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Blockchain-based Proportional Fair Scheduling for Multi-Operator O-RAN
arXiv:2609.07473v1 [cs.NI] 7 Sep 2026
Kun Huang, Xintong Ling, Member, IEEE, Meining Wu, Jiaheng Wang, Senior Member, IEEE, Zhi Ding, Fellow, IEEE, and Xiqi Gao, Fellow, IEEE Abstract—The openness and disaggregation of Open radio access network (O-RAN) facilitate resource sharing and coordination across networks, creating new demands for efficient and trustworthy cross-operator scheduling. However, such scheduling is beyond the scope and capability of conventional proportional fair scheduling (PFS), which lacks mechanisms for establishing trust among independent operators. To fulfill this gap, we propose the blockchain-based proportional fair scheduling (BC-PFS) that enables trustworthy inter-network coordination and resource pooling across operators in O-RAN. Specifically, we design four core smart contracts including registration, status reporting, scheduling, and settlement contracts with corresponding Solidity implementations to ensure trustworthy on-chain execution. Theoretically, to evaluate the BC-PFS performance, we develop an analytical framework to derive the user average throughput via both probabilistic and ordinary differential equation (ODE) approaches, and provide a simplified closed-form solution. Based on the above performance assessment, we quantify the pooling effect in O-RAN achieved through trustworthy cross-operator collaboration via BC-PFS, and point out that this effect grows monotonically in both the numbers of operator networks and users. Simulations validate the theoretical analysis and show the performance of the BC-PFS in O-RAN. Index Terms—Blockchain, O-RAN, proportional fairness, trustworthiness, user scheduling, wireless communications.
I. I NTRODUCTION As the telecommunication industry transitions from 5G to the upcoming 6G, the growing diversity of applications demands seamless communication with guaranteed quality of service (QoS) across heterogeneous networks [1]. Open radio access network (O-RAN) has emerged as a next-generation network architecture to meet these requirements through openness, disaggregation, intelligence, and programmability [2], [3]. By leveraging network function virtualization and open interfaces, O-RAN breaks the closed nature of traditional architectures and enables broader participation from multiple operators, vendors, and third-party applications. The growing demand for spectrum and RAN infrastructure sharing in ORAN necessitates efficient and trustworthy coordination across different networks and operators [4]–[6]. However, the trust foundation for this collaboration is often limited in O-RAN due to the disaggregated network K. Huang, X. Ling, M. Wu, J. Wang, X. Gao are with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China (e-mail: [email protected], [email protected], [email protected], [email protected], and [email protected]). K. Huang, X. Ling, J. Wang, and X. Gao are also with the Purple Mountain Laboratories, Nanjing 210023, China. J. Wang is also with the Cyber Science and Engineering, Southeast University. Z. Ding is with the Department of Electrical and Computer Engineering, University of California at Davis, Davis, CA 95616 USA (e-mail: [email protected]).
components, multi-vendor implementations, and separate network management [7]. Specifically, cross-operator collaborative scheduling requires multiple stakeholders to trustfully exchange user and resource information and jointly execute their scheduling decisions. The reliability of coordinated decisions and their execution is also challenging for different parties to verify. Moreover, geographically fragmented and heterogeneous networks further increase the difficulty of tracing and auditing the scheduling process. Therefore, the lack of trust mechanisms constrains reliable coordination across networks, ultimately degrading overall network performance. These challenges underscore the necessity for a scheduling paradigm that ensures both distributed resource orchestration and trustworthy inter-network cooperation within O-RAN. As the default downlink scheduling for 4G, proportional fair scheduling (PFS), can balance system throughput and user fairness [8] and has demonstrated robust performance under diverse network conditions [8], [9]. To extend PFS beyond isolated networks, early multi-cell PFS approaches proposed in [10] coordinated collaborative decisions across base stations with shared user information. Subsequent studies further developed joint PFS strategies that maximize the product of user priorities and extended them to multi-user selection scenarios [11]–[13]. However, these schemes inherently rely on information sharing among base stations or networks, and generally assume trustworthy execution among participating entities and truthful information exchange. Yet this assumption is hard to sustain in multi-operator O-RAN, where participants lack an inherent trust foundation for collaborative scheduling. As a result, the cross-network coordination process is vulnerable to potential manipulation risks and unreliable execution. Remark that blockchain provides a promising trust infrastructure for trustworthy inter-network collaboration through consensus-based validation and smart contract execution [14]. By maintaining a shared, tamper-resistant, and traceable ledger, blockchain offers multiple operators a consistent and verifiable view of coordination without relying on intermediaries [15]. Consensus mechanisms establish agreement on scheduling rules and states, while smart contracts enable automatic and transparent execution of coordination strategies [16]. These capabilities have enabled blockchain to support dynamic user scheduling and resource sharing across operators for O-RAN [17]–[20]. Recent studies have explored blockchain to facilitate crossoperator resource coordination and user association. Research in [17], [20]–[23] enables flexible cross-operator spectrum access through blockchain-based decentralized coordination, while works like [24], [25] utilized smart contracts for dynamic spectrum pricing and demand-driven resource alloca-
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tion. Beyond architectural designs, several studies have sought to improve network performance through mathematical modeling of user association [26]–[28]. The study in [29] introduced a proof-of-strategy consensus mechanism that maximizes the global efficiency with optimal user association. To ensure QoS during spectrum access, [22] employed a repeated gametheoretic framework to establish the trust relationship between users and service providers. Similar optimization perspectives can be found in [30], [31], where auctions and game theory are employed to refine association strategies. Although these blockchain-based schemes improve coordination transparency and auditability, they often rely on complex iterative algorithms, which pose challenges for deployment in large-scale and geographically distributed O-RAN. Therefore, it calls for a low-complexity on-chain scheduling scheme. Meanwhile, most modeling studies lack analytical frameworks to quantitatively characterize the performance gains of cross-operator scheduling through blockchain. The performance improvement under blockchain-enhanced cooperation remains unclear. In summary, despite these advances, several critical limitations remain. On the one hand, existing cross-operator user scheduling algorithms often lack explicit trust mechanisms for inter-network cooperation, making reliable coordination difficult to sustain in multi-operator O-RAN. On the other hand, most blockchain-based scheduling approaches lack tight integration with the physical layer and often involve high computational complexity. Furthermore, existing studies provide limited analysis of cross-operator scheduling enabled by blockchain. The fundamental questions remain unanswered, particularly regarding how to facilitate efficient and trustworthy cross-operator scheduling and how many improvements can be attained through blockchain-enhanced O-RAN. To fulfill the above gaps, we propose the blockchain-based proportional fair scheduling (BC-PFS) which integrates PFS to enable trustworthy cross-operator user scheduling for ORAN. BC-PFS preserves the low-complexity and responsive adaptation of conventional PFS, and meanwhile it establishes trusted coordination and verifiable execution among operators. Moreover, we develop systematic analytical frameworks to assess the system performance and quantify the pooling effect of O-RAN via BC-PFS. Ultimately, we aim to address the fundamental questions about efficiency, fairness, and measurable utility improvements in O-RAN via blockchain-enabled scheduling across operators. The main contributions of this paper are summarized as follows: • We propose the BC-PFS that enables trustworthy user scheduling for multi-operator O-RAN. Specifically, we design the BC-PFS with four core smart contracts, namely registration, status reporting, scheduling, and settlement contracts. • We establish a comprehensive analytical framework for the BC-PFS performance evaluation and derive the user throughput through both probabilistic modeling and ordinary differential equation (ODE) approaches. Furthermore, we provide a simplified solution in a closed form by introducing a relatively weak assumption. • We quantitatively characterize the pooling effect introduced by BC-PFS in O-RAN. We prove that the pooling
Regional O-Cloud
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Global data
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O-CU
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…
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Fig. 1. Illustration of the BC-PFS for multi-operator O-RAN.
effect is monotonically increasing with respect to the numbers of operator networks and users per network, demonstrating consistent performance improvement from the network openness. • Through a series of simulations, we validate our theoretical derivations and demonstrate the superiority of the BC-PFS compared to conventional non-cooperative approaches. We provide the BC-PFS implementation based on Solidity and JavaScript at https://github.com/ kunhuangseu/BC-PFS. The remainder of this paper is organized as follows. Section II outlines the system model. Section III revisits the PFS algorithm. Section IV presents the proposed BC-PFS. Section V provides the mathematical performance analysis. Section VI quantitatively evaluates the pooling effect. Section VII presents the simulations. Finally, Section VIII concludes the paper. II. S YSTEM MODEL A. Multi-operator O-RAN We consider a K-operator O-RAN, each serving N users. As illustrated in Fig. 1, O-RAN consists of functional entities deployed across edge and regional O-RAN cloud (O-Cloud). At edge O-Cloud, each operator maintains logically isolated ORAN distributed unit (O-DU) and O-RAN centralized unit (OCU) instances. The O-DU performs radio resource scheduling and connects to O-RAN radio units (O-RUs), which provide radio access to user equipment (UEs). At the regional OCloud, the near-real-time RAN intelligent controller (Near-RT RIC) provides programmable control and coordination, while the non-real-time RIC (Non-RT RIC) and service management and orchestration (SMO) support long-term orchestration, policy management, and network optimization. The scheduling is based on a time-division allocation framework, where time is partitioned into discrete, uniformly spaced slots of duration. We assume that the channel coherence time is larger than a slot time. Let rnk (t) denote the instantaneous rate provided by operator k to user n at time slot t. Every operator k has the corresponding licensed spectrum to provide services, and the instantaneous rates rnk1 (t) and rnk2 (t) for any two distinct operators k1 ̸= k2 are mutually independent. Furthermore, µnk (t) represents the throughput achieved by user n from operator k up to time slot t.
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In the considered multi-operator O-RAN, a user can dynamically access multiple operators for service, while each operator’s channel can serve only one user in each time slot. This association is captured by a binary user-network association indicator Ink (t) ∈ {0, 1}, where Ink (t) = 1 indicates user n accesses the channel of operator k during slot t, while Ink (t) = 0 indicates the absence of scheduling for user n by operator k in that slot. Consequently,Pthe achievable rate K for user n at slot t is given by rn (t) = k=1 rnk (t) Ink (t). The cumulative throughput for user n aggregates PKcontributions from all operators and is given by µn (t) = k=1 µnk (t). B. Wireless Channel Model In this work, the wireless channel is modeled as the wellknown Rayleigh fading by assuming that the signal envelope follows a Rayleigh distribution [32]. It can effectively characterize rich scattering environments lacking line-of-sight propagation paths. According to the Rayleigh fading model, for user n, its instantaneous signal-to-noise ratio (SNR) can be modeled as an exponentially distributed random variable with 1 the probability density function, fSNRn (x) = γn exp − γxn , where γn represents the average SNR for user n. Based on Shannon’s formula, the instantaneous transmission rate rn can be expressed as: rn = B log2 (1 + SNRn ) ,
(1)
with B denoting the channel bandwidth. The statistical characteristics of the rate are given by: Z ∞ r̄n = B log2 (1 + γn x) · exp (−x) dx, (2) Z0 ∞ 2 σn2 = (B log2 (1 + γn x)) · exp (−x) dx − r̄n2 . (3) 0
The average r̄n and variance σn2 characterize the long-term average performance and random fluctuations of the user’s transmission rate, respectively. The logarithmic rate model in (1) can be simplified to a linear rate approximation through a first-order Taylor expansion of Shannon’s formula under low-SNR conditions. This linear model has been widely used in the analysis of user scheduling [33], [34] and offers a practical trade-off between accuracy and computational simplicity for low-SNR regimes. III. R EVISIT P ROPORTIONAL FAIR S CHEDULING In wireless networks, user scheduling fundamentally addresses a constrained optimization problem aimed at selecting proper users to serve in limited wireless resources to maximize network performance while adhering to fairness and feasibility requirements [35], [36]. This optimization framework typically involves defining criteria that quantitatively characterize key system performance metrics, including throughput and fairness. Among various fairness criteria, proportional fairness has emerged as a pivotal concept due to its ability to achieve a balance between network throughput and fairness. We define the long-term average throughput of user n, denoted by µ̄n = lim E [µn (t)], as the limit of the expected t→∞
instantaneous throughput over time. In a single operator network with N users, as formally proposed by [37], proportional fairness is achieved by a feasible throughput allocation {µ̄∗n , n = 1, 2, . . . , N } such that for any other feasible allocation {µ̄n , n = 1, 2, . . . , N }, the sum of proportional changes satisfies: N X (µ̄n − µ̄∗n ) /µ̄∗n ≤ 0. (4) n=1
This definition implies that any unilateral improvement in a user’s throughput must be offset by a disproportionate reduction in others’ allocations, thereby discouraging spectral monopolization. The intrinsic value of proportional fairness lies in its dual emphasis on efficiency and equity, ensuring that users with weaker channel conditions are not perpetually marginalized while maintaining high aggregate throughput. The relationship between proportional fairness and optimization theory is established through resource allocation problems. As proven in [37], the solution inherently satisfies the proportional fairness criterion with the logarithmic utility function. In this case, the PFS corresponds to solving the following optimization problem: maximize
N X
ln (µ̄n ) ,
(5)
n=1
which is subject to system feasibility constraints. The logarithmic utility function inherently prioritizes fairness by assigning diminishing marginal returns to throughput increases. This property ensures that modest improvements for users with historically low throughput yield higher utility than marginal gains for users already operating at high rates. In a single-operator network, the PFS operates as follows. The operator acquires users’ instantaneous rates {rn , n = 1, 2, . . . , N } through feedback mechanisms, where users periodically report their channel state information (CSI). The operator then selects the user maximizing the ratio of the transmission rates to throughput {µn , n = 1, 2, . . . , N }. Mathematically, user selection follows proportional fair rule: rn (t + 1) . 1≤n≤N µn (t)
n∗ (t + 1) = argmax
(6)
After user selection, the operator allocates spectrum resources and configures modulation schemes to maximize the selected user’s data rate. Data transmission starts with dynamic parameter adjustments based on real-time CSI updates. Simultaneously, the selected user’s average throughput updates to incorporate the current rate via: µn (t + 1) = (1 − α) µn (t) + αrn (t + 1) In (t + 1) .
(7)
The smoothing factor α ∈ (0, 1) controls the throughput adaptation rate. Smaller α yields gradual throughput adjustments, while larger α enables faster responses to channel fluctuations. The basic goal of PFS is to dynamically adapt resource allocation over time, ensuring that long-term average throughput converges to the optimal solution. By integrating realtime channel state information with historical throughput data, these algorithms strike a balance between exploiting favorable channel conditions and achieving fairness.
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However, while proved effective in single-operator environments, the PFS faces inevitable challenges in multi-operator ORAN due to the absence of trust. In O-RAN, coordinated user scheduling across operators is significant to enable efficient resource utilization. Serving as the foundational prerequisite for such cooperation, trust among operators ensures reliable commitment to scheduling patterns and prevents deviations from agreed policies. Without robust trusted multi-operator collaboration mechanisms, scheduling policies in existing approaches may fail in practice due to opportunistic or malicious behaviors like free-riding or strategic non-compliance, which result in globally suboptimal performance. Therefore, the gap in current research highlights the critical need for novel scheduling paradigms that enable trustworthy multi-operator collaboration within the open and disaggregated architecture of O-RAN. IV. B LOCKCHAIN - BASED P ROPORTIONAL FAIR S CHEDULING A. Overview To overcome these limitations of traditional user scheduling in Section III, we integrate blockchain into the PFS framework and design the BC-PFS to bridge the trust gap in multioperator collaboration and preserve the fairness-efficiency balance. This section presents the BC-PFS scheme for trustworthy and efficient user association in multi-operator O-RAN. To support the proposed BC-PFS, we introduce a two-layer blockchain architecture into O-RAN, consisting of multiple edge chains and a global chain (See Fig. 1). The edge chain is deployed at the edge O-Cloud and corresponds to a sub-region, defined as a geographical service area covered by a group of O-DUs from multiple operators. It coordinates cross-operator user scheduling within a sub-region. The global chain is deployed in proximity to the SMO and aggregates information from edge chains to support overall network management across sub-regions. For user scheduling, each O-DU collects user states and submits the required scheduling information to the edge chain. Based on the reported user states and historical throughput, the edge chain executes PFS algorithm via smart contracts and delivers the scheduling decisions to the corresponding ODUs, which then translate these decisions into radio resource allocation for transmission. Meanwhile, the edge chain periodically uploads aggregated state information and historical transaction records to the global chain, providing consistent global state view and auditability. The global chain is responsible for transaction settlement, spectrum management, and network coordination. Moreover, based on the scheduling information maintained by global chain, the SMO and Non-RT RIC can further perform long-term policy optimization. In principle, BC-PFS is the collaborative scheduling of multiple operators. Specifically, each operator independently schedules one user per time slot. Operator k schedules user n∗k at slot t + 1 according to the proportional fair rule: rnk (t + 1) , for k = 1, . . . , K. µn (t) 1≤n≤KN
n∗k (t + 1) = argmax
(8)
After user selection, the throughput is updated iteratively to reflect the latest resource allocations: µn (t + 1) = (1 − α) µn (t) + α
K X
rnk (t + 1) Ink (t + 1) .
k=1
(9) Remark that this approach can achieve stationary average throughput by dynamically adapting to channel variations and adjusting scheduling decisions over time. By integrating instantaneous channel conditions with long-term performance, it ensures that the average throughput converges to the globally optimal solution of the maximization problem [33]: maximize
KN X
ln (µ̄n ) .
(10)
n=1
In the proposed BC-PFS, the blockchain serves as a core to coordinate cross-operator scheduling for O-RAN. First, it provides a transparent platform for inter-operator spectrum pooling in O-RAN, enhancing the capability of multiple operators to trustfully share and aggregate licensed frequency bands across networks. Second, the edge chain collects authenticated user states, executes the PFS rule through smart contracts, and records scheduling decisions immutably, ensuring verifiable cross-operator user scheduling in a sub-region. Third, the global chain aggregates service and transaction records from edge chains to support overall coordination, service settlement, and network management. Therefore, blockchain enables cross-network resource pooling and trustworthy collaborative scheduling. By ensuring both transparent execution and traceable records of scheduling, it establishes a reliable trust foundation for inter-operator collaboration. Enhanced by blockchain, BC-PFS ensures that all participants operate under verifiable protocols through automated execution of smart contracts, while enforcing agreement on real-time scheduling decisions via consensus mechanisms. B. Workflow In this subsection, we present the complete cycle in Fig. 2(a) to illustrate the proposed BC-PFS in O-RAN. 1) User and operator registration. The workflow initiates with user and operator registration. After user accesses the network, the O-DU collects user’s registration information and submits it to the edge chain. The registration contract verifies the user identity and records the authenticated identities onchain. Meanwhile, each operator registers with the global chain through the SMO. The global chain and edge chains then synchronize necessary registration information. 2) User status report. Following registration, users continuously measure their channel conditions and periodically upload status reports including critical metrics such as CSI. The O-DU preprocesses these reports and submits required scheduling information to the edge chain. The status reporting contract aggregates and validates reports and updates the latest user states stored on-chain, ensuring scheduling decisions are based on up-to-date channel conditions. 3) User scheduling. UE with uplink (UL) transmission demand sends scheduling request (SR) and buffer status report
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UL grant (DCI 0_x) DL assignment (DCI 1_x) Aggregated service records 4. Transaction settlement Settlement contract
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(a) On-chain smart contracts Addresses of operators 𝑘 and users 𝑛
Latest rate 𝑟𝑛𝑘
Status reporting contract
Registration contract
Scheduling results 𝐼𝑛𝑘
Scheduling contract Scheduling decision Submit report
Operator register SMO/NonRT RIC
Policy optimize
UE register Near-RT RIC
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Fig. 2. Implementation of the BC-PFS in O-RAN. (a) Brief workflow of the BC-PFS. (b) Interactions among smart contracts and network entities.
(BSR) to the O-DU for indication, while downlink (DL) transmission demands are available at the network side. During each scheduling interval, the scheduling contract retrieves the latest user states, and queries the historical throughput records from the previous validated block. It then executes the PFS algorithm using the instantaneous rates and historical throughputs to determine the scheduled users and update throughput records. The scheduling decision is delivered to the corresponding O-DU, which performs the resource allocation for the selected users and conveys the resulting UL grant and DL assignment to the UEs through downlink control information (DCI) formats 0_x and 1_x, respectively. 4) Settlement and optimization. Upon scheduling, the edge chain aggregates service records and periodically synchronizes them with the global chain, where the settlement contract calculates cross-operator service charges according to predefined pricing and settlement rules. All settlement records, including billing details and service quality metrics, are permanently recorded on-chain, forming verifiable network service credentials. Meanwhile, the Non-RT RIC analyzes long-term service data from the global chain to optimize network policies and deliver to the Near-RT RIC, which further optimizes scheduling parameters for execution by O-DU. The above process enables trustworthy user scheduling across network operators through secure verification, immutable records, and traceable auditing. In particular, BC-PFS
is built on the permissioned blockchain jointly maintained by the participating operators, and can use the consensus protocols such as Raft [38]. This design is suitable for the multioperator scenario, since cross-operator coordination requires controlled membership. Particularly, the consensus process based on Raft incurs relatively low energy consumption and is favorable for resource-constrained wireless deployments. Cryptographic techniques, particularly digital signatures, can ensure the integrity and authenticity of all status reports and transactions, preventing unauthorized modifications, and enabling secure verification of participants’ identities throughout the scheduling process. Smart contract can automatically enforce scheduling decisions and payments. Thanks to the low complexity of the PFS, the corresponding smart contracts can be easily implemented based on users’ instantaneous rates and historical throughputs in the current and previous blocks. The online incremental update style of the PFS also fits well with on-chain operations via smart contracts since the blockchain information is also updated incrementally. Moreover, the scheduling operations and transactions are immutably recorded on the blockchain, which ensures transparency and establishes trust between different networks. Note that the BC-PFS requires the consensus among different operators, which inevitably introduces extra delay. More specifically, the scheduling interval is determined by the block time, i.e., the average time to create a new block which contains the most recent scheduling result. If the channel coherence time is much shorter than the block time, or equivalently the scheduling interval, the BC-PFS will rely on outdated channel states, which could largely degrade the performance. Therefore, permissioned ledgers with short block time are more preferred over public permissionless blockchains like Bitcoin whose block time is around 10 minutes. For instance, Kaspa [39] achieves the block time on the order of 100 ms, and Quorum [40] can be configured to generate blocks every 50 ms. Furthermore, the project of MegaEth [41] can achieve the block time of 10 ms and further claims it can be reduced to around 1 ms, approaching real-time operations. These lowlatency blockchains can achieve the block time close to or even shorter than the channel coherence time in scenarios with slow or moderate mobility (10-50 ms) [42]. (As a reference, a pedestrian walking at 3 km/h has the coherence time around 76 ms when the carrier frequency 2 GHz is used.) Admittedly, in high-mobility cases, the consensus process may have a nonnegligible impact on the performance of BC-PFS. C. Implementation via Smart Contracts In this subsection, we present the realization of BC-PFS via smart contracts. Fig. 2(b) depicts the interactions among smart contracts and O-RAN entities, where smart contracts execute the core business logic of BC-PFS by handling registration, user status reporting, scheduling, and settlement. The design and functionality of four core smart contracts are as follows. Furthermore, we provide Solidity codes in terms of the four core smart contracts in Algorithms 1-4, respectively. 1 1 The repository address of the source code is given in the Introduction.
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Algorithm 1 Registration Contract
Algorithm 3 Scheduling Contract
1: struct Info {
1: mapping(address => address) public selectedUser;
2: uint id; 3: bool isOperator; } 4: uint private nextOpId = 1; 5: uint private nextUserId = 1; 6: mapping(address => Info) public registry; 7: event RegistrationSuccess(address user, uint id, bool
2: mapping(address => uint) public throughput; 3: uint public constant alphaInv = 10000; 4: mapping(address => mapping(address => bool)) public
isOperator); 8: function register(address user, bytes memory proof, bool isOperator) public { 9: // Identity validation 10: require(validProof(proof, isOperator), "Invalid credentials"); 11: // Registration 12: if (isOperator) { 13: registry[user] = Info(nextOpId++, true); 14: } else { 15: registry[user] = Info(nextUserId++, false); } 16: emit RegistrationSuccess(user, registry[user].id, isOperator); }
6: event Scheduled(address[] operators, address[]
Algorithm 2 Status Reporting Contract 1: mapping(address => mapping(address => uint[])) public
operatorUserRates; 2: event ReportSubmitted(address user, address operator,
uint timestamp, uint rate); 3: function submitReport(address user, address operator,
bytes memory csi) public { // CSI validation require(validCSI(csi), "Invalid CSI data"); // Rate estimation uint rate = rateEstimation(csi); 8: // Rate update 9: operatorUserRates[user][operator].push(rate); 10: emit ReportSubmitted(user, operator, block.timestamp, rate); } 4: 5: 6: 7:
•
The registration contract (see Algorithm 1) processes both user and operator registration requests. During registration, the contract first verifies two critical conditions: the entity ID must be unregistered and the provided credentials must be validated according to the validProof function. For successful registrations, the contract updates the registry mapping with a tuple containing the generated ID and role flag, then emits a RegistrationSuccess event containing the address, assigned ID, and role type.
The status reporting contract (see Algorithm 2) processes real-time channel state updates from registered users. The contract validates incoming CSI through the validCSI function before processing, rejecting reports with invalid data formats. Approved reports undergo rate estimation through the rateEstimation function before being permanently recorded in the operatorUserRates mapping, with successful updates triggering ReportSubmitted events. • The scheduling contract (see Algorithm 3) implements •
schedulingMatrix; 5: mapping(address => mapping(address => uint)) public
allocatedRate; selectedUser); 7: function updateScheduling() public { 8: // User selection via (8) 9: address[] memory users = getAllUsers(); 10: address[] memory operators = getAllOperators(); 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30: 31: 32: 33: 34: 35: 36: 37:
for (uint k = 0; k < operators.length; k++) { address op = operators[k]; address prev = selectedUser[op]; schedulingMatrix[prev][op] = false; allocatedRate[prev][op] = 0; uint maxPriority = 0; uint bestLatestRate = 0; selectedUser[op] = address(0); for (uint n = 0; n < users.length; n++) { uint latestRate = StatusReporting.getLatestRate(users[n], op); uint priority = latestRate / throughput[users[n]]; if (priority > maxPriority) { maxPriority = priority; selectedUser[op] = users[n]; bestLatestRate = latestRate;}} schedulingMatrix[selectedUser[op]][op] = true; allocatedRate[selectedUser[op]][op] = bestLatestRate;} // Throughput update via (9) for (uint n = 0; n < users.length; n++) { uint totalAllocated = 0; for (uint k = 0; k < operators.length; k++) { address op = operators[k]; if (schedulingMatrix[users[n]][op]) { notify(users[n], operators[k]); totalAllocated += allocatedRate[users[n]][op];}} throughput[users[n]] = ((alphaInv - 1) * throughput[users[n]] + totalAllocated) / alphaInv; } emit Scheduled(operators, selectedUser); }
the BC-PFS algorithm to execute user scheduling. The contract is initiated by evaluating all possible useroperator pairs through a two-stage process. First, it calculates priority scores for each pair by dividing the user’s instantaneous transmission rate by their historical throughput average, simultaneously updating the schedulingMatrix. Each operator then selects the highest-priority user via (8). Second, each user’s throughput record is updated with newly allocated rates according to (9). Upon selection, the contract notifies participants of scheduling results and emits Scheduled events to provide an immutable on-chain record of all allocation decisions.
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Algorithm 4 Settlement Contract 1: event PaymentProcessed(address user, address operator,
uint cost); 2: // Notification of scheduling details 3: function processSettlement() public {
address[] memory users = getAllUsers(); address[] memory operators = getAllOperators(); for (uint n = 0; n < users.length; n++) { for (uint k = 0; k < operators.length; k++) { bool scheduled = Scheduling.getSchedulingMatrix (users[n], operators[k]); 9: if (scheduled) { 10: (uint duration, uint bandwidth) = getServiceParameters(users[n], operators[k]); 11: settleService(users[n], operators[k], duration, bandwidth); }}}} 12: // Service settlement 13: function settleService(address user, address operator, uint duration, uint bandwidth) public { 14: uint cost = calculateCost(operator, duration, bandwidth); 15: processPayment(user, operator, cost); 16: emit PaymentProcessed(user, operator, cost); }} 4: 5: 6: 7: 8:
Notably, the canonical representations µ and r characterize the time-independent essential features of these quantities by abstracting away their temporal dependencies. This convergence result provides fundamental insights. First, the weak convergence property demonstrates that the stochastic BC-PFS process ultimately follows the deterministic ODE solution, with the distribution of µnk (t) converging to the ODE’s trajectory despite short-term random fluctuations. Therefore, the long-term average throughput µ̄nk equals the ODE’s limit point and µnk (t) converges to µ̄nk when t → ∞. Second, the ODE system admits a unique limit point, independent of initial conditions, which implies the uniqueness of µ̄nk . This convergence enables our subsequent steady-state analysis. From Lemma 1, we take expectations on both sides of (9), which yields E [µn (t + 1)] = (1 − α) E [µn (t)] + α
K X
E [rnk (t + 1) Ink (t + 1)] . (13)
k=1
At the long-term steady state, this simplifies to µ̄n = lim E [µn (t + 1)] t→∞
•
The settlement contract (see Algorithm 4) performs financial settlement based on the scheduling results. The process is initiated through the processSettlement function, which systematically scans the schedulingMatrix to identify active allocations. Each validated scheduling transaction triggers the core settleService function, which dynamically computes service costs using operator-specific pricing metrics in calculateCost. The contract automatically initiates fund transfers via processPayment while broadcasting transaction confirmation through PaymentProcessed events.
= lim
t→∞
t→∞
µ̇nk = h̄nk (µ) − µnk , n = 1, ..., KN, k = 1, ..., K,
(11)
where µ = {µnk , n = 1, ..., KN, k = 1, ..., K} and h̄nk (µ) represents the stationary expectation of the rate provided by operator k to user n under the event rµnk > rµmk , ∀m ̸= n: n m rnk rmk h̄nk (µ) =E rnk | > , ∀m ̸= n, m ≤ KN . (12) µn µm
k=1
0
(15)
Moreover, under the scheduling rule (8), we have: µ̄n =
(16)
K Z ∞ X k=1
0
A. Analysis on Throughput
Lemma 1. Under stationary user rate conditions, we define µ and r as the canonical representations of µ (t) and r (t) , respectively. The user throughput µnk (t) generated by the scheduling algorithm in (8) and (9) weakly converges to the limit point of the following ODE system.
(14)
·Pr (Ink (t + 1) = 1 |rnk (t + 1) = x )) dx.
t→∞
In this section, we would like to quantitatively evaluate the performance of the proposed BC-PFS in O-RAN through rigorous mathematical analysis. The PFS framework exhibits well-established weak convergence properties [10]. The following lemma characterizes the asymptotic behavior of PFS:
E [rnk (t + 1) Ink (t + 1)] .
k=1
By using Bayes’ theorem, the average throughput can be expanded as: K Z ∞ X µ̄n = lim (xfrnk (x)
lim
V. P ERFORMANCE A NALYSIS
K X
xfrnk (x)
KN Y
Pr
m=1, m̸=n
rmk (t + 1) x > µn (t) µm (t)
By taking the limit, (16) can be written as: K Z ∞ KN X Y µ̄m x dx, µ̄n = xfrnk (x) Frmk µ̄n 0 m=1,
dx.
(17)
k=1
m̸=n
where fr (x) and Fr (x) represent the probability density function and the cumulative distribution function of user rate r, respectively. The mathematical structure of (17) reveals the intricate relationship between average throughput and instantaneous rates in BC-PFS. For Rayleigh fading channels, [34] established performance bounds under proportional fairness as follows. Lemma 2. In Rayleigh fading environments, given two users i and j with r̄i ≤ r̄j , under the definition of proportional µ̄ r̄ σ fairness, σji ≤ µ̄ji ≤ r̄ji , where µ̄i and µ̄j represent the average throughput of users i and j, respectively. These bounds demonstrate that average throughput ratios are constrained by both average rate differences and channel
8
variability. Lemma 2 can be extended to multi-operator scenarios. We assume that user n’s average rate r̄nk provided by operator k is mainly determined by the user itself. This assumption is realistic when the large-scale fading depending on the user’s location is dominant over the channel quality. That is, the transmission rates provided by different operators r̄nk exhibit minor fluctuations around user n’s average rate over K networks r̄Kn . According to our assumption and Lemma 2, under the linear rate approximation model over Rayleigh fading, the relationship between the average throughput of different users satisfies r̄n µ̄n = . (18) µ̄m r̄m
B. ODE Approach Remark that the above results can also be derived equivalently via ODE. The user throughput µnk (t) in the BC-PFS weakly converges to the limiting solution of a coupled ODE system. We will show the derivation in the simplest non-trivial case with two operator networks and one user per network, i.e., K = 2 and N = 1. The expected instantaneous rates decompose as r̄1 = r̄11 + r̄12 and r̄2 = r̄21 + r̄22 , while the total throughput components satisfy µ1 = µ11 + µ12 and µ2 = µ21 + µ22 . Under the linear rate model assumption, by computing h̄ (µ) according to Lemma 1, the ODE system has the following form: µ2 1 r̄11 2 − µ11 , µ̇ = r̄ − 11 11 µ1 µ2 r̄11 + r̄21 µ2 1 r̄12 2 − µ12 , µ̇ = r̄ − 12 12 µ1 µ + 2
This equality provides a theoretical foundation for deriving (17), and reduces the computational complexity of solving inter-network scheduling optimization problems by directly relating average throughput ratios to channel rate. For user n, the average throughput µ̄n expressed in (17) can be rewritten as: µ̄n = (19) x KN K Z ∞ exp − X r̄nk x Y xr̄m 1 − exp − dx. r̄nk r̄mk r̄n 0 m=1, k=1
m̸=n
The result in (19) provides a semi-closed-form solution to describe the average throughput for the BC-PFS in ORAN. We would like to further expand the integral into a summation form via the inclusion-exclusion principle, yielding a more precise expression at the cost of higher computational complexity: Z ∞ K X X 1 |S| exp − x x µ̄n = (−1) r̄nk 0 r̄nk k=1 S∈P(An ) !! X r̄m ·exp −x dx, (20) r̄mk r̄n m∈S
where An = {1, ..., KN } \ {n} denotes the complete set of all user indices excluding user n and P (An ) represents its power set containing all possible subsets.2 Meanwhile, S is an arbitrary subset from an element in P (An ), which is a user subset, and |S| indicates the number of elements in S. Upon substituting the expanded summation form into the integral, the average throughput is derived as: !−2 K X r̄m X X (−1)|S| 1 + . (21) µ̄n = r̄nk r̄nk r̄mk r̄n k=1 S∈P(An )
m∈S
This formulation provides a complete characterization of user average throughput in the blockchain-enhanced O-RAN, and quantifies how multi-operator O-RAN aggregation affects system performance. The accuracy of (19) and (21) is verified by the simulations in Section VII. Note that for large-scale scenarios with numerous operator networks or users per network, (21) may still require large computational overheads. 2 For instance, P ({1, 2}) ={∅, {1} , {2} , {1, 2}}.
r̄12
r̄22 2
µ2 r̄21 2 − µ21 , µ̇ = r̄ − 21 21 µ1 µ2 r̄11 + r̄21 µ2 2 µ̇22 = r̄22 − µ1 r̄22µ2 2 − µ22 . r̄12 + r̄22
The equilibrium solutions can be obtained by setting µ̇nk = 0. By using (18), we derive the expressions for the average throughput: ! 2 2 r̄1 r̄1 r̄ r̄ µ̄1 = r̄1 − r̄1 11r̄2 2 + r̄1 12r̄2 2 , r̄11 + r̄21 r̄12 + r̄22 ! 2 2 r̄2 r̄2 r̄21 r̄22 µ̄2 = r̄2 − r̄1 + r̄2 2 + r̄1 + r̄2 2 . r̄11
r̄21
r̄12
r̄22
The above result has the same form as (21). We can generalize the derivation to K-operator networks by constructing and solving corresponding ODEs. The generalization to KN users and K operator networks requires constructing conditional expectations h̄nk (µ) and solving the corresponding high-dimensional ODE system. Similarly, we can obtain the universal analytical solution: 2 r̄n |S| r̄nk µ̄ = PK P , (−1) n
k=1
S∈P(An )
r̄ = PK r̄ . n k=1 nk
P r̄m r̄n m∈S r̄mk r̄nk +
2
This comprehensive solution captures several crucial aspects of the multi-operator environment in the BC-PFS. The double summation structure reflects the hierarchical nature of resource competition, where users contend both within and across operator networks. Besides, the denominator terms precisely quantify the cumulative impact from competing users, with m each r̄r̄mk term representing the normalized competitive pressure from user m through operator k. Notably, this result is exactly equivalent to the result in (21), which establishes theoretical consistency between the probabilistic and ODE approaches to modeling the BC-PFS in O-RAN.
9
VI. P OOLING E FFECT
C. Simplified Solution While the approaches in Sections V-A and V-B provide theoretical assessment, they involve high-dimensional coupling terms that introduce significant computational complexity. To develop a more tractable analytical framework, we would like to simplify µ̄nk in (19) for analyzing the impact of x , µ̄nk blockchain. By applying variable substitution y = r̄nk can be transformed to:
After obtaining the simplified closed-form solution in (25), we would like to quantify the performance of BC-PFS in ORAN through a systematic utility analysis. Recall that the PFS is equivalent to maximizing the network utility U [33], given by KN X U≜ ln (µ̄n ) . (26)
µ̄nk = Z ∞ r̄nk yexp (−y)
For an isolated network where operators perform noncooperative PFS, the average throughput per user reduces to νn N ω (N ). Thus, the baseline network utility is KN X νn ω (N ) Uiso = . (27) ln N n=1
0
KN Y
m=1,m̸=n
(22) r̄nk r̄m y dy. 1 − exp − r̄mk r̄n
1 According to the above assumption, we denote νn = K r̄n as user n’s average rate over K networks, and the transmission rates provided by different operators r̄nk exhibit minor fluctuan | ≤ δ, where δ is the deviation of tions around νn , i.e., | r̄nkν−ν n user rate from the average. As a result, the average throughput µ̄nk for user n can be simplified to µ̄snk , given by: Z ∞ KN −1 µ̄nk ≈νn yexp (−y) (1 − exp (−y)) dy ≜ µ̄snk . 0
(23) By substituting z = 1 − exp (−y) with Taylor expansion, we can obtain the simplified solution µ̄snk as: µ̄snk =νn
Z 1
(− ln (1 − z)) z KN −1 dz = νn
0 ∞
νn X = KN i=1
1 1 − i i + KN
=
∞ X
1 i (i + KN ) i=1
νn ω (KN ) , KN
(24)
PKN where ω (KN ) ≜ i=1 1i . Consequently, we obtain the simplified average throughput of user n in the compact expression: µ̄sn =
νn ω (KN ) . N
(25)
This result in (25) shows that the user throughput of BC-PFS primarily depends on the average user rate νn , the number of operator networks K, and the number of users per network N . Meanwhile, the factor ω (KN ) explicitly quantifies the advantages of resource pooling and inter-network cooperation in O-RAN as network becomes more open, i.e., the total user number KN increases. This closed-form solution provides valuable insights for performance analysis on the BC-PFS across multi-operator networks. More importantly, the above simplified user throughput serves as a performance lower bound in fact. If we take the channel variability among operator networks into consideration, the rate fluctuation across different operator networks can provide extra multi-network diversity gain, analogous to multiuser diversity gain. Consequently, the actual performance of the BC-PFS in O-RAN will exceed the value in (25). Our simulation results in Section VII illustrate the close approximation of our simplified solution in (25) and also verify (25) as a lower bound.
n=1
Through BC-PFS, the average throughput per user is approximated as νNn ω (KN ). Consequently, the blockchain-enhanced cooperative O-RAN achieves a higher network utility, given by KN X νn ω (KN ) . (28) Ucoop = ln N n=1 We define their gap as the pooling gain ψ (N, K): ω (KN ) ψ (N, K) ≜Ucoop − Uiso = KN ln . ω (N )
(29)
Apparently, the pooling gain ψ (N, K) is determined by the numbers of operator networks K and users per network N . Through harmonic number expansion, we can show that the pooling gain is always positive, i.e., ! PKN ψ (N, K) = KN ln 1 +
1 i=N +1 i
ω (N )
> 0.
(30)
Mathematically, we prove that the O-RAN using BC-PFS outperforms the isolated networks based on non-cooperative PFS. Since the above results are based on the assumption without considering multi-network diversity, the simplified solution in (25), in fact, is a conservative lower bound. In other words, the actual network utility will be further enhanced by the diversity on the network level, amplifying the pooling gain beyond ψ (N, K) in (30). Mathematically, the following theorem points out that the pooling gain ψ (N, K) can be enlarged by both network size and the number of users. Theorem 1. For N = 1, 2, ... and K = 2, 3, ..., the pooling gain ψ (N, K) is strictly monotonically increasing with respect to the number of networks K and the number of users per network N . Proof. We first prove the monotonicity in K. For fixed N ≥ 1, consider the difference with respect to K: ψ (N, K + 1) − ψ (N, K) (31) ω (KN ) ω ((K + 1) N ) = (K + 1) N ln − KN ln ω (N ) ω (N ) ω ((K + 1) N ) ω ((K + 1) N ) = N ln + K ln . ω (N ) ω (KN )
10
ψ (N + 1, K) − ψ (N, K) (32) ω (K (N + 1)) ω (KN ) = K (N + 1) ln − N ln ω (N + 1) ω (N ) PK 1 KN +i 1 + i=1 ω(KN ) + ln ω (KN ) . = K (N + 1) ln 1 ω (N ) 1 + (N +1)ω(N )
5.8
User 1 throughput (Kbps)
Since ω (·) is an increasing function, both logarithmic terms in (31) are positive, which ensures that ψ (N, K + 1) − ψ (N, K) > 0. Hence, ψ (N, K) strictly increases in K. Consider the difference of ψ (N, K) with respect to N :
5.6 Accurate: =0.1 Accurate: =0.05 Simplified Simulation: =0.1 Simulation: =0.05
5.5 3150
PK
1 i=1 KN +i
1 ≥ (N +1)ω(N prove the positivity im) , we can 1+a > mediately. Otherwise, we have ln 1+b = ln 1 + a−b 1+b a − b for 0 ≤ a < b, and we can further obtain
3175
ψ (N + 1, K) − ψ (N, K) K (N + 1) X 1 1 ω (KN ) >K − + ln ω (KN ) i=1 KN + i ω (N ) ω (N ) 1 1 ω (KN ) >K − + ln > 0. ω (KN ) ω (N ) ω (N )
(33) !
Therefore, ψ (N, K) is also strictly monotonically increasing with N . Theorem 1 indicates that the pooling gain ψ (N, K) is always positive and also exhibits monotonic increase in both the number of networks and users. It points out the positive impact of cross-operator resource pooling of O-RAN enhanced by BC-PFS, since BC-PFS helps O-RAN to establish the trust required for reliable coordination across operators. Moreover, the pooling effect can be further enhanced in a more open network with a larger size. On the one hand, the monotonicity with respect to N indicates that, as the number of users per network increases, the gains from the BC-PFS become more pronounced. This characteristic is mainly gained from multiuser diversity, while BC-PFS in O-RAN further amplifies this effect through inter-network collaborative scheduling. On the other hand, the monotonicity with respect to the number of networks K reflects the diversity on the network level. The BC-PFS breaks through the resource limitations of isolated single networks and enhances the openness of O-RAN. VII. S IMULATION AND ANALYSIS In this section, we present the simulation results to support our analysis and conclusions. We model the wireless channel as a block-fading Rayleigh channel with the channel coherence time of 50 ms, and the scheduling interval is also set to 50 ms. The throughput update window is set to α1 = 10000, which ensures that the tracking parameter α is sufficiently small to achieve both fair scheduling and accurate long-term throughput measurements. We adopt the bandwidth of 1 MHz for every operator, with average SNR values across all users ranging from –20 dB to –10 dB. First, we illustrate the average throughput obtained from the accurate solution (21), the simplified solution (25), and the simulation results, to show the impact of the average rate
3200
3225
3250
3275
3300
Simulation time (s)
ω(KN )
(a)
6.7
User 1 throughput (Kbps)
If
5.7
6.6
6.5 Accurate: =0.1 Accurate: =0.05 Simplified Simulation: =0.1 Simulation: =0.05
6.4 3100
3125
3150
3175
3200
3225
3250
Simulation time (s)
(b)
Fig. 3. Simulation, accurate solutions, and simplified solutions of user throughput of the BC-PFS across K operators. (a) K = 2. (b) K = 5.
fluctuation on system performance. Fig. 3(a) and Fig. 3(b) present a comparative analysis of throughput performance for multi-operator networks with configurations of K = 2 and K = 5, respectively. The average throughput is evaluated under two average rate distributions δ = 0.05 and δ = 0.1, with 10 users per network. The accurate analytical solution in (21) can well characterize the average throughput from simulation, which validates the accuracy of our theoretical analysis. Notably, for concentrated rate distributions, i.e., smaller δ, the simplified solution exhibits negligible deviation from both simulations and the accurate solutions, confirming its practical effectiveness under realistic conditions consistent with the assumptions in Section V-C. Fig. 4 illustrates the average throughput under varying average rate distributions in multi-operator O-RAN, where each network serves 10 users and the number of operator networks is set to K = 2, K = 3, and K = 4. The results demonstrate that under uniform rate distribution with zero fluctuation, i.e. δ = 0, the throughput reaches its lower bound, which aligns with the theoretical conclusion presented in Section V-C. As the fluctuation amplitude of the average rate increases, the user throughput generally exhibits an upward trend, reflecting the enhanced channel diversity gain. Notably, as the number of operator networks increases, the deviation
11
8.5 8 7.5
Accurate Simplified
K=4 K=3 K=2
7 6.5 6 5.5
17.5 PFS 17 Max-sum scheduling
5 4.5
Max-min fair scheduling
0
0.2
0.4
0.6
0.8
Fig. 4. Comparison of simplified solutions, accurate solutions, and simulations under different rate distributions.
Throughput boundary BC-PFS
10
8 User 2 max Max-sum scheduling PFS Max-min fair scheduling
6
4 Throughput region of 2 single-operator scheduling User 1 max 0
0
2
4
16.5 17
17.5
18
18.5
19
19.5
Network 1 utility
/
User 2 throughput (Kbps)
BC-PFS
18
Network 2 utility
User 1 throughput (Kbps)
18.5
Accurate Simulation Simplified
6
8
10
User 1 throughput (Kbps)
Fig. 5. Throughput region: case of two users.
between the simulation and the simplified solution diminishes for the same fluctuation amplitude. This arises from the spatial averaging effect driven by the law of large numbers. With more operator networks, the actual user rate is tightly clustered around the average. Therefore, for larger values of K, the assumption of small fluctuations in the average rate is more strongly met, which improves the accuracy of the simplified solution. This consistency underscores the robustness of the simplified solution in large-scale O-RANs, where interoperator coordination becomes more efficient. Fig. 5 compares the throughput performance of traditional scheduling strategies and the BC-PFS for O-RAN. The light blue area reflects the achievable region of two users’ throughput under the scenario where networks operate independently. Each user can achieve their maximum throughput under exclusive spectrum access, with these individual maximums defining the boundary points of the throughput region. For traditional benchmark schemes, each operator independently serves its own local users without cross-operator coordination. The benchmark schemes differ in their scheduling rules. Specifically, the PFS selects the user with the largest ratio of the instantaneous rate to the historical average throughput, max-sum scheduling selects the user with the largest instantaneous rate, and max-min fair scheduling prioritizes the user
Fig. 6. Utility region: case of two networks.
with the lowest long-term average throughput. These strategies all operate along the throughput boundary, rendering them Pareto optimal, a state where no user’s throughput can be improved without degrading others. In contrast, the BC-PFS in O-RAN enables trustworthy cooperation among multiple operators through blockchain and achieves higher average user throughput than single-network scheduling. This gain is further amplified as the number of cooperating networks increases, which highlights the superiority of the BC-PFS in facilitating trustworthy inter-network coordination. Concerning the performance advantage of the BC-PFS for O-RAN, Fig. 6 illustrates the utility regions achieved by various scheduling approaches in a two-operator case, where each operator serves two users. For non-cooperative scheduling strategies, as theoretically expected, the PFS achieves superior utility performance compared to both max-sum scheduling and max-min fair scheduling, confirming its well-known proportional fairness property. More importantly, the BC-PFS shows significant utility improvement over these conventional noncooperative scheduling schemes, demonstrating the advantage of blockchain-enabled coordination. The simulation further reveals that the network utility of the BC-PFS in O-RAN grows significantly with increasing rate fluctuation range, namely δ. This occurs because larger rate variations enhance multi-operator diversity opportunities, which the BC-PFS effectively harnesses through its internetwork coordination. This observation also aligns with our theoretical analysis in Section VI, where we established that (28) represents the lower bound of network utility, i.e., the red point in Fig. 6. Notably, for non-cooperative scheduling among independent networks, each operator operates in isolation and their scheduling decisions cannot exploit diversity from rate variations, resulting in lower network utility. Furthermore, Fig. 7 compares the network utility achieved by different scheduling approaches under various network configurations. Fig. 7(a) demonstrates the variation of network utility with the number of operator networks K under a fixed N = 10. The results show that in traditional non-cooperative scheduling frameworks, the PFS consistently maintains optimal utility performance due to its proportional fairness property, which defines the performance upper bound
12
Theory Simulation
160
150
Pooling gain
Network utility
200
BC-PFS PFS Max-sum scheduling Max-min fair scheduling
200
100 Maximum utility of non-cooperative scheduling
120
80
40
50 2
3
4
5
0
6
Number of operator networks
2
3
4
5
6
Number of operator networks
(a)
Fig. 8. Simulation and theoretical results of pooling gain.
150
BC-PFS PFS Max-sum scheduling Max-min fair scheduling
100
100 Maximum utility of non-cooperative scheduling 50 10
20
300
90
Number of users per network
Network utility
200
30
40
50
Number of users per network
(b)
80
250
70 200
60 50
150
40
100
30 50
20 10
0
2
4
6
8
10
Number of operator networks
Fig. 7. Impact of network parameters on network utility. (a) Number of operator networks. (b) Number of users per network.
Fig. 9. Pooling gain under different network settings.
for non-cooperative strategies. In contrast, the BC-PFS in ORAN demonstrates significant performance advantages across different K through blockchain-enabled inter-network coordination. Notably, the performance gap between the BC-PFS and PFS monotonically increases with the growing number of participating networks, which validates the scaling effect of multi-network cooperation. Fig. 7(b) focuses on the impact of the number of users per network N on network utility U with a fixed K = 2. Similar to the pattern observed in Fig. 7(a), while the utility of all scheduling schemes improves with increasing N , the BC-PFS consistently maintains a stable performance lead, and this advantage becomes more pronounced as the user scale expands. This O-RAN performance enhancement via BC-PFS can be recognized as the pooling gain. Fig. 8 presents the pooling gain under varying numbers of operators K and users per network N . The results demonstrate strong agreement between theoretical and simulated values. Specifically, when maintaining constant rate fluctuation conditions, the pooling gain exhibits monotonic improvement with both increasing K and N , which is consistent with Theorem 1. These findings not only validate our theoretical framework, but also reveal that the pooling effect becomes increasingly pronounced with growing openness.
Furthermore, Fig. 9 visualizes the pooling gain’s dependence on the number of users per network and network numbers through a contour plot, where darker shades indicate higher gains. This gradient-based representation confirms the monotonic growth property derived in Section VI. The pooling gain consistently intensifies as network expands. Notably, the contour spacing becomes progressively uniform at larger network scales. The initially nonlinear growth of pooling gain gradually converges toward linearity as network expands, which can be analytically characterized by (29). When N and K are small, the logarithmic term dominates, creating ) nonlinear growth; as network scale increases, the ratio ω(KN ω(N ) approaches a constant via harmonic number approximation, causing ψ (N, K) to exhibit quasi-linear scaling with respect to KN . These insights show that trustworthy resource pooling in O-RAN achieved through BC-PFS can yield predictable, near-linear performance gains as the network scales. VIII. C ONCLUSION This work has presented the BC-PFS, a promising solution to establishing trustworthy cross-operator user scheduling in O-RAN. By integrating blockchain with the PFS, we have developed a decentralized framework based on smart contracts to execute the PFS on-chain, enhancing trust and
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traceability for O-RAN resource management. Meanwhile, we have established both accurate and simplified closed-form expressions for average throughput, offering key insights into system performance. Our quantitative analysis shows that the pooling gain increases monotonically with both the number of networks and users. Simulations have validated the theoretical analysis and confirmed the superiority of the BC-PFS over non-cooperative approaches for O-RAN. R EFERENCES [1] Y. Fu, X. Wang, and F. Fang, “Multi-objective multi-dimensional resource allocation for categorized QoS provisioning in beyond 5G and 6G radio access networks,” IEEE Trans. Commun., vol. 72, no. 3, pp. 1790–1803, Mar. 2024. [2] O-RAN Alliance, “O-RAN: Towards an open and smart RAN,” White Paper, Oct. 2018. [Online]. Available: https://mediastorage.o-ran.org/ white-papers/O-RAN.White-Paper-2018-10.pdf [3] M. Polese, L. Bonati, S. D’Oro, S. Basagni, and T. Melodia, “Understanding O-RAN: Architecture, interfaces, algorithms, security, and research challenges,” IEEE Commun. Surv. Tutorials, vol. 25, no. 2, pp. 1376–1411, Jun. 2023. [4] O-RAN Alliance, “Spectrum sharing based on shared O-RUs,” O-RAN Next Generation Research Group Research Report, Oct. 2023. [Online]. Available: https://mediastorage.o-ran.org/ngrg-rr/ nGRG-RR-2023-05-Spectrum_Sharing_with_Shared_O-RU-v1_0.pdf [5] A. Damnjanovic, D. Knisley, A. Saurabh, R. Prakash, X. Zhang, and S. Chen, “Spectrum sharing with O-RAN architecture,” in Proc. IEEE Int. Symp. Dyn. Spectr. Access Netw. (DySPAN), Washington, DC, USA, May 2024, pp. 108–113. [6] F. Javed, J. Mangues-Bafalluy, E. Zeydan, and L. Blanco, “Trustworthy reputation for federated learning in O-RAN using blockchain and smart contracts,” IEEE Open J. Commun. Soc., vol. 6, pp. 1343–1362, Feb. 2025. [7] L. Giupponi and F. Wilhelmi, “Blockchain-enabled network sharing for O-RAN in 5G and beyond,” IEEE Netw., vol. 36, no. 4, pp. 218–225, Aug. 2022. [8] D. Astely, E. Dahlman, A. Furuskär, Y. Jading, M. Lindström, and S. Parkvall, “LTE: the evolution of mobile broadband,” IEEE Commun. Mag., vol. 47, no. 4, pp. 44–51, Apr. 2009. [9] M. E. Haque, F. Tariq, M. R. A. Khandaker, K.-K. Wong, and Y. Zhang, “A survey of scheduling in 5G URLLC and outlook for emerging 6G systems,” IEEE Access, vol. 11, pp. 34 372–34 396, Apr. 2023. [10] H. Zhou, P. Fan, and J. Li, “Global proportional fair scheduling for networks with multiple base stations,” IEEE Trans. Veh. Technol., vol. 60, no. 4, pp. 1867–1879, Feb. 2011. [11] J. Gu, S. J. Bae, S. F. Hasan, and M. Y. Chung, “Heuristic algorithm for proportional fair scheduling in D2D-cellular systems,” IEEE Trans. Wireless Commun., vol. 15, no. 1, pp. 769–780, Jan. 2016. [12] X. Li, R. Shankaran, M. A. Orgun, G. Fang, and Y. Xu, “Resource allocation for underlay D2D communication with proportional fairness,” IEEE Trans. Veh. Technol., vol. 67, no. 7, pp. 6244–6258, Jul. 2018. [13] M. Zhang, Y. Guo, L. Salaün, C. W. Sung, and C. S. Chen, “Proportional fair scheduling for downlink mmwave multi-user MISO-NOMA systems,” IEEE Trans. Veh. Technol., vol. 71, no. 6, pp. 6308–6321, Jun. 2022. [14] T. Faisal, M. Dohler, S. Mangiante, and D. R. Lopez, “BEAT: Blockchain-enabled accountable and transparent network sharing in 6G,” IEEE Commun. Mag., vol. 60, no. 4, pp. 52–56, 2022. [15] M. Wu, X. Ling, J. Wang, Y. Le, K. Huang, B. Cao, Y. Huang, D. Niyato, Z. Ding, and X. You, “Blockchain-driven resource management in wireless communications and networks: Models, approaches, and applications,” IEEE Commun. Surv. Tutorials, vol. 28, pp. 2306–2344, Aug. 2026. [16] H. Xu, Z. Zhou, L. Zhang, Y. Sun, and C.-L. I, “BE-RAN: Blockchainenabled Open RAN for 6G with DID and privacy-preserving communication,” in Proc. IEEE Global Commun. Conf. Workshops (GLOBECOM Workshops), Cape Town, SA, Dec. 2024, pp. 1015–1021. [17] X. Ling, J. Wang, T. Bouchoucha, B. C. Levy, and Z. Ding, “Blockchain radio access network (B-RAN): Towards decentralized secure radio access paradigm,” IEEE Access, vol. 7, pp. 9714–9723, Jan. 2019. [18] X. Ling, Y. Le, J. Wang, Y. Huang, and X. You, “Trust and trustworthiness in information and communications technologies,” IEEE Wireless Commun., vol. 32, no. 2, pp. 84–92, Apr. 2025.
[19] B. Chen, X. Ling, W. Cao, J. Wang, and Z. Ding, “Analysis of channel uncertainty in trusted wireless services via repeated interactions,” IEEE J. Sel. Areas Commun., vol. 43, no. 6, pp. 2248–2265, Jun. 2025. [20] Z. Wang, B. Cao, C. Liu, C. Xu, and L. Zhang, “Blockchain-based fog radio access networks: Architecture, key technologies, and challenges,” Digital Commun. Networks, vol. 8, no. 5, pp. 720–726, Oct. 2022. [21] X. Ling, J. Wang, Y. Le, Z. Ding, and X. Gao, “Blockchain radio access network beyond 5G,” IEEE Wireless Commun., vol. 27, no. 6, pp. 160– 168, Dec. 2020. [22] W. Cao, X. Ling, J. Wang, Z. Ding, and X. Gao, “A framework for QoSguaranteed fast access services in blockchain radio access network,” IEEE Trans. Wireless Commun., vol. 23, no. 4, pp. 2711 – 2725, Apr. 2024. [23] X. Ling, Y. Le, S. Chen, J. Wang, and X. Zhou, “Blockchain-enabled decentralized services and networks: Assessing roles and impacts,” IEEE J. Sel. Areas Commun., vol. 43, no. 6, pp. 2141–2154, Jun. 2025. [24] G. Femenias, M. Francisca Hinarejos, F. Riera-Palou, J.-L. FerrerGomila, and A. Jaume-Barceló, “Dynamic spectrum sharing in a blockchain enabled network with multiple cell-free massive MIMO virtual operators,” IEEE Access, vol. 12, pp. 70 615–70 633, May 2024. [25] A. Al-Khatib, H. Hadi, H. Timinger, and K. Moessner, “Blockchainempowered resource trading for optimizing bandwidth reservation in vehicular networks,” IEEE Access, vol. 12, pp. 90 084–90 098, Jun. 2024. [26] L. Qian, C. Liu, and J. Zhao, “User connection and resource allocation optimization in blockchain empowered metaverse over 6G wireless communications,” IEEE Trans. Wireless Commun., vol. 24, no. 1, pp. 19–34, 2025. [27] H. Zhang, S. Leng, and H. Chai, “A blockchain enhanced dynamic spectrum sharing model based on proof-of-strategy,” in Proc. IEEE Int. Conf. Commun. (ICC), Dublin, IE, Jul. 2020, pp. 1–6. [28] Z. Yu, Z. Chang, T. Mikkonen, V. Frascolla, and S. Mumtaz, “Blockchain cooperative spectrum management for Wi-Fi and LTE-unlicensed coexistence networks,” IEEE Trans. Commun., vol. 74, pp. 6409–6425, Mar. 2026. [29] H. Zhang, S. Leng, Y. Wei, and J. He, “A blockchain enhanced coexistence of heterogeneous networks on unlicensed spectrum,” IEEE Trans. Veh. Technol., vol. 71, no. 7, pp. 7613–7624, Apr. 2022. [30] R. Zhu, H. Liu, L. Liu, X. Liu, W. Hu, and B. Yuan, “A blockchainbased two-stage secure spectrum intelligent sensing and sharing auction mechanism,” IEEE Trans. Ind. Inf., vol. 18, no. 4, pp. 2773–2783, Aug. 2022. [31] D. Ayepah-Mensah, G. Sun, G. O. Boateng, S. Anokye, and G. Liu, “Blockchain-enabled federated learning-based resource allocation and trading for network slicing in 5G,” IEEE/ACM Trans. Networking, vol. 32, no. 1, pp. 654–669, Feb. 2024. [32] T. Rappaport, Wireless Communications: Principles and Practice, 2nd ed. USA: Prentice Hall PTR, Dec. 2001. [33] H. Kushner and P. Whiting, “Convergence of proportional-fair sharing algorithms under general conditions,” IEEE Trans. Wireless Commun., vol. 3, no. 4, pp. 1250–1259, Jul. 2004. [34] E. Liu, Q. Zhang, and K. K. Leung, “Asymptotic analysis of proportionally fair scheduling in Rayleigh fading,” IEEE Trans. Wireless Commun., vol. 10, no. 6, pp. 1764–1775, Jun. 2011. [35] S. C. Liew and Y. J. Zhang, “Proportional fairness in multi-channel multi-rate wireless networks-part I: The case of deterministic channels with application to AP association problem in large-scale WLAN,” IEEE Trans. Wireless Commun., vol. 7, no. 9, pp. 3446–3456, Sep. 2008. [36] H. Ma, J. Cheng, and X. Wang, “Proportional fair secrecy beamforming for MISO heterogeneous cellular networks with wireless information and power transfer,” IEEE Trans. Commun., vol. 67, no. 8, pp. 5659–5673, Aug. 2019. [37] F. Kelly, “Charging and rate control for elastic traffic,” Eur. Trans. Telecommun., vol. 8, no. 1, pp. 33–37, Jan. 1997. [38] H. Luo, X. Yang, H. Yu, G. Sun, B. Lei, and M. Guizani, “Performance analysis and comparison of nonideal wireless PBFT and RAFT consensus networks in 6G communications,” IEEE Internet Things J., vol. 11, no. 6, pp. 9752–9765, Mar. 2024. [39] Kaspa. [Online]. Available: https://kaspa.org/ [40] A. Baliga, I. Subhod, P. Kamat, and S. Chatterjee, “Performance evaluation of the quorum blockchain platform,” arXiv:1809.03421, Oct. 2018. [Online]. Available: https://arxiv.org/abs/1809.03421 [41] MegaEth. [Online]. Available: https://www.megaeth.com/ [42] H. Jung, T. â. Kwon, K. Cho, and Y. Choi, “REACT: Rate adaptation using coherence time in 802.11 WLANs,” Comput. Commun., vol. 34, no. 11, pp. 1316–1327, Jul. 2011.