Method of scheduling successive tasks subject only to timing constraints

0Associated
Cases 
0Associated
Defendants 
0Accused
Products 
79Forward
Citations 
0
Petitions 
1
Assignment
First Claim
1. Method for scheduling successive tasks by means of a computer, said tasks being subject only to timing constraints, a timing constraint requiring that the execution start time be in at least one predetermined time interval relative to an absolute reference time;
 said method comprising the following successive steps in this order;
calculating for each task upper and lower limits of the interval in which execution of that task must start;
constructing a first series in which all said tasks are scheduled in increasing order of their lower limit, and are scheduled in increasing order of their upper limit when several tasks have a same lower limit;
constructing a second series in which all said tasks are scheduled in increasing order of their upper limit, and are scheduled in decreasing order of their lower limits when several tasks have a same upper limit and have different lower limits;
constructing a current permutation, first by scheduling all said tasks in the order of said first series;
verifying if said current permutation satisfies all said constraints supplying to said tasks, the tasks being considered one by one in the order corresponding to said current permutation, to check whether each task satisfies all the constraints applying to said task;
concluding that the scheduling succeeds if all said constraints are satisfied;
otherwise, determining in said current permutation the first illplaced task for which a constraint is not satisfied;
determining in said second series a candidate task immediately following said illplaced task in said second series that also precedes said illplaced task in said current permutation, said candidate being a task which has already been verified, all the tasks following said candidate task in said current permutation being not considered as satisfying all the constraints, any more;
verifying that if said candidate task is shifted to a position immediately after said illplaced task all said constraints applying to all said tasks shifted in this way are then satisfied; and
if at least one constraint is not satisfied, concluding that said candidate task is not suitable and then determining in said second series another candidate task and repeating the previous verification; and
, if this is not possible, concluding that the scheduling fails;
if all said constraints are satisfied, concluding that the scheduling succeeds.
1 Assignment
0 Petitions
Accused Products
Abstract
A method of scheduling successive tasks subject only to timing constraints calculates for each task upper and lower limits of the interval in which execution of that task must start. It then constructs a first series in which all the tasks are scheduled in increasing order of their lower limit and a second series in which all the asks are scheduled in increasing order of their upper limit, before constructing an initial permutation by scheduling all the tasks in the order of the first series and verifying if the initial permutation satisfies all the constraints. If not all the constraints are satisfied, the method determines in the initial permutation the first illplaced task for which a constraint is not satisfied and a candidate task in the second series immediately preceding the illplaced task in the second series in the current permutation. It then verifies that if the candidate task is shifted in the current permutation to a position immediately after the illplaced task all the constraints applying to all the tasks shifted in this way are then satisfied.
79 Citations
Method for preventing process collision of micro controller  
Patent #
US 7,954,103 B2
Filed 08/03/2006

Current Assignee
Wistron Corporation

Sponsoring Entity
Wistron Corporation

Method and system for performing heuristic constraint simplification  
Patent #
US 7,793,242 B2
Filed 11/15/2007

Current Assignee
GlobalFoundries Inc.

Sponsoring Entity
International Business Machines Corporation

Computation of travel routes, durations, and plans over multiple contexts  
Patent #
US 7,739,040 B2
Filed 06/30/2006

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

Method and system for performing heuristic constraint simplification  
Patent #
US 7,788,616 B2
Filed 11/15/2007

Current Assignee
GlobalFoundries Inc.

Sponsoring Entity
International Business Machines Corporation

Inferring road speeds for contextsensitive routing  
Patent #
US 7,706,964 B2
Filed 06/30/2006

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

Method and apparatus for synchronization of periodic processes  
Patent #
US 7,712,099 B2
Filed 11/02/2004

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

DESTINATION AUCTIONED THROUGH BUSINESS OF INTEREST  
Patent #
US 20090157540A1
Filed 12/14/2007

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

GENERATIONAL INTELLIGENT NAVIGATION SYNCHRONIZATION OR UPDATE  
Patent #
US 20090157498A1
Filed 12/14/2007

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

ROUTE REWARD AUGMENTATION  
Patent #
US 20090210302A1
Filed 02/19/2008

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

LOAD BALANCE PAYMENT  
Patent #
US 20090210242A1
Filed 02/19/2008

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

SAFE ROUTE CONFIGURATION  
Patent #
US 20090210142A1
Filed 02/19/2008

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

Collaborative route planning for generating personalized and contextsensitive routing recommendations  
Patent #
US 7,610,151 B2
Filed 06/27/2006

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

Computing and harnessing inferences about the timing, duration, and nature of motion and cessation of motion with applications to mobile computing and communications  
Patent #
US 7,617,042 B2
Filed 06/30/2006

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

INFERRING ROAD SPEEDS FOR CONTEXTSENSITIVE ROUTING  
Patent #
US 20080004789A1
Filed 06/30/2006

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

Method and system for performing heuristic constraint simplification  
Patent #
US 7,315,996 B2
Filed 09/22/2005

Current Assignee
GlobalFoundries Inc.

Sponsoring Entity
International Business Machines Corporation

ROUTE PLANNING WITH CONTINGENCIES  
Patent #
US 20080004802A1
Filed 06/30/2006

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

COMPUTATION OF TRAVEL ROUTES, DURATIONS, AND PLANS OVER MULTIPLE CONTEXTS  
Patent #
US 20080004794A1
Filed 06/30/2006

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

COMPUTING AND HARNESSING INFERENCES ABOUT THE TIMING, DURATION, AND NATURE OF MOTION AND CESSATION OF MOTION WITH APPLICATIONS TO MOBILE COMPUTING AND COMMUNICATIONS  
Patent #
US 20080004793A1
Filed 06/30/2006

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

Method and System for Performing Heuristic Constraint Simplification  
Patent #
US 20080066034A1
Filed 11/15/2007

Current Assignee
GlobalFoundries Inc.

Sponsoring Entity
GlobalFoundries Inc.

Method and System for Performing Heuristic Constraint Simplification  
Patent #
US 20080066033A1
Filed 11/15/2007

Current Assignee
GlobalFoundries Inc.

Sponsoring Entity
GlobalFoundries Inc.

Providing predictable scheduling of programs using repeating precomputed schedules on discretely scheduled and/or multiprocessor operating systems  
Patent #
US 7,409,689 B2
Filed 10/31/2005

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

INSTRUCTION AND TRAINING TOOL  
Patent #
US 20080216012A1
Filed 02/29/2008

Current Assignee
United Space Alliance LLC

Sponsoring Entity
United Space Alliance LLC

Method for the RealTime Analysis of a System  
Patent #
US 20080276247A1
Filed 03/03/2006

Current Assignee
INCHRON GMBH

Sponsoring Entity
INCHRON GMBH

Providing predictable scheduling of programs using repeating precomputed schedules on discretely scheduled and/or multiprocessor operating systems  
Patent #
US 7,171,669 B2
Filed 05/07/2004

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

Providing predictable scheduling of programs using repeating precomputed schedules on discretely scheduled and/or multiprocessor operating systems  
Patent #
US 7,162,720 B2
Filed 05/10/2004

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

Method for handling byte code in a configurable manner  
Patent #
US 7,185,102 B1
Filed 07/11/2000

Current Assignee
Sun Microsystems Incorporated

Sponsoring Entity
Sun Microsystems Incorporated

Method and system for performing heuristic constraint simplification  
Patent #
US 20070067746A1
Filed 09/22/2005

Current Assignee
GlobalFoundries Inc.

Sponsoring Entity
GlobalFoundries Inc.

Program execution service windows  
Patent #
US 20070150815A1
Filed 12/22/2005

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

Variable speed printing  
Patent #
US 20070248374A1
Filed 04/19/2007

Current Assignee
OcTechnologies B.V.

Sponsoring Entity
OcTechnologies B.V.

COLLABORATIVE ROUTE PLANNING FOR GENERATING PERSONALIZED AND CONTEXTSENSITIVE ROUTING RECOMMENDATIONS  
Patent #
US 20070299599A1
Filed 06/27/2006

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

METHOD FOR PREVENTING PROCESS COLLISION OF MICRO CONTROLLER  
Patent #
US 20070288928A1
Filed 08/03/2006

Current Assignee
Wistron Corporation

Sponsoring Entity
Wistron Corporation

Method for notifying an individual of a previously scheduled event  
Patent #
US 6,144,942 A
Filed 04/28/1998

Current Assignee
Round Rock Research LLC

Sponsoring Entity
Mei California Inc.

Method for flow scheduling  
Patent #
US 6,738,972 B1
Filed 12/30/1999

Current Assignee
OpenTV Inc.

Sponsoring Entity
OpenTV Inc.

Method and apparatus for providing enhanced pay per view in a video server  
Patent #
US 6,012,080 A
Filed 03/27/1996

Current Assignee
ATT Inc.

Sponsoring Entity
ATT Inc.

Providing predictable scheduling of programs using repeating precomputed schedules on discretely scheduled and/or multiprocessor operating systems  
Patent #
US 6,745,222 B1
Filed 07/08/1999

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

Providing predictable scheduling of programs using a repeating precomputed schedule  
Patent #
US 6,317,774 B1
Filed 01/09/1997

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

Providing predictable scheduling of programs using a repeating precomputed schedule  
Patent #
US 6,490,612 B1
Filed 05/04/2000

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

Providing predictable scheduling of programs using repeating precomputed schedules on discretely scheduled and/or multiprocessor operating systems  
Patent #
US 7,000,232 B2
Filed 05/17/2004

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

Method and apparatus for providing enhanced pay per view in a video server employing a coarsegrained striping scheme  
Patent #
US 5,964,829 A
Filed 03/27/1996

Current Assignee
ATT Inc.

Sponsoring Entity
ATT Inc.

Providing predictable scheduling of programs using repeating precomputed schedules on discretely scheduled and/or multiprocessor operating systems  
Patent #
US 7,155,724 B2
Filed 05/17/2004

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

Providing predictable scheduling of programs using a repeating precomputed schedule  
Patent #
US 6,718,360 B1
Filed 05/04/2000

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

Providing predictable scheduling of programs using repeating precomputed schedules on discretely scheduled and/or multiprocessor operating systems  
Patent #
US 7,107,593 B2
Filed 05/13/2004

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

Providing predictable scheduling of programs using repeating precomputed schedules on discretely scheduled and/or multiprocessor operating systems  
Patent #
US 20060053423A1
Filed 10/31/2005

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

Automatic planning and cueing system and method  
Patent #
US 7,027,996 B2
Filed 03/07/2002

Current Assignee
Attention Control Systems Inc.

Sponsoring Entity
Attention Control Systems Inc.

Method and device for reconstructing the process sequence of a control program  
Patent #
US 7,086,054 B2
Filed 12/28/2001

Current Assignee
Robert Bosch GmbH

Sponsoring Entity
Robert Bosch GmbH

Method and apparatus for synchronization of periodic processes  
Patent #
US 20050132372A1
Filed 11/02/2004

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

Providing predictable scheduling of programs using repeating precomputed schedules on discretely scheduled and/or multiprocessor operating systems  
Patent #
US 20040210904A1
Filed 05/13/2004

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

Providing predictable scheduling of programs using repeating precomputed schedules on discretely scheduled and/or multiprocessor operating systems  
Patent #
US 20040216121A1
Filed 05/17/2004

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

Providing predictable scheduling of programs using repeating precomputed schedules on discretely scheduled and/or multiprocessor operating systems  
Patent #
US 20040210901A1
Filed 05/10/2004

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

Providing predictable scheduling of programs using repeating precomputed schedules on discretely scheduled and/or multiprocessor operating systems  
Patent #
US 20040210900A1
Filed 05/07/2004

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

Time scheduling method and scheduler for a modular engine that negotiates among modules timing offers which include a timing range  
Patent #
US 6,633,990 B1
Filed 03/15/2000

Current Assignee
OcTechnologies B.V.

Sponsoring Entity
OcTechnologies B.V.

Computer system process scheduler determining and executing processes based upon changeable priorities  
Patent #
US 6,668,269 B1
Filed 12/29/1999

Current Assignee
Fujitsu Limited

Sponsoring Entity
Fujitsu Limited

Internetbased appointment scheduling  
Patent #
US 20020143600A1
Filed 03/08/2002

Current Assignee
Dugan Health LLC

Sponsoring Entity
Dugan Health LLC

Method and device for reconstructing the process sequence of a control program  
Patent #
US 20020152256A1
Filed 12/28/2001

Current Assignee
Robert Bosch GmbH

Sponsoring Entity
Robert Bosch GmbH

Method of determining a schedule, scheduler and system  
Patent #
US 20020156669A1
Filed 02/26/2002

Current Assignee
US Philips Corporation

Sponsoring Entity
US Philips Corporation

Multimode mobile telephone apparatus  
Patent #
US 6,317,582 B1
Filed 05/04/2000

Current Assignee
Kyocera Corporation

Sponsoring Entity
Kyocera Corporation

Method and system for schedule and task management  
Patent #
US 6,035,278 A
Filed 07/08/1997

Current Assignee
Facebook Inc.

Sponsoring Entity
Netscape Communications Corporation

Representing work practices  
Patent #
US 5,745,113 A
Filed 04/03/1996

Current Assignee
Verizon Patent and Licensing Incorporated

Sponsoring Entity
RESEARCH ON LEARNING INSTITUTE FOR, BELL ATLANTIC SCIENCE AND TECHNOLOGY INC.

Route planning with contingencies  
Patent #
US 8,126,641 B2
Filed 06/30/2006

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

Computation of travel routes, durations, and plans over multiple contexts  
Patent #
US 8,090,530 B2
Filed 01/22/2010

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

Automated resource planning tool and user interface  
Patent #
US 8,103,533 B2
Filed 08/24/2006

Current Assignee
United Space Alliance LLC

Sponsoring Entity
United Space Alliance LLC

Universal software architecture for decision support  
Patent #
US 8,170,980 B1
Filed 10/10/2003

Current Assignee
NAVY SECRETARY OF THE UNITED STATES

Sponsoring Entity
NAVY SECRETARY OF THE UNITED STATES

Method for the realtime capability analysis of a system by selectively using approximated or actual system expenses for jobs  
Patent #
US 8,185,900 B2
Filed 03/03/2006

Current Assignee
INCHRON GMBH

Sponsoring Entity
INCHRON GMBH

Variable speed printing  
Patent #
US 8,231,287 B2
Filed 04/19/2007

Current Assignee
OcTechnologies B.V.

Sponsoring Entity
OcTechnologies B.V.

Stowage and center of gravity verification and assessment tool  
Patent #
US 8,301,473 B2
Filed 02/29/2008

Current Assignee
United Space Alliance LLC

Sponsoring Entity
United Space Alliance LLC

Distributing data via a network including timing information  
Patent #
US 8,312,167 B2
Filed 06/06/2005

Current Assignee
Oracle America Inc.

Sponsoring Entity
Oracle America Inc.

Computation of travel routes, durations, and plans over multiple contexts  
Patent #
US 8,473,197 B2
Filed 12/15/2011

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

Program execution service windows  
Patent #
US 8,495,613 B2
Filed 12/22/2005

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Corporation

Collaborative route planning for generating personalized and contextsensitive routing recommendations  
Patent #
US 8,718,925 B2
Filed 05/14/2009

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

PROGRAM EXECUTION SERVICE WINDOWS  
Patent #
US 20140165051A1
Filed 07/22/2013

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

Employee scheduling and schedule modification method and apparatus  
Patent #
US 8,788,308 B1
Filed 03/29/2004

Current Assignee
West Corporation

Sponsoring Entity
West Corporation

Route monetization  
Patent #
US 8,793,066 B2
Filed 12/14/2007

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

GENERATING MULTIPLE OPTIMAL DAILY SCHEDULES FOR MULTIPLE TIME PERIODS IN A DAY AND FOR MULTIPLE DAILY PATTERNS  
Patent #
US 20150012323A1
Filed 07/02/2013

Current Assignee
Oracle International Corporation

Sponsoring Entity
Oracle International Corporation

Computation of travel routes, durations, and plans over multiple contexts  
Patent #
US 9,008,960 B2
Filed 06/19/2013

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

Program execution service windows  
Patent #
US 9,195,450 B2
Filed 07/22/2013

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

Providing user controlled ability to determine data level of detail in a graph  
Patent #
US 9,208,591 B2
Filed 04/18/2013

Current Assignee
International Business Machines Corporation

Sponsoring Entity
International Business Machines Corporation

Computing and harnessing inferences about the timing, duration, and nature of motion and cessation of motion with applications to mobile computing and communications  
Patent #
US 9,398,420 B2
Filed 01/06/2014

Current Assignee
Microsoft Technology Licensing LLC

Sponsoring Entity
Microsoft Technology Licensing LLC

Methods, systems, and apparatus for scheduling appointments  
Patent #
US 9,842,317 B2
Filed 05/27/2013

Current Assignee
Brandon Jordan

Sponsoring Entity
Brandon Jordan

Computer method and apparatus for automated scheduling  
Patent #
US 10,241,654 B2
Filed 12/20/2013

Current Assignee
Dassault Systemes Americas Corp.

Sponsoring Entity
Dassault Systemes Americas Corp.

No References
1 Claim
 1. Method for scheduling successive tasks by means of a computer, said tasks being subject only to timing constraints, a timing constraint requiring that the execution start time be in at least one predetermined time interval relative to an absolute reference time;
said method comprising the following successive steps in this order; calculating for each task upper and lower limits of the interval in which execution of that task must start; constructing a first series in which all said tasks are scheduled in increasing order of their lower limit, and are scheduled in increasing order of their upper limit when several tasks have a same lower limit; constructing a second series in which all said tasks are scheduled in increasing order of their upper limit, and are scheduled in decreasing order of their lower limits when several tasks have a same upper limit and have different lower limits; constructing a current permutation, first by scheduling all said tasks in the order of said first series; verifying if said current permutation satisfies all said constraints supplying to said tasks, the tasks being considered one by one in the order corresponding to said current permutation, to check whether each task satisfies all the constraints applying to said task; concluding that the scheduling succeeds if all said constraints are satisfied; otherwise, determining in said current permutation the first illplaced task for which a constraint is not satisfied; determining in said second series a candidate task immediately following said illplaced task in said second series that also precedes said illplaced task in said current permutation, said candidate being a task which has already been verified, all the tasks following said candidate task in said current permutation being not considered as satisfying all the constraints, any more; verifying that if said candidate task is shifted to a position immediately after said illplaced task all said constraints applying to all said tasks shifted in this way are then satisfied; and if at least one constraint is not satisfied, concluding that said candidate task is not suitable and then determining in said second series another candidate task and repeating the previous verification; and
, if this is not possible, concluding that the scheduling fails;if all said constraints are satisfied, concluding that the scheduling succeeds.
1 Specification
1. Field of the Invention
The invention concerns a method of scheduling successive tasks by means of a computer by determining a task execution order and an execution start time for each task, no two tasks ever being executed simultaneously. This scheduling is based on a plurality of constraints that the tasks must satisfy. This process is more particularly concerned with applications in which there is only one type of constraint, namely timing constraints: the execution of a task must begin at a time within at least one predetermined time interval relative to an absolute time reference. There are no constraints applying directly to the order of the tasks. The process naturally concerns also applications in which all constraints can be regarded as equivalent to timing constraints relative to an absolute reference.
The method of the invention is applicable in particular to tasks that must be executed successively because they are executed by single means capable of executing only one task at a time, for example: a machine tool, a data bus, a team of workers. In the field of electronic data processing, the method can be applied to the management of a plurality of predetermined tasks to be executed successively in the same processor or on the same bus. In the field of industrial process control, the method can be applied in particular to the management of a socalled field bus used to transmit information successively in accordance with a predetermined series.
2. Description of the Prior Art
The prior art includes many scheduling methods:
socalled polynomial or critical path methods;
linear programming methods, especially the simplex method on which the PROLOG III language is based;
dynamic programming methods that can be applied only to relatively small problems; and
heuristic methods that use some algorithms employing the above methods but further reduce the number of cases to be verified by simplifying certain constraints; the resulting solution is then less than optimal.
The prior art methods have two drawbacks: they require a long computation time since they systematically verify a very large number of permutations before giving a solution. The computation time is usually proportional to the factorial of the number of tasks to be scheduled.
To schedule repetitive tasks the prior art methods determine the duration of a macrocycle equal to the lowest common multiple of all the task periods and the duration of a microcycle equal to the highest common denominator of all the task periods, and then look for a permutation of the tasks such that all the constraints are satisfied simultaneously, trying out all possible permutations until one verifying this condition is found, the verification being carried out microcycle by microcycle. If a conflict appears within a microcycle the permutation currently being verified is abandoned and another is tried. The work done in connection with verification of that permutation during previous microcycles becomes of no utility since all the constraints previously satisfied are called into question again.
The prior art methods are therefore somewhat impractical for use in industrial applications.
An object of the invention is to propose a scheduling method that is free of these drawbacks so that a solution to a static scheduling problem is obtained faster, and also to enable dynamic scheduling problems to be handled, i.e. to make it possible to carry out rescheduling as and when the number of tasks to be scheduled and/or the constraints applying to those tasks change. Dynamic scheduling can be beneficial in scheduling machining tasks on a machine tool, for example, if the products to be manufactured are highly diverse; for scheduling aircraft takeoffs and landings on a runway; for scheduling tasks on a data bus or processor; etc.
The invention consists in a method for scheduling successive tasks by means of a computer, said tasks being subject only to timing constraints, a timing constraint requiring that the execution start time be in at least one predetermined time interval relative to an absolute reference time;
said method comprising the following successive steps in this order:
calculating for each task upper and lower limits of the interval in which execution of that task must start;
constructing a first series in which all the tasks are scheduled in increasing order of their lower limit and are scheduled in increasing order of their upper limits when several tasks have a same lower limit;
constructing a second series in which all the tasks are scheduled in increasing order of their upper limit and are scheduled in decreasing order of their lower limits when several tasks have a same upper limit and different lower limits;
constructing an initial permutation by scheduling all the tasks in the order of the first series;
verifying if said current permutation satisfies all the constraints supplying to the tasks, the tasks being considered one by one in the order corresponding to said current permutation to check whether each task satisfies all the constraints applying to said task;
concluding that the scheduling succeeds if all the constraints are satisfied;
otherwise, determining in said current permutation the first illplaced task for which a constraint is not satisfied;
determining in the second series a candidate task immediately following the illplaced task in said second series that also precedes the illplaced task in the current permutation, said candidate being a task which has already been verified, all the tasks following said candidate task in said current permutation being not considered as satisfying all the constraints, any more;
verifying that if the candidate task is shifted to a position immediately after the illplaced task all the constraints applying to all the tasks shifted in this way are then satisfied; and
if at least one constraint is not satisfied, concluding that the candidate task is not suitable and then determining in the second series another candidate ask and repeating the previous verification; and, if this is not possible, concluding that the scheduling fails;
if all the constraints are satisfied, concluding that the scheduling succeeds.
This method has the advantage of being particularly fast since if the scheduling of a task fails it shifts one or more tasks preceding the task in respect of which scheduling has failed without systematically calling into question again all the constraints already satisfied, and therefore without calling into question again all of the work already done. This significantly reduces the computation time compared to any method that systematically explores all scheduling possibilities.
The method is also faster than the prior art methods in the case of repetitive tasks since the permutations are verified in a single microcycle. Finally, this method can be applied to scheduling a system in which a plurality of tasks can be carried out in parallel. It then consists in breaking the system down into a plurality of subsystems in which the tasks must all be executed successively and applying the method of the invention to each of these subsystems.
The method of the invention will be more clearly understood and other features of the invention will emerge from the following description of one embodiment of the invention and the accompanying drawings.
FIG. 1 shows the block schematic of a system in which the dynamic scheduling method of the invention is implemented.
FIGS. 2 and 3 show one step of the method of the invention adapted to take into account the periodic nature of certain tasks.
FIGS. 4 through 7 show the basic principles of the method of the invention.
FIG. 8 shows the flowchart of one embodiment of the invention.
FIGS. 9 through 16 show timing diagrams illustrating the use of the flowchart shown in FIG. 8.
FIGS. 17 through 26 show one example of the use of the method of the invention to schedule tasks.
The method of the invention can be used for static or dynamic scheduling. Its reduction in computation time is particularly advantageous in dynamic scheduling, however, since it allows changing constraints and/or tasks to be dealt with in real time.
FIG. 1 shows the block schematic of one embodiment of the device implementing the method of the invention for dynamic scheduling of industrial production tasks, for example.
The device includes a computer OR connected to a production system SY. The production system SY supplies to an input of the computer OR the parameters needed to determine a schedule: the identifier ID_{K} of each task and, if appropriate, the period T_{K} of each task and the definition R_{L} of each constraint to be satisfied. These parameters are supplied each time that a change occurs in the nature of the task to be executed and/or in the constraints.
The computer then determines a new permutation PER and the production system SY then executes the tasks in accordance with that new permutation.
The tasks are usually executed by various devices of the system SY. The computer OR must be programmed to transmit to each device information indicating to it the times at which it can execute a task. The person skilled in the art will know how to program the computer OR to implement the method of the invention and to transmit information to each device and this programming will not be described beyond the description of the method itself. Likewise, the person skilled in the art will know what hardware to use to connect a computer OR to a system SY comprising various devices adapted to execute respective different tasks.
The method of the invention is not limited to static scheduling applications. The same steps can perform dynamic scheduling in real time.
Each constraint applies only to the execution start time, which muse be within a given interval [t_{min}, t_{max} ]. Execution can continue beyond the limit of an interval, but it muse respect the fixed duration. If a plurality of timing constraints apply to the same task, they can be replaced by a single constraint for which the interval [t_{min}, t_{max} ] is obtained from the respective intervals of chose constraints by application of the AND operator.
A preliminary step of the method of the invention reduces the number of permutations to be verified by verifying the scheduling of tasks only during a microcycle which is the highest common denominator of all the task repetition periods. If a permutation is found such that if all the tasks are executed within the same microcycle and satisfy all the constraints, then that permutation will not lead to any conflict during any of the microcycles constituting a macrocycle, since the worst case scenario is that in which all the tasks occur in the same microcycle because multiples of their periods coincide.
FIG. 2 and 3 shows this preliminary step of the method of the invention. Note that in the case where one or more tasks are not to be regarded a priori as repetitive, it is sufficient to assign them an arbitrary common period, but one that facilitates the determination of a microcycle. It is therefore sufficient to determine the value of the highest common denominator of the period of the repetitive tasks and then to choose a multiple of that value to constitute common periods for all the nonrepetitive tasks.
FIGS. 2 and 3 relate to the example in which six periodic tasks T1 through T6 with respective periods of 10 ms, 20 ms, 30 ms, 40 ms, 50 ms and 40 ms have to be scheduled. The execution time is the same for all the tasks and is equal to 1 ms.
In FIG. 2 the time intervals [t_{min}, t_{max} ] for each task are shown crosshatched.
These intervals are as follows:
______________________________________TA1 [t.sub.min, t.sub.max ] = [0, 4 ms] modulo 10 msTA2 [10, 13 ms] modulo 20 msTA3 [20, 23 ms] modulo 30 msTA4 [30, 32 ms] modulo 40 msTA5 [40, 41 ms] modulo 50 msTA6 [20, 22 ms] modulo 40 ms______________________________________
These tasks have a macrocycle equal to the lowest common multiple of their period, which is 600 ms. A macrocycle can be divided into 60 microcycles each of 10 ms. Two repetitive tasks can be in conflict if they have to be executed during the same microcycle. For example, the tasks TA1, TA3 and TA6 must be executed during the same microcycle [20 ms, 30 ms] then [120 ms, 140 ms], etc.
To determine a schedule that suits the worst case scenario, it is sufficient to consider a microcycle in which all tasks without exception must be executed once. In practise, the tasks do not all occur in the same microcycle. An imaginary case in which this event occurs is considered below.
Time shifting can shift the execution time of each task into the interval of the same microcycle [0, 10 ms]. It is defined by the formula:
t.sub.st.sup.Kimag =t.sub.st.sup.K k.PGCD
where
t_{st}^{Kimag} is the imaginary execution start time of a task K;
t_{st}^{K} is the real execution start time of the task K;
PGCD is the duration of a microcycle, i.e. the highest common denominator of the periods; and
k is an integer such that the imaginary execution start time t_{st}^{Kimag} of the task K is shifted into the required microcycle, in this instance the microcycle [0, 10 ms].
The interval [T_{min}^{K}, T_{max}^{K} ] defined by all the constraints applying to the task K is therefore also shifted by k.PGCD.
The scheduling method of the invention is then applied in this interval [0, 10 ms]. When a permutation satisfying all the constraints has been determined, the actual execution times t_{st}^{K} are deduced from the imaginary execution start times using the formula:
t.sub.st.sup.K =t.sub.st.sup.Kimag +k.PGCD modulo the period of the task K.
FIG. 3 shows the resulting scheduling of the tasks over the interval 0 to 100 ms, the interval 100 ms to 600 ms not being shown but being scheduled in a similar way. Each execution interval is shown in black. Consider, for example, the interval 20 ms to 30 ms in which there can be conflict between the tasks TA1, TA3 and TA6. The task TA1 is executed in the interval 23 ms to 24 ms. The task TA3 is executed in the interval 22 ms to 23 ms. The task TA6 is executed in the interval 21 ms to 22 ms. Thus TA1 and TA3 or TA6 are never executed simultaneously, regardless of the microcycle in question, which is one of the 60 microcycles constituting the macrocycle [0, 600 ms].
in an application in which some tasks can be executed simultaneously, this preliminary step further consists in grouping the tasks into a plurality of independent subsets, if possible, each subset containing only tasks related by constraints. Scheduling a plurality of subsets independently is faster than scheduling a single more complex set.
FIGS. 4 through 7 show the basic principles of the method of the invention for scheduling all the tasks of a set of tasks in which the number of permutations to be verified is minimized. This procedure succeeds if it determines a permutation constituting a schedule satisfying all the constraints applying to the tasks in
FIG. 4 shows a timing diagram illustrating a first basic principle of the method of the invention. This first basic principle imposes priority execution of tasks for which the time interval starts the soonest, i.e. with the smallest value t_{min}. In practise the initial permutation of the tasks, i.e. the first permutation to be verified, is constituted by a series MINSERIES, in which the tasks are arranged in increasing order of their t_{min} values.
The rectangles IDA and IDB in FIG. 4 show time intervals respectively assigned to the execution of a task A and to the execution of a task B. Constraints require that execution of task A start within an interval [t_{min}^{A}, t_{max}^{A} ] or at worst at time t_{max}^{A}. They require execution of task B start within an interval [t_{min}^{B}, t_{max}^{B} ] or, at worst, at time T_{max}^{B}. In this example t_{min}^{A} is before t_{min}^{B}. The first basic principle is to execute task A first, shifting the interval in which this execution occurs as close as possible to the lower limit t_{min}^{A} ; then executing task B during an interval IDB starting as close as possible to the lower limit T_{min}^{B} without overlapping the interval IDA. In this example the interval IDA has a duration δ^{A} which overlaps the interval [T_{min}^{B}, T_{max}^{B} ] and consequently the intervals IDA and IDB will be as close together as possible by choosing:
t.sub.st.sup.B =t.sub.min.sup.A +δ.sup.A
FIG. 5 shows, for the same example, the consequences of failure to apply this basic principle, i.e. of having task B executed before task A. The execution interval IDB then starts as soon as possible, i.e. at the lower limit of the interval [T_{min}^{B}, T_{max}^{B} ]. The execution interval IDA should start after the end of the execution interval IDB in order not to overlap the interval IDB, but in the example the execution interval IDB has a length such that it goes beyond the upper limit t_{max}^{A} of the interval in which starting execution of task A is permitted. At best, task A could start at time t_{st}^{A} =t_{max}^{A}, but then the two execution intervals IDB and IDA overlap. The overlap area is crosshatched in FIG. 5.
The example thus shows that failing to respect the first basic principle increases the probability that some constraints will not be satisfied.
FIG. 6 shows a timing diagram illustrating a second basic principle of the method of the invention. This basic principle chooses the task to be given execution priority from a plurality of tasks for which the lower limit t_{min} has the same value. In this case, this second basic principle imposes priority execution of the task for which the upper limit t_{max} has the smallest value.
In example shown in FIG. 6, two tasks A and B have the same lower limit t_{min}^{A} =t_{min}^{B} for the time interval in which execution must start. Constraints imposed by the existence of other execution intervals that are not shown prohibit execution of tasks A and B before a time t_{0} after t_{min}^{A} =t_{min}^{B} , in a time interval shown crosshatched. Thus the execution intervals IDA and IDB can only start from this time t_{0}. The upper limit T_{max}^{B} for task B is greater than the upper limit t_{max}^{A} for task A. In accordance with the second basic principle, task A takes priority for execution. Task B is then executed from a time:
t.sub.st.sup.B =t.sub.min.sup.A +δ.sup.A
Thus the two execution intervals IDA and IDB occur one after the other with no particular problem due to the upper limit T_{max}^{B}.
FIG. 7 shows, for the same example, a conflict that is more likely to occur if the second basic principle is not respected. Task B is executed first, from time t_{st}^{B} which coincides with the end t_{0} of the prohibited interval shown by the crosshatching. Task A must then be executed from a time t_{st}^{A} =T_{min}^{B} +δ^{A} to prevent the intervals IDA and IDB overlapping, but this time t_{st}^{A} would then be in a prohibited area, shown crosshatched in the figure after the upper limit t_{max}^{A} of the interval in which execution should start. There is therefore a constraint that is not satisfied. This example shows that failing to respect the second basic principle increases the probability that some constraints will not be satisfied.
FIG. 8 shows a flowchart of a preferred embodiment of the method of the invention. It starts with a step E21 to determine the series MINSERIES comprising all the tasks scheduled in increasing order of their lower limit t_{min}. This series will be used to apply the first basic principle stated above. To maximize the probability of finding quickly a permutation that satisfies all the constraints applying to all the tasks, the initial permutation is constituted by MINSERIES and the permutations verified thereafter, in the event of failure, are deduced from MINSERIES by successive modifications.
Step E21 also determines the series MAXSERIES comprising all the tasks in increasing order of their upper limit t_{max}. This series will be used to apply the second basic principle stated above if the initial permutation has to be modified.
The next step E22 verifies the current permutation, i.e. verifies if it satisfies all the constraints applying to all the tasks. At the beginning of this step the current permutation is the initial permutation determined by step E21.
This verification consists in taking each successive task in the order of the current permutation and verifying that the execution interval [t_{st}, t_{end} ] imposed by the position occupied by that task in the current permutation is compatible with the time interval [t_{min}, t_{max} ] imposed by the constraints that apply to that task. The task for which this verification is in progress is called the current task. If the result of the verification is positive for each of the tasks, this indicates that the current permutation succeeds (S).
If the verification finds at least one task whose position in the permutation imposes starting execution at a time t_{st} that is not in the time interval [t_{min}, t_{max} ] imposed by the constraints, that task is regarded as illplaced since there is a conflict between two constraints. When step E22 finds a task (X) that is the first illplaced task in the current permutation, it draws the conclusion (R) that it is necessary to look for a task that is a candidate for shifting to constitute a new permutation. The method then executes a step E23 which looks for the task immediately following the current task X in the series MAXSERIES. If there is no such task, step E23 fails (F).
If step E23 finds a candidate task Q immediately following the current task X in the series MAXSERIES, the next step E24 of the method verifies that the task Q has already been considered to be wellplaced during an earlier step E21. All the tasks considered to be wellplaced are those which have a position below that of the current task X, since the verification of step E22 is carried out according to increasing positions in the current permutation. Consequently, step E24 simply verifies that the candidate task Q precedes the current task X in the current permutation. If step E24 determines that the task Q was not considered to be wellplaced, then step E23 is repeated to find another candidate task immediately following the task Q in MAXSERIES.
On the other hand, if the candidate task Q was considered to be wellplaced, the next step E25 of the method compares the execution start time t_{st}^{Q} of the task Q to the upper limit t_{max}^{X} of the time interval corresponding to the current task X.
If t_{st}^{Q} ≧t_{max}^{X}, the outcome is that denoted α. The next step is then a step E29 which shifts the task Q in the current permutation to insert it between the current task X and the task after the task X in the current permutation. The tasks that were between Q and X, and the task X itself, are shifted one position downwards to fill in the space left free by Q. Consequently, the task Q subsequently occupies the position that was previously that of X. The method then repeats step E22 to verify if the new current permutation obtained in this way satisfies all the constraints. Note that the task Q and all the other tasks that followed it have been shifted. They can no longer be considered to have been wellplaced as they do not necessarily satisfy all the constraints in their respective new positions. The tasks below Q retain the same position, and it is therefore not necessary to verify that they satisfy all the constraints applying to them, and therefore the work of verifying those constraints is not done again.
if t_{st}^{Q} <t_{max}^{X}, the outcome is that denoted α (see below) and the next step of the method is step E26.
FIG. 9 shows the outcome α, for example. The execution intervals are shown in dashed outline if a conflict prevents execution and in full outline otherwise.
Consider a task X in position PSi of the current permutation and a task Q considered to be wellplaced and in position PSj of the current permutation. The position PSi imposes the start of execution of task X at a time t_{st} (PSi) that is beyond the limit t_{max}^{X}. The position PSj imposes the start of execution of task Q at a time t_{st} (PSi) which in this example is beyond the time t_{max}^{X}. Since the time t_{st}^{Q} (PSi) is beyond the time t_{max}^{X}, the limit t_{max}^{Q} is therefore a fortiori greater than or equal to t_{max}^{X} and therefore has a chance of being near t_{st} (PSi), the ideal being that t_{st} (PSi) falls short of t_{max}^{Q}.
FIG. 10 shows that by shifting the tasks Q and X respectively into the positions PSi1 and PSi, the probability that the time t_{st} (PSi1) falls in the interval [T_{min}^{X}, t_{max}^{X} ] is greater than the probability that the time t_{st} (PSi) had of falling in this same interval, because t_{st} increases systematically with the position PS. The figure shows that the end t_{max}^{X} of the segment corresponding to task X has moved closer to an execution interval, namely that which starts at t_{st} (PSi1), and therefore has a greater chance of intersecting with that interval.
The end t_{max}^{Q} of the segment corresponding to the task Q has a nonnull chance of intercepting the rectangle representing the execution interval starting at time t_{st} (PSi). Consequently, the new permutation has a greater chance than the old one of satisfying all the constraints and it is therefore beneficial to try out this new permutation. In the example shown, t_{st} (PSi1) is beyond t_{max} and there is therefore still a conflict in respect of the task X. It is therefore necessary to make one or more further modifications to the current permutation.
In this example, t_{st} (PSi) is beyond t_{max}^{Q} and there is therefore also a conflict in respect of the task after the shift. However, in other examples corresponding to the outcome α it may be that the constraints applying to X and those applying to Q are satisfied simultaneously. Consequently, it is beneficial to verify any new permutation corresponding to the out come α.
In the case of the outcome α, on the other hand, further verifications are needed before it is possible to conclude that the task Q is a worthwhile candidate task. The next step of the method is then step E26 that compares the time t_{end}^{Q} of end of execution of the task Q to the upper limit t_{max}^{X} of the interval in which execution of the task X must start. If t_{end}^{Q} ≧t_{max}^{X}, the outcome is that denoted β and the next step of the method is the step E30 which shifts the task Q to insert it after the task X. The contrary outcome is that denoted β. Other verifications are needed before it is possible to conclude that the task Q is a worthwhile candidate task.
FIGS. 11 and 12 show the outcome β. FIG. 11 shows a task X in the position PSi and a task Q in the position PSj such that the outcome α and the outcome β apply: the time t_{end}^{Q} of end of execution of the task Q, i.e. the time t_{end} (PSi) imposed by the position PSj of Q in the current permutation, is beyond the time t_{max}^{X}.
FIG. 12 shows that by displacing the tasks Q and X to the positions PSi1 and PSi, respectively, the probability that the time t_{st} (PSi1) falls in the interval [T_{min}^{X}, t_{max}^{X} ] is greater than the probability that the time t_{st} (PSi) had of falling in that same interval, because t_{st} increases systematically with the position PS. What is more, the time t_{end}^{Q} =t_{end} (PSi) is beyond the limit t_{max}^{X} and there is no conflict. This means that t_{max}^{Q} is either greater than t_{max}^{X} or slightly less than t_{max}^{X}, being at most at a distance equal to the interval [t_{st} (PSj), t_{end} (PSj)]. There is therefore little chance that t_{max}^{Q} is greater than t_{max}^{X} and therefore in this case there is little chance that t_{max}^{Q} will be beyond t_{st} (PSi) after the shift. It is therefore worthwhile to try out a permutation corresponding to the outcome β, which will not necessarily be a waste of time.
In the example shown in FIG. 12, the shifting of Q and X is not sufficient for all the constraints on Q and X to be satisfied. The current permutation needs to be modified a little more.
In the case of the outcome β, step E27 compares the upper limit t_{max}^{Q} of the time interval in which execution of task Q must start to the upper limit t_{max}^{X} of the time interval in which execution of task X must start. If t_{max}^{Q} >t_{max}^{X}, the outcome is denoted η and the next step of the method is then a step E28. The contrary outcome is that denoted η and step E23 is then repeated since there is no chance that the candidate task Q is worthwhile.
FIGS. 13 and 14 show the outcome η by means of an example in which the limit t_{max}^{Q} is before the time t_{max}^{X}.
FIG. 13 shows the tasks X and Q shifted into the positions PSi1 and PSi, respectively. This figure shows that the time t_{max}^{Q} is closer to the time t_{st} (PSi) than it was to the time t_{st} (PSj) but there is no chance of the task Q being executed since t_{max}^{Q} is still farther from t_{st} (PSi) than t_{max}^{Q} was. Since there is a 100% chance of the task Q being illplaced, it is not worthwhile to attempt any modification of the current permutation and for this reason the next step of the method is a step E23 that looks for another candidate task for shifting.
In the case of outcome η, step E28 compares the lower limit t_{min}^{Q} of the interval in which execution of task Q must start with the lower limit T_{min}^{X} of the time interval in which the execution of task X must start.
The objective of this step E28 is to verify that the task Q is before the current task X in the series MINSERIES so that the task Q can be moved after the task X. Otherwise (outcome α) the next step of the method is a step E23 that looks for another task that is a candidate for shifting.
In the case of the outcome χ, the next step of the method is a step E29 that shifts the task Q after the task X; step E22 is then repeated to verify if all the constraints applying to the tasks of the current layer are satisfied.
FIG. 15 shows an example in which the outcomes α, β, χ, η apply simultaneously.
FIG. 16 shows the tasks X and Q shifted into the positions PSi1 and PSi, respectively, in an example in which the constraints are such that they are satisfied after this shifting: t_{st} (PSi) is in the interval [t_{min}^{Q}, t_{max}^{Q} ] and t_{st} (PSi1) is in the interval [T_{min}^{X}, t_{max}^{X} ].
FIGS. 17 through 26 show the implementation of step E2 for scheduling a layer comprising thirteen tasks: A, B, C, D, E, G, J, K, L, N, P, S, T.
FIG. 17 shows on a time scale from 0 to 14 ms the position of the execution intervals respectively corresponding to thirteen positions PS1 through PS13. The tasks are executed in the order of the positions PS1 through PS13 and each execution interval has a duration equal to 1 ms.
Each task must satisfy one or more constraints that result in a single constraint. The start of the execution interval (crosshatched rectangle in FIG. 17) must be in a given time interval (bold segments in FIG. 18). In the worst case it can begin at the upper limit.
Step E21 determines the series MINSERIES comprising all the tasks of the current layer scheduled in increasing order of their lower limit t_{min}.
MINSERIES=N, J, S, D, E, B, T, A, K, P, L, C, G.
In FIG. 18 the thirteen tasks are shown in the same order as MINSERIES, along the ordinate axis. To maximize the probability of quickly finding a permutation that satisfies all the constraints applying to the tasks of the layer in question, the socalled initial permutation verified first is that consisting of MINSERIES, and those verified thereafter, in the event of failure, are deduced from MINSERIES by successive modifications.
Step E21 further determines the series MAXSERIES comprising all the paths of the current layer scheduled in order of increasing t_{max} values. This series will be used to apply the second basic principle stated above if the initial permutation has to be modified. In this example:
MAXSERIES=E, D, A, K, P, B, L, C, G, T, J, S, N
A step E22 then verifies the current permutation, i.e. verifies if it satisfies all the constraints applying to the tasks of the layer in question. At the start of step E2 the current permutation is the initial permutation determined by step E21.
The verification is performed successively for each task, in the order of the current permutation: N, J, S, D, E . . ., G. If the result of the verification is positive for a task, that task is considered to be wellplaced in the permutation, but its position can be called into question again subsequently if this is needed to satisfy other constraints.
Comparing FIGS. 17 and 18 shows that there is no problem in executing the tasks N, J, S and D, respectively, during the execution intervals shown in FIG. 17. They are therefore all considered to be wellplaced, the current permutation is:
N, J, S, D, E, B, T, A, K, P, L, C, G.
The tasks underlined are those considered to be wellplaced.
FIG. 19 shows the first conflict encountered in verifying the initial permutation. Crosshatched rectangles in FIG. 19 represent execution intervals for which there is no conflict between the constraint and a dashed outline black rectangle shows the execution interval that is causing a conflict. It corresponds to the position PS5 currently occupied by the task E. This execution interval does not intersect the interval in which the execution of the task E must start. Step E23 therefore looks in the series MAXSERIES for a task following the task E, i.e. such that the limit t_{max} has a higher value.
MAXSERIES=E, D, A, K, P, B, L, C, G, T, J, S, N
Step E23 finds the task D. Step E24 verifies that it is considered to be wellplaced, by verifying that its position is below the position of the current task E in the current permutation. Step E25 then arrives at a conclusion α. Step E26 then arrives at a conclusion β. Step E29 then shifts D into the position POS5 and E moves back into the position POS4. Step E22 verifies that the constraints applying to E and the subsequent tasks are satisfied but finds that the constraint applying to D are no longer satisfied. The permutation tried out is not suitable. It is not retained at the new current permutation.
FIG. 19 shows that this modification cannot succeed because the interval corresponding to D has an upper limit which is not higher than that of the interval corresponding to E. They are in fact exactly the same. To prevent unnecessary verification of new permutations it is possible to take precautions when constructing MAXSERIES during step E21:
If two tasks of the same layer have a common time interval constraint [t_{min}, t_{max} ], it is possible to put them in two different positions in the series MINSERIES that constitutes the initial permutation and in the series MAXSERIES. In the example shown in FIGS. 17 through 26 there are at least two possibilities for constructing the series MAXSERIES since it is possible to permutate D and E:
D, E, A, K, P, B, L, C, G, T, J, S, N
E, D, A, K, P, B, L, C, G, T, J, S, N
It would have been preferable to choose the order E, D that is the opposite of the order D, E in which they are taken to constitute the initial permutation.
MINSERIES=N, J, S, D, E, B, T, A, K, P, L, C, G.
Step E23 then finds as successive candidate tasks the tasks A, K, P, B, L, C, G, T, J and step E24 retains task J.
It can be shown that, as a general rule, if the masks of the layer to be scheduled have to satisfy only timing constraints, it is preferable to place in a reverse order all the tasks having the same interval [t_{min}, t_{max} ] regardless of their number. On the other hand, if both timing constraints and sequence constraints apply, it is preferable to place them in the same order.
Consider the remainder of this example, retaining the initial choice of MAXSERIES:
E, D, A, K, P, B, L, C, G, T, J, S, N
The current permutation is:
N, J, S, E, D, B, T, A, K, P, L, C, G.
After this unfortunate attempt to modify the permutation, step E23 finds as successive candidate tasks the tasks A, K, F, B, L, C, G, T and these are rejected by step E24 Step E23 then finds task J.
FIG. 20 shows this new attempt at modification. Step E24 verifies that task J is considered to be wellplaced. Step E25 reaches the conclusion α. Step E26 reaches the conclusion β. Step E27 and E28 arrive at the conclusions η and χ. Consequently step E29 places task J in position PS5 in place of E. Task E, D and S move back one place: S to position PS2, D to position PS3 and E to position PS4. The other tasks do not change place.
Step E22 then verifies that the tasks that have been shifted satisfy all the constraints applying to them, starting with the shifted task in the lowest position: S, then E, then D, then J. It then verifies that there is no conflict between the successive constraints for tasks B, T, A, K.
FIG. 21 shows the new current permutation. Note that the execution interval for task K starts at the exact time which is the end of the interval in which execution of task K should start. There is no conflict, but the constraints are satisfied. The current permutation is:
N, S, E, D, J, B, T, A, K, P, L, C, G.
FIG. 22 shows that a conflict occurs for task P. Step E23 is then repeated to determine in the series MAXSERIES the task immediately following the task P. It finds task B. Step E24 verifies that task B is considered to be wellplaced. Steps E25 and E26 reach conclusions α and then β. Step E27 is then executed. This finds that T_{max}^{B} =t_{max}^{P} and consequently arrives at the conclusion η. Step E23 is then repeated to find another candidate task in the series MAXSERIES. The segment representing the time interval corresponding to B would not intersect with the execution interval [9 ms, 10 ms] corresponding to the position POS10 if task B were shifted to that position. Consequently, the constraints that apply to B would not be satisfied.
Step E23 then finds the successive tasks L, C, G but step E24 rejects them as they are not considered to be wellplaced in the permutation. Finally, steps E23 and E24 find task T. Steps E25 through E28 successively arrive at the conclusions α, β, η, χ. Step E29 shifts T to position POS10. Tasks P, K, A move back into positions POS9, POS8, POS7, respectively.
Step E22 then verifies that the constraints applying to the shifted tasks A, K, P, T and the subsequent tasks are satisfied. The new current permutation is:
N, S, E, D, J, B, A, K, P, T, L, C, G.
As shown in FIG. 23, step E22 then finds that the constraints applying to task L are not satisfied.
Step E23 then finds task T and step E24 verifies that it is considered to be wellplaced. Step E25 and the subsequent steps can then be executed. They arrive at conclusions α, β then η and χX. Step E29 can then be executed. It places task T in the position POS11 that was occupied by task L, which moves back one place. Step E22 verifies that the constraints applying to the shifted tasks L and T are satisfied. Consequently, the current permutation becomes:
N, S, E, D, J, V, A, K, P, L, T, C, G.
As shown in FIG. 24, there is a conflict between the constraints applying to task C. Step E23 determines the task following task C in the series MAXSERIES. It finds task G but step E24 finds that task G is not considered to be wellplaced in the permutation. Step E23 is repeated and determines another task T preceding task C in the series MAXSERIES. Step E24 verifies that task T is considered to be well placed in the permutation. Step E25 arrives that the conclusion α since t_{st}^{T} =t_{max}^{C}. Step 29 is therefore executed immediately and places task T after task C in the permutation, which is the same thing as permutating the positions of C and T. Step E22 then verifies that all the constraints applying to the shifted tasks C and T are satisfied.
The new current permutation is:
N, S, E, D, J, B, A, K, P, L, C, T, G.
As shown in FIG. 25, step E22 finds that there is a conflict between the constraints applying to task G. Step E23 determines a task T immediately following task G in the series MAXSERIES. Step E24 verifies that task T is considered to be wellplaced in the permutation. Step E25 arrives at the conclusion α since t_{st}^{T} =t_{max}^{G}. Step E29 places T after G in the permutation, which is the same thing as permutating T and G. Step E22 then verifies that all the constraints applying to the shifted tasks G and T are satisfied. The new current permutation is:
N, S, E, D, J, B, A, K, P, L, C, G, T.
All the tasks are considered to be wellplaced because all the constraints are satisfied, as shown in FIG. 26, and consequently step E22 succeeds (S).