[34] Churchill Ruel Vance, Brown James Ward. Complex variables and applications. McGraw-Hill; 1989. ISBN 978-0-07-010905-6.

Spiros Louvros holds the position of Assistant Professor, Computer & Informatics Engineering Department, TEI of WesternGreece, Hellas. He holds Bachelor in Physics from University of Crete, Hellas and Master (MSc) from University of Cranfield, U.K.In 2004 received his PhD from University of Patras, Hellas. Current research interests are in telecommunication traffic engi-neering, wireless networks, Mobility management & optimization.

Michael Paraskevas holds a diploma in electrical engineering and PhD in digital signal processing from University of Patras,Greece. He is Assistant Professor at Computer & Informatics Engineering Department, TEI of Western Greece and Director ofDirectorate of Greek School Network, Computer Technology Institute and Press ‘‘Diophantus’’. Current research interests are insignal theory, DSP, analog and digital communications, next generation networks, e-government and e-learning services.


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Analytical average throughput and delay estimations for LTE