CN108009024A - Distributed game task discharging method in Ad-hoc cloud environments - Google Patents
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F9/00—Arrangements for program control, e.g. control units
- G06F9/06—Arrangements for program control, e.g. control units using stored programs, i.e. using an internal store of processing equipment to receive or retain programs
- G06F9/46—Multiprogramming arrangements
- G06F9/48—Program initiating; Program switching, e.g. by interrupt
- G06F9/4806—Task transfer initiation or dispatching
- G06F9/4843—Task transfer initiation or dispatching by program, e.g. task dispatcher, supervisor, operating system
- G06F9/485—Task life-cycle, e.g. stopping, restarting, resuming execution
- G06F9/4856—Task life-cycle, e.g. stopping, restarting, resuming execution resumption being on a different machine, e.g. task migration, virtual machine migration
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F9/00—Arrangements for program control, e.g. control units
- G06F9/06—Arrangements for program control, e.g. control units using stored programs, i.e. using an internal store of processing equipment to receive or retain programs
- G06F9/46—Multiprogramming arrangements
- G06F9/50—Allocation of resources, e.g. of the central processing unit [CPU]
- G06F9/5005—Allocation of resources, e.g. of the central processing unit [CPU] to service a request
- G06F9/5027—Allocation of resources, e.g. of the central processing unit [CPU] to service a request the resource being a machine, e.g. CPUs, Servers, Terminals
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- G06F2209/00—Indexing scheme relating to G06F9/00
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- G—PHYSICS
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- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F2209/00—Indexing scheme relating to G06F9/00
- G06F2209/50—Indexing scheme relating to G06F9/50
- G06F2209/509—Offload
Abstract
The invention discloses one kind to be based on distributed game task discharging method in Ad Hoc cloud environments, including:The transmit power and the task amount of processing speed and task unloading equipment setting constraints between equipment are assisted according to task unloading equipment and task;So that system overall tasks disposal ability C is maximized under the constraints;Distribution task is unloaded according to C is maximized;Distribution game task discharging method according to the present invention, it is possible to achieve choose suitable unloading target device and the purpose of optimum distribution resource and task in systems, be finally reached the maximization of system processing power.
Description
Technical Field
The invention relates to the field of communication, in particular to a distributed game task unloading method in an Ad-hoc cloud environment of a self-organizing network, which adopts a distributed game algorithm to distribute resources.
Background
With the development of science and technology, mobile intelligent terminals gradually come into the sight of people, and the related applications are more and more extensive and play more and more important roles in the daily life of people. The intelligent terminal can help a user to process more and more heavy data tasks, but the resources of the intelligent terminal cannot be improved from the date of factory delivery, the limitation of the processing capacity of the intelligent terminal cannot meet the rapidly-increasing requirement of people, and the user cannot be updated and updated anytime and anywhere. Therefore, in order to improve processing efficiency and reduce power consumption, a resource sharing scheme based on mobile cloud computing is beginning to be widely adopted.
In mobile cloud computing, clouds capable of handling tasks offloaded from mobile terminals include remote clouds, local clouds, and Ad-Hoc clouds. In remote clouds, mobile devices typically offload computing tasks to a remote cloud computing center or server via a cellular network. Remote clouds generally have strong computing power, but communication costs are high and latency is large; the local cloud can directly unload the computing task to a PC or a server near the mobile device through Wi-Fi to be executed, the local cloud has the dual advantages of strong computing capability and large communication bandwidth, however, in many wireless environments, because the local cloud device with strong functions is not deployed, the user resources are limited, and the remote cloud center is too far away, so that the problems of high transmission delay, high cost and the like exist. Therefore, under the condition of limited resources, the idle resources of the neighbor users are shared for cooperative processing through task segmentation and unloading, mobile cloud computing can be realized in an Ad-Hoc mode, and the processing capacity of the mobile users is improved. Therefore, the concept of Ad-Hoc cloud is proposed in the prior art, and is used as an extension of mobile cloud computing in an infrastructure-free wireless environment, so that mobile users share resources and cooperatively process tasks through mutual cooperation. Meanwhile, the industry also proposes related concepts such as moving edge calculation and fog calculation for similar application scenarios, and is widely recognized. However, the prior art does not mention the problems of how to select a proper mobile user, how to reasonably allocate resources for traffic offloading, how to allocate tasks and the like in an Ad-Hoc cloud environment.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a distributed game task unloading method in an Ad-hoc cloud environment, which can realize the purposes of selecting proper task unloading target equipment in a system and optimally allocating resources and tasks.
The invention discloses a distributed game task unloading method in an Ad-hoc cloud environment of a self-organizing network, which comprises the following steps of:
setting constraint conditions according to the sending power and the processing rate between the task unloading equipment and the task assisting equipment and the task quantity of the task unloading equipment;
maximizing the overall task processing capacity C of the system under the constraint condition;
the allocation tasks are offloaded according to maximize C.
Preferably, the maximizing the overall task processing capacity C of the system under the constraint condition includes:
obtaining a global optimal solution according to the resource allocation global optimization;
performing task allocation according to the global optimal solution to obtain the minimum effective processing duration of the unloading task;
and obtaining the overall task processing capacity C of the maximized system according to the global optimal solution and the minimum effective processing duration.
Preferably, the constraint condition includes:
wherein, P ij,t Representing the transmission power, P, from device i to device j ji,t Represents the transmission power from device j to i; r ij,p Representing the processing rate, R, from device i to device j ji,p Represents the processing rate from device j to i; m is a group of ij Representing the amount of tasks offloaded from device i to device j;represents the average transmit power of the device i,represents the average processing rate of device i;represents the average transmit power of the device j,represents the average processing rate of device j;representing the average task size of device i.
Preferably, the obtaining a global optimal solution according to the resource allocation global optimization includes:
wherein the content of the first and second substances,respectively representing a selling price of transmission power from a device i to a device j and a selling price of transmission power from the device j to the device i;a selling price representing a selling price from the device i to the device j and a selling price representing a processing rate from the device j to the device i, respectively; p ij,t ,P ji,t Represents the transmission power from device i to device j and from device j to i, respectively; r ij,p ,R ji,p Respectively representing the processing rate from device i to device j and the processing rate from device j to i; w represents a bandwidth; g ij Represents the channel gain between devices i and j; n is a radical of ij Representing the noise power between devices i and j; beta is a ij Selling prices for the processing rates from device i to device j;respectively representing a transmission power threshold from device i to device j and a transmission power threshold from device j to i;is the result of the optimization of the processing rate from device j to device i.
Preferably, the global optimization of resource allocation includes that, considering the transmission rate and the processing rate, the maximum effective unloading rate R obtained by resource allocation is:
wherein N represents the number of devices; r is ij,p And R ji,p Respectively representing the processing rate from device i to device j and the processing rate from device j to i; r ij,t And R ji,t The transmission rate from device i to device j and the transmission rate from device j to device i, respectively; r ii,p Representing the processing rate of device i at the local node.
Preferably, the task allocation according to the global optimal solution, and obtaining the minimum effective processing duration of the offload task includes:
wherein N represents the number of devices, T ij For the total duration of the task to be offloaded from i to j, R ij For the total rate from device i to j, M ij In order to off-load the amount of tasks,representing the average task size of device i.
Preferably, the offloading assigning the task according to the maximization C includes outputting a task of device i to device j according to the following formula:
wherein R is ij Representing the total rate, R, from device i to device j i Representing the total rate at device i, M i Task that indicates that device i needs to be offloaded, M ij Representing the tasks of device i through device j.
Compared with the prior art, the distributed game task unloading method can achieve the purposes of selecting appropriate unloading target equipment in the system and optimally allocating resources and tasks, and finally achieves the maximization of the processing capacity of the system.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the embodiments of the present invention will be briefly described below. However, the following figures are some examples of the invention, and it will be obvious to a person skilled in the art that other embodiments can be obtained from these figures without inventive effort.
FIG. 1 is a schematic diagram of a system model of the present invention;
FIG. 2 is a schematic flow diagram of a preferred embodiment of a distributed game task offloading method in an Ad-Hoc cloud environment of the present invention;
FIG. 3 is a flowchart of an embodiment of maximizing the overall task processing capacity C of the system under the constraint of the present invention;
FIG. 4 is a diagram illustrating a comparison of system capacity simulation between a distributed game task offloading strategy and a centralized strategy, a greedy strategy, and a local processing strategy in accordance with the present invention;
FIG. 5 is a diagram illustrating a comparison of delay simulation between a distributed game task offloading strategy and a centralized strategy, a greedy strategy, and a local processing strategy according to the present invention;
FIG. 6 is a diagram illustrating comparison of energy consumption simulation between a distributed game task offloading strategy and a centralized strategy, a greedy strategy, and a local processing strategy according to the present invention;
FIG. 7 is a simulation comparison diagram of algorithm time complexity between a distributed game task offloading strategy and a centralized strategy, a greedy strategy, and a local processing strategy according to the present invention.
Detailed Description
Preferred embodiments of the present invention will be described below with reference to the accompanying drawings. However, it will be appreciated by those skilled in the art that the present invention may be embodied in different forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art.
Game theory, a mathematical tool, has been widely used to observe the behavior between different objects, and the interaction between different strategies, in order to maximize the utility of an individual. Therefore, in order to realize resource sharing among mobile devices, the invention provides a distributed scheme based on game theory. Suitable assistant(s) are found for task offloading by allocating resources and tasks appropriately. Firstly, an effective distributed method is used, an approximately centralized optimal solution is obtained, secondly, a resource allocation algorithm based on a game theory is designed for task transmission and processing, and finally, task unloading is carried out according to a resource allocation strategy.
As shown in FIG. 1, for the sake of generality, assume that in an Ad-Hoc cloud environment, there are N mobile devices (using MU) i ,i∈[1,N]Representation). Wherein, the task uninstalling device MU 1 Computationally intensive task M to be processed 1 Depending only on the capabilities of the devices themselves to accomplish these tasks may exhaust their resources or have significant computational delay, resulting in a poor user experience. At the same time, the nearby task assistance device MU 2 、MU 3 Has abundant resources. MU (Multi-user) 1 Offloading its own tasks to the MU over a premium wireless network (e.g., wi-Fi) 2 、MU 3 ,MU 2 、MU 3 The task is completed by utilizing the self computing power, and the result is fed back to the MU 1 And finishing the calculation processing work of the task.
Assuming that the task processing capacity of the whole system is C, a higher C value indicates that the system has higher available task processing capacity, and thus becomes the most important target in Ad-Hoc mobile cloud computing. The C-maximization problem is defined as the optimization problem (P1), namely:
max{C} (1)
P ij,t and P ji,t Representing the transmission power from device i to device j and from device j to i, respectively; r ij,p And R ji,p Respectively representing the processing rates from device i to device j and from device j to i;andrespectively representing the average transmission power and the average processing rate of the device i;andrespectively representing the average transmission power and the average processing rate of the device j; m is a group of ij Representing the amount of tasks offloaded from device i to device j; andrepresenting the average task volume of device i.
Where constraints (2), (3), (4), (5) indicate that the sum of the allocated transmission powers and the sum of the allocated processing rates cannot exceed the threshold for each mobile device, constraint (6) indicates that all tasks should be completely offloaded or processed locally.
According to the idea of the invention, the problem P1 is further decomposed into two parts, namely a resource allocation optimization problem P1 'and a task allocation optimization problem P1' for optimization.
Resource allocation problem P1' optimization: considering the transmission rate and the processing rate of the optimized device, the maximum effective unloading rate R obtained through resource allocation is:
s.t.(2),(3),(4),(5)。
wherein R is ij,p And R ji,p Respectively representing the processing rates from device i to device j and from device j to i; r ij,t And R ji,t The transfer rates from device i to device j and from device j to device i, respectively. When i ≠ j, i will task M ij Sending to j; if i = j, i is processed at the local node at the processing rate R ii,p To process task M ii 。
Task allocation problem P1' optimization: the minimum effective offload duration for processing offload tasks is mainly considered to be obtained by task allocation,
s.t.(6)。
wherein, T ij A minimum effective duration of task offloading for the two devices; m ij To offload the task volume; r ij Is the maximum effective unload rate;
next, a method of dealing with the buy-sell game will be described to solve the problem P1'. And then designing a distributed algorithm to realize an optimal balanced configuration strategy of the resources. And finally, according to the configuration of the resources, proposing a task allocation scheme.
R ij Is a convex function of the transmission power. R can be obtained based on the Shannon formula ij And R ji Are each R ij,t And a convex function of rj, t. While R is ij And is a convex function of the transmission power.
Since the constraints (2), (3), (4), (5) are convex functions, the optimization problem P1' can be solved using the Lagrangian multiplier method, where the Lagrangian factor λ 1 ,λ 2 ,λ 3 ,λ 4 >, 0 corresponds to the constraints of transmission power and processing rate in (2), (3), (4) and (5), respectively. Equations (10) to (23) can be obtained as follows by solving the convex function problem using the KKT condition.
Separately obtaining P from L ij,t 、P ji,t 、R ji,p 、R ii,p 、λ 1 、λ 2 、λ 3 、λ 4 The partial derivatives were made equal to 0 and the results obtained after finishing were as follows:
wherein, the first and the second end of the pipe are connected with each other,are respectively the optimal solutions of the formulas (11) to (18), andare all larger than zero (see formula (23));andrepresents the maximum transmission power from i to j and j to i, respectively;andrepresents the maximum transmission power from i to j and j to i, respectively;representing a maximum processing rate of the local node;is the maximum of the Lagrange factor
By solving the global optimization problem P1', an optimal resource allocation scheme for transmission power and processing rate can be obtained. However, it is very difficult to collect all necessary information, and it is also not feasible to construct a centralized node for computing from a global perspective in an Ad-Hoc mobile cloud environment. Next, using a buy-sell game, the global optimization problem P1 'is converted into a distributed optimization problem P2, and a distributed algorithm is designed to implement the optimal solution of P1'.
The utility of buyer i is
Wherein alpha is ij And alpha ji Selling prices of transmission power from i to j and selling prices of transmission power from j to i, respectively; beta is a ij Selling prices, beta, for processing rates from i to j ii Indicating the processing rate selling price processed at the local node i itself.
Therefore, to obtain the maximumAre respectively paired withCalculating P ij,t ,P ji,t ,R ji,p ,R ii,p First order partial derivatives and making them equal to 0 yield equations (25) to (28) as follows:
here, theThe solutions of equations (25) to (28) are also the optimal resource requirements.
When the KKT condition of P1' is observed, it can be seen that
Equations (11) to (14) are equivalent to equations (25) to (28). I.e. the optimal resource demand is at the buyer As it is equivalent in P1'.
From equation (8), since Rij = Rii, p when i = j, it can be seen thatIs a linear function of ri, p. Therefore, ifIs the maximum effective processing rate that can be provided. OtherwiseIn summary, the optimal resource requirement can be obtained by equation (33).
When i seeks help from j, i is buyer j is seller. On the other hand, i is also a seller for itself, since i needs to buy transmission power to itself first to send its task to j. At the same time, i may handle some tasks at the local node, which also requires the use of i's own processing resources. Thus, each mobile device is both a buyer seeking to offload and a seller providing services.
The seller analyzes as follows
Wherein f () represents a variableα ji 、β ji 、W、G ij 、N、R ji,t 、R ji,p Substituting into the effective expression (24)) to calculate the maximum transmission powerAndthe expression of (1); w is the bandwidth;andrepresents the maximum transmission power from i to j and j to i, respectively;andrepresents the optimal selling prices of the transmission power from the device i to the device j and from the device j to the device i respectively;an optimal selling price representing a processing rate from the device i to the device j;andrepresents the maximum transmission power from i to j and j to i, respectively;representing a maximum processing rate of the local node; g ij And G ji Channel gains from i to j and from j to i, respectively; n represents the noise power; beta is a ji The selling price is the processing rate from j to i.
The optimization scheme (maximizing seller utility) for seller j is as follows
The constraint condition is
Using the Lagrange multiplier method, one can obtain
Based on the use of KKT conditions in the seller j optimization problem, one can obtain
α ji =μ 1 (38)
β ji =μ 2 (39)
In the same manner, the metrics for sellers j through i may be changed to obtain α ij ,Pij,β ij Rij, p is shown below.
α ij =μ 3 (45)
β ij =μ 4 (46)
Wherein, mu 1 Represents a selling price of transmission power from the devices j to i; mu.s 2 Representing devices j to devicePreparing a selling price of the processing rate of i; mu.s 3 Represents the selling price of transmission power, mu, from device i to j 4 A selling price representing a processing rate from the device i to j;represents the optimal selling price of the transmission power from the devices j to i;an optimal selling price representing a processing rate from the device j to the device i;represents the optimum selling price of transmission power from the devices i to j,an optimal selling price representing a processing rate from the devices i to j; as is apparent from the formulae (11) to (14), if α ij =λ 1 ,α ji =λ 2 ,β ji =λ 3 ,β ii =λ 4 (when i = j in equation (46)), it is exactly the seller-optimal problem in P1'. In this case, the buy-sell game is equivalent to the optimization problem (P1') of the resource allocation. Therefore, the optimal selling price of the seller and the resource demand of the buyer can be calculated, and then the globally optimal solution of P1' is obtained as follows.
Wherein g { } denotes a variableW、G ij 、N ij 、β ij 、P ji,t 、R ji,p Calculating the optimal price after substituting seller utility expressionAbout variablesW、G ij 、N ij 、β ij 、P ji,t 、R ji,p The expression of (1);respectively representing the optimal selling prices of the transmission power from the device i to the device j and from the device j to the device i;an optimal selling price representing a processing rate from the device i to the device j;the optimal selling price represents the task processing rate of the equipment; beta is a beta ij Selling prices for the processing rates from device i to device j; p is ij,t ,P ji,t Representing the transmission power from device i to device j and from device j to i, respectively; r ij,p ,R ji,p Respectively representing the processing rates from device i to device j and from device j to i; r ij,t ,R ji,t Respectively representing the transmission rates from device i to device j and from device j to i;the optimization result of the processing rate from the device j to the device i is obtained; w represents a bandwidth; g ij Represents the channel gain between devices i and j; n is a radical of ij Representing the noise power between devices i and j;representing the maximum transmission power between devices i and j, respectively.
For this reason, if the seller offers its optimal seller price to sell its resources, the buyer will issue a request for optimal resource demand accordingly. Thus, the optimization (P1') of the global resource configuration can yield an approximate result based on the result of the buy-sell game (P2).
Next, the present invention provides a distributed algorithm to implement the buy-sell game algorithm, and by updating the sale price and the resource demand, the optimized solution of the resource allocation in P1' is implemented. Algorithm 1 (resource allocation algorithm) determines the resource allocation for transmission and processing.
Then, based on the resource allocation result, the invention designs a task allocation algorithm in algorithm 2 (task allocation algorithm) to determine how many tasks should be allocated to the local and helper nodes respectively for the problem P1 ".
Fig. 4-7 show a delay comparison graph, an energy consumption comparison graph, a system capacity comparison graph, and an algorithm time complexity comparison graph between the distributed game task offloading strategy and the centralized strategy, the greedy strategy, and the local processing strategy (without task migration) according to the present invention.
As can be seen from fig. 4, the system capacity and the task amount of each task allocation strategy are independent of the allocated power and processing capacity. Since the centralized strategy is considered globally, the impact of the transmission power and the terminal processing capacity is integrated, thereby obtaining the maximum system capacity. Meanwhile, the distributed game task unloading strategy is inferior and is closest to the optimal global distribution result, and the greedy strategy is low in system capacity due to the fact that more resources are excessively pursued to be distributed for task unloading and the resources of the nodes are not fully utilized. In contrast to the greedy strategy, the local processing strategy has the lowest system capacity because it processes tasks by itself only because of resource constraints.
As can be seen from fig. 5, the task offloading delay of the centralized policy is the minimum, and then the distributed game task offloading policy, the greedy policy, and the local processing policy of the present invention are respectively used, which is mainly because the offloading delay of the task is mainly related to the system capacity, and the larger the system capacity is, the smaller the task offloading delay is.
As can be seen from fig. 6, although the centralized strategy can achieve the maximum system capacity, it also results in higher energy consumption than the distributed betting task offloading strategy of the present invention.
Fig. 7 shows the algorithm time complexity corresponding to the four strategies, and it can be seen that the complexity of the distributed game task offloading strategy of the present invention is significantly reduced compared to the centralized strategy.
Compared with the existing centralized strategy, greedy strategy and local processing strategy, the distributed game task unloading strategy has better comprehensive performance in the aspects of time delay, energy consumption, system capacity and algorithm time complexity and has stronger applicability. The distributed game task unloading method can achieve the purposes of selecting appropriate unloading target equipment in the system and optimally distributing resources and tasks, and finally achieves the maximization of the processing capacity of the system.
The above-mentioned embodiments, which further illustrate the objects, technical solutions and advantages of the present invention, should be understood that the above-mentioned embodiments are only preferred embodiments of the present invention, and should not be construed as limiting the present invention, and any modifications, equivalents, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (7)
1. A distributed game task unloading method in an Ad-hoc network cloud environment is characterized by comprising the following steps:
setting constraint conditions according to the sending power and the processing rate between the task unloading equipment and the task assisting equipment and the task quantity of the task unloading equipment;
maximizing the overall task processing capacity C of the system under the constraint condition;
the allocation tasks are offloaded according to maximize C.
2. The distributed gaming task offloading method of claim 1 wherein maximizing overall system task processing capacity C under the constraint comprises:
obtaining a global optimal solution according to the resource allocation global optimization;
distributing tasks according to the global optimal solution to obtain the minimum effective processing duration of the unloading tasks;
and acquiring the overall task processing capacity C of the maximized system according to the global optimal solution and the minimum effective processing duration.
3. The distributed gaming task offloading method of claim 2 wherein the constraints comprise:
wherein, P ij,t Representing the transmission power, P, from device i to device j ji,t Represents the transmission power from device j to i; r ij,p Representing the processing rate, R, from device i to device j ji,p Represents the processing rate from device j to i; m ij Representing the amount of tasks offloaded from device i to device j;represents the average transmit power of the device i,represents the average processing rate of device i;represents the average transmit power of the device j,represents the average processing rate of device j;representing the average task size of device i.
4. The distributed gaming task offloading method in an Ad-hoc cloud environment of claim 2 wherein the obtaining a global optimal solution based on resource allocation global optimization comprises:
wherein the content of the first and second substances,respectively representing a selling price of transmission power from the device i to the device j and a selling price of transmission power from the device j to the device i;a selling price representing a selling price from the device i to the device j and a selling price representing a processing rate from the device j to the device i, respectively; p ij,t ,P ji,t Represents the transmission power from device i to device j and from device j to i, respectively; r ij,p ,R ji,p Respectively representing the processing rate from device i to device j and the processing rate from device j to i; w represents a bandwidth; g ij Represents the channel gain between devices i and j; n is a radical of ij Representing the noise power between devices i and j; beta is a ij Selling prices for the processing rates from device i to device j;respectively representing a transmission power threshold from device i to device j and a transmission power threshold from device j to i;is the result of the optimization of the processing rate from device j to device i.
5. The distributed gaming task offloading method of claim 2, wherein the global optimization of resource allocation comprises considering a transmission rate and a processing rate, and a maximum effective offloading rate R obtained by resource configuration is:
wherein N represents the number of devices; r ij,p And R ji,p Respectively representing the processing rate from device i to device j and the processing rate from device j to i; r ij,t And R ji,t The transmission rate from device i to device j and the transmission rate from device j to device i, respectively; r ii,p Representing the processing rate of device i at the local node.
6. The distributed gaming task offloading method in an Ad-hoc cloud environment of claim 2, wherein the task allocation according to the global optimal solution, and obtaining a minimum effective processing duration for offloading tasks comprises:
wherein N represents the number of devices, T ij For the total duration of the task to be offloaded from i to j, R ij For the total rate from device i to j, M ij In order to off-load the amount of tasks,representing the average task volume of device i.
7. The distributed gaming task offloading method of claim 1 wherein said offloading tasks according to maximize C comprises exporting tasks from device i to device j according to the following formula:
wherein R is ij Representing the total rate, R, from device i to device j i Representing the total rate at device i, M i Task that indicates that device i needs to be offloaded, M ij Representing the tasks of device i through device j.
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