Martingale-Based Resource Partitioning Bounds for URLLC and eMBB Network Slicing in 5G

Martingale-Based Resource Partitioning Bounds for URLLC and eMBB Network Slicing in 5G 1st Hanbing Li School of Computer Science and Artificial Intelligence Shandong Normal University Ji’nan, 250358, China 2026317277@stu.sdnu.edu.cn

arXiv:2610.36602v1 [cs.NI] 27 Oct 2026

2nd Yantong Wang∗

School of Computer Science and Artificial Intelligence Shandong Engineering Research Center for Multimodal Computing and Intelligent Decision Making Shandong Normal University Ji’nan, 250358, China yantong@sdnu.edu.cn

Abstract—Network slicing is a key enabler for the coexistence of ultra-reliable low-latency communications (URLLC) and enhanced mobile broadband (eMBB) in 5G networks. However, allocating limited radio resources to heterogeneous slices while satisfying stringent quality-of-service (QoS) requirements remains challenging, particularly in the ultra-reliable regime where conventional effective-capacity-based approaches may lead to excessive resource allocation. To address this issue, this paper proposes a martingale-based resource block (RB) allocation framework for coexisting URLLC and eMBB slices. By constructing exponential martingale processes using the moment generating functions (MGFs) of the arrival and service processes and applying Doob’s maximal inequality, explicit non-asymptotic upper bounds are derived for the URLLC delay violation probability and the eMBB backlog violation probability. Based on the resulting theoretical stability conditions, a low-complexity bisection algorithm is developed to determine the optimal martingale parameters, followed by a deterministic search algorithm to obtain the minimum RB requirements for both slices. Numerical results show that the analytical bounds effectively upper-bound the Monte Carlo simulation results. At a target violation probability of 10−6 for each slice, the proposed framework requires 44 RBs in total, satisfying the system capacity constraint of 50 RBs. Compared with the conventional effective-capacity baseline requiring 64 RBs, the proposed framework reduces total RB consumption by approximately 31.2% relative to the conventional effectivecapacity baseline. Index Terms—Network Slicing, URLLC, eMBB, Martingale Theory, Resource Partitioning, Statistical QoS Bounds

I. Introduction Network slicing is an important technology in fifthgeneration (5G) and beyond wireless networks for supporting diverse vertical industry applications, enabling isolated bearing and resource sharing for heterogeneous services by constructing logically independent network slices over a shared physical infrastructure [1]. In practical deployments, ultra-reliable low-latency communications This work is sponsored by Shandong Provincial Natural Science Foundation, China under Grant ZR2023QF084.

(URLLC) and enhanced mobile broadband (eMBB) represent two typical service scenarios, and ensuring their efficient coexistence is of great significance for satisfying the quality-of-service (QoS) requirements of different services [2]. However, these two services impose significantly different QoS requirements and compete for limited radio resources: URLLC traffic is highly sensitive to latency and reliability, demanding strict low-latency and highreliability constraints, whereas eMBB services primarily target high throughput while maintaining stable service performance in dynamic traffic environments [3]. Under limited radio spectrum resources, the dual stochasticity of traffic arrivals and wireless services further increases the complexity of queuing QoS characterization and resource provisioning. How to accurately characterize the queuing dynamics coupled by such dual stochasticity, and accordingly achieve efficient resource block (RB) allocation for coexisting URLLC and eMBB slices, remains a vital research problem in 5G radio resource management [4]. Existing studies employ analytical methods, such as effective capacity (EC) and stochastic network calculus (SNC), as well as Monte Carlo simulations to analyze statistical QoS under stochastic traffic. EC characterizes the relationship between service statistics and queueing QoS [5], while conventional EC analysis is closely related to asymptotic queueing results based on large deviations theory [6]. Although EC provides a tractable approach for statistical QoS analysis, its asymptotic formulation does not directly focus on finite-time QoS guarantees. MGFbased SNC approaches provide analytical QoS bounds, but may rely on union bounds to combine multiple violation events, which can lead to loose bounds [7]. Such conservatism may further affect resource allocation efficiency when these bounds are used to determine resource requirements. Meanwhile, standard Monte Carlo simulations provide empirical estimates of QoS performance, but require a large number of samples to reliably estimate

rare-event probabilities, making them challenging to apply to events on the order of 10−6 [8]. Therefore, existing approaches face challenges in simultaneously providing finite-time QoS guarantees and efficient resource allocation under stringent reliability requirements. These limitations motivate the development of finite-time QoS probability bounds that can directly determine the minimum RB allocation satisfying stringent QoS constraints. To address the aforementioned issues, this paper proposes a martingale-based resource block (RB) allocation framework for the coexistence of URLLC and eMBB slices. The main contributions are summarized as follows. First, based on the moment generating functions (MGFs) of the arrival and service processes, we construct exponential martingales and supermartingales to characterize queueing behavior and derive explicit non-asymptotic upper bounds on URLLC delay violation probabilities and eMBB backlog violation probabilities using Doob’s maximal inequality [9]. Second, we formulate the RB allocation problem in terms of the resulting QoS feasibility conditions and develop a low-complexity bisection algorithm to determine the martingale parameters, followed by a linear search algorithm to identify the minimum number of RBs required to satisfy the QoS constraints of both slices. Finally, numerical results demonstrate the effectiveness of the proposed framework. When the target violation probabilities of both slices are set to 10−6 , the proposed scheme requires 44 RBs, satisfying the system capacity constraint of 50 RBs. Compared with a baseline scheme incorporating a finite-blocklength penalty factor of β = 0.82, which requires 64 RBs, the proposed scheme reduces total RB consumption by approximately 31.2%. II. System Model A. Network Scenario

Nu

gNB

B. Traffic Model Due to the heterogeneous traffic characteristics in 5G networks, URLLC traffic is modeled as discrete packet arrivals, whereas eMBB traffic is represented by its data volume in bits [10]. For consistent queueing analysis, the URLLC packet arrivals are converted into bits using the packet size, so that the aggregate arrival processes of both slices are expressed in a common unit [11]. We assume the number of packets sending to URLLC user k ∈ Uu follows a two-state ON-OFF process. In the ON state, packets arrive at a constant rate λu (packets/s); in the OFF state, no packet arrives. In time slot t, the number of arriving packets of user k is { λu Ts , state ON; au,k (t) = (1) 0, state OFF, with steady-state ON probabilities πon and OFF 1 − πon , respectively. For tractability of the martingale analysis, we assume that the ON/OFF states are independent across slots. Each URLLC packet consists of Lu bits, and the aggregate URLLC arrival in slot t (in bits) is Au (t) = Lu

Ku ∑

au,k (t).

URLLC

eMBB

Fig. 1: Network Scenario. As shown in figure 1, we consider the downlink of a single-cell orthogonal frequency division multiple access (OFDMA) system, in which a 5G base station (gNB) serves Ku URLLC users and Ke eMBB users over a total bandwidth of B Hz. The bandwidth is divided into N resource blocks (RBs), each of bandwidth ∆f = B/N Hz. Network slicing partitions the RBs into two slices, in which the URLLC slice is allocated Nu RBs with bandwidth

(2)

k=1

The moment generating function (MGF) of the per-user per-slot arrival (in bits) becomes [ ] E eθau,k (t)Lu = πon eθλu Ts Lu + (1 − πon ). (3) For Ku i.i.d. users, the aggregate MGF is [ ]Ku φA,u (θ) = πon eθλu Ts Lu + (1 − πon ) .

Ne RBs

Bu = Nu ∆f , while the eMBB slice is scheduled Ne RBs with bandwidth Be = Ne ∆f , and there is Nu + Ne ≤ N . A time-slotted structure is adopted with duration Ts for slice-level scheduling. Moreover, we assume block fading, i.e., the channel gain is constant within one slow slot and varies independently across slots.

(4)

Each eMBB user k ∈ Ue receives traffic at a mean rate λe (bits/s). The amount of traffic for user k in slow slot t follows a Poisson distribution with mean λe Ts , whose MGF of the per-user per-slot arrival is [ ] ( ( )) E eθae,k (t) = exp λe Ts eθ − 1 . (5) For Ke i.i.d. users, the aggregate MGF is ( ( )) φA,e (θ) = exp Ke λe Ts eθ − 1 .

(6)

C. Service Model According to the Shannon formula, the achievable service rate per RB for slice i ∈ {u, e} in slot t is ci (t) = ∆f Ts log2 (1 + γi (t)) ,

(7)

where γi (t) denotes the instantaneous signal-to-noise ratio (SNR) of the served user in slot t. Under block fading,

we model γi (t) = ρi gi (t), where gi (t) is the normalized channel power gain with E[gi (t)] = 1 considering the exponential distribution for Rayleigh fading and ρi is the average SNR. Given Ni RBs allocated to slice i, the per-slot service rate of the slice is Si (t) = Ni ci (t) = Ni ∆f Ts log2 (1 + ρi gi (t)) .

(8)

Lemma 1. Under Rayleigh fading with normalized channel gain i.i.d. gi (t) ∼ Exp(1), the service MGF becomes i θi φS,i (θi , Ni ) = e1/ρi ρ−α Γ(1 − αi θi , 1/ρi ) , i

(9)

where the∫upper incomplete gamma function is defined as ∞ Γ(a, x) = x ta−1 e−t dt and αi = Ni ∆f Ts / ln 2. Proof. By eq.(8), hence e−θSi (t) = (1 + ρi gi (t))−αi θ with αi = Ni ∆f Ts / ln 2. Since gi (t) is i.i.d., we omit the slot index t hereafter. Then [ ] [ ] φS,i (θi , Ni ) = E e−θi Si (t) = E (1 + ρi gi )−αi θi ∫ ∞ = (1 + ρi g)−αi θi e−g dg (10) 0 ∫ e1/ρi ∞ −αi θi −u/ρi = u e du (11) ρi 1 ∫ ∞ i θi = e1/ρi ρ−α v (1−αi θi )−1 e−v dv. (12) i 1/ρi

Eq.(10) is the definition of expectation with fgi (g) = e−g ; eq.(11) holds considering u = 1 + ρi g; and eq.(12) applies the substitution v = u/ρi . To align with the form Γ(a, x), let a = 1 − αi θ, x = 1/ρi yield eq.(9). Notice 1/ρi > 0 holds. D. QoS Requirements As for URLLC, the delay violation probability should satisfy P (Wu ≥ du ) ≤ εu ,

(13)

where Wu is the queueing delay of URLLC traffic, du is the delay budget in slots, and εu is the target violation probability. Unlike URLLC, whose QoS is latency-critical and is thus captured by the delay violation probability in eq.(13), eMBB traffic is throughput-centric and can tolerate queueing delays; we therefore characterize the eMBB QoS requirement by the backlog violation probability as follows P (Be ≥ Qe ) ≤ εe ,

(14)

where Be is the queue backlog of the eMBB slice, Qe is the backlog threshold, and εe is the relevant violation probability.

III. Martingale-Based QoS Bounds A. Martingale Construction For each network slice i ∈ {u, e}, we assume the arrival process Ai (t) and the service process Si (t) are mutually independent within each time slot t. Furthermore, the joint random vectors {(Ai (t), Si (t))}t≥1 are independent and identically distributed (i.i.d.) across time slots, and the initial queue backlog Bi (0) is independent of future arrival and service processes. All required MGF are assumed to be finite. Lemma 2. Let Θi = {θi > 0 : φA,i (θi )φS,i (θi , Ni ) ≤ 1} denote the martingale domain of slice i. Then Θi is nonempty if and only if E[Ai (n)] < E[Si (n)]. Proof. Let f (θi , Ni ) = ln φA,i (θi ) + ln φS,i (θi , Ni ), which is strictly convex and satisfies f (0, Ni ) = 0 and f ′ (0, Ni ) = E[Ai (t)] − E[Si (t)]. If E[Ai (t)] < E[Si (t)], then f ′ (0, Ni ) < 0, so f (ϵ, Ni ) < 0 for sufficiently small ϵ > 0, implying ϵ ∈ Θi . Conversely, if f ′ (0, Ni ) ≥ 0, strict convexity implies f (θi , Ni ) > f ′ (0, Ni )θi ≥ 0 for all θi > 0, so φA,i (θi )φS,i (θi , Ni ) > 1 and Θi = ∅. ■ To analyze the queue backlog, consider the backlog Bi (t) of slice i at time slot t, whose evolution follows the onestep Lindley recursion: Bi (t) = max (0, Bi (t − 1) + Ai (t) − Si (t)) .

(15)

By recursively expanding this relation backward to the initial time slot 0, and assuming the system starts with an empty queue (i.e., Bi (0) = 0), we obtain ( ) t ∑ Bi (t) = max 0, max (Ai (m) − Si (m)) . (16) 1≤k≤t

m=t−k+1

For a given time slot t ≥ 1, define the time-reversed cumulative net arrival process for 1 ≤ k ≤ t as Wi (k) = ∑ t m=t−k+1 (Ai (m) − Si (m)) (with Wi (0) = 0). Since we assume an empty initial queue Bi (0) = 0, the queue backlog can be compactly expressed as: Bi (t) = max Wi (k). 0≤k≤t

(17)

Furthermore, since the arrival process Ai (t) and the service process Si (t) are mutually independent within each time slot, the MGF of the single-slot net arrival Ai (t) − Si (t) can be factorized as [ ] [ ] [ ] E eθi (Ai (t)−Si (t)) = E eθi Ai (t) E e−θi Si (t) = φA,i (θi )φS,i (θi , Ni ). Accordingly, we let ci = φA,i (θi )φS,i (θi , Ni ) > 0. Lemma 3. For a given time slot t ≥ 1 and any θi ∈ Θi , the process Xi (k) = eθi [Wi (k)] c−k i ,

0≤k≤t

(18)

is a non-negative supermartingale with respect to the backward filtration F̃k = σ(Ai (t), Si (t), . . . , Ai (t − k + 1), Si (t − k + 1)) (where F̃0 = {∅, Ω}).

Proof. By definition, for 1 ≤ k ≤ t, Wi (k) = Wi (k − 1) + Ai (t − k + 1) − Si (t − k + 1). Since Wi (k − 1) is F̃k−1 measurable, and (Ai (t−k +1), Si (t−k +1)) is independent of F̃k−1 due to the independent and identically distributed (i.i.d.) nature across time slots, we have θi [Wi (k−1)] E[Xi (k) | F̃k−1 ] = c−k i e [ ] [ ] · E eθi Ai (t−k+1) E e−θi Si (t−k+1) θi [Wi (k−1)] = c−k ci = Xi (k − 1). i e

Since Xi (k) > 0 and its expectation is finite (i.e., integrability holds with E[Xi (k)] = E[Xi (0)] = 1 < ∞), and it satisfies the martingale equality, it is a non-negative martingale and hence a non-negative supermartingale with initial state Xi (0) = 1. ■ B. Backlog and Delay Violation Bounds Theorem 1. For slice i ∈ {u, e} and any θi ∈ Θi , the queue backlog violation probability bound is given by P (Bi (t) ≥ Qi ) ≤ e−θi Qi ,

∀Qi ≥ 0.

(19)

Proof. From (17) we have Bi (t) = max0≤k≤t Wi (k). Since ci ≤ 1 for any θi ∈ Θi (which implies c−k ≥ 1 for i all 0 ≤ k ≤ t), we obtain the following event inclusions: { } {Bi (t) ≥ Qi } = max Wi (k) ≥ Qi 0≤k≤t { } = max eθi [Wi (k)] ≥ eθi Qi 0≤k≤t { } ⊆ max Xi (k) ≥ eθi Qi . 0≤k≤t

Applying Doob’s maximal inequality to the non-negative supermartingale Xi (k) yields ( ) P (Bi (t) ≥ Qi ) ≤ P max Xi (k) ≥ eθi Qi 0≤k≤t

≤ E[Xi (0)]e−θi Qi = e−θi Qi , which establishes the theorem. ■ Theorem 2. For any θu ∈ Θu , the delay violation probability bound of the URLLC slice is given by P (Wu ≥ du ) ≤ [φS,u (θu , Nu )]du .

(20)

Proof. Under First-In-First-Out (FIFO) scheduling, a packet arriving at slot t waits for at least du slots if and only if the current backlog Bu (t) is no less than the cumulative service over the subsequent du slots, i.e., ∑t+d u Su (m). {Wu ≥ du } = {Bu (t) ≥ Z}, where Z = m=t+1 Since Bu (t) depends on the history up to slot t while Z depends on future slots, Bu (t) is independent of Z.

Conditioning on Z = z with probability density function fZ (z) and applying Theorem 1, we obtain ∫ ∞ P (Wu ≥ du ) = P (Bu (t) ≥ Z | Z = z)fZ (z) dz ∫0 ∞ ≤ e−θu z fZ (z) dz 0 ] [ ∑t+du = E[e−θu Z ] = E e−θu m=t+1 Su (m) . Since the service process {Su (m)} is i.i.d. across time slots, the MGF of the sum factorizes into the product of individual service MGFs, yielding E[e−θu Z ] = [φS,u (θu , Nu )]du . ■ C. Bandwidth Allocation Combining the violation probability bounds in Theorems 1 and 2 with the slice QoS targets in eq.(13) and eq.(14), we obtain the following resource allocation result. Theorem 3. Let Θi (Ni ) denote the martingale domain for a given allocation Ni . The minimum numbers of RBs that guarantee the URLLC and eMBB QoS are } { [ ] 1 ∗ , −du ln φS,u (θu , Nu ) ≥ ln Nu = min Nu : max ϵu θu ∈Θu (Nu ) (21) { } 1 Ne∗ = min Ne : max θe Qe ≥ ln , (22) ϵe θe ∈Θe (Ne ) and the corresponding minimum bandwidth demands are Bu∗ = Nu∗ ∆f and Be∗ = Ne∗ ∆f . Proof. From eq.(20), the URLLC delay requirement eq.(13) is satisfied whenever there exists θu ∈ Θu (Nu ) such that [φS,u (θu , Nu )]du ≤ ϵu . Taking the natural logarithm on both sides and rearranging the terms yields −du ln φS,u (θu , Nu ) ≥ ln(1/ϵu ), which leads to eq.(21).The eMBB allocation in eq.(22) follows analogously from eq.(19) and eq.(14). ■ IV. Resource Partitioning Algorithm Theorem 3 decouples the statistical QoS requirements of the two slices into independent minimum bandwidth demands Nu∗ and Ne∗ . Since both objective functions inside the maxima in eq.(21) and eq.(22) are strictly increasing with respect to θi > 0, their maxima are inevitably attained at the boundary of the martingale domain, i.e., θi∗ (Ni ) = max{θi > 0 : φA,i (θi )φS,i (θi , Ni ) ≤ 1}. The resource partitioning problem then evaluates whether the available N RBs can accommodate both slices based on θi∗ (Ni ). If no feasible (Nu , Ne ) exists, the system is QoSinfeasible, requiring additional bandwidth or relaxed QoS targets. As proved in Lemma 2, f (θi , Ni ) is strict convex in θi with f (0, Ni ) = 0 and f ′ (0, Ni ) < 0 in the stable regime; it first decreases below 0 and then rises back to 0 at θi∗ (Ni ) by verifying limθi →∞ f (θi , Ni ) = ∞, making bisection applicable. The bisection procedure is summarized in Algorithm 1. Since increasing Ni reduces the service MGF and hence enlarges Θi (Ni ), the sets in eq.(21)–eq.(22)are monotone

Algorithm 1: Bisection for θi∗ (Ni ) Require: Allocation Ni , tolerance δ, start-point θ0 Ensure: Optimal parameter θi∗ (Ni ) given Ni 1: θlow ← 0, θhigh ← θ0 2: while f (θhigh ; Ni ) ≤ 0 do 3: θhigh ← 2 θhigh 4: end while 5: while θhigh − θlow > δ do 6: θmid ← (θlow + θhigh )/2 7: if f (θmid ; Ni ) ≤ 0 then 8: θlow ← θmid 9: else 10: θhigh ← θmid 11: end if 12: end while 13: return (θlow + θhigh )/2

exponential (Poisson or ON–OFF arrival MGF), both O(1). The total complexity per slice is O(N log(1/δ)). In practice, a small safety margin θi ← (1 − η)θi∗ with η ≪ 1 is applied to avoid numerical boundary violations where φA,i φS,i slightly exceeds 1 due to floating-point errors. V. NUMERICAL INVESTIGATIONS We evaluate the proposed martingale bounds and resource allocation algorithms via Monte Carlo simulations over 3 × 105 slots. System parameters are listed in Table I. A conventional effective capacity model with a finite-blocklength penalty factor β = 0.82 serves as the baseline. Because the baseline relies on asymptotic approximations and lacks Doob’s maximal inequality to bound the supremum of net arrivals, it overestimates tail violation probabilities and causes resource over-allocation. TABLE I: Simulation Parameters [12]

and the minima are well defined. Then the minimum Ni∗ can be found by incrementally increasing Ni and checking the QoS constraint. The allocation algorithm is given in Algorithm 2.

Parameter Total Bandwidth Total Resource Blocks Slot Duration Number of Users URLLC Traffic Params eMBB Arrival Rate Average SNR QoS Thresholds Algorithm Parameters

Symbol B N Ts Ku , K e πon , λu , Lu λe ρu , ρ e du , Q e δ, θ0 , η

Value 10 MHz 50 RBs 0.5 ms 10, 10 0.2, 104 pkt/s, 160 bits 600 kbps 10 dB, 10 dB 4 slots, 12 kbits 10−7 , 10−6 , 10−5

'HOD\9LRODWLRQ3UREDELOLW\P(Wu du )

Algorithm 2: Martingale-Based Minimum RB Allocation for URLLC and eMBB Slices Require: QoS targets (du , ϵu ), (Qe , ϵe ); SNR ρu , ρe ; Total capacity N A. URLLC Delay Violation Probability Bound Ensure: Minimum required allocations Nu∗ , Ne∗ , or Infeasible 1: for each slice i ∈ {u, e} do  3URSRVHG0DUWLQJDOH%RXQGdu = 4VORWV 2: Calculate the minimum stable RBs %DVHOLQH%RXQG = 0.82   Ni,min = ⌊E[Ai ]/E[ci ]⌋ + 1 according to Lemma 2 0RQWH&DUOR6LPdu = 4VORWV 7DUJHW u = 10 3 3: Initialize search starting point: Ni ← Ni,min   4: while true do 5: Obtain the optimal martingale parameter θi∗ via   Algorithm 1 6: Calculate the exponent Fi (Ni ) based on (21) or   (22) accordingly 7: if Fi (Ni ) ≥ ln(1/ϵi ) then   8: Ni∗ ← Ni 9: break           10: end if $OORFDWHG5HVRXUFH%ORFNVNu 11: Ni ← Ni + 1 12: end while Fig. 2: URLLC delay violation probability P (Wu ≥ du ) 13: end for versus allocated RBs Nu (du = 4 slots). 14: if (Nu∗ + Ne∗ ) ≤ N then 15: return Nu∗ , Ne∗ Fig. 2 plots P (Wu ≥ du ) versus Nu ∈ [6, 20] for du = 4 16: else slots, where Nu,min = 6 is the stability threshold from 17: return Infeasible Lemma 2. The proposed bound upper-bounds the simu18: end if lation curve and decays faster than the baseline. To reach The outer loop of Algorithm 2 iterates at most N times per slice, and each bisection run requires O(log(1/δ)) evaluations of f (θi ; Ni ). Each evaluation involves one upper incomplete gamma function (Lemma 1) and one

ϵu = 10−6 , the proposed bound requires Nu∗ = 20 RBs, while the baseline remains at 1.0 × 10−5 . Moreover, the Monte Carlo estimate becomes zero for Nu ≥ 15 because no violations are observed in 3 × 105 simulated slots . This illustrates the finite-sample resolution limit of Monte

Carlo simulation in the rare-event region. While Theorem 2 directly computes the bound 10−6 via Algorithm 1. B. eMBB Backlog Violation Probability Bound

%DFNORJ9LRODWLRQ3UREDELOLW\P(Be Qe )



3URSRVHG0DUWLQJDOH%RXQGQe = 12NELWV %DVHOLQH%RXQG = 0.82 0RQWH&DUOR6LPQe = 12NELWV 7DUJHW e = 10 3

   

proposed scheme, even at the stringent target ϵ = 10−6 , the total demand Nu∗ + Ne∗ reaches 44 RBs, which is below the physical capacity limit of N = 50 RBs. Due to looser analytical bounds, the baseline exceeds the 50RB limit when ϵ < 10−5 (making the system infeasible under the considered capacity constraint.). At ϵ = 10−6 , the proposed method drastically reduces the total required RBs from 64 to 44, reducing the required RBs from 64 for the baseline to 44, corresponding to a 31.2% reduction in RB consumption relative to the baseline..

 

VI. Conclusions

 

In this paper, we proposed a martingale-based resource block (RB) allocation framework for coexisting URLLC and eMBB slices in 5G wireless networks. By constructing exponential martingale processes based on the MGFs of the arrival and service processes and applying Doob’s maximal inequality, we derived explicit non-asymptotic upper bounds on the URLLC delay violation probability and the eMBB backlog violation probability. Building upon the theoretical stability thresholds, we designed a low-complexity bisection algorithm to compute the optimal martingale parameters and a deterministic search procedure to determine the minimum required RBs. Numerical results demonstrated that the proposed analytical bounds effectively upper-bound the Monte Carlo simulation results while avoiding the finite-sample resolution limits encountered in rare-event simulation of finite-sample simulations in rare-event regimes. At the reliability target of 10−6 , the proposed scheme requires a total of 44 RBs, satisfying the physical capacity constraint of 50 RBs, and reduces total RB consumption by 31.2% compared to the 64 RBs required by the conventional effective capacity baseline.

   















$OORFDWHG5HVRXUFH%ORFNVNe



Fig. 3: eMBB backlog violation probability P (Be ≥ Qe ) versus allocated RBs Ne (Qe = 12 kbits). Fig. 3 shows P (Be ≥ Qe ) versus Ne ∈ [11, 24] for Qe = 12 kbits, starting from the stability threshold Ne,min = 11 . Throughout the range, the proposed bound is tighter than the baseline . To reach the stringent eMBB target ϵe = 10−6 , the proposed martingale method requires only Ne∗ = 24 RBs to successfully cross the threshold. In contrast, the baseline curve remains strictly above the target line at this allocation, failing to guaranty the required reliability. Similarly, the Monte Carlo estimate becomes zero for Ne ≥ 18 because no violations are observed in the finite simulation sample, while Theorem 1 provides a deterministic evaluation up to 10−6 .

0LQLPXP5HTXLUHG5HVRXUFH%ORFNVNi*

C. Minimum RB Allocation and Feasibility 4R6,QIHDVLEOH5HJLRQ > 505%V 6\VWHP&DSDFLW\N = 505%V %DVHOLQH7RWDO5%V = 0.82 3URSRVHG7RWDO5%VNu* + Ne*  3URSRVHGH0%%Ne*  3URSRVHG85//&Nu*

 

References

6DYLQJV YV5%V

    10 1

10 2

10 3

10 4

10 5

7DUJHW9LRODWLRQ3UREDELOLW\ u = e =

10 6

Fig. 4: Minimum required RBs Ni∗ and system feasibility versus target violation probability ϵu = ϵe = ϵ. Fig. 4 compares the minimum required RBs from Algorithm 2 for ϵu = ϵe = ϵ ∈ [10−6 , 10−1 ]. Under the

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