Multiprocessor Scheduling of Sensor Transactions for Real-Time Data Quality Maintenance

In cyber-physical systems, sensor transactions should be eﬀectively scheduled to maintain the temporal validity of real-time data objects. Previous studies on sensor transaction scheduling mainly focus on uniprocessor systems. In this paper, we study the problem of data quality-based scheduling of sensor transactions on multiprocessor platforms. The data quality is deﬁned to describe the validity degree of real-time data objects. Two methods, named the Partitioned Scheduling for Quality Maximization (P-QM) and the improved P-QM scheduling (IP-QM), are proposed. P-QM maximizes the data quality by judiciously determining the preallocated computation time of each sensor transaction and assigns the transactions to diﬀerent processors. IP-QM improves the data quality obtained from P-QM by adaptively executing transaction instances on each processor based on the current status of the system. It is demonstrated through experiments that IP-QM can provide higher data quality than P-QM under diﬀerent system workloads.


Introduction
Cyber-physical systems (CPS) feature a tight combination of the computational and physical elements of the systems [1][2][3].
ey are widely used in applications that need to process real-time data in a timely manner. Example applications include road traffic control and industry process control [4][5][6]. e real-time data objects model the current status of entities in a system environment. Different from the traditional data objects, their values may become invalid with the passage of time. Associated with each real-time data object is a validity interval that specifies the lifetime of its current value. If this lifetime does not expire, the data object is temporally valid. Otherwise, it becomes invalid and a new data value needs to be installed.
In CPS, sensor transactions are generated to sample the status of external entities and update the corresponding data values. ey should be effectively scheduled to maintain the temporal validity of real-time data objects. is scheduling problem consists of two issues. at is, the determination of the release time and the deadline of each transaction instance and the scheduling of these instances. Various methods have been proposed to solve this problem. Some examples are the More-Less scheme (ML), the deferrable scheduling algorithm with fixed priority (DS-FP), and HS EDF [7][8][9]. ese methods aim to reduce the update workload while providing a complete guarantee on temporal validity, i.e., guaranteeing that the data objects are valid all the time. e update workload reduction allows a system to consume less energy. In addition, it leaves more processor resources to other types of transactions, such as user transactions and triggered transactions.
For many systems, providing completely guaranteed temporal validity for real-time data objects could be difficult. At first, the user transactions in the system may have strict timeliness requirements. ey compete with the sensor transactions for the same set of resources for execution. To guarantee temporal validity, more resources should be given to sensor transactions. is could lead to high deadline miss ratio of user transactions. Secondly, the system workload can be highly dynamic. e computation time of a transaction in the worst case can be much larger than that in the normal case. e arrivals of transaction instances may be aperiodical and unpredictable. As a result, data validity violations may occur during system overloads. To tackle these problems, several quality of service-(QoS-) based methods have been proposed to schedule sensor transactions and user transactions to maintain the quality of real-time data objects and the quality of transactions [10][11][12][13][14][15][16][17][18][19][20][21][22].
Current methods for sensor transaction scheduling are mainly restricted to uniprocessor systems. In this paper, we study the problem of data quality-based scheduling of sensor transactions on multiprocessor platforms. We consider the partitioned scheduling approach. On each processor, the earliest deadline first (EDF) scheme is adopted. e major contributions of the paper are as follows: (i) A definition of data quality is presented to describe the validity degree of real-time data objects. e definition considers the validity of individual data objects and the validity of correlated data object sets. (ii) Two scheduling methods, named the Partitioned Scheduling for Quality Maximization (P-QM) and the improved P-QM scheduling (IP-QM), are proposed to maximize the data quality. (iii) Experiments are conducted to evaluate the performance of the proposed methods. e results show that IP-QM outperforms P-QM in terms of the average quality of individual data objects and the average quality of correlated sets. e rest of the paper is organized as follows. Section 2 reviews previous studies on temporal validity maintenance. Section 3 describes the system model. e definition of data quality is also presented in this section. e details of the P-QM method and the IP-QM method are presented in Section 4 and Section 5, respectively. Performance studies are given in Section 6. Finally, Section 7 concludes the paper.

Related Studies
In recent years, there have been a number of studies on maintaining temporal validity of real-time data. ML adopts the periodic task model and the deadline monotonic scheme (DM) [7]. In ML, a sensor transaction's period is set to be no shorter than its relative deadline. e sum of the period and the deadline is equal to the validity interval. e Half-Half scheme (HH) can be viewed as a special case of ML [23]. DS-FP adopts a sporadic task model [8]. It reduces the processor workload by judiciously deferring the release times of transaction instances. Two extensions of DS-FP were presented in [24] to reduce the online scheduling overhead. e basic idea is to produce a hyperperiod of the DS-FP schedule and repeat the hyperperiod infinitely. A necessary and sufficient schedulability condition for DS-FP in discrete time systems was proposed in [25]. e problem of temporal validity maintenance under dynamic priority scheduling was first studied in [9]. ree algorithms were proposed to derive the periods and deadlines of sensor transactions under the earliest deadline first scheme (EDF). A two-phase algorithm was proposed in [26] to reduce the searching cost of period and deadline assignment under EDF. Li et al. [27] presented two methods to maintain temporal validity for EDF when transmission delays are considered. DS-FP was extended in [25] to be a dynamic priority scheduling algorithm by applying EDF to schedule update instances. e problem of scheduling tasks with both maximum distance constraints (i.e., temporal validity constraints) and minimum distance constraints was investigated in [28]. Jha et al. [29] investigated how to maintain the mutual temporal consistency of real-time data objects. Han et al. studied the problem of maintaining temporal validity in the presence of mode changes [30]. Two algorithms were presented to search for proper mode switch points. ese studies are all limited to uniprocessor systems.
In [31], Li et al. proposed several algorithms to partition a set of sensor transactions on multiprocessors under EDF and DM. e resource augmentation bounds of these algorithms were also derived. e global EDF algorithm and the half-half principle were applied in [32] to satisfy the data validity constraints. e energy-aware real-time data processing problem on multicore platforms was studied in [33]. Efficient techniques were proposed to maintain temporal validity while reducing energy cost. e studies described above focus on providing a complete guarantee on temporal validity. e co-scheduling problem of periodic application transactions and update transactions was investigated in [10][11][12]. e aim is to meet the deadlines of all the application transactions while maximizing the quality of data objects. e algorithms based on fixed priority scheduling scheme, EDF and DS-LALF, were presented, respectively. In [13], a set of extensions of ML was proposed to achieve the trade-off between QoS of temporal validity and the number of supported transactions. Labrinidis and Roussopoulos studied the problem of maintaining the freshness of views on a web server [14]. A quality-aware update scheduling algorithm was proposed based on the popularity of views. In [15], a QoS management architecture was proposed to support the desired QoS by applying feedback control, admission control, and flexible freshness management. On-demand schemes were proposed in [16] to skip the updates with similar data object values. Amirijoo et al. employed the notion of imprecise computation for QoS specification and management [17]. Differentiated data service approaches for transaction classes with diverse importance and QoS requirements were proposed in [18]. An effective approach was presented in [19] to decrease both the deadline miss ratio and power consumption by realtime query aggregation and data freshness adaptation. All the above QoS-based methods are designed for uniprocessor systems.
In [20], a scheduling framework was presented to assign update jobs to multiple tracks and schedule them on each track, with the objective of minimizing the total staleness of data objects in a streaming warehouse. A track here represents a fraction of the computing resources. Bateni et al. proposed an update scheduling algorithm upon multiprocessors that has bounded stretch and weighted staleness under the quasiperiodic model [21]. In [22], Kang and Chung proposed a multiple inputs/multiple outputs (MIMO) feedback control method to support the timeliness of data-intensive tasks running on multicore-based embedded platforms. Although these studies have also studied the data quality maintenance upon multiprocessors, their definitions of data quality are different from ours. In [20,21], the data quality is defined in terms of the data staleness. A data value will become stale after its generation.
e staleness increases linearly with the passage of time until a new value is installed. In [22], the data quality is defined as the ratio of the number of fresh data objects to the total number of data objects. In our work, however, a data value remains valid during its validity interval. e data quality is then defined based on the validity of individual data objects and the validity of correlated data object sets.

System Model
Let X � X i n i�1 denote a set of real-time data objects and Γ � τ i n i�1 a set of sensor transactions. e data object . τ i consists of a sequence of instances (update jobs), in which each samples a value of X i and installs it. e jth instance of τ i is denoted as τ i,j . We assume that the jitter between the sampling time and the release time of an instance is zero. g i (t) is the probability density function that describes the distribution of the actual computation times of τ i 's instances. Transactions in Γ are indexed according to nondecreasing order of their validity intervals, i.e., e transaction set Γ is scheduled upon a multiprocessor platform Π � π i m i�1 . In general, there are three approaches to schedule Γ: partitioned approach, global approach, and hybrid approach [34]. In the partitioned approach, each transaction is assigned to a processor and is always executed on it. In the global approach, an instance that has been preempted on one processor can resume its execution on a different processor. e hybrid approach is the combination of partitioned and global approach. It can be further classified into semipartitioned approach and cluster-based approach. In this paper, the partitioned approach is adopted. On each processor, the EDF scheme is used to schedule the transaction instances. Definition 1. X i is temporally valid at time t if, for its update job finished latest before t, the sampling time of this job (t r ) plus V i is not less than t, i.e., t r + V i ≥ t [7].
According to Definition 1, if X i 's current value is sampled at t c , it should be updated before t c + V i . Otherwise, it will become invalid. However, when the actual computation time of the corresponding update job is large, the update may not be finished before t c + V i due to the lack of processor resources.
otherwise. e quality of X i , q(X i ), is defined as follows: A correlated data object set is a set of real-time data objects whose values are used together to compute the corresponding derived data or to make decisions. It is often required that a certain percentage of the data objects in the correlated set are valid to produce the result with sufficient is set by users based on application requirements. e quality of Y k , q(Y k ), is defined as follows: e overall data quality of the system, q(X), is defined as follows: (3) Sensor transactions should be effectively scheduled on Π to maximize q(X). is scheduling problem consists of two subproblems. e first is to assign the sensor transactions to processors in a data quality-aware manner. Notice that, different from the traditional real-time tasks, the periods and deadlines of sensor transactions are unknown. ey must also be derived during the assignment. e second is to determine which instances can be executed in the system and how to execute them. Table 1 summarizes the major symbols that are used in the paper.

The P-QM Method
In this section, we present the P-QM method. Instead of maximizing q(X) directly, P-QM tries to maximize an approximated overall data quality. Let C p � C p i n i�1 denote the preallocated computation times of transactions in Γ. e actual computation time of instance τ i,j is denoted as c i,j . e quality of X i with respect to C p i , q(X i | C p i ), is defined as the probability that c i,j is no larger than C p i , i.e., Pr(c i,j ≤ C p i ). For a correlated set Y k , let C p (k) denote the preallocated computation times of the corresponding transactions. e quality of Y k with respect to C p (k), q(Y k | C p (k)), is defined as follows: In equation (4), can be viewed as the probability that the number of data objects in Y k that are valid at a time instant is no less than N th (Y k ) under C p (k). e overall data quality of the system with respect to C p , q(X | C p ), is defined as follows: P-QM maximizes q(X | C p ) by judiciously determining C p and assigning the transactions to processors. A transaction assignment method under the given C p is presented at Mobile Information Systems 3 first.
en, based on this method, a greedy heuristic is presented to determine C p in order to maximize q(X | C p ).

Assigning Transactions to Processors.
In P-QM, sensor transactions are assigned to processors in a way that the temporal validity of data objects in X is completely guaranteed under the given C p . Let P i and D i denote the period and the deadline of transaction τ i , respectively. e assignment problem is described as follows.
Given the transaction set Γ with C p and the multiprocessor platform Π, assign each τ i to a processor in Π and derive P i and D i , such that the following constraints are satisfied: (1) Validity constraints: ∀i, Feasibility constraints: ∀j, 1 ≤ j ≤ m, the transactions assigned to π j with given preallocated computation times and derived deadlines and periods are feasible by using EDF scheduling Notice that if the above constraints are all satisfied, then the temporal validity of data objects is guaranteed. is is because for each X i , its jth (j ≥ 2) value sampled at (j − 1)P i is updated by instance τ i,j which will finish before (j − 2)P i + V i . is means a value is certain to be refreshed before its validity interval expires.
Next, we present the algorithm used in P-QM to solve the assignment problem. Let Δ i and λ i denote the cumulative density and the density of e following conditions are checked at first: If equation (6) or equation (7) holds, the assignment mode is set to be the restricted mode. In this mode, transaction τ i 's deadline is restricted to be no larger than V i /2. Otherwise, it is set to be the unrestricted mode. In this mode, τ i 's deadline can be larger than V i /2 but should be no larger than V i . Suppose the first i transactions have been successfully assigned to processors. Let Γ k denote the set of transactions that are assigned to processor π k . e utilization Otherwise, it is computed as follows: e period of τ i is set to be V i − D i . τ i is assigned to the first processor π k such that the following conditions are satisfied: e assignment fails if no such processor exists. is process is described in Algorithm 1. Notice that both the calculation of τ i 's deadline and the check of condition 2 can be carried out in an incremental way. us, the time required to assign τ i to a processor is O(m). e check of equations (6) and (7) takes O(n) time. erefore, the time complexity of Algorithm 1 is O(mn + n). e following theorem shows the correctness of the algorithm. Theorem 1. If Algorithm 1 succeeds, then the temporal validity of data objects in X is completely guaranteed.
Proof. For each τ i ∈ Γ, when it is assigned to a processor, its deadline D i is set to be no less than C p i and no larger than V i − C p i due to equation (8) and condition 1 of the algorithm. Its period P i is set to be V i − D i . us, the validity constraints and the deadline constraints are satisfied. We only need to show the feasibility constraints are also satisfied. e first transaction τ 1 is assigned to processor π 1 with deadline C p 1 and period V 1 − C p 1 . Obviously, it is EDFschedulable. Suppose that the first i − 1 transactions have been assigned to processors and the transactions on each processor are EDF-schedulable. Consider that transaction τ i is assigned to processor π k . e transaction set on processors except π k are not affected by τ i ; thus, they are still EDF-schedulable. If τ i is the first transaction assigned to π k , then obviously it is EDF-schedulable. Let h * (τ i , t) denote the approximated demand bound of Notice that D i is larger than the deadlines of transactions that are assigned to π k prior to τ i . According to [35], e overall data quality of the system λ i e density of τ i Δ i e cumulative density of τ i λ max e maximum density among λ 1 , λ 2 , . . ., λ n 4 Mobile Information Systems Let Γ k,j denote the transactions in Γ k ∪ τ i that are assigned to π k before τ j . Based on equation (8), one has Equation (10) means the transaction set on π k is EDFschedulable after τ i is added. By induction, the feasibility constraints are satisfied when all transactions have been assigned to processors. □ Theorem 2. Algorithm 1 succeeds if equation (6) or equation (7) holds.
Proof. At first, we consider the case in which equation (6) holds and equation (7) does not hold. Suppose in this case, transaction τ i fails to be assigned to any processor. en, on each processor, one or both of the conditions of the algorithm are not satisfied. Let Π 1 denote the set of processors on which condition 1 is not satisfied and Π 2 denote the set of processors on which condition 1 is satisfied and condition 2 is not satisfied. |Π 1 | + |Π 2 | � m. e transactions assigned to processors in Π 1 and Π 2 are denoted as Γ 1 and Γ 2 , respectively. For each π k ∈ Π 1 , it must be According to eorem 1, equation (11) implies Notice that, for each τ j ∈ Γ k , V j /2 ≤ P j ≤ V j − C p j . Summing over all processors in Π 1 and making some transformations, we obtain For each π k ∈ Π 2 , it must be Summing over all processors in Π 2 and making some transformations, we obtain If both |Π 1 | and |Π 2 | are larger than zero, then based on equations (13) and (15) and the definition of Δ max and λ sum , we obtain Since equation (7) does not hold, it must be Input： Γ, C p , m Output： e assigned processor of each transaction 1. assignment mode ⟵ restricted if equation (6) or (7) holds. Otherwise assignment mode ⟵ unrestricted; 2. for i � 1 to n do 3. for j � 1 to m do 4.
if condition 1 and condition 2 are satisfied then 7.
Γ is infeasible on Π under C p , return failure; 13. end if 14. end for 15. return success; ALGORITHM 1: Transaction assignment.

Mobile Information Systems
If |Π 2 | � 0, we also obtain equation (17). If |Π 1 | � 0, then λ sum > 1/2, otherwise equation (15) does not hold. We then obtain Both equations (17) and (18) contradict the assumption that equation (6) holds. en, we consider the case in which equation (7) holds. Suppose that the first i − 1 transactions have been successfully assigned to processors. For each π k ∈ Π, it must be Equation (19) means the deadline of τ i is not larger than V i /2; thus, condition 1 of the algorithm holds. Condition 2 of the algorithm also holds since equation (7) holds implies λ sum ≤ (1/2). erefore, each transaction can be assigned to a processor.
Notice that there may exist some transactions in Γ with λ i � (1/2). Let Γ h denote the set of such transactions. If |Γ h | > m, or |Γ h | � m and n ≥ |Γ h | + 1, then Γ is infeasible on Π since a processor can only accommodate just one transaction from Γ h . If |Γ h | < m, then eorem 2 can be applied to Γ/Γ h and m − |Γ h |.

Determining the Preallocated Computation Times.
e problem of determining the preallocated computation times C p is formulated as an optimization problem: subject to (1) Feasibility constraint: the transactions in Γ can be successfully assigned to processors under C p by Algorithm 1 (2) Computation time constraint: ∀i, l the preallocated computation time of X k l . Given two sets of preallocated computation times C p1 (k) and C p2 (k) of Y k . If for each l, Proof. Let q(i, j, 1) and q(i, j, 2) denote the quality of X k l i l�1 with a valid threshold of j under C p1 (k) and C p2 (k), respectively. For each i, 1 ≤ i ≤ |Y k |, it must be (1) For all j, 1 ≤ j ≤ i, q(i, j, 1) < q(i, j − 1, 1) and q(i, j, 1) can be calculated as follows: q(i, j, 2) can also be calculated in the same way. By applying equation (21), we obtain In equation (22), (1) and (2), respectively. Δ i− 1,j− 1 and Δ i− 1,j are no smaller than 0 due to the assumption on X k l i− 1 l�1 and (3). erefore, q(i, j, 2) ≥ q(i, j, 1). By induction, we know that According to eorem 3, q(X | C p ) will increase by increasing the preallocated computation times of some transactions. Based on this observation, P-QM adopts a greedy heuristic to determine C p . For each τ i , the heuristic sets the initial value of C p i to be its minimum computation time at the start. It then selects one transaction at a time to increase its preallocated computation time.
Let Δq i denote the increase in the overall data quality due to an increase of current C p i by the step size of δ. Δq i is computed as follows: To determine whether increasing C p i by δ is feasible, the following condition is checked at first: If equation (24) holds, then increasing C p i by δ is infeasible. To see why, let us suppose increasing C p k by δ is feasible; then, for each τ j (j ≠ i), there must exist at least one 6 Mobile Information Systems instance of τ j finished in any interval with length V j − C p j , otherwise the data object X j will be invalid at some time points. is is also true for τ i with C p i + δ. e total time required to execute the instances should not exceed the platform capacity. us, it must be Dividing both sides of the negation of equation (25) by , we obtain equation (24). Notice that if equation (24) holds, then τ i cannot be selected to increase C p i in the future. Equation (24) can be evaluated in constant time since the value of the sum term in it can be obtained in the previous step.
If equation (24) does not hold, the following condition is checked: If equation (26) holds, then increasing C p i by δ is feasible according to equation (6) and eorem 2. Equation (26) can be checked in an incremental way; thus, the time required is O(1). If equation (26) does not hold, Algorithm 1 is applied. If no transaction can be selected in a step or every transaction's preallocated computation time has reached its maximum value, the algorithm terminates and returns the current C p . e selection process is described in Algorithm 2.
More sophisticated optimization algorithms can be used for our optimization problem to obtain better solutions, such as the evolutionary algorithm and particle swarm optimization [36,37]. However, as the number and the worst case computation times of sensor transactions can be large, these optimization algorithms may coverage slowly. e usage of these algorithms for our optimization problem will be left as our future work. In addition, as will be illustrated in the experiment section, by applying the adaptive instance execution method on the transactions with preallocated computation times obtained from the greedy heuristic, the data quality can be greatly improved.
After determining the preallocated computation times and assigning transactions to processors, the system will execute the transaction instances to update the corresponding real-time data objects. In P-QM, when instance τ i,l arrives, if c i,l is no larger than C p i , then τ i,l is admitted, otherwise it is dropped. e admitted instances are scheduled using the EDF scheme.

The IP-QM Method
IP-QM also uses Algorithms 1 and 2 to determine C p i and assign transactions to processors. It further improves the data quality by executing transaction instances adaptively on each processor. Figure 1 shows the adaptive execution of instances on a processor.
As shown in Figure 1, when instance τ i,l arrives at processor π k , the instance dropper determines whether τ i,l is admitted or not. If τ i,l 's actual computation time c i,l is no larger than C p i , then τ i,l is admitted. Otherwise, the following condition is checked: In equation (27), c r (s, r i,l ) is the total remaining computation time of τ s 's instances with release times no larger than r i,l and deadlines no larger than d i,l . d i,l is the (absolute) deadline of τ i,l , d i,l � (l − 1)P i + D i . E(s) is the average computation time of an instance of τ s : N(s, r i,l ) is the number of τ s 's instances with release times larger than r i,l and deadlines no larger than d i,l : τ i,l is admitted if equation (27)

Experiment Evaluation
is section presents the results obtained from performance studies on the proposed methods. e performance metrics used in the experiments are the average quality of individual data objects (ADQ_IND), the average quality of correlated sets (ADQ_COR), and the average update workload (AUW). e definition of ADQ_IND and ADQ_COR are given by equations (1) and (2). Let W denote the simulation time and L p the total time that processor π p is busy executing the transaction instances: e parameters and default settings used in the experiments are presented in Table 2. e validity interval of a data object is uniformly distributed in [2000,4000]. e computation time of a sensor transaction is generated following the normal distribution with mean computation time EC i and standard deviation DC given in Table 2. EC i itself is uniformly selected in [10,20]. e number of data objects in a correlated set is uniformly distributed in [2,8]. ese objects are randomly selected from the data object set. e threshold of a correlated set is set to be N c k · P th , where P th indicates the percentage of data objects in Y k that are permitted to be invalid at a time point. Figure 2 shows the average quality of individual data objects when the number of sensor transactions varies from 100 to 200. e number of processors is 2. P th is set to be 0.4. When the number of transactions is no larger than 120, the ADQ_IND of P-QM and IP-QM are equal to 1. Both methods' ADQ_IND decrease as the number of transactions increases. However, the ADQ_IND of IP-QM keeps higher than that of P-QM. is is because for an incoming instance with actual computation time larger than the preallocated computation time, IP-QM will accept it for execution if equation (27) is satisfied, while P-QM always drops it. In addition, IP-QM will drop some instances that are already in  8 Mobile Information Systems the system. As discussed in Section 5, this kind of dropping does not affect the validity of data objects. However, it can leave more processor resources to instances with long execution times. Notice that the difference in the data quality between the two methods increases as the number of transactions increases. When the number reaches 200, the difference goes to about 0.37. Figure 3 shows the average quality of correlated sets. It can be seen that the ADQ_COR of IP-QM is constantly higher than that of P-QM. For example, when the number of transactions is 180, the ADQ_COR of IP-QM is about 0.98, while the ADQ_COR of P-QM is about 0.87. Both methods' ADQ_COR decrease with the increase in the number of transactions. However, IP-QM's ADQ_COR drops much slower than P-QM. One observation from Figures 2 and 3 is that the ADQ_COR is higher than the ADQ_IND under both methods. e reason is that the threshold of a correlated set is not large, which leads to a high probability that the correlated set is valid at a time point while the individual data objects in it are not. In addition, since the data objects in a correlated set are selected randomly from the whole data object set, the average valid ratio of them is very close to the average valid ratio of all data objects. Figure 4 shows the average update workload. It can be seen that when the number of sensor transactions is larger than 120, the AUW of IP-QM is higher than that of P-QM. In addition, the AUW of IP-QM always goes up when the number of transactions increases, while the AUW of P-QM   Mobile Information Systems goes up before the number reaches 140 and drops slowly after that. is is because in P-QM, when there are a large number of transactions in the system, to accommodate them, the preallocated computation times of them are decreased. is leads to more rejected instances which cancel out the increase in the system workload. In IP-QM, however, many of the instances rejected by P-QM can be accepted and finished due to the admission and dropping rules. ese extra instances contribute to the higher average quality of individual data objects and correlated sets. Figures 5 and 6 show the average quality of individual data objects and correlated sets under different setting of P th , respectively. e number of sensor transactions is fixed at 200. It can be seen that the ADQ_IND and the ADQ_COR of IP-QM are higher than that of P-QM. For example, when P th is 0.3, the ADQ_IND and the ADQ_COR of IP-QM are about 0.87 and 0.92, while the ADQ_IND and the ADQ_COR of P-QM are about 0.53 and 0.58. ere is not much changes in the ADQ_IND of both methods since the average system workload under different values of P th are very close to each other when the number of transactions is fixed. Meanwhile, both methods' ADQ_COR increase as the value of P th increases. is is because a larger P th means fewer data objects in a correlated set are required to be valid; thus, there are more chances to access a correlated set that is valid under both methods.  Experiments were repeated for systems with 4 processors. Figures 7 and 8 show partial results. e number of transactions varies from 220 to 320. P th is set to be 0.4. It can be seen that IP-QM still outperforms P-QM in terms of ADQ_IND and ADQ_COR. In addition, the performance degradation of both methods becomes much slower when more processor resources are available.

Conclusions
is paper studies the problem of data quality-based scheduling of sensor transactions on multiprocessor    platforms. A definition of data quality is given to describe the validity degree of real-time data objects. Two methods, P-QM and IP-QM, are proposed. P-QM maximizes the data quality by judiciously determining the preallocated computation times of sensor transactions and assigning the transactions to processors. IP-QM improves the data quality obtained from P-QM by adaptively executing transaction instances on each processor. Experiment results show that IP-QM outperforms P-QM in terms of the average quality of individual data objects and the average quality of correlated sets. In this work, the partitioned approach is adopted for transaction scheduling. In the future, we will study how to use the global approach and the hybrid approach to schedule the transactions so that the real-time data quality can be effectively maintained.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.