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SDN-based Resource Allocation in Edge and Cloud Computing Systems: An Evolutionary Stackelberg Differential Game Approach

Jun Du, Chunxiao Jiang, Abderrahim Benslimane, Song Guo, Yong Ren

arXiv:2109.12543v1eess.SY

TL;DR

Growing computation-heavy applications create resource-management challenges for 5G networks, while ECC systems require mechanisms for dynamic access and resource sharing. The paper combines an SDN-based architecture with evolutionary user selection and a Stackelberg differential game for cloud–edge resource trading. It derives dynamic pricing and allocation strategies and reports equilibrium, convergence, and stable user-selection behavior.

  • Problem

    5G networks face increasing computation-heavy applications, requiring ECC resource management that supports low-latency, on-demand services and time-varying user tasks.

  • Method

    The paper combines SDN-based ECC management with evolutionary-game service selection and Stackelberg differential-game pricing and resource sharing.

  • Results

    The derived strategies provide optimal dynamic resource pricing and allocation, while analysis and simulations reveal equilibrium, convergence, and evolutionary stable states.

  • Takeaways & Limitations

    The integrated mechanism supports dynamic coordination of user service selection and cloud–edge resource sharing for time-varying computational tasks.

Abstract

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Recently, the boosting growth of computation-heavy applications raises great challenges for the Fifth Generation (5G) and future wireless networks. As responding, the hybrid edge and cloud computing (ECC) system has been expected as a promising solution to handle the increasing computational applications with low-latency and on-demand services of computation offloading, which requires new computing resource sharing and access control technology paradigms. This work establishes a software-defined networking (SDN) based architecture for edge/cloud computing services in 5G heterogeneous networks (HetNets), which can support efficient and on-demand computing resource management to optimize resource utilization and satisfy the time-varying computational tasks uploaded by user devices. In addition, resulting from the information incompleteness, we design an evolutionary game based service selection for users, which can model the replicator dynamics of service subscription. Based on this dynamic access model, a Stackelberg differential game based cloud computing resource sharing mechanism is proposed to facilitate the resource trading between the cloud computing service provider (CCP) and different edge computing service providers (ECPs). Then we derive the optimal pricing and allocation strategies of cloud computing resource based on the replicator dynamics of users' service selection. These strategies can promise the maximum integral utilities to all computing service providers (CPs), meanwhile the user distribution can reach the evolutionary stable state at this Stackelberg equilibrium. Furthermore, simulation results validate the performance of the designed resource sharing mechanism, and reveal the convergence and equilibrium states of user selection, and computing resource pricing and allocation.

I. INTRODUCTION

The paper addresses growing computation demands in 5G networks by combining edge/cloud computing with SDN-based resource management. It proposes evolutionary and Stackelberg game mechanisms to coordinate user selection, pricing, and resource sharing.

  • Motivation: Computation-heavy applications are increasing sharply in 5G and future wireless networks, creating greater computational requirements.Examples include blockchain mining, interactive gaming, virtual reality, and video services.
  • Motivation: Hybrid edge and cloud computing provides low-latency, on-demand computation offloading, while edge nodes complement distant cloud resources.Cloud nodes may produce long response latency because they are physically far from users; edge computing places resources closer to the network edge.
  • Challenges: Dynamic user selection and bidirectional interactions can complicate resource sharing and cause congestion among computing providers.The system must handle service subscription, task uploading, and service response across cloud and edge providers.
  • Approach: The proposed SDN architecture uses global system information to dynamically adjust resource sharing and computation offloading for time-varying user demand.SDN separates infrastructure and resource management while enabling centralized system-state collection and control.
  • Approach: A hierarchical framework combines evolutionary user selection with Stackelberg differential-game resource sharing between the cloud provider and edge providers.The framework targets dynamic pricing, allocation, and user service requirements across heterogeneous computing providers.
  • Evaluation: The analysis derives equilibrium and evolutionary stable states and uses simulations to examine convergence of user selection, pricing, and allocation.The paper reports higher integral utilities and faster decision convergence than traditional static strategies.

B. Control Plane

The control plane separates resource management from the infrastructure and exchanges subscription, capacity, pricing, and request information with the management plane. This architecture supports dynamic resource decisions and cloud–edge sharing in the ECC system.

  • Architecture: The SDN architecture separates computing resource management from infrastructure through infrastructure, control, and management planes.This separation forms a hierarchical game-based cloud-resource market in the control plane.
  • Information flow: Control-plane request-analysis controllers collect user subscriptions and local computational power from computing providers in real time.The collected information is sent to the management plane for decision-making.
  • Decision flow: The management plane determines dynamic cloud-resource prices and the computational power requested by edge providers.These strategies are returned to the control plane to guide resource sharing between the cloud and edge providers.
  • System model: The system models one remote cloud provider and multiple edge providers serving users through access points, gateways, and base stations.Users can request edge resources for nearby service or cloud resources for highly complex computational tasks.
  • Resource sharing: Edge providers request a time-varying proportion of the cloud provider’s computational power after observing its announced unit price.The allocation state includes each edge provider’s request and the cloud provider’s remaining computational power.
  • Population model: Users are represented by population shares across providers, with edge and cloud shares summing to one.Computational power is treated as computing frequency or speed, measurable as computing times per unit time.

B. Hierarchical Game Framework

The paper formulates coupled service selection and resource trading as a hierarchical dynamic game. Evolutionary dynamics model users’ provider choices, while a Stackelberg differential game models cloud pricing and edge resource requests.

  • Game motivation: User service selection, cloud pricing, and edge resource requests are time-varying and mutually interdependent.When many users select one provider, the computational power received by each user decreases, encouraging movement toward alternatives.
  • Framework: The hierarchical game has a user level for evolutionary service selection and a resource level for Stackelberg pricing and sharing.The two levels jointly represent users’ dynamic choices and provider interactions over limited computational power.
  • User layer: Users select providers according to access prices and received computational power, which varies with provider load and resource decisions.Access prices remain fixed, whereas received computational power changes over time.
  • Resource layer: The cloud provider dynamically sets the resource price, and edge providers dynamically choose resource requests to respond to demand.The resulting non-cooperative Stackelberg differential game lets providers optimize their own utilities.
  • Evolutionary dynamics: Users initially choose providers randomly or by experience and periodically adapt through learning and imitation.Evolutionary game theory is used to model the resulting service-selection dynamics and investigate evolutionary stable states.
  • Evolutionary model: The evolutionary game treats user devices as players whose strategies are selecting an edge provider or the cloud provider.The population distribution state records the shares of users selecting each provider.

4) Utility:

The model uses replicator dynamics to represent time-varying user service selection and establishes existence and uniqueness of the resulting population trajectory under resource-allocation controls.

  • Replicator dynamic: Replicator dynamics model how users switch among computing-service providers over time according to utility differences.The number selecting an ECP increases when its utility exceeds the population’s expected utility, and decreases otherwise.
  • Replicator dynamic: The population state starts from x(0) = x0, while δ > 0 controls the frequency of strategy adaptation.
  • Existence and uniqueness: Under measurable resource-allocation controls, the user population trajectory exists uniquely for all t ∈ [0, ∞).
  • Existence and uniqueness: Continuity, measurability, and a global Lipschitz condition establish global existence and uniqueness of the evolutionary population system.

C. Analysis of Evolutionary Stable State (ESS)

The paper analyzes evolutionary stability in the user-selection dynamics and connects the resulting stable state to Nash equilibrium and the hierarchical pricing-and-allocation game.

  • ESS definition: An ESS resists sufficiently small mutant populations by giving non-mutants higher expected utility than mutants.
  • ESS interpretation: ESS is a refinement of Nash equilibrium because it addresses deviations by a set of players rather than one user.
  • ESS definition: A larger εx indicates that the ESS can resist a larger proportion of users adopting mutant strategies.
  • Stability analysis: For any initial population state, the replicator dynamics are globally asymptotically stable and converge to the game’s ESS.
  • Stackelberg game: The CCP sets the resource price first, ECPs respond with resource requests, and the CCP dynamically learns their expected best responses.

1) Maximization of Integral Utility for ECPs:

The ECP optimization problem balances user-service revenue, cloud-resource charges, and resource-capacity mismatch while tracking the evolving user population.

  • ECP utility: The ECP utility combines economic revenue, charges for CCP computational power, and a penalty for resource-demand mismatch.The mismatch term compares users’ nominal accessible computing rate with available service capacity.
  • ECP optimization: ECPs maximize integral utility by choosing requested computational power subject to the population state generated by the evolutionary game.
  • ECP optimization: The discount rate ρ > 0 determines how future ECP utilities contribute to the integral objective.
  • CCP optimization: The CCP separately optimizes pricing to obtain revenue from subscribed users and ECPs while minimizing performance-discrepancy costs.

B. Open-Loop Stackelberg Equilibrium Solutions

The open-loop Stackelberg solution first optimizes each ECP's resource request under the CCP's price, then characterizes equilibrium through Hamiltonian and Pontryagin conditions. The resulting ECP allocation decreases as the CCP's price increases.

  • Open-loop game formulation: ECPs choose resource-request strategies after observing the CCP's pricing strategy in the open-loop Stackelberg sequence.The CCP then selects pricing based on the ECPs' resource-request responses over the finite horizon [0,T].
  • Equilibrium definition: An open-loop Stackelberg equilibrium consists of optimal CCP pricing and ECP request strategies given the other players' strategies.The strategy profile is defined as Φ∗(t) ≜ {p∗(t), r∗(t)}.
  • ECP optimality conditions: Pontryagin’s Maximum Principle characterizes candidate ECP strategies by requiring them to maximize each ECP’s Hamiltonian.The analysis establishes Hamiltonian systems and costate dynamics for the ECP optimization problems.
  • ECP optimal strategy: The optimal computational power request for each ECP is obtained from the Hamiltonian optimality conditions and the associated costate variables.The derived request uses the population state and an N-dimensional vector q_n(x).
  • ECP pricing response: Optimal computational power requests and allocations for ECPs decrease as the CCP-determined price p(t) increases.This relation follows from the optimal solutions summarized in the ECP lemma.

2) Open-loop Stackelberg Equilibrium of CCP:

The CCP’s open-loop equilibrium is derived with dynamic optimal control after incorporating the ECPs’ optimal responses. The resulting pricing strategy is characterized through a Hamiltonian and is an open-loop Stackelberg equilibrium.

  • CCP optimality conditions: The CCP’s open-loop equilibrium solutions are characterized using Pontryagin’s Maximum Principle for its dynamic pricing problem.The formulation introduces CCP costate functions associated with the population distribution state.
  • CCP Hamiltonian: The CCP Hamiltonian incorporates pricing, population state, ECP resource requests, and CCP costate variables.Its notation includes Λ(t), M(t), and Ψ(t) as state-related and costate quantities.
  • CCP pricing strategy: The optimal CCP pricing strategy is p∗(t) ≜ f_p(x(t), Λ(t), M(t), Ψ(t), t).This strategy is obtained from the CCP Hamiltonian conditions and constitutes an open-loop Stackelberg equilibrium.
  • Leader optimization: The CCP first substitutes the ECPs’ optimal response into its Hamiltonian before deriving the pricing optimality conditions.The resulting Hamiltonian is stated to be concave in p(t), yielding a unique optimal pricing strategy.
  • Solution derivation: The pricing strategy is obtained by taking the first derivative of the CCP Hamiltonian with respect to p(t) and solving the resulting conditions.The derivation also computes the elements of M(t) and Ψ(t).

3) Open-loop Stackelberg Equilibrium Solutions:

The complete open-loop Stackelberg equilibrium combines the optimal CCP price, ECP requests, population state, and costate dynamics. These coupled equations form a two-point boundary value problem whose solution yields the optimal controls.

  • Equilibrium controls: The equilibrium controls are the CCP price p∗(t) and ECP requests r∗_n(t), obtained from the respective optimal strategies.They are represented by the functions f_p and f_r of the state, costates, and time.
  • Coupled dynamics: Substituting the optimal controls into the state and costate equations produces a dynamic control system for x∗(t), Λ∗_n(t), M∗(t), and Ψ∗(t).The resulting system couples the population distribution with all associated costate variables.
  • Equilibrium computation: Solving the resulting two-point boundary value problem provides the optimal controls and the open-loop Stackelberg game equilibrium.The equilibrium is denoted Φ∗(t) ≜ {p∗(t), r∗(t)}.

VI. SIMULATION RESULTS

The simulation evaluates evolutionary service selection and the proposed Stackelberg resource pricing and allocation mechanism in a hybrid edge/cloud system. The setup contains one CCP, multiple ECPs, and 100 randomly distributed user devices.

  • Simulation evaluation: The simulations analyze evolutionary-game service selection and evaluate the proposed pricing and allocation mechanism using MATLAB2019b.The evaluation begins with a scenario setup for the ECC system.
  • Scenario setup: The simulated ECC system includes a single CCP and multiple ECPs that access the CCP’s computing resource.The CPs provide edge and cloud computing services within the ECC system.
  • Scenario setup: K = 100 user devices are randomly distributed within the coverage of the ECC system.These users receive services from the simulated cloud and edge computing providers.

A. Evolution of Population Distribution

User service selections converge to equilibrium under replicator dynamics, while cloud capacity, prices, learning rate, and provider populations shape distribution, resource requests, utilities, and convergence speed.

  • User proportions converge to an equilibrium where no user changes service selection strategy.
  • At Rc = 5kH/s, lowering pc to 0.2 attracts more users directly to the CCP, whereas pc = 0.5 shifts users toward ECPs.At the higher price, the CCP shares all its computing resource with ECPs and derives utility mainly from resource sharing.
  • With N = 6 ECPs, providers with larger user populations request and receive more cloud resource, increasing utilities for users and providers.ECP 5 and ECP 6 receive the largest shares, while ECP 1 and ECP 2 receive the least.
  • Replicator dynamics equalize individual user utilities at equilibrium.
  • Increasing δ accelerates replicator-dynamics convergence, and OLSEC converges faster than the static SSEC scheme.OLSEC incorporates dynamic learning and prediction of CP strategies, whereas SSEC relies only on user selection strategies.

B. Dynamic Pricing and Allocation of Computing Resource

The proposed mechanism studies dynamic cloud-resource pricing and allocation under changing capacity, prices, and delayed population information. It derives an SDN-based ECC resource-management framework with game-theoretic control and establishes convergence and stability conditions.

  • At equilibrium, increasing CCP capacity lowers the optimal cloud-resource price and increases the resource proportion retained by the CCP.
  • Pricing and allocation strategies converge to the Stackelberg equilibrium.
  • Population-information delay τx affects delayed replicator dynamics, with stable evolutionary states guaranteed when τx < π/2Θ.The analysis tests τx = 0.7 and τx = 1.7 under specified system parameters.
  • The SDN-based ECC architecture supports on-demand resource management for time-varying computational tasks in 5G heterogeneous networks.
  • The Stackelberg differential-game mechanism derives pricing and allocation strategies that maximize integral utilities for the CCP and ECPs while user selection reaches an evolutionary stable state.
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