Keenan Scalpo’s barber shop had two chairs; but only one chair was operated, since Scalpo worked alone. He never had anyone waiting for a haircut, but he managed to keep satisfactorily and profitably busy, because he found that the next customer tended to enter the shop reasonably soon after he finished with the previous one. Actually, he was idle exactly 50 per cent of the time.
When his nephew, J. Brainwash Klipp, was graduated with the degree A.B. (awful barber) from the tonsorial curriculum at Science Tech., young Brainy asked if he could join his uncle and make it a two-barber shop. Uncle Scalpo thought that this was ridiculous since there couldn’t possibly be any heads for Brainy to beautify since there never was more than one customer in the shop.
Brainy, who had studied queuing theory, argued the case of voluntary truncation and asked for a two-week trial period. Uncle consented and he installed young Brainwash in the other chair. Lo and behold, they were both kept reasonably busy, but, still, the seats for waiting customers were never used.
a) If either barber can service 4 customers per hour, and if they are both idle now 40 per cent of the time, what percentage of their potential business do they lose as a consequence of voluntary truncation? (No one enters unless at least one barber is free.) Assume that the system is Poisson-exponential.