// 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 (b'=2); [send2] (b=1|b=2) & (y1 (b'=2); [send3] (b=1|b=2) & (y1 (b'=2); [send4] (b=1|b=2) & (y1 (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 (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 (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 (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);