Question

1.) Suppose jobs arrive at a single-machine workstation at a rate of 20 per hour and the average process time is 2.5 minutes. a.) (2 points) What is the utilization of the machine? b.) Suppose that inter arrival and process times are exponential, i.) (2 points) What is the average time a job spends at the station (i.e. waiting plus process time)?

Answer 1

Answer:

a) ρ = 0.8333

b)W=0.249 hr

W= 14.94 min

Explanation:

Job arrive in rate = λ

λ =  20 per hour

Service rate = μ

μ = 60 /2.5  per hour

μ  =24 per hour

a)

Utilization of machine ρ

${\rho}=\dfrac{\lambda }{\mu }$

By putting the values

${\rho}=\dfrac{\lambda }{\mu }$

${\rho}=\dfrac{20 }{24}$

ρ = 0.8333

b)

Time spend at station W

$W=\dfrac{L_q}{\lambda }+\dfrac{1}{\mu }$

$L_q=\dfrac{\rho^2}{1-\rho}$

$L_q=\dfrac{0.8333^2}{1-0.833}$

Lq=4.15

Now by putting the values

$W=\dfrac{4.15}{20 }+\dfrac{1}{24 }$

W=0.249 hr

W= 14.94 min