Analytical average throughput and delay estimations for LTE

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Analytical average throughput and delay estimations for LTE
uplink cell edge users q
Spiros Louvros ⇑
, Michael Paraskevas
Computer & 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 2013
Received in revised form 13 March 2014
Accepted 18 March 2014
Available online 10 May 2014
a b s t r a c t
Estimating average throughput and packet transmission delay for worst case scenario (cell
edge users) is crucial for LTE cell planners in order to preserve strict QoS for delay sensitive
applications. Cell planning techniques emphasize mostly on cell range (coverage) and
throughput predictions but not on delay. Cell edge users mostly suffer from throughput
reduction due to bad coverage and consequently unexpected uplink transmission delays.
To estimate cell edge throughput a common practice on international literature is the
use of simulation results. However simulations are never accurate since MAC scheduler
is a vendor specific software implementation and not 3GPP explicitly specified. This paper
skips simulations and proposes an IP transmission delay and average throughput analytical
estimation using mathematical modeling based on probability delay analysis, thus offering
to cell planners a useful tool for analytical estimation of uplink average IP transmission.
Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Nowadays IP based multi-service wireless cellular networks mobile handsets are requesting reliable data transmission
from QoS perspective point of view [1–4]. In 3GPP standards four negotiated QoS profiles are defined based on four existing
QoS 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 to
previous mobile broadband networks (HSPA, 3G), pose strong demands on throughput and latency, requesting new multiple
access techniques over air interface and simplified network architecture [6,7]. Using OFDM/SC-FDMA technology a minimum
group of 12 sub-carriers of total 180 kHz bandwidth is known as Resource Block (RB). In a frequency-time domain resource
grid 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 Transmission
Time Interval (TTI)) in time domain.
From cell planning perspective uplink is always the weakest link in the power-link budget and throughput analysis, for
both outdoor and indoor to outdoor coverage. MAC scheduler, residing in eNodeB, is responsible for dynamically allocating
uplink/downlink resources [8]. The primary goal of uplink scheduler is the ability to allocate an appropriate amount of
consecutive resources in the SC-FDMA with the appropriate transport format, modulation to appropriately map symbols
http://dx.doi.org/10.1016/j.compeleceng.2014.03.008
0045-7906/Ó 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: splouvros@gmail.com (S. Louvros), mparask@teimes.gr (M. Paraskevas).
Computers and Electrical Engineering 40 (2014) 1552–1563
Contents lists available at ScienceDirect
Computers and Electrical Engineering
journal homepage: www.elsevier.com/locate/compeleceng

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) or
interference (Interference limited environment), due to low scheduling decisions of the uplink scheduler. A lot of research
has been performed on international literature regarding ICI and scheduling decisions focusing on throughput estimations
and 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 proposed
on [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 is
not standardized by 3GPP; it is rather left on vendor (Ericsson, Nokia, HUAWEI, etc.) implementations trying to make more
efficient 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 for
uplink transmission Link Adaptation block is implemented on user equipment (UE). In order to depict the MAC functionality
from 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 power
and 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 reduce
the 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 since
the 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 delay
as a function of scheduler resource allocation decisions and channel conditions. Prediction evaluation is considered to be
split 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 other
words 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 research
papers exist in international literature using either theoretical simulations or analytical probabilistic models trying to
combine packet delay and resource allocation principles. In [15] a semi-analytical macroscopic probabilistic model has
been 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 with
MAC scheduler decisions and throughput. End-to-end QoS performance of Bandwidth and QoS Aware (BQA) scheduler
for 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 of
resources allocated from MAC decisions they do not consider the reality since allocation of resources from MAC scheduler
is vendor specific and only vendor official simulators [19] or drive tests could depict the reality; consequently there is not
much work on such a topic on international literature. One important such drive test reference is on [20] which will be
used 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 Quality
Index (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 analytical
mathematical modeling. Few papers exist on international literature. One very good research paper is [21] where authors
have derived a mathematical model for delay estimations. An oldest approach [22] indicates also an innovative algorithm
to 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 other
papers 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 and
most of predicted results are generated from public LTE simulators not following vendor specific solutions; thus estimations
are not accurate for specific network equipments. This paper proposes an analytical mathematical model to predict buffer
delay 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 uplink
throughput 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 one
important Theorem, is proposed calculating the probability of n packets existing in the system either in scheduled blocks
or in the transmission buffer. On Section 3 an explicit calculation for non-delay probability on UE buffer is proposed and
a mathematical Theorem is also stated. On Section 4 an overall uplink average IP throughput formula, considering uplink
air interface transmission delay as input, is proposed for cell planning analytical predictions. Applications on cell planning
and parameter justifications are analytically presented on Section 5 and final conclusions on Section 6. Finally on Appendices
A and B formal mathematical proofs on delay probabilities for Lemma and Theorems of Sections 2 and 3 are explicitly
provided.
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 into
MAC segments and then properly scheduled over SBs on air interface resources [23]. Each MAC packet is supposed to be
transmitted 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 do
consider IP segmented packets arriving from upper layers to MAC layer where a single server, known in our case as MAC
scheduler 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 n
arrived packets in the system and the occupied resources are less than maximum m channels (SB) available in the radio
interface, otherwise queue contains IP packets. IP packets arrival process is considered to be Poisson with k packet rate of
arrival. Service time lo is considered to be constant for all parallel channels and the reasoning behind constant service time
is the small deviations in transmission delays due mostly on processor load fluctuations. It has to be clear that transit time
effects are neglected on this analysis since there are no transit effects when scheduler operates as a continuous scheduling
process. Fig. 1 presents the mathematical model in block diagram format. Considering queue equilibrium, mathematical
analysis considers always m k. Define pn the probability of existing specifically n packets in both queue and service at a
given time s and pn the probability that no more than n packets exists in the model at given time s. Since service time is
considered to be constant a good assumption might be to consider typical unit of time to be the service time lo. Following
Lemma 1 and Theorem 1 provide the probability that specifically n packets exist in the system at the unit of time. Proofs are
analytically 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 Á
kn
n!
eÀk
þ
Xn
k¼0
pmþk Á
knÀk
ðn À kÞ!
eÀk
À pm Á
kn
n!
eÀk
; 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 at
the 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 in
Appendix A):
PðzÞ ¼
X1
n¼0
pnzn
¼
ðk À mÞðz À z1Þðz À z2Þ . . . ðz À zmÀ1Þðz À 1Þ
ð1 À z1Þð1 À z2Þ . . . ð1 À zmÀ1Þ½1 À zmekð1ÀzÞŠ
; 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 pmÀ1 ¼
XmÀ1
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-delay
probability 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
À
X1
k¼1
1
k
1À
XmÀ1
l¼0
ðkkÞl
l!
eÀkk
#
; ð4Þ
4. LTE air interface total delay analysis
IP packets, arriving on MAC scheduler, are segmented into MAC packet segments (SDU) completely transmitted over air
interface 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 using
the uplink packet physical channel PUCCH) [24–27]. In order to proceed further with our analytical model a TCP/UDP IP
packet of MI variable bits and average hMIi bits per packet is considered to be segmented into total hMI/Mmaci number of
MAC packets of variable length Mmac (bits per packet), containing a fixed number of Mover header bits per packet [15]. Total
average 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 Repeat
Request (HARQ) [26–28]. Indeed real radio channel conditions with dispersive channel characteristics introduce ISI and thus
Bit Error Rate (BER) on the receiver especially in low SNR cellular areas [29–32]. In this scenario we also consider corrupted
packets to be uncorrelated between each other; thus if one MAC packet is corrupted and retransmission is requested, next
MAC packet of the TCP/IP original packet could be also corrupted or not, without any previous memory of the previous
packet condition. Assuming that the average number of MAC retransmissions is nmac, average TCP/IP packet transmission
delay time could be estimated as:
Tretr
delay
D E
¼
ð1 þ nmacÞhMIi þ ð1 þ nmacÞhdMI=Mmacei Á Mover
M Á N Á nTTI
Ts þ nTs þ ð1 À 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 constraint
that 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 the
number of antenna ports (in case of MIMO implementation). Factor (1 À Po) is the delay probability in the UE transmission
buffer for a MAC packet. Finally n is an integer indicating the number of TTIs one MAC packet is not scheduled by scheduler
in a total scheduling period and Ts is TTI duration of 1 ms; depends mainly on service QCI, on CQI reports, on UE transmitter
mean 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 ¼
hMIi
Tretr
delay
D E ¼
hMIi
ð1þnmacÞhMIiþð1þnmacÞhdMI=MmaceiÁMover
MÁNÁnTTI
þ n þ ð1 À 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 the
MAC packet Mmac., considering also LTE MAC Scheduler priority rules estimated to be [15]:
nmac ¼
1 À ð1 À 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 estimated
is parameter v which influences scheduling and delay over air interface. 3GPP standards do not provide any strict restriction
on maximum number of retransmissions, leaving it on vendor specific firmware implementation. According to cell planning
considerations maximum number of retransmissions could be estimated indirectly by considering 3GPP specifications on
QoS restrictions. Indeed following 3GPP standards there is always a strict delay restriction on LTE services regarding the
maximum cell range with a restricted delay time smax TTI ms depending on service [15,23]. Hence due to HARQ function
one 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 À nTs
Ts
; ð8Þ
Substituting (8) into (7) we have the estimated average number of retransmissions [15]:
nmac ¼
1 À ð1 À ð1 À pbÞMmac
Þ
v
ð1 À pbÞMmac
¼
1 À ð1 À ð1 À pbÞMmac
Þ
ð1 À pbÞMmac
smaxÀnTs
Ts
; ð9Þ
where pb is defined as the average bit error probability of MAC packet bits. Average bit error probability could be estimated
by real drive tests or LTE radio simulations, as evaluated on [15]; it depends explicitly on SINR in the cell planning area and is
affected 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 3GPP
MAClayer uplink mapping, hdMI/Mmacei could be estimated considering also that MAC payload carried in one subframe of an
uplink RB will vary depending on the coding and modulation scheme selected from Link Adaptation algorithm. 3GPP define
precisely the corresponding data rate at MAC Layer [24]. As an example Fig. 2 illustrates three modulation schemes in worst
channel conditions (cell edge users).
Considering the worst scenario for uplink user on cell edge, Link Adaptation Block will decide on QPSK modulation
scheme with Transmission Diversity spatial mode. Following Fig. 2 Mmac = 96 bits per TTI; thus MI/Mmac = (1500 Â 8)/
96 = 125 and hdMI/Mmacei= 125 MAC packet segments per IP packet. Moreover due to Transmission Diversity spatial mode
M = 1. Mover is the estimated overhead due to RLC/MAC packet formation. RLC/MAC overhead on LTE, based on 3GPP MAC
standards [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 on
vendor specific software, average number of allocated SB in all possible cell ranges of LTE coverage could be only estimated
either 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 resource
blocks, is estimated as:
c0;target ¼
PUE=ðLpath Á NÞ
ðNRB þ IRBÞ
) N ¼
PUE
Lpath Á ðNRB þ IRBÞ Á 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.

Lpath½dBŠ ¼
39 þ 20log10 d½mŠð Þ; 10 m d 6 45 m
À39 þ 67log10 d½mŠð Þ; d 45 m
'
; ð11Þ
NRB per resource block is considered to be the background wideband noise, calculated as À111.44 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 either
by drive tests or by simulations. A good approximation for cell edge scenario might be in the range of [À90, . . . , À70] dB [32].
Number of transmitted bits per SB, nTTI, could be easily calculated for worst case cell edge UEs. From Fig. 2, Link Adaptation
block will allocate QPSK modulation which implies 2 bits per symbol together with TX diversity. One SB on a sub-frame of
1 ms contains 14 Â 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 À 24) Â 2 = 288 bits/ms.
Number of channels m in (4) depends on available allocated bandwidth on cell. Fig. 4 defines the number of available
radio resources (channels) per allocated cell bandwidth, based on 3GPP [26]. Finally, considering the overall transmission
delay in (5), the number of TTIs one MAC packet is not scheduled by scheduler n has to be estimated. This is indeed hidden
inside the algorithm of vendor specific MAC Scheduler functionality; thus direct calculation is impossible. Following then
Ref. [21] simulations, average scheduling delay for normal load (number of available users) conditions is considered to be
in 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 frequency
band of 2.6 GHz, hMii = 1500 bytes = (1500 Â 8) bits, pb = 0.1, worst case 3GPP specs [24] provide Mmac = 96 bits and N = 1 for
co,target = À5 dB and Mover = 24 bits, smax = 0.1 s (conversational voice or live video streaming), n = 5, M = 1 (SISO scenario
without diversity), nTTI = 288 bits, cell range d = 500 m, NRB + IRB = À80 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 simulation
estimations; 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 analytical
MAC number of retransmissions and the allocation of N resource blocks. However it provides indeed a good estimation
for cell planning initial calculations.
6. Conclusions
Cell coverage affects the scheduler decisions and thus the user throughput due to degraded CQI reports in bad channel
condition areas. Scheduler is vendor specific implementation and it is difficult to use analytical models in order to estimate
average uplink transmission rate. Cell planners are very much interested in predicting MAC scheduler decisions in order to
tune properly cell ranges and expected delays. In this paper an analytical mathematical method, based on delay probabilities
and 3GPP QoS standards, has been demonstrated to facilitate the estimation of average uplink throughput. Model is based on
IP 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 service
conditions. 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 be
implemented; moreover a detailed calculation of number of allocated resource blocks N vs. SINR, BER or cell distance should
be 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 predictions
and 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 in
service po could be calculated as the intersection of (the probability pm that no more than zero packets exist in the queue as
long as m packets exist in the server at the beginning of unit of time) and (the probability (Poisson distribution) of zero
arrivals during the considered time interval), that is:
p0 ¼ pm eÀk
¼ pm Á eÀk
; 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 keÀk
Þ [ ðpmþ1 eÀk
Þ ¼ pm Á keÀk
þ pmþ1 Á eÀk
; ðA:2Þ
Considering the general case, the overall probability pn that specifically n packets exists in the system at the unit of time
equals:
pn ¼ pm Á
kn
n!
eÀk
þ pmþ1 Á
knÀ1
ðn À 1Þ!
eÀk
þ ::: þ pnþmeÀk
¼ pm Á
kn
n!
eÀk
þ
Xn
k¼0
pmþk Á
knÀk
ðn À kÞ!
eÀk
À pm Á
kn
n!
eÀk
; ðA:3Þ
Proof of Theorem 1. Expanding into Laurent series P(z):
PðzÞ ¼
X1
n¼0
pnzn
¼ pmeÀk
X1
n¼0
ðkzÞn
n!
þ eÀk
X1
n¼0
kn
Xn
k¼0
pmþkznÀk
zk
kk
ðn À kÞ!
!
À pmeÀk
X1
n¼0
ðkzÞn
n!
)
PðzÞ ¼ ðpm À pmÞekð1ÀzÞ
þ eÀk
X1
n¼0
kn
Xn
k¼0
pmþkznÀk
zk
kk
ðn À kÞ!
!
;
By definition of pn and pm obviously pm ¼
Pm
n¼0pn, hence:
PðzÞ ¼
Xm
n¼0
pn À pm
!
ekð1ÀzÞ
þ eÀk
X1
n¼0
kn
Xn
k¼0
pmþkznÀk
zk
kk
ðn À kÞ!
!
) PðzÞ ¼ pmÀ1 Á ekð1ÀzÞ
þ eÀk
Á
X1
n¼0
kn
Xn
k¼0
pmþkznÀk
zk
kk
ðn À kÞ!
!
; ðA:4Þ
Following the summations and after appropriate mathematical calculations, considering also PmðzÞ ¼
Pm
n¼0pnzn
as the def-
inition of finite Laurent series, Eq. (4) is then simplified into:
PðzÞ ¼
PmðzÞ À pmzm
1 À zmekð1ÀzÞ
; 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 also
bounded into the unit circle on the complex space jzj 6 1. Since jpmzm
j 6 jPm(z)j on jzj 6 1 then both have same number of
zeroes 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 À z1Þðz À z2Þ . . . ðz À zmÞ
1 À zmekð1ÀzÞ
; ðA:6Þ
Considering (3) and the nominator of (A.6) it could be shown that z = 1 is a root; indeed:
limz!1 PmðzÞ À pmzm
ð Þ ¼ limz!1
Xm
n¼0
pnzn
À pmzm
!
¼
Xm
n¼0
pn À pm ¼ 0; ðA:7Þ
consequently (A.6) could be rewritten as
PðzÞ ¼
Aðz À z1Þðz À z2Þ . . . ðz À zmÀ1Þ Á ðz À 1Þ
1 À zmekð1ÀzÞ
; ðA:8Þ
Total probability condition for P(z) holds:
limz!1PðzÞ ¼ limz!1
X1
n¼0
pnzn
¼
X1
n¼0
pn ¼ 1 ) A ¼
k À m
ð1 À z1Þð1 À z2Þ . . . ð1 À zmÀ1Þ
;

Finally using the Laurent series:
PðzÞ ¼
X1
n¼0
pnzn
¼
ðk À mÞðz À z1Þðz À z2Þ . . . ðz À zmÀ1Þðz À 1Þ
ð1 À z1Þð1 À z2Þ . . . ð1 À zmÀ1Þ 1 À zmekð1ÀzÞ½ Š
; ðA:9Þ
Appendix B
Proof of Theorem 2. Indeed we could use complex analysis, starting from (3) and the following observation:
pm ¼ pm À pmÀ1 ¼
Xm
n¼0
pn À
XmÀ1
n¼0
pn; ðB:1Þ
Considering Laurent series expansion function H(z):
HðzÞ ¼
X1
k¼0
pkzk
; ðB:2Þ
Combining (3) and (B.1) and taking into account that pÀ1 is meaningless in our analysis:
pm ¼ pm ÀpmÀ1 )
X1
n¼0
pnzn
¼
X1
n¼0
pnzn
À
X1
n¼0
pnÀ1zn
) PðzÞ ¼ HðzÞÀ
X1
l¼À1
plzlþ1
) PðzÞ ¼ HðzÞÀz
X1
l¼0
plzl
) HðzÞ ¼
PðzÞ
1Àzð Þ
;
ðB:3Þ
Substituting (A.9) into (B.1) then:
HðzÞ ¼
X1
k¼0
pkzk
¼
ðk À mÞðz À z1Þðz À z2Þ . . . ðz À zmÀ1Þ
ð1 À z1Þð1 À z2Þ . . . ð1 À zmÀ1Þ½1 À zmekð1ÀzÞŠ
; ðB:4Þ
Differentiating (m À 1) times with respect to z, dividing by factor (m À 1)! and setting z = 0, non-delay probability could
be calculated as:
P0 ¼ pmÀ1 ¼
ðk À mÞ
XmÀ1
l¼1
ð1 À zlÞ
; ðB:5Þ
To calculate roots z1, z2, . . . , zmÀ1, we have to rely into complex analysis and the generalized argument theorem from
complex calculus [34]. We shall select function f(z) as f(z) = log(z À 1) and we do select an analytical function inside a contour
C in the z-plane which should have number of poles and zeroes inside the contour. We do select an exponential function
which has m multiple z = 0 poles inside the contour C and z1, z2, . . . , zmÀ1 zeroes:
hðzÞ ¼ 1 À
ekz
ekzm
¼ 1 À
X1
n¼0
ðkzÞn
n!
ekzm
; ðB:6Þ
Following the generalized argument theorem we integrate over the contour area C:
1
2pi
Z
C
fðzÞh
0
ðzÞ=hðzÞdz ¼
1
2pi
Z
C
logðz À 1Þh
0
ðzÞ=hðzÞdz ¼ Àpi þ
XmÀ1
l¼1
logð1 À zlÞ; ðB:7Þ
Taking logarithmic function of P0 on (B.5), substituting to (B.7) and integrating by parts:
1
2pi
Z
C
logðz À 1Þh
0
ðzÞ=hðzÞdz ¼ Àpi þ logðm À kÞ À logðP0Þ ¼
1
2pi
½logðz À 1Þ log hŠ

C
À
1
2pi
Z
C
log h
z À 1
Á 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 also
at center at z = 1, as described in Fig. 6. We then have to calculate the integral over C1, C2 and remaining line paths among
these circles. Starting with the integrals over contour area C1 the extreme points of calculation have to be deﬁned as a circle
with extreme polar coordinate points (R, h = 0) and (R, h = 2p). Then considering Fig. 6, expressing the circle in complex polar
coordinates: z À 1 = Reih
) log(z À 1) = log R + ih and for function h(z) from (B.6):
log hðzÞ ¼ log 1 À
ekðzÀ1Þ
zm

¼ log 1 À
ekReih
ð1 þ Reih
Þ
m
!
; ðB:9Þ
From (B.8) and considering the contour C1 in extreme points:

1
2pi

2p
0
À
1
2pi
Z
C
log h
z À 1
Á dz ¼ Àpi þ logðm À kÞ À logðP0Þ; ðB:10Þ
Substituting polar coordinates into (B.10) and expanding complex exponential with Euler formula:
1
2pi

2p
0
¼
1
2pi
ðlog R þ ihÞ log 1 À
ekRfcos hþi sin hg
ð1 þ Rfcos h þ i sin hgÞ
m
!

2p
0
¼ log 1 À
ekR
ð1 þ RÞm

; ðB:11Þ
Hence considering (B.10) and (B.11) the contribution of contour area C1 will be:
1
2pi
Z
C1
logðz À 1Þ
h
0
h
dz ¼
1
2pi

C1
À
1
2pi
Z
C1
log h
z À 1
Á dz )
1
2pi
Z
C1
logðz À 1Þ
h
0
h
dz
¼ log 1 À
ekR
ð1 þ RÞm

À
1
2pi
Z
C1
log h
z À 1
Á 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). We
do start our analysis from the general form of generalized argument theorem:
1
2pi
Z
logðz À 1Þh
0
ðzÞ=hðzÞdz ¼
1
2pi
Z z¼1þR
z¼1þr
logðz À 1Þh
0
ðzÞ=hðzÞdz

h¼0
þ
Z z¼1þr
z¼1þR
logðz À 1Þh
0
ðzÞ=hðzÞdz

h¼2p
ðB:13Þ
¼
1
2pi

h¼0
À
1
2pi
Z z¼1þR
z¼1þr
log h
z À 1
dz þ
1
2pi

h¼2p
À
1
2pi
Z z¼1þr
z¼1þR
log h
z À 1
dz; ðB:14Þ
Substituting polar coordinates:
1
2pi
Z
logðz À 1Þh
0
ðzÞ=hðzÞdz ¼
1
2pi
logðR þ ihÞ Á log 1 À
ekReih
ð1 þ Reih
Þ
m
! #

h¼0
À
1
2pi
Z z¼1þR
z¼1þr
log 1 À ekReih
ð1þReih
Þ
m

Reih
dz
þ
1
2pi
logðr þ ihÞ Á log 1 À
ekreih
ð1 þ reihÞ
m
! #

h¼2p
þ
1
2pi
Z z¼1þR
z¼1þr
log 1 À ekReih
ð1þReih
Þ
m

Reih
dz;
ðB:15Þ
Eliminating same factors and using Euler expansion in (B.15) the contribution of remainings into the integral over contour
C is:
1
2pi
Z
logðz À 1Þh
0
ðzÞ=hðzÞdz ¼ À log 1 À
ekR
ð1 þ RÞm

þ log 1 À
ekr
ð1 þ rÞm

; ðB:16Þ
Final contribution will be the other contour area C2. To proceed we consider again (B.8) and taking into account polar
coordinates for internal circle, z = 1 + reih
, ﬁnally we get:
1
2pi
Z
C2
logðz À 1Þ
h
0
h
dz ¼
1
2pi
Z
C2
logðreih
Þ
h
0
ð1 þ reih
Þ
hð1 þ reihÞ
dð1 þ reih
Þ ¼
1
2p
Z h¼0
h¼2p
½log r þ ihŠreih h
0
ð1 þ reih
Þ
hð1 þ 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 À k)(reih
) + O(r); consequently from (B.17):
limz!1h
0
ðzÞ=hðzÞ $ 1=ðz À 1Þ ¼
1
reih
)
1
2p
Z h¼0
h¼2p
½log r þ ihŠreih h
0
ð1 þ reih
Þ
hð1 þ reihÞ
dh ¼
1
2p
Z h¼0
h¼2p
½log r þ ihŠdh
¼ Àpi À log r; ðB:18Þ
and
limz!1hðzÞ $ ðz À 1Þh
0
ðz ¼ 1Þ ¼ ðm À kÞðz À 1Þ ) logðhð1 þ reih
ÞÞ ¼ logðm À kÞ þ logðreih
Þ
¼ logðm À kÞ þ log r þ ih ) log r ¼ limz!1 logðhð1 þ reih
ÞÞ À logðm À kÞ À ih; ðB:19Þ
Substituting (B.19) to (B.18) then:
1
2p
Z h¼0
h¼2p
½log r þ ihŠdh ¼ Àpi À limz!1 logðhð1 þ reih
ÞÞ þ logðm À kÞ À ih
Â Ã

h ¼ 0
z ! 1
¼ Àpi À logðhð1 þ reih
ÞÞ þ logðm À kÞ;
ðB:20Þ

Combining (B.8), (B.12), (B.16) and (B.20) we finally get:
1
2pi
Z
C
logðz À 1Þh
0
ðzÞ=hðzÞdz ¼ log 1 À
ekR
ð1 þ RÞm

À
1
2pi
Z
C1
log h
z À 1
Á dz À log 1 À
ekR
ð1 þ RÞm

þ log 1 À
ekr
ð1 þ rÞm

À pi À limz!1 logðhð1 þ reih
ÞÞ þ logðm À kÞ À ih
¼ Àpi þ logðm À kÞ À logðP0Þ; ðB:21Þ
Hence combining (B.21) with (B.9):
1
2pi
Z
C1
log h
z À 1
Á dz ¼ logðP0Þ ¼
1
2pi
Z
C1
1
z À 1
Á log 1 À
ekðzÀ1Þ
zm

dz; ðB:22Þ
Using Laurent series expansion around z = 1 with convergence inside the circle ekðzÀ1Þ
zm

1:
logðP0Þ ¼ À
1
2pi
Z
C1
1
z À 1
Á
X1
k¼1
ekkðzÀ1Þ
kz
m
!
dz ¼ À
X1
k¼1
1
2pki
Z
C1
1
z À 1
Á
ekkðzÀ1Þ
kz
m

dz
!
; ðB:23Þ
To calculate above integral we do use residues theorem, hence:
1
2pi
Z
C1
1
z À 1
Á
ekkðzÀ1Þ
zm

dz ¼ 1 À
XmÀ1
l¼0
ðkkÞ
l
l!
eÀkk
; ðB:24Þ
And finally the non-delay probability equals:
logðP0Þ ¼ À
X1
k¼1
1
k
1 À
XmÀ1
l¼0
ðkkÞ
l
l!
eÀkk
#
) P0 ¼ 10
À
X1
k¼1
1
k
1À
XmÀ1
l¼0
kkð Þl
l!
eÀkk
#
; ðB:25Þ
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Spiros Louvros holds the position of Assistant Professor, Computer Informatics Engineering Department, TEI of Western
Greece, 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 of
Directorate of Greek School Network, Computer Technology Institute and Press ‘‘Diophantus’’. Current research interests are in
signal theory, DSP, analog and digital communications, next generation networks, e-government and e-learning services.

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