Shoppers arrive at a retail store at an average of 10 per minute (Poisson) where the...

Question

Question

Shoppers arrive at a retail store at an average of 10 per minute (Poisson) where the service rate is almost 207 per hour (Poisson).
1. What is the average number of shoppers in the system with 3 cashiers? (10 pts)

What is the minimum number of cashiers needed to keep the average time in the system under three minutes? (10 pts)

Answer 1

Answer #1

average arrival rate lambda = 10x60 =600 per hour

average service rate mu = 207 per hour

rho = 600/207 = 2.898

Probability that there is not customer in the system

P(0) = 1/ [ 1+rho+(rho)²/2 + (rho)³ xmu / 2 x( 3xmu-lambda)]

= 1/ [ 1+2.898+4.199+119.95] =0.0078

Lq = 0.0078 x lambda x mu x (rho)³ / 2 x ( 3xmu-lambda)²

= 0.0078 x600x207 x24.338 / 2 x 441

= 26.73

Waiting time in line Wq= Lq / lambda= 26.73 / 600 =0.04455 hours = 2.67 min.

Time in system = Wq +1/mu = 0.04455 +1/207 =0.04938 hours

= 2.962 min

Hence 3 persons can keep the time below 3 min,

Answer 2

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