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Pricing and Resource Allocation via Game Theory for a Small-Cell Video Caching System
Jun Li, He Chen, Youjia Chen, Zihuai Lin, Branka Vucetic, Lajos Hanzo
TL;DR
The paper tackles commercial pricing and resource allocation for small-cell video caching, where repeated on-demand requests create cellular traffic and back-haul redundancy. It models direct SBS delivery with stochastic geometry and optimizes NSP–VR interactions using a Stackelberg game. Analytical probabilities closely match Monte-Carlo simulations, while the framework evaluates pricing and SBS allocation under different storage and popularity conditions.
Problem
Commercial small-cell caching requires joint treatment of video placement, SBS rental pricing, and resource allocation among NSPs and VRs.
Method
The paper models MUs and SBSs as independent PPPs, derives direct-download probabilities, and formulates a Stackelberg game over SBS pricing and allocation.
Results
The analytical results closely match Monte-Carlo simulations, and the framework quantifies pricing and SBS resource allocation.
Takeaways & Limitations
Treating SBS storage as a game-theoretic resource links leasing prices, VR allocations, storage size, and video popularity within one profit model.
Abstract
from arXiv · showhide
Evidence indicates that downloading on-demand videos accounts for a dramatic increase in data traffic over cellular networks. Caching popular videos in the storage of small-cell base stations (SBS), namely, small-cell caching, is an efficient technology for reducing the transmission latency whilst mitigating the redundant transmissions of popular videos over back-haul channels. In this paper, we consider a commercialized small-cell caching system consisting of a network service provider (NSP), several video retailers (VR), and mobile users (MU). The NSP leases its SBSs to the VRs for the purpose of making profits, and the VRs, after storing popular videos in the rented SBSs, can provide faster local video transmissions to the MUs, thereby gaining more profits. We conceive this system within the framework of Stackelberg game by treating the SBSs as a specific type of resources. We first model the MUs and SBSs as two independent Poisson point processes, and develop, via stochastic geometry theory, the probability of the specific event that an MU obtains the video of its choice directly from the memory of an SBS. Then, based on the probability derived, we formulate a Stackelberg game to jointly maximize the average profit of both the NSP and the VRs. Also, we investigate the Stackelberg equilibrium by solving a non-convex optimization problem. With the aid of this game theoretic framework, we shed light on the relationship between four important factors: the optimal pricing of leasing an SBS, the SBSs allocation among the VRs, the storage size of the SBSs, and the popularity distribution of the VRs. Monte-Carlo simulations show that our stochastic geometry-based analytical results closely match the empirical ones. Numerical results are also provided for quantifying the proposed game-theoretic framework by showing its efficiency on pricing and resource allocation.
I. INTRODUCTION
The paper frames small-cell caching as a commercial resource-allocation problem: an NSP leases SBS storage to VRs, who cache popular videos for faster local delivery to MUs. It combines stochastic-geometry modeling with a Stackelberg game to study pricing, allocation, storage, and popularity.
- Motivation: Small-cell caching places popular videos closer to MUs, reducing transmission latency, redundant back-haul traffic, and macro-cell video traffic.These benefits motivate caching at SBSs in heterogeneous networks.
- Research problem: The paper addresses pricing and storage-rental decisions because existing wireless-caching research mainly emphasizes data placement and downloading delay.The commercial setting introduces revenues for VRs and leasing opportunities for NSPs.
- System model: The commercial system includes an NSP, multiple VRs, and MUs, with the NSP leasing SBSs and VRs using them to cache popular videos.The system focuses on negotiation over SBS rental and subsequent video delivery.
- Game formulation: A Stackelberg game treats the NSP as leader setting SBS prices and VRs as competing followers selecting rented SBS fractions.The resulting framework formulates joint profit maximization and investigates a non-convex equilibrium problem.
- Analytical model: MUs and SBSs are modeled as independent PPPs, and stochastic geometry quantifies the probability that an MU obtains a requested video directly from an SBS.The model also represents video popularity through request probabilities and groups files according to SBS storage capacity.
C. Video Placement and Download
The placement and delivery model assigns SBS storage to video groups and uses stochastic geometry to derive direct-download probabilities and profit components. These probabilities then support the NSP’s leasing and back-haul savings model and the VRs’ surcharge and rental-cost models.
- Video Placement and Download: Each VR receives a fraction of uniformly distributed SBSs, modeled as a thinned HPPP with intensity τvλ.The fraction vector τ specifies the SBS allocation among VRs.
- Video Placement and Download: Each SBS caches one of F video groups, while an MU requesting a group searches the nearest covering SBS allocated to the relevant VR.The file groups contain Q videos, matching the SBS storage capacity.
- Profit model: The NSP profit combines SBS leasing income with reduced back-haul costs, while each VR’s profit combines local-downloading surcharge revenue with SBS rental costs.All profits are expressed per unit area and unit period.
- Direct downloading probability: Theorem 1 derives the probability that an MU obtains a requested video directly from an SBS memory.This event probability is the basis for the subsequent profit calculations.
- Direct downloading probability: The direct-download probability is independent of transmit power and SBS intensity, and increasing storage size Q increases the probability.The reported interpretation follows from the relationship between Q and the number of video groups F.
IV. PROBLEM FORMULATION
The paper formulates commercial small-cell caching as a Stackelberg game in which the NSP prices SBS leasing and VRs choose rented fractions. Stochastic-geometry success probabilities support profit models whose equilibrium is obtained by solving follower and leader optimization problems.
- A. Stackelberg Game Formulation: The system models the NSP as leader and VRs as competing followers, with the NSP setting SBS prices before VRs choose rental fractions.
- 1) Optimization Formulation of the Leader:: The NSP maximizes profit by selecting a price vector because each VR’s rented SBS fraction depends on its charged price.
- 2) Optimization Formulation of the Followers:: Each VR optimizes its rented SBS fraction by balancing additional surcharge revenue against the cost of renting more SBSs.
- B. Stackelberg Equilibrium: The Stackelberg equilibrium combines leader and follower optima so that neither the NSP nor the VRs benefit from deviating.
- B. Stackelberg Equilibrium: The equilibrium is found by solving the VR non-cooperative subgame for best responses, then solving the NSP’s pricing problem using those responses.
- V. GAME THEORETIC OPTIMIZATION: For a fixed price, each VR’s optimization is concave in its rental fraction and therefore admits a KKT-based optimal solution.
- V. GAME THEORETIC OPTIMIZATION: A VR opts out when its price reaches the stated threshold, while the NSP’s resulting pricing problem is non-convex because participation indicators are binary.
A. Special Case: ξv = 1, ∀v
The special case assumes every VR participates, derives pricing under that condition, and identifies storage thresholds governing whether the assumption is valid. As storage shrinks, competition can exclude less popular VRs.
- A. Special Case: ξv = 1, ∀v: The all-participating case formulates the NSP’s pricing problem under ξv = 1 for every VR and derives its optimal solution.
- A. Special Case: ξv = 1, ∀v: The derived pricing solution is valid under a storage-size constraint that is necessary and sufficient for all VRs to participate.
- A. Special Case: ξv = 1, ∀v: The threshold ensuring universal participation increases exponentially with γ/3, reflecting the effect of increasingly uneven VR popularity.
- A. Special Case: ξv = 1, ∀v: Because storage satisfies Q ≤ N, sufficiently high popularity concentration can force some least-popular VRs out of the game.
- A. Special Case: ξv = 1, ∀v: The quantities Uv increase strictly with VR index and partition storage regimes according to how many of the most popular VRs can remain active.
- A. Special Case: ξv = 1, ∀v: When Uv < Q ≤ Uv+1, the NSP can retain at most the v most popular VRs in an optimal solution.
- A. Special Case: ξv = 1, ∀v: For each retained-VR case, the paper derives a corresponding candidate price vector under the stated storage regime.
C. General Case
The general solution handles uncertain participation by partitioning storage regimes and comparing candidate solutions for different numbers of active VRs. The resulting optimal prices are piece-wise in storage size and, with follower responses, form the Stackelberg equilibrium.
- C. General Case: The general problem treats storage regimes Uv < Q ≤ Uv+1 separately, allowing only the corresponding number of most popular VRs to participate.
- C. General Case: For each possible number of active VRs, the NSP solves the corresponding pricing problem and selects the best candidate among those solutions.
- C. General Case: The optimal price vector is a piece-wise function of SBS storage size Q.
- C. General Case: The selected price vector uses the candidate formula indexed by the number of participating VRs.
- C. General Case: The NSP’s centralized algorithm obtains the optimal price vector, which together with follower rental responses constitutes the Stackelberg equilibrium.
- C. General Case: The equilibrium NSP profit increases exponentially with the VR popularity parameter γ.
VI. DISCUSSIONS OF OTHER SCHEMES
The paper compares non-uniform pricing with uniform pricing and global optimization schemes.
- VI. DISCUSSIONS OF OTHER SCHEMES: The discussion considers uniform pricing and global optimization as two alternatives to the non-uniform pricing scheme.
A. Uniform Pricing Scheme
The uniform pricing scheme assigns the same SBS rental price to every VR and is optimized through the associated profit and storage constraints. Compared with non-uniform pricing, it requires more storage to retain all VRs but maximizes back-haul cost reduction.
- A. Uniform Pricing Scheme: Uniform pricing imposes one common rental price on all VRs and is analyzed through a corresponding optimization problem.
- A. Uniform Pricing Scheme: The storage required to accommodate every VR is larger under uniform pricing than under non-uniform pricing.
- A. Uniform Pricing Scheme: The minimum storage requirement under uniform pricing increases exponentially with γ/2.
- A. Uniform Pricing Scheme: Uniform pricing is inferior for maximizing NSP profit but optimal for maximizing back-haul cost reduction when follower allocations are optimal.
B. Global Optimization Scheme
The paper compares uniform and non-uniform pricing through global profit optimization and validates its direct-downloading analysis with Monte-Carlo simulations. The uniform scheme coincides with the global solution for maximizing total profit and back-haul cost reduction, while the analytical probability closely matches simulations.
- B. Global Optimization Scheme: The global optimization maximizes the combined profit of the NSP and VRs, with SBS allocation as the only optimization variable.
- B. Global Optimization Scheme: Uniform pricing and global optimization yield the same SBS allocation, maximizing both total profit SGLB and back-haul cost reduction SBH.
- B. Global Optimization Scheme: The direct-downloading probability simulations closely match the analytical results across storage sizes Q = 10, 50, 100, 500 and SBS intensities λ = 10, 20, 30.
- B. Global Optimization Scheme: Increasing storage size raises the probability of direct downloading, while SBS intensity does not affect that probability in the evaluated model.
B. Impact of the VR Preference Parameter γ
The preference parameter γ controls popularity unevenness, participation, revenues, and the storage needed to retain all VRs. Non-uniform pricing retains more participants and yields higher NSP profit, whereas uniform pricing yields higher combined profit and back-haul savings.
- B. Impact of the VR Preference Parameter γ: Uniform pricing requires more storage than non-uniform pricing to retain all VRs, with the gap widening as γ increases.
- B. Impact of the VR Preference Parameter γ: For γ > 0.66 under uniform pricing and γ > 0.98 under non-uniform pricing, the storage requirement exceeds N and unpopular VRs are excluded.
- B. Impact of the VR Preference Parameter γ: As γ increases, both schemes retain fewer VR participants, while non-uniform pricing consistently retains more VRs than uniform pricing.
- B. Impact of the VR Preference Parameter γ: Both schemes’ revenues increase exponentially with γ; non-uniform pricing maximizes NSP profit, whereas uniform pricing maximizes SGLB and SBH.
- C. Impact of the Storage Size Q: Increasing Q enables more VRs to participate, with non-uniform pricing accommodating more VRs for a given storage size.
- C. Impact of the Storage Size Q: For γ = 1, both schemes’ revenues increase with Q; non-uniform pricing has higher NSP profit, while uniform pricing has higher SGLB and SBH.
APPENDIX A PROOF OF THEOREM 1
The appendix derives the probability that a typical MU downloads a requested file from the nearest SBS caching the relevant VR’s file group. It combines nearest-SBS distance, interference, and channel-success calculations under thinned Poisson processes.
- APPENDIX A PROOF OF THEOREM 1: SBSs serving a VR and file group are modeled as a thinned homogeneous PPP, which determines the nearest-SBS distance distribution.
- APPENDIX A PROOF OF THEOREM 1: The event probability concerns a typical MU connecting to the nearest SBS that caches the requested file group for a given VR.
- APPENDIX A PROOF OF THEOREM 1: The resulting probability averages the SINR success condition over the nearest-SBS distance and interference under the interference-dominant assumption.
- APPENDIX A PROOF OF THEOREM 1: The derivation separates interference from SBSs outside and inside the relevant thinned process and multiplies the corresponding expectations.
APPENDIX B PROOF OF LEMMA 2
The proof applies KKT conditions to establish the optimal solution of the concave optimization problem for SBS leasing.
- KKT conditions are applied to the objective function and its constraints to characterize the optimal solution.
- The proof uses complementary-slackness relations and nonzero pricing variables to simplify the KKT system.
- The resulting conditions complete the derivation of the optimal solution for the optimization problem.
APPENDIX C PROOF OF THEOREM 2
The proof establishes the condition under which the proposed solution is both necessary and sufficient, then analyzes which video retailers participate in the game.
- A sufficient condition guarantees that the solution in Eq. (22) is optimal and that ξv = 1 for all video retailers.
- The proof establishes necessity by showing that excluding a video retailer yields an optimal solution contradicting Eq. (22).
- The condition A(δ,α)−C(δ,α)+1 is necessary for the proposed solution in Eq. (22), completing the optimality characterization.
- Because qv1 < qv2 for adjacent retailers, the proof derives Uv1 > Uv2, establishing an ordering of their utilities.
- When participation is limited, retaining the most popular retailers is optimal; adding extra retailers cannot satisfy the required condition in the stated interval.