Question

Jobs arrive at a facility at an average rate of $5$ in an $8$ hour shift. The arrival of the jobs follows Poisson distribution. The average service time of a job on the facility is $40$ minutes. The service time follows exponential distribution. Idle time (in hours) at the facility per shift will be _______.

• (A) $\frac{5}{7}$
• (B) $\frac{14}{3}$
• (C) $\frac{7}{5}$
• (D) $\frac{10}{3}$

Answer 1

The Correct Option is B

Explanation:
Given:Arrival rate $(\lambda )=\frac{5}{8}\frac{\text{ Jobs }}{\text{ Hour }}$Service time for one job is $40$ minTherefore, Service rate $(\mu )=\frac{3}{2}\frac{\text{ Jobs }}{\text{ Hour }}$Utilization factor $\rho =\frac{\lambda }{\mu }$Idle time $=1-\rho$$\lambda =\frac{5}{8}\frac{\text{ Jobs }}{\text{ Hour }}$ $\mu =\frac{3}{2}\frac{\text{ Jobs }}{\text{ Hour }}$ $\rho =\frac{\lambda }{\mu }=\frac{\frac{5}{8}}{\frac{3}{2}}=\frac{5}{12}$ Idle time $=1-\rho$$=1-\frac{5}{12}$$=\frac{7}{12}$ hourTherefore, Idle time for $8$ hour shift $=\frac{7}{12}\times 8$$=\frac{14}{3}$ hours.Hence, the correct option is (B).