Pricing-Driven Service Caching and Task Offloading in Mobile Edge Computing

I. INTRODUCTION

• The BS jointly chooses cached programs and prices under storage and computation constraints, while WDs make coupled offloading decisions with private information unknown to one another.The interaction is modeled as a two-stage dynamic game of incomplete information; the BS maximizes expected profit in Stage I, and WDs minimize their costs in Stage II.Source: §I · pp. 3–4
• At Bayesian equilibrium, each WD follows a threshold-based offloading policy, and the BS jointly optimizes caching and prices using uniform or differentiated pricing.The threshold depends on program prices, BS CPU frequency, and statistical characteristics of WDs’ private information; differentiated pricing charges the same price for programs with the same computational workload.Source: §I · p. 4

A. Related Work

• Prior work studied service caching and decentralized MEC pricing separately, while this paper jointly coordinates caching, pricing, and offloading under incomplete information.The cited Stackelberg and pricing studies assumed programs were cached and WD private information was known to all.Source: A. Related Work · pp. 5–6

II. SYSTEM MODEL

• The system comprises M single-antenna WDs and a single-antenna BS with limited computation and storage, and each WD chooses local execution or offloading if its required program is cached.Each WD has one task requiring one of N service programs; a WD whose program is not cached must compute locally.Source: §II · pp. 6–7

A. BS’ Service Caching Model

• The BS selectively caches service programs because its storage is limited, and caching decisions account for program size and acquisition cost.A cached program is required for the BS to serve tasks that use it.Source: A. BS’ Service Caching Model · p. 7

B. WDs’ Communication Model with BS

• WDs use orthogonal uplink channels of equal bandwidth, and offloading transmission time depends on input-data size and the wireless link.Tasks using the same program may have different input and output data; downloading computation results is assumed negligible relative to uploading inputs.Source: B. WDs’ Communication Model with BS · pp. 7–8

C. Computation Model at the BS and the WDs

• The edge server divides its total CPU frequency equally among parallel VMs, assigning one VM to each offloaded task.The model assumes each VM executes one task.Source: C. Computation Model at the BS and the… · p. 8
• The number of offloaded tasks is random because it depends on WDs’ decisions, and edge-processing time increases with that number.The number of offloaded tasks is defined by the offloading decisions.Source: C. Computation Model at the BS and the… · p. 9

III. TWO-STAGE DYNAMIC GAME FORMULATION UNDER INCOMPLETE INFORMATION FOR MEC SERVICE PROVISION

• The BS leads by choosing service caching and program prices in Stage I; WDs follow by making individual offloading decisions in Stage II.WDs’ choices reflect announced prices and competition for the BS’s limited computation resource.Source: §III · p. 9, Figure 2 · p. 9
• The authors analyze the BS–WD interaction by backward induction.Source: §III · p. 9

A. WDs’ Problem Formulation in Stage II

• Each WD minimizes total cost, defined as task execution delay plus payment to the BS for offloading.The BS charges for program task execution per CPU cycle.Source: A. WDs’ Problem Formulation in Stage II · pp. 9–10
• Because a WD cannot observe others’ private task information or offloading choices, it does not know the total number of offloaded tasks and must reason under incomplete information.The paper derives the Bayesian equilibrium to characterize each WD’s best response.Source: A. WDs’ Problem Formulation in Stage II · p. 10

B. BS’ Problem Formulation in Stage I

• The BS maximizes expected profit: WD payments minus program-acquisition costs, serving type-j tasks only when program j is cached.The expected profit accounts for program-type distributions and WDs’ Stage II offloading decisions.Source: B. BS’ Problem Formulation in Stage I · pp. 10–11

IV. ANALYSIS OF THE WDS’ OFFLOADING EQUILIBRIUM IN STAGE II

• With other users’ decisions known, a WD is more willing to offload when local execution takes longer, upload delay is shorter, or fewer WDs compete for edge computation.In this complete-information case, offloading is preferred when payment is below the difference between local execution time and edge-computing time.Source: §IV · p. 12
• Under incomplete information, WDs’ private θ_i values are i.i.d. with a commonly known regular distribution, and program popularity q_j is estimated from historical data.WDs also do not know other users’ task types, making their required offloading prices unknown.Source: §IV · p. 12
• At Bayesian equilibrium, a WD offloads if and only if its private term θ_i meets the threshold π_ϕi + δ*(π), where δ*(π) is common to all WDs.The policy compresses private information into a comparison with the program price plus the equilibrium decision parameter.Source: §IV · pp. 12–13
• The equilibrium parameter δ*(π) is unique and can be found by bisection; each WD compares its private term with its own program price plus δ*(π).The analysis establishes uniqueness through monotonicity and proposes a search over the parameter’s interval.Source: §IV · p. 13
• A WD’s decision threshold rises with its own program’s price and falls when other programs’ prices rise, so the BS can encourage type-j offloading by lowering price j or raising other prices.Source: §IV · pp. 13–14
• The decision threshold decreases as edge CPU frequency f_c increases and increases as the number of users M grows, making offloading more likely with greater capacity and less likely with more competition.More users compete for limited computation resources, raising the risk of longer edge-computing latency.Source: §IV · p. 14

V. UNIFORM PRICING TO COPE WITH MEC SERVICE CACHING

• The BS sets one common unit price for all programs while optimizing prices and binary caching decisions in Stage I.Identical pricing simplifies the analysis of Problem (P1).Source: §V · p. 14

• Under uniform pricing, each program’s tasks are offloaded with probability 1−F(δ*(π)+π), and price optimization decouples from caching by maximizing the price-revenue term.Source: A. BS’ Profit Maximization in Problem… · pp. 14–15
• The optimal uniform price π* is the unique solution to the pricing condition and can be found by bisection over a feasible range [0, φ].The proof establishes uniqueness by showing the pricing function is increasing and has a root; φ is chosen sufficiently large.Source: A. BS’ Profit Maximization in Problem… · p. 15
• After finding the optimal uniform price, the remaining caching optimization is a standard knapsack problem solvable in pseudo-polynomial time.The paper gives MATLAB’s kp01(·) package as an example toolbox.Source: A. BS’ Profit Maximization in Problem… · p. 16

B. Uniform Distribution of θi in (12)

• Under θ_i uniformly distributed on [θ, θ̄], Proposition 5.2 gives the optimal uniform price; the analysis derives it using a closed-form expression for δ(π).The assumption θ̄ > 1/f_c avoids the case in which no WDs offload.Source: B. Uniform Distribution of θi in (12) · pp. 16–17
• The BS’s optimal uniform price increases with computation capacity f_c, because greater capacity raises WDs’ willingness to offload.The BS can therefore charge more to obtain higher profit.Source: B. Uniform Distribution of θi in (12) · p. 17
• The optimal uniform price also increases with the upper bound θ̄, since a larger average θ_i raises WDs’ offloading probability.A larger average θ_i reflects a greater difference between local execution time and the relevant offloading quantity, allowing the BS to increase price for higher profit.Source: B. Uniform Distribution of θi in (12) · p. 17
• In the uniform-pricing scheme, the BS first caches the optimal program set, then announces one price for all programs.The section notes that program differences could motivate differentiated prices, which it examines next.Source: B. Uniform Distribution of θi in (12) · p. 17

A. BS’ Profit Maximization in Problem (P1)

• At the Stage II Bayesian equilibrium, type-j task offloading probability is 1 − F(δ∗(π) + πj), which informs the BS’s Stage I profit maximization.Source: A. BS’ Profit Maximization in Problem… · p. 17
• Prices and caching are coupled in Problem (P3), so the BS alternates price optimization for a feasible cache set with cache selection to maximize expected profit.For a moderate number of programs, it can enumerate feasible cache decisions; for larger problems, it can use meta-heuristics such as Gibbs sampling or particle swarm optimization.Source: A. BS’ Profit Maximization in Problem… · pp. 18–19
• When cached programs have equal workloads, the BS sets the same optimal price per CPU cycle for each program.Source: A. BS’ Profit Maximization in Problem… · p. 20
• With very large BS computational capacity, optimal prices depend only on the WDs’ valuation PDF and CDF, while communication delay can still deter offloading.Source: A. BS’ Profit Maximization in Problem… · p. 20
• Under uniform valuations, when two cached programs have different workloads, the program with the larger workload has a lower unit price.The section explains that a lower unit price can encourage more workload offloading from that program.Source: A. BS’ Profit Maximization in Problem… · p. 21
• Under uniform valuations, a program’s optimal price rises with its popularity if it has the larger workload; if it has the smaller workload, the direction depends on the larger-workload program’s popularity.When the other program is not popular, the smaller-workload program’s price falls as its popularity rises; otherwise, its price rises.Source: A. BS’ Profit Maximization in Problem… · pp. 21–22

VII. SIMULATION RESULTS

• The simulations set each program’s acquisition cost to r_j = 1, assume equal program sizes c_j, and draw θ_i uniformly from −10 × 10^-8 to 10 × 10^-8.With equal program sizes, the BS’s caching space C directly determines the number of cached programs.Source: §VII · p. 22

• Type-1 offloading probability falls as π1 rises but rises as π2 rises, while type-2 offloading probability moves oppositely because higher π2 frees computation capacity.Source: A. WDs’ Task Offloading Behaviors in… · p. 22, Figure 3 · p. 23
• When π2 = 2 × 10−8, type-2 offloading probability becomes fixed when π1 is high (e.g., above 4 × 10−8), because type-1 offloading falls to zero.Source: A. WDs’ Task Offloading Behaviors in… · p. 22, Figure 3 · p. 23
• With equal prices of 5 × 10−8 and popularity {qj} = [0.5, 0.5], the task types have equal offloading probabilities, which rise with fc and fall as M increases.Source: Figure 4 · p. 23, A. WDs’ Task Offloading Behaviors in… · p. 23

B. BS’ Service Caching and Pricing Strategies in Stage I

• With equal workloads, the differentiated scheme charges the same price as uniform pricing when both programs are cached; below popularity 0.1 for program 1, it is not cached and is priced to prevent its offloading.The reported price in the latter case is π1 = 10 × 10−8.Source: B. BS’ Service Caching and Pricing… · p. 23, Figure 5 · p. 24
• With workloads [200, 300] Mcycles, the BS always caches program 2; when q1 exceeds 0.5, it caches both, prices lower-workload program 1 higher, and lowers both prices as q1 increases.The passage attributes caching program 2 to its greater total revenue per type-2 task.Source: B. BS’ Service Caching and Pricing… · p. 24, Figure 5 · p. 24
• As edge computation capability rises, the uniform price and the differentiated price for the larger-workload program rise, while the lower-workload program’s price falls; at sufficiently high capability, differentiated prices converge.The uniform price lies between the differentiated prices.Source: Figure 6 · p. 24, B. BS’ Service Caching and Pricing… · p. 25

C. Performance Comparison Between Uniform and Differentiated Pricing Algorithms

• Expected BS profit increases as edge-server CPU frequency fc or WD count M rises under both uniform and differentiated pricing; uniform pricing achieves most differentiated-pricing profit, though the gap widens with M.Source: C. Performance Comparison Between… · p. 25

VIII. CONCLUSIONS

• The proposed two-stage incomplete-information game models a BS optimizing caching and program prices under storage and computation limits, followed by WDs choosing offloading decisions for their own costs.In Stage II, WDs estimate other WDs’ decisions.Source: §VIII · pp. 25–26
• The authors derive a threshold-based Bayesian-equilibrium offloading strategy and use predicted equilibrium decisions to develop uniform and differentiated BS pricing algorithms; simulations validate the analysis and mechanism’s effectiveness.Source: §VIII · p. 26

APPENDIX A PROOF OF PROPOSITION 4.1

• The proof of Proposition 4.1 starts from Φ(πj, δ∗(πj)) = 0 and applies the implicit function theorem to derive a comparative-static expression.The excerpt then turns to the relation between δ∗+πj and another program's price πk, but does not show the direction of that relation.Source: Appendix A · p. 26
• The Proposition 4.1 proof also examines how δ∗+πj relates to πk for k ∈ N \ j, without the excerpt specifying the result.Source: Appendix A · p. 26

APPENDIX B PROOF OF PROPOSITION 4.2

• For Proposition 4.2, implicit differentiation shows that δ∗ and δ∗+πj decrease as computation cost fc increases, for a given price πj.Source: Appendix B · p. 26
• For Proposition 4.2, δ∗ and δ∗+πj increase with the number of devices M, for a given price πj.Source: Appendix B · p. 27

APPENDIX C PROOF OF PROPOSITION 6.2

• Because ωeq(π) is increasing in π, with ωeq(0) < 0 and ωeq(π) → +∞ as π → +∞, there exists a unique π∗ > 0 satisfying ωeq(π∗) = 0.Source: Appendix C · p. 27
• The optimality argument substitutes (28) into ωd(πj) for each cached program j and then uses equal prices and workloads; the supplied excerpt cuts off at “we have L1π∗”.Source: Appendix C · p. 27

APPENDIX D PROOF OF PROPOSITION 6.3

• The common equilibrium parameter δ* is expressed in terms of the two programs’ popularities, prices, workload bounds, and number of devices.The expression is given as [(M −1)(qj + qk) + 1](¯θ −θ) −(M −1)[qj(πj −θ) + qk(πk −θ)].Source: Appendix D · p. 28
• The optimal prices for programs j and k are specified by coupled expressions, each depending on the other program’s price and both programs’ parameters.These are equations (30) and (31), respectively.Source: Appendix D · p. 28
• Combining the two price equations yields closed-form expressions for the optimal prices and a further workload-dependent relation.The derivation is given in equations (32) and (33), followed by an expression involving the workload difference.Source: Appendix D · p. 28

APPENDIX E PROOF OF PROPOSITION 6.4

• Appendix E differentiates the optimal price expression with respect to program j’s popularity qj to analyze its response to popularity.The derivative is calculated from equation (32).Source: Appendix E · p. 28
• When Lj > Lk, the derivative of program j’s price with respect to qj is positive for every qk > 0.Source: Appendix E · p. 28
• When Lj < Lk, the analysis considers the condition 0 < qk < 2fc(¯θ−θ)Lj (Lk−Lj)(M−1).The supplied passage ends before stating the derivative’s sign under this condition.Source: Appendix E · p. 28