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A Mathematical Task Dispatching Model in Wireless Sensor Actor Networks

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In this paper, we propose a mathematical task dispatching model to shorten the total tasks completion time, i.e. make-span, in Wireless Sensor Actor Networks. We formulate the allocation of tasks as a Mixed Integer Nonlinear Programming (MINLP) problem with the goal of minimizing the completion time of tasks that have been allocated to actors but have not yet been dispatched to actors for execution in the networks. The proposed approach calculates the optimized dispatching rate of tasks that lead to minimum make-span. It is shown that this dispatching rate can also enlarge the network lifetime. Experimental results with a prototyped simulation of the proposed approach show shorter make-span and longer network lifetime compared to when one of the three famous task allocation algorithms, namely, the min-min, opportunistic load balancing (OLB), and stochastic allocation algorithms, is used.

1. Introduction

A group of wirelessly communicative sensor nodes and actor nodes that respectively collect environmental information and behave in reply to sensory information build up an individual kind of wireless network called Wireless Sensor Actor Networks (WSANs). The constituent parts of WSANs can be configured differently according to the requirements of applications and existing technologies. In this paper, we consider only WSANs with a semi-automated architecture wherein all sensor nodes transmit their sensory information to a single node in the network named the network sink. This singleton node is more powerful than sensor nodes and actor nodes, and it is made responsible for getting sensory information and determining appropriate tasks (actions) to be done by actors.

One of the main challenges of WSANs is to efficiently use all its capabilities at its disposal to satisfy the quality as well as functional requirements of running applications. In WSANs with semi-automated architecture, this challenge can be partly resolved by the singleton sink node if it can select the most proper set of actors to perform tasks using quality parameters such as reliability, make-span, and completion time of tasks.

To make efficient use of the capabilities of WSANs, the sink must decide on the most appropriate group of actors to perform the tasks using quality parameters such as make-span, network lifetime, and reliability of services. Therefore, the sink must figure out an efficient dispatching rate to distribute tasks to the related actors considering the limited size queue (buffer) of each actor and the fact that an actor cannot receive any more tasks when its buffer is full. The challenge for the sink is thus to find a dispatching rate that minimizes the completion time of tasks that are waiting in the associated queues of actors to be executed by actors.

Min-min, opportunistic load balancing (OLB), and stochastic allocation are three popular examples of task allocation algorithms that are generally used in distributed systems like WSANs. Load balancing is the main goal of OLB achieved by keeping all actors as busy as possible. This algorithm schedules the tasks based on minimum estimated completion time of tasks in arbitrary order. The min-min algorithm considers the approximate execution and completion time of all tasks on each actor and only then repetitively assigns a task with lowest completion time to an actor with minimum execution time. The stochastic allocation algorithm is very simple and allocates tasks to available resources (actors) stochastically. This algorithm does not consider any binding such as execution time of tasks and current situation of resources (idle/busy). The main advantage of this algorithm is its simplicity and can be implemented very fast.

However, existing WSAN scheduling algorithms suppose unbounded queues that are reasonably unrealistic. In this paper, we consider the limitation on the size of queues and yet attempt to reduce the completion time of allocated tasks to each actor that are not performed by the actor in such a way that the make-span is minimized. To attain this goal, it is necessary for the sink to get an approximation of the ability of each actor to calculate an appropriate dispatching rate for that actor and guarantee that the demanded quality parameters of running application are met.

2. Related Works

M. Sharifi et al. have presented an energy and time aware approach to assign tasks to actors in WSANs. They figure out the capability of actors to perform tasks and use this information to assign tasks to actors in such a way to lessen the make-span of tasks. They reported 45% improvement in the network make-span compared to when they use the OLB algorithm. Their approach provides a suitable tradeoff between completion times of all tasks and a balanced load on actors, but it ignores the limitation on the size of actors’ buffers.

Farias et al. proposed a task scheduling algorithm for WSANs to improve the energy efficiency and so, increasing the network lifetime. To reach this goal, their algorithm tries to utilize the characteristics of applications with common tasks and avoid repeating tasks unnecessarily. However, their approach can increase the total remaining energies of actuators, but neither make-span nor reliability of services has been considered by their algorithm.

Shu et al. presented an energy aware scheduling algorithm to maximize the network lifetime while making strict sensing guarantees in the WSN. In order to verify their algorithm, they performed an in-depth evaluation of its performance via large-scale simulations and reported an average of 39.2% improvement of network lifetime over the baseline method. The main drawback of their algorithm is that neither the reliability of services nor the execution deadline for applications was considered in their work.

Okhovvat et al. have proposed a starvation free, time and energy aware scheduling algorithm called Scate. This algorithm allows concurrent executions of any mix of small and large tasks and yet prevents probable starvation of tasks. Reducing the total completion time of tasks and increasing the residual energies of actors simultaneously was the dual objective of Scate. The main drawback of their algorithm is that it does not guarantee the execution deadline for applications.

Momeni et al. have proposed a mathematical approach to reduce average number of waiting tasks in WSANs. They calculate the best rate of dispatching of tasks by the network sink to allocated actors through a steady state analysis and showed that their approach reduces the mean number of waiting tasks. In their approach reducing the make-span did not consider explicitly, but they believe that their approach may reduce the total tasks completion time too.

Byun and So have proposed an epidemic-inspired algorithm for data dissemination in WSANs that considers the delay requirements and try to decrease energy consumption. They used a mathematical analysis to predict and support the demanded performance of an application. Their approach controls the infectivity rate that results in an adaptive number of active/sleep nodes. They asserted that their approach can reduce the energy consumption while achieving application delay requirements.

Given this background on task allocation, in this paper we present a mathematical model using queuing theory to reduce the mean number of allocated tasks awaiting execution by actors in WSANs.

3. Assumptions

We have considered a semi-automated WSAN with a single network sink and m actors Aj (j=1,…, m) that should perform n tasks Ti (i = 1, . . . , n). In such a network, a schedule for each task is an allocation of one or more time slots to one or more actors. This scheduling problem is known as an NP-complete problem. In this paper, the aim of our approach is defined to decrease the completion time of tasks allocated to each actor in order to minimize the make-span. This goal is achieved by the calculating of capability of each actor at the time of assignment of tasks such as its current task load, and its speed in executing tasks.

We have further assumed that tasks are independent and sensors transmit their gathered data from physical environment to the sink. The sink figures out the appropriate actions (tasks) and then dispatches them to actors to be performed. Tasks are non-preemptive and their generation process follows a Poisson distribution.

4. Proposed Approach

We compute the make-span as the sum of the completion time of allocated tasks to actors. Each of the actors is modeled by a M/M/1/K queuing system wherein tasks arrive at actor Ai with λi rate and are executed with µi rate. To reduce the make-span, we should adjust the dispatching rate of tasks to actor properly. As we assumed that the queue of each actor has limited capacity K, system will reach to a steady state and hence, there is no need to consider the relation λ < µ that has to be considered if the queues had assumed to be boundless.

In our proposed model, tasks are generated by the sink based on the received sensory information and then are assigned to appropriate actors. Appropriate actors are the actors that can finish tasks sooner and hence, minimize the completion time of tasks. These actors are determined by the sink in the proposed approach.

It is assumed that the generation rate of tasks (λ) follows a Poisson process and based on the splitting Poisson distribution, allocated actors receive the tasks with λi rate. This shown by relation (4.1) for n actors:


Since the main objective of the presented approach is to minimize the completion time of tasks that should be done by the actors, the dispatching rate of tasks to each actor have to be estimated appropriately. In fact, our approach aims to find the best dispatching rate λi (i=1 to n), to minimize the completion time of tasks waiting to be executed by the actors. Thus, each actor is modeled as an M/M/1/k queue wherein the interval time between the allocations of two consecutive tasks and also the service times is an exponential process. Figure 2, shows the continuous time Markov chain (CTMC) model of actor Ai as a M/M/1/K queue. Each ellipse denotes a state of actor Ai, and the number inside of each ellipse shows the number of existing tasks in the queue of actor Ai.

To get a steady state analysis of CTMC shown in Figure 2, we use following relations wherein πi denotes the steady state probability of existing tasks in state i. In these relations, λi denotes the rate of arrival tasks at state i, and µi is the service rate of actor Ai. Table 1 shows the notations we have used in defining the relations (4.2)-(4.19).

λi¬ .π0 = µi.π1

π1( λi¬ + µi ) = λi¬ .π0+ µi.π2

π2(λi¬ + µi) = λi¬ .π1+ µi.π3 (4.2)




λi¬ .πk-1 = µi.πk

Table 1. Notations used in the relations

Term Definition Term Definition

N Number of tasks Li Number of tasks in the actor Ai

K Size of queue of each actor LQi Number of waiting tasks in the queue of actor i

λi Arrival rate of tasks to actor i Wi The time that actor i finishes its assigned tasks

πi Steady state probability of existing tasks in state i WQi waiting time of tasks in actor i.

µi Service rate of actor i WQTotal Total waiting time of all tasks

Α A constant number that is greater than zero m Number of actors

As shown by (4.3), the total probability is always equal to 1 and hence, πo can be computed by (4.4):



Since each πn is a function of π0, every πn is greater than zero if and only if π0 is bigger than zero. According to (4.5), π0 and then all πn are positive.


α: Constant

The probability of steady state πn for each state of actor Ai is calculated by (4.6) using (4.2) and (4.3).


Using (4.4) and (4.6), we deduce (4.7):


Since it has been assumed that each actor process and carry out the tasks consecutively, if T tasks are in the buffer of an actor, T-1 tasks are waiting. We have considered that the queue of each actor has a limited capacity k and hence, it makes the system comes to a steady state. We can therefore calculate the number of tasks in the queue of actor Ai by (4.8) in which Li is the number of assigned tasks to the actor Ai, and LQi is the number of waiting tasks in the queue of that actor.


To compute the spent time of tasks, we used the Little theorem [21] and therefore, we get (4.9) in which W denotes spent time of tasks in an actor, L denotes the queue size of that actor, and λ denotes the arrival rate of tasks to that actor.

L= W. λ → W =L ∕ λ (4.9)

Relations (4.8) and (4.9) result in relation (4.10):


To figure out WQi, Wi should be calculated. To do this, both equality and inequality of λi and µi are studied. In the case of inequality, (4.11) can calculate Wi.


We can derive (4.12) and (4.13) from (4.11):



Simplification of (4.13) results in (4.14).


Finally, (4.15) can be derived from (4.11), (4.12), (4.13), and (4.14). We use (4.15) to determine the spent of allocated tasks to actor Ai.


In the case that λi and µi are equal, (4.16) gives the number of tasks allocated to Ai. In the other words, if the arrival rate of tasks to an actor is the same as the service rate by that actor the relation (4.16) can be used.


After solving and simplifying (4.11), (4.12), (4.13) (4.14), (4.15) and (4.16), we derive (4.17) that calculates the mean waiting time of tasks in actor Ai.

To calculate the total completion time of tasks that should be performed by actors in the WSAN, we apply (4.17) to compute the completion time of tasks in each actor. The total completion time of tasks that should be accomplished by actors is thus given by (4.18):


We can finally use (4.17), (4.18) and (4.19) to formulate the main goal of this paper, which is to minimize the overall completion times of all tasks in the network, i.e. make-span, presuming that the capacity of the all actors are the same and equal to k.

The goal will be as (4.19) where k, , are constants, 0 ˂ , 0 ≤ λi and where m is the total number of actors. It should be noted that if λi comes to zero, the actor Ai will be unavailable and the relation (4.19) is be applied to available actors.

5. Experimental Results

To show the efficiency of our approach we conducted our simulations using MATLAB in a typical scenario. We evaluated the proposed approach in compare with three well-known task allocation algorithms, namely, the min-min, OLB, and stochastic allocation algorithms in terms of total completion time of tasks and lifetime of actors. In addition, to study the effect of scale on the efficiency of our approach, we performed simulations in both large and small scales in two different settings. In the small scale, we assumed a 2D space, square field, 10m 10m, containing 100 sensor nodes with 1 meter transmission range, and 7 actor nodes. We have assumed that the tasks to be executed by actors were independent and that actors could browse the whole network with no restrictions on routing hops. The primary energy of each actor is assumed to be the same as others and equal to 25 Joules. The bandwidth of nodes is assumed to be 250 Kb/s.

In the large scale, we assumed a 2D space, square field, 100m 100m containing 10000 sensors with 1 meter transmission range, and 20 actor nodes. The primary energy of each actor was assumed to be the same as others and equal to 25 Joules. The bandwidth of nodes is assumed to be 250 Kb/s.

As the network sink is usually faster than actors and it has fewer faults (or ideally has no faults at all), we have assumed that the queue of the sink never overloads. The size of queue of each actor was assumed to be 10 and to simplify we assumed that the sizes of all tasks were the same. To have a better evaluation, actors are chosen from three different categories with fast, medium and slow service rates. ¬We further assumed that each actor runs only a single task at any time and tasks were independent and sensors transmitted their collected information from environment to the sink and the sink allocated tasks to each actor with proper rate.

It is important to note that using the proposed approach in larger scales with more sensor/actor nodes will lead to similar results. In fact, our choice of semi-automated architecture for WSANs does not confine the applicability of the proposed approach to real large-scale WSANs. As stated by Liu and others, some large-scale WSANs may be single-hop in terms of wireless communication for transmitting information. A sink can be mobile and get close to sensors so that transmission of data could be done in a single hop. In other examples, embedded sensors may move toward a stationary sink. For example, sensors can be embedded into actors to trace their locations over time. When the actor approaches a fixed sink, collected information can be transmitted.

The make-span of the network under four task allocation approach in both small scale and large scale settings, respectively. In the small scale wherein the required time to transmit data between sink and actors is not much compared to execution time of tasks on the actors, the min-min algorithm results in less make-span compared to OLB and stochastic allocation while the proposed approach results in the best make-span.”

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A Mathematical Task Dispatching Model In Wireless Sensor Actor Networks. (2019, Jun 27). GradesFixer. Retrieved January 25, 2022, from
“A Mathematical Task Dispatching Model In Wireless Sensor Actor Networks.” GradesFixer, 27 Jun. 2019,
A Mathematical Task Dispatching Model In Wireless Sensor Actor Networks. [online]. Available at: <> [Accessed 25 Jan. 2022].
A Mathematical Task Dispatching Model In Wireless Sensor Actor Networks [Internet]. GradesFixer. 2019 Jun 27 [cited 2022 Jan 25]. Available from:
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