Two 1 man barber shops sit side by side in Dubkirk Square Ea

Two 1 man barber shops sit side by side in Dubkirk Square. Each can hold a max of 4 people, and any potential customer who finds a shop full will not wait for a haircut. Barber1 charges $11 per hair cut and takes an avg of 12 min to complete a hair cut. Barber2 charges $5 per haircut and takes an average of 6 min to complete a hair cut. An avg of 10 potential customers per hour arrive at each barber shop. Of course, a potential customer becomes an actual customer only if he finds that the shop is not full. Assuming that inter-arrival times and haircut times are exponential, which barber will earn more money?

Solution

In both cases

P_o + P₁ + P₂ + P₃ + P₄ = 1

P_n = P_o(/)ⁿ

So = /

P_o(1+ + ² + ³ + ⁴) = 1

=> P_o = 1/(1+ + ² + ³ + ⁴)

For first barber with $11 charge

= 10 hr^-1

= 60/12 = 5 hr^-1

= 2

P_o = 1/(1+2+4+8+16) = 0.032

P₄ = ⁴ P_o = 16/31 = 0.516

Effective arrival (\')=*(1-P₄) = 10*(1-0.516) = 4.84 hr^-1

Money earned = 4.84*11 = $ 53.22

For second barber with $5 charge

=10 hr^-1

= 60/6 = 10 hr^-1

= 1

P_o = 1/(1+1+1+1+1) = 0.2

P₄ = 0.2

Effective arrival (\')=*(1-P₄) = 10*(1-0.2) = 8 hr^-1

Money earned = 8*5 = $ 40

So first barber will erarn more money than second one.