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Title: Rate Monotonic Schedulability Tests Using Period-Dependent Conditions
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Keywords: Real-time systems; Periodic scheduling; Rate monotonic analysis; Schedulability test.
Corresponding Author: Ms. Wan-Chen Lu,
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First Author: Wan-Chen Lu
Order of Authors: Wan-Chen Lu; Kwei-Jay Lin; Hsin-Wen Wei; Wei-Kuan Shih
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Abstract: Feasibility and schedulability problems have received considerable attention from the real-time systems research community in recent decades. Since the publication of the Liu and Layland bound, many researchers have tried to improve the schedulability bound of the RM scheduling. The LL bound does not make any assumption on the relationship between any of the task periods. In this paper we consider the relative period ratios in a system. By reducing the difference between the smallest and the largest virtual period values in a system, we can show that the RM schedulability bound can be improved significantly. This research has also proposed a system design methodology to improve the schedulability of real time system with a fixed system load.
Paper Title: Rate Monotonic Schedulability Tests Using Period-Dependent Conditions Author List: Wan-Chen Lu, Kwei-Jay Lin, Hsin-Wen Wei, Wei-Kuan Shih
Addresses and Affiliations of Authors:
W.-C. Lu, H.-W. Wei, W.-K. Shih
Department of Computer Science, National Tsing Hua University, Hsinchu, Taiwan, ROC; e-mail: {wanchen, bertha, wshih}@rtlab.cs.nthu.edu.tw
K.-J. Lin
Department of Electrical Engineering and Computer Science, University of California, Irvine, CA; e-mail: klin@uci.edu
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Rate Monotonic Schedulability Tests Using Period-Dependent Conditions
Abstract Feasibility and schedulability problems have received considerable attention from
the real-time systems research community in recent decades. Since the publication of the Liu and Layland bound, many researchers have tried to improve the schedulability bound of the RM scheduling. The LL bound does not make any assumption on the relationship between any of the task periods. In this paper we consider the relative period ratios in a system. By reducing the difference between the smallest and the largest virtual period values in a system, we can show that the RM schedulability bound can be improved significantly. This research has also proposed a system design methodology to improve the schedulability of real time system with a fixed system load.
Keywords Real-time systems．Periodic scheduling．Rate monotonic analysis．Schedulability
test
1. Introduction
The design of single-processor real-time systems remains an important issue due to the development of many embedded controllers and sensor devices (Liu and Lee, 2002; Tan and Mooney, 2004). Embedded devices usually have a limited computing power yet must deliver a real time response. They often use simple schedulers such as fixed priority scheduling. How to achieve the best performance under a known resource constraint and how to keep the cost as low as possible are important for many embedded devices designed for the large consumer
This paper is an extended version of the paper “Rate Monotonic Schedulability Conditions Using Relative Period Ratios”
that appeared in the 12th IEEE International Conference on Embedded and Real-Time Computing Systems and Applications.
* Manuscript
or sensor market.
For a periodic task set, each periodic task τi is defined with two parameters (ci, pi), where ci and pi are the worst case computation time and the period of task τi respectively. The utilization factor of task τi is defined by ui = ci/pi. The schedulability of periodic real-time tasks using the Rate Monotonic (RM) fixed priority scheduling algorithm can be checked by summing the utilization factors of all tasks in a system. If the load (i.e. total utilization) of the system is less than or equal to a specified bound, the system is guaranteed to be schedulable.
Liu and Layland (1973) have proposed the first and the most well-known RM bound (referred as the LL bound in this paper). The LL bound is in1uin(21/n-1) where n is the number of periodic tasks in the system. When the number of tasks n approaches ∞, the LL bound is ln 2
0.693.
The LL bound provides a simple guideline when we design real-time systems. Given a real-time task set, if the load is larger than the LL bound, the schedulability of the task set may not be guaranteed. For example, if the total system load is 0.8 and the number of tasks in the system is more than 2, RM scheduling may not guarantee that all jobs can meet their deadlines. However, it is well known that the LL bound is pessimistic and may be lower than the actual achievable bound given a specific system. For example, the task set T = {(4, 16), (3, 17) (3, 18), (2, 19), (2, 20)} has a total load of 0.798, which is higher than the LL bound for 5 tasks (0.743), yet it is schedulable.
Since the publication of the LL bound in Liu and Layland (1973), many researchers have tried to improve the schedulability bound of the RM algorithm. One known result is that if all periods are multiples of any smaller period in a system (called the simply periodic system), the schedulability bound can be as high as 1 (Liu, 2000). It has also been observed that the number of tasks can be reduced by forming groups of tasks with the harmonic period
property; in that case, it is the number of groups that decides the LL bound (Kuo et al., 2003).
The schedulability bound is also improved by making conditions on the relationship between any of the periods in Lauzac et al. (2003). A polynomial time checking algorithm has also been designed to compare a system being designed with its simply periodic counterpart (Han and Tyan, 1997). An exact analysis has been proposed in Audsley et al. (1993) using the worst case response time recurrence analysis. However, the exact analysis is time consuming and does not provide a simple methodology on how to improve the schedulability of a system.
Many others have studied the RM scheduling bounds extensively (Lehoczky et al., 1989;
Burchard et al., 1995; Bini et al., 2003).
Therefore, the first and probably the most important issue for designing an efficient real-time system is to select the best applicable schedulability bound. In this paper, we propose a method to improve the RM schedulability bound by considering the relative period values in a system as in Lauzac et al. (2003). Suppose z1 and z2 are the smallest and largest period ratios in the system respectively. When the number of tasks in a system approaches infinity, the schedulability bound derived in Lauzac et al. (2003) is 2z1- ln z1-1. In this paper, we improve the bound to 2z1+1/z2+(ln z2 - ln z1)-2. Moreover, when the number of tasks in a system is n, the schedulability bound derived in Lauzac et al. (2003) is 2z1 - 1 + (n-1)((1/z1)1/(n-1)-1), and that derived by our method is z1 + 1/z2 - 2 + (n-2)((z2/z1)1/(n-2)-1).
For the general case, we define virtual period vi for each task τi as p p pn i i. Assume there are n periodic tasks in a system; let the smallest virtual period ratio z1 = min1in-1{vi/pn} and the largest virtual period ratio z2 = max1in-1{vi/pn}. By reducing the difference between the smallest and largest virtual period ratios in a system, we can show that the RM schedulability bound can be improved significantly. Therefore, this paper presents a bound q
= f(z1, z2) which is a function of the smallest and largest virtual period ratios. If the condition min1in-1{vi/pn}z1 and max1in-1{vi/pn}z2 is satisfied and the total system load is less
than or equal to q, it can be shown that all tasks in the system are RM schedulable. For example, when z1 is 0.6 and z2 is 0.65, q is 0.818504. When z1 is 0.9 and z2 is 0.95, q is as large as 0.906699 regardless of the number of tasks in a system. These q bounds are significantly better than the LL bound of ln 2.
The main contribution of this paper is as follows:
1. We have provided the RM bound as a function of z1 and z2, where z1 is the ratio of the smallest virtual period to the largest period and z2 is the ratio of the largest virtual period to the largest period in a system;
2. Given a desirable schedulability bound and a known longest period in a system, we can identify the acceptable period range for all other tasks.
The remainder of this paper is organized as follows. In Section 2, we define some terminology and provide a motivation for the proposed RM bound. Section 3 presents a RM bound according to the ratios of the smallest and largest virtual periods to pn. Section 4 provides a methodology for real-time system design by adjusting the periods of all tasks except τn. The paper is concluded in Section 5.
2. Definitions and Motivation
Before we discuss our new result, we first present some formal definitions about RM scheduling.
Definition 1. (Liu and Layland, 1973) Let T = {τ1, τ2, ..., τn} be a set of n periodic tasks. Each task τi(i = 1, ..., n) is a tuple (ci, pi), where ci and pi are the maximum computation time and the period of task τi, respectively. The utilization of τi, denoted by ui, is equal to ci/pi. The total utilization of T is denoted as U(T). Without loss of generality, we may assume that tasks in T are indexed in the order of increasing periods, and hence task τn has the longest period of all tasks.
Definition 2. (Liu, 2000) Job Ji,j: Ji,j is the jth job of periodic task τi. That is, Ji,j is released at
time (j-1)pi and has a deadline at time jpi, for j1. Note that the feasible scheduling interval of Ji,jis the time period ((j-1)pi, jpi), and therefore, the first job of task τi has period (0,pi).
Definition 3. (Liu, 2000) Current Period at time t: At any time t, the current period of a task is the period that begins before t and ends at or after t. That is, for task τi, the current period of τi is the time interval (t p pi ip ,i tp pi i]. Note that the period starting at time t is not a current period of time t.
Definition 4. (Liu, 2000) Current Jobs at time t: At any time t, the current job of a task is the job that is released at the beginning of its current period.
Here is an example to illustrate the definition of current jobs. Let T = {τ1 = (1,2), τ2 = (2,7)}
be a set of periodic tasks. At time 7, the current periods of τ1 and τ2 are (6,8] and (0,7], respectively, and their current jobs are J1,4 and J2,1, respectively, as illustrated in Fig. 1.
2 3
1
0 4 5 6 7
J1,1
10 9 8
J2,1
J1,2 J1,3 J1,4 J1,5
Fig. 1 Current jobs of τ1and τ2 at time 7
Definition 5. Critical Jobs: The critical jobs are defined specifically at time pn, the largest period in a system. At pn, the current jobs of all tasks, excluding Jn,1, are called the critical jobs of the system. Please note that every task except τn has a critical job. Also note that the critical jobs are identified on time pn.
Definition 6. Saturated Schedule and Saturated Task Set: Let S be an RM schedule corresponding to a specific task set. We say that S is a saturated schedule if no idle time exists in the time interval (0,pn] in S and all jobs meet deadlines. The task set running on a saturated schedule is called a saturated task set.
For a saturated schedule, if the utilization of task τn is increased by any (however small) value, τn will miss its deadline.
Definition 7. RM Bound (RMB) and the RMB Task Set: Any task set with a total utilization less than or equal to RMB is guaranteed schedulable. A saturated task set with a total utilization equal to RMB is called the RMB task set.
Suppose that is the universal set of task sets, formed by all periodic task sets with all possible utilizations and all possible combinations of periods. schedulableis a subset of , including all task sets that are schedulable under the RM policy. The conditions that a task set is a RMB task set, denoted by Trmb, are as follows:
1. Trmb is a saturated task 2. Trmb belongs to schedulable
3. Trmb has the smallest total utilization among all task sets belonging to schedulable
The LL bound is shown in Liu and Layland (1973) to be the RMB defined in Definition 7.
This RMB is the most general RMB since all possible task sets are considered. However, if we constrain our consideration to some task sets with a specific condition, we can define the conditional RMB as follows.
Definition 8. Conditional RM Bound under Condition C (CRMB(C)) and the CRMB(C) Task Set: Given a condition C about a periodic task set (e.g. “p10.6pn and pn-10.8pn”), the CRMB(C) is the RMB of all task sets that satisfy condition C. A saturated task set meeting condition C and with a total utilization equal to CRMB(C) is called the CRMB task set.
For example, CRMB(p1>0.5pn) is equal to the LL bound. In this paper, we will first present our study on deriving CRMB(p1z1pn, pn-1z2pn). With that new bound and given a fixed system load, we then present a real-time system design methodology to meet the condition C so that CRMB(C)the system load. In other words, the system may be refined to satisfy the condition C in order to guarantee that the CRMB(C) is larger than or equal to the given system load.
3. The Derivation of the CRMB Formula
3.1. CRMB for p1>pn/2
Just like the derivation of the LL bound, we also need the worst case (or hard-to-schedule) task set to derive our CRMB(C). For the LL bound, the hard-to-schedule case happens on the tasks set with p1>pn/2. In this paper, we will show that the hard-to-schedule case happens when p1z1pn and pn-1z2pn where z1>1/2 and z21. That is, to find the CRMB(C), we only need to consider the task set with p1z1pn and pn-1z2pn. In the next subsection, we will show how to handle the case when some periods are relatively small (i.e. piz1pn where z11/2 for some task τi).
Our study is motivated by the observation that many jobs actually use more processor time than they are supposed to (according to their utilizations) in RM schedules. For task τn with a deadline pn, all other tasks’ current jobs at time pn may take some extra time and utilization.
(The current job of τn does not consume extra processor time.) These are called the critical jobs (defined earlier) at time pn. We define another concept, called the gross utilization, to reflect how the total CPU time before pn is distributed among all tasks.
Definition 9. Gross Utilization: For a task set T, let S be the actual RM schedule. The gross utilization of task τi, denoted by yi, is defined by ei/pn where ei is the total scheduled time of τi
during (0,pn] in S.
For example, let T = {τ1, τ2, ..., τn} be a set of periodic tasks. The computation time of τ1 is 20, and the period of τ1 and τn are 80 and 100, respectively. Using the RM scheduling, τ1 is scheduled for 40 time units before time 100. Therefore, y1 = 40/100 = 0.4 while u1 = 0.25 (less than y1).
Definition 10. Excessive Utilization: For a task set T, the excessive utilization wi of task τi is equal to yi - ui.
The excessive utilization w1 of task τ1 in the last example is 0.4 - 0.25 = 0.15.
Let S be a RM schedule for a task set T. In S, each task τi has two types of utilization ui
and yi. ui is the actual utilization of τi and is independent to the actual schedule. yi is the gross utilization of τi and depends on the actual schedule. When the excessive utilization wi of all tasks is zero, the total utilization can be as high as 1 and no task will miss its deadline.
If the critical job of a task τi is scheduled at or after pn, or the critical job scheduled in (0,pn] is not larger than (pn p p p )un i i i, the excessive utilization wi of this task may be zero, or even less than 0. Since we are only interested in the schedulability bound, we assume that the excessive utilization is 0 whenever it is less than 0.
Suppose the smallest system period ratio is z1, i.e. z1p1/pn. Let us now assume 1/2 < z11.
(z1 with a smaller value will be discussed in the next subsection). Since p1z1pn, pn-1z2pn
and pn-1 ... p2p1, all critical jobs will be scheduled after the beginning of the 2nd period of τ1 (i.e. after z1pn), either in (z1pn, pn] or after pn. We will show that when z1 increases or z2
decreases, CRMB(p1z1pn, pn-1z2pn) increases. For simplicity, we will use CB(z1, z2) to denote CRMB(p1z1pn, pn-1z2pn) for the rest of this paper.
We first assume that the number of tasks in a task set approaches infinity. We will remove this condition later in the section. Without loss of generality, we assume that each task has a unit-length computation time. (The unit-length computation time can be set to any small value for discussion purposes.)
Lemma 1. Suppose a saturated task set T is a CB(z1, z2) task set and T satisfies the condition p1z1pn and pn-1z2pn. Then, in the saturated schedule of T, jobs scheduled in (p1,pn] must be critical jobs of tasks in T.
Proof: Assume this lemma is incorrect. That is, there exists an x in (p1,pn], and the job scheduled in x is not a critical job. So, we can increase the deadlines of jobs scheduled in (p1,
x-1] by one and this task set is also a saturated task set. After this deadline modification, we have a task set with a smaller utilization. In this way, we have found a new saturated task set with a smaller total utilization and we get a new CB(z1, z2) with a lower total utilization. This is a contradiction. Therefore, jobs scheduled in (p1,pn] must be critical jobs of tasks in T. □
Lemma 2. Suppose the saturated task set T is a CB(z1, z2) task set and T meets the condition p1z1pn and pn-1z2pn. In a saturated schedule of T, if the critical job of τk at pn is scheduled at x in (p1,pn], the period of τk is x.
Proof: According to Lemma 1, we know that all critical jobs of T at pn are executed between z1pn and pn. (Fig. 2 shows the RM schedule of critical jobs.) In the saturated schedule of T, if the period of τk is not equal to x, pk must be less than x; otherwise, the critical job of τk cannot be scheduled in x. If we set pk be equal to x, the schedule remains the same; but uk will be decreased, and we find a new saturated task set with a smaller total utilization. That is, we get a new CB(z1, z2) with a lower total utilization. This is a contradiction. Therefore, the period of τk must be x. □
...
p1
p2
p3
pn-1
pn
2p1
...
Fig. 2 Critical jobs scheduling
Theorem 1. Given that z1 = p1/pn and z2 = pn-1/pn, CB(z1, z2) = 2z1+1/z2+(ln z2- ln z1)-2.
Proof: We know that p1z1pn, and in a saturated task set, each task has unit-length
computation time. According to Lemma 1 and Lemma 2, we know that in the schedule of the CB(z1, z2) task set, jobs scheduled in (p1,pn] must be critical jobs, and if the critical job of τk at pn is scheduled in x, x belongs to (p1,pn], then, the period of τk is x. Since p1z1pn and pn-1z2pn, in the worst case, p1 = z1pn and pn-1 = z2pn. Therefore, in the worst case, there is one unit critical job of τi scheduled in (z1pn, z2pn] where i = 1 to n-2. For τi, there are two units scheduled in the time interval (0,pn]. Therefore, ui = 1/pi, but yi = 2/pn. The excessive utilization is yi - ui = 2/pn - 1/pi. When the unit-length of computation time is sufficiently small (that is, the number of tasks n approaches infinity), the total excessive utilization for τ1,
..., τn-2 in the saturated task set is:
(the total gross utilization of τ1, τ2, ..., τn-2) - (the total utilization of τ1, τ2, ..., τn-2), i.e.
2(z2- z1)- 2
1(1 )
z
x z x dx
= 2(z2- z1)-(ln z2- ln z1).
And the excessive utilization for τn-1 in the saturated task set is (the total gross utilization of τn-1) - (the total utilization of τn-1)
= 2(1-z2) - (pn-pn-1)/pn-1 = 3-2z2-1/z2.
Thus, CB(z1, z2) is 1- (the total excessive utilization), i.e.
1-((2(z2- z1)-(ln z2-ln z1))+(3-2z2-1/z2)) = 2z1+1/z2+(ln z2- ln z1)-2. □
From Theorem 1, if z1 = 0.55 and z2 = 0.6, the CB(0.55, 0.6) is equal to 2(0.55)+1/0.6+(ln0.6 - ln0.55)-2 = 0.853678, or if z1 = 0.55 and z2 = 1, the CB(0.55, 1) is equal to 2(0.55)+1/1+(ln1 - ln0.55)-2 = 0.697837. Table 3 shows the values of CB(z1, z2) where z1 and z2 = 0.55, 0.6, ..., 1. It is obvious that on condition that z1 is fixed, if z2
decreases, CB(z1, z2) increases. Note that the gray area in Table 1 shows the schedulability bound proposed in Lauzac et al. (2003) which only considers the condition of z1 (i.e. 0.55, 0.6,
..., 1) on large number of tasks. Therefore, a higher schedulability bound could be achieved
by adjusting the value of z2.
Table 1 CB(z1, z2) on large number of tasks
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
0.55 0.918182 0.853678 0.805516 0.769733 0.743488 0.724693 0.711789 0.703588 0.699175 0.697837 0.6 0.866667 0.818504 0.782722 0.756477 0.737682 0.724777 0.716576 0.712164 0.710826 0.65 0.838462 0.802679 0.776434 0.757639 0.744735 0.736534 0.732121 0.730783
0.7 0.828571 0.802326 0.783531 0.770627 0.762426 0.758013 0.756675
0.75 0.833333 0.814539 0.801634 0.793433 0.789020 0.787682
0.8 0.850000 0.837095 0.828894 0.824482 0.823144
0.85 0.876471 0.868270 0.863857 0.862519
0.9 0.911111 0.906699 0.905361
0.95 0.952632 0.951293
1 1.000000
The bound is higher when the number of tasks is n < ∞. The CB(z1, z2) for n tasks is given
in Theorem 2. Note that when n = 2, z1= z2= p1/p2.
Theorem 2. Given that z1 = p1/pn, z2 = pn-1/pn, CB(z1, z2) = 2z1+1/z2 -2 + (n-2)((z2/z1)1/(n-2)-1) where n is the number of tasks in a system.
Proof: For a hard-to-schedule task set (Liu and Layland, 1973), it has been shown that for k =
1, 2, ..., n-2, pk+1= pk(1 + uk) (see Fig. 2). Moreover, since p1z1pnand pn-1z2pn, we know that pnp1/z1 and pn-1z2pn. Combining these two expressions, we have the following inequality:
p1(1+u1)p2(1+u2)...pn-2(1+un-2)pn-1pnp2p3...pn-1(z2pn)(p1/z1)
(1+u1)(1+u2)...(1+un-2)z2/z1
Treating u1, u2, ..., un-2 as variables, it has been shown in Liu (2000) that (1 + (u1+u2+...+un-2)/(n-2))n-2(1+u1)(1+u2)...(1+un-2)
Thus we can obtain the following result
(1 + (u1+u2+...+un-2)/(n-2))n-2z2/z1
u1+u2+...+un-2(n-2)((z2/z1)1/(n-2)-1)
Moreover, the utilization of τn-1in a hard-to-schedule task set is (pn-pn-1)/pn-1, that is, 1/z2-1.
Since z1is always larger than 1/2, all tasks except τnhave two jobs scheduled in (0, pn]. The utilization of τnin a hard-to-schedule task set is 1 - (the total gross utilization of τ1, τ2, ..., τn-1)
= 1 - 2(1 - z1) = 2z1-1. Therefore, CB(z1, z2) = u1+u2+...+un-2+un-1+un
(n-2)((z2/z1)1/(n-2)-1)+(1/z2-1)+(2z1-1)
= 2z1+1/z2 -2 + (n-2)((z2/z1)1/(n-2)-1)□
Table 2 shows the values of CB(z1, z2) (where z1 and z2 = 0.55, 0.6, ..., 1) when the number of periodic tasks in a system is 3. The gray area in Table 2 is close to the schedulability bound derived in Lauzac et al. (2003) which only considers the condition of z1
(i.e. 0.55, 0.6, ..., 1).
Table 2 CB(z1, z2) on the system of 3 tasks
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
0.55 0.918182 0.857576 0.820280 0.801299 0.796970 0.804545 0.821925 0.847475 0.879904 0.918182 0.6 0.866667 0.821795 0.795238 0.783333 0.783333 0.793137 0.811111 0.835965 0.866667 0.65 0.838462 0.805495 0.787179 0.780769 0.784163 0.795726 0.814170 0.838462
0.7 0.828571 0.804762 0.792857 0.790756 0.796825 0.809774 0.828571
0.75 0.833333 0.816667 0.809804 0.811111 0.819298 0.833333
0.8 0.850000 0.838971 0.836111 0.840132 0.850000
0.85 0.876471 0.869935 0.870279 0.876471
0.9 0.911111 0.908187 0.911111
0.95 0.952632 0.952632
1 1.000000
In Theorem 1, we have proved the CB(z1, z2) bound when the number of tasks approaches
∞. In the following, we prove that if the number of tasks n < ∞, and the total utilization is less
than the CB(z1, z2) bound, the task set is also schedulable.
Lemma 3. Given a task set T with n tasks, let T’ be the task set created by dividing each task
in T into subtasks with unit-length computation time. The period of each subtask remains the same as that of its corresponding tasks in T. T’ is schedulable if and only if T is schedulable.
Proof. The RM scheduling policy is a fixed priority policy. Since τ1 in T has the smallest period among all tasks in T, it has the highest priority. All subtasks corresponding to τ1 in T’
also have the highest priority among all tasks in T’. Therefore, the time interval assigned to τ1
in the schedule of T is the same as the time interval used by all subtasks of τ1 in the schedule of T’. This is also true for τ2, τ3, ..., τn. Therefore, there exists a schedule of T such that T is schedulable if and only if there exists a schedule of T’ such that T’ is schedulable. □
Although in our proof, each task is divided into small subtasks; it is for the proof discussion only, not for actually scheduling these tasks.
Theorem 3. Let CB(z1, z2) be the RM bound for the case that the number of tasks n approaches infinity. T is a task set with the number of tasks being a fixed value n. If U(T)CB(z1, z2), then T is schedulable.
Proof: We create a new task set T’ by dividing each task in T into small subtasks with
unit-length computation time (or we can set the unit to be a sufficiently small value). After T’
is constructed, the number of tasks n approaches infinity and we perform the schedulability test on T’. Since U(T’) = U(T), if the U(T)CB(z1, z2), then U(T’)CB(z1, z2). Therefore, T’
is schedulable. From Lemma 3, we know that T is also schedulable. □
From Theorem 3, we know that the schedulability bound for the case that n approaches infinity is also a schedulability bound for any given n, the number of tasks. Therefore, if the number of tasks in the system is unknown, to simplify the on-line schedulability testing, we can apply the schedulability bound for the case that the number of tasks n approaches infinity.
3.2. CRMB for the General Case
For the case that some tasks have their periods less than or equal to pn/2, we can show that CB(z1, z2) is actually the same. Since p1pn/2, the definition of the ratios z1 and z2 needs to be changed to reflect the actual schedule. Under this condition, we use the critical jobs defined in Section 2 to define the z1 and z2. Let ki be p pn i and ki2. It is obvious that, the ready time of the critical job of τi is kipi. Therefore, we define the virtual period of each task as follows.
Definition 11. Virtual Period of Task τi: A virtual period of τi, denoted by vi, is the ready time of the critical job of τi. That is, vi =p p pn i i.
Note that for the case discussed in Section 3.1, all tasks have a virtual period equal to their real period. Therefore, all pi (for i = 1, ..., n) used in the discussions of previous sections may be replaced by their virtual period (i.e. vi). After the periods replacement, we need to find the smallest and the largest values among all virtual periods and the values of z1 and z2 are modified to be:
z1 = min1in-1{vi/pn} z2 = max1in-1{vi/pn}
The above formula provides us a new insight when scheduling real-time systems. That is, we only need to consider the ready times of all critical jobs, or the virtual periods. If the actual periods of tasks are small, only the ready times of all critical jobs need to be examined.
All of them must satisfy the constraint defined in Theorem 1 and Theorem 2. This idea will
be discussed in more details in Section 4.
In this section, we consider the case that p1pn/2. p1pn/2 implies that it is possible that p2 pn/2. Under this condition, we need to make a transformation for all tasks with relative small periods in the system to find the CB(z1, z2). In the following theorem, we will derive the formula for the general task sets.
Theorem 4. Let T be a task set with n periodic tasks. Suppose z1 = min1in-1{vi/pn} and z2 = max1in-1{vi/pn}. T is RM schedulable if U(T)CB(z1, z2) = 2z1+1/z2+(ln z2- ln z1)-2.
Proof: For the case that all tasks have virtual periods equal to their actual periods, the
discussion in Section 3.1 provides the proof. Here we only consider the task sets where some tasks have relatively small periods so that their virtual periods are not their actual periods.
Assume the theorem is not correct. That is, there is a task set T with p1p2 ... pjpn/2, U(T)CB(z1, z2) = 2z1+1/z2+(ln z2- ln z1)-2 where z1 = min1in-1{vi/pn}, z2 = max1in-1
{vi/pn} and T is not schedulable. Since T is not schedulable, pn must have missed its deadline in a saturated schedule from 0 to pn.
We will transform T by increasing those small task periods so that all periods pi>pn/2. The transformation of τ1 = (c1, p1), for example, is done as follows. Let k =p pn 1and k2. We divide τ1 into two tasks, denoted by A1 and B1 where A1 = (c1,kp1) and B1 = ((k-1)c1,kp1).
Furthermore, we delay the ready time of B1 to time p1. Let us call the new task set created by this transformation as Tnew. In Tnew, uA1+uB1 = c1/(kp1)+ (k-1)c1/(kp1) = c1/p1 = u1. Both A1
and B1 have a period that is the virtual period v1 of τ1 and U(Tnew) = U(T). Fig. 3 shows the transformation when k = 3. In Fig. 3(a), we can see the virtual period v1 of τ1 is 3p1. In Fig.
3(b), τ1 is partitioned into A1 and B1. Now, the period of A1 is v1 > pn/2. The critical job of A1
is the same as τ1. However, the critical job of B1 is not scheduled in the time interval (v1,pn]
so that the excessive utilization of B1 is zero. If T is not schedulable, Tnew still is not schedulable since the total execution time needed by all critical jobs at pn remains the same.
In the new task set Tnew, if there is another task with a period less than or equal to pn/2, we perform the same transformation on the task and the newly created task set will remain unschedulable. In this way, we can perform the transformation one by one on all tasks τi with pi pn/2, and the new task set created by each transformation remains unschedulable. The final task set after all transformations are done, denoted by Tnew-complete, is still unschedulable.
Moreover, all tasks in Tnew-complete have a period vi > pn/2 and z1 is equal to min1in-1{vi/pn}, z2 is equal to max1in-1{vi/pn}. But according to Theorem 1, we know that Tnew-complete is schedulable if U(Tnew-complete) = U(T)CB(z1, z2) = 2z1 + 1/z2 + (ln z2- ln z1) - 2, where 1/2 <
z1z21. This is a contradiction. Therefore, the theorem is correct. □
p1
p2
p3
p4
p5
3p1
2p1 4p1
0
(a) The saturated task set under p1pn/2
p2
p3
p4
p5
B1
4p1
= A1
3p1
p1
0
(b) The transformation of for τ1
Fig. 3 CB under p1pn/2
Here is an example to show what the value of CB(z1, z2) is for a task set under p1pn/2. Let
T = {τ1, τ2, τ3} be a set of periodic tasks. The period of τ1, τ2 and τ3 are 3, 4, and 10 respectively. Since p1/p3 and p2/p3 are less than 1/2, the virtual period of τ1 is set to 3p1 = 9 and the virtual period of τ2 is set to 2p1 = 8. To calculate the CB(z1, z2) of T, the virtual periods, 9 and 8, are used. Therefore, the CB(z1, z2) of T is equal to CRMB(min(9,8)10z1, max(9,8)10z2) = CRMB(810z1, 910z2) = CB(0.8, 0.9) = 2(0.8)+1/0.9+ (ln(0.9)- ln (0.8))-2 = 0.8288. Hence, if U(T)0.8288, all tasks in T are schedulable.
4. System Design Methodology
The schedulability bound CB(z1, z2) described in the previous section allows us to achieve a higher schedulability bound if a system has a high period ratio z1 and a low period ratio z2
(when the number of tasks in a task set approaches infinity). Therefore, we can use the CB(z1, z2) bound to check for schedulability in those systems. In addition, it also provides system designers the possibility to explore and to adjust task periods in a system in order to achieve a higher schedulability bound.
Suppose there are n tasks in a system and q is the current system load. The maximum period among all tasks, pn, is also known. If the periods of all other tasks can be modified, our goal is to set them in such a way that the schedulability of all tasks is guaranteed. For designing the system efficiently, we assume the value of z2 is equal to 1 when finding the CRMB for a system. It should be obvious that as long as the threshold z1pn can be determined and if all task periods are set to be larger than or equal to the threshold, the task set is guaranteed RM schedulable.
Unfortunately, the inverse function for the CB(z1, z2) function in Theorem 2 is very difficult to derive. We thus use a binary search algorithm to find the threshold z1pn where 1/2<z11. The binary search algorithm FIND-THRESHOLD is shown as follows.
Algorithm 1: FIND-THRESHOLD
Input: q and pn Output: threshold
L = 1/2; R = 1;
while (R-L)>1/pndo z1 = (L+R)/2;
if CB(z1, 1) = 2z1+1/z2+(ln z2 - ln z1)-2 < q then L = z1;
else
R = z1; end if end while
return threshold = Rpn;
In this algorithm, the system load q and the period of τn (i.e. pn) are given. The interval (L, R) is the current range for the binary search. When z1 = (L+R)/2 has a CB(z1, 1) smaller than the required system load, we should find a larger z1 so that the guaranteed CB(z1, 1) can be increased. The search range will be reduced to ((L+R)/2, R). Otherwise, the current search range will be reduced to (L, (L+R)/2). The search continues until the range is smaller than 1/pn. The threshold produced by the algorithm is the minimum value for all task periods except τn. Please note that in the binary search process, we always keep CB(R, 1)q.
For example, assume T = {τ1, τ2, τ3} is a set of periodic tasks and the total system load of T is 0.8. If p3 is equal to 100, we can design the periods of τ1 and τ2 when T needs to be guaranteed schedulable by setting a suitable period threshold. The threshold of T produced by the search is 77.34. If the virtual periods of τ1 and τ2 are selected to be larger than 77.34 but smaller than 100, all tasks in T will be schedulable.
5. Conclusions
In this paper, we present the RM schedulability bound as a function of z1 and z2, the ratios of the smallest and the largest virtual periods to the largest period in a periodic real-time system.
By reducing the difference between the smallest virtual period and the largest virtual period,
a periodic task system can achieve a higher RM schedulability bound. This new insight provides a new system design perspective to adjust task periods. We plan to use this result to improve the schedulability bound for multiprocessor real time systems.
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