"""One room, random arrivals: the average wait in line, in visit-lengths.
Each patient's wait follows from the one before (Lindley's recursion):
    wait[n+1] = max(0, wait[n] + visit[n] - gap[n+1])
No queueing formula is used. Save as queue_sim.py and run: python3 queue_sim.py
It takes a few minutes; the page reports the five runs at 90% busy."""
import random

def mean_wait(busy, fixed, patients, seed):
    rng = random.Random(seed)
    wait = total = 0.0
    for _ in range(patients):
        total += wait
        visit = 1.0 if fixed else rng.expovariate(1.0)   # mean visit length = 1
        gap = rng.expovariate(busy)                      # mean gap between arrivals = 1/busy
        wait = max(0.0, wait + visit - gap)
    return total / patients

if __name__ == '__main__':
    N = 2_000_000
    for busy in (0.8, 0.9, 0.95):
        for fixed in (False, True):
            runs = [mean_wait(busy, fixed, N, seed) for seed in range(1, 6)]
            print(f'busy {busy:.2f}  {"fixed " if fixed else "random"} visits  seeds 1-5: '
                  + '  '.join(f'{r:.2f}' for r in runs) + f'   mean {sum(runs)/5:.2f}')