J. Cent. South Univ. (2020) 27: 3364-3374

DOI: https://doi.org/10.1007/s11771-020-4552-2

Temporal consistency maintenance on multiprocessor platforms with instance skipping

BAI Tian(白天), LI Zhi-jie(李志杰), FAN Bo(范波)

School of Information Science and Engineering, Hunan Institute of Science and Technology,Yueyang 414000, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract: Maintaining temporal consistency of real-time data is important for cyber-physical systems. Most of the previous studies focus on uniprocessor systems. In this paper, the problem of temporal consistency maintenance on multiprocessor platforms with instance skipping was formulated based on the (m,k)-constrained model. A partitioned scheduling method SC-AD was proposed to solve the problem. SC-AD uses a derived sufficient schedulability condition to calculate the initial value of m for each sensor transaction. It then partitions the transactions among the processors in a balanced way. To further reduce the average relative invalid time of real-time data, SC-AD judiciously increases the values of m for transactions assigned to each processor. Experiment results show that SC-AD outperforms the baseline methods in terms of the average relative invalid time and the average valid ratio under different system workloads.

Key words: cyber-physical systems; sensor transactions; multiprocessor scheduling; temporal consistency

Cite this article as: BAI Tian, LI Zhi-jie, FAN Bo. Temporal consistency maintenance on multiprocessor platforms with instance skipping [J]. Journal of Central South University, 2020, 27(11): 3364-3374. DOI: https://doi.org/10.1007/ s11771-020-4552-2.

1 Introduction

Cyber-physical systems (CPSs) are used in many application domains that need to process real-time data in a timely manner [1-3]. Examples of real-time data include the position of a vehicle and the temperature of an engine. A real-time data object is temporally consistent (or temporally valid) if its latest value truly reflects the current status of the corresponding entity in the system environment [4]. Sensor transactions are generated to sample the status of entities and refresh the data values. They should be effectively scheduled to maintain the temporal consistency of real-time data; otherwise, the systems cannot make responses to environment change timely.

In recent years, there have been a number of works on temporal consistency maintenance. XIONG et al [4] proposed the More-Less scheme (ML) which adopts the periodic task model and the deadline monotonic scheme (DM) to schedule sensor transactions. XIONG et al [5] proposed the DS-FP method which reduces the processor workload by judiciously deferring the release times of transaction instances. The DS-FP method was extended in Ref. [6] to be a dynamic priority scheduling algorithm named DS-EDF by applying the earliest deadline first (EDF) policy for update jobs scheduling. XIONG et al [7] proposed three algorithms to derive the periods and deadlines of sensor transactions under EDF. LI et al [8] proposed a two-phase algorithm to reduce the searching cost of period and deadline assignment under EDF. HAN et al [9] studied the problem of maintaining temporal consistency in the presence of mode changes. Two mode switch points searching algorithms were presented. All above methods are designed for uniprocessor systems.

LI et al [10] proposed several algorithms to partition a set of sensor transactions on multiprocessors under EDF and DM. The resource augmentation bounds of these algorithms were also derived. ZHOU et al [11] proposed efficient methods to maintain temporal consistency while reducing energy cost on multicore platforms. The co-scheduling of update and control transactions on multiprocessor platforms was studied in Ref. [12]. Several partitioned scheduling algorithms were given. KANG [13] adopted the global EDF algorithm and the Half-Half scheme (HH) to satisfy the data validity constraints.

The works described above consider providing a complete guarantee on temporal consistency. This may be hard to achieve when the system workload is unpredictable or there is severe resource competition between different types of transactions. The feedback control method was applied in Refs. [14-18] to support the desired quality of data objects and quality of user transactions. XIONG et al [19] proposed a set of extensions of ML to achieve the trade-off between QoS of temporal consistency and the number of supported transactions. Scheduling algorithms, based on fixed priority scheduling, EDF and DS-LALF, were presented in Refs. [20-22] to meet the deadlines of all user transactions while maximizing the quality of data objects. KANG [23] presented an effective approach to decrease both the deadline miss ratio and power consumption by real-time query aggregation and data freshness adaptation. These works are all limited to uniprocessor systems.

In this paper, the problem of maintaining temporal consistency of real-time data objects on multiprocessor platforms is studied. The (m, k)- constrained model is used for instance skipping to provide a certain degree of temporal consistency. The major contributions of the paper are as follows:

1) The problem of temporal consistency maintenance on multiprocessor platforms with instance skipping was formulated based on the (m, k)-constrained model.

2) A partitioned scheduling method SC-AD was proposed to solve the problem. SC-AD minimizes the average relative invalid time of real-time data by judiciously determining the values of m for each sensor transaction and assigning the transactions to processors.

3) The performance of the proposed method was evaluated through experiments. It is shown that SC-AD outperforms the baseline methods in terms of the average relative invalid time and the average valid ratio.

2 System model

Let X={X_i}_1≤i≤n denote a set of real-time data and Г={τ_i}_1≤i≤n a set of sensor transactions. The data object X_i is associated with a validity interval V_i. Sensor transaction τ_i is used to update the value of X_i. The computation time of τ_i is C_i. The jitter between the sampling time and the release time of a transaction instance is assumed to be zero. Transactions are sorted by the shortest validity first (SVF) scheme [4], i.e., V_i ≤ V_j if i < j.

The sensor transactions are scheduled on a multiprocessor platform Π={π_i}_1≤i≤b. The partitioned approach is considered [24]. That is, a transaction is assigned to a processor and is always executed on it. On each processor, the fixed priority scheduling method is adopted to schedule the transactions. τ_i ’s priority is larger than τ_j if i

Definition 1: A real-time data object X_i at time t_c is temporally consistent if (for its instance τ_i,j finished last before t_c) the sampling time r_i,j plus V_i is not less than t_c, i.e., r_i,j+V_i≥t_c [5].

If the temporal consistency of real-time data cannot be fully guaranteed, a promising approach is to skip some transaction instances in a way that the user transactions can access valid data objects in a large part of time. The (m, k)-constrained model is a good choice [25, 26]. In this model, for transaction τ_i, m_i out of any k_i consecutive instances’ deadlines must be met. By changing the value of m_i, one can obtain different degrees of temporal consistency. There are two major patterns proposed in literature: the evenly-distributed-mandatory pattern and the deeply-red pattern. The former one is used in this paper since it exhibits a relatively good schedulability and stability [26]. According to the evenly-distributed-mandatory pattern, an instance τ_i,j is accepted if:

Otherwise, it can be skipped. For τ_i, let P_i and D_i denote its period and relative deadline, respectively. The total invalid time for k_i consecutive instances of τ_i is (k_i–m_i)P_i. The relative invalid time of X_i with respect to m_i, RIT(X_i,m_i), is defined as:

                   (1)

The average relative invalid time of the real-time data set X with respect to M={m_i}_1≤i≤n, ARIT(X, M), is defined as:

               (2)

Then the problem of temporal consistency maintenance on multiprocessor platforms with instance skipping (TCM-IS) is described as follows:

Given a sensor transaction set Г scheduled on Π. τ_i is represented by a three-tuple (C_i, V_i, k_i). Determine P_i, D_i and m_i for each τ_i, so that ARIT(X, M) is minimized, subject to: 1) For each τ_i, 2) For each τ_i, and 3) For each π_i, the set of transactions assigned to π_i is feasible under the fixed priority scheduling.

3 Partitioned scheduling method

In this section, the partitioned scheduling method SC-AD is presented to solve the TCM-IS problem. SC-AD uses HH scheme to assign periods and deadlines to sensor transactions. That is, for each τ_i, both P_i and D_i are set to be V_i/2.

3.1 Determining initial value of M

Let λ_i=C_i/V_i and denote the density and the cumulative density of τ_i, respectively. Let λ_max=max{λ_i|1≤i≤n} and Δ_max=max{Δ_i|1≤i≤n} denote the maximum density and maximum cumulative density among transactions in Г, respectively. According to Ref. [26], the request bound of τ_i in [0, t], R (i,t), is:

The following theorem gives a sufficient schedulability condition for Г on Π under any valid partitioning scheme with respect to the given M.

Theorem 1: For the given M, Г is schedulable on Π if

       (3)

Proof: Suppose that τ_i is the first transaction that cannot be assigned to any processor and inequality (3) holds for Г. Let Г_k denote the transactions that have been assigned to processor π_k. It must be:

                     (4)

By taking an upper bound of the left hand side of inequality (4), one has:

             (5)

Summing over all b processors and dividing both sides by V_i/2 yields:

   (6)

From inequality (6), one can obtain:

This contradicts inequality (3). So, the theorem is proved.

SC-AD uses theorem 1 to determine the initial value of M. It transforms the TCM-IS problem to the following optimization problem:

subject to: 1) and 2) Inequality (3) is satisfied.

The number of transactions and the value of k_i may be large. Thus the exhaustive searching method could be inefficient. A greedy method is given to solve the above problem. The initial value of each m_i is set to 1. Then at each step a transaction is selected to increase its value of m_i. Let M_l denote the largest value of m_i/k_i after l steps. The cost of increasing τ_i’s value of m_i by the step size of δ at the l+1-th step, c(i), is:

            (7)

In Eq. (7), I_i is an indicator function whose value is 1 if (m_i+δ)/k_i >M_l and 0, otherwise. The utility of τ_i, u(i), is δ/(2c(i)).

In a step, transaction τ_i is successfully selected if τ_i has the largest utility and inequality (3) is satisfied after the increase of τ_i’s value of m_i by δ. If no transaction can be selected due to the dissatisfaction of inequality (3) or every transaction’s value of m_i reaches the maximum value, the process ends. This process is described in Algorithm 1.

Algorithm 1 Calculation of m_i

Notice that inequality (3) can be evaluated in an incremental way. Let S_l denote the value of the numerator of the right hand side of inequality (3) after l steps. If τ_i is selected to increase its value of m_i, then one just needs to check the following condition:

Then S_l+1 is set to S_l+2c(i) if τ_i has the largest utility and the above condition holds. Thus inequality (3) can be evaluated in constant time and each step takes O(n) time. There are at most steps. Therefore, the overall time complexity of Algorithm 1 is

3.2 Partitioning transactions among processors

After obtaining the initial value of M, SC-AD assigns the transactions to processors. For τ_i, define AR (i, t) and FR (i) as follows:

To determining whether τ_i is feasible on π_k , the following condition is checked:

             (8)

The time required for the check is O(|Г_k|). It’s not hard to see from the proof of theorem 1 that if inequality (8) is satisfied, then τ_i is feasible on π_k. Otherwise, the worst case response time of τ_i with respect to Г_k, R_i,k, is calculated. Notice that R_k (i, t) is non-decreasing with t given m_i and P_i, thus R_i,k can be computed in an iterative way. Letdenote the value of R_i,k after the l-th iteration. Then:

              (9)

Theorem 2: A lower bound of R_i,k , LB(R_i,k ), is if

Proof: R_i,k is no less than C_i, so we only need to consider the case in which the value of LB(R_i,k) obtained by theorem 2 is larger than C_i. For τ_i, we show that following condition holds:

              (10)

If 0i/2, then:

.

Inequality (10) holds since R(i,t)=C_i in this case.If t>V_i/2, let t=(l+x)V_i/2, l ≥1, 0≤x<1, then:

Inequality (10) holds since R(i, t)≥C_i in this case. For a time point t, 0i,k), we have:

.

If the inequality condition of theorem 2 holds, by applying inequality (10), we obtain:

.

Thus the theorem is proved.

Notice that if the inequality condition of theorem 2 doesn’t hold, then for every time point t, 0i/2, it must be:

.

From the proof of theorem 2, we know that τ_i is certain to be infeasible on π_k .

LB(R_i,k) can be obtained during the evaluation of inequality (8). Thus there is no extra cost for computing LB(R_i,k). The initial value of R_i,k is set as LB(R_i,k). R_i,k is then computed using Eq. (9) iteratively. The iteration continues until If≤V_i/2 at this time, then τ_i is feasible on π_k. If there is one value of R_i,k that is larger than V_i/2 during the iteration, τ_i is considered to be infeasible on π_{k .} The computation of R_i,k takes O(|Г_k|V_i) time.

Let λ(k) denote the sum of densities of transactions assigned to π_k. SC-AD assigns the transactions to processors in a balanced way. For τ_i , the processor π_kthat is selected to accommodate it is:

π_k=argmin{λ(l)|τ_i is feasible on π_l, 1≤l≤b}.

The processor allocation is described in Algorithm 2. Notice that the transaction set can be successfully allocated among processors under SC-AD if Algorithm 1 succeeds. If Algorithm 1 returns failure, then Algorithm 2 will be invoked for the transaction set with each m_i being set to 1. If the partition still fails, the transaction set is considered to be infeasible on Π. The time required for the assignment of τ_i is O(biV_i). The overall time complexity of Algorithm 2 is O(bn²V_max) where V_max=max{V_i|1≤i≤n}.

Algorithm 2 Processor allocation

3.3 Further reducing ARIT

SC-AD tries to further decrease the average relative invalid time by adjusting the values of m_i for some transactions. The adjustment starts from the first processor. On each processor, several rounds of adjustment are conducted. In each round, the adjustment is conducted from the transaction with the lowest priority to the transaction with the highest priority.

Suppose that transactions on each processor are sorted in decreasing order of their priorities. For simplicity, we reuse τ_i to denote the i-th transaction on π_k. The worst case response time of τ_i in the lth round, after τ_s (i>s) is selected for adjustment, is denoted as R_i,k(l,s). When τ_s is selected, m_s is increased by δ, and then R_i,k(l, s) of all τ_i (i>s) are recalculated. If there is τ_a with R_a,k(l, s)>V_a/2, the adjustment for τ_s fails, and the next transaction is considered.

Theorem 3: If i=s+1, the initial value of R_i,k(l+1,s),can be set to be:

Otherwise, can be set to be:

max{R_i-1,k(l+1,s)+C_i, R_i,k(l+1,s+1), LB(R_i,k(l+1, s)}.

Proof: Let R^l,s(i, t) denote the request bound of τ_i in [0, t] in the l-th round, after τ_s’s value of m_s is increased. For a time point t, 0≤ti_,k(l,1):

.

In addition, R^l+1,s(s,t)≥R^l,1(s,t). Thus, R_i,k(l+1, s) is no less than R_i,k(l,1). The initial value of R_i,k(l+1,s) can be set to be R_i,k(l, 1).

If i>s+1, then the value of R_i,k(l, 1) is updated when the (s+1)th transaction’s value of m_s+1 is increased. With the same reasoning as above, we know that R_i,k(l+1, s) is no less than R_i,k(l+1, s+1). In addition, R_i-1,k(l+1, s) has been derived. For a time point t, 0≤t≤R_i-1,k(l+1, s):

Thus R_i,k(l+1, s)≥R_i-1,k(l+1,s)+C_i. From theorem 2 we know that LB(R_i,k(l+1, s)) is also an lower bound of R_i,k(l+1, s). The theorem is proved.

Theorem 4: The adjustment for τ_i on π_k is infeasible, if

                     (11)

Proof: If the adjustment for τ_i succeeds, then for each τ_j (j≠i), m_j instances are finished in an interval with length k_jV_j/2. For τ_i, m_i+δ instances are finished in an interval with length k_iV_i/2. It must be:

.

This implies:

.

Thus the adjustment for τ_i fails if the above inequality does not hold. The theorem is proved.

The adjustment process is described in Algorithm 3. According to theorem 3, the initial value of R_i,k(l,s) is non-decreasing in l and s. Let O(i) denote τ_i’s index in Г. The total number of feasibility checks for τ_i after a higher priority transaction is selected for adjustment is The total time required for the checks is If the adjustment for τ_s fails and τ_a(a>s) is the first transaction with R_a,k(l,s)>V_a/2, and then for each τ_i (so_(i)) must be added. Let S_F(k) denote the set of transactions on π_k that need to take the extra time into account. The time required for the adjustment of values of m_i is calculated as follows:

.

Thus the overall time complexity of Algorithm 3 is

Algorithm 3 Adjustment of m_i

Example 1: Consider a set of transactions {τ_i}_1≤i≤6 scheduled on {π_i}_1≤i≤2. The parameters of transactions are given in Table 1. The initial values of m_i of transactions obtained by SC-AD are {2,1,1,1,2,1}. The transactions are then allocated to processors. Transactions τ₁, τ₄ and τ₅ are assigned to π₁, and the rest are assigned to π₂. The values of m_i of transactions on π₁ and π₂ are adjusted. For π₁, the order of adjustment is τ₅, τ₄, τ₁ in each round. m₅ can be set to 6 since it has no impact on other transactions. The values of m₄ and m₁ can be increased to their maximum values since lower priority transactions remain feasible after adjustment. The similar process is conducted for transactions on π₂. The final values of m_i of transactions are {6,1,5,3,6,5}. Figure 1 shows the schedule generated by SC-AD in [0,40].

Table 1 Parameters of transactions

Figure 1 Schedule generated by SC-AD

4 Performance evaluation

In this section, an experimental evaluation of the proposed method was presented. The baseline methods SC-P and SC-G were given for comparison. SC-P uses Algorithm 1 to obtain the values of m_i. It then uses First-Fit scheme (FF) to assign the transactions to processors. SC-G determines the values of m_i in a greedy way. At each step, transaction τ_i is selected to increase m_i by δ if τ_i has the largest value of ((m_i+δ)c_i/k_iV_i) and the transaction set Г is feasible after this increase. The feasibility of Г is determined by trying to partition the transactions among the processors. Inequality (8) is used to determine whether a transaction can be accommodated to a processor.

The performance metrics include the average relative invalid time (ARIT), the average valid ratio(AVR) and the average workload (AW). ARIT is given by Eq. (2). Let T denote the simulation time, L(i), the total valid time of X_i during T, then:

.

Let W(i) denote the total busy time of processor π_i during T, then:

.

A summary of the parameters and default settings used in experiments are presented in Table 2. Systems with two and four processors were studied. The validity interval of a real-time data object is uniformly distributed between 2000 and 4000 ms. The computation time of a sensor transaction is uniformly distributed between 10 and 20 ms. The value of parameter k_i is randomly selected in Refs. [10, 20].

Table 2 Parameters and settings

Figure 2 shows the average relative invalid time of the methods when the number of sensor transactions varies from 140 to 240. The number of processors is 2. It can be seen that the ARIT of SC-AD is constantly lower than that of SC-P or SC-G. For example, when the number of transactions is 180, the ARIT of SC-AD is about 2.29, while the ARIT of SC-P and SC-G is about 6.13 and 5.77, respectively. This is because after assigning the transactions to processors, SC-AD adjusts the values of m_i for transactions on each processor according to the worst case response times of them. This further decreases the ARIT. As the number of transactions increases, the processor resources left for the methods to increase the values of m_i become less, thus the ARIT of the methods increases and the difference between them decreases. This is also verified in Figure 2. When the number of transactions reaches 240, SC-P always returns failure, and thus its ARIT becomes zero. However, SC-AD and SC-G can still provide certain degree of consistency. At this time, the ARIT of SC-AD is still lower than that of SC-G.

Figure 2 Average relative invalid time comparison (b=2)

Figure 3 shows the average valid ratio of the methods. It can be seen that the AVR of SC-AD is always higher than that of SC-P or SC-G. This is because according to Figure 2, the average value of m_i under SC-AD is larger than that under the other two methods. This means that more instances will be finished under SC-AD. If instance τ_i,j is finished at f_i,j, the data object X_i will be valid in [f_i,j, r_i,j +V_i]. Thus on the average the total valid time of a data object under SC-AD is larger than that under SC-P and SC-G. With the increase of transaction set size, the m_i becomes smaller; therefore, all methods’ AVR decrease.

Figure 4 shows the average workload of the methods. Since the average value of m_i under SC-AD is larger than that under SC-P or SC-G, the AW of SC-AD is also higher than that of SC-P or SC-G. This is verified by Figure 4. For example, when the transaction set size is 140, the AW of SC-AD is about 0.68 while the AW of SC-P and SC-G is about 0.25 and 0.27, respectively. All methods’ AW decrease with the increase of transaction set size.

Figure 3 Average valid ratio comparison (b=2)

Figure 4 Average workload comparison (b=2)

Figure 5 shows the average relative invalid time of the methods under different values of k_i. The number of transactions is fixed at 200. It can be seen that the ARIT of SC-AD is always lower than that of SC-P or SC-G. In addition, with the increase in the value of k_i, all methods’ ARIT increase. The reason is that under different values of k_i, the average system workloads are very similar due to the fixed number of transactions. This means that the resources left for a method to increase m_i are also nearly the same. Therefore, a larger k_i leads to a larger value of ARIT.

Figure 6 shows the average valid ratio of the methods. It can be seen that the AVR of SC-AD is higher than that of SC-P or SC-G. According to Figure 5, the average ratios between m_i and k_i of a method are very similar under different k_i, and thus the differences of AVRs under each method are not larger. This is also verified in Figure 6.

Figure 5 Average relative invalid time comparison (b=2, n=200)

Figure 6 Average valid ratio comparison (b=2, n=200)

Figures 7 and 8 show the average relative invalid time and the average valid ratio of the methods when the number of sensor transactions varies from 280 to 380. The number of processors is 4. It can be seen that the ARIT of SC-AD is still lower than that of SC-P or SC-G, and the AVR of SC-AD is larger than that of SC-P or SC-G. Experiments were also repeated for 8 processors; similar results were observed and are omitted here for brevity.

Figure 7 Average relative invalid time comparison (b=4)

Figure 8 Average valid ratio comparison (b=4)

5 Conclusions

In this paper, the problem of maintaining temporal consistency of real-time data on multiprocessor platforms with instance skipping was studied. This problem was formulated based on the (m,k)-constrained model. A partitioned scheduling method SC-AD was proposed. SC-AD judiciously determines the value of m for each transaction and allocates transactions among processors. It further increases the values of m for transactions on each processor. It is demonstrated through experiments that compared with the baseline methods, SC-AD archives lower average relative invalid time and higher average valid ratio.

Contributors

BAI Tian provided the method and wrote the first draft of the manuscript; LI Zhi-jie conducted the literature review and edited the draft of manuscript; FAN Bo analyzed the experimental results and edited the draft of manuscript.

Conflict of interest

BAI Tian, LI Zhi-jie, and FAN Bo declare that they have no conflict of interest.

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(Edited by YANG Hua)

中文导读

多处理器平台上考虑实例丢弃的数据时态一致性维护

摘要：维护实时数据的时态一致性对于信息物理融合系统来说非常重要。已有的研究工作大多针对单处理器平台来进行。基于(m, k)约束模型定义了多处理器平台上考虑实例丢弃的时态一致性维护问题，提出了一种解决该问题的方法SC-AD。SC-AD利用推导出的传感器事务集可调度的充分条件来确定各传感器事务参数m的初始值，然后据此进行多处理器事务分派。SC-AD合理地增加各处理器上已分派事务参数m的值以进一步降低实时数据的平均相对无效时间。实验结果表明，在不同的系统负载下，SC-AD在平均相对无效时间和平均有效率等性能指标上都要优于基线算法。

关键词：信息物理融合系统；传感器事务；多处理器调度；时态一致性

Foundation item: Project(2020JJ4032) supported by the Hunan Provincial Natural Science Foundation of China

Received date: 2020-03-18; Accepted date: 2020-08-06

Corresponding author: BAI Tian, PhD, Lecturer; Tel: +86-18390188018; E-mail: baitiannobel@163.com; ORCID: https://orcid.org/ 0000-0003-0540-5472