1006 lines
119 KiB
Plaintext
1006 lines
119 KiB
Plaintext
Journal of Systems Architecture 160 (2025) 103360
|
||
|
||
|
||
Contents lists available at ScienceDirect
|
||
|
||
|
||
Journal of Systems Architecture
|
||
journal homepage: www.elsevier.com/locate/sysarc
|
||
|
||
|
||
|
||
|
||
Collaborative optimization of offloading and pricing strategies in dynamic
|
||
MEC system via Stackelberg game
|
||
Jing Mei a , Cuibin Zeng a , Zhao Tong a ,1 ,∗, Longbao Dai a , Keqin Li b ,2
|
||
a College of Information Science and Engineering, Hunan Normal University, Changsha, Hunan, 410081, China
|
||
b
|
||
Department of Computer Science, State University of New York, New Paltz, NY, 12561, USA
|
||
|
||
|
||
|
||
ARTICLE INFO ABSTRACT
|
||
|
||
Keywords: The rapid advancement of 5G technology has indirectly propelled the growth of connected devices within the
|
||
Mobile edge computing Internet of Things (IoT). Within the IoT domain, mobile edge computing (MEC) has demonstrated potential in
|
||
Energy harvesting task processing. However, as computational services expand, the reliable determination of user offloading
|
||
Lyapunov optimization
|
||
strategies and the rational establishment of service prices offered by servers to users continue to present
|
||
Stackelberg game
|
||
challenging research directions. The primary focus of this paper revolves around task offloading in the MEC
|
||
system, encompassing numerous user terminal devices that support energy harvesting (EH), a MEC server and
|
||
a central cloud server. The optimization goals are to maximize the utilities for both users and the MEC server
|
||
by adjusting offloading and pricing strategies. To guarantee the task queue’s stability within the system and
|
||
achieve a reasonable allocation of system resources, we propose a dynamic task offloading approach rooted in
|
||
Lyapunov optimization theory and Stackelberg game theory. In this algorithm, the MEC server takes on the role
|
||
of the leader, while each user terminal device acts as the follower. Aiming at the game equilibrium existence of
|
||
the algorithm, a series of mathematical analysis is carried out. Additionally, we conduct extensive simulation
|
||
experiments to validate the proposed algorithm’s effectiveness. The proposed algorithm achieves improvements
|
||
in user utility, with a 6.43% increase compared to the average time-constrained task offloading (ATCTO)
|
||
scheme, a 61.80% improvement over the local-only processing (LOP) scheme, and a 23.97% enhancement
|
||
over the genetic algorithm (GA) scheme. Meanwhile, it achieves a task queue backlog reduction of 50.00%
|
||
compared to ATCTO, 70.00% compared to LOP and 15.28% compared to GA.
|
||
|
||
|
||
|
||
1. Introduction data privacy and security.
|
||
Energy consumption for local computing and wireless transmission
|
||
With the proliferation of 5G technology, its high speed and low tasks on user terminal devices (e.g., wearable devices, tablets, and
|
||
latency enable real-time connectivity, and the IoT stands as one of smartphones) is sourced from their internal batteries. For convenience,
|
||
the beneficiaries in the era of 5G. According to [1], it is forecasted these user terminal devices will be referred to as ‘‘users’’. Due to limited
|
||
that the number of IoT devices will exceed 75 billions by 2025. This size of users, long lifetime battery might not be appropriate, while
|
||
explosive growth will significantly amplify the scale of mobile data smaller-capacity battery may require frequent replacements, causing
|
||
traffic. Effectively managing the surge in mobile data traffic can be significant inconvenience. Fortunately, in recent years, EH technologies
|
||
achieved through mobile data offloading [2]. However, relying solely have garnered significant attention. These technologies enable users
|
||
on cloud computing is difficult to improve user experience quality. to harvest energy from the nature (e.g., solar), and store it in bat-
|
||
It is the increasing user demand for quality of experience that drives teries. By integrating rechargeable batteries with energy harvesting
|
||
the development of MEC. MEC moves computing power and resources technologies, the frequency of battery replacements can be notably
|
||
closer to users, usually on edge devices of the network. This aids in
|
||
reduced. In the future, the IoT could potentially integrate various
|
||
reducing data transmission latency and supporting applications with
|
||
energy harvesting methods [3]. Therefore, the integration of energy
|
||
strong real-time requirements. Moreover, edge data processing reduces
|
||
harvesting technologies into MEC holds practical significance.
|
||
the need to transmit sensitive data to remote servers, thereby enhancing
|
||
|
||
|
||
∗ Corresponding author.
|
||
E-mail addresses: jingmei1988@163.com (J. Mei), zeng1941183190@gmail.com (C. Zeng), tongzhao@hunnu.edu.cn (Z. Tong), awaken6758@gmail.com
|
||
(L. Dai), lik@newpaltz.edu (K. Li).
|
||
1
|
||
Member, IEEE.
|
||
2
|
||
Fellow, IEEE.
|
||
|
||
https://doi.org/10.1016/j.sysarc.2025.103360
|
||
Received 3 September 2024; Received in revised form 7 January 2025; Accepted 31 January 2025
|
||
Available online 12 February 2025
|
||
1383-7621/© 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
|
||
J. Mei et al. Journal of Systems Architecture 160 (2025) 103360
|
||
|
||
|
||
In order to flexibly respond to the changing demands between user • We account for the practical limitation of the MEC server’s com-
|
||
devices and MEC server, this study considers a dynamic MEC system. puting capacity within each time slot, determined by its CPU
|
||
In this system, factors such as the arrival of tasks and energy collection frequency. If the server cannot process all accumulated tasks from
|
||
will change with the change of time. The system can dynamically adjust users, it requests processing from the central cloud server at a
|
||
the task offloading strategy and the service pricing strategy of the cost.
|
||
MEC server according to real-time conditions, thereby improving user Next, we will introduce the remaining structure of this paper.
|
||
utility and MEC server utility. The authors considered the dynamic MEC Section 2 delves into the related work. Section 3 defines the system
|
||
system in [4], but did not set the services provided by the MEC server model and presents the optimization problems. Section 4 describes
|
||
as paid services. In contrast, [5] explored the pricing issue in a dynamic the analysis and solution of the objective optimization problems. Sub-
|
||
system, but the study implemented a unified service pricing for all user sequently, we introduce a multi-device task offloading and pricing
|
||
devices and failed to consider the differences between user devices. mechanism algorithm (MDTOPMA) in Section 5. Section 6 evaluates
|
||
Given the heterogeneity of user devices, we introduced a differentiated the algorithm’s performance through experiments. The final section
|
||
pricing strategy. concludes by summarizing the contributions of this paper.
|
||
In this paper, we explore a dynamic MEC system enabled with EH,
|
||
which is composed of multiple user terminal devices, a MEC server, and 2. Related work
|
||
a central cloud. Each user possesses the capability to harness energy
|
||
from the surrounding natural environment and store it in their built-in Numerous investigations have delved into the domain of computa-
|
||
tion offloading within the context of MEC. Ning et al. [8] presented
|
||
rechargeable batteries. Users can offload tasks to the MEC server when
|
||
an energy-efficient scheduling framework for vehicle networking with
|
||
needed, and the MEC server can further offload tasks to the central
|
||
support for Multi-Access Edge Computing, aimed at minimizing the
|
||
cloud as required. In this study, the MEC server’s computational capac-
|
||
energy consumption of roadside units (RSUs) while meeting task la-
|
||
ity is limited, necessitating revenue generation through user charges.
|
||
tency requirements. The authors proposed a heuristic algorithm to
|
||
Generally, the energy consumed by users for wireless transmission tasks
|
||
address this issue. Mao et al. [9] focused on the joint optimization
|
||
is lower than that required for performing equivalent-sized local com-
|
||
of execution delay and device energy consumption, proposing a low-
|
||
puting tasks. In order to decrease energy consumption, users are more
|
||
complexity suboptimal algorithm based on an alternating minimization
|
||
likely to opt for the method of remote task offloading. Nevertheless, due strategy. The algorithm minimized the weighted sum of execution
|
||
to the fact that the MEC server does not offer services free of charge, delay and device energy consumption by adjusting task offloading
|
||
each user needs to strike a balance between task offloading and service scheduling and transmission power allocation. Zhao et al. [10] pro-
|
||
pricing. The main purpose of our research is to determine the optimal posed an energy-efficient offloading algorithm to save mobile device
|
||
distribution of tasks between local computing and remote offloading energy while meeting application response time requirements. Chen
|
||
for users, while also determining suitable service pricing for individual et al. [11] investigated energy-efficient task offloading in MEC and
|
||
users for the MEC server. proposed a dynamic offloading algorithm that guarantees the average
|
||
Given that the optimization objective of this paper involves a dy- queue length. Li et al. [12] proposed a computational offloading mech-
|
||
namic long-term MEC scenario, i.e., the parameters and conditions anism based on a two-stage Stackelberg game and used two dynamic
|
||
(e.g., energy harvesting, task arrival) may change over time, mak- iterative algorithms to solve the utility optimization problem in the
|
||
ing the solution of the problem extremely challenging. To simplify game. Although the above studies have their own merits, they do not
|
||
the problem, the authors in [5–7] focused their target optimization consider the computing power of the device itself.
|
||
on a short time slot, defined as a brief period during which sys- To overcome this deficiency, in recent years, many studies have
|
||
tem conditions are assumed to be relatively stable. However, focusing begun to consider introducing local computing resources. Based on
|
||
only on a single time slot may cause system instability problems, the size of the offloaded data, Hu et al. [13] determined the tasks
|
||
since it is difficult to cope with rapid changes. To overcome this that necessitate local handling. They proposed a MEC system energy
|
||
difficulty, we employ Lyapunov optimization theory to transform the consumption optimization problem and solved it in two stages. Wang
|
||
long-term optimization problem into a sequence of short-term optimiza- et al. [14] divided the computing task into two parts, one of which
|
||
tion problems, enabling the achievement of long-term optimization would be used for local computing. Their research intended to minimize
|
||
objectives through the resolution of multiple short-term optimization the AP’s overall energy consumption, however they did not consider
|
||
the energy consumption resulting from the MEC server’s computational
|
||
tasks. To enhance the efficiency and rationality of resource allocation,
|
||
tasks. They omitted for the energy consumption consumed by the MEC
|
||
we introduce Stackelberg game theory. In this framework, the MEC
|
||
server during task processing.
|
||
server acts as a leader, formulating differentiated pricing strategies to
|
||
To more comprehensively address issues such as task offloading and
|
||
guide the offloading decisions of multiple user devices. By combin-
|
||
energy management, and to achieve optimal performance with limited
|
||
ing Lyapunov optimization theory with Stackelberg game theory, we
|
||
resources, game theory is an appropriate approach. There is currently
|
||
can effectively address the complexities in dynamic environments and
|
||
a lot of work taking game theory into MEC. Li et al. [6] described the
|
||
achieve long-term optimization objectives. interaction between mobile device (MD) and edge cloud server (ECS)
|
||
This research work makes the following main contributions: in the process of computing load as a Stackelberg game and confirmed
|
||
the equilibrium of this game. The authors in [5] proposed an optimal
|
||
• We explore task offloading and pricing in a multi-user environ- resource purchase strategy with a set price, and proposed the optimal
|
||
ment with a single MEC server, incorporating energy harvesting pricing for edge cloud computing resources utilizing the Stackelberg
|
||
(EH) for users to utilize renewable energy for task computation. game model. Liu et al. [7] proposed the problem of transmission
|
||
• We utilize Lyapunov optimization theory to adapt to external power offloading optimization and edge cloud pricing in mobile edge
|
||
changes by breaking down long-term goals into short-term objec- computing systems, and adopted the offloading strategy and price
|
||
tives, stabilizing the task queue while optimizing performance. control (OSPC) algorithm based on Stackelberg game to solve it. Bishoyi
|
||
et al. [15] presented a distributed algorithm utilizing the alternating
|
||
• We propose a Stackelberg game model for task offloading and direction multiplier method (ADMM) to solve the Stackelberg game
|
||
pricing, where users consider energy use, queue length, and pric- they proposed. Although Bishoyi et al. considered the Stackelberg
|
||
ing, while the MEC server applies differentiated pricing based on game, they only considered the optimization of the problem within one
|
||
offloaded tasks. time slot. Zeng et al. [16] introduced a reward system to incentivize
|
||
|
||
2
|
||
J. Mei et al. Journal of Systems Architecture 160 (2025) 103360
|
||
|
||
Table 1
|
||
Difference between our scheme and the main related schemes.
|
||
|
||
Scheme Local computation MEC energy consumption Multiple time slots Differentiated pricing
|
||
|
||
[5] S.-H.Kim et al. ✓ × × ×
|
||
[6] M.Li et al. ✓ × × ✓
|
||
[7] X.Liu et al. ✓ × × ✓
|
||
[8] Z.Ning et al. × ✓ × ×
|
||
[9] Y.Mao et al. × × × ×
|
||
[10] X.Zhao et al. × ✓ × ×
|
||
[11] Y.Chen et al. × × ✓ ×
|
||
[12] F.Li et al. × ✓ × ✓
|
||
[13] X.Hu et al. ✓ × × ×
|
||
[14] F.Wang et al. ✓ × ✓ ×
|
||
[15] P.K.Bishoyi et al. ✓ ✓ × ✓
|
||
[16] F.Zeng et al. ✓ × × ✓
|
||
Our Scheme ✓ ✓ ✓ ✓
|
||
|
||
|
||
|
||
volunteer vehicles participating in Vehicular Edge Computing (VEC)
|
||
offloading and devised an enhanced genetic algorithm to explore the
|
||
optimal strategy for the VEC server. However, they also did not account
|
||
for changes in the number of tasks over time.
|
||
Among most of the studies mentioned before, some studies failed to
|
||
fully leverage local computing resources, some did not account for the
|
||
energy consumption associated with the MEC server’s computations,
|
||
and some only focused on the optimization of single slot goals (see
|
||
Table 1). To cope with these situations, each user in this paper supports
|
||
local computing through EH technology, while the MEC server consid-
|
||
ers computing energy consumption as its own overhead. In order to
|
||
achieve a reasonable allocation of resources among users and the MEC
|
||
server and tackle the long-term optimization problem, we combine the
|
||
Stackelberg game with Lyapunov optimization.
|
||
|
||
3. System model and problem formulation
|
||
|
||
In this section, we will provide an overview of the MEC system
|
||
architecture and various computation models. Based on these compu-
|
||
tational models, the optimization problems for both users and the MEC
|
||
server will be deduced. Fig. 1. A mobile edge computing system architecture.
|
||
|
||
|
||
3.1. Mobile edge computing system architecture
|
||
|
||
We investigate a system architecture consisting of three layers. MEC server’s computational capacity is exceeded. We consider the cen-
|
||
The first layer, called the user terminal device layer, consists of 𝑛 tral cloud to possess formidable computational capabilities, enabling it
|
||
user devices (e.g., wearables, tablets, smartphones), indexed by 𝑁 = to handle a significant amount of tasks independently.
|
||
{1, 2, … , 𝑛}. Each user is equipped with an energy harvesting device Various uncertainties and interferences exist in practical application
|
||
that enables them to collect energy from the surrounding environment environments. To enhance system stability, we consider a time-slot-
|
||
to power their own operations. The second layer is known as the MEC based system and partition time into equidistant time slots. The system
|
||
service layer, consisting of a single MEC server co-located with a base is indexed by 𝑡 ∈ = {0, 1, 2, … , 𝑇 } with slot length 𝜏.
|
||
station. Lastly, the third layer is denoted as the central cloud service
|
||
layer, consisting of a cloud server. The three-layer system architecture
|
||
is illustrated in Fig. 1. 3.2. Computing task and task queue model
|
||
In this architecture, users can transmit their computational tasks to
|
||
the base station via a wireless network using the time division multiple For each user 𝑖, at the start of the 𝑡th time slot, the user’s application
|
||
access (TDMA) protocol. The base station forwards the tasks to the MEC requests a set of tasks for computation. The size of the tasks is the task
|
||
server for computation. Once tasks are completed, the MEC server sends arrival rate 𝑎𝑖 (𝑡). The tasks received during the current time slot can
|
||
the results back to the base station, which then delivers them back to only be handled in future time slots. In order to achieve more flexible
|
||
the users. If the cumulative tasks offloaded by users exceed the MEC task offloading, we assume that the computational tasks of the users
|
||
server’s computational capacity, the excess tasks are offloaded to the follow a data partitioning model [4,17], where the data bits of the
|
||
central cloud via a wired network, and the results are subsequently re- computational tasks are independent and can be arbitrarily partitioned
|
||
turned by the central cloud. Excess tasks can be generated by adjusting into multiple independent subtasks. Each user’s tasks are stored in the
|
||
{ }
|
||
the relevant parameters, such as the task arrival rate, and available task queue 𝑸 = 𝑄1 , 𝑄2 , … , 𝑄𝑛 .
|
||
computational resources. By adjusting these parameters, the total of- Let 𝑄𝑖 (𝑡) represent the tasks remaining incomplete for the 𝑖th user
|
||
floading demand can be controlled, simulating the scenario where the in the preceding 𝑡 time slots. The amount of tasks in the queue can be
|
||
|
||
3
|
||
J. Mei et al. Journal of Systems Architecture 160 (2025) 103360
|
||
|
||
|
||
adjusted using the following formula. 3.3.4. Transmission rate model
|
||
𝑄𝑖 (𝑡 + 1) = max{0, 𝑄𝑖 (𝑡) − 𝑞𝑖 (𝑡)} + 𝑎𝑖 (𝑡), (1) In the system architecture, data exchange between the users and
|
||
the MEC server occurs through wireless networks. During this data
|
||
where 𝑞𝑖 (𝑡) denotes the total tasks processed by user 𝑖 during the 𝑡th exchange process, we consider Shannon’s formula as the calculation
|
||
time slot. 𝑞𝑖 (𝑡) can be modeled as 𝑞𝑖 (𝑡) = 𝑞𝑖0 (𝑡) + 𝑞𝑖1 (𝑡), where 𝑞𝑖0 (𝑡) formula for the channel’s data transfer rate. Similar to [11], the trans-
|
||
denotes the quantity of tasks processed locally by user 𝑖, and 𝑞𝑖1 (𝑡) mission rate can be modeled as
|
||
signifies the quantity of tasks offloaded remotely by user 𝑖. 𝑞𝑖 (𝑡) satisfies ( )
|
||
𝑝𝑖 ℎ𝑖
|
||
𝑟𝑡𝑟
|
||
𝑖 = 𝑤 log2 1 + 𝑤𝑛 , (10)
|
||
0 ≤ 𝑞𝑖 (𝑡) ≤ 𝑄𝑖 (𝑡), 𝑡 ∈ . (2) 0
|
||
|
||
where 𝑤 signifies the base station’s bandwidth; 𝑛0 stands for the noise
|
||
According to the definition of queue stability in [18], the constraint power density; and ℎ𝑖 indicates the channel gain.
|
||
for task queue stability is given as According to the transmission time and transmission rate, the of-
|
||
E{𝑄𝑖 (𝑡)} floading task size 𝑞𝑖1 (𝑡) = 𝑇𝑖𝑡𝑟 (𝑡)𝑟𝑡𝑟
|
||
𝑖 can be deduced.
|
||
lim = 0, 𝑡 ∈ . (3)
|
||
𝑡→+∞ 𝑡
|
||
3.4. Mobile edge server processing model
|
||
|
||
3.3. Local execution and communication model
|
||
When users offload tasks to the MEC server, the server receives and
|
||
3.3.1. Local computing energy consumption model processes these tasks, thereby resulting in energy consumption. The
|
||
Local task execution involves users using their available processing energy consumption of the MEC server can be modeled as
|
||
resources to handle tasks, leading to local energy consumption. This ∑
|
||
𝑛
|
||
|
||
local energy consumption is dynamic, as it varies based on offloading 𝐻 𝑒 (𝑡) = 𝑘𝑒 (𝑓 𝑒 )2 𝑐
|
||
𝑞𝑖1 (𝑡), (11)
|
||
𝑖=1
|
||
strategy. Same as [19], we model it as
|
||
where 𝑘𝑒 is the capacitance switching coefficient of the MEC server;
|
||
𝑢
|
||
𝐻𝑖0 (𝑡) = 𝑘𝑢𝑖 [𝑓𝑖𝑢 ]2 𝑞𝑖0 (𝑡)𝑑𝑖 , (4) 𝑐 (𝑡)
|
||
𝑓 𝑒 represents the CPU computing frequency of the MEC server; 𝑞𝑖1
|
||
where 𝑘𝑢𝑖 is the capacitance switching coefficient, which is dependent denotes the cycle count of the offloaded task for user 𝑖 and can be
|
||
on the chip architecture [19], and the superscript 𝑢 is employed to 𝑐 (𝑡) = 𝑞 (𝑡)𝑑 .
|
||
described as 𝑞𝑖1 𝑖1 𝑖
|
||
denote user-specific parameters, distinguishing them from the parame- Considering the limited resources of the MEC server, it may not be
|
||
ters denoted by the same symbol for the MEC server; 𝑓𝑖𝑢 represents the able to fully process all the tasks offloaded by users. Consequently, the
|
||
CPU computing frequency of user 𝑖, while 𝑑𝑖 represents the computing MEC server is required to upload the excess tasks to the central cloud
|
||
density of user 𝑖. for synchronous processing. Let 𝑞 𝑒 represent the processing capacity of
|
||
Given the computation frequency 𝑓𝑖𝑢 and the time slot length 𝜏, the the MEC server within a time slot, i.e., the maximum number of task
|
||
local computing tasks are constrained as
|
||
cycles it can handle. When the amount of task cycles offloaded by users
|
||
𝑓𝑖𝑢 𝜏 exceeds the MEC server’s computational capacity, the excess portion is
|
||
0 ≤ 𝑞𝑖0 (𝑡) ≤ ,𝑡 ∈ . (5)
|
||
𝑑𝑖 transferred to the central cloud. Consequently, the MEC server’s energy
|
||
consumption model is expressed as
|
||
|
||
3.3.2. Transmission energy consumption model 𝐻 𝑒 (𝑡) = 𝑘𝑒 (𝑓 𝑒 )2 min {𝑞 𝑒 , 𝑞 𝑐 (𝑡)} , (12)
|
||
Let 𝑝𝑖 represent the transmission power of user 𝑖. When users offload ∑𝑛 𝑐
|
||
where 𝑞 𝑒 = 𝑓 𝑒 𝜏 and 𝑞 𝑐 (𝑡) = 𝑖=1 𝑞𝑖1 (𝑡).
|
||
tasks to the MEC server, they incur transmission energy consump-
|
||
tion [19], which is modeled as
|
||
𝑢 3.5. The utility optimization problem
|
||
𝐻𝑖1 (𝑡) = 𝑇𝑖𝑡𝑟 (𝑡)𝑝𝑖 , (6)
|
||
|
||
where 𝑇𝑖𝑡𝑟 (𝑡) represents the duration of task transmission for user 𝑖. 3.5.1. MEC server utility optimization problem
|
||
The task transmission time does not exceed one time slot and
|
||
The MEC server generates costs while processing tasks. It is assumed
|
||
satisfies
|
||
that the MEC server obtains revenue by pricing data per cycle. We
|
||
0 ≤ 𝑇𝑖𝑡𝑟 (𝑡) ≤ 𝜏 , 𝑡 ∈ . (7) adopt a differential pricing approach. Let 𝑅𝑒𝑖 (𝑡) represent the fee that
|
||
user 𝑖 needs to pay to the MEC server for offloading data per cycle and
|
||
{ }
|
||
define 𝑹(𝒕) = 𝑅𝑒1 (𝑡), 𝑅𝑒2 (𝑡), … , 𝑅𝑒𝑛 (𝑡) . Let 𝜋 𝑒 (𝑡) represent the revenue
|
||
Based on Eqs. (4) and (6), the overall energy consumption 𝐻𝑖𝑢 (𝑡) is
|
||
derived as 𝐻𝑖𝑢 (𝑡) = 𝐻𝑖0
|
||
𝑢 (𝑡) + 𝐻 𝑢 (𝑡). obtained from processing all user tasks by the MEC server, which is
|
||
𝑖1 ∑
|
||
denoted as 𝜋 𝑒 (𝑡) = 𝑛𝑖=1 𝑅𝑒𝑖 (𝑡)𝑞𝑖1
|
||
𝑐 (𝑡). Let 𝑐 𝑒 represent the cost incurred by
|
||
|
||
3.3.3. Energy harvesting model the MEC server for each unit of energy consumption. When the amount
|
||
We consider the user’s energy harvesting to follow the HUS strat- of tasks offloaded by all users exceeds the processing capacity of the
|
||
egy [20], where the energy collected during the current time slot is MEC server, the MEC server needs to pay a fee to the central cloud for
|
||
only available for use in subsequent time slots. Let 𝑒𝑖 (𝑡) represent the handling. Let 𝑞 𝑟 (𝑡) = max {0, 𝑞 𝑐 (𝑡) − 𝑞 𝑒 } represent the tasks redirected to
|
||
energy harvested by user 𝑖, and 𝐵𝑖 (𝑡) represent the remaining energy in the central cloud and 𝑅𝑐 represent the cost incurred by the MEC server
|
||
the battery. The battery energy update is modeled as: for offloading data to the central cloud per cycle. The optimization
|
||
𝐵𝑖 (𝑡 + 1) = min{max{𝐵𝑖 (𝑡) − 𝐻𝑖𝑢 (𝑡), 0} + 𝑒𝑖 (𝑡), 𝐵𝑖max }, (8) problem 𝑬𝟏 of the MEC server can be modeled as
|
||
1 ∑
|
||
𝑇 −1
|
||
where 𝐵𝑖max represents the maximum capacity of the battery. 𝑬𝟏 ∶ max 𝑠 = lim E{𝜋 𝑒 (𝑡) − 𝜖 𝑒 (𝑡)} (13)
|
||
𝑹(𝒕) 𝑇 →+∞ 𝑇
|
||
The overall energy consumption generated by user 𝑖 cannot surpass 𝑡=0
|
||
the remaining battery energy, i.e., s.t.0 ≤ 𝑅𝑒𝑖 (𝑡) ≤ 𝑅𝑒_max
|
||
𝑖 (𝑡), 𝑖 ∈ 𝑁 , 𝑡 ∈ , (14)
|
||
0 ≤ 𝐻𝑖𝑢 (𝑡) ≤ 𝐵𝑖 (𝑡), 𝑡 ∈ . (9)
|
||
where 𝜖 𝑒 (𝑡) = 𝑐 𝑒 𝐻 𝑒 (𝑡) + 𝑅𝑐 𝑞 𝑟 (𝑡); 𝑅𝑒_max
|
||
𝑖 (𝑡) represents the maximum price
|
||
per cycle of data charged by the MEC server to user 𝑖.
|
||
|
||
4
|
||
J. Mei et al. Journal of Systems Architecture 160 (2025) 103360
|
||
|
||
|
||
3.5.2. User utility optimization problem 4.2. Problem transformation based on Lyapunov optimization
|
||
A higher total task quantity processed by users in a time slot implies
|
||
a reduced task queue, thereby increasing user satisfaction. We adopt the Since the initial optimization problem involves the situation in
|
||
logarithmic utility function [21], modeling it as the long-term range, and the parameters such as energy acquisition
|
||
∑
|
||
1 and task arrival will change over time, this makes it complicated to
|
||
𝜔𝑖 (𝑡) = 𝜒 log2 (1 + 𝑞𝑖𝑗 (𝑡)), (15) solve the initial problem directly. However, Lyapunov optimization
|
||
𝑗=0 can transform this long-term problem into multiple tractable short-
|
||
where 𝜒 is a weight parameter. term problems, so that only these short-term problems need to be dealt
|
||
In general, for a same task, the energy consumption resulting from with, and the original difficult problem can be solved more efficiently.
|
||
local execution exceeds that of transmitting the same quantity of of- Moreover, through Lyapunov optimization, the user’s task queue can
|
||
remain stable (i.e., satisfying Eq. (3)). Therefore Lyapunov optimization
|
||
floaded tasks [22]. Therefore, we consider that offloading an appro-
|
||
theory is employed.
|
||
priate amount of tasks remotely is advantageous for reducing energy
|
||
The Lyapunov function for user 𝑖 can be defined as
|
||
consumption. Consequently, we incorporate the energy saved by of-
|
||
1
|
||
floading tasks compared to local processing into the user utility model. 𝐿𝑖 (𝑡) ≜ [𝑄𝑖 (𝑡)]2 . (19)
|
||
2
|
||
The saved energy is modeled as
|
||
𝑞 (𝑡)𝑝
|
||
𝜓𝑖 (𝑡) = 𝑘𝑢𝑖 [𝑓𝑖𝑢 ]2 𝑞𝑖1 (𝑡)𝑑𝑖 − 𝑖1 𝑡𝑟 𝑖 . (16) Based on [24], the Lyapunov drift for user 𝑖 can be defined as
|
||
𝑟𝑖 { }
|
||
𝛥𝑖 (𝑡) ≜ E 𝐿𝑖 (𝑡 + 1) − 𝐿𝑖 (𝑡)|𝑄𝑖 (𝑡) . (20)
|
||
Considering that the MEC server does not provide services to users
|
||
for free, user 𝑖 is required to pay a certain cost to the MEC server, To balance queue stability and user utility optimization, we intro-
|
||
determined by the offloaded task cycle count. The offloading cost is duce the concept of drift-plus-penalty function, which makes a trade-off
|
||
modeled as between task queue stability and user utility. By incorporating an
|
||
𝑐𝑖 (𝑡) = 𝑅𝑒𝑖 (𝑡)𝑞𝑖1
|
||
𝑐
|
||
(𝑡). (17) additional penalty term into the Lyapunov function, we can optimize
|
||
the user utility while satisfying the queue stability condition.
|
||
To ensure queue stability, it is necessary to minimize Lyapunov
|
||
The user’s utility is related to the total quantity of tasks handled by drift. However, the objective is to maximize user utility, we transform
|
||
the user and the energy savings achieved. The user’s costs are related to the maximization of the user utility function into the minimization of
|
||
the total energy consumption and the offloading costs. Therefore, the the user loss function (i.e., the negative of the user utility function).
|
||
user’s utility optimization problem 𝑷 𝟏 can be described as The drift-plus-penalty function for user 𝑖 is represented as 𝛥𝑖 (𝑡) −
|
||
{ }
|
||
𝑉 E 𝑢𝑖 (𝑡)|𝑄𝑖 (𝑡) , where V is a non-negative weight parameter.
|
||
1 ∑
|
||
𝑇 −1
|
||
𝑷𝟏 ∶ max 𝑢𝑖 = lim E{𝑢𝑖 (𝑡)} (18) When considering Lyapunov drift-plus-penalty function, the lack of
|
||
𝑇𝑖𝑡𝑟 (𝑡),𝑞𝑖0 (𝑡) 𝑇 →+∞ 𝑇
|
||
𝑡=0 information about the following time slot (i.e., 𝐿𝑖 (𝑡 + 1)) makes direct
|
||
s.t.(2), (3), (5), (7) and (9), solving challenging. To remove the reliance on future information, we
|
||
use scaling to get an upper bound on this function.
|
||
where 𝑢𝑖 (𝑡) = 𝜋𝑖𝑢 (𝑡) − 𝜖𝑖𝑢 (𝑡), 𝜋𝑖𝑢 (𝑡) = 𝜔𝑖 (𝑡) + 𝜓𝑖 (𝑡), 𝜖𝑖𝑢 (𝑡) = 𝐻𝑖𝑢 (𝑡) + 𝜆𝑐𝑖 (𝑡), and
|
||
𝜆 is a weight parameter. Theorem 1. When a control parameter 𝑉 > 0 is chosen, and considering
|
||
{ }
|
||
that both 𝑞𝑖 (𝑡) ∈ 0, 1, … , 𝑄𝑖 (𝑡) and 𝑎𝑖 (𝑡) ∈ {0, 1, … , 𝑎max }, we obtain
|
||
4. Problem analysis and solution { }
|
||
𝛥𝑖 (𝑡) − 𝑉 E 𝑢𝑖 (𝑡)|𝑄𝑖 (𝑡)
|
||
{ }
|
||
≤ 𝑧 − E 𝑞𝑖 (𝑡)[𝑄𝑖 (𝑡) + 𝑎𝑖 (𝑡)]|𝑄𝑖 (𝑡)
|
||
In this section, the game relationship and related processes between { }
|
||
the users and the MEC server will be introduced in detail. To transform − 𝑉 E 𝑢𝑖 (𝑡)|𝑄𝑖 (𝑡) , (21)
|
||
long-term optimization problem into multiple short-term optimization
|
||
where 𝑧 = 21 {[𝑄𝑖 (𝑡)]2 + (𝑎max )2 } + 𝑄𝑖 (𝑡)𝑎𝑖 (𝑡).
|
||
problem and ensure the task queue’s stability, the Lyapunov optimiza-
|
||
tion theory will be adopted. The optimal strategies of both users and
|
||
MEC will be taken into consideration. Proof. Taking the square of both sides of Eq. (1) and we find that
|
||
(𝑚𝑎𝑥[𝑥, 0])2 < 𝑥2 for any 𝑥 ∈ R. Therefore, the inequality can be
|
||
4.1. The game relationship between users and MEC server calculated as
|
||
{ }2
|
||
[𝑄𝑖 (𝑡 + 1)]2 = max{0, 𝑄𝑖 (𝑡) − 𝑞𝑖 (𝑡)} + 𝑎𝑖 (𝑡)
|
||
Without adequate incentive measures, the MEC server may be less
|
||
≤ [𝑄𝑖 (𝑡) − 𝑞𝑖 (𝑡)]2 + [𝑎𝑖 (𝑡)]2
|
||
willing to participate in computation offloading [23]. To incentivize the
|
||
MEC server, we employ a strategy rooted in Stackelberg game theory + 2𝑎𝑖 (𝑡)[𝑄𝑖 (𝑡) − 𝑞𝑖 (𝑡)]
|
||
to enable multiple users and the MEC server to both achieve their = [𝑄𝑖 (𝑡)]2 + [𝑞𝑖 (𝑡)]2 + [𝑎𝑖 (𝑡)]2
|
||
respective benefits. In this game process, the MEC server plays the role − 2𝑄𝑖 (𝑡)𝑞𝑖 (𝑡) + 2𝑎𝑖 (𝑡)[𝑄𝑖 (𝑡) − 𝑞𝑖 (𝑡)]
|
||
of the leader, while users act as followers. In the 𝑡th time slot, firstly,
|
||
≤ 2[𝑄𝑖 (𝑡)]2 + (𝑎max )2 − 2𝑄𝑖 (𝑡)𝑞𝑖 (𝑡)
|
||
the MEC server will provide each user with an initial service quotation
|
||
𝑅𝑒𝑖 (𝑡). Secondly, each user is required to make task offloading strategy + 2𝑎𝑖 (𝑡)[𝑄𝑖 (𝑡) − 𝑞𝑖 (𝑡)]. (22)
|
||
(i.e., 𝑞𝑖0 (𝑡) and 𝑇𝑖𝑡𝑟 (𝑡)) based on the price. Subsequently, the MEC
|
||
server will update the corresponding prices based on the user’s remote
|
||
Based on Definition (19) and the aforementioned inequality, the
|
||
offloading task 𝑞𝑖1 (𝑡) and its own computational capacity. Following
|
||
Lagrangian function for the time slot 𝑡 + 1 is computed as
|
||
this, users will update their task offloading decisions. As this process
|
||
1
|
||
continues, through multiple iterative steps, until a balance is reached 𝐿𝑖 (𝑡 + 1) = [𝑄𝑖 (𝑡 + 1)]2
|
||
2
|
||
between users’ task offloading decisions and the MEC server prices, the (𝑎max )2
|
||
iteration for the current time slot concludes. At this point, a Stackelberg ≤ [𝑄𝑖 (𝑡)]2 + − 𝑄𝑖 (𝑡)𝑞𝑖 (𝑡) (23)
|
||
2
|
||
equilibrium is achieved between users and the MEC server. + 𝑎𝑖 (𝑡)[𝑄𝑖 (𝑡) − 𝑞𝑖 (𝑡)].
|
||
|
||
5
|
||
J. Mei et al. Journal of Systems Architecture 160 (2025) 103360
|
||
|
||
|
||
|
||
Based on Definitions (19) and (20), as well as Eq. (23), the inequal- The first-order partial derivative of 𝑖 with respect to 𝑞𝑖0 (𝑡) is given
|
||
ity can be derived as by the following expression
|
||
1 𝜕𝑖 𝜕𝑖 (𝑡)
|
||
𝛥𝑖 (𝑡) ≤ {[𝑄𝑖 (𝑡)]2 + (𝑎max )2 } = + 𝜇1 + 𝜇2 𝑘𝑢𝑖 (𝑓𝑖𝑢 )2 𝑑𝑖 + 𝜇5 − 𝜇6 . (31)
|
||
2 𝜕 𝑞𝑖0 (𝑡) 𝜕 𝑞𝑖0 (𝑡)
|
||
{ }
|
||
+ 𝑄𝑖 (𝑡)𝑎𝑚𝑎𝑥
|
||
𝑖 (𝑡) − E 𝑞𝑖 (𝑡)[𝑄𝑖 (𝑡) + 𝑎𝑖 (𝑡)]|𝑄𝑖 (𝑡) . (24)
|
||
∗ (𝑡)
|
||
Solving for 𝜕𝑖 ∕𝜕 𝑞𝑖0 (𝑡) = 0, the optimal local task workload 𝑞𝑖0
|
||
} { can be calculated as
|
||
By adding the expression −𝑉 E 𝑢𝑖 (𝑡)|𝑄𝑖 (𝑡) to both sides of Eq. (24), 𝑉𝜒
|
||
∗
|
||
we can deduce Eq. (21). 𝑞𝑖0 (𝑡) = − 1, (32)
|
||
𝑔 ln 2
|
||
Taking into account the impact of task queue length on user expe- where 𝑔 = [𝜇1 + 𝜇5 − 𝜇6 − 𝑄𝑖 (𝑡) − 𝑎𝑖 (𝑡)] + 𝑘𝑢𝑖 (𝑓𝑖𝑢 )2 𝑑𝑖 [𝜇2 + 𝑉 ].
|
||
rience, we incorporate the task queue length into the consideration of
|
||
user utility through Lyapunov optimization. According to Theorem 1, 4.3.2. Optimal offloading task strategy
|
||
the original problem 𝑷 𝟏 is transformed into 𝑷 𝟐, described as Compute the first-order partial derivative of 𝑖 with respect to 𝑇𝑖𝑡𝑟 (𝑡),
|
||
{ } 𝜕𝑖 𝜕𝑖 (𝑡)
|
||
𝑷 𝟐 ∶ min 𝑖 (𝑡) = − 𝑞𝑖 (𝑡)[𝑄𝑖 (𝑡) + 𝑎𝑖 (𝑡)] − 𝑉 𝑢𝑖 (𝑡) (25) = + 𝜇1 𝑟𝑡𝑟 (33)
|
||
𝑇𝑖𝑡𝑟 (𝑡),𝑞𝑖0 (𝑡) 𝑡𝑟 𝑖 + 𝜇2 𝑝𝑖 + 𝜇3 − 𝜇4 .
|
||
𝜕 𝑇𝑖 (𝑡) 𝜕 𝑇𝑖𝑡𝑟 (𝑡)
|
||
s.t.(2), (5), (7) and (9).
|
||
By solving for 𝜕𝑖 ∕𝜕 𝑇𝑖𝑡𝑟 (𝑡) = 0, the optimal task transmission time
|
||
𝑇𝑖𝑡𝑟 (𝑡) can be obtained as
|
||
𝑉𝜒 1
|
||
4.3. Optimal strategy for users 𝑇𝑖∗ (𝑡) = − , (34)
|
||
𝑦 ln 2 𝑟𝑡𝑟
|
||
𝑖
|
||
|
||
Each user in the system can collect a certain amount of energy where 𝑦 = [𝑉 𝜆𝑅𝑒𝑖 (𝑡)𝑑𝑖 − 𝑄𝑖 (𝑡) − 𝑎𝑖 (𝑡) + 𝜇1 − 𝑉 𝑘𝑢𝑖 (𝑓𝑖𝑢 )2 𝑑𝑖 ]𝑟𝑡𝑟
|
||
𝑖 + [2𝑉 + 𝜇2 ]𝑝𝑖 +
|
||
during each time slot and store it in a battery. Users can use this 𝜇3 − 𝜇4 .
|
||
battery energy for local task processing or task offloading to the MEC To ensure that the user 𝑖 maintains an transmission time 𝑇𝑖∗ (𝑡) ≥ 0,
|
||
server. Each user must address two essential inquiries: (i) How many we solve for 𝑇𝑖∗ (𝑡) = 0 to obtain the maximum price the user 𝑖 can
|
||
tasks should be processed locally in each time slot; (ii) How much task accept, denoted as 𝑅𝑒_max
|
||
𝑖 (𝑡).
|
||
transfer time is required in each time slot. 𝑒_max 𝜒 𝜇4 − 𝜇3 − 𝑝𝑖 (2𝑉 + 𝜇2 )
|
||
𝑅𝑖 (𝑡) = +
|
||
𝜆𝑑𝑖 ln 2 𝑉 𝜆𝑑𝑖 𝑟𝑡𝑟
|
||
𝑖
|
||
4.3.1. Optimal local task strategy 𝑄𝑖 (𝑡) + 𝑎𝑖 (𝑡) − 𝜇1 + 𝑉 𝑘𝑢𝑖 (𝑓𝑖𝑢 )2 𝑑𝑖
|
||
Referring to (25), we calculate the first-order partial derivative of + . (35)
|
||
𝑉 𝜆𝑑𝑖
|
||
𝑖 (𝑡) with respect to 𝑞𝑖0 (𝑡) as
|
||
𝜕𝑖 (𝑡) By referring to Eq. (34), we can obtain the optimal amount of
|
||
= −[𝑄𝑖 (𝑡) + 𝑎𝑖 (𝑡)] ∗ (𝑡)) for user 𝑖 within the 𝑡th time slot.
|
||
𝜕 𝑞𝑖0 (𝑡) remote offloading tasks (i.e., 𝑞𝑖1
|
||
𝜒
|
||
− 𝑉[ − 𝑘𝑢𝑖 [𝑓𝑖𝑢 ]2 𝑑𝑖 ], (26) ∗
|
||
𝑞𝑖1 (𝑡) = 𝑇𝑖∗ (𝑡)𝑟𝑡𝑟
|
||
𝑖 . (36)
|
||
(1 + 𝑞𝑖0 (𝑡)) ln 2
|
||
and the second-order partial derivative of 𝑖 (𝑡) with respect to 𝑞𝑖0 (𝑡) is
|
||
computed as
|
||
4.4. Optimal strategy for MEC server
|
||
𝜕 2 𝑖 (𝑡) 𝑉𝜒
|
||
= . (27)
|
||
𝜕 𝑞𝑖0 (𝑡)2 [1 + 𝑞𝑖0 (𝑡)]2 ln 2 According to the dynamic programming theory, it breaks down the
|
||
problem into a sequence of sub-problems, and uses the properties of
|
||
Since Eq. (27) is non-negative, 𝑖 (𝑡) exhibits convexity concerning overlapping sub-problems to reduce the amount of computation. In the
|
||
𝑞𝑖0 (𝑡). same way, in long-term optimization problem, each time slot can be
|
||
Following Eq. (25), we calculate the first-order partial derivative of considered a stage, and by making optimal strategies in each stage,
|
||
𝑖 (𝑡) with respect to 𝑇𝑖𝑡𝑟 (𝑡) as optimal result can be achieved in the long run. According to [17], the
|
||
𝜕𝑖 (𝑡) optimization problem 𝑬𝟏 can be transformed into 𝑬𝟐 as
|
||
= −[𝑄𝑖 (𝑡) + 𝑎𝑖 (𝑡)]𝑟𝑡𝑟
|
||
𝑖
|
||
𝜕 𝑇𝑖𝑡𝑟 (𝑡) 𝑬𝟐 ∶ max (𝑡) = 𝜋 𝑒 (𝑡) − 𝜖 𝑒 (𝑡) (37)
|
||
𝑹(𝒕)
|
||
𝜒 𝑟𝑡𝑟
|
||
𝑖 (𝑡)
|
||
− 𝑉[ + 𝑘𝑢𝑖 [𝑓𝑖𝑢 ]2 𝑟𝑡𝑟
|
||
𝑖 (𝑡)𝑑𝑖 + 𝐶], (28) s.t.(14).
|
||
(1 + 𝑞𝑖1 (𝑡)) ln 2
|
||
where 𝐶 = −2𝑝𝑖 − 𝜆𝑖 𝑅𝑒𝑖 (𝑡)𝑟𝑡𝑟 𝑖 𝑑𝑖 ; and the second-order partial derivative
|
||
of 𝑖 (𝑡) with respect to 𝑇𝑖𝑡𝑟 (𝑡) is calculated as For the MEC server, the optimal price (i.e., 𝑅𝑒𝑖 (𝑡)) for each user
|
||
needs to be determined. By combining Eqs. (36) and (37), the first-order
|
||
𝜕 2 𝑖 (𝑡) 𝑉 𝜒(𝑟𝑡𝑟𝑖 (𝑡))
|
||
2
|
||
= . (29) partial derivative of (𝑡) with respect to 𝑅𝑒𝑖 (𝑡) is calculated as
|
||
𝑡𝑟
|
||
𝜕 𝑇𝑖 (𝑡)2 (1 + 𝑞𝑖1 (𝑡))2 ln 2
|
||
⎧𝑞 ∗ (𝑡)𝑑𝑖 if𝑞 𝑒 < 𝑞 𝑐 (𝑡);
|
||
⎪ 𝑖1
|
||
Since Eq. (29) is non-negative, 𝑖 (𝑡) is also convex with respect to ⎪ 𝑉 𝜆𝑑𝑖 [𝑟𝑖 ] 𝜒 𝑒
|
||
2 2 𝑡𝑟 2
|
||
𝑐
|
||
𝑇𝑖𝑡𝑟 (𝑡). In conclusion, 𝑖 (𝑡) is a convex function with respect to 𝑞𝑖0 (𝑡) and 𝜕(𝑡) ⎪− 𝑦2 ln 2 [𝑅𝑖 (𝑡) − 𝑅 ],
|
||
𝑒 = ⎨ (38)
|
||
is also convex with respect to 𝑇𝑖𝑡𝑟 (𝑡), and the constraints of problem 𝑷 𝟐 𝜕 𝑅𝑖 (𝑡) ⎪𝑞 ∗ (𝑡)𝑑 if𝑞 𝑒 ≥ 𝑞 𝑐 (𝑡);
|
||
𝑖1 𝑖
|
||
are affine. Therefore, 𝑷 𝟐 can be solved using the method of Lagrange ⎪ 2 2 𝑡𝑟 2
|
||
⎪− 𝑉 𝜆𝑑𝑖 [𝑟𝑖 ] 𝜒 [𝑅𝑒 (𝑡) − 𝑐 𝑒 𝑘𝑒 (𝑓 𝑒 )2 ],
|
||
multipliers. Let 𝝁 = {𝜇1 , 𝜇2 , … , 𝜇6 } denote the Lagrange multipliers. ⎩ 𝑦2 ln 2 𝑖
|
||
Similar to [25], the Lagrangian function of Eq. (25) is expressed as and the second-order partial derivative of (𝑡) with respect to 𝑅𝑒𝑖 (𝑡) is
|
||
𝑖 (𝑞𝑖0 (𝑡), 𝑇𝑖𝑡𝑟 (𝑡), 𝝁) = 𝑖 (𝑡) + 𝜇1 [𝑞𝑖 (𝑡) − 𝑄𝑖 (𝑡)] calculated as
|
||
⎧ 2𝑉 2 𝜆𝑑𝑖2 [𝑟𝑡𝑟𝑖 ]2 𝜒 𝑧1
|
||
+ 𝜇2 [𝐻𝑖𝑢 (𝑡) − 𝐵𝑖 (𝑡)] + 𝜇3 [𝑇𝑖𝑡𝑟 (𝑡) − 𝜏] 𝜕 2 (𝑡) ⎪ , if𝑞 𝑒 < 𝑞 𝑐 (𝑡);
|
||
𝑦3 ln 2
|
||
𝑓 𝑢𝜏 =⎨ (39)
|
||
− 𝜇4 𝑇𝑖𝑡𝑟 + 𝜇5 [𝑞𝑖0 (𝑡) − 𝑖 ] − 𝜇6 𝑞𝑖0 (𝑡). 𝜕 𝑅𝑒𝑖 (𝑡)2 ⎪ 2𝑉 2 𝜆𝑑𝑖2 [𝑟𝑡𝑟𝑖 ]2 𝜒 𝑧2
|
||
(30) , if𝑞 𝑒 ≥ 𝑞 𝑐 (𝑡);
|
||
𝑑𝑖 ⎩ 𝑦3 ln 2
|
||
|
||
|
||
6
|
||
J. Mei et al. Journal of Systems Architecture 160 (2025) 103360
|
||
|
||
|
||
where 𝑧1 = [𝑄𝑖 (𝑡) + 𝑎𝑖 (𝑡) − 𝜇1 + 𝑉 𝑘𝑢𝑖 (𝑓𝑖𝑢 )2 𝑑𝑖 − 𝑉 𝜆𝑅𝑐𝑖 (𝑡)𝑑𝑖 ]𝑟𝑡𝑟
|
||
𝑖 − [2𝑉 + 𝜇2 ]𝑝𝑖 − Proof. By taking the first-order partial derivative of Eq. (34) with
|
||
2
|
||
𝜇3 + 𝜇4 and 𝑧2 = [𝑄𝑖 (𝑡) + 𝑎𝑖 (𝑡) − 𝜇1 + 𝑉 𝑘𝑢𝑖 (𝑓𝑖𝑢 )2 𝑑𝑖 − 𝑉 𝜆𝑐 𝑒 𝑘𝑒 (𝑓 𝑒 )2 𝑑𝑖 ]𝑟𝑡𝑟
|
||
𝑖 − respect to 𝑅𝑒𝑖 (𝑡), 𝜕 𝑇𝑖∗ (𝑡)∕𝜕 𝑅𝑒𝑖 (𝑡) = −𝑉 2 𝜒 𝜆𝑑𝑖 𝑟𝑡𝑟
|
||
𝑖 ∕ (𝑦 ln 2) is obtained. It is
|
||
[2𝑉 + 𝜇2 ]𝑝𝑖 − 𝜇3 + 𝜇4 . evident that 𝜕 𝑇𝑖∗ (𝑡)∕𝜕 𝑅𝑒𝑖 (𝑡) < 0, indicating that 𝑇𝑖𝑡𝑟 (𝑡) decreases as 𝑅𝑒𝑖 (𝑡)
|
||
Due to 𝑦 > 0 and the numerator of the second derivative of (𝑡) increases. This implies that as the price of the MEC server increases,
|
||
with respect to 𝑅𝑒𝑖 (𝑡) being constant when the MEC server makes game users are less willing to offload tasks.
|
||
decisions, the sign of the second derivative depends on the sign of the
|
||
numerator. Consequently, in any iteration, (𝑡) is either a concave or
|
||
Lemma 4. If the optimal task transmission time 𝑇𝑖∗ (𝑡) of user 𝑖 is fixed,
|
||
convex function relative to 𝑅𝑒𝑖 (𝑡). If (𝑡) exhibits convexity concerning ( )
|
||
the 𝑅𝑒𝑖 (𝑡) of the MEC server takes the maximum value at 𝑅∗𝑖 (𝑡).
|
||
𝑅𝑒𝑖 (𝑡), then the optimal solution of 𝑅𝑒𝑖 (𝑡) is at one of the endpoints of the
|
||
constraint range. Therefore, we can deduce that the optimal solution for
|
||
𝑅𝑒𝑖 (𝑡), denoted as 𝑅∗𝑖 (𝑡), is calculated as Proof. Based on the optimal strategy for the MEC server discussed
|
||
𝑅∗𝑖 (𝑡) = 𝑅𝑖𝑒_max (𝑡). (40) in this section of the paper, it can be concluded that the optimization
|
||
function (i.e., (𝑡)) of the MEC server is either concave or convex with
|
||
respect to the variable 𝑅𝑒𝑖 (𝑡) in each iteration. When the optimization
|
||
If (𝑡) exhibits concavity concerning 𝑅𝑒𝑖 (𝑡), and it is straightforward function (𝑡) is convex, the efficiency of the MEC server reaches its
|
||
to ascertain that the constraints of optimization problem 𝑬𝟐 are affine, maximum at 𝑅𝑒𝑖 (𝑡) = 𝑅∗𝑖 (𝑡), as shown in Eq. (40). Similarly, when (𝑡)
|
||
then we can employ the Lagrange multiplier method to obtain the is concave, the efficiency of the MEC server reaches its maximum at
|
||
solution for 𝑅∗𝑖 (𝑡). The Lagrangian function for Eq. (37) is expressed 𝑅𝑒𝑖 (𝑡) = 𝑅∗𝑖 (𝑡), as indicated by Eq. (42). According to Definition 1,
|
||
as 𝑅𝑆𝑖 𝐸 (𝑡) = 𝑅∗𝑖 (𝑡).
|
||
𝑒 (𝑹(𝒕), 𝜼) = (𝑡) − 𝜂1 𝑅𝑒𝑖 (𝑡) + 𝜂2 [𝑅𝑒𝑖 (𝑡) − 𝑅𝑒_max (𝑡)], (41) ( )
|
||
𝑖 In conclusion, 𝑇𝑖∗ (𝑡), 𝑅∗𝑖 (𝑡) represents the optimal decision for task
|
||
transmission time and price, and it is also the Stackelberg equilibrium
|
||
where 𝜼 = {𝜂1 , 𝜂2 } denotes the Lagrange multipliers, with each multi- ( )
|
||
plier being non-negative. solution 𝑇𝑖𝑆 𝐸 (𝑡), 𝑅𝑆𝑖 𝐸 (𝑡) .
|
||
By solving for 𝜕𝑒 ∕𝜕 𝑅𝑒𝑖 (𝑡) = 0, the optimal processing price 𝑅∗𝑖 (𝑡) is
|
||
calculated as 5. Multi-device task offloading and pricing mechanism algorithm
|
||
⎧ ∗ 𝑦2 ln 2 𝑒 𝑐
|
||
⎪[𝑞𝑖1 (𝑡)𝑑𝑖 − 𝜂1 + 𝜂2 ] 𝑉 2 𝜆𝑑 2 (𝑟𝑡𝑟 )2 𝜒 if𝑞 < 𝑞 (𝑡); We will delve into the updates of Lagrange multipliers and MEC
|
||
⎪ 𝑖 𝑖
|
||
server prices in this section. At the end of this section, pseudocode for
|
||
⎪+𝑅𝑐 ,
|
||
𝑅∗𝑖 (𝑡) = ⎨ 𝑦2 ln 2
|
||
(42) the algorithm that describes the game process between users and the
|
||
∗ 𝑒 𝑐
|
||
⎪[𝑞𝑖1 (𝑡)𝑑𝑖 − 𝜂1 + 𝜂2 ] 𝑉 2 𝜆𝑑 2 (𝑟𝑡𝑟 )2 𝜒 if𝑞 ≥ 𝑞 (𝑡); MEC server is provided.
|
||
⎪ 𝑖 𝑖
|
||
⎪+𝑐 𝑒 𝑘𝑒 (𝑓 𝑒 )2 ,
|
||
⎩ 5.1. Lagrange multiplier update strategy
|
||
|
||
According to Eqs. (32) and (34), the optimal local task workload and
|
||
4.5. Stackelberg equilibrium analysis task transmission time can be determined, respectively. As the results
|
||
( are obtained using the Lagrange multiplier method, it is necessary to
|
||
In this section, we demonstrate that the optimal strategy 𝑇𝑖∗ (𝑡), 𝑅∗𝑖 update these multipliers to ensure the satisfaction of constraints during
|
||
(𝑡)), 𝑖 ∈ 𝑁 between the users and the MEC server is the Stakelberg the optimization process. A standard subgradient method is employed
|
||
equilibrium solution. We consider the MEC server as a leader in a to update the Lagrange multipliers (i.e., 𝝁). The update method is
|
||
Stackelberg game, and users as followers. For simplicity, the game equi- shown as
|
||
librium within a time slot will be analyzed. The Stackelberg equilibrium { [ ]}+
|
||
is defined in the following. 𝜇1 = 𝜇1 + 𝛼 𝑞𝑖 (𝑡) − 𝑄𝑖 (𝑡) ,
|
||
{ [ 𝑢 ]}+
|
||
( 𝑆𝐸 ) 𝜇2 = 𝜇2 + 𝛼 𝐻𝑖 (𝑡) − 𝐵𝑖 (𝑡) ,
|
||
Definition 1. 𝑇𝑖 (𝑡), 𝑅𝑆𝑖 𝐸 (𝑡) is a Stackelberg equilibrium solution { [ ]}+
|
||
𝜇3 = 𝜇3 + 𝛼 𝑇𝑖∗ (𝑡) − 𝜏 ,
|
||
when the price 𝑅∗𝑖 (𝑡) of leader is determined, and 𝑇𝑖𝑆 𝐸 (𝑡) satisfies [ ]+
|
||
( ) { ( 𝑡𝑟 )} 𝜇4 = 𝜇4 − 𝛼 𝑇𝑖∗ (𝑡) , (45)
|
||
𝑖 𝑇𝑖𝑆 𝐸 (𝑡) = inf 𝑖 𝑇𝑖 (𝑡) , ∀𝑡 ∈ , (43) { [
|
||
𝑇𝑖min (𝑡)≤𝑇𝑖𝑡𝑟 (𝑡)≤𝑇𝑖max (𝑡) 𝑓 𝑢 𝜏 ]}+
|
||
𝜇5 = 𝜇5 + 𝛼 𝑞𝑖0 ∗
|
||
(𝑡) − 𝑖 ,
|
||
and when the task transmission time 𝑇𝑖∗ (𝑡) is determined, and 𝑅𝑆𝑖 𝐸 (𝑡) 𝑑𝑖
|
||
[ ∗
|
||
] +
|
||
satisfies 𝜇6 = 𝜇6 − 𝛼 𝑞𝑖0 (𝑡) ,
|
||
( ) { ( 𝑒 )}
|
||
𝑅𝑆𝑖 𝐸 (𝑡) = sup 𝑅𝑖 (𝑡) , ∀𝑡 ∈ . (44) where 𝛼 represents the iteration step size and [𝑥]+ = max{0, 𝑥}.
|
||
𝑅min 𝑒 max (𝑡)
|
||
𝑖 (𝑡)≤𝑅𝑖 (𝑡)≤𝑅𝑖 The local task workload and task transmission time for each user are
|
||
( ) computed by updating the Lagrange multipliers and the price in each
|
||
Next, it will be verified whether the optimal solution 𝑇𝑖∗ (𝑡), 𝑅∗𝑖 (𝑡) iteration.
|
||
( 𝑆𝐸 )
|
||
is the Stackelberg equilibrium solution 𝑇𝑖 (𝑡), 𝑅𝑆𝑖 𝐸 (𝑡) .
|
||
5.2. Price update strategy
|
||
( )
|
||
Lemma 2. If the price 𝑅∗𝑖 (𝑡) of the leader is fixed, the function 𝑖 𝑇𝑖𝑡𝑟 (𝑡)
|
||
∗
|
||
of user 𝑖 takes the minimum value at 𝑇𝑖 . In one iteration, when the function (𝑡) exhibits convexity con-
|
||
cerning the price 𝑅𝑒𝑖 (𝑡), the optimal price 𝑅∗𝑖 (𝑡) = 𝑅𝑒_max
|
||
𝑖 (𝑡). However,
|
||
when the function (𝑡) is concave with respect to the price 𝑅𝑒𝑖 (𝑡), it is
|
||
Proof. Based on Eq. (29), 𝑖 (𝑡) exhibits convexity concerning 𝑇𝑖𝑡𝑟 (𝑡), difficult to compute the optimal price 𝑅∗𝑖 (𝑡) based on 𝜕(𝑡)∕𝜕 𝑅𝑒𝑖 (𝑡) = 0.
|
||
indicating that the function 𝑖 (𝑡) attains its minimum value at 𝑇𝑖∗ (𝑡). To address this, the gradient ascent method is employed to update the
|
||
According to Definition 1, 𝑇𝑖∗ (𝑡) is the 𝑇𝑖𝑆 𝐸 (𝑡). price 𝑅𝑒𝑖 (𝑡), using the first-order partial derivative of the MEC server
|
||
utility function with respect to the price as the Marginal Utility [26] to
|
||
update the price. The update expression can be derived as
|
||
Lemma 3. For users, the optimal task transmission time 𝑇𝑖∗ (𝑡) decreases
|
||
𝜕(𝑡)
|
||
with the increased price 𝑅𝑒𝑖 (𝑡). 𝑅𝑒_𝜅+1 (𝑡) = 𝑅𝑒_𝜅
|
||
𝑖 (𝑡) + 𝛽 , (46)
|
||
𝑖
|
||
𝜕 𝑅𝑒_𝜅
|
||
𝑖 (𝑡)
|
||
|
||
7
|
||
J. Mei et al. Journal of Systems Architecture 160 (2025) 103360
|
||
|
||
|
||
where 𝛽 represents the iteration step size and 𝜅 represents the number Algorithm 2 Multi-Device Task Offloading and Pricing Mechanism
|
||
of iterations within the current time slot. Algorithm (MDTOPMA)
|
||
Due to the constraint (14), after one iteration, the service price of 1: Input:𝑘𝑢𝑖 , 𝑓𝑖𝑢 , 𝑑𝑖 , 𝜏, 𝑝𝑖 , 𝐵𝑖max , 𝑤, ℎ𝑖 , 𝑛0 , 𝑘𝑒 , 𝑓 𝑒 , 𝑐 𝑒 , 𝑅𝑐 , 𝑖 ∈ 𝑁;
|
||
the MEC server is calculated as 2: Output:optimal solution 𝑞𝑖0 ∗ (𝑡), 𝑇 ∗ (𝑡), 𝑅∗ (𝑡), 𝑖 ∈ 𝑁, 𝑡 ∈ , 𝑡𝑜𝑡𝑎𝑙𝐼 𝑡;
|
||
𝑖 𝑖
|
||
{ { }} 3: initial 𝜆, 𝛼, 𝛽, 𝑄𝑖 (0), 𝐵𝑖 (0), 𝝁 ← [𝜇1 , ..., 𝜇6 ], 𝝈 ← [𝜎1 , ..., 𝜎6 ];
|
||
𝑅𝑖𝑒_𝜅+1 (𝑡) = min 𝑅𝑒_max
|
||
𝑖 (𝑡), max 0, 𝑅𝑖𝑒_𝜅 (𝑡) . (47) 𝒑𝒓𝒆
|
||
4: 𝝁 ← 𝝁;
|
||
5: for all 𝑖 ∈ 𝑁 do
|
||
Finally, the price of the (𝜅 + 1)-th iteration within the 𝑡th time slot 6: Calculate 𝑟𝑡𝑟 𝑖 by Eq. (10);
|
||
can be expressed as 7: end for
|
||
⎧𝑅𝑒_𝜅 (𝑡), 2 (𝑡)
|
||
if 𝜕𝑅𝜕 𝑒_𝜅
|
||
8: for 𝑡 ∈ do
|
||
⎪ 𝑖 2 ≤ 0;
|
||
𝑒_𝜅+1 𝑖 (𝑡) 9: 𝑖𝑡 ← 0;
|
||
𝑅𝑖 (𝑡) = ⎨ (48)
|
||
𝑒_max 𝜕 2 (𝑡) while 𝑖𝑡 ≤ 𝑡𝑜𝑡𝑎𝑙𝐼 𝑡 or 𝝁 − 𝝁𝒑𝒓𝒆 ≥ 𝝈 do
|
||
⎪𝑅𝑖 (𝑡), if 𝜕𝑅𝑒_𝜅 (𝑡)2 > 0; 10:
|
||
⎩ 𝑖 11: for all 𝑖 ∈ 𝑁 do
|
||
12: Calculate 𝑞𝑖0 (𝑡) and 𝑇𝑖𝑡𝑟 (𝑡) according to
|
||
To provide a clearer description of the price updating process, the
|
||
13: Eq. (32) and Eq. (34) respectively;
|
||
pseudocode for price updating is presented in the Alg. 1.
|
||
14: 𝑡𝑖𝑚𝑒 ← 𝑡𝑖𝑚𝑒 + 𝑇𝑖𝑡𝑟 (𝑡);
|
||
Algorithm 1 Price Update Algorithm 15: end for
|
||
16: if 𝑡𝑖𝑚𝑒 > 𝜏 then
|
||
1: Input:𝑘𝑢𝑖 , 𝑓𝑖𝑢 , 𝑑𝑖 , 𝜏, 𝑝𝑖 , 𝝁, 𝑟𝑡𝑟 𝑒 𝑒 𝑒 𝑐 𝑐
|
||
𝑖 , 𝑘 , 𝑉 , 𝜆, 𝜒, 𝑓 , 𝑐 , 𝑅 , 𝑞 , 𝑖 ∈ 𝑁; 17: for all 𝑖 ∈ 𝑁 do
|
||
2: Output:𝑅𝑒𝑖 (𝑡), 𝑖 ∈ 𝑁; 18: 𝑇𝑖𝑡𝑟 (𝑡) ← 𝜏 𝑇𝑖𝑡𝑟 (𝑡)∕𝑡𝑖𝑚𝑒;
|
||
3: for all 𝑖 ∈ 𝑁 do 19: end for
|
||
4: Calculate 𝜕(𝑡)∕𝜕 𝑅𝑒𝑖 (𝑡) based on Eq. (38); 20: end if
|
||
5: Calculate 𝜕 2 (𝑡)∕𝜕 𝑅𝑒𝑖 (𝑡)2 based on Eq. (39); 21: for 𝑖 ∈ 𝑁 do
|
||
6: Calculate 𝑅𝑖𝑒_ max (𝑡) based on Eq. (40); 22: 𝑐 (𝑡) ← 𝑇 𝑡𝑟 (𝑡)𝑟𝑡𝑟 ∕𝑑 ;
|
||
𝑞𝑖1 𝑖 𝑖 𝑖
|
||
7: Calculate 𝑅𝑒_𝑖𝑡 𝑖 (𝑡) based on Eq. (47); 23: 𝑞 𝑐 ← 𝑞 𝑐 + 𝑞𝑖1 𝑐 (𝑡), 𝑖 ← 𝑖 + 1;
|
||
8: Calculate 𝑅𝑒𝑖 (𝑡) based on Eq. (48); 24: end for
|
||
9: end for 25: Call Algorithm 1;
|
||
26: 𝝁𝒑𝒓𝒆 ← 𝝁, calculate 𝝁 by Eq. (45);
|
||
5.3. Multi-device task offloading and pricing mechanism algorithm 27: 𝑖𝑡 ← 𝑖𝑡 + 1;
|
||
28: end while
|
||
∗ (𝑡) ← 𝑞 (𝑡), 𝑇 ∗ (𝑡) ← 𝑇 𝑡𝑟 (𝑡), 𝑅∗ (𝑡) ← 𝑅𝑒 (𝑡);
|
||
𝑞𝑖0
|
||
By integrating the content in Section 4 and Section 5 of this pa- 29: 𝑖0 𝑖 𝑖 𝑖 𝑖
|
||
per, the optimization of 𝑞𝑖0 (𝑡), 𝑇𝑖𝑡𝑟 (𝑡), and 𝑅𝑒𝑖 (𝑡) for each time slot is 30: end for
|
||
formulated. The implementation process of the proposed Multi-Device
|
||
Task Offloading and Pricing Mechanism Algorithm is outlined in the
|
||
Alg. 2. The core of this algorithm lies in lines 8 to 22 of the pseu-
|
||
of [2000, 6000] bits [11]. The bandwidth of the base station 𝑤 is
|
||
docode. In pseudocode, lines 8 to 13 describe how each user calculates
|
||
set to 20 MHz [27]. The task computation density 𝑑𝑖 is uniformly
|
||
local processing tasks and transfer time based on price. Lines 14 to
|
||
distributed within [2,12]×103 cycles/bit. The time slot duration 𝜏 is
|
||
18 demonstrate the redistribution of transmission time for each user
|
||
configured as 0.1 s [28]. 𝜆 is set to 100. The noise power density
|
||
using time slot constraints when the total transmission time of all
|
||
𝑛0 is set to 10−9 W/Hz. The channel gain adheres to an exponential
|
||
users exceeds one time slot. Lines 19 to 22 illustrate how the MEC
|
||
distribution, denoted as E(1) [29]. The harvested energy 𝑒𝑖 (𝑡) is uni-
|
||
server updates the prices based on each user’s offloading tasks. The
|
||
formly distributed within [0, 20] mJ/s [30]. For the MEC server, the
|
||
time complexity of the algorithm primarily arises from the iterative
|
||
CPU computing frequency 𝑓 𝑒 is set to 3 GHz [27] and the capacitance
|
||
computation of task offloading and pricing for each user device. In each
|
||
switching coefficient 𝑘𝑒 is set to 10−28 [19]. All the simulations are
|
||
time slot, the time complexity is 𝑂(𝐷 ⋅ 𝑁), where 𝐷 is the number of
|
||
performed on a workstation PC with an Intel i5 12600KF processor,
|
||
iterations and 𝑁 is the number of users.
|
||
16 GB of memory, and Windows 10 operating system.
|
||
|
||
6. Performance evaluation 6.2. Game convergence simulation experiments
|
||
|
||
In order to assess the validity of MDTOPMA, we will organize three In this simulation, we primarily focus on the convergence of the
|
||
distinct sets of simulations. First of all, we demonstrate the convergence Stackelberg game and the stability of the system. For ease of obser-
|
||
of the game through iterative experiments within a time slot. And the vation, there are 4 users considered in this simulation. In Fig. 2, we
|
||
stability of the queue is validated through experiments spanning mul- plot the differentiated prices of the MEC server, tasks offloaded by
|
||
tiple time slots. Second, under the premise of confirming the existence users, tasks processed locally by users, and user utility as the number of
|
||
of game equilibrium, we will focus on parameter tuning experiments iterations accumulates. Fig. 2(a) shows that the prices of the MEC server
|
||
to evaluate the impact of different parameters on performance. Finally, tend to stabilize after a certain number of iterations. In the figure, the
|
||
an empirical study is conducted to compare it with the benchmark final price obtained by each user through the game is different. This
|
||
schemes. is attributed to the heterogeneity among users, for instance, a certain
|
||
user with higher energy consumption when processing tasks locally
|
||
6.1. Simulation setting tends to lean towards offloading tasks. By observing Eq. (42), it can be
|
||
seen that 𝜕 2 𝑅∗𝑖 (𝑡)∕𝜕 𝑞𝑖1
|
||
∗ (𝑡)2 > 0, the MEC server will increase the service
|
||
|
||
For each user, the CPU computing frequency 𝑓𝑖𝑢 is uniformly dis- price of the user accordingly. In addition, since each user’s task arrival
|
||
tributed within [0.9, 1.2] GHz [15] and the capacitance switching coef- and energy collection may be different, this will also result in varying
|
||
ficient 𝑘𝑢𝑖 is distributed within [10−28 , 10−27 ] [18,19]. The transmission task size offloaded by each user, consequently leading to differentiated
|
||
power 𝑝𝑖 is uniformly distributed within [100,150] mW [11]. The task pricing by the MEC server. Fig. 2(b) and Fig. 2(c) demonstrate that
|
||
arrival quantity 𝑎𝑖 (𝑡) follows a uniform distribution within the range the offloaded tasks and locally processed tasks for each user tend to
|
||
|
||
8
|
||
J. Mei et al. Journal of Systems Architecture 160 (2025) 103360
|
||
|
||
|
||
|
||
|
||
Fig. 2. Price, offloaded tasks, locally processed tasks, and user utility versus iteration.
|
||
|
||
|
||
|
||
|
||
Fig. 3. Price, offloaded tasks, locally processed tasks, and queue backlog versus time.
|
||
|
||
|
||
|
||
9
|
||
J. Mei et al. Journal of Systems Architecture 160 (2025) 103360
|
||
|
||
|
||
|
||
|
||
Fig. 5. Average remote offloaded tasks with different values of 𝑎𝑖 (𝑡).
|
||
|
||
|
||
|
||
|
||
a state of continuous ups and downs. But overall, the task queue is
|
||
gradually stabilizing. This is consistent with what we expect to achieve
|
||
with Lyapunov optimization.
|
||
Observing both Fig. 2 and Fig. 3, it is evident that the game
|
||
converges in both short-term and long-term scenarios, ensuring the
|
||
long-term stability of the MEC system.
|
||
|
||
6.3. Parameter tuning simulation experiments
|
||
|
||
In this simulation, multiple experiments are conducted to demon-
|
||
strate the dynamic task offloading and update of prices with consid-
|
||
eration of 20 users. In Fig. 4, the average processed tasks and queue
|
||
backlog for users are plotted with varying parameter 𝑉 . Fig. 4(a)
|
||
depicts how the average processed tasks correlates with the parameter
|
||
Fig. 4. Average processed tasks and queue backlog vary with different values of 𝑉 . 𝑉 . The observed phenomenon shows that as 𝑉 decreases, the average
|
||
processing tasks present a downward trend. According to Eq. (25), as
|
||
the parameter 𝑉 decreases, the proportion of −𝑄𝑖 (𝑡) in the minimization
|
||
Eq. (25) increases, which means that the task queue backlog will show
|
||
an upward trend. Therefore, it can be deduced that as V decreases,
|
||
stabilize after a certain number of iterations. The trend in Fig. 2(b) the average processing tasks of users will show a downward trend.
|
||
exhibits a characteristic of initially increasing and then decreasing. This aligns with the findings exhibited in Fig. 4(a). Fig. 4(b) illustrates
|
||
This phenomenon stems from the relatively low initial prices depicted how the average queue backlog varies with the parameter 𝑉 . As 𝑉
|
||
in Fig. 2(a), resulting in a sharp increase in the tasks offloaded by increases, we can observe a decreasing trend in the queue backlog. This
|
||
users. However, as prices rise, the offloaded tasks gradually decrease, phenomenon aligns with the trend of increasing processed tasks shown
|
||
aligning with the anticipated trend. Additionally, Lemma 3 can be in Fig. 4(a).
|
||
used to substantiate this observation. If the MEC server offers a higher Fig. 5, the average remote offloaded tasks for users are plotted with
|
||
price to a particular user, that user typically reduces the amount of varying 𝑎𝑖 (𝑡). It can be observed that as the task arrival rate increases,
|
||
tasks offloaded, such as user 3. Based on the three previous figures in the average number of tasks offloaded by users also increases. This is
|
||
Fig. 2, it can be inferred that user utility also tends to stabilize with the because with an increase in the task arrival rate, the backlog in the task
|
||
number of iterations, which aligns with the results shown in Fig. 2(d). queue also grows. To maintain queue stability, users tend to offload
|
||
According to the results in Fig. 2, it can be confirmed that there exists a more tasks.
|
||
Stackelberg equilibrium between the users and the MEC server within In Fig. 6, we plot the offloaded tasks, prices, and the MEC server util-
|
||
a time slot. ity with varying values of the parameter 𝜆. In Fig. 6(a), as 𝜆 increases,
|
||
In Fig. 3, we present the variations of the MEC server prices, user the convergence price of the MEC server decreases. This is because
|
||
offloaded tasks, user local processing tasks, and the backlog in the user 𝜆 is weighted on the offloading cost function (i.e., 𝑐𝑖 (𝑡)) of users.
|
||
task queue with varying time slots. Fig. 3(a) illustrates the gradual With increase of 𝜆, the proportion of user offloading costs increases,
|
||
stabilization of differential prices provided by the MEC server to every leading to a decreased inclination among users for task offloading.
|
||
user during long-term evolution. Due to the heterogeneity of users and Consequently, MEC server stimulate task offloading by reducing prices.
|
||
the varying task arrival rates and energy harvesting conditions in each In Fig. 6(b), as the value of 𝜆 decreases, users tend to offload more
|
||
time slot, the prices calculated through the game may also differ for tasks. This is because a lower 𝜆 results in a smaller proportion of user
|
||
each user. Fig. 3(b) depicts the increasing trend of remote offloaded offloading costs, leading to an increased desire for task offloading.
|
||
tasks for each user starting from zero and gradually decreasing there- Fig. 6(c) illustrates that the effectiveness of the MEC server increases
|
||
after. This pattern can be attributed to the initially low prices shown in with lower 𝜆. This experimental result can be calculated based on the
|
||
Fig. 3(a), which encourage users to offload more tasks. However, as the experimental data in Fig. 6(a) and Fig. 6(b).
|
||
prices subsequently rise, users’ inclination for offloading diminishes,
|
||
leading to a reduction in the amount of offloaded tasks. Fig. 3(c) and 6.4. Comparison with benchmark schemes
|
||
Fig. 3(d) illustrate the steady state of user local processing tasks and
|
||
user task queue during long-term evolution, respectively. The arrival To further evaluate the MDTOPMA′ s performance, there are three
|
||
of tasks in each time slot is uncertain, so the queue in Fig. 3(d) shows schemes compared with MDTOPMA:
|
||
|
||
10
|
||
J. Mei et al. Journal of Systems Architecture 160 (2025) 103360
|
||
|
||
|
||
|
||
|
||
Fig. 6. Price, offloaded tasks, and the MEC server utility with different values of 𝜆.
|
||
|
||
|
||
|
||
|
||
Fig. 7. User utility and task queue backlog with four different schemes.
|
||
|
||
|
||
• Local-Only Processing (LOP) scheme [19]: In each time slot, may still choose to remotely offload many tasks, resulting in reduced
|
||
users solely process all tasks locally, i.e., 𝑇𝑖∗ (𝑡) = 0. utilities.
|
||
• Average Time-Constrained Task Offloading (ATCTO) scheme:
|
||
In each time slot, the channel transmission time for each user’s of- 7. Conclusion
|
||
floaded tasks must not surpass a predefined threshold, i.e., 𝑇𝑖𝑡𝑟 (𝑡) ≤
|
||
𝜏∕𝑛. The ATCTO scheme is inspired by the Equal Allocation Strat- In this paper, we study a task offloading problem for a MEC system
|
||
egy from [11], which emphasizes fairly distributing the offloading consisting of three layers of cloud–edge-device, where each user ter-
|
||
time among users. minal device supports EH. While solving the optimization problem, the
|
||
• Genetic Algorithm (GA) scheme: In this study, we use a genetic Lyapunov optimization theory is applied to convert the long-term prob-
|
||
algorithm for comparison and initialize a population of 20 indi- lem into problem of each time slot and stabilize the task queue of each
|
||
viduals, each representing offloading strategies for users and a user. In order to reasonably allocate resources between the users and
|
||
pricing strategy for the MEC server. After multiple iterations, we the MEC server, the Stackelberg game theory is employed to regulate
|
||
select the individual with the highest fitness for comparison. resources. Combining the above two theories, we apply the MDTOPMA
|
||
to solve the optimal offloading strategy of each user and the pricing
|
||
In Fig. 7, the average user utility and average queue backlog are strategy of the MEC server in each time slot. The simulation experiment
|
||
plotted alongside different algorithm schemes. In the experimental results indicate that, when compared to other algorithms, MDTOPMA
|
||
results, our MDTOPMA surpasses the other schemes in user utility not only enhances user benefits but also reduces the backlog of user
|
||
and task queue backlog. In Fig. 7(a), it can be observed that the task queues. In the simulation experiment of parameter performance
|
||
improvement in user utility by LOP is significantly smaller than other optimization, the adjustment of parameter value also leads to better
|
||
schemes. This is because LOP only allows users to process tasks locally, choice of offloading decision and pricing decision.
|
||
neglecting the computing capacity of the MEC server, and the energy
|
||
consumption of remote offloading is generally lower than the energy CRediT authorship contribution statement
|
||
consumption generated by local computing tasks. Fig. 7(b) demon-
|
||
strates that the task queue backlog of LOP is significantly higher than Jing Mei: Writing – review & editing, Investigation, Funding ac-
|
||
that of our proposed algorithm. This is mainly due to the fact that LOP quisition, Formal analysis. Cuibin Zeng: Writing – original draft, Re-
|
||
does not utilize the MEC server’s computational resources. In Fig. 7, sources, Methodology. Zhao Tong: Writing – review & editing, Method-
|
||
the performance of ATCTO is less than our scheme. For instance, if a ology, Funding acquisition, Conceptualization. Longbao Dai: Writing –
|
||
user calculates the optimal transmission time T, such that 𝑇𝑖∗ (𝑡) > 𝜏∕𝑛, review & editing, Investigation, Conceptualization. Keqin Li: Writing –
|
||
but is constrained by 𝑇𝑖𝑡𝑟 (𝑡) ≤ 𝜏∕𝑛, it results in the user being unable to review & editing, Conceptualization.
|
||
achieve the optimal performance. Although ATCTO allows each user to
|
||
have a transmission time ranging from 0 to 𝜏∕𝑛, ensuring that each user Declaration of competing interest
|
||
has a fair opportunity to utilize network resources, achieving better
|
||
performance is challenging. In Fig. 7, GA’s task queue performance is The authors declare that they have no known competing finan-
|
||
better than ATCTO, but the user benefit is lower than ATCTO. This cial interests or personal relationships that could have appeared to
|
||
is due to the randomness of GA. Even when the price is high, users influence the work reported in this paper.
|
||
|
||
11
|
||
J. Mei et al. Journal of Systems Architecture 160 (2025) 103360
|
||
|
||
|
||
Acknowledgments [19] M. Guo, W. Wang, X. Huang, Y. Chen, L. Zhang, L. Chen, Lyapunov-based partial
|
||
computation offloading for multiple mobile devices enabled by harvested energy
|
||
in mec, IEEE Int. Things J. 9 (11) (2022) 9025–9035, http://dx.doi.org/10.1109/
|
||
The authors would like to thank the anonymous reviewers for
|
||
JIOT.2021.3118016.
|
||
their valuable comments and suggestions. This work was supported by [20] C. Qiu, Y. Hu, Y. Chen, Lyapunov optimized cooperative communications with
|
||
the Program of National Natural Science Foundation of China (grant stochastic energy harvesting relay, IEEE Int. Things J. 5 (2) (2018) 1323–1333,
|
||
No. 62072174, 61502165), Provincial Natural Science Foundation of http://dx.doi.org/10.1109/JIOT.2018.2793850.
|
||
[21] B. Cao, S. Xia, J. Han, Y. Li, A distributed game methodology for crowdsensing
|
||
Hunan, China (grant No. 2020JJ5370, 2022JJ40278, 2023JJ30083),
|
||
in uncertain wireless scenario, IEEE Trans. Mob. Comput. 19 (1) (2020) 15–28,
|
||
Scientific Research Fund of Hunan Provincial Education Department, http://dx.doi.org/10.1109/TMC.2019.2892953.
|
||
China (grant No. 22A0026, 22A0592), Graduate Research Innovation- [22] M. Tao, K. Ota, M. Dong, H. Yuan, Stackelberg game-based pricing and offloading
|
||
Program of Hunan Province (Grant No. CX20240548). in mobile edge computing, IEEE Wirel. Commun. Lett. 11 (5) (2022) 883–887,
|
||
http://dx.doi.org/10.1109/LWC.2021.3138938.
|
||
[23] Y. Liu, C. Xu, Y. Zhan, Z. Liu, J. Guan, H. Zhang, Incentive mechanism for
|
||
Data availability
|
||
computation offloading using edge computing: A stackelberg game approach,
|
||
Comput. Netw. 129 (DEC.24) (2017) 399–409.
|
||
Data will be made available on request. [24] Y. Li, S. Xia, M. Zheng, B. Cao, Q. Liu, Lyapunov optimization-based trade-
|
||
off policy for mobile cloud offloading in heterogeneous wireless networks, IEEE
|
||
Trans. Cloud Comput. 10 (1) (2022) 491–505, http://dx.doi.org/10.1109/TCC.
|
||
References 2019.2938504.
|
||
[25] X. Chen, Z. Lu, W. Ni, X. Wang, F. Wang, S. Zhang, S. Xu, Cooling-aware
|
||
optimization of edge server configuration and edge computation offloading for
|
||
[1] Ana Bera, 80 IoT statistics (infographic), 2019, https://safeatlast.co/blog/iot-
|
||
wirelessly powered devices, IEEE Trans. Veh. Technol. 70 (5) (2021) 5043–5056,
|
||
statistics/, Website.
|
||
http://dx.doi.org/10.1109/TVT.2021.3076057.
|
||
[2] K.P. Naveen, R. Sundaresan, Double-auction mechanisms for resource trading
|
||
[26] M.S.S. Rao, S.A. Soman, Marginal pricing of transmission services using min–
|
||
markets, IEEE/ACM Trans. Netw. 29 (3) (2021) 1210–1223, http://dx.doi.org/
|
||
max fairness policy, IEEE Trans. Power Syst. 30 (2) (2015) 573–584, http:
|
||
10.1109/TNET.2021.3058251.
|
||
//dx.doi.org/10.1109/TPWRS.2014.2331424.
|
||
[3] D. Ma, G. Lan, M. Hassan, W. Hu, S.K. Das, Sensing, computing, and commu-
|
||
[27] L. Chen, S. Zhou, J. Xu, Computation peer offloading for energy-constrained
|
||
nications for energy harvesting iots: A survey, IEEE Commun. Surv. & Tutorials
|
||
mobile edge computing in small-cell networks, IEEE/ACM Trans. Netw. 26 (4)
|
||
22 (2) (2020) 1222–1250, http://dx.doi.org/10.1109/COMST.2019.2962526.
|
||
(2018) 1619–1632, http://dx.doi.org/10.1109/TNET.2018.2841758.
|
||
[4] J. Peng, H. Qiu, J. Cai, W. Xu, J. Wang, D2d-assisted multi-user cooperative
|
||
[28] Y. Dai, K. Zhang, S. Maharjan, Y. Zhang, Deep reinforcement learning for
|
||
partial offloading, transmission scheduling and computation allocating for mec,
|
||
stochastic computation offloading in digital twin networks, IEEE Trans. Ind.
|
||
IEEE Trans. Wirel. Commun. 20 (8) (2021) 4858–4873, http://dx.doi.org/10.
|
||
Inform. 17 (7) (2021) 4968–4977, http://dx.doi.org/10.1109/TII.2020.3016320.
|
||
1109/TWC.2021.3062616. [29] J. Mei, L. Dai, Z. Tong, X. Deng, K. Li, Throughput-aware dynamic task offloading
|
||
[5] S.-H. Kim, S. Park, M. Chen, C.-H. Youn, An optimal pricing scheme for under resource constant for mec with energy harvesting devices, IEEE Trans.
|
||
the energy-efficient mobile edge computation offloading with ofdma, IEEE Netw. Serv. Manag. (2023) 1, http://dx.doi.org/10.1109/TNSM.2023.3243629.
|
||
Commun. Lett. 22 (9) (2018) 1922–1925, http://dx.doi.org/10.1109/LCOMM. [30] X. Lyu, W. Ni, H. Tian, R.P. Liu, X. Wang, G.B. Giannakis, A. Paulraj, Op-
|
||
2018.2849401. timal schedule of mobile edge computing for internet of things using partial
|
||
[6] M. Li, Q. Wu, J. Zhu, R. Zheng, M. Zhang, A computing offloading game for information, IEEE J. Sel. Areas Commun. 35 (11) (2017) 2606–2615, http:
|
||
mobile devices and edge cloud servers, Wirel. Commun. Mob. Comput. 2018 //dx.doi.org/10.1109/JSAC.2017.2760186.
|
||
(2018) 1–10.
|
||
[7] Z. Liu, J. Fu, Y. Zhang, Computation offloading and pricing in mobile edge
|
||
computing based on stackelberg game, Wirel. Netw. 27 (7) (2021) 4795–4806, Jing Mei received the Ph.D degree in computer science
|
||
http://dx.doi.org/10.1007/s11276-021-02767-z. from Hunan University, China, in 2015. She is currently
|
||
[8] Z. Ning, J. Huang, X. Wang, J.J.P.C. Rodrigues, L. Guo, Mobile edge computing- an associate professor in the College of Information Sci-
|
||
enabled internet of vehicles: Toward energy-efficient scheduling, IEEE Netw. 33 ence and Engineering in Hunan Normal University. Her
|
||
(5) (2019) 198–205, http://dx.doi.org/10.1109/MNET.2019.1800309. research interests include cloud computing, fog computing
|
||
[9] Y. Mao, J. Zhang, K.B. Letaief, Joint task offloading scheduling and transmit and mobile edge computing, high performance computing,
|
||
power allocation for mobile-edge computing systems, in: 2017 IEEE Wireless task scheduling and resource management, etc. She has
|
||
Communications and Networking Conference, WCNC, 2017, pp. 1–6, http://dx. published more than 30 research articles in international
|
||
doi.org/10.1109/WCNC.2017.7925615. conference and journals, such as IEEE Transactions on Com-
|
||
[10] X. Zhao, L. Zhao, K. Liang, An energy consumption oriented offloading algorithm puters, IEEE Transactions on Parallel and Distributed System,
|
||
for fog computing, in: Quality, Reliability, Security and Robustness in Heteroge- IEEE Transactions on Service Computing, Cluster Computing,
|
||
neous Networks: 12th International Conference, QShine 2016, Seoul, Korea, July Journal of Grid Computing, Journal of Supercomputing.
|
||
7–8, 2016, Proceedings 12, Springer, 2017, pp. 293–301.
|
||
[11] Y. Chen, N. Zhang, Y. Zhang, X. Chen, W. Wu, X. Shen, Energy efficient
|
||
Cuibin Zeng received the B.S. degree in computer science
|
||
dynamic offloading in mobile edge computing for internet of things, IEEE Trans.
|
||
and technology from Jishou University, Jishou, China, in
|
||
Cloud Comput. 9 (3) (2021) 1050–1060, http://dx.doi.org/10.1109/TCC.2019.
|
||
2022. He is currently pursuing the M.S. degree at the
|
||
2898657.
|
||
[12] F. Li, H. Yao, J. Du, C. Jiang, Y. Qian, Stackelberg game-based computation College of Information Science and Engineering, Hunan
|
||
offloading in social and cognitive industrial internet of things, IEEE Trans. Ind. Normal University, Changsha, China. His research focuses
|
||
Inform. 16 (8) (2020) 5444–5455, http://dx.doi.org/10.1109/TII.2019.2961662. on resource scheduling and price allocation in mobile edge
|
||
[13] X. Hu, K.-K. Wong, K. Yang, Wireless powered cooperation-assisted mobile computing.
|
||
edge computing, IEEE Trans. Wirel. Commun. 17 (4) (2018) 2375–2388, http:
|
||
//dx.doi.org/10.1109/TWC.2018.2794345.
|
||
[14] F. Wang, J. Xu, X. Wang, S. Cui, Joint offloading and computing optimization in
|
||
wireless powered mobile-edge computing systems, IEEE Trans. Wirel. Commun.
|
||
17 (3) (2018) 1784–1797, http://dx.doi.org/10.1109/TWC.2017.2785305. Zhao Tong received the Ph.D degree in computer science
|
||
[15] P.K. Bishoyi, S. Misra, Enabling green mobile-edge computing for 5g-based from Hunan University, Changsha, China in 2014. He was a
|
||
healthcare applications, IEEE Trans. Green Commun. Netw. 5 (3) (2021) visiting scholar at the Georgia State University from 2017 to
|
||
1623–1631, http://dx.doi.org/10.1109/TGCN.2021.3075903. 2018. He is currently an associate professor at the College
|
||
[16] F. Zeng, Q. Chen, L. Meng, J. Wu, Volunteer assisted collaborative offloading of Information Science and Engineering of Hunan Normal
|
||
and resource allocation in vehicular edge computing, IEEE Trans. Intell. Transp. University, the young backbone teacher of Hunan Province,
|
||
Syst. 22 (6) (2021) 3247–3257, http://dx.doi.org/10.1109/TITS.2020.2980422. China. His research interests include parallel and distributed
|
||
[17] S. Xia, Z. Yao, Y. Li, S. Mao, Online distributed offloading and computing computing systems, resource management, big data and
|
||
resource management with energy harvesting for heterogeneous mec-enabled iot, machine learning algorithm. He has published more than 25
|
||
IEEE Trans. Wirel. Commun. 20 (10) (2021) 6743–6757, http://dx.doi.org/10. research papers in international conferences and journals,
|
||
1109/TWC.2021.3076201. such as IEEE-TPDS, Information Sciences, FGCS, NCA, and
|
||
[18] Q. Zhang, L. Gui, F. Hou, J. Chen, S. Zhu, F. Tian, Dynamic task offloading JPDC, PDCAT, etc. He is a senior member of the China
|
||
and resource allocation for mobile-edge computing in dense cloud ran, IEEE Computer Federation (CCF) and a Member of the IEEE.
|
||
Int. Things J. 7 (4) (2020) 3282–3299, http://dx.doi.org/10.1109/JIOT.2020.
|
||
2967502.
|
||
|
||
|
||
12
|
||
J. Mei et al. Journal of Systems Architecture 160 (2025) 103360
|
||
|
||
|
||
Longbao Dai received the B.S. degree in computer science machine learning, intelligent and soft computing. He has
|
||
and technology from Hunan University of Science and authored or co-authored over 850 journal articles, book
|
||
Engineering, Yongzhou, China, in 2021. He is currently chapters, and refereed conference papers, and has received
|
||
working toward the M.S. degree at the College of Infor- several best paper awards. He holds over 70 patents an-
|
||
mation Science and Engineering, Hunan Normal University, nounced or authorized by the Chinese National Intellectual
|
||
Changsha, China. His research interests focus on distributed Property Administration. He is among the world’s top 5
|
||
parallel computing, modeling and resource pricing and allo- most influential scientists in parallel and distributed com-
|
||
cation in mobile edge computing systems, and combinatorial puting in terms of both single–year impact and career–long
|
||
optimization. impact based on a composite indicator of Scopus citation
|
||
database. He has chaired many international conferences.
|
||
He is currently an associate editor of the ACM Comput-
|
||
ing Surveys and the CCF Transactions on High Performance
|
||
Keqin Li is a SUNY Distinguished Professor of Computer
|
||
Computing. He has served on the editorial boards of the
|
||
Science with the State University of New York. He is
|
||
IEEE Transactions on Parallel and Distributed Systems, the IEEE
|
||
also a National Distinguished Professor with Hunan Uni-
|
||
Transactions on Computers, the IEEE Transactions on Cloud
|
||
versity, China. His current research interests include cloud
|
||
Computing, the IEEE Transactions on Services Computing, and
|
||
computing, fog computing and mobile edge computing,
|
||
the IEEE Transactions on Sustainable Computing. He is an
|
||
energy–efficient computing and communication, embed-
|
||
IEEE Fellow and an AAIA Fellow. He is also a Member
|
||
ded systems and cyber–physical systems, heterogeneous
|
||
of Academia Europaea (Academician of the Academy of
|
||
computing systems, big data computing, high–performance
|
||
Europe).
|
||
computing, CPU–GPU hybrid and cooperative computing,
|
||
computer architectures and systems, computer networking,
|
||
|
||
|
||
|
||
|
||
13
|
||
|