control
Uwe Resource R. Zimmer The Australian National University
8
Real-Time & Embedded Systems Real-Time & Embedded Systems
References for this chapter
[Ada95RM] (link to on-line version) Ada Working Group ISO/IEC JTC1/SC 22/WG 9 Ada 95 Reference Manual Language and Standard Libraries ISO/IEC 8652:1995(E) with COR.1:2000, June 2001 [Mercer97] Clifford W. Mercer Operating system resource reservation for realtime and multimedia applications Ph.D. thesis CMU-CS-97-155, June 1997, Pittsburgh, Pennsylvania 15213-3890
[Murthy2001] C. Siva Ram Murthy, G. Manimaran [Bloom79] Resource Management in Real-time Systems Toby Bloom and Networks Evaluating synchronization mechanisms MIT Press, Cambridge, Massachusetts, Proceedings of the seventh ACM Symposium London, England on Operating systems principles, 1979 [Burns01] Alan Burns and Andy Wellings Real-Time Systems and Programming Languages Addison Wesley, third edition, 2001 all references and links are available on the course page
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Real-Time & Embedded Systems Real-Time & Embedded Systems
Resource control
Topics in real-time resource control
from synchronization primitives and schedulers to resource management:
Toby Blooms evaluation criteria for synchronization primitives Resource atomicity, liveliness, and double interaction Resource reclaiming (C. Siva Ram Murthy, G. Manimaran) Resource reservation schemes (Clifford W. Mercer)
(not covered here: general dead-lock prevention / avoidance / detection / recovery algorithms operating systems course)
2002 Uwe R. Zimmer, The Australian National University Page 670 of 769 (chapter 8: to 725)
Real-Time & Embedded Systems Real-Time & Embedded Systems
Evaluating synchronization mechanisms
Categorizing resource/service requests
(based on Toby Bloom) Service requests can be categorized by: their type (read requests might be treated very differently from update request) their time (often: by their order or relative time only) their attributes, parameters, and the priority of the calling process (this includes timing constraints) the synchronization state of the resource (states which refer to the synchronisation aspect including timing constraints) the internal state of the resource (states which refer to the actual contents and available resources including timing constraints)
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Evaluating synchronization mechanisms
Categorizing resource synchronization methods
(based on Toby Bloom) Two (contradicting?) criteria:
Expressive power
are all (required) forms of synchronization available? can all timing requirements be expressed?
Ease of use
how error-prone are the constructs? how easy can basic methods be combined to complex resource control systems?
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Evaluating synchronization mechanisms
Accepting or Avoiding?
Requests which cannot be fullled right now, can be handled via
Conditional wait
accept all calls and suspend the threads internally all threads are immediately inside the synchronized server client threads are released from the server, only when the request is completed (can be overcome)
Avoidance synchronisation
suspend tasks on the level of guards all threads are at the borders of the synchronized server threads can easily revoke their requests
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Evaluating synchronization mechanisms
Handling resource requests
Required features: Handling request types by priorities Handling threads by priorities Handling threads in order or by their timing constraints Handling requests by client-attributes Handling requests by server state
(Ada95, Occam2) (most rt-systems) (most systems) (Real-time Java) (mostly: call needs to be accepted rst) (Ada95, Occam2)
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Evaluating synchronization mechanisms
Handling requests by types
WHILE TRUE PRI ALT ALT i=0 FOR max update [i] ? object ALT j=0 FOR max modify [j] ? object pragma Queuing_Policy (Priority_Queing); protected Resource_Manager is entry Update (); entry Modify (); end Resource_Manager;
serves clients with higher priority rst serves entries in order of declaration
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Evaluating synchronization mechanisms
Handling requests by types
WHILE TRUE PRI ALT ALT i=0 FOR max update [i] ? object ALT j=0 FOR max modify [j] ? object pragma Queuing_Policy (FIFO_Queing); protected Resource_Manager is entry Update (); entry Modify (); end Resource_Manager; protected body Resource_Manager is
how to control the order of requests regardless of their types? how to control permission depending on call-parameters?
entry Update () when is entry Modify () when and UpdateCount = 0 is end Resource_Manager;
serves entries in dened order serves clients in FIFO-order (disregarding priorities)
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Evaluating synchronization mechanisms
Handling requests by parameters
protected body resource_control is entry allocate(size : instances_of_resource) when resources_free >= size is begin resource_free := resource_free - size; end allocate; procedure free(size : instances_of_resource) is begin resource_free := resource_free + size; end free; end resource_control;
NOT VALID in ADA!
SR [Andrews and Olsson 1993] allows for such an direct access in most other synchronization environments: accept all and then conditional wait or requeue
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Handling requests by parameters (using wrappers)
package Resource_Manager is Max_Resources : constant Integer := 100; type Resource_Range is new Integer range 1..Max_Resources; subtype Instances_Of_Resource is Resource_Range range 1..50; procedure Allocate (Size : Instances_Of_Resource); procedure Free (Size : Instances_Of_Resource); end Resource_Manager; package body Resource_Manager is
Manager is informed about the request attributes rst entry family
task Manager is entry Sign_In (Size : Instances_Of_Resource); entry Allocate (Instances_Of_Resource); entry Free (Size : Instances_Of_Resource); end Manager;
procedure Allocate (Size : Instances_Of_Resource) is begin Manager.Sign_In (Size); double interaction is hidden Manager.Allocate (Size); end Allocate; procedure Free (Size : Instances_Of_Resource) is begin Manager.Free (Size); end Free;
Real-Time & Embedded Systems Real-Time & Embedded Systems
Handling requests by parameters (using wrappers)
package Resource_Manager is Max_Resources : constant Integer := 100; type Resource_Range is new Integer range 1..Max_Resources; subtype Instances_Of_Resource is Resource_Range range 1..50; procedure Allocate (Size : Instances_Of_Resource); procedure Free (Size : Instances_Of_Resource); end Resource_Manager; package body Resource_Manager is
Manager can apply any policy to accept the Allocate entries entry family
task Manager is entry Sign_In (Size : Instances_Of_Resource); entry Allocate (Instances_Of_Resource); entry Free (Size : Instances_Of_Resource); end Manager;
procedure Allocate (Size : Instances_Of_Resource) is begin Manager.Sign_In (Size); double interaction is hidden Manager.Allocate (Size); end Allocate; procedure Free (Size : Instances_Of_Resource) is begin Manager.Free (Size); end Free;
Real-Time & Embedded Systems Real-Time & Embedded Systems
Evaluating synchronization mechanisms
Handling requests by types, attributes, and in a global order
Lack of expressive power (e.g. in Ada95) may lead to:
Double Interactions
e.g. register all requests rst, then serve the individual types in a global order e.g. announce the parameters rst, then serve the individual types based in parameters Requests are no longer atomic! Server deadlocked, when wrongly assuming that the client is going to make the second call Client deadlocked, when wrongly assuming that the client died and is not going to make the second call
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Evaluating synchronization mechanisms
Handling requests by types, attributes, and in a global order
Lack of expressive power (e.g. in Ada95) may lead to:
Double Interactions
Ways out:
Dene the double interaction by means of atomic actions and make this known to the underlying synchronization methods. Assume that the client will never die during a double interaction sequence Eliminate the double interaction by means of a attributed, single request type and requeuing
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Evaluating synchronization mechanisms
Handling requests by types, attributes, and in a global order
type Request_Kinds is (Allocate_Req, Expand_Req, Free_Req); type Resource_Range is type Resource_Range_Groups is (small, medium, large); protected Resource_Control is entry Resource_Request (Kind : Request_Kinds; Amount : Resource_Range); private entry Allocate_Sign_In entry Allocate entry Expand_Sign_In entry Expand entry Free end Resource_Control; (Amount : Resource_Range); (Resource_Range_Groups); (Amount : Resource_Range); (Resource_Range_Groups); (Amount : Resource_Range);
Server has full control over the types, parameters, and orders
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Evaluating synchronization mechanisms
Handling requests by types, attributes, and in a global order
type Request_Kinds is (Allocate_Req, Expand_Req, Free_Req); type Resource_Range is The clients are providing type Resource_Range_Groups is (small, medium, large);
all information
protected Resource_Control is entry Resource_Request (Kind : Request_Kinds; Amount : Resource_Range); private entry Allocate_Sign_In entry Allocate entry Expand_Sign_In entry Expand entry Free end Resource_Control; (Amount : Resource_Range); (Resource_Range_Groups); (Amount : Resource_Range); (Resource_Range_Groups); (Amount : Resource_Range);
Server has full control over the types, parameters, and orders
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Evaluating synchronization mechanisms
Handling requests by types, attributes, and in a global order
type Request_Kinds is (Allocate_Req, Expand_Req, Free_Req); type Resource_Range is The clients are providing type Resource_Range_Groups is (small, medium, large);
all information
protected Resource_Control is entry Resource_Request (Kind : Request_Kinds; Amount : Resource_Range); private entry Allocate_Sign_In entry Allocate entry Expand_Sign_In entry Expand entry Free end Resource_Control; (Amount : Resource_Range); (Resource_Range_Groups); The protected object is arranging the
suspending (Amount : Resource_Range); queues accordingly (Resource_Range_Groups);(requeue-facility)
(Amount : Resource_Range);
Server has full control over the types, parameters, and orders
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Evaluating synchronization mechanisms
Handling requests by types, attributes, and in a global order
type Request_Kinds is (Allocate_Req, Expand_Req, Free_Req); type Resource_Range is type Resource_Range_Groups is Is the client going (small, medium, large); protected Resource_Control is
to loose all control?
entry Resource_Request (Kind : Request_Kinds; Amount : Resource_Range); private entry Allocate_Sign_In entry Allocate entry Expand_Sign_In entry Expand entry Free end Resource_Control; (Amount : Resource_Range); (Resource_Range_Groups); (Amount : Resource_Range); (Resource_Range_Groups); (Amount : Resource_Range);
Server has full control over the types, parameters, and orders
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Evaluating synchronization mechanisms
Handling requests by types and in a global order
requeue with abort
With a standard requeue statement: any outstanding timeout is cancelled the thread is no longer abortable clients losing control stemming from an ATC statement, or a timed entry-call the server can rely on the client thread no being revoked. With a requeue with abort statement: all timeouts are maintained allows the client to still revoke the call maintains client side control
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requeue can also lead to external entries! aborts need to be considered carefully
Real-Time & Embedded Systems Real-Time & Embedded Systems
Evaluating synchronization mechanisms
Categorizing resource/service requests
(based on Toby Bloom) Service requests can be categorized by: their type their time (often: by their order or relative time only) their attributes, parameters, and the priority of the calling process the synchronization state of the resource the internal state of the resource The real-time perspective:
take special care of failing tasks (atomic actions, deadlocks) determine and handle timing constraints in resource requests
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Real-Time & Embedded Systems Real-Time & Embedded Systems
Resource Reclaiming [Murthy2001]
Motivation for resource reclaiming
1. Worst case assumptions give schedulable systems, but might leave only a few spare resources. 2. Resources might not be actually used at run-time. 3. Some aspects of reliability in a real-time system rely directly on the amount of spare resources.
Resource reclaiming may enhance the systems reliability
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Resource Reclaiming [Murthy2001]
Resource reclaiming properties
Correctness:
maintain the feasibility!
Inexpensiveness:
resource reclaiming overhead need to be small in comparison to the possible gains
Bounded complexity:
resource reclaiming should be included in the tasks worst case computation time complexity needs to be bound by a constant
Effectiveness:
improve the systems actual reliability, thus e.g. more failures can be handled by applying resource reclaiming
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Resource Reclaiming [Murthy2001]
Expanded task-model
Each task t i has the following attributes: T i : cycle time E i : ready time D i : deadline C i : worst case computation time C i : actual computation time R i : worst case response time a set of resource conicts: t i t j , i.e. t i or t j requires a resource exclusively. a set of precedence constraints: t i < t j , i.e. t i completes always before t j may start.
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Further assumptions: n processors available tasks cannot migrate at most one task per processor task-queues are in shared memory tasks are not pre-empted
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Resource Reclaiming [Murthy2001]
More terminology
Feasible (prerun) schedule S :
taking into account timing, resource, precedence constraints, and worst case computation times.
Postrun schedule S :
starting from S and considering the actual computation times into account.
Start and nish times:
the scheduled start st i and nish times ft i as from the feasible prerun schedule S , and the actual start st i and nish times ft i as depicted in the postrun schedule S of the task t i .
Correct postrun schedule:
a postrun schedule is considered correct iff t i Q : ( st i st i ) ( ft i d i ) .
Passing tasks:
a task t i passed a task t j iff ( st i < st j ) ( ft j < st i ) , i.e. the strict order in S is not maintained.
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Real-Time & Embedded Systems Real-Time & Embedded Systems
Resource Reclaiming [Murthy2001]
Resource reclaiming algorithms
Two extreme versions: Dispatching according to the feasible prerun schedule S , i.e. no reclaiming at all resource reclaiming cost is zero. Global re-scheduling, whenever reclaiming is requested, or at each release of a resource, i.e. optimal reclaiming can be applied only, if the reclaiming cost is smaller than the gained resources Optimal scheduling of dynamically arriving non-pre-emptive tasks on a multi-processor environment
NP-hard
all practical re-scheduling algorithms are approximating. The come in two classes: Algorithms without passing bounded complexity Algorithms with passing in general: O(log n), but bounded with restricted passing
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Resource Reclaiming [Murthy2001]
Resource reclaiming from independent tasks
trivial: apply a greedy strategy, which dispatches tasks, whenever there are runable tasks.
Prerun schedule S
t10 t1 t2 t6 t10 1 5 10 t7 t11 15 20 25 t8 t12 30 35 t6 t7 t2 t3 t8 t11 t3 t4 t12 t5 t9 t13 40 45 50 t t4 t13 t5 t9
P1 P2 P3
t1
Feasible prerun schedule S
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Resource Reclaiming [Murthy2001]
Resource reclaiming from independent tasks
trivial: apply a greedy strategy, which dispatches tasks, whenever there are runable tasks.
Postrun schedule S
t10 t1 t2 t6 t10 1 5 10 t7 t11 15 20 25 t8 t12 30 35 t6 t7 t2 t3 t8 t11 t3 t4 t12 t5 t9 t13 40 45 50 t t4 t13 t5 t9
P1 P2 P3
t1
Postrun schedule S without resource reclaiming
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Resource Reclaiming [Murthy2001]
Resource reclaiming from independent tasks
trivial: apply a greedy strategy, which dispatches tasks, whenever there are runable tasks.
Reclaimed resources
t10 t1 t3 t7 t11 5 10 t12 15 t6 t7 t4 t8 t13 20 25 30 35 40 45 50 t t2 t8 t11 t5 t9 t3 t12 t4 t13 t5 t9
P1 P2 P3
t1
t2 t6 t10
1
Postrun schedule S with resource reclaiming for independent tasks
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Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
greedy reclaiming
Postrun schedule S
t10 t1 t2 t6 t10 1 5 10 t7 t11 15 20 25 t8 t12 30 35 t6 t7 t2 t3 t8 t11 t3 t4 t12 t5 t9 t13 40 45 50 t t4 t13 t5 t9
P1 P2 P3
t1
Postrun schedule S without resource reclaiming Tasks t 4 t 7 ; t 4 t 9 ; t 7 t 9 ; t 7 t 12 ; t 9 t 12 have conicting resource locks
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Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
greedy reclaiming
Runtime anomaly
t10 t1 t3 t7 t11 5 10 t12 15 t13 20 25 30 35 40 45 50 t t8 t6 t7 t2 t8 t11 t4 t3 t5 t9 t12 t4 t13 t5 t9
P1 P2 P3
t1
t2 t6 t10
1
Postrun schedule S without greedy resource reclaiming Tasks t 4 t 7 ; t 4 t 9 ; t 7 t 9 ; t 7 t 12 ; t 9 t 12 have conicting resource requests
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Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
basic reclaiming: look for simultaneous idling
Postrun schedule S
t10 t1 t2 t6 t10 1 5 10 t7 t11 15 20 25 t12 t8 30 35 t6 t7 t2 t3 t8 t11 t3 t4 t12 t5 t9 t13 40 45 50 t t4 t13 t5 t9
P1 P2 P3
t1
Postrun schedule S without resource reclaiming Tasks t 4 t 7 ; t 4 t 9 ; t 7 t 9 ; t 7 t 12 ; t 9 t 12 have conicting resource locks
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Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
basic reclaiming: look for simultaneous idling
Basic reclaiming
t10 t1 t2 t6 t10 1 5 10 t7 t11 15 20 t8 t12 25 30 t6 t7 t3 t2 t8 t11 t4 t3 t5 t9 t13 35 40 45 50 t t12 t4 t13 t5 t9
P1 P2 P3
t1
Postrun schedule S without basic resource reclaiming Tasks t 4 t 7 ; t 4 t 9 ; t 7 t 9 ; t 7 t 12 ; t 9 t 12 have conicting resource locks
2002 Uwe R. Zimmer, The Australian National University
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Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
early start algorithm Detect overlaps in the prerun schedule S: t <i = { t j ft j < st i } t >i = { t j st j > ft i } t ~i = { t j ( ( t j t <i ) ( t j t >i ) ) } all tasks which overlap with t i in S Detect tasks overlapping with t i on processor k and order all sets Allow tasks in t ~i to be executed simultaneously and ensure that they do not overlap with tasks out of t <i or t >i . Complexity O ( m 2 ) ; with m processors.
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Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
early start algorithm
Postrun schedule S
t10 t1 t2 t6 t10 1 5 10 t7 t11 15 20 25 t8 t12 30 35 t6 t7 t2 t3 t8 t11 t3 t4 t12 t5 t9 t13 40 45 50 t t4 t13 t5 t9
P1 P2 P3
t1
Postrun schedule S without resource reclaiming Tasks t 4 t 7 ; t 4 t 9 ; t 7 t 9 ; t 7 t 12 ; t 9 t 12 have conicting resource locks
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Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
early start algorithm
Early start resource reclaiming
t10 t1 t2 t6 t10 1 5 10 t7 t11 15 20 t6 t7 t3 t8 t12 25 30 t2 t8 t11 t4 t3 t5 t9 t13 35 40 45 50 t t12 t4 t13 t5 t9
P1 P2 P3
t1
Postrun schedule S without early start resource reclaiming Tasks t 4 t 7 ; t 4 t 9 ; t 7 t 9 ; t 7 t 12 ; t 9 t 12 have conicting resource locks
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Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
Restriction vectors (RV) algorithm (Shen, Ramamritham, Stankovic, 93)
Restriction vector (RV):
RV i [ j ] = t k t <i ( j ) ( t l t <i ( j ) st l > st k ) if j = proc ( i ) t m t <i ( j ) ( t m < t i t m t i ) ( t l t <i ( j ) ( st l > st m ) ( t l < t i t l t i ) ) if j proc ( i ) no such task
Completion bit matrix (CBM):
1 iff task t has completed its scheduled execution in processor j i CMB [ i, j ] = 0 otherwise
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Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
Restriction vectors (RV) algorithm (Shen, Ramamritham, Stankovic, 93) compute the RV i ( j ) by checking the k most recent tasks in t <i ( j ) for any task t i next to be scheduled on processor j : fetch the most recent CBM if t l RV i ( j ) CBM ( i, j ) = 1 then start t i else idle until the next CBM update. The algorithm is heuristic in the sense that it is only checking the k most recent tasks in t <i ( j ) The complexity is O ( m 2 ) since m processors need to check m RV-entries bounded
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Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
Restriction vectors (RV) algorithm (Shen, Ramamritham, Stankovic, 93)
Prerun schedule S
t10 t1 t2 t6 t10 1 5 10 t7 t11 15 20 25 t8 t12 30 35 t6 t7 t2 t3 t8 t11 t3 t4 t12 t5 t9 t13 40 45 50 t t4 t13 t5 t9
P1 P2 P3
t1
Tasks t 4 t 7 ; t 4 t 9 ; t 7 t 9 ; t 7 t 12 ; t 9 t 12 have conicting resource locks. Tasks t 10 < t 8 ; t 10 < t 4 ; t 8 < t 9 ; t 8 < t 13 ; t 1 < t 2 ; t 1 < t 3 ; t 2 < t 12 ; t 3 < t 12 ; t 11 < t 12 have precedence relations.
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Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
Restriction vectors (RV) algorithm (Shen, Ramamritham, Stankovic, 93)
Prerun schedule S
t10 t1 t2 t6 t10 1 5 10 t7 t11 15 20 25 t8 t12 30 35 t6 t7 t2 t3 t8 t11 t3 t4 t12 t5 t9 t13 40 45 50 t t4 t13 t5 t9
P1 P2 P3
t1
RVs: t 1 : [ -, -, - ] ; t 6 : [ -, -, - ] ; t 11 : [ -, -, t 10 ] ;
t 2 : [ t 1, - , - ] ; t 7 : [ -, t 6 , - ] ;
t 3 : [ t 2, - , - ] ; t 8 : [ -, t 7, t 10 ] ;
t 4 : [ t 3, t 7, t 10 ] ; t 9 : [ t 4, t 8, t 12 ] ;
t 5 : [ t 4, -, - ] ; t 10 : [ -, -, - ] ;
t 12 : [ t 3, t 7, t 11 ] ; t 13 : [ -, t 8, - ]
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Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
Restriction vectors (RV) algorithm (Shen, Ramamritham, Stankovic, 93)
Postrun schedule S
t10 t1 t2 t6 t10 1 5 10 t7 t11 15 20 25 t8 t12 30 35 t6 t7 t2 t3 t8 t11 t3 t4 t12 t5 t9 t13 40 45 50 t t4 t13 t5 t9
P1 P2 P3
t1
RVs: t 1 : [ -, -, - ] ; t 6 : [ -, -, - ] ; t 11 : [ -, -, t 10 ] ;
t 2 : [ t 1, - , - ] ; t 7 : [ -, t 6 , - ] ;
t 3 : [ t 2, - , - ] ; t 8 : [ -, t 7, t 10 ] ;
t 4 : [ t 3, t 7, t 10 ] ; t 9 : [ t 4, t 8, t 12 ] ;
t 5 : [ t 4, -, - ] ; t 10 : [ -, -, - ] ;
t 12 : [ t 3, t 7, t 11 ] ; t 13 : [ -, t 8, - ]
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Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
Restriction vectors (RV) algorithm (Shen, Ramamritham, Stankovic, 93)
Restriction vector based resource reclaiming
t10 t1 t3 t7 t11 5 10 15 t8 t12 20 t6 t7 t4 t2 t8 t11 t5 t9 t13 25 30 35 40 45 50 t t3 t12 t4 t13 t5 t9
P1 P2 P3
t1
t2 t6 t10
1
RVs: t 1 : [ -, -, - ] ; t 6 : [ -, -, - ] ; t 11 : [ -, -, t 10 ] ;
t 2 : [ t 1, - , - ] ; t 7 : [ -, t 6 , - ] ;
t 3 : [ t 2, - , - ] ; t 8 : [ -, t 7, t 10 ] ;
t 4 : [ t 3, t 7, t 10 ] ; t 9 : [ t 4, t 8, t 12 ] ;
t 5 : [ t 4, -, - ] ; t 10 : [ -, -, - ] ;
t 12 : [ t 3, t 7, t 11 ] ; t 13 : [ -, t 8, - ]
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Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
Restriction vectors (RV) algorithm (Shen, Ramamritham, Stankovic, 93)
Proof of Correctness
Lemma: Given a feasible prerun schedule S : if t i ( st i > st i ) then passing must have occurred. Proof: Assuming that no passing occurred, then all t t <i have been dispatched before t i and all t t >i are only dispatched after t i completed. By denition of a feasible schedule all t t ~i do not interfere with t i and can thus by no means delay the execution of t i . Therefore st i st i
2002 Uwe R. Zimmer, The Australian National University Page 709 of 769 (chapter 8: to 725)
Real-Time & Embedded Systems Real-Time & Embedded Systems
Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
Restriction vectors (RV) algorithm (Shen, Ramamritham, Stankovic, 93)
Proof of Correctness
Theorem: The RV-algorithm gives a correct postrun schedule S . Proof: By the above lemma, passing occurred if S is incorrect, i.e. t i, t j ( st j > ft i ) ( st j < st i ) ( st i > st i ) . Two cases need to be distinguished: case 1: t i and t j have resource or precedence conicts, then t i is directly or transitively included in the restriction vector RV j . Therefore this case of passing is prevented by the RV-algorithm. case 2: t i and t j have no resource or precedence conicts. In this case t j cannot delay the execution of t i by means of passing and the postrun schedule S would be correct still. Therefore the RV-algorithm allows for restricted forms of passing only, which does not corrupt the correctness of the postrun schedule S .
2002 Uwe R. Zimmer, The Australian National University Page 710 of 769 (chapter 8: to 725)
Real-Time & Embedded Systems Real-Time & Embedded Systems
Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
Restriction vectors (RV) algorithm with task migration (Manimaran, Murthy, 97)
Restriction vector (RV) with static processor assignment:
RV i [ j ] = t k t <i ( j ) ( t l t <i ( j ) st l > st k ) if j = proc ( i ) t m t <i ( j ) ( t m < t i t m t i ) ( t l t <i ( j ) ( st l > st m ) ( t l < t i t l t i ) ) if j proc ( i ) no such task
Restriction vector (RV) with dynamic processor assignment:
t t <i ( j ) ( t m < t i t m t i ) ( t l t <i ( j ) ( st l > st m ) ( t l < t i t l t i ) ) if t m exists RV i [ j ] = m no such task
2002 Uwe R. Zimmer, The Australian National University Page 711 of 769 (chapter 8: to 725)
Real-Time & Embedded Systems Real-Time & Embedded Systems
Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
Restriction vectors (RV) algorithm with task migration (Manimaran, Murthy, 97)
Prerun schedule S
t10 t1 t2 t6 t10 1 5 10 t7 t11 15 20 25 t8 t12 30 35 t6 t7 t2 t3 t8 t11 t3 t4 t12 t5 t9 t13 40 45 50 t t4 t13 t5 t9
P1 P2 P3
t1
RVs: t 1 : [ -, -, - ] ; t 6 : [ -, -, - ] ; t 11 : [ -, -, - ] ;
t 2 : [ t 1, - , - ] ; t 7 : [ -, -, - ] ;
t 3 : [ t 1, - , - ] ; t 8 : [ -, -, t 10 ] ;
t 4 : [ -, t 7, t 10 ] ; t 9 : [ t 4, t 8, t 12 ] ;
t 5 : [ -, -, - ] ; t 10 : [ -, -, - ] ;
t 12 : [ t 3, t 7, t 11 ] ; t 13 : [ -, t 8, - ]
Page 712 of 769 (chapter 8: to 725)
2002 Uwe R. Zimmer, The Australian National University
Real-Time & Embedded Systems Real-Time & Embedded Systems
Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
Restriction vectors (RV) algorithm with task migration (Manimaran, Murthy, 97)
Prerun schedule S
t10 t1 t2 t6 t10 1 5 10 t7 t11 15 20 25 t8 t12 30 35 t6 t7 t2 t3 t8 t11 t3 t4 t12 t5 t9 t13 40 45 50 t t4 t13 t5 t9
P1 P2 P3
t1
RVs: t 1 : [ -, -, - ] ; t 6 : [ -, -, - ] ; t 11 : [ -, -, - ] ;
t 2 : [ t 1, - , - ] ; t 7 : [ -, -, - ] ;
t 3 : [ t 1, - , - ] ; t 8 : [ -, -, t 10 ] ;
t 4 : [ -, t 7, t 10 ] ; t 9 : [ t 4, t 8, t 12 ] ;
t 5 : [ -, -, - ] ; t 10 : [ -, -, - ] ;
t 12 : [ t 3, t 7, t 11 ] ; t 13 : [ -, t 8, - ]
Page 713 of 769 (chapter 8: to 725)
2002 Uwe R. Zimmer, The Australian National University
Real-Time & Embedded Systems Real-Time & Embedded Systems
Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
Restriction vectors (RV) algorithm with task migration (Manimaran, Murthy, 97)
Postrun schedule S with task migration
t10 t1 t3 t7 t11 5 10 15 t8 t12 20 25 t6 t7 t4 t2 t8 t11 t3 t5 t9 t13 30 35 40 45 50 t t12 t4 t13 t5 t9
P1 P2 P3
t1
t2 t6 t10
1
RVs: t 1 : [ -, -, - ] ; t 6 : [ -, -, - ] ; t 11 : [ -, -, - ] ;
t 2 : [ t 1, - , - ] ; t 7 : [ -, -, - ] ;
t 3 : [ t 1, - , - ] ; t 8 : [ -, -, t 10 ] ;
t 4 : [ -, t 7, t 10 ] ; t 9 : [ t 4, t 8, t 12 ] ;
t 5 : [ -, -, - ] ; t 10 : [ -, -, - ] ;
t 12 : [ t 3, t 7, t 11 ] ; t 13 : [ -, t 8, - ]
Page 714 of 769 (chapter 8: to 725)
2002 Uwe R. Zimmer, The Australian National University
Real-Time & Embedded Systems Real-Time & Embedded Systems
Resource Reclaiming [Murthy2001]
Resource reclaiming from interdependent tasks
Restriction vectors (RV) algorithm with task migration (Manimaran, Murthy, 97)
Postrun schedule S with task migration
t10 t1 t3 t6 t7 t4 t12 t11 5 10 t7 15 t8 20 25 t2 t8 t11 t3 t5 t13 t9 30 35 40 45 50 t ...