=Paper= {{Paper |id=Vol-2332/paper-01-001 |storemode=property |title= An Accurate Model of the 3GPP NR Access Point Service Process |pdfUrl=https://ceur-ws.org/Vol-2332/paper-01-001.pdf |volume=Vol-2332 |authors=Vyacheslav O. Begishev,Eduard S. Sopin,Dmitri A. Moltchanov,Andrey K. Samuylov,Irina A. Gudkova,Konstantin E. Samouylov }} == An Accurate Model of the 3GPP NR Access Point Service Process == https://ceur-ws.org/Vol-2332/paper-01-001.pdf

UDC 519.217
An Accurate Model of the 3GPP NR Access Point Service
Process
Vyacheslav O. Begishev* , Eduard S. Sopin* , Dmitri A. Moltchanov†* ,
Andrey K. Samuylov†* , Irina A. Gudkova*‡ , Konstantin E. Samouylov*‡
*
Department of Applied Probability and Informatics
Peoples’ Friendship University of Russia (RUDN University)
Miklukho-Maklaya str. 6, Moscow, 117198, Russia
†
Department of Electronics and Communications Engineering
Tampere University of Technology
Korkeakoulunkatu str. 10, Tampere, 32720, Finland
‡
Federal Research Center “Computer Science and Control” of the Russian Academy of
Sciences (FRC CSC RAS)
44-2 Vavilov St, Moscow, 119333, Russian Federation
Email: begishev_vo@rudn.university, sopin_es@rudn.university, dmitri.moltchanov@tut.fi,
gudkova_ia@rudn.university, andrey.samuylov@tut.fi, samuylov_ke@rudn.university

The service process of sessions in 3GPP New Radio (NR) wireless access systems operating
in millimeter wave frequency band is heavily affected by the dynamic blockage of propagation
paths between user equipment (UE) and access point (AP). Although the ability of UEs’
transievers to operate over reflected propagation path components partially compensates
for this phenomenon, it simultaneously leads to the dynamic fluctuations in the amount of
resources requested during the session lifetime to support the required bitrate. In our study,
we formulate an accurate model of the 3GPP NR AP service process by taking into account
time-varying changes in the amount of requested resources caused by dynamic blockage of
propagation paths. The derived metrics of interest includes new and ongoing session drop
probabilities as well as the system resource utilization. The presented numerical results
indicate that the presence of blockage events decreases the probability of session drops upon
arrival at the expense of increasing blocking probability during the service process. However,
it does not drastically affect the system resource utilization.

Key words and phrases: 3GPP New Radio (NR), signal-to-interference ratio, mmWave.

Copyright © 2018 for the individual papers by the papers’ authors. Use permitted under the CC-BY license —
https://creativecommons.org/licenses/by/4.0/. This volume is published and copyrighted by its editors.
In: K. E. Samouylov, L. A. Sevastianov, D. S. Kulyabov (eds.): Selected Papers of the 12th International Workshop on
Applied Problems in Theory of Probabilities and Mathematical Statistics (Summer Session) in the framework of the
Conference “Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems”,
Lisbon, Portugal, October 22–27, 2018, published at http://ceur-ws.org
Begishev V. O. et al. 5

1. Introduction
In December 2017 3GPP has rectified the Phase 1 LTE-anchored New Radio (NR)
access technology. As the 3GPP efforts will now continue towards stand alone NR
technology the emphasis of the research community is shifting towards the efficient use
of the newly standardized system.
Performance of the mmWave deployments has been recently assessed using the tools
of stochastic geometry. Applying the Campbell theorem for functionals over point
processes, the moments of aggregate interference in THz and mmWave systems in
presence of molecular absorption, human-body blockage, and directional transmit and
receive antennas have been derived in [1]. Using the Taylor expansion approximation,
the authors then extended their analysis to the moments of signal-to-interference ratio
(SIR) in [2]. Particularly, the authors in [3] obtained the probability density function
(pdf) of SIR for mmWave systems operating at 28 GHz. The pdfs of interference and
SIR in the absence of blockage have been reported in [4]. The SIR distribution is
further contributed by [5], where the authors introduced a simple model of atmospheric
absorption that assumes a constant attenuation coefficient as well as disregards the
effect of blockage. An upper bound on mmWave system capacity in presence of dynamic
blockage has been obtained in [6].
The stochastic geometry approach allows to characterize wireless specifics of mmWave
communications leaving the question of traffic dynamics in mmWave access networks
unanswered. Acknowledging this problem, studies addressing both stochastic compo-
nents started to appear recently, see, e.g., [7–9]. However, most of these models take
simplified assumptions about propagation phenomenon and user equipment (UE) opera-
tion assuming that blockage of the line-of-sight path always leads to outage conditions
with the currently serving AP.
In this paper, we use the tools of queuing theory to develop an accurate model of the
session service process at a 3GPP NR access point (AP). To capture random locations
of UE in the service area, the resource requirements are assumed to be random variable.
Futhermore, to account for blockage of propagation paths between AP and UE we
assume that there is external process of blockage. The target metrics of interest are new
and ongoing sessions drop probabilities as well as the system resource utilization.

2. System Model
We consider a single 3GPP NR AP. The amount of resources available at AP is
𝐵 Hz. Consider a multiserver queuing system with 𝑁 servers, where arriving sessions
require a server and random amount of resources of total volume 𝑀 . The session arrival
process is homogenous Poisson with intensity 𝜆. Each session, upon arrival, requeres
radnom amount of resources from the system, 𝑅, with cumulative distribution function
(CDF) 𝐹𝑅 (𝑥). The random nature of resource requests stems from random locations
of UE in the AP service zone. Given a certain session rate, the set of modulation and
coding schemes (MCS) for 3GPP NR available in 3GPP Relese 15, 3GPP multipath
propagation model, and distribution of users in the AP service zone CDF 𝐹𝑅 (𝑥) can be
found using conventional methods of stochastic geometry [7, 8].
We use the standardized 3GPP stochastic multipath propagation model specified
in [10] that assumes that the received power at the UE consists of power coming from
a number of rays, including LoS path and several reflected components. At any given
instant of time UE is associuated with the ray having the highest power. Furthermore,
rays are assumed to be blocked by a crowd moving around active UEs. Using the results
of [11, 12] one may approximatethe blockage process of rays using Poisson process with
intensity 𝜇. Resource requirements of sessions are independent identically distributed
random variables, independent of arrival and serving processes, which are determined
by probability distribution {𝑝𝑚 }, 𝑚 ≥ 0.
6 APTP+MS’2018

rB a
rB
Access point rB
a
rB
User equipment xN
Active link (LoS non-blocked)
Active link (LoS blocked) ~ a 2  users

rB x4
rB rB
xN 2
r x1 xi

rB rB Radio resources

xj
rB

Figure 1. The considered street 3GPP deployment with connectivity operation.

Server Radio
Initial arrivals (mmWave APs) resources Successful
sessions
 
1
R
 ..
.
N Interrupted
sessions

Figure 2. An illustration of the proposed queuing network model.

According to the abovementioned discussion each active session in the system is
associated with a homogeneous Poisson process of events with intensity 𝜇.1 Upon each
event the active session changes its resource requirements by drawing them from the
CDF 𝐹𝑅 (𝑥). If the new resource requirements is smaller, the session continues service
at AP. alternatively, if the new resource requirements are higher the session might be
dropped if there is insufficient amount of free resources at the AP current. Thus, in the
considered system, a session can be dropped upon arrival or during the service process.
We are interested in drop probabilities and new and already accepted sessions as well as
in system resource utilization.
1
In what follows, these events are referred to as “signals”.
Begishev V. O. et al. 7

3. The Queuing Framework
For the analysis of the described queuing system we use the simplified approach,
which is common for the queues with random resource requirements. Particularly, instead
of keeping track of the amount of resources occupied by each session we track only total
amount of resources occupied by all the sessions in the system. While this approach
allows to significantly reduce the complexity of the analysis, in the simplified system we
do not know the exact number of resources that should be released at the departure
time instant of a session or at the arrival time instant of a signal. To alleviate this
shortcoming and decide upon the amount ot resources released we use the Bayes law.
Particularly, if 𝑛 sessions in the system occupy 𝑚 resources, then the probability that 𝑖
(𝑛−1) (𝑛) (𝑛)
resources are released on the departure time instant is 𝑝𝑖 𝑝𝑚−𝑖 /𝑝𝑚 , where 𝑝𝑚 is the
probability that 𝑛 sessions occupy 𝑚 resources. The sought probability is estimated as
in (1)
𝑚
(𝑛)
∑︁
𝑝𝑚 = 𝑝𝑖 𝑝𝑛−1
(𝑚−𝑖)
, 𝑛 ≥ 2. (1)
𝑖=0
(1)
where 𝑝𝑚 = 𝑝𝑚 , 𝑚 ≥ 0.
Resource reallocation at the arrival time instant of a signal is performed similarly, i.e.,
if 𝑛 sessions in the system occupy 𝑚 resources, then the probability that 𝑖 resources are
(𝑛−1) (𝑛)
released upon signal arrival is 𝑝𝑖 𝑝𝑚−𝑖 /𝑝𝑚 . Thus, after arrival of a signal 𝑛 sessions
(𝑛−1) (𝑛)
occupy 𝑘 resources with probability [𝑝𝑖 𝑝𝑚−𝑖 /𝑝𝑚 ]𝑝𝑘−𝑚+𝑖 , 𝑘 ≤ 𝑀 and with probability
(𝑛−1) (𝑛) ∑︀ −𝑚+𝑖
[𝑝𝑖 𝑝𝑚−𝑖 /𝑝𝑚 ](1 − 𝑀 𝑘=0 𝑝𝑘 ) new resource requirements of the session cannot be
fulfilled and it is lost.
Behavior of the system can be described by the stochastic process 𝑋(𝑡) = (𝜉(𝑡), 𝛿(𝑡)),
where 𝜉(𝑡) denotes number of sessions in the system and 𝛿(𝑡) is the total amount of
occupied resources. The set of states is provided in (2)
{︁ }︁
(𝑛)
⋃︁
𝑆= 𝑆𝑛 , 𝑆𝑛 = (𝑛, 𝑚) : 0 ≤ 𝑚 ≤ 𝑀, 𝑝𝑚 > 0 . (2)
0≤𝑛≤𝑁

Arrange the states in 𝑆𝑛 in increasing order of the amount of the occupied resource
and denote 𝐼(𝑛, 𝑚) the sequence number of the state (𝑛, 𝑚). The stationary probabilities
(3) of 𝑋(𝑡) are written as

𝑞𝑛 (𝑚) = lim 𝑃 {𝜉(𝑡) = 𝑛, 𝛿(𝑡) = 𝑚} , (𝑛, 𝑚) ∈ 𝑆𝑛 , (3)
𝑡→∞

The system of equilibrium equations takes the form as in (4)

𝑀 𝑀
(︃ )︃ (︃ )︃
∑︁ ∑︁ ∑︁ ∑︁
𝜆 𝑝𝑚 𝑞0 (0) = 𝜇 𝑞1 (𝑚) + 𝛾 1− 𝑝𝑖 , (4)
𝑚=0 𝑚:(1,𝑚)∈𝑆1 𝑚:(1,𝑚)∈𝑆1 𝑞1 (𝑚) 𝑖=0

⎛ ⎞
𝑀
∑︁−𝑗 ∑︁
⎝𝜆 𝑝𝑗 + 𝑛𝜇 + 𝑛𝛾 ⎠ 𝑞𝑛 (𝑚) = 𝜆 𝑞𝑛−1 (𝑚 − 𝑗)𝑝𝑗 +
𝑗=0 𝑗:(𝑛−1,𝑚−𝑗)∈𝑆𝑛−1
(𝑛)
∑︁ 𝑝𝑗 𝑝𝑚
+ (𝑛 + 1)𝜇 𝑞𝑛+1 (𝑚 + 𝑗) (𝑛+1) +
𝑗:(𝑛+1,𝑚+𝑗)∈𝑆𝑛+1 𝑝 𝑚+𝑗
8 APTP+MS’2018

𝑚𝑖𝑛(𝑗,𝑚) (𝑛−1)
∑︁ ∑︁ 𝑝𝑗−𝑖 𝑝𝑖
+ 𝑛𝛾 𝑞𝑛 (𝑗) (𝑛)
𝑝𝑚−𝑖 +
𝑗:(𝑛,𝑗)∈𝑆𝑛 𝑖=0 𝑝𝑗
⎞ ⎛
𝑀 −𝑚 (𝑛)
∑︁ ∑︁ 𝑝𝑗 𝑝𝑚
+ (𝑛 + 1)𝛾 𝑞𝑛+1 (𝑚 + 𝑗) (𝑛+1) · 1 −
⎝ 𝑝𝑗 ⎠ ,
𝑗:(𝑛+1,𝑚+𝑗)∈𝑆𝑛+1 𝑝𝑚+𝑗 𝑗=0

(𝑛, 𝑚) ∈ 𝑆𝑛 , 1 ≤ 𝑛 ≤ 𝑁 − 1,

∑︁
(𝑁 𝜇 + 𝑁 𝛾) 𝑞𝑁 (𝑚) = 𝜆 𝑞𝑁 −1 (𝑚 − 𝑗)𝑝𝑗 +
𝑗:(𝑁 −1,𝑚−𝑗)∈𝑆𝑁 −1

𝑚𝑖𝑛(𝑗,𝑚) (𝑁 −1)
∑︁ ∑︁ 𝑝𝑗−𝑖 𝑝𝑖
+𝑁 𝛾 𝑞𝑁 (𝑗) (𝑁 )
𝑝𝑚−𝑖 , (𝑁, 𝑚) ∈ 𝑆𝑁 .
𝑗:(𝑁,𝑗)∈𝑆𝑁 𝑖=0 𝑝𝑗

The system of equations (4) - (5) is complemented with the normalization conditions
and then solved numerically. Since all state transitions take place between either states
from one substate 𝑆𝑛 or states from adjacent substates 𝑆𝑛 and 𝑆𝑛−1 , the generator
matrix of 𝑋(𝑡) can be represented in block-tridiagonal form simplifying the solution.
Using the stationary probabilities, one can evaluate the main performance measures
of the system: the average number of occupied resources 𝑏 in (5), session blocking
probability 𝜋𝑑 (the probability that a session is lost upon arrival) in (6) and session
blocking probability during the service time, 𝜋𝑡 (the probability that a session is dropped
upon signal arrival) in (7): ∑︁
𝑏= 𝑚𝑞𝑛 (𝑚), (5)
(𝑛,𝑚)∈𝑆

∑︁ 𝑀 −𝑚
∑︁
𝜋𝑑 = 1 − 𝑞𝑛 (𝑚) 𝑝𝑗 , (6)
(𝑛,𝑚)∈𝑆,𝑛<𝑁 𝑗=0
⎛ ⎞
𝑚 𝑝 𝑝(𝑛−1) 𝑀 −𝑚+𝑗
∑︁ ∑︁ 𝑗 𝑚−𝑗 ∑︁
𝜋𝑡 = 𝑞𝑛 (𝑚) ⎝1 − 𝑝𝑖 ⎠ . (7)
(𝑛)
(𝑛,𝑚)∈𝑆 𝑗=0 𝑝𝑚 𝑖=0

We specifically note that the ongoing session drop probability is interpreted as the
fraction of signals that lead to the drop of ongoing session.

4. Numerical Results
In this section we provide sample illustrative results. The default system parameters
used in what follows are provided in Table 1 below.
The response of the 3GPP NR system service process to input system parameters is
illustrated in Figs. 3-6. Analyzing the behavior of the blocking probabilities upon arrival
and during the service time as a function of the session arrival intensity one may observe
that over the considered interval both curves are characterized by exponential behavior.
Furthermore for the chosen value of system parameters the blocking probability upon
session arrival is much higher than the blocking probability of session during the service
time. To reveal the detailed behavior of these two metrics consider blocking probabilities
upon arrival and during the session time as a function of signal intensity illustrated in
Fig. 4. As one may observe, the increase in the signals intensity leads to the decrease in
Begishev V. O. et al. 9

Table 1
The default system parameters.

Parameter Value

Number of resource blocks per timeframe 100
Number of servers available 100
Sessions request distribution geometric
Mean session request size (RBs) 2
Sessions arrival intensity 1, 1.1,...,2
Session service intensity 0.05



Figure 3. Blocking probability as a function of the session arrival intensity.

the blocking probability of ongoing sessions. The probability that a session is blocked
upon arrival decreases as well. The underlying reason for this behavior is that the
increase in the signals intensity results in more sessions dropped during the service
process due to insufficient amount of resources thus leaving more resources “on average”
for new session arrivals. In extreme case when the signals intensities is very high, almost
all the sessions are admitted in the system and then eventually dropped during the
service process. Alternatively letting the signals intensity approaching zero no losses
during the service time are experienced.
Fig. 5 shows the system resource utilization as a function of the signals intensity.
As one may observe, for rather wide range of signals intensity the resource utilization
remains almost the same. In spite of this behavior, more resources are wasted as signal
intensity increases as more sessions leave the system prior to service completions. Thus,
aside from classic systems, where system utilization is one of the critical performance
indicators for systems provided in prospective 5G systems one has to consider more
advanced metrics that quantify not only resource utilization but a fraction of resources
wasted due to partial service.
Finally, Fig. 6 shows the new and ongoing session drop probability as a function of
mean session size. Recall that keeping the session arrival rate constant while increasing
10 APTP+MS’2018



Figure 4. Blocking probability as a function of the signal intensity.



Figure 5. System resource utilization as a function of the signal intensity.

the mean session size we increase the offered traffic load to the size. Thus, expectedly,
both probabilities are characterized by the increasing behavior. Similarly to previous
illustrations the new session loss probability is higher than the ongoing session drop
probability.
Begishev V. O. et al. 11

Figure 6. Blocking probability as a function of the mean session request size.

5. Conclusions
In this paper we have developed an accurate model for the 3GPP NR system
service process. The proposed models not only accounts for inherently variable session
resource requirements induced by random rate requests and random location of users but
captures the blockage of propagation path between UEs and AP. The latter is modeled
by introducing an external process of events causing resource re-allocations for sessions
already accepted to the system.
The developed model allows for systematic analysis of 3GPP NR AP service process
in various deployments. The sample numerical results have shown that the presence
of external process of signals modeling the blockage process decreases the probability
of session drops upon arrival at the expense of increasing blocking probability during
the service process. At the same time, it does not drastically affect the system resource
utilization.

Acknowledgments
The publication has been prepared with the support of the “RUDN University
Program 5-100” and funded by RFBR according to the research projects No.18-07-00576,
18-37-00380. This work has been developed within the framework of the COST Action
CA15104, Inclusive Radio Communication Networks for 5G and beyond (IRACON).

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