Consideriamo un esempio "fantastico", tratto dalla rivista Scientific American.
ogni secondo arriva uno zombie di fronte a un muro lungo 1 in cui è praticata un'apertura ampia w. Gli zombie che non passano attraverso l'apertura, dopo la facciata contro il muro si rialzano e si predispongono a ritentare l'avventura, per cui il flusso di zombie è senza fine, e sempre con lo stesso regime. Inoltre:
(1) le posizioni lungo il muro in cui arrivano gli zombie hanno distribuzione uniforme
(non viene privilegiata alcuna parte del muro), per cui, ovunque sia collocata l'apertura, per essa c'è un flusso
stazionario di zombie (la media teorica del numero di zombie Nw che passano in un intervallo di tempo fissato
è proporzionale a w:
esiste una costante positiva λ tale che
(2) la posizione di arrivo di ogni zombie è indipendente da quella di ciascuno dei precedenti, cioè siamo di fronte a un flusso senza memoria;
(3) è trascurabile la probabilità che due o più zombie arrivino praticamente
nella stessa posizione, cioè, al rimpicciolire dell'apertura, la differenza relativa tra
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Si può simulare il fenomeno usando generatore di numeri pseudocasuali
( if rnd < w then U = 1 else U = 0 U = 1 indica il passaggio dello zombie per l'apertura |
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Infatti il generatore di numeri pseudocasuali verifica (1) e (2), come abbiamo già osservato, e (come si può controllare sperimentalmente) verifica anche (3).
Sviluppando questa idea è stato realizzato un programma (qui redatto in Basic) che simula il fenomeno nel caso in cui w = 1/10. Man mano che arrivano gli zombie esso calcola i tempi di attesa tra un passaggio per l'apertura e il passaggio successivo. Ecco il programma e un suo possibile esito:
1 ' w: fraz. del muro aperta; t: tempo tra 2 passaggi; ore: durata del fenomeno 2 ' "if then else" vale 1 se lo zombie passa per l'apertura 3 ' tz: variabile indiciata in cui sono classificati i tempi tra i passaggi 10 w=0.1 : dim tz(10000) : ore = 8.5 : sec = 60*60*ore : i=1 : n=0 20 if i <= sec then gosub 100 else END 30 goto 20 100 t=0 : j=0 : while j<1 110 if rnd < w then j=1 else j=0 120 t=t+1 : i=i+1 130 wend 140 n=n+1 : tz(n)=t : print t;", "; 150 if n/20 = int(n/20) then print ' ogni 20 uscite vado a capo 160 return
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Mettendo questi dati nello script "istogramma" posso ottenere:
Se calcoliamo quanti zombie passano per l'apertura in 60 secondi otteniamo (simulando per qualche ora) con questo programma otteniamo un esito simile a quello riportato sotto:
1 ' nz: variabile indiciata in cui sono messi ogni 60 sec quanti zombie passano 10 w=0.1 : dim nz(10000) : ore = 8.5 : sec = 60*60*ore : i=1 : u=0 20 if i <= sec then gosub 500 else goto 40 30 goto 20 40 for k=1 to r 50 print nz(k);", "; : if k/20 = int(k/20) then print 60 next 70 end 500 t=0 505 j=0 : while j<1 510 if rnd < w then j=1 else j=0 515 t=t+1 : r=i/60 : u=u+j 520 if r = int(r) then nz(r)=u : if r = int(r) then u=0 525 i=i+1 : wend 530 n=n+1 535 return
1, 6, 6, 6, 1, 2, 4, 8, 4, 11, 6, 4, 5, 5, 3, 5, 4, 6, 8, 4, 10, 5, 3, 4, 4, 7, 4, 5, 5, 5, 2, 6, 10, 7, 7, 6, 1, 6, 8, 7, 11, 6, 4, 9, 8, 5, 5, 7, 3, 11, 8, 7, 6, 5, 5, 8, 6, 2, 9, 5, 8, 7, 9, 5, 10, 6, 3, 8, 3, 7, 11, 6, 5, 5, 4, 6, 4, 7, 3, 5, 4, 5, 5, 10, 4, 7, 5, 3, 6, 10, 9, 1, 4, 3, 10, 4, 4, 5, 7, 5, 6, 5, 6, 3, 4, 1, 9, 5, 6, 6, 4, 5, 7, 4, 4, 11, 13, 2, 6, 9, 4, 6, 6, 4, 5, 6, 3, 5, 7, 5, 4, 4, 3, 9, 8, 6, 7, 8, 3, 9, 4, 10, 3, 5, 4, 7, 6, 7, 4, 4, 4, 5, 8, 6, 4, 2, 8, 11, 6, 2, 5, 7, 3, 11, 5, 1, 11, 3, 11, 3, 6, 2, 5, 5, 7, 7, 6, 8, 5, 4, 7, 9, 7, 5, 5, 6, 5, 4, 3, 10, 3, 8, 5, 6, 4, 6, 5, 17, 9, 7, 7, 7, 10, 3, 3, 8, 6, 4, 6, 10, 0, 5, 5, 5, 5, 7, 5, 6, 8, 6, 9, 7, 5, 6, 5, 6, 4, 6, 7, 5, 8, 8, 9, 7, 5, 6, 5, 6, 4, 4, 6, 4, 5, 10, 10, 4, 4, 8, 4, 10, 8, 4, 10, 6, 8, 10, 7, 6, 7, 8, 5, 7, 6, 6, 4, 5, 6, 3, 9, 7, 8, 11, 4, 6, 8, 4, 6, 8, 5, 9, 7, 5, 9, 7, 8, 5, 6, 2, 8, 6, 4, 10, 3, 6, 6, 7, 6, 7, 12, 8, 6, 7, 9, 6, 7, 4, 9, 8, 4, 5, 7, 9, 8, 3, 9, 6, 5, 8, 8, 4, 4, 7, 10, 3, 3, 2, 8, 3, 7, 5, 7, 3, 8, 5, 3, 10, 5, 5, 7, 5, 6, 5, 8, 7, 10, 4, 9, 5, 7, 6, 4, 6, 6, 3, 6, 6, 7, 7, 9, 8, 5, 7, 6, 5, 6, 4, 8, 4, 7, 9, 6, 3, 6, 5, 7, 5, 12, 7, 4, 7, 4, 3, 3, 6, 8, 5, 6, 3, 6, 6, 8, 8, 4, 5, 3, 9, 4, 2, 6, 7, 9, 5, 7, 6, 6, 6, 9, 5, 4, 9, 6, 4, 8, 3, 3, 5, 8, 5, 4, 5, 3, 5, 3, 12, 5, 5, 3, 9, 3, 4, 9, 3, 6, 5, 7, 7, 11, 5, 4, 5, 7, 10, 7, 7, 7, 8, 7, 5, 5, 6, 12, 6, 6, 5, 7, 6, 8, 4, 5, 6, 6, 3, 9, 5, 6, 6, 3, 7, 9, 5, 3, 5, 8, 6, 7, 4, 5, 4, 6, 7, 4, 7, 5, 6, 6, 6, 8, 6, 4, 3, 6, 7, 5, 4, 7, 9, 5, 9, 9, 8, 6, 6, 8, 9, 6, 10, 6, 5, 5, 9
Mettendo questi dati nello script "istogramma" posso ottenere:
Come si vede, il primo istogramma, dei tempi di attesa, ha, grosso modo, andamento decrescente, simile all'istogramma della
(
)
distribuzione esponenziale negativa. Quello degli zombi che passano ogni minuto ha un andamento a campana asimmetrica, che ha qualche somiglianza con una binomiale
().
