19/04/2016. What does it mean? Response-Time Analysis of Conditional DAG Tasks in Multiprocessor Systems. What does it mean? What does it mean?

Transcription

1 9/04/20 What does it mean? «Response-time analysis» Response-Time Analysis of Conditional DAG Tasks in Multiprocessor Systems «conditional» «DAG tasks» «multiprocessor systems» Alessandra Melani 2 What does it mean? What does it mean? «Response-time analysis» «conditional» «DAG tasks» «Response-time analysis DAG:» Directed Acyclic Graph «conditional» «DAG tasks» «multiprocessor systems» If-then-else statements Switch statements «multiprocessor systems» 3 4 In other words We will analyze a multiprocessor real-time systems by means of a schedulability test based on response-time analysis Parallel task models Many parallel programming models have been proposed to support parallel computation on multiprocessor platforms (e.g., OpenMP, OpenCL, Cilk, Cilk Plus, Intel TBB) assuming Global Fixed Priority or Global EDF scheduling policies and assuming a parallel task model (i.e., a task is modelled as a Directed Acyclic Graph - DAG) Early real-time scheduling models: each recurrent task is completely sequential Recently, more expressive execution models allow exploiting task parallelism 5

2 9/04/20 Fork-join Synchronous-parallel Each task is an alternating sequence of sequential and parallel segments Every parallel segment has a degree of parallelism (number of processors) Generalization of the fork-join model Allows consecutive parallel segments Allows an arbitrary degree of parallelism of every segment Synchronization at segment boundaries: a sub-task in the new segment may start only after completion of all sub-tasks in the previous segment 7 8 DAG Directed acyclic graph (DAG), cp-dag Conditional - parallel DAG (cp-dag),,,,, ; Generalization of the previous two models Every node is a sequential sub-task Arcs represent precedence constraints between sub-tasks Two types of nodes Regular: all successors must be executed in parallel Conditional: to model start/end of a conditional construct (e.g., if-then-else statement) Each node has a WCET, In this lecture, we will focus on this task model 9 Conditional pairs Why this restriction?, form a conditional pair is a starting conditional node is the joining point of the conditional branches starting at Restriction: there cannot be any connection between a node belonging to a branch of a conditional statement (e.g., ) and nodes outside that branch (e.g., ), including other branches of the same statement It does not make sense for to wait for if is executed Analogously, cannot be connected to since only one is executed Violation of the correctness of conditional constructs and the semantics of the precedence relation 2 2

3 9/04/20 Formal definition () Formal definition (2) Let, be a pair of conditional nodes in a DAG,. The pair, is a conditional pair if the following hold: Suppose there are exactly outgoing arcs from to the nodes,,,,forsome. Then there are exactly incoming arcs into in, from some nodes,,, For each,2,,, let and denote all the nodes and arcs on paths reachable from that do not include. By definition, is the sole source node of the DAG,. It must hold that is the sole sink node of., 3 4 Formal definition (3) How is parallel code structured? It must hold that for all,,. Additionally, with the exception of,, there should be no arcs in into nodes in from nodes not in, for each,2,,. That is, \, should hold for all. 5 #pragma omp parallel num_threads(n) { #pragma omp master { #pragma omp task { // T 0 if (condition) { #pragma omp task { // T } } else { #pragma omp task { // T 2 } #pragma omp task { // T 3 } #pragma omp task { // T 4 } } }}} if (condition) {} else {} Which branch leads to the worst-case response-time? T 2 T 3 T 4 T Which branch leads to the WCRT? processor Upper branch Which branch leads to the WCRT? 3 processors Upper branch T if (condition) {} T else {} 2 Lower branch 2 processors 8 if (condition) {} else {} T T 2 T 3 Lower branch T 3 T 4 Upper branch T 4 3 processors + interfering task Lower branch Upper branch Lower branch

4 9/04/20 Lesson learnt System model Depending on the number of processors and on the interfering tasks, it is not obvious to identify the branch leading to the WCRT if (condition) {} else {} It makes sense to account for the different execution flows by enriching the task model conditional-parallel tasks (cp-tasks) τ, expressed as cp-dags in the form, platform composed of identical processors sporadic arrival pattern (minimum inter-arrival time between jobs of task τ ) Why don t we do it also with sequential tasks? Only the longest path matters Conditional branches are already incorporated in the notion of WCET if (condition) {} else {} constrained relative deadline Problem Schedulability analysis for cp-tasks, globally scheduled on m identical processors with any work-conserving algorithm (including G-FP and G-EDF) 9 20 Quantities of interest. Chain (or path). Chain (or path) of a cp-task A chain (or path) of a cp-task τ is a sequence of nodes λ,,,, such that,,,,,. 3. Volume 4. Worst-case workload 5. Critical chain Chain (or path) A chain (or path) of a cp-task τ is a sequence of nodes λ,,,, such that,,,,,. The longest path of a cp-task τ is any source-sink chain of the task that achieves the longest length The length of the chain, denoted by λ, is the sum of the WCETs of all its nodes: λ, also represents the time required to execute it when the number of processing units is infinite (large enough to allow maximum parallelism) Necessary condition for feasibility:

5 9/04/20 How to compute the longest path? How to compute the longest path?. Find a topological order of the given cp-dag A topological order is such that of there is an arc from to in the cp-dag, then appears before in the topological order can be done in Example: for this cp-dag possible topological orders are,,,,,,,,,,,,,,,,,,,,,,,, 2. For each vertex, of the cp-dag in the topological order, compute the length of the longest path ending at, by looking at its incoming neighbors and adding, to the maximum length recorded for those neighbors If, has no incoming neighbors, set the length of the longest path ending at, to, Example: For, record For, record 2 For, record 5 For, record For, record max 5, 25 2 How to compute the longest path? 3. Finally, the longest path in the cp-dag may be obtained by starting at the vertex, with the largest recorded value, then repeatedly stepping backwards to its incoming neighbor with the largest recorded value, and reversing the sequence found in this way Example: recorded values Starting at and stepping backward we find the sequence,,,,, The longest path is then,,,,, Complexity of the longest path computation: In the absence of conditional branches, the volume of a task is the worst-case execution time needed to complete it on a dedicated single-core platform It can be computed as the sum of the WCETs of all its vertices: 0 3. Volume,, It also represents the maximum amount of workload generated by a single instance of a DAGtask Worst-case workload 4. Worst-case workload In the presence of conditional branches, the worst-case workload of a task is the worst-case execution time needed to complete it on a dedicated single-core platform, over all combination of choices for the conditional branches It also represents the maximum amount of workload generated by a single instance of a cp-task In this example, the worst-case workload is given by all the vertices except, since the branch corresponding to yields a larger workload 29 How can it be computed? reverse topological order takes the element of the permutation S takes the accumulated worst-case workload from till the end of the cp-dag if the vertex has some successors if the vertex is the head node of a conditional pair is the successor of achieving the largest partial workload is merged into if instead the vertex is a regular one the workload contribution of all successors is merged into the worst-case workload accumulated by the source vertex is returned as output 30 5

6 9/04/20 4. Worst-case workload 5. Critical chain What is the complexity of this algorithm? Given a set of cp-tasks and a (work-conserving) scheduling algorithm, the critical chain λ of a cp-task τ is the chain of vertices of τ that leads to its worst-case response-time How can it be identified? set operations Any of them may require to compute, which has cost The time complexity is then We should know the worst-case instance of τ (i.e., the job of τ that has the largest response-time in the worst-case scenario) Then we should take its sink vertex, and recursively pre-pend the last to complete among the predecessor nodes, until the source vertex, has been included in the chain Key observation: the critical chain is unknown, but is always upperbounded by the longest path of the cp-task! 3 32 Critical interference To find the response-time of a cp-task, it is sufficient to characterize the maximum interference suffered by its critical chain The critical interference, imposed by task τ on task τ is the cumulative workload executed by vertices of τ while a node belonging to the critical chain of τ is ready to execute but is not executing Work-conserving schedulers Global schedulers are typically work-conserving (e.g., Global FP/EDF) Property: a ready job cannot execute only if all m processors are busy m τ i τ τ 2 τ 5 τ τ 5 τ 4 τ i τ 2 τ 7 τ 8 τ i r i r i + R i i Critical chain τ Critical interference of k τ on τ m τ τ τ 4 i τ 2 τ 2 τ 5 i τ 5 τ 7 τ 8 i, We can safely assume that the interference is distributed across all m processors Critical interference : total interference suffered by task τ, : total interference of task τ on task τ Types of interference We need to deal with two types of interference: o : from other tasks in the system; analogous to the classic notion o Intra-task interference: from vertices of the same task on itself; peculiar to parallel tasks only m τ τ τ τ τ τ 4 i 2 τ 2 5 i 5 τ 7 τ 8 i Interfering Interfering (i.e., not critical), λ + λ + For any work-conserving algorithm!, 35 Interfered Interfered (i.e., critical) λ λ +, λ,, Intra-task int. inter-task int. 3

7 9/04/20 Caused by other cp-tasks executing in the system Finding it exactly is difficult We need to find an upper-bound on the workload of an interfering task in the scheduling window, In the sequential case (global multiprocessor scheduling): Carry-in job Body jobs Carry-out job Sequential case The first job of τ starts executing as late as possible, with a starting time aligned with the beginning of the scheduling window Later jobs are executed as soon as possible Parallel case This scenario may not give a safe upper-bound on the interfering workload. Why? What is the scenario that maximizes the interfering workload? 37 Shifting right the scheduling window may give a larger interfering workload! 38 Pessimistic assumption Each interfering job of task τ executes for its worst-case workload The carry-in and carry-out contributions are evenly distributed among all processors Distributing them on less processors cannot increase the workload within the window Other task configurations cannot lead to a higher workload within the window Lemma: An upper-bound on the workload of an interfering task τ in a scheduling window of length is given by / min, Proof: The maximum number of carry-in and body instances within the window is / / Proof (continued): Each of the / instances contributes for The portion of the carry-out job included in the window is / min, Intra-task interference It is the interference from vertices of the same task on itself Who is interfering and who is interfered? The interfered contribution is the critical chain Critical chain: chain that leads to the WCRT of the cp-task At most processors may be occupied by the carry-out job The carry-out job cannot execute for more than units Critical chain longest path Longest path is time-units Critical chain can be either or If (c) {} else {} T T 2 T 3 T 4 endif

8 9/04/20 Intra-task interference Putting things together Simple upper-bound λ λ, λ, λ λ Length of the longest path,, λ critical chain Schedulability condition Given a cp-task set globally scheduled on processors, an upper-bound on the response-time of a task τ can be derived by the fixed-point iteration of the following expression, starting with : Global FP, Decreasing priority order 0, Global EDF, / min, Putting things together Reference Global FP The fixed-point iteration updates the bounds in decreasing priority order, starting from the highest priority task, until either: one of the response-time bounds exceeds the task relative deadline (negative schedulability result); A. Melani, M. Bertogna, V. Bonifaci, A. Marchetti- Spaccamela, G. Buttazzo, Response-Time Analysis of Conditional DAG Tasks in Multiprocessor Systems, Proceedings of the 27 th Euromicro Conference on Real- Time Systems (ECRTS 205) OR no more update is possible (positive schedulability result), i.e., : Global EDF Multiple rounds may be needed 45 4 Schedulability example Solution sketch Global FP m= Task is schedulable min High-priority task D = 35 T = 37 Len = 28 W = 37 Low-priority task D = 39 T = 229 Len = 37 W = Task 2 is schedulable

9 9/04/20 Thank you! Alessandra Melani 49 9