Analytical Modelling of NB-IoT Uplink Resource Split Between NPRACH and NPUSCH for Device Density Optimization

Engineering and Technology Journal e-ISSN: 2456-3358 Volume 10 Issue 12 December-2025, Page No.-8241-8246 DOI: 10.47191/etj/v10i12.24, I.F. – 8.482 © 2025, ETJ 8241 ETJ Volume 10 Issue 12 December 2025, Hussein N. Fadhel Analytical Modelling of NB-IoT Uplink Resource Split Between NPRACH and NPUSCH for Device Density Optimization Hussein N. Fadhel Northern Technical University, Iraq, Kirkuk 36001 ABSTRACT: Narrowband Internet of Things (NB-IoT) networks must support large numbers of sporadically active devices within a narrow uplink bandwidth. A key configuration choice is the split between random-access resources, Narrowband Physical Random Access Channel (NPRACH), and data-channel resources, Narrowband Physical Uplink Shared Channel (NPUSCH), which we capture with a single parameter, 𝑟UL, defined as the fraction of uplink resources reserved for NPRACH. We develop a stylized analytical model that couples a multichannel slotted-ALOHA description of NPRACH with a simple NPUSCH capacity model and use it to characterize how the parameter 𝑟UL affects random-access collisions, data-channel utilization, and end-to-end reliability. For representative single-cell parameters, the analysis indicates that very small values of 𝑟UL, for example, below about 0.10, drive NPRACH into an overloaded regime with per-attempt collision probabilities exceeding 99 percent, while an intermediate range between 0.25 and 0.40 yields a stable operating region. Packet-level simulations for three traffic profiles, 30-byte, 150-byte, and 450-byte packets, confirm these trends and quantify their impact, relative to baseline configurations resembling typical deployments, tuning 𝑟UL within the stable region increases the maximum supportable device density by approximately 61 to 238 percent, about 1.6 to 3.4 times, at a 99 percent packet-delivery target. These gains are achieved purely through software-level reconfiguration of the NPRACH/NPUSCH split, without changes to hardware or spectrum. KEYWORDS: Narrowband IoT, resource allocation, device density optimization, collision analysis, NPRACH, NPUSCH, massive IoT. I. INTRODUCTION The Internet of Things (IoT) is expected to connect tens of billions of devices, many of which generate small, sporadic uplink messages that must be delivered reliably and at low cost. Narrowband IoT (NB-IoT) is the 3GPP cellular technology designed to meet this demand, offering wide-area coverage, extended battery life, and low device complexity by operating in only 180 kHz of spectrum [1]-[3]. Despite rapid standardization and commercial deployment, both analytical studies and measurement driven evaluations indicate that existing NB-IoT networks can operate well below their theoretical capacity when the uplink access procedure and resource allocation are not carefully tuned [3]- [6]. A central design challenge in NB-IoT is the allocation of uplink resources between random access and data transmission. In the uplink, the Narrowband Physical Random Access Channel (NPRACH) is used for contentionbased access and initial synchronization, while the Narrowband Physical Uplink Shared Channel (NPUSCH) carries user data [1], [7]. We focus on a single dimensionless parameter, denoted by 𝑟UL, that captures this allocation, where, 𝑟UL is the fraction of uplink resources devoted to NPRACH, with the remaining fraction 1−𝑟UL available for NPUSCH. This abstraction is in line with recent work on uplink resource partitioning and NPRACH-aware allocation schemes [8], [9]. This single factor induces a fundamental trade-off. If 𝑟UL is too small, NPRACH opportunities are too sparse to accommodate the offered traffic, leading to heavy randomaccess contention, frequent preamble collisions, and access failures. If 𝑟UL is too large, excessive resources are reserved for NPRACH, starving NPUSCH and causing congestion, delays, or drops on the data channel. Detailed models of the NB-IoT random-access procedure including stochasticgeometry and Markov-chain analyses with coverageenhancement groups [10], [11], packet-level evaluations of dense public and private deployments [4], [12], and survey work on congestion management [5], [13] all point to random access as a primary bottleneck in dense NB-IoT networks. At the same time, operators often configure NPRACH and NPUSCH parameters according to spectrum constraints and qualitative rules of thumb rather than explicit end-to-end optimization, leaving room for improvement in how 𝑟UL is chosen in practice [3]-[6]. Recent work has explored data-driven and reinforcementlearning-based approaches to tune NB-IoT parameters adaptively, including random access configuration, access barring, scheduling policies, and uplink resource allocation [4]-[16]. These methods are promising for complex, time- “Analytical Modelling of NB-IoT Uplink Resource Split Between NPRACH and NPUSCH for Device Density Optimization” 8242 ETJ Volume 10 Issue 12 December 2025, Hussein N. Fadhel varying environments, and surveys of Random Access (RA) congestion-control schemes increasingly emphasize learningbased adaptations [13]. However, such approaches offer limited analytical insight into the underlying capacity limits and failure mechanisms. In particular, there is comparatively little work that couples a tractable random-access model with an explicit representation of NPUSCH utilization, so as to expose simple operating regions and failure boundaries as functions of 𝑟UL and the traffic load, and to provide closedform design rules that can guide both static configuration and adaptive control. In this paper, we develop a stylized but transparent analytical framework that links NB-IoT random access and datachannel utilization through a multichannel slotted-ALOHA model of NPRACH. We treat the random-access stage as a slotted-ALOHA system with multiple orthogonal preambles, derive the load and collision probability as functions of 𝑟UL and the arrival rate, and couple the resulting successful-access rate to a simple model of NPUSCH capacity that captures the impact of packet size and coverage-enhancement level. The framework focuses on a single NB-IoT cell with homogeneous Poisson arrivals, a fixed coverageenhancement mix, and a heuristic mapping from NPUSCH utilization to congestion probability. These assumptions enable closed-form expressions while intentionally abstracting away backoff dynamics, access class barring, and multi-cell effects. Within this setting, we study how to choose 𝑟UL so as to maximize the supported device density under a target reliability constraint. Our contributions are threefold. First, we formulate an analytical model that couples NPRACH contention and NPUSCH utilization through a single control parameter 𝑟UL, making the main NB-IoT uplink bottlenecks explicit in terms of random-access load, collision probability, and datachannel occupancy. Second, we use this model to delineate qualitative operating regions, including collapse, congestionlimited, and stable regimes, in the (𝑟UL, λ) plane, where 𝜆 denotes the aggregate arrival rate of devices (devices/s), and to identify a practical range of 𝑟UL values that keeps the system within the stable region under the assumed traffic model. Third, we validate the analytical predictions with packet-level simulations and quantify the achievable devicedensity improvements relative to typical baseline configurations, observing gains between approximately 61% and 238% (1.6–3.4 times more devices per km2) across three representative traffic profiles. We translate these results into simple design guidelines that allow network operators to treat 𝑟UL as an explicit tuning factor and to complement more detailed optimization and learning-based control schemes [14]-[16]. The rest of this paper is organized as follows. Section II presents the system model and mathematical analysis framework, deriving the impact of 𝑟UL on random access and NPUSCH utilization. Section III validates the analysis with packet-level simulations and discusses the resulting device density gains. Section IV concludes the paper and outlines directions for future work, including extensions to multi-cell scenarios and adaptive tuning of 𝑟UL and related NB-IoT configuration parameters. II. SYSTEM MODEL AND ANALYTICAL FRAMEWORK A. System Model, Assumptions, and Metrics We consider a single NB-IoT cell in which devices generate uplink reports according to a homogeneous Poisson process with rate 𝜆 (devices/s). The uplink follows the tandem structure of NPRACH for initial access and NPUSC for data transmission. The resource-splitting parameter 𝑟UL governs how much of the uplink resource budget is devoted to NPRACH versus NPUSCH. We define: 𝑟𝑈𝐿 =𝑁𝑃𝑅𝐴𝐶𝐻 𝑟𝑒𝑠𝑜𝑢𝑟𝑐𝑒𝑠 (𝑁𝑃𝑅𝐴𝐶𝐻 𝑟𝑒𝑠𝑜𝑢𝑟𝑐𝑒𝑠 + 𝑁𝑃𝑈𝑆𝐶𝐻 𝑟𝑒𝑠𝑜𝑢𝑟𝑐𝑒𝑠),(1) so that 𝑟𝑈𝐿 is the fraction of uplink resources reserved for NPRACH, and the remaining fraction 1−𝑟UL is available for NPUSCH. Increasing 𝑟UL makes random access more frequent but reduces the capacity of NPUSCH. Decreasing 𝑟UL frees resources for NPUSCH but increases contention on NPRACH. To keep the analysis tractable and aligned with our goal of obtaining closed-form expressions, we adopt a stylized model with the following simplifying assumptions, (i) a single cell with no inter-cell interference and homogeneous Poisson arrivals, (ii) a multichannel slotted-ALOHA description of NPRACH with 𝑚= 48 orthogonal preambles and independent preamble choices, without modelling detailed backoff dynamics or access-class barring, (iii) an NPUSCH service model based on a per-second Resource-Unit (RU) budget and an aggregate utilization metric, rather than a detailed scheduler, and (iv) a fixed distribution over Coverage-Enhancement (CE) levels and a heuristic mapping from NPUSCH utilization to congestion probability. These assumptions intentionally abstract away fine-grained protocol features in exchange for analytical transparency. Within this framework, our primary performance metric is the maximum device density (devices/km2) that can be supported while meeting a target end-to-end success probability 𝑃succ ≥ 𝑃tar (with 𝑃tar = 0.99 in our examples). At the analytical level, we track three quantities: (i) the random access collision probability 𝑃coll on NPRACH, (ii) the congestion probability 𝑃cong on NPUSCH, and (iii) the combined per-attempt failure probability 𝑃failure resulting from both contention and congestion. B. Random-Access Analysis via Multichannel Slotted ALOHA In the NB-IoT configuration considered, each Random Access Opportunity (RAO) offers 𝑚 = 48 orthogonal “Analytical Modelling of NB-IoT Uplink Resource Split Between NPRACH and NPUSCH for Device Density Optimization” 8243 ETJ Volume 10 Issue 12 December 2025, Hussein N. Fadhel preambles. The nominal RAO period for the baseline configuration is 𝛿RAO. Allocating a fraction 𝑟UL of the uplink resource budget to NPRACH effectively scales the RAO density, yielding 𝑇RAO =𝛿RAO 𝑟UL . (2) Smaller values of 𝑟UL correspond to sparser RAOs and thus higher contention. Let 𝜆 be the aggregate arrival rate of new reports (devices/s). Under the slotted-ALOHA model with independent choices of preamble and RAO, the expected load per preamble in each RAO is 𝐺 = 𝜆𝛿RAO 𝑟UL , (3) which is dimensionless and can be interpreted as the average number of contending devices per preamble per RAO. Under the standard independence assumption, the probability that a given preamble experiences no collision, i.e., exactly one device selects it, is 𝐺𝑒−𝐺, and the collision probability experienced by an individual device is 𝑃coll = 1− 𝑒−𝐺, 𝑆(𝐺)= 𝐺𝑒−𝐺, (4) where 𝑆(𝐺) is the normalized slotted-ALOHA throughput. As 𝐺 increases beyond 1, the throughput saturates and then decreases, while 𝑃coll approaches 1, indicating an overloaded random-access channel. For a representative configuration with λ= 50 devices/s, 𝑚 = 48 preambles, and 𝛿RAO = 0.64 s, low values of 𝑟UL correspond to large 𝐺 and near-certain collisions. For example, at 𝑟UL = 0.05 we obtain 𝐺 ≈13.33 and 𝑃coll > 0.999, effectively making random access unusable. At 𝑟UL= 0.30, 𝐺 ≈2.22 and 𝑃coll ≈0.89, which is still high at the per attempt level but lies near the high-efficiency region of the slotted-ALOHA curve. TABLE I RANDOM-ACCESS OPERATING POINT VERSUS 𝒓UL (REPRESENTATIVE PARAMETERS) 𝒓UL Load 𝑮 𝑷coll 𝑺(𝑮) Zone (Interpretive) 0.05 13.33 >99.9% 0.000 Collapse 0.10 6.67 99.9% 0.008 Collapse / Critical 0.15 4.44 98.8% 0.053 High-load 0.20 3.33 96.4% 0.119 Marginal 0.25 2.67 93.1% 0.184 Stable (Conservative) 0.30 2.22 89.1% 0.245 Stable 0.35 1.90 85.0% 0.286 Stable / Recommended 0.40 1.67 81.2% 0.313 Stable / Recommended 0.45 1.48 77.3% 0.329 Stable 0.50 1.33 73.6% 0.353 High-efficiency Table I summarizes the random-access operating point for a range of 𝑟UL values under this parameter set. The “Collapse” zone corresponds to very high collision probability and negligible throughput. The “Stable” zone combines high but manageable collision probabilities with the high-throughput region of the slotted-ALOHA curve. These qualitative zones will later be compared against simulation results in Section III. The table corresponds to the parameters 𝜆 = 50 devices/s, 𝛿RAO = 0.64 s, 𝑚= 48. 𝑆(𝐺) = 𝐺𝑒−𝐺 is the normalized slotted-ALOHA throughput. Zone labels are qualitative and specific to this parameter set. They are used to interpret operating regions rather than to claim universal thresholds. C. Data Transmission Phase and NPUSCH Utilization Devices that successfully complete NPRACH proceed to NPUSCH for data transmission. Under the slotted-ALOHA model, the aggregate rate of successful random-access attempts is 𝜆𝑠 = 𝑚 𝐺𝑒−𝐺 𝑇𝑅𝐴𝑂 = 𝑚 𝑆(𝐺) 𝑇𝑅𝐴𝑂 [devices/s], (5) where 𝐺 and 𝑇𝑅𝐴𝑂 are given by (2)–(3). This is the arrival rate into the NPUSCH service system. Each device carries a packet of size 𝐿 bytes and belongs to a coverage-enhancement level CE ∈ {0,1,2}. In line with NBIoT transport block sizing, we approximate the number of resource units (RUs) required for a packet as 𝑁RU = ⌈𝐿 32⌉𝑅CE [RUs/packet], (6) where 𝑅CE ∈ {1,8,64} for CE ∈ {0,1,2}, respectively. Let P(CE = 𝑖) denote the fraction of devices in each CE level. In our numerical examples we use a fixed distribution representative of urban deployments, leading to 𝐸[𝑁𝑅𝑈] = ⌈𝐿 32⌉ ∑P(CE = 𝑖)𝑅CE 𝑖 =𝑖. (7) The total NPUSCH RU budget per second is denoted by 𝑁total. Allocating a fraction 𝑟UL of the uplink to NPRACH leaves 𝐶NPUSCH = 𝑁total(1−𝑟UL) [RUs /s] (8) for data transmission. The resulting NPUSCH utilization is 𝜌NPUSCH = 𝜆𝑠𝐸[𝑁RU] 𝐶NPUSCH . (9) This quantity captures how close NPUSCH is to saturation as a function of 𝑟UL, the arrival rate, and the traffic mix. D. End-to-End Failure Probability and Optimization The end-to-end per-attempt failure probability combines contention failures on NPRACH and congestion failures on NPUSCH. We approximate it as 𝑃failure = 𝑃coll + (1−𝑃coll)𝑃cong, (10) “Analytical Modelling of NB-IoT Uplink Resource Split Between NPRACH and NPUSCH for Device Density Optimization” 8244 ETJ Volume 10 Issue 12 December 2025, Hussein N. Fadhel where 𝑃coll is given by (4) and 𝑃cong is the probability that a device that has gained access to NPUSCH fails to be served within the latency or buffer constraints of the application. In principle, 𝑃cong could be derived from a detailed queueing model of the NPUSCH scheduler. To keep the framework simple and analytically transparent, we instead adopt a heuristic mapping from utilization to congestion: 𝑃cong =max {0,𝜌NPUSCH −0.8 0.2 }, (11) so that 𝑃cong= 0 for 𝜌NPUSCH ≤0.8 and increases linearly to 1 as 𝜌NPUSCH approaches 1. The threshold value 0.8 is chosen to reflect the utilization range where queueing delays begin to grow rapidly in M/D/1-type systems. The linear mapping itself is a modeling choice that is later calibrated against simulation in Section III. Equations (4)–(11) together define 𝑃failure as a function of 𝑟UL, the arrival rate, and the traffic mix. In practice, NB-IoT devices are allowed multiple random-access attempts. Our simulator implements these attempts and their timers explicitly, while the analytical model uses 𝑃failure as a firstorder approximation of the per-attempt failure probability. The reliability constraint 𝑃succ = 1 − 𝑃failure ≥𝑃tar, combined with a traffic model and a deployment geometry, yields a maximum sustainable device density for each value of 𝑟UL. Rather than claiming universal boundaries, we interpret the results of this analytical model as a way to delineate qualitative operating regions in the (𝑟UL, λ) plane and to identify practical ranges of 𝑟UL that keep the system out of collapse and congestion-dominated regimes for the scenarios of interest. These regions are compared against packet-level simulation results in Section III, which both validate the analytical trends and quantify how they translate into concrete device-density gains. III. VALIDATION AND RESULTS A. Simulation Setup We validate the analytical framework of Section II through a discrete-event NB-IoT uplink simulator that implements the NPRACH and NPUSCH procedures, the coverage enhancement levels, and the retransmission rules defined in the 3GPP specifications [1], [4], [7]. New reports are generated according to a homogeneous Poisson process with rate λ and immediately trigger a random-access attempt on NPRACH. Collisions are resolved exactly as in the analytical model. In each RAO, devices choose among the 𝑚= 48 orthogonal preambles uniformly at random; an NPRACH attempt succeeds if and only if it is the unique use of its preamble in that RAO. Upon successful random access, devices enter the NPUSCH scheduler, which allocates RUs according to the coverage-enhancement level and packet size. The simulator tracks both contention failures on NPRACH and congestion-induced failures on NPUSCH, so that the measured metrics are directly comparable with those defined in the mathematical analysis. Unless stated otherwise, we consider a single-cell deployment with 𝜆 = 50 devices/s, 𝛿RAO = 0.64 s, and an RU budget consistent with a single-carrier NB-IoT configuration. The coverage-enhancement distribution is fixed to the representative mix used in the analytical model, yielding the same mean RU requirement 𝐸[𝑁RU]. Each configuration is simulated for 400 s, and multiple independent runs are averaged so that 95% confidence intervals on 𝑃succ are within a few percentage points and are omitted from the plots for clarity. A packet is declared successful if it is delivered within at most 𝐴max random-access attempts and without exceeding an NPUSCH delay budget of 10 s, otherwise it is counted as a failure. The resulting overall success probability 𝑃succ is compared against the reliability target 𝑃tar (here, 𝑃tar = 0.99) when computing the maximum sustainable device density. B. Optimization Results Table II summarizes the network capacity obtained from simulation for three packet sizes (30 B, 150 B, and 450 B) and several values of 𝑟UL. For each packet size we select a baseline configuration that is representative of typical deployments (for example, 𝑟UL= 0.15 for 30 B and 150 B traffic and 𝑟UL = 0.20 for 450 B), and we report the corresponding device density (devices/km2) at which the endto-end success probability 𝑃succ remains above 0.99. For all configurations, the table reports the device density and the per-attempt collision rate measured from simulation, together with the relative improvement with respect to the baseline configuration. TABLE II: COMPREHENSIVEDEVICEDENSITYOPTIMIZATIONRESULTS Packet Size 𝒓UL Device Density (devices/k m2) Improveme nt vs. baseline Collision Rate 30 B 0.15 485 baseline 99% 0.25 834 +72% 93% 0.35 1,120 +131% 82% 0.40 1,316 +171% 80% 150 B 0.15 331 baseline 99% 0.20 634 +91% 96% 0.30 960 +190% 89% 0.35 1,120 +238% 82% 450 B 0.20 254 baseline 96% 0.30 409 +61% 89% 0.40 485 +91% 80% 0.45 520 +105% 78% The table reveals three consistent patterns. First, moving from the baseline configuration to the optimized 𝑟UL often approximately doubles or more than doubles the sustainable “Analytical Modelling of NB-IoT Uplink Resource Split Between NPRACH and NPUSCH for Device Density Optimization” 8245 ETJ Volume 10 Issue 12 December 2025, Hussein N. Fadhel device density. The improvements range from +61% (large packets) to +238% (mixed traffic) while still satisfying the same reliability target. Second, although the optimized points operate with very high random-access collision rates, between 78% and 93%, the overall success probability remains above 0.99 because devices are allowed multiple access attempts and NPUSCH remains uncongested. Third, the optimal 𝑟UL shifts gradually with packet size, smaller packets favor larger values of𝑟UL, whereas large packets benefit from leaving more resources for NPUSCH. These shifts are consistent with the analytical trade-off between collision probability and NPUSCH utilization derived in Section II. C. Collision Zones and Capacity Optimization To quantify how the analytical collision regions translate into practice, Figure 1 plots the per-attempt collision probability as a function of 𝑟UL for the representative parameter set. The solid curve shows the multichannel slotted-ALOHA prediction from (4), and the markers overlay simulation results. Figure 1. Random-access collision probability as a function of 𝒓UL for the representative configuration (𝝀 = 𝟓𝟎 devices/s, 𝜹RAO = 0.64 s, 𝒎= 𝟒𝟖). The curve shows the multichannel slotted-ALOHA prediction (4), and markers show simulation results. Shaded regions illustrate collapse (red), marginal (yellow), and stable (green) operating zones. The background shading highlights three operating zones: a red “collapse” zone for 𝑟UL <0.10, where 𝑃coll >99% and throughput is negligible. A yellow marginal zone for 0.10≤ 𝑟UL <0.25 where collisions remain above 93%, and a green stable zone for 0.25≤𝑟UL ≤0.40 where collisions are still high but random access remains feasible. Simulation points track the analytical curve closely, typically within a few percentage points, supporting the collision component of the mathematical model. Figure 2 complements the table by plotting the sustainable device density as a function of 𝑟UL for each packet size. For each configuration, the device density corresponds to the maximum value at which the simulated end-to-end success probability satisfies 𝑃succ ≥0.99. The shaded vertical band marks the practical optimization zone 0.25≤𝑟UL ≤0.40, where all three traffic profiles achieve near-maximal capacity while remaining in the stable collision region. The annotations highlight, for example, the +171% gain for 30 B packets at 𝑟UL = 0.40 and the +238% gain for 150 B packets at 𝑟UL = 0.35, relative to their respective baselines in Table II. Figure 2. Network capacity (devices/km2) versus 𝒓UL for three packet sizes. Each point gives the maximum sustainable device density for which 𝑷succ ≥ 0.99in simulation. The shaded region indicates the practical optimization zone where all traffic profiles achieve nearmaximal capacity while remaining in the stable collision region. Across the configurations in Table II and Figure 2, the practical range 0.25≤𝑟UL ≤0.40 consistently delivers large gains while keeping the system within the stable collision region and below the NPUSCH congestion threshold. The exact optimum depends on the traffic mix and packet size, but the overall pattern is robust: values of 𝑟UL around 0.30–0.35 strike a good compromise between collision probability and data-channel utilization. The simulations therefore corroborate the analytical trade-offs derived in Section II and indicate that many NB-IoT deployments can significantly increase device density through software-level reconfiguration of the NPRACH/NPUSCH resource split, without changing hardware or spectrum. CONCLUSION This paper presented an analytical framework and supporting simulation study for understanding how the NB-IoT uplink resource split between NPRACH and NPUSCH, captured by the parameter 𝑟UL, affects random-access collisions, datachannel utilization, and end-to-end reliability. By modeling NPRACH as a multichannel slotted-ALOHA system and NPUSCH as a resource-unit budget driven by the successful random-access rate, we derived closed-form expressions for the collision probability, NPUSCH utilization, and a firstorder approximation of the overall failure probability as functions of 𝑟UL and the offered load. Under a single-cell setting with homogeneous Poisson arrivals, a fixed coverage-enhancement mix, and a heuristic mapping from utilization to congestion probability, the analysis delineates qualitative operating regions in the (𝑟UL, λ) plane. Low values of 𝑟UL lead to random-access overload, “Analytical Modelling of NB-IoT Uplink Resource Split Between NPRACH and NPUSCH for Device Density Optimization” 8246 ETJ Volume 10 Issue 12 December 2025, Hussein N. Fadhel high values of 𝑟UL limit NPUSCH, and an intermediate range around 0.25 ≤ 𝑟UL ≤ 0.40 yields a stable region where random access remains feasible and NPUSCH utilization stays below the congestion threshold for the traffic profiles we consider. Packet-level simulations confirm these trends and quantify their impact: relative to baseline configurations resembling typical deployments, tuning 𝑟UL within this range supports approximately 61% to 238% more devices per km2 (about 1.6–3.4 times) while maintaining 𝑃succ ≥0.99, despite high per attempt collision probabilities. 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