Statistical QoS provisioning for MTC networks under finite blocklength
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Abstract
This paper analyzes the effective capacity of delayconstrained machinetype communication (MTC) networks operating in the finite blocklength regime. First, we derive a closedform mathematical approximation for the effective capacity in quasistatic Rayleigh fading channels. We characterize the optimum error probability to maximize the concave effective capacity function with reliability constraint and study the effect of signaltointerferenceplusnoise ratio (SINR) variations for different delay constraints. The tradeoff between reliability and effective capacity maximization reveals that we can achieve higher reliability with limited sacrifice in effective capacity specially when the number of machines is small. Our analysis reveals that SINR variations have less impact on effective capacity for strict delayconstrained networks. We present an exemplary scenario for massive MTC access to analyze the interference effect proposing three methods to restore the effective capacity for a certain node which are power control, graceful degradation of delay constraint, and joint compensation. Joint compensation combines both power control and graceful degradation of delay constraint, where we perform the maximization of an objective function whose parameters are determined according to the delay and SINR priorities. Our results show that networks with stringent delay constraints favor power controlled compensation, and compensation is generally performed at higher costs for shorter packets.
Keywords
Effective capacity Machinetype communication Finite blocklength Ultrareliable communication1 Introduction
Modern communication systems are becoming an indispensable part of our lives. Driven by the demands of users for extra services, the fifth generation (5G) of mobile communication is expected to introduce new features such as ultrareliable lowlatency communications (URLLC) and massive machinetype communication (mMTC) [1, 2, 3, 4, 5, 6]. These features may serve many yet unforeseen applications to enable the Internet of Things (IoT). IoT aims at bringing connectivity to anything that can benefit from internet connection [7]. URLLC has emerged to provide solutions for reliable and lowlatency transmissions in wireless systems. The design of URLLC systems imposes strict quality of service (QoS) constraints to fulfill very low latency in the order of milliseconds with expected reliability of higher than 99.9% [1, 4]. In [8], Schulz et al. discussed the reliability requirements for different IoT applications. According to their study, latency bounds range from 1 ms in factory automation to 100 ms in road safety. In addition, the packet loss rate constraints range from 10^{−9} in printing machines to 10^{−3} for traffic efficiency. Such requirements are far more stringent than the ones in the current LongTerm Evolution (LTE) standards [9].
The mMTC refers to networks that can support a variety of connected smart devices at the same time with the same base station (BS). It obligates a certain level of connectivity to a machine via ultrareliable communication (URC) over relatively long term (> 10 ms) [1]. The number of connected devices is expected to cross the 28 billion border by 2021, where a single macrocell may need to uphold 10,000 or more devices in the future [10, 11]. Moreover, the traffic behavior of MTC is quite different from the HTC (humantype communication), where [12]:

MTC is coordinated (i.e, there are simultaneous access attempts from many machine reacting to the same events), while HTC is uncoordinated.

MTC uses short as well as small number of packets.

MTC traffic is real as well as nonrealtime, periodic, and eventdriven.

MTC QoS requirement is different from HTC (i.e., different reliability and latency requirements).
In this context, MTC has gained an increasing interest in recent years via employing new multiple radio access technologies and efficient utilization of spectrum resources to improve reliability and robustness [13, 14, 15]. Another research topic that has gathered much attention is the cooperative transmission in MTC, where in [11], the authors proposed a locationbased cooperative strategy to reduce the error outage probability, but without study of the latency aspects.
Traditional communication systems are based on Shannon theoretic models and utilize metrics such as channel capacity or ergodic capacity [16]. Unlike classical systems, URLLC networks are designed to communicate on short packets in order to satisfy extremely low latency in realtime applications and emerging technologies such as ehealth, industrial automation, and smart grids whenever data sizes are reasonably small such as sensor readings or alarm notifications. In the finite blocklength regime, the length of metadata is of comparable size with the length of data. Such demands stimulated a revolutionary trend in information theory studying communication at finite blocklength (FB) [17, 18, 19]. In this context, conventional metrics (e.g., channel capacity or ergodic capacity) become highly suboptimal [17]. For this reason, the maximum achievable rate for quasistatic fading channels was characterized in [19] as a function of blocklength and error probability ε. In [9], the authors analyzed the effect of using smaller resource blocks on error probability bounds in OFDM. The effect of relaying of blocklengthlimited packets was studied and compared to direct transmission in [20, 21] where the authors concluded that relaying is more efficient than direct transmission in the FB regime specially with an average channel state information (CSI). Furthermore, the authors of [3] introduced a pernode throughput model for additive white Gaussian noise (AWGN) and quasistatic collision channels. Therein, average delay is considered, and interference is treated as AWGN.
To model the delay requirements in URLLC and MTC networks, we resort to the effective capacity (EC) metric which was introduced in [22]. It indicates the maximum possible arrival rate that can be supported by a network with a target delay constraint. In [18], the authors considered quasistatic Rayleigh fading channels and introduced a statistical model for a single node effective rate in bits per channel use (bpcu) for a certain error probability and delay exponent which reflects the latency requirement. However, throughout the paper, a closedform expression for the EC was not provided. Exploiting the EC theory, the authors of [23] characterized the latencythroughput tradeoff for cellular networks. In [24], Musavian and LeNgoc analyzed the EC maximization of secondary node with some interference power constraints for primary node in a cognitive radio environment with interference constraints. Three types of constraints were imposed, namely average interference power, peak interference power, and interference power outage. The fundamental tradeoff between EC and consumed power was studied in [25] where the authors suggested an algorithm to maximize the EC subject to power constraint for a single node scenario. In [26], we studied the pernode EC in MTC networks operating in quasistatic Rayleigh fading proposing three methods to alleviate interference, namely power control, graceful degradation of delay constraint, and the joint method. To the best of our knowledge, EC for FB packet transmission in multinode MTC scenario has not been investigated until part of the work in this journal was presented in [26], which will be depicted here with extra details.
Based on its intuition, the EC theory provides a mathematical framework to study the interplay among transmit power, interference, delay, and the achievable rate for different wireless channels. In this paper, we derive a mathematical expression for EC in quasistatic Rayleigh fading for delayconstrained networks. Our results depict that a system can achieve higher reliability with a negligible sacrifice in its EC. We consider dense MTC networks and characterize the effect of interference on their EC. We propose three methods to allow a certain node maintain its EC which are (i) power control, (ii) graceful degradation of delay constraint, and (iii) joint model. Power control depends on increasing the power of a certain node to recover its EC which in turn degrades the SINR of other nodes. Our analysis proves that SINR variations have limited effect on EC in networks with stringent delay limits. Hence, the side effect of power control is worse for less stringent delay constraints and vice versa. We illustrate the tradeoff between power control and graceful degradation of delay constraint. Furthermore, we introduce a joint model which combines both of them. The operational point to determine the amount of compensation performed by each of the two methods in the joint model is determined by maximization of an objective function leveraging the network performance.
The motivation beyond this paper is to provide a solid understanding of the tradeoff between power, delay, and reliability in MTC networks in the finite blocklength regime. Our objective is to pave the road for utilizing short packets in 5G and machinetype networks. Extra plots that were not present in [26] are illustrated to provide the reader with full understanding of the objective function in joint compensation and the compensation process itself. Moreover, we extend the analysis in [26] by solving the optimization problem to obtain the optimum error probability which maximizes the EC in the ultrareliable region. We also characterize the tradeoff between reliability and EC which shows that we can obtain a huge gain in reliability in return for a slight reduction in EC.
List of abbreviations and symbols
bpcu  Bits per channel use 
EC  Effective capacity 
Max  Maximize 
 Probability density function 
QoS  Quality of service 
SINR  Signaltointerferenceplusnoise ratio 
s.t  Subject to 
URC  Ultrareliable communication 
C(ρh^{2})  Shannon capacity 
D _{max}  Maximum delay 
\(\mathbb {E}[ \ ]\)  Expectation of 
EC  Effective capacity 
EC _{ max}  Maximum effective capacity 
\(\mathfrak {L\left (\epsilon,\lambda \right)}\)  Lagrangian function 
N  Number of nodes 
Pr()  Probability of 
\(P_{\text {out\_delay}}\)  Delay outage probability 
Q(x)  Gaussian Qfunction 
Q^{−1}(x)  Inverse Gaussian Qfunction 
T _{ f}  Blocklength 
V(ρh^{2})  Channel dispersion 
e  Exponential Euler’s number 
h^{2}  Fading coefficient 
ln  Natural logarithm to the base e 
log2  Logarithm to the base 2 
r  Normalized achievable rate 
w  Additive while Gaussian noise vector 
x _{ n}  Transmitted signal vector of node n 
y _{ n}  Received signal vector of node n 
z  Fading random variable 
α  Collision loss factor 
α _{ c}  Compensation loss factor 
\(\alpha _{c_{o}}\)  Operational point of compensation loss factor 
α _{ t}  Total loss 
γ _{ c}  Compensation gain 
θ  Delay exponent 
ε  Error probability 
ε _{ t}  Target error probability 
ε ^{∗}  Optimum error probability 
η _{ α}  Compensation loss priority factor 
η _{ θ}  Delay priority factor 
ρ  Signaltonoise ratio 
ρ _{ c}  Compensation SNR 
\(\rho _{c_{o}}\)  Operational point of compensation SNR 
ρ _{ i}  Signaltointerferenceplusnoise ratio 
ρ _{ s}  SINR of other noncompensating nodes 
2 Preliminaries
2.1 Network model
Based on our network model, the received vector \(\mathbf {y}_{n}\in \mathbb {C}^{n}\) of node n is given by:
where \(\mathbf {x}_{n} \in \mathbb {C}^{n}\) is the transmitted packet of node n, and h_{n} is the fading coefficient for node n which is assumed to be quasistatic with Rayleigh distribution. This implies that the fading coefficient h_{n} remains constant for each block of T_{f} channel uses which span the whole packet duration and changes independently from one block to another. The index s includes all N−1 interfering nodes which collide with node n, and w is the additive complex Gaussian noise vector whose entries are defined to be circularly symmetric with unit variance. Given the signaltonoise ratio ρ of a single node, the signaltointerferenceplusnoise ratio of any node n is
To simplify the analysis, we assume that (i) each node always has a packet to transmit (buffer is always nonempty), (ii) all nodes are equidistant from the common controller (i.e., same path loss), and (iii) the fading coefficients h_{s} are independent and identically distributed and perfectly known to the receiver. Thus, as the number of nodes increases, the sum of Rayleighdistributed fading envelopes of N−1 interfering nodes becomes \(\sum _{s} h_{s}^{2}\approx N1\) [27], and the interference resulting from nodes in set s can be modeled as in [3] where (2) reduces to:
Note that CSI acquisition in this setup is not trivial, and its cost is negligible whenever the channel remains constant over multiple symbols. Additionally, as in [3], we aim to provide a performance benchmark for such networks without interference coordination.
2.2 Communication at finite blocklength
In this section, we present the notion of FB transmission, in which short packets are conveyed at rate that depends not only on the SNR, but also on the blocklength and the probability of error ε [17]. In this case, ε has a small value but not vanishing. For error probability ε∈[0,1], the normalized achievable rate in bits per channel use is given by:
where
is Shannon’s channel capacity for sufficiently long packets, while
denotes the channel dispersion which appears for relatively short packets (T_{f}<2000 channel uses) [18], \(Q(t)=\int _{t}^{\infty }\frac {1}{\sqrt {2 \pi }}e^{\frac {s^{2}}{2}} ds\) is the Gaussian Qfunction, and Q^{−1}(t) represents its inverse, ρ_{i} is the SINR, and h^{2} is the fading envelope.
The channel is assumed to be Rayleigh quasistatic fading where the fading coefficients remain constant over T_{f} symbols which spans the whole packet duration. For Rayleigh channels [28], the fading coefficients Z=h^{2} have the following probability density function distribution:
2.3 Effective capacity
The concept of EC indicates the capability of communication nodes to exchange data with maximum rate and certain latency constraint and thus guarantees QoS by capturing the physical and link layer aspects. A statistical delay violation model implies that an outage occurs when a packet delay exceeds a maximum delay bound D_{max}, and its probability is defined as [22]:
where Pr(·) denotes the probability of a certain event. Conventionally, a network’s tolerance to long delay is measured by the delay exponent θ. The network has more tolerance to large delays for small values of θ (i.e., θ→0), while for large values of θ, it becomes more delay strict. For the infinite blocklength model, the EC capacity is defined as:
In quasistatic fading, the channel remains constant within each transmission period T_{f} [21], and the EC is subject to the finite blocklength error bounds and thus according to [18] can be written as:
where
In [18, 29], the effective capacity is statistically studied for single node scenario in block fading but never to a closed form expression. It has been proven that the EC is concave in ε and, hence, has a unique maximizer. In what follows, we shall represent the EC expression for quasistatic Rayleigh fading.
3 Effective capacity analysis under finite blocklength
Lemma 1
The effective capacity of a certain node communicating in a quasistatic Rayleigh fading channel is approximated by:
with
where \(d=\frac {\theta T_{f}}{\ln (2)}\). Also, let \(c=\theta \sqrt {T_{f}} Q^{1}(\epsilon)\log _{2}e\) and \(x=\sqrt {1\frac {1}{(1+\rho _{i} z)^{2}}}\), and Γ(·,·) is the upper incomplete gamma function ([30], § 8.3502).
Proof
Please refer to Appendix 1. □
Lemma 2
Proof
The expectation in (10) is shown to be convex in ε in [18] independent of the distribution of channel coefficients z=h^{2}. Thus, it has a unique minimizer ε^{∗} which is consequently the EC maximizer given by (14). □
Note that \(c=\theta \sqrt {T_{f}} Q^{1}(\epsilon)\log _{2}e\) is not a function of z, so it can be taken out of the integration which simplifies the optimization problem. To obtain the maximum pernode effective capacity EC_{max}, we simply insert ε^{∗} into (12).
4 Maximization of effective capacity in the ultrareliable region
Taking a closer look at Fig. 2, we observe the tradeoff between the pernode EC and error probability ε. It is apparent that we can earn a lower error probability ε by sacrificing only a small amount of EC. For example, we observe that for the fivenode network operating in quasistatic Rayleigh fading channel, if we tolerate a decrease in the EC from 0.11 to 0.1 bpcu, the error probability ε improves to 10^{−3} instead of 2.5×10^{−2}. Thus, sacrificing only 9% of the EC maximum value boosts the error probability by nearly 1250% and hence leads to a dramatic enhancement of reliability. This observation is an open topic for analysis of the EC ε tradeoff with a target of maximizing the EC with some error constraint reflecting the reliability guarantees.
5 Methods
Given that all nodes transmit at the same time slot, the controller attempts decoding the transmitted symbols arriving from all of them. When the controller decodes one node’s data, the other streams appear as interference to it [3]. For this model, imagine that a node needs to raise its EC in order to meet its QoS constraint. At first glance, applying successive interference cancelation at the base station would seem to be an attractive solution. However, this will result in extra delay for lower priority nodes where the decoder must wait for the higher priority packets to be decoded first to perform interference cancelation which dictates parallel decoding [32]. We study the interference alleviation scenarios for one node at a certain time slot, while other nodes’ packets also are transmitted and decoded at the same time as a lower bound worse case.
5.1 Power control
The method of power control is based on increasing the SNR of node n to allow it recover from the interference effect. Let ρ_{c} be the new SNR of node n, while the other nodes still transmit with SNR equal to ρ. Then, we equate the SINR equation in (3) to the case where no collision occurs (N=1) to obtain:
When a certain node transmits with SNR of ρ_{c}, its EC is the same as in the case when transmitting with SNR equals to ρ while other nodes are silent. The method of power control is simple; however, it causes extra interference into the other nodes due to the power increase of the recovering node.
From (24), we define the SINR of other nodes colliding in the same network (nodes in set s) after the compensation of one node as:
Now, we are interested in comparing the pernode EC in three cases: (i) no collision, (ii) collision without compensation, and (iii) collision with compensation of one node. Here, we look for the maximum achievable pernode EC in each case. Define the collision loss factor α as the ratio between the maximum effective capacity EC_{max} in case of collision and in case of no collision as:
where ε^{∗} and \(\epsilon _{i}^{*}\) are the optimal error probabilities for the cases of no collision (N=1) and collision without compensation, respectively, and both are obtained from (19).
To determine the effect of compensation of one node on the other nodes, we define the compensation loss factor α_{c} as the ratio between maximum EC of other nodes (set s) in case of one node compensation and in case of no compensation, that is:
where \(\epsilon _{s}^{*}\) is the optimum error probability obtained from (14) when the SINR is set to ρ_{s}. To understand the effect of increased interference on the network performance, we study the effect of SINR variations on EC for different delay constraints.
Proposition 1
SINR variations have comparably limited effect on EC when the delay constraint becomes more strict and vice versa.
Proof
Please refer to Appendix 2. □
5.2 Graceful degradation of the delay constraint
where \(\epsilon _{i}^{*}\) is the maximizer of EC for the parameters ρ_{i} and θ_{i}, and \(\epsilon _{i}^{*}\) is the optimum error probability for ρ_{i} and θ_{i}. The solution of (30) renders the necessary value of θ_{i} to compensate for the EC decrease due to collision in this case. Notice that (30) can be solved numerically to obtain the necessary value of θ_{2} to compensate for the rate decrease due to collision in this case.
5.3 Joint compensation model
where \(\epsilon _{s_{o}}^{*}\) is the optimum error probability obtained from (14) for the parameters \(\rho _{s_{o}}\) and θ_{1}. \(\alpha _{c_{o}}\) is considered to be the loss factor caused by the part of compensation performed via power control.
From (33), we compute the necessary value of θ_{2} to continue the compensation process via graceful degradation of the delay constraint.
where the solution to this problem gives the optimum operational point which can be found from (31), (32), and (33).
6 Results and discussion
7 Conclusions
In this work, we presented a detailed analysis of the EC for delayconstrained MTC networks in the finite blocklength regime. For quasistatic Rayleigh fading channels, we proposed an approximation for the EC and defined the optimum error probability. We characterized the optimization problem to maximize EC with error constraint which showed that there is a relatively small sacrifice in EC for high SINR. Our analysis indicated that SINR variations have minimum effect on EC under strict delay constraints. In a dense MTC network scenario, we illustrated the effect of interference on EC. We proposed power control as an adequate method to restore the EC in networks with less stringent delay constraints. Another method is graceful degradation of delay constraint, where we showed that a very limited extension in the delay bound could successfully recover the EC. Joint compensation emerges as a combination between these two methods, where an operational point is selected to maximize an objective function according to the networks design aspects. Finally, we concluded that for high values of θ, fixedrate transmission performs strictly better. As future work, we aim to analyze the impact of imperfect CSI on the EC and coordination algorithms that maximize EC with fairness constraints.
8 Appendix 1: Proof of Lemma 1
After manipulating (42) and inserting it into (10), we obtain (12).
9 Appendix 2: Proof of Proposition 1
which is strictly negative and thus validating our proposition.
Notes
Funding
This work is partially supported by Aka Project SAFE (Grant no. 303532) and by Finnish Funding Agency for Technology and Innovation (Tekes), Bittium Wireless, Keysight Technologies Finland, Kyynel, MediaTek Wireless, and Nokia Solutions and Networks.
Availability of data and materials
The paper is selfcontained, since we provide a mathematical framework which can be reproduced with the details provided in Sections 2 to 5, and in Section 5, numerical results and parameter settings are described in details.
Authors’ contributions
MS derived the equations and performed the system simulations. ED revised the equations and contributed to the writing of the introduction, system model, and conclusion. HA supervised and reviewed the paper, while ML directed and supervised the research. All authors participated in this work, and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer institutional affiliations.
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