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Added some csma examples of different sizes. 

Former-commit-id: e77375c9e5
tempestpy_adaptions
dehnert 11 years ago
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  1. 129
      examples/mdp/csma/csma2_6.nm
  2. 130
      examples/mdp/csma/csma3_2.nm
  3. 132
      examples/mdp/csma/csma3_4.nm
  4. 134
      examples/mdp/csma/csma3_6.nm
  5. 135
      examples/mdp/csma/csma4_2.nm
  6. 137
      examples/mdp/csma/csma4_4.nm
  7. 139
      examples/mdp/csma/csma4_6.nm

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// CSMA/CD protocol - probabilistic version of kronos model (3 stations)
// gxn/dxp 04/12/01
mdp
// note made changes since cannot have strict inequalities
// in digital clocks approach and suppose a station only sends one message
// simplified parameters scaled
const int sigma=1; // time for messages to propagate along the bus
const int lambda=30; // time to send a message
// actual parameters
const int N = 2; // number of processes
const int K = 6; // exponential backoff limit
const int slot = 2*sigma; // length of slot
const int M = 63; // max number of slots to wait
//const int lambda=782;
//const int sigma=26;
//----------------------------------------------------------------------------------------------------------------------------
// the bus
module bus
b : [0..2];
// b=0 - idle
// b=1 - active
// b=2 - collision
// clocks of bus
y1 : [0..sigma+1]; // time since first send (used find time until channel sensed busy)
y2 : [0..sigma+1]; // time since second send (used to find time until collision detected)
// a sender sends (ok - no other message being sent)
[send1] (b=0) -> (b'=1);
[send2] (b=0) -> (b'=1);
// a sender sends (bus busy - collision)
[send1] (b=1|b=2) & (y1<sigma) -> (b'=2);
[send2] (b=1|b=2) & (y1<sigma) -> (b'=2);
// finish sending
[end1] (b=1) -> (b'=0) & (y1'=0);
[end2] (b=1) -> (b'=0) & (y1'=0);
// bus busy
[busy1] (b=1|b=2) & (y1>=sigma) -> (b'=b);
[busy2] (b=1|b=2) & (y1>=sigma) -> (b'=b);
// collision detected
[cd] (b=2) & (y2<=sigma) -> (b'=0) & (y1'=0) & (y2'=0);
// time passage
[time] (b=0) -> (y1'=0); // value of y1/y2 does not matter in state 0
[time] (b=1) -> (y1'=min(y1+1,sigma+1)); // no invariant in state 1
[time] (b=2) & (y2<sigma) -> (y1'=min(y1+1,sigma+1)) & (y2'=min(y2+1,sigma+1)); // invariant in state 2 (time until collision detected)
endmodule
//----------------------------------------------------------------------------------------------------------------------------
// model of first sender
module station1
// LOCAL STATE
s1 : [0..5];
// s1=0 - initial state
// s1=1 - transmit
// s1=2 - collision (set backoff)
// s1=3 - wait (bus busy)
// s1=4 - successfully sent
// LOCAL CLOCK
x1 : [0..max(lambda,slot)];
// BACKOFF COUNTER (number of slots to wait)
bc1 : [0..M];
// COLLISION COUNTER
cd1 : [0..K];
// start sending
[send1] (s1=0) -> (s1'=1) & (x1'=0); // start sending
[busy1] (s1=0) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // detects channel is busy so go into backoff
// transmitting
[time] (s1=1) & (x1<lambda) -> (x1'=min(x1+1,lambda)); // let time pass
[end1] (s1=1) & (x1=lambda) -> (s1'=4) & (x1'=0); // finished
[cd] (s1=1) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // collision detected (increment backoff counter)
[cd] !(s1=1) -> (s1'=s1); // add loop for collision detection when not important
// set backoff (no time can pass in this state)
// probability depends on which transmission this is (cd1)
[] s1=2 & cd1=1 -> 1/2 : (s1'=3) & (bc1'=0) + 1/2 : (s1'=3) & (bc1'=1) ;
[] s1=2 & cd1=2 -> 1/4 : (s1'=3) & (bc1'=0) + 1/4 : (s1'=3) & (bc1'=1) + 1/4 : (s1'=3) & (bc1'=2) + 1/4 : (s1'=3) & (bc1'=3) ;
[] s1=2 & cd1=3 -> 1/8 : (s1'=3) & (bc1'=0) + 1/8 : (s1'=3) & (bc1'=1) + 1/8 : (s1'=3) & (bc1'=2) + 1/8 : (s1'=3) & (bc1'=3) + 1/8 : (s1'=3) & (bc1'=4) + 1/8 : (s1'=3) & (bc1'=5) + 1/8 : (s1'=3) & (bc1'=6) + 1/8 : (s1'=3) & (bc1'=7) ;
[] s1=2 & cd1=4 -> 1/16 : (s1'=3) & (bc1'=0) + 1/16 : (s1'=3) & (bc1'=1) + 1/16 : (s1'=3) & (bc1'=2) + 1/16 : (s1'=3) & (bc1'=3) + 1/16 : (s1'=3) & (bc1'=4) + 1/16 : (s1'=3) & (bc1'=5) + 1/16 : (s1'=3) & (bc1'=6) + 1/16 : (s1'=3) & (bc1'=7) + 1/16 : (s1'=3) & (bc1'=8) + 1/16 : (s1'=3) & (bc1'=9) + 1/16 : (s1'=3) & (bc1'=10) + 1/16 : (s1'=3) & (bc1'=11) + 1/16 : (s1'=3) & (bc1'=12) + 1/16 : (s1'=3) & (bc1'=13) + 1/16 : (s1'=3) & (bc1'=14) + 1/16 : (s1'=3) & (bc1'=15) ;
[] s1=2 & cd1=5 -> 1/32 : (s1'=3) & (bc1'=0) + 1/32 : (s1'=3) & (bc1'=1) + 1/32 : (s1'=3) & (bc1'=2) + 1/32 : (s1'=3) & (bc1'=3) + 1/32 : (s1'=3) & (bc1'=4) + 1/32 : (s1'=3) & (bc1'=5) + 1/32 : (s1'=3) & (bc1'=6) + 1/32 : (s1'=3) & (bc1'=7) + 1/32 : (s1'=3) & (bc1'=8) + 1/32 : (s1'=3) & (bc1'=9) + 1/32 : (s1'=3) & (bc1'=10) + 1/32 : (s1'=3) & (bc1'=11) + 1/32 : (s1'=3) & (bc1'=12) + 1/32 : (s1'=3) & (bc1'=13) + 1/32 : (s1'=3) & (bc1'=14) + 1/32 : (s1'=3) & (bc1'=15) + 1/32 : (s1'=3) & (bc1'=16) + 1/32 : (s1'=3) & (bc1'=17) + 1/32 : (s1'=3) & (bc1'=18) + 1/32 : (s1'=3) & (bc1'=19) + 1/32 : (s1'=3) & (bc1'=20) + 1/32 : (s1'=3) & (bc1'=21) + 1/32 : (s1'=3) & (bc1'=22) + 1/32 : (s1'=3) & (bc1'=23) + 1/32 : (s1'=3) & (bc1'=24) + 1/32 : (s1'=3) & (bc1'=25) + 1/32 : (s1'=3) & (bc1'=26) + 1/32 : (s1'=3) & (bc1'=27) + 1/32 : (s1'=3) & (bc1'=28) + 1/32 : (s1'=3) & (bc1'=29) + 1/32 : (s1'=3) & (bc1'=30) + 1/32 : (s1'=3) & (bc1'=31) ;
[] s1=2 & cd1=6 -> 1/64 : (s1'=3) & (bc1'=0) + 1/64 : (s1'=3) & (bc1'=1) + 1/64 : (s1'=3) & (bc1'=2) + 1/64 : (s1'=3) & (bc1'=3) + 1/64 : (s1'=3) & (bc1'=4) + 1/64 : (s1'=3) & (bc1'=5) + 1/64 : (s1'=3) & (bc1'=6) + 1/64 : (s1'=3) & (bc1'=7) + 1/64 : (s1'=3) & (bc1'=8) + 1/64 : (s1'=3) & (bc1'=9) + 1/64 : (s1'=3) & (bc1'=10) + 1/64 : (s1'=3) & (bc1'=11) + 1/64 : (s1'=3) & (bc1'=12) + 1/64 : (s1'=3) & (bc1'=13) + 1/64 : (s1'=3) & (bc1'=14) + 1/64 : (s1'=3) & (bc1'=15) + 1/64 : (s1'=3) & (bc1'=16) + 1/64 : (s1'=3) & (bc1'=17) + 1/64 : (s1'=3) & (bc1'=18) + 1/64 : (s1'=3) & (bc1'=19) + 1/64 : (s1'=3) & (bc1'=20) + 1/64 : (s1'=3) & (bc1'=21) + 1/64 : (s1'=3) & (bc1'=22) + 1/64 : (s1'=3) & (bc1'=23) + 1/64 : (s1'=3) & (bc1'=24) + 1/64 : (s1'=3) & (bc1'=25) + 1/64 : (s1'=3) & (bc1'=26) + 1/64 : (s1'=3) & (bc1'=27) + 1/64 : (s1'=3) & (bc1'=28) + 1/64 : (s1'=3) & (bc1'=29) + 1/64 : (s1'=3) & (bc1'=30) + 1/64 : (s1'=3) & (bc1'=31) + 1/64 : (s1'=3) & (bc1'=32) + 1/64 : (s1'=3) & (bc1'=33) + 1/64 : (s1'=3) & (bc1'=34) + 1/64 : (s1'=3) & (bc1'=35) + 1/64 : (s1'=3) & (bc1'=36) + 1/64 : (s1'=3) & (bc1'=37) + 1/64 : (s1'=3) & (bc1'=38) + 1/64 : (s1'=3) & (bc1'=39) + 1/64 : (s1'=3) & (bc1'=40) + 1/64 : (s1'=3) & (bc1'=41) + 1/64 : (s1'=3) & (bc1'=42) + 1/64 : (s1'=3) & (bc1'=43) + 1/64 : (s1'=3) & (bc1'=44) + 1/64 : (s1'=3) & (bc1'=45) + 1/64 : (s1'=3) & (bc1'=46) + 1/64 : (s1'=3) & (bc1'=47) + 1/64 : (s1'=3) & (bc1'=48) + 1/64 : (s1'=3) & (bc1'=49) + 1/64 : (s1'=3) & (bc1'=50) + 1/64 : (s1'=3) & (bc1'=51) + 1/64 : (s1'=3) & (bc1'=52) + 1/64 : (s1'=3) & (bc1'=53) + 1/64 : (s1'=3) & (bc1'=54) + 1/64 : (s1'=3) & (bc1'=55) + 1/64 : (s1'=3) & (bc1'=56) + 1/64 : (s1'=3) & (bc1'=57) + 1/64 : (s1'=3) & (bc1'=58) + 1/64 : (s1'=3) & (bc1'=59) + 1/64 : (s1'=3) & (bc1'=60) + 1/64 : (s1'=3) & (bc1'=61) + 1/64 : (s1'=3) & (bc1'=62) + 1/64 : (s1'=3) & (bc1'=63) ;
// wait until backoff counter reaches 0 then send again
[time] (s1=3) & (x1<slot) -> (x1'=x1+1); // let time pass (in slot)
[time] (s1=3) & (x1=slot) & (bc1>0) -> (x1'=1) & (bc1'=bc1-1); // let time pass (move slots)
[send1] (s1=3) & (x1=slot) & (bc1=0) -> (s1'=1) & (x1'=0); // finished backoff (bus appears free)
[busy1] (s1=3) & (x1=slot) & (bc1=0) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // finished backoff (bus busy)
// once finished nothing matters
[time] (s1>=4) -> (x1'=0);
endmodule
//----------------------------------------------------------------------------------------------------------------------------
// construct further stations through renaming
module station2=station1[s1=s2,x1=x2,cd1=cd2,bc1=bc2,send1=send2,busy1=busy2,end1=end2] endmodule
//----------------------------------------------------------------------------------------------------------------------------
// reward structure for expected time
rewards "time"
[time] true : 1;
endrewards
//----------------------------------------------------------------------------------------------------------------------------
// labels/formulae
label "all_delivered" = s1=4&s2=4;
label "one_delivered" = s1=4|s2=4;
label "collision_max_backoff" = (cd1=K & s1=1 & b=2)|(cd2=K & s2=1 & b=2);

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// CSMA/CD protocol - probabilistic version of kronos model (3 stations)
// gxn/dxp 04/12/01
mdp
// note made changes since cannot have strict inequalities
// in digital clocks approach and suppose a station only sends one message
// simplified parameters scaled
const int sigma=1; // time for messages to propagate along the bus
const int lambda=30; // time to send a message
// actual parameters
const int N = 3; // number of processes
const int K = 2; // exponential backoff limit
const int slot = 2*sigma; // length of slot
const int M = 3; // max number of slots to wait
//const int lambda=782;
//const int sigma=26;
//----------------------------------------------------------------------------------------------------------------------------
// the bus
module bus
b : [0..2];
// b=0 - idle
// b=1 - active
// b=2 - collision
// clocks of bus
y1 : [0..sigma+1]; // time since first send (used find time until channel sensed busy)
y2 : [0..sigma+1]; // time since second send (used to find time until collision detected)
// a sender sends (ok - no other message being sent)
[send1] (b=0) -> (b'=1);
[send2] (b=0) -> (b'=1);
[send3] (b=0) -> (b'=1);
// a sender sends (bus busy - collision)
[send1] (b=1|b=2) & (y1<sigma) -> (b'=2);
[send2] (b=1|b=2) & (y1<sigma) -> (b'=2);
[send3] (b=1|b=2) & (y1<sigma) -> (b'=2);
// finish sending
[end1] (b=1) -> (b'=0) & (y1'=0);
[end2] (b=1) -> (b'=0) & (y1'=0);
[end3] (b=1) -> (b'=0) & (y1'=0);
// bus busy
[busy1] (b=1|b=2) & (y1>=sigma) -> (b'=b);
[busy2] (b=1|b=2) & (y1>=sigma) -> (b'=b);
[busy3] (b=1|b=2) & (y1>=sigma) -> (b'=b);
// collision detected
[cd] (b=2) & (y2<=sigma) -> (b'=0) & (y1'=0) & (y2'=0);
// time passage
[time] (b=0) -> (y1'=0); // value of y1/y2 does not matter in state 0
[time] (b=1) -> (y1'=min(y1+1,sigma+1)); // no invariant in state 1
[time] (b=2) & (y2<sigma) -> (y1'=min(y1+1,sigma+1)) & (y2'=min(y2+1,sigma+1)); // invariant in state 2 (time until collision detected)
endmodule
//----------------------------------------------------------------------------------------------------------------------------
// model of first sender
module station1
// LOCAL STATE
s1 : [0..5];
// s1=0 - initial state
// s1=1 - transmit
// s1=2 - collision (set backoff)
// s1=3 - wait (bus busy)
// s1=4 - successfully sent
// LOCAL CLOCK
x1 : [0..max(lambda,slot)];
// BACKOFF COUNTER (number of slots to wait)
bc1 : [0..M];
// COLLISION COUNTER
cd1 : [0..K];
// start sending
[send1] (s1=0) -> (s1'=1) & (x1'=0); // start sending
[busy1] (s1=0) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // detects channel is busy so go into backoff
// transmitting
[time] (s1=1) & (x1<lambda) -> (x1'=min(x1+1,lambda)); // let time pass
[end1] (s1=1) & (x1=lambda) -> (s1'=4) & (x1'=0); // finished
[cd] (s1=1) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // collision detected (increment backoff counter)
[cd] !(s1=1) -> (s1'=s1); // add loop for collision detection when not important
// set backoff (no time can pass in this state)
// probability depends on which transmission this is (cd1)
[] s1=2 & cd1=1 -> 1/2 : (s1'=3) & (bc1'=0) + 1/2 : (s1'=3) & (bc1'=1) ;
[] s1=2 & cd1=2 -> 1/4 : (s1'=3) & (bc1'=0) + 1/4 : (s1'=3) & (bc1'=1) + 1/4 : (s1'=3) & (bc1'=2) + 1/4 : (s1'=3) & (bc1'=3) ;
// wait until backoff counter reaches 0 then send again
[time] (s1=3) & (x1<slot) -> (x1'=x1+1); // let time pass (in slot)
[time] (s1=3) & (x1=slot) & (bc1>0) -> (x1'=1) & (bc1'=bc1-1); // let time pass (move slots)
[send1] (s1=3) & (x1=slot) & (bc1=0) -> (s1'=1) & (x1'=0); // finished backoff (bus appears free)
[busy1] (s1=3) & (x1=slot) & (bc1=0) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // finished backoff (bus busy)
// once finished nothing matters
[time] (s1>=4) -> (x1'=0);
endmodule
//----------------------------------------------------------------------------------------------------------------------------
// construct further stations through renaming
module station2=station1[s1=s2,x1=x2,cd1=cd2,bc1=bc2,send1=send2,busy1=busy2,end1=end2] endmodule
module station3=station1[s1=s3,x1=x3,cd1=cd3,bc1=bc3,send1=send3,busy1=busy3,end1=end3] endmodule
//----------------------------------------------------------------------------------------------------------------------------
// reward structure for expected time
rewards "time"
[time] true : 1;
endrewards
//----------------------------------------------------------------------------------------------------------------------------
// labels/formulae
label "all_delivered" = s1=4&s2=4&s3=4;
label "one_delivered" = s1=4|s2=4|s3=4;
label "collision_max_backoff" = (cd1=K & s1=1 & b=2)|(cd2=K & s2=1 & b=2)|(cd3=K & s3=1 & b=2);

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// CSMA/CD protocol - probabilistic version of kronos model (3 stations)
// gxn/dxp 04/12/01
mdp
// note made changes since cannot have strict inequalities
// in digital clocks approach and suppose a station only sends one message
// simplified parameters scaled
const int sigma=1; // time for messages to propagate along the bus
const int lambda=30; // time to send a message
// actual parameters
const int N = 3; // number of processes
const int K = 4; // exponential backoff limit
const int slot = 2*sigma; // length of slot
const int M = 15; // max number of slots to wait
//const int lambda=782;
//const int sigma=26;
//----------------------------------------------------------------------------------------------------------------------------
// the bus
module bus
b : [0..2];
// b=0 - idle
// b=1 - active
// b=2 - collision
// clocks of bus
y1 : [0..sigma+1]; // time since first send (used find time until channel sensed busy)
y2 : [0..sigma+1]; // time since second send (used to find time until collision detected)
// a sender sends (ok - no other message being sent)
[send1] (b=0) -> (b'=1);
[send2] (b=0) -> (b'=1);
[send3] (b=0) -> (b'=1);
// a sender sends (bus busy - collision)
[send1] (b=1|b=2) & (y1<sigma) -> (b'=2);
[send2] (b=1|b=2) & (y1<sigma) -> (b'=2);
[send3] (b=1|b=2) & (y1<sigma) -> (b'=2);
// finish sending
[end1] (b=1) -> (b'=0) & (y1'=0);
[end2] (b=1) -> (b'=0) & (y1'=0);
[end3] (b=1) -> (b'=0) & (y1'=0);
// bus busy
[busy1] (b=1|b=2) & (y1>=sigma) -> (b'=b);
[busy2] (b=1|b=2) & (y1>=sigma) -> (b'=b);
[busy3] (b=1|b=2) & (y1>=sigma) -> (b'=b);
// collision detected
[cd] (b=2) & (y2<=sigma) -> (b'=0) & (y1'=0) & (y2'=0);
// time passage
[time] (b=0) -> (y1'=0); // value of y1/y2 does not matter in state 0
[time] (b=1) -> (y1'=min(y1+1,sigma+1)); // no invariant in state 1
[time] (b=2) & (y2<sigma) -> (y1'=min(y1+1,sigma+1)) & (y2'=min(y2+1,sigma+1)); // invariant in state 2 (time until collision detected)
endmodule
//----------------------------------------------------------------------------------------------------------------------------
// model of first sender
module station1
// LOCAL STATE
s1 : [0..5];
// s1=0 - initial state
// s1=1 - transmit
// s1=2 - collision (set backoff)
// s1=3 - wait (bus busy)
// s1=4 - successfully sent
// LOCAL CLOCK
x1 : [0..max(lambda,slot)];
// BACKOFF COUNTER (number of slots to wait)
bc1 : [0..M];
// COLLISION COUNTER
cd1 : [0..K];
// start sending
[send1] (s1=0) -> (s1'=1) & (x1'=0); // start sending
[busy1] (s1=0) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // detects channel is busy so go into backoff
// transmitting
[time] (s1=1) & (x1<lambda) -> (x1'=min(x1+1,lambda)); // let time pass
[end1] (s1=1) & (x1=lambda) -> (s1'=4) & (x1'=0); // finished
[cd] (s1=1) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // collision detected (increment backoff counter)
[cd] !(s1=1) -> (s1'=s1); // add loop for collision detection when not important
// set backoff (no time can pass in this state)
// probability depends on which transmission this is (cd1)
[] s1=2 & cd1=1 -> 1/2 : (s1'=3) & (bc1'=0) + 1/2 : (s1'=3) & (bc1'=1) ;
[] s1=2 & cd1=2 -> 1/4 : (s1'=3) & (bc1'=0) + 1/4 : (s1'=3) & (bc1'=1) + 1/4 : (s1'=3) & (bc1'=2) + 1/4 : (s1'=3) & (bc1'=3) ;
[] s1=2 & cd1=3 -> 1/8 : (s1'=3) & (bc1'=0) + 1/8 : (s1'=3) & (bc1'=1) + 1/8 : (s1'=3) & (bc1'=2) + 1/8 : (s1'=3) & (bc1'=3) + 1/8 : (s1'=3) & (bc1'=4) + 1/8 : (s1'=3) & (bc1'=5) + 1/8 : (s1'=3) & (bc1'=6) + 1/8 : (s1'=3) & (bc1'=7) ;
[] s1=2 & cd1=4 -> 1/16 : (s1'=3) & (bc1'=0) + 1/16 : (s1'=3) & (bc1'=1) + 1/16 : (s1'=3) & (bc1'=2) + 1/16 : (s1'=3) & (bc1'=3) + 1/16 : (s1'=3) & (bc1'=4) + 1/16 : (s1'=3) & (bc1'=5) + 1/16 : (s1'=3) & (bc1'=6) + 1/16 : (s1'=3) & (bc1'=7) + 1/16 : (s1'=3) & (bc1'=8) + 1/16 : (s1'=3) & (bc1'=9) + 1/16 : (s1'=3) & (bc1'=10) + 1/16 : (s1'=3) & (bc1'=11) + 1/16 : (s1'=3) & (bc1'=12) + 1/16 : (s1'=3) & (bc1'=13) + 1/16 : (s1'=3) & (bc1'=14) + 1/16 : (s1'=3) & (bc1'=15) ;
// wait until backoff counter reaches 0 then send again
[time] (s1=3) & (x1<slot) -> (x1'=x1+1); // let time pass (in slot)
[time] (s1=3) & (x1=slot) & (bc1>0) -> (x1'=1) & (bc1'=bc1-1); // let time pass (move slots)
[send1] (s1=3) & (x1=slot) & (bc1=0) -> (s1'=1) & (x1'=0); // finished backoff (bus appears free)
[busy1] (s1=3) & (x1=slot) & (bc1=0) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // finished backoff (bus busy)
// once finished nothing matters
[time] (s1>=4) -> (x1'=0);
endmodule
//----------------------------------------------------------------------------------------------------------------------------
// construct further stations through renaming
module station2=station1[s1=s2,x1=x2,cd1=cd2,bc1=bc2,send1=send2,busy1=busy2,end1=end2] endmodule
module station3=station1[s1=s3,x1=x3,cd1=cd3,bc1=bc3,send1=send3,busy1=busy3,end1=end3] endmodule
//----------------------------------------------------------------------------------------------------------------------------
// reward structure for expected time
rewards "time"
[time] true : 1;
endrewards
//----------------------------------------------------------------------------------------------------------------------------
// labels/formulae
label "all_delivered" = s1=4&s2=4&s3=4;
label "one_delivered" = s1=4|s2=4|s3=4;
label "collision_max_backoff" = (cd1=K & s1=1 & b=2)|(cd2=K & s2=1 & b=2)|(cd3=K & s3=1 & b=2);

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// CSMA/CD protocol - probabilistic version of kronos model (3 stations)
// gxn/dxp 04/12/01
mdp
// note made changes since cannot have strict inequalities
// in digital clocks approach and suppose a station only sends one message
// simplified parameters scaled
const int sigma=1; // time for messages to propagate along the bus
const int lambda=30; // time to send a message
// actual parameters
const int N = 3; // number of processes
const int K = 6; // exponential backoff limit
const int slot = 2*sigma; // length of slot
const int M = 63; // max number of slots to wait
//const int lambda=782;
//const int sigma=26;
//----------------------------------------------------------------------------------------------------------------------------
// the bus
module bus
b : [0..2];
// b=0 - idle
// b=1 - active
// b=2 - collision
// clocks of bus
y1 : [0..sigma+1]; // time since first send (used find time until channel sensed busy)
y2 : [0..sigma+1]; // time since second send (used to find time until collision detected)
// a sender sends (ok - no other message being sent)
[send1] (b=0) -> (b'=1);
[send2] (b=0) -> (b'=1);
[send3] (b=0) -> (b'=1);
// a sender sends (bus busy - collision)
[send1] (b=1|b=2) & (y1<sigma) -> (b'=2);
[send2] (b=1|b=2) & (y1<sigma) -> (b'=2);
[send3] (b=1|b=2) & (y1<sigma) -> (b'=2);
// finish sending
[end1] (b=1) -> (b'=0) & (y1'=0);
[end2] (b=1) -> (b'=0) & (y1'=0);
[end3] (b=1) -> (b'=0) & (y1'=0);
// bus busy
[busy1] (b=1|b=2) & (y1>=sigma) -> (b'=b);
[busy2] (b=1|b=2) & (y1>=sigma) -> (b'=b);
[busy3] (b=1|b=2) & (y1>=sigma) -> (b'=b);
// collision detected
[cd] (b=2) & (y2<=sigma) -> (b'=0) & (y1'=0) & (y2'=0);
// time passage
[time] (b=0) -> (y1'=0); // value of y1/y2 does not matter in state 0
[time] (b=1) -> (y1'=min(y1+1,sigma+1)); // no invariant in state 1
[time] (b=2) & (y2<sigma) -> (y1'=min(y1+1,sigma+1)) & (y2'=min(y2+1,sigma+1)); // invariant in state 2 (time until collision detected)
endmodule
//----------------------------------------------------------------------------------------------------------------------------
// model of first sender
module station1
// LOCAL STATE
s1 : [0..5];
// s1=0 - initial state
// s1=1 - transmit
// s1=2 - collision (set backoff)
// s1=3 - wait (bus busy)
// s1=4 - successfully sent
// LOCAL CLOCK
x1 : [0..max(lambda,slot)];
// BACKOFF COUNTER (number of slots to wait)
bc1 : [0..M];
// COLLISION COUNTER
cd1 : [0..K];
// start sending
[send1] (s1=0) -> (s1'=1) & (x1'=0); // start sending
[busy1] (s1=0) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // detects channel is busy so go into backoff
// transmitting
[time] (s1=1) & (x1<lambda) -> (x1'=min(x1+1,lambda)); // let time pass
[end1] (s1=1) & (x1=lambda) -> (s1'=4) & (x1'=0); // finished
[cd] (s1=1) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // collision detected (increment backoff counter)
[cd] !(s1=1) -> (s1'=s1); // add loop for collision detection when not important
// set backoff (no time can pass in this state)
// probability depends on which transmission this is (cd1)
[] s1=2 & cd1=1 -> 1/2 : (s1'=3) & (bc1'=0) + 1/2 : (s1'=3) & (bc1'=1) ;
[] s1=2 & cd1=2 -> 1/4 : (s1'=3) & (bc1'=0) + 1/4 : (s1'=3) & (bc1'=1) + 1/4 : (s1'=3) & (bc1'=2) + 1/4 : (s1'=3) & (bc1'=3) ;
[] s1=2 & cd1=3 -> 1/8 : (s1'=3) & (bc1'=0) + 1/8 : (s1'=3) & (bc1'=1) + 1/8 : (s1'=3) & (bc1'=2) + 1/8 : (s1'=3) & (bc1'=3) + 1/8 : (s1'=3) & (bc1'=4) + 1/8 : (s1'=3) & (bc1'=5) + 1/8 : (s1'=3) & (bc1'=6) + 1/8 : (s1'=3) & (bc1'=7) ;
[] s1=2 & cd1=4 -> 1/16 : (s1'=3) & (bc1'=0) + 1/16 : (s1'=3) & (bc1'=1) + 1/16 : (s1'=3) & (bc1'=2) + 1/16 : (s1'=3) & (bc1'=3) + 1/16 : (s1'=3) & (bc1'=4) + 1/16 : (s1'=3) & (bc1'=5) + 1/16 : (s1'=3) & (bc1'=6) + 1/16 : (s1'=3) & (bc1'=7) + 1/16 : (s1'=3) & (bc1'=8) + 1/16 : (s1'=3) & (bc1'=9) + 1/16 : (s1'=3) & (bc1'=10) + 1/16 : (s1'=3) & (bc1'=11) + 1/16 : (s1'=3) & (bc1'=12) + 1/16 : (s1'=3) & (bc1'=13) + 1/16 : (s1'=3) & (bc1'=14) + 1/16 : (s1'=3) & (bc1'=15) ;
[] s1=2 & cd1=5 -> 1/32 : (s1'=3) & (bc1'=0) + 1/32 : (s1'=3) & (bc1'=1) + 1/32 : (s1'=3) & (bc1'=2) + 1/32 : (s1'=3) & (bc1'=3) + 1/32 : (s1'=3) & (bc1'=4) + 1/32 : (s1'=3) & (bc1'=5) + 1/32 : (s1'=3) & (bc1'=6) + 1/32 : (s1'=3) & (bc1'=7) + 1/32 : (s1'=3) & (bc1'=8) + 1/32 : (s1'=3) & (bc1'=9) + 1/32 : (s1'=3) & (bc1'=10) + 1/32 : (s1'=3) & (bc1'=11) + 1/32 : (s1'=3) & (bc1'=12) + 1/32 : (s1'=3) & (bc1'=13) + 1/32 : (s1'=3) & (bc1'=14) + 1/32 : (s1'=3) & (bc1'=15) + 1/32 : (s1'=3) & (bc1'=16) + 1/32 : (s1'=3) & (bc1'=17) + 1/32 : (s1'=3) & (bc1'=18) + 1/32 : (s1'=3) & (bc1'=19) + 1/32 : (s1'=3) & (bc1'=20) + 1/32 : (s1'=3) & (bc1'=21) + 1/32 : (s1'=3) & (bc1'=22) + 1/32 : (s1'=3) & (bc1'=23) + 1/32 : (s1'=3) & (bc1'=24) + 1/32 : (s1'=3) & (bc1'=25) + 1/32 : (s1'=3) & (bc1'=26) + 1/32 : (s1'=3) & (bc1'=27) + 1/32 : (s1'=3) & (bc1'=28) + 1/32 : (s1'=3) & (bc1'=29) + 1/32 : (s1'=3) & (bc1'=30) + 1/32 : (s1'=3) & (bc1'=31) ;
[] s1=2 & cd1=6 -> 1/64 : (s1'=3) & (bc1'=0) + 1/64 : (s1'=3) & (bc1'=1) + 1/64 : (s1'=3) & (bc1'=2) + 1/64 : (s1'=3) & (bc1'=3) + 1/64 : (s1'=3) & (bc1'=4) + 1/64 : (s1'=3) & (bc1'=5) + 1/64 : (s1'=3) & (bc1'=6) + 1/64 : (s1'=3) & (bc1'=7) + 1/64 : (s1'=3) & (bc1'=8) + 1/64 : (s1'=3) & (bc1'=9) + 1/64 : (s1'=3) & (bc1'=10) + 1/64 : (s1'=3) & (bc1'=11) + 1/64 : (s1'=3) & (bc1'=12) + 1/64 : (s1'=3) & (bc1'=13) + 1/64 : (s1'=3) & (bc1'=14) + 1/64 : (s1'=3) & (bc1'=15) + 1/64 : (s1'=3) & (bc1'=16) + 1/64 : (s1'=3) & (bc1'=17) + 1/64 : (s1'=3) & (bc1'=18) + 1/64 : (s1'=3) & (bc1'=19) + 1/64 : (s1'=3) & (bc1'=20) + 1/64 : (s1'=3) & (bc1'=21) + 1/64 : (s1'=3) & (bc1'=22) + 1/64 : (s1'=3) & (bc1'=23) + 1/64 : (s1'=3) & (bc1'=24) + 1/64 : (s1'=3) & (bc1'=25) + 1/64 : (s1'=3) & (bc1'=26) + 1/64 : (s1'=3) & (bc1'=27) + 1/64 : (s1'=3) & (bc1'=28) + 1/64 : (s1'=3) & (bc1'=29) + 1/64 : (s1'=3) & (bc1'=30) + 1/64 : (s1'=3) & (bc1'=31) + 1/64 : (s1'=3) & (bc1'=32) + 1/64 : (s1'=3) & (bc1'=33) + 1/64 : (s1'=3) & (bc1'=34) + 1/64 : (s1'=3) & (bc1'=35) + 1/64 : (s1'=3) & (bc1'=36) + 1/64 : (s1'=3) & (bc1'=37) + 1/64 : (s1'=3) & (bc1'=38) + 1/64 : (s1'=3) & (bc1'=39) + 1/64 : (s1'=3) & (bc1'=40) + 1/64 : (s1'=3) & (bc1'=41) + 1/64 : (s1'=3) & (bc1'=42) + 1/64 : (s1'=3) & (bc1'=43) + 1/64 : (s1'=3) & (bc1'=44) + 1/64 : (s1'=3) & (bc1'=45) + 1/64 : (s1'=3) & (bc1'=46) + 1/64 : (s1'=3) & (bc1'=47) + 1/64 : (s1'=3) & (bc1'=48) + 1/64 : (s1'=3) & (bc1'=49) + 1/64 : (s1'=3) & (bc1'=50) + 1/64 : (s1'=3) & (bc1'=51) + 1/64 : (s1'=3) & (bc1'=52) + 1/64 : (s1'=3) & (bc1'=53) + 1/64 : (s1'=3) & (bc1'=54) + 1/64 : (s1'=3) & (bc1'=55) + 1/64 : (s1'=3) & (bc1'=56) + 1/64 : (s1'=3) & (bc1'=57) + 1/64 : (s1'=3) & (bc1'=58) + 1/64 : (s1'=3) & (bc1'=59) + 1/64 : (s1'=3) & (bc1'=60) + 1/64 : (s1'=3) & (bc1'=61) + 1/64 : (s1'=3) & (bc1'=62) + 1/64 : (s1'=3) & (bc1'=63) ;
// wait until backoff counter reaches 0 then send again
[time] (s1=3) & (x1<slot) -> (x1'=x1+1); // let time pass (in slot)
[time] (s1=3) & (x1=slot) & (bc1>0) -> (x1'=1) & (bc1'=bc1-1); // let time pass (move slots)
[send1] (s1=3) & (x1=slot) & (bc1=0) -> (s1'=1) & (x1'=0); // finished backoff (bus appears free)
[busy1] (s1=3) & (x1=slot) & (bc1=0) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // finished backoff (bus busy)
// once finished nothing matters
[time] (s1>=4) -> (x1'=0);
endmodule
//----------------------------------------------------------------------------------------------------------------------------
// construct further stations through renaming
module station2=station1[s1=s2,x1=x2,cd1=cd2,bc1=bc2,send1=send2,busy1=busy2,end1=end2] endmodule
module station3=station1[s1=s3,x1=x3,cd1=cd3,bc1=bc3,send1=send3,busy1=busy3,end1=end3] endmodule
//----------------------------------------------------------------------------------------------------------------------------
// reward structure for expected time
rewards "time"
[time] true : 1;
endrewards
//----------------------------------------------------------------------------------------------------------------------------
// labels/formulae
label "all_delivered" = s1=4&s2=4&s3=4;
label "one_delivered" = s1=4|s2=4|s3=4;
label "collision_max_backoff" = (cd1=K & s1=1 & b=2)|(cd2=K & s2=1 & b=2)|(cd3=K & s3=1 & b=2);

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// CSMA/CD protocol - probabilistic version of kronos model (3 stations)
// gxn/dxp 04/12/01
mdp
// note made changes since cannot have strict inequalities
// in digital clocks approach and suppose a station only sends one message
// simplified parameters scaled
const int sigma=1; // time for messages to propagate along the bus
const int lambda=30; // time to send a message
// actual parameters
const int N = 4; // number of processes
const int K = 2; // exponential backoff limit
const int slot = 2*sigma; // length of slot
const int M = 3; // max number of slots to wait
//const int lambda=782;
//const int sigma=26;
//----------------------------------------------------------------------------------------------------------------------------
// the bus
module bus
b : [0..2];
// b=0 - idle
// b=1 - active
// b=2 - collision
// clocks of bus
y1 : [0..sigma+1]; // time since first send (used find time until channel sensed busy)
y2 : [0..sigma+1]; // time since second send (used to find time until collision detected)
// a sender sends (ok - no other message being sent)
[send1] (b=0) -> (b'=1);
[send2] (b=0) -> (b'=1);
[send3] (b=0) -> (b'=1);
[send4] (b=0) -> (b'=1);
// a sender sends (bus busy - collision)
[send1] (b=1|b=2) & (y1<sigma) -> (b'=2);
[send2] (b=1|b=2) & (y1<sigma) -> (b'=2);
[send3] (b=1|b=2) & (y1<sigma) -> (b'=2);
[send4] (b=1|b=2) & (y1<sigma) -> (b'=2);
// finish sending
[end1] (b=1) -> (b'=0) & (y1'=0);
[end2] (b=1) -> (b'=0) & (y1'=0);
[end3] (b=1) -> (b'=0) & (y1'=0);
[end4] (b=1) -> (b'=0) & (y1'=0);
// bus busy
[busy1] (b=1|b=2) & (y1>=sigma) -> (b'=b);
[busy2] (b=1|b=2) & (y1>=sigma) -> (b'=b);
[busy3] (b=1|b=2) & (y1>=sigma) -> (b'=b);
[busy4] (b=1|b=2) & (y1>=sigma) -> (b'=b);
// collision detected
[cd] (b=2) & (y2<=sigma) -> (b'=0) & (y1'=0) & (y2'=0);
// time passage
[time] (b=0) -> (y1'=0); // value of y1/y2 does not matter in state 0
[time] (b=1) -> (y1'=min(y1+1,sigma+1)); // no invariant in state 1
[time] (b=2) & (y2<sigma) -> (y1'=min(y1+1,sigma+1)) & (y2'=min(y2+1,sigma+1)); // invariant in state 2 (time until collision detected)
endmodule
//----------------------------------------------------------------------------------------------------------------------------
// model of first sender
module station1
// LOCAL STATE
s1 : [0..5];
// s1=0 - initial state
// s1=1 - transmit
// s1=2 - collision (set backoff)
// s1=3 - wait (bus busy)
// s1=4 - successfully sent
// LOCAL CLOCK
x1 : [0..max(lambda,slot)];
// BACKOFF COUNTER (number of slots to wait)
bc1 : [0..M];
// COLLISION COUNTER
cd1 : [0..K];
// start sending
[send1] (s1=0) -> (s1'=1) & (x1'=0); // start sending
[busy1] (s1=0) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // detects channel is busy so go into backoff
// transmitting
[time] (s1=1) & (x1<lambda) -> (x1'=min(x1+1,lambda)); // let time pass
[end1] (s1=1) & (x1=lambda) -> (s1'=4) & (x1'=0); // finished
[cd] (s1=1) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // collision detected (increment backoff counter)
[cd] !(s1=1) -> (s1'=s1); // add loop for collision detection when not important
// set backoff (no time can pass in this state)
// probability depends on which transmission this is (cd1)
[] s1=2 & cd1=1 -> 1/2 : (s1'=3) & (bc1'=0) + 1/2 : (s1'=3) & (bc1'=1) ;
[] s1=2 & cd1=2 -> 1/4 : (s1'=3) & (bc1'=0) + 1/4 : (s1'=3) & (bc1'=1) + 1/4 : (s1'=3) & (bc1'=2) + 1/4 : (s1'=3) & (bc1'=3) ;
// wait until backoff counter reaches 0 then send again
[time] (s1=3) & (x1<slot) -> (x1'=x1+1); // let time pass (in slot)
[time] (s1=3) & (x1=slot) & (bc1>0) -> (x1'=1) & (bc1'=bc1-1); // let time pass (move slots)
[send1] (s1=3) & (x1=slot) & (bc1=0) -> (s1'=1) & (x1'=0); // finished backoff (bus appears free)
[busy1] (s1=3) & (x1=slot) & (bc1=0) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // finished backoff (bus busy)
// once finished nothing matters
[time] (s1>=4) -> (x1'=0);
endmodule
//----------------------------------------------------------------------------------------------------------------------------
// construct further stations through renaming
module station2=station1[s1=s2,x1=x2,cd1=cd2,bc1=bc2,send1=send2,busy1=busy2,end1=end2] endmodule
module station3=station1[s1=s3,x1=x3,cd1=cd3,bc1=bc3,send1=send3,busy1=busy3,end1=end3] endmodule
module station4=station1[s1=s4,x1=x4,cd1=cd4,bc1=bc4,send1=send4,busy1=busy4,end1=end4] endmodule
//----------------------------------------------------------------------------------------------------------------------------
// reward structure for expected time
rewards "time"
[time] true : 1;
endrewards
//----------------------------------------------------------------------------------------------------------------------------
// labels/formulae
label "all_delivered" = s1=4&s2=4&s3=4&s4=4;
label "one_delivered" = s1=4|s2=4|s3=4|s4=4;
label "collision_max_backoff" = (cd1=K & s1=1 & b=2)|(cd2=K & s2=1 & b=2)|(cd3=K & s3=1 & b=2)|(cd4=K & s4=1 & b=2);

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// CSMA/CD protocol - probabilistic version of kronos model (3 stations)
// gxn/dxp 04/12/01
mdp
// note made changes since cannot have strict inequalities
// in digital clocks approach and suppose a station only sends one message
// simplified parameters scaled
const int sigma=1; // time for messages to propagate along the bus
const int lambda=30; // time to send a message
// actual parameters
const int N = 4; // number of processes
const int K = 4; // exponential backoff limit
const int slot = 2*sigma; // length of slot
const int M = 15; // max number of slots to wait
//const int lambda=782;
//const int sigma=26;
//----------------------------------------------------------------------------------------------------------------------------
// the bus
module bus
b : [0..2];
// b=0 - idle
// b=1 - active
// b=2 - collision
// clocks of bus
y1 : [0..sigma+1]; // time since first send (used find time until channel sensed busy)
y2 : [0..sigma+1]; // time since second send (used to find time until collision detected)
// a sender sends (ok - no other message being sent)
[send1] (b=0) -> (b'=1);
[send2] (b=0) -> (b'=1);
[send3] (b=0) -> (b'=1);
[send4] (b=0) -> (b'=1);
// a sender sends (bus busy - collision)
[send1] (b=1|b=2) & (y1<sigma) -> (b'=2);
[send2] (b=1|b=2) & (y1<sigma) -> (b'=2);
[send3] (b=1|b=2) & (y1<sigma) -> (b'=2);
[send4] (b=1|b=2) & (y1<sigma) -> (b'=2);
// finish sending
[end1] (b=1) -> (b'=0) & (y1'=0);
[end2] (b=1) -> (b'=0) & (y1'=0);
[end3] (b=1) -> (b'=0) & (y1'=0);
[end4] (b=1) -> (b'=0) & (y1'=0);
// bus busy
[busy1] (b=1|b=2) & (y1>=sigma) -> (b'=b);
[busy2] (b=1|b=2) & (y1>=sigma) -> (b'=b);
[busy3] (b=1|b=2) & (y1>=sigma) -> (b'=b);
[busy4] (b=1|b=2) & (y1>=sigma) -> (b'=b);
// collision detected
[cd] (b=2) & (y2<=sigma) -> (b'=0) & (y1'=0) & (y2'=0);
// time passage
[time] (b=0) -> (y1'=0); // value of y1/y2 does not matter in state 0
[time] (b=1) -> (y1'=min(y1+1,sigma+1)); // no invariant in state 1
[time] (b=2) & (y2<sigma) -> (y1'=min(y1+1,sigma+1)) & (y2'=min(y2+1,sigma+1)); // invariant in state 2 (time until collision detected)
endmodule
//----------------------------------------------------------------------------------------------------------------------------
// model of first sender
module station1
// LOCAL STATE
s1 : [0..5];
// s1=0 - initial state
// s1=1 - transmit
// s1=2 - collision (set backoff)
// s1=3 - wait (bus busy)
// s1=4 - successfully sent
// LOCAL CLOCK
x1 : [0..max(lambda,slot)];
// BACKOFF COUNTER (number of slots to wait)
bc1 : [0..M];
// COLLISION COUNTER
cd1 : [0..K];
// start sending
[send1] (s1=0) -> (s1'=1) & (x1'=0); // start sending
[busy1] (s1=0) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // detects channel is busy so go into backoff
// transmitting
[time] (s1=1) & (x1<lambda) -> (x1'=min(x1+1,lambda)); // let time pass
[end1] (s1=1) & (x1=lambda) -> (s1'=4) & (x1'=0); // finished
[cd] (s1=1) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // collision detected (increment backoff counter)
[cd] !(s1=1) -> (s1'=s1); // add loop for collision detection when not important
// set backoff (no time can pass in this state)
// probability depends on which transmission this is (cd1)
[] s1=2 & cd1=1 -> 1/2 : (s1'=3) & (bc1'=0) + 1/2 : (s1'=3) & (bc1'=1) ;
[] s1=2 & cd1=2 -> 1/4 : (s1'=3) & (bc1'=0) + 1/4 : (s1'=3) & (bc1'=1) + 1/4 : (s1'=3) & (bc1'=2) + 1/4 : (s1'=3) & (bc1'=3) ;
[] s1=2 & cd1=3 -> 1/8 : (s1'=3) & (bc1'=0) + 1/8 : (s1'=3) & (bc1'=1) + 1/8 : (s1'=3) & (bc1'=2) + 1/8 : (s1'=3) & (bc1'=3) + 1/8 : (s1'=3) & (bc1'=4) + 1/8 : (s1'=3) & (bc1'=5) + 1/8 : (s1'=3) & (bc1'=6) + 1/8 : (s1'=3) & (bc1'=7) ;
[] s1=2 & cd1=4 -> 1/16 : (s1'=3) & (bc1'=0) + 1/16 : (s1'=3) & (bc1'=1) + 1/16 : (s1'=3) & (bc1'=2) + 1/16 : (s1'=3) & (bc1'=3) + 1/16 : (s1'=3) & (bc1'=4) + 1/16 : (s1'=3) & (bc1'=5) + 1/16 : (s1'=3) & (bc1'=6) + 1/16 : (s1'=3) & (bc1'=7) + 1/16 : (s1'=3) & (bc1'=8) + 1/16 : (s1'=3) & (bc1'=9) + 1/16 : (s1'=3) & (bc1'=10) + 1/16 : (s1'=3) & (bc1'=11) + 1/16 : (s1'=3) & (bc1'=12) + 1/16 : (s1'=3) & (bc1'=13) + 1/16 : (s1'=3) & (bc1'=14) + 1/16 : (s1'=3) & (bc1'=15) ;
// wait until backoff counter reaches 0 then send again
[time] (s1=3) & (x1<slot) -> (x1'=x1+1); // let time pass (in slot)
[time] (s1=3) & (x1=slot) & (bc1>0) -> (x1'=1) & (bc1'=bc1-1); // let time pass (move slots)
[send1] (s1=3) & (x1=slot) & (bc1=0) -> (s1'=1) & (x1'=0); // finished backoff (bus appears free)
[busy1] (s1=3) & (x1=slot) & (bc1=0) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // finished backoff (bus busy)
// once finished nothing matters
[time] (s1>=4) -> (x1'=0);
endmodule
//----------------------------------------------------------------------------------------------------------------------------
// construct further stations through renaming
module station2=station1[s1=s2,x1=x2,cd1=cd2,bc1=bc2,send1=send2,busy1=busy2,end1=end2] endmodule
module station3=station1[s1=s3,x1=x3,cd1=cd3,bc1=bc3,send1=send3,busy1=busy3,end1=end3] endmodule
module station4=station1[s1=s4,x1=x4,cd1=cd4,bc1=bc4,send1=send4,busy1=busy4,end1=end4] endmodule
//----------------------------------------------------------------------------------------------------------------------------
// reward structure for expected time
rewards "time"
[time] true : 1;
endrewards
//----------------------------------------------------------------------------------------------------------------------------
// labels/formulae
label "all_delivered" = s1=4&s2=4&s3=4&s4=4;
label "one_delivered" = s1=4|s2=4|s3=4|s4=4;
label "collision_max_backoff" = (cd1=K & s1=1 & b=2)|(cd2=K & s2=1 & b=2)|(cd3=K & s3=1 & b=2)|(cd4=K & s4=1 & b=2);

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examples/mdp/csma/csma4_6.nm
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// CSMA/CD protocol - probabilistic version of kronos model (3 stations)
// gxn/dxp 04/12/01
mdp
// note made changes since cannot have strict inequalities
// in digital clocks approach and suppose a station only sends one message
// simplified parameters scaled
const int sigma=1; // time for messages to propagate along the bus
const int lambda=30; // time to send a message
// actual parameters
const int N = 4; // number of processes
const int K = 6; // exponential backoff limit
const int slot = 2*sigma; // length of slot
const int M = 63; // max number of slots to wait
//const int lambda=782;
//const int sigma=26;
//----------------------------------------------------------------------------------------------------------------------------
// the bus
module bus
b : [0..2];
// b=0 - idle
// b=1 - active
// b=2 - collision
// clocks of bus
y1 : [0..sigma+1]; // time since first send (used find time until channel sensed busy)
y2 : [0..sigma+1]; // time since second send (used to find time until collision detected)
// a sender sends (ok - no other message being sent)
[send1] (b=0) -> (b'=1);
[send2] (b=0) -> (b'=1);
[send3] (b=0) -> (b'=1);
[send4] (b=0) -> (b'=1);
// a sender sends (bus busy - collision)
[send1] (b=1|b=2) & (y1<sigma) -> (b'=2);
[send2] (b=1|b=2) & (y1<sigma) -> (b'=2);
[send3] (b=1|b=2) & (y1<sigma) -> (b'=2);
[send4] (b=1|b=2) & (y1<sigma) -> (b'=2);
// finish sending
[end1] (b=1) -> (b'=0) & (y1'=0);
[end2] (b=1) -> (b'=0) & (y1'=0);
[end3] (b=1) -> (b'=0) & (y1'=0);
[end4] (b=1) -> (b'=0) & (y1'=0);
// bus busy
[busy1] (b=1|b=2) & (y1>=sigma) -> (b'=b);
[busy2] (b=1|b=2) & (y1>=sigma) -> (b'=b);
[busy3] (b=1|b=2) & (y1>=sigma) -> (b'=b);
[busy4] (b=1|b=2) & (y1>=sigma) -> (b'=b);
// collision detected
[cd] (b=2) & (y2<=sigma) -> (b'=0) & (y1'=0) & (y2'=0);
// time passage
[time] (b=0) -> (y1'=0); // value of y1/y2 does not matter in state 0
[time] (b=1) -> (y1'=min(y1+1,sigma+1)); // no invariant in state 1
[time] (b=2) & (y2<sigma) -> (y1'=min(y1+1,sigma+1)) & (y2'=min(y2+1,sigma+1)); // invariant in state 2 (time until collision detected)
endmodule
//----------------------------------------------------------------------------------------------------------------------------
// model of first sender
module station1
// LOCAL STATE
s1 : [0..5];
// s1=0 - initial state
// s1=1 - transmit
// s1=2 - collision (set backoff)
// s1=3 - wait (bus busy)
// s1=4 - successfully sent
// LOCAL CLOCK
x1 : [0..max(lambda,slot)];
// BACKOFF COUNTER (number of slots to wait)
bc1 : [0..M];
// COLLISION COUNTER
cd1 : [0..K];
// start sending
[send1] (s1=0) -> (s1'=1) & (x1'=0); // start sending
[busy1] (s1=0) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // detects channel is busy so go into backoff
// transmitting
[time] (s1=1) & (x1<lambda) -> (x1'=min(x1+1,lambda)); // let time pass
[end1] (s1=1) & (x1=lambda) -> (s1'=4) & (x1'=0); // finished
[cd] (s1=1) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // collision detected (increment backoff counter)
[cd] !(s1=1) -> (s1'=s1); // add loop for collision detection when not important
// set backoff (no time can pass in this state)
// probability depends on which transmission this is (cd1)
[] s1=2 & cd1=1 -> 1/2 : (s1'=3) & (bc1'=0) + 1/2 : (s1'=3) & (bc1'=1) ;
[] s1=2 & cd1=2 -> 1/4 : (s1'=3) & (bc1'=0) + 1/4 : (s1'=3) & (bc1'=1) + 1/4 : (s1'=3) & (bc1'=2) + 1/4 : (s1'=3) & (bc1'=3) ;
[] s1=2 & cd1=3 -> 1/8 : (s1'=3) & (bc1'=0) + 1/8 : (s1'=3) & (bc1'=1) + 1/8 : (s1'=3) & (bc1'=2) + 1/8 : (s1'=3) & (bc1'=3) + 1/8 : (s1'=3) & (bc1'=4) + 1/8 : (s1'=3) & (bc1'=5) + 1/8 : (s1'=3) & (bc1'=6) + 1/8 : (s1'=3) & (bc1'=7) ;
[] s1=2 & cd1=4 -> 1/16 : (s1'=3) & (bc1'=0) + 1/16 : (s1'=3) & (bc1'=1) + 1/16 : (s1'=3) & (bc1'=2) + 1/16 : (s1'=3) & (bc1'=3) + 1/16 : (s1'=3) & (bc1'=4) + 1/16 : (s1'=3) & (bc1'=5) + 1/16 : (s1'=3) & (bc1'=6) + 1/16 : (s1'=3) & (bc1'=7) + 1/16 : (s1'=3) & (bc1'=8) + 1/16 : (s1'=3) & (bc1'=9) + 1/16 : (s1'=3) & (bc1'=10) + 1/16 : (s1'=3) & (bc1'=11) + 1/16 : (s1'=3) & (bc1'=12) + 1/16 : (s1'=3) & (bc1'=13) + 1/16 : (s1'=3) & (bc1'=14) + 1/16 : (s1'=3) & (bc1'=15) ;
[] s1=2 & cd1=5 -> 1/32 : (s1'=3) & (bc1'=0) + 1/32 : (s1'=3) & (bc1'=1) + 1/32 : (s1'=3) & (bc1'=2) + 1/32 : (s1'=3) & (bc1'=3) + 1/32 : (s1'=3) & (bc1'=4) + 1/32 : (s1'=3) & (bc1'=5) + 1/32 : (s1'=3) & (bc1'=6) + 1/32 : (s1'=3) & (bc1'=7) + 1/32 : (s1'=3) & (bc1'=8) + 1/32 : (s1'=3) & (bc1'=9) + 1/32 : (s1'=3) & (bc1'=10) + 1/32 : (s1'=3) & (bc1'=11) + 1/32 : (s1'=3) & (bc1'=12) + 1/32 : (s1'=3) & (bc1'=13) + 1/32 : (s1'=3) & (bc1'=14) + 1/32 : (s1'=3) & (bc1'=15) + 1/32 : (s1'=3) & (bc1'=16) + 1/32 : (s1'=3) & (bc1'=17) + 1/32 : (s1'=3) & (bc1'=18) + 1/32 : (s1'=3) & (bc1'=19) + 1/32 : (s1'=3) & (bc1'=20) + 1/32 : (s1'=3) & (bc1'=21) + 1/32 : (s1'=3) & (bc1'=22) + 1/32 : (s1'=3) & (bc1'=23) + 1/32 : (s1'=3) & (bc1'=24) + 1/32 : (s1'=3) & (bc1'=25) + 1/32 : (s1'=3) & (bc1'=26) + 1/32 : (s1'=3) & (bc1'=27) + 1/32 : (s1'=3) & (bc1'=28) + 1/32 : (s1'=3) & (bc1'=29) + 1/32 : (s1'=3) & (bc1'=30) + 1/32 : (s1'=3) & (bc1'=31) ;
[] s1=2 & cd1=6 -> 1/64 : (s1'=3) & (bc1'=0) + 1/64 : (s1'=3) & (bc1'=1) + 1/64 : (s1'=3) & (bc1'=2) + 1/64 : (s1'=3) & (bc1'=3) + 1/64 : (s1'=3) & (bc1'=4) + 1/64 : (s1'=3) & (bc1'=5) + 1/64 : (s1'=3) & (bc1'=6) + 1/64 : (s1'=3) & (bc1'=7) + 1/64 : (s1'=3) & (bc1'=8) + 1/64 : (s1'=3) & (bc1'=9) + 1/64 : (s1'=3) & (bc1'=10) + 1/64 : (s1'=3) & (bc1'=11) + 1/64 : (s1'=3) & (bc1'=12) + 1/64 : (s1'=3) & (bc1'=13) + 1/64 : (s1'=3) & (bc1'=14) + 1/64 : (s1'=3) & (bc1'=15) + 1/64 : (s1'=3) & (bc1'=16) + 1/64 : (s1'=3) & (bc1'=17) + 1/64 : (s1'=3) & (bc1'=18) + 1/64 : (s1'=3) & (bc1'=19) + 1/64 : (s1'=3) & (bc1'=20) + 1/64 : (s1'=3) & (bc1'=21) + 1/64 : (s1'=3) & (bc1'=22) + 1/64 : (s1'=3) & (bc1'=23) + 1/64 : (s1'=3) & (bc1'=24) + 1/64 : (s1'=3) & (bc1'=25) + 1/64 : (s1'=3) & (bc1'=26) + 1/64 : (s1'=3) & (bc1'=27) + 1/64 : (s1'=3) & (bc1'=28) + 1/64 : (s1'=3) & (bc1'=29) + 1/64 : (s1'=3) & (bc1'=30) + 1/64 : (s1'=3) & (bc1'=31) + 1/64 : (s1'=3) & (bc1'=32) + 1/64 : (s1'=3) & (bc1'=33) + 1/64 : (s1'=3) & (bc1'=34) + 1/64 : (s1'=3) & (bc1'=35) + 1/64 : (s1'=3) & (bc1'=36) + 1/64 : (s1'=3) & (bc1'=37) + 1/64 : (s1'=3) & (bc1'=38) + 1/64 : (s1'=3) & (bc1'=39) + 1/64 : (s1'=3) & (bc1'=40) + 1/64 : (s1'=3) & (bc1'=41) + 1/64 : (s1'=3) & (bc1'=42) + 1/64 : (s1'=3) & (bc1'=43) + 1/64 : (s1'=3) & (bc1'=44) + 1/64 : (s1'=3) & (bc1'=45) + 1/64 : (s1'=3) & (bc1'=46) + 1/64 : (s1'=3) & (bc1'=47) + 1/64 : (s1'=3) & (bc1'=48) + 1/64 : (s1'=3) & (bc1'=49) + 1/64 : (s1'=3) & (bc1'=50) + 1/64 : (s1'=3) & (bc1'=51) + 1/64 : (s1'=3) & (bc1'=52) + 1/64 : (s1'=3) & (bc1'=53) + 1/64 : (s1'=3) & (bc1'=54) + 1/64 : (s1'=3) & (bc1'=55) + 1/64 : (s1'=3) & (bc1'=56) + 1/64 : (s1'=3) & (bc1'=57) + 1/64 : (s1'=3) & (bc1'=58) + 1/64 : (s1'=3) & (bc1'=59) + 1/64 : (s1'=3) & (bc1'=60) + 1/64 : (s1'=3) & (bc1'=61) + 1/64 : (s1'=3) & (bc1'=62) + 1/64 : (s1'=3) & (bc1'=63) ;
// wait until backoff counter reaches 0 then send again
[time] (s1=3) & (x1<slot) -> (x1'=x1+1); // let time pass (in slot)
[time] (s1=3) & (x1=slot) & (bc1>0) -> (x1'=1) & (bc1'=bc1-1); // let time pass (move slots)
[send1] (s1=3) & (x1=slot) & (bc1=0) -> (s1'=1) & (x1'=0); // finished backoff (bus appears free)
[busy1] (s1=3) & (x1=slot) & (bc1=0) -> (s1'=2) & (x1'=0) & (cd1'=min(K,cd1+1)); // finished backoff (bus busy)
// once finished nothing matters
[time] (s1>=4) -> (x1'=0);
endmodule
//----------------------------------------------------------------------------------------------------------------------------
// construct further stations through renaming
module station2=station1[s1=s2,x1=x2,cd1=cd2,bc1=bc2,send1=send2,busy1=busy2,end1=end2] endmodule
module station3=station1[s1=s3,x1=x3,cd1=cd3,bc1=bc3,send1=send3,busy1=busy3,end1=end3] endmodule
module station4=station1[s1=s4,x1=x4,cd1=cd4,bc1=bc4,send1=send4,busy1=busy4,end1=end4] endmodule
//----------------------------------------------------------------------------------------------------------------------------
// reward structure for expected time
rewards "time"
[time] true : 1;
endrewards
//----------------------------------------------------------------------------------------------------------------------------
// labels/formulae
label "all_delivered" = s1=4&s2=4&s3=4&s4=4;
label "one_delivered" = s1=4|s2=4|s3=4|s4=4;
label "collision_max_backoff" = (cd1=K & s1=1 & b=2)|(cd2=K & s2=1 & b=2)|(cd3=K & s3=1 & b=2)|(cd4=K & s4=1 & b=2);
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