I Introduction
The support for Internet of Things (IoT) plays a major role in the evolution of wireless/cellular systems, including the upcoming 5G communication systems [1]. There are two general classes of IoT transmission technologies: (1) Lowpower widearea (LPWA) communication technologies (e.g., LoRa [2], Sigfox [3], IEEE 802.11ah) that operate in unlicensed spectra, and offer lowcost devices and ease of network deployment; (2) Cellular IoT technologies (e.g., LTE Cat1, LTE Cat0, LTEM CatM1, NBIoT) that operate in licensed spectra. Among these, NBIoT is a recent technology that has gained a significant momentum, as observed by the fast standardization during 2016 [4] and the increasing number of deployments.
NBIoT is designed to accommodate a massive number of lowthroughput, lowcost, and delaytolerant devices. Similarly to LTE networks, each device registers at the network through an Access Reservation Protocol (ARP), i.e., random access, in which time and frequency misalignments can be adjusted. The preamble sequence is transmitted at the first step of the ARP, however, the preamble structure is no longer based on ZadoffChu sequence. The preamble design and detection algorithm for the NBIoT ARP was presented in [5], while an overview of the NBIoT air interface was given in [6].
The performance of the ARP significantly degrades due to preamble misdetections and collisions. The preambles in NBIoT systems were designed with a goal of extending coverage and reducing the occurrence of misdetections. Yet, the number of devices within cell coverage is expected to grow, leading to an increasing number of preamble collisions. This motivates us to configure the preamble structure by considering the effect of collisions.
This letter presents an enhanced ARP with a partial preamble transmission (PPT) mechanism that leverages the tradeoff between misdetections and collisions. We can significantly improve the performance of the ARP by puncturing the preamble sequence through the proposed PPT mechanism. It is worth noting that the proposed ARP requires only minor modifications on how the preambles are transmitted and detected, which can be easily implemented in NBIoT systems.
Ii Access Reservation Protocol in NBIoT
In NBIoT systems, the ARP consists of 5 steps, and the detail of each step is described as follows:

Step 1: The device selects one of the available preamble sequences, and transmits it in the Narrowband Physical Random Access Channel (NPRACH);

Step 2: The eNodeB detects the preamble sequences and responds to the detected preamble sequences by sending a Random Access Response (RAR), which includes the index of the preamble sequence, the time alignment (TA) offset and an uplink grant;

Step 3: The device proceeds with the signaling information exchange on the resources indicated by the RAR, termed the RRC Connection Request;

Step 4: The eNodeB acknowledges the signaling information received from the device with the RRC Connection Setup message;

Step 5: Finally, the device transmits its data concatenated with the RRC Connection Setup Complete message.
NBIoT can be implemented with a kHz bandwidth, composed of subcarriers with each subcarrier spacing of kHz, which can be configured for the NPRACH; allowing 12, 24, 36, or 48 orthogonal RA preamble sequences to be available. A preamble sequence is composed of multiple symbol groups as shown in Fig. 1, where a single symbol group consists of a cyclic prefix (CP) and symbols. The rationale behind this structure is to reduce the relative CP overhead, and, thus, should be carefully set to be sufficiently small so that the channel condition remains the same within each of the symbol groups. All of symbols have the same value of “1”. symbol groups configure a basic unit for repetition, which can be repeated times, where , . Therefore, the length of a preamble sequence, , can be represented as . and are commonly set to 5 and 4, respectively.
Each symbol group uses a single carrier, and hops across frequency to facilitate to estimate uplink timing alignment at the eNodeB. Thus, selecting a preamble sequence implies that each device selects a hopping pattern, and
represents a mapping function from a preamble index to the set of subcarrier indices which are used by the corresponding preamble sequence. The subcarrier index used by the th symbol group of the th preamble sequence is denoted as , , and, thus, . Finally, NBIoT networks can support up to 3 coverage classes, which can be configured with different values of , in order to support specific coverage requirements. However, in this letter, we focus on a single coverage class.Iii Proposed Access Reservation Protocol with a Partial Preamble Transmission Mechanism
Iiia Partial Preamble Transmission Mechanism
The key idea of a partial preamble transmission (PPT) mechanism is to allow each device to transmit a fraction of a preamble sequence, which is called a partial preamble sequence (PPS). According to the PPS configuration, the NPRACH can be virtually divided into multiple subNPRACHs, each of which is named a partial unit, where a single PPS is transmitted. We note that this approach does not alter the intrinsic structure of the protocol, and is instead a reconfiguration of the protocol.
In the baseline ARP, when the amount of NPRACH resources is determined, the length of the preamble sequence is configured as , where represents the number of repetitions. However, in the proposed scheme, a PPS with a length of can be configured as , with , where represents the number of repetitions in the PPSs. In this case, the eNodeB can configure nonoverlapping PPSs, where . In Fig. 2, we show three examples of the configuration of PPSs when . Note that when , the proposed ARP utilizes the assigned NPRACH in the same way as in the baseline ARP.
This partitioning of the preamble sequences may lead to degradation in the detection performance, while allowing the same preamble sequence to be reused up to times by multiple devices, thus reducing the occurrence of collisions. In essence, we establish a tradeoff relationship between the occurrence of misdetections and collisions. A decrease in the detection performance of the PPSs can be to some extent compensated by suitably increasing the PPS transmit power.
IiiB An Enhanced Access Reservation Protocol with a Partial Preamble Transmission Mechanism
The proposed ARP with the PPT mechanism mainly differs from the baseline NBIoT ARP at the first two steps as follows:

Step 1: Each device randomly selects an index of preamble sequence among preambles, and randomly selects a partial unit among the available partial units. Each device transmits a PPS on the selected partial unit.

Step 2: The eNodeB determines which PPSs are received. The eNodeB accumulates the received power spread over each of the partial units, and compares it with the predefined detection threshold, , at every partial unit^{1}^{1}1The number of detection events is increased by G times, however, the number of correlations per detection event remains the same, and, thus, this is not a severe burden for the eNodeB from the detection complexity perspective.. If a certain PPS is detected, then the eNodeB transmits the RAR, which consists of an index of preamble, an index of partial unit, a TA offset, and an uplink grant. Each device uses both the index of preamble and the index of partial unit to identify the destination of the RAR.
Iv Performance Analysis
We now mathematically characterize the detection and collision probabilities associated with the proposed ARP; and formulate an optimization problem where the objective is to maximize the ARP success probability.
Iva System Model
We focus on a single NBIoT cell, consisting of an eNodeB and IoT devices, which attempt to access the network through the ARP. Let denote the number of IoT devices which attempt the ARP in a single ARP session. Let denote the number of preambles configured in a single NPRACH. We assume that each device performs an openloop power control to compensate for the path loss. Thus, the channel between each device and the eNodeB can be modeled as a singletap flat fading channel, where the channel coefficient follows a Rayleigh distribution, i.e., . For simplicity, we assume that the channel does not vary in a block of symbol groups, i.e., a single basic unit, but varies independently over the blocks.^{2}^{2}2Note that under the block fading channel model where is any positive integer, our mathematical analysis is still applicable.
Let denote the received signal of a tagged preamble sequence, which is simultaneously utilized by devices. It can be expressed as , for , , and , where represents the th received symbol in the th symbol group at the th repetition. can be represented as:
(1) 
where , , , and represent the received power per symbol at the eNodeB, the channel coefficient between the eNodeB and the th device among devices which use the tagged preamble sequence, the th transmitted symbol in the th symbol group at the th repetition, and the Gaussian noise with zero
mean and variance of
, respectively.IvB Normalized Received power
To be able to perform the decision whether a certain PPS has been transmitted or not, the eNodeB needs to accumulate the received power of the PPS spread over the multiple symbol groups corresponding to the sequence. The normalized received power of a tagged PPS, which is transmitted simultaneously by IoT devices, is represented as:
(2) 
where represents the correlation between and given by , where denotes the complex conjugate. The term can be expressed as:
(3) 
where follows the same distribution as , and, thus, [7]. Therefore,
(4) 
where
represents a gamma distribution with shape
and rate .IvC Falsealarm and misdetection probabilities
A falsealarm occurs when a PPS that was not transmitted by any device, is detected to be active at the eNodeB. The falsealarm probability, , is expressed as:
(5)  
where
represents the cumulative distribution function (CDF) of a gamma distribution,
. Note that can be expressed as , where and represent the lower incomplete gamma function and the gamma function, respectively.A misdetection occurs when a desired PPS is not detected, and its probability, , can be expressed as:
(6)  
where represents a scaling factor which is defined as , while represents the probability that
IoT devices simultaneously utilize the same tagged PPS, which is modeled by a binomial distribution as:
(7) 
IvD Collision probability
A collision occurs when two or more IoT devices select the same PPS. When a tagged IoT device selects a PPS, the average collision probability of the tagged IoT device, , can be expressed as:
(8) 
IvE Optimal PPS configuration
The occurrence of misdetections and collisions affects the ARP performance ^{3}^{3}3The limited amount of available uplink/downlink resources can also affect the ARP performance, yet the same level of performance degradation would be observed in the baseline protocol. Therefore, in this letter, our focus is solely on the effects associated with the expansion of the contention space through the PPT mechanism.. We evaluate this performance through the ARP success probability, , which is expressed as:
(9) 
where both and are functions of , and, thus, we can formulate an optimization problem to find the optimal configuration of the PPSs as follows:
(10) 
The optimal value of can be found numerically, while assuming that , , , and an estimate of the number of contending devices is available.
V Numerical Results
We present the numerical results related to the detection performance and success probability of the proposed ARP. We have performed simulations using Matlab and the system parameters used during the evaluation are listed in Table I.


Parameters  Values 


,  5, 4 
Number of preambles ()  12 
Number of repetitions in baseline ARP ()  
Number of repetitions in proposed ARP ()  , 
Number of devices per a single ARP session ()  1 10 
Detection threshold ()  5 15 dB 
SNR ()  10, 5 dB 

We first evaluate the impact of the PPT mechanism on the detection performance metrics for a single device, i.e. .
In Fig. 3, we depict and for varying values. Generally, the decreases and the increases as the value increases when the value is given. Furthermore, a higher leads to a lower and . The former is due to noise averaging; the latter is due to a higher diversity gain. In the proposed ARP, only a fraction of a preamble sequence is transmitted by each device, therefore the eNodeB is not able to fully exploit the diversity associated with the original preamble sequence, and, thus, the detection performance degrades (i.e., a higher is observed). However, if each device transmits the PPS with a higher transmit power, then the can be decreased. In practice, the preamble detector aims to provide a constant , e.g., if we set a target , i.e., , to , then the detection thresholds should be set to dB and dB, for and , respectively.
Fig. 4 shows the performance of the proposed ARP for varying the value of PPSs when , dB, and . Decreasing the implies that the eNodeB can generate more contention resources from a single original preamble sequence while slightly sacrificing the detection performance. As the increases, the decreases, however, the increases. When the is equal to the , the performance becomes the same as that of the baseline ARP. Due to this tradeoff relationship according to how to utilize the given NPRACH resources, there exists an optimal point for maximizing the ARP success probability.
Table II shows the solutions of the optimization problem in Eq. (10) for varying and when the parameters are given as , , and . The baseline ARP can guarantee extremely low misdetection probabilities, since a sufficient number of repetitions are used to improve the detection performance. As a result, the ARP success probability is affected mostly by the collision probability. On the other hand, in the proposed ARP, we can adjust the configuration of the PPSs. Therefore, when the system load is light, then it mitigates the misdetection probability. However, when the system load becomes heavy, it mitigates the collision probability even though the detection performance degrades, and, thus, the ARP success probability can be drastically improved, compared to that of the baseline ARP. Note that if we consider more constraints on either or then the may change.


, ,  


Parameters  Conventional ARP  Proposed ARP  


1  64  0  0  100  64  0  0  100  
100  64  0  100  
2  8.3  0  91.7  16  2.08  97.9  
91.7  32  4.2  95.4  
5  29.4  0  70.6  8  4.10  94.5  
70.6  32  15.7  84.0  
10  54.3  0  45.7  8  8.99  89.7  
45.7  16  17.3  72.3  

Vi Conclusions
We proposed an enhanced access reservation protocol (ARP) with a partial preamble transmission (PPT) mechanism for NBIoT systems. The proposed ARP can mitigate the collision probability while slightly sacrificing the detection performance. We mathematically analyzed our proposed ARP in terms of the false alarm and misdetection probabilities, and collision probability. We also investigated the tradeoff relationship between the misdetection probability and the collision probability, and found an optimal resource utilization strategy according to the system loads. Through extensive simulations, we verified that the proposed ARP outperforms the conventional NBIoT ARP.
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