UML’03 Stochastic Evaluation of the 1st Axxom Case Study Holger Hermanns Yaroslav S. Usenko...

The 1st AXXOM Case Study revisited

Dr.-Ing Holger Hermanns, Universiteit Twente

Demonstration Model of a Laquer Production

Quality Check

Pre-dispersion/Dispersion

Dose Spinner

Axxom Software AG

Pipeless Plant

Mixing vessels move between stationsPlant topology (paths, collisions) is not described and not modelled here.Multiple equal resource instances

Sebastian Panek

Problem Statement


Axxom case study -questions

Angelika MaderUniversity of Twente

Question 3: AVAILABILITY

problem: machines break down for a certain amount of time (say 50%)

solution: extend the production time of a product requiring this machine (say factor 2)

question: is this the right way to deal with this kind of probabilities?

Performance & Availability FactorsPerformance factor: reflects unpredictable perturbations of the production process.

A performance factor of 0.8 extends the occupation times used for planning by a factor of 1/0.8, i.e., 1.25.


Our ViewBoth the performance and the availability factors relate to unplanned or unplannable pertubations of the production process.

They reflect random influences with partially known characteristics.

This holds in particular for the performance factor, and to a lesser extent for the availability factor.


Behaviour of a single machine0 6 14 20 28 34 42 48 56Scheduled behaviour:


Our ApproachDevelop a model reflecting the stochastic perturbations.

Use this model to study a-priori computed valid schedules.

Quantify the risk to violate the schedule, andto miss deadlines.

Note: different valid schedules may differ w.r.t. these risks.

This provides means to rank valid schedules.

We exercise this approach using Modest.


MoDeST (1)Semantics of MoDeST: STA, stochastic timed automata, a model made up fromtimed automata with deadlines [Alur/Dill, Bornot/Sifakis]stochastic automata [DArgenio/Brinksma/Katoen]probabilistic automata [Segala/Lynch]u(y240)get_prodno_pricecashset_price tauy:=0w(y120)A specification formalism for Modelling and Description of Stochastic Timed sytems


MoDeST (2): syntaxu(y240)get_prodno_pricecashset_price tauy:=0w(y120)xd:=U[10,20]x:=0u(xxd)w(xxd)


A schedule, produced by A. Mader29 jobs, grouped into 3 job types,each job type is composed of multiple partially concurrent tasks, running on 11 different machines.each job has a deadline of 336 hrs (2 weeks)


Machines in Modestprocess TP2_machine()// Disperser{clock w,y;float br,r,work,deadline; do{::when (TP2_work>0) act_work_TP2 {=work=TP2_work, TP2_work=0, deadline=TP2_deadline,w=0, //current working timebr=Rand(...) //next time to break=};do{:: when (cjobs>=deadline) act_error_TP2 {= TP2_done=2 =}; break:: when (br>=work && w>=work) act_done_TP2 {= TP2_done=1 =}; break:: when (br=br) act_break_TP2 {= work-=br, r=Rand(...), y=0 =};when(y==r) act_repaired_TP2 {= w=0, br=Rand(...) =}}}}


Jobs (type 2)

process Job_type2(int number, float starttime, float earliesttime, float deadline){int mv=0,ds=0;// which mixing vessel and dose spinner will we get?clock c;

when(cjobs==starttime) tau {= ii+=1 =}; // starting time according to the schedule

// disperser for 27when(TP2_lock==0) tau {= TP2_lock=1, TP2_deadline=deadline-49-26-2, TP2_work=27 =};when(TP2_done>0) tau {= TP2_done=0, TP2_lock=0 =};

// Lock an UNI mixing vesselalt{:: when(MVU1_lock==0) tau {= mv=1, MVU1_lock=1 =}:: when(MVU2_lock==0) tau {= mv=2, MVU2_lock=1 =}};// Two parallel activities:par{... };...// are we on time?alt{ :: when(cjobsdeadline) tau }}


Schedule violations vs. deadline missesschedule violation risk:It seems wise to


The systempar{ :: ABF1_machine() :: ABF2_machine() :: TP2_machine() :: DOK1_machine() :: DOK2_machine() :: DVT1_machine( :: BR1_machine() :: HDL1_machine() :: MVU1_machine() :: MVU2_machine() :: MVM1_machine() :: MVM2_machine() :: MVM3_machine()

:: do {:: tau {= i+=1, d=0, cjobs=0 =}; par{:: Job_type1(17, js17, 101, 101+336):: Job_type2(15, js15, 52, 52+336):: Job_type2( 5, js5, 191, 191+336):: Job_type2(14, js14, 274, 274+336):: Job_type2(18, js18, 278, 278+336):: Job_type2( 4, js4, 388, 388+336):: Job_type3(28, js28, 276, 276+336) }; INC_d(d) }}


Intermezzo: Stochastic PerturbationsIt is natural to interpret the availability/performance factor as the ratio of time the system is available/performing.

In the dependability context this ratio arises as:

So a factor of, say, 0.8 relates MTBF and MTTR:

If MTBF and MTTR are given, the best probabilistic approx- imation is obtained with negative exponential distributions, para- metrized with these mean durations.

Unfortunately, the means are not given, only their ratio.Mean time between failuresMean time to repair


MOTORMoDeSTcompilerDiscreteeventsimulator


What we considered


What we consideredCaution: Unfair comparisonA six-hour disperserjob scheduled for 14 hoursbut not in these schedules


Some comparative resultsJob success probability


Chart1

0000

0000

0.00662970050.00557162030.00627659840.0036389293

0.0439921680.03921093180.04304496630.0635314592

0.31762079980.29565449620.28358976540.371411162

0.42880835040.42813748180.44115243220.393709487

0.18901516220.21460024540.21056484330.1570413135

0.01328770630.01618015850.01472583290.0100217035

Schedule 1

Schedule 2

Schedule 3

Schedule 4

Successful jobs

Sheet1

1234

d00.0000%0.0000%0.0000%0.0000%

d10.0302%0.0025%0.0395%0.0133%

d22.8997%2.7797%3.0487%2.9398%

d313.0893%12.4745%12.3431%15.5660%

d430.3550%27.9802%28.1208%32.8763%

d535.0828%37.5270%37.4022%32.5296%

d616.6978%16.8824%17.2630%14.6327%

d71.7804%2.2890%1.7179%1.3775%

dd64.9650%65.7182%65.4910%63.4790%

job426.5086%26.8350%24.9791%26.5560%

job579.1144%81.0578%80.9478%73.6024%

job1448.5887%50.4772%53.0869%44.6654%

job1586.2647%86.8094%85.7221%87.8626%

job1799.6965%99.7122%99.6615%99.1208%

job1814.9257%15.5889%14.4609%13.2889%

job2899.6064%99.4976%99.5288%99.2047%

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Sheet1

0000

0000

0000

0000

0000

0000

0000

0000

0000

1

2

3

4

Done Jobs

0000

0000

0000

0000

0000

0000

0000

Individual Jobs

Some comparative resultsJob success probabilitypace


Chart1

0.68409042280.69353513630.69287515340.6666037965

0.64964985820.6571823970.65490976380.6347895285

0.60073805930.60547686170.60202728750.5903952891

0.56650523780.57178718130.5761679550.5554239009

0.61611678610.61304466550.60990051510.6017072629

Schedule 1

Schedule 2

Schedule 3

Schedule 4

Sheet1

1.00000.31600.10000.03160.0100

Schedule 10.68410.64960.60070.56650.6161

Schedule 20.69350.65720.60550.57180.6130

Schedule 30.69290.65490.60200.57620.6099

Schedule 40.66660.63480.59040.55540.6017

7.0000

Sheet1

0.68409042280.69353513630.69287515340.6666037965

0.64964985820.6571823970.65490976380.6347895285

0.60073805930.60547686170.60202728750.5903952891

0.56650523780.57178718130.5761679550.5554239009

0.61611678610.61304466550.60990051510.6017072629

Schedule 1

Schedule 2

Schedule 3

Schedule 4

Sheet2

0

0

0

0

Sheet3

A detailled view into schedule 1pace


Chart1

0.25471850050.93347036930.52454513390.96858057280.99975669270.10746758160.9995877438

0.26508623680.79114434120.48588737070.86264711220.9969645160.14925707730.9960642612

0.29148977340.65739718720.44415151660.73859907640.95071063380.18462143570.9377439078

0.34700791150.59708636260.41513221570.64620498470.8672250510.23742142840.855029481

0.42372276070.61541409270.50876778060.69789645790.85400489330.37135568130.8413089223

job4

job5

job14

job15

job17

job18

job28

Individual job success probability

1

1.00000.31600.10000.03160.0100

hit68.4090%64.9650%60.0738%56.6505%61.6117%

no jobs0.0000%0.0000%0.0429%0.2895%0.5607%

1 jobs0.0000%0.0302%0.7752%2.6526%3.3086%

2 jobs0.6630%2.8997%7.7600%12.6971%12.9881%

3 jobs4.3992%13.0893%20.5165%22.8063%16.9134%

4 jobs31.7621%30.3550%30.1174%26.3516%18.7088%

5 jobs42.8808%35.0828%25.2051%20.1249%18.7639%

6 jobs18.9015%16.6978%12.6060%11.1813%18.3297%

7 jobs1.3288%1.7804%2.9119%3.8280%10.3398%

job425.4719%26.5086%29.1490%34.7008%42.3723%

job593.3470%79.1144%65.7397%59.7086%61.5414%

job1452.4545%48.5887%44.4152%41.5132%50.8768%

job1596.8581%86.2647%73.8599%64.6205%69.7896%

job1799.9757%99.6965%95.0711%86.7225%85.4005%

job1810.7468%14.9257%18.4621%23.7421%37.1356%

job2899.9588%99.6064%93.7744%85.5029%84.1309%

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1

000000000

000000000

000000000

000000000

000000000

hit

no jobs

1 jobs

2 jobs

3 jobs

4 jobs

5 jobs

6 jobs

7 jobs

Done jobs

0000000

0000000

0000000

0000000

0000000

job4

job5

job14

job15

job17

job18

job28

Individual jobs

Compare:


ConclusionStochastic machine model for 1st Axxom case. Allows ranking of schedules.

Provides deeper insight into schedule particularities.

Depends on more than just the perf./avail. factors.

Motor is an early prototype. First (semi-)serious application.


PerspectiveWell, at least Motor has quite some potential for a similar improvement in the next decade.9Kim G. Larsenb UPPAAL 1995 -20001994199519961997199819992kDec96Sep98Every 9 month10 times better performance!


UML’03 Stochastic Evaluation of the 1st Axxom Case Study Holger Hermanns Yaroslav S. Usenko...

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