A firm has s single mechanist in a repair shop. He works eight hours a day, and on an average four machines break each day. It takes on the average one hour to repair the machine. Using Poisson exponential model, determine: - 1. The expected number of machines in the repair shop - 2. The expected number of machines in the shop on which the mechanist has not started to work - 3. The average down time (waiting for repairs or undergoing repairs) per machine - 4. The average time a machine waits for service -

Mumbai University > MECH > Sem 7 > Operations Research

Marks: 10 M

Year: May 2012

Mean arrival rate: $λ = 1/2 h$; Mean service rate: $μ = 1/1 h$

Traffic intensity (or utilization factor): $ρ = \dfracλ μ = \dfrac{1/2}{1} = \dfrac12$

1. The expected number of machines in the repair shop:

   $Ls = \dfrac{λ}{μ - λ} = \dfrac{1/2}{1 - 1/2} = 1$

2. The expected number of machines in the shop on which the mechanist has not started to work:

   $Lq = \dfrac{ρ^2}{1-ρ} = \dfrac{(1/2)^2}{1-1/2} = 0.5$

3. The average down time (waiting for repairs or undergoing repairs) per machine:

   $Ws = \dfrac{1}{μ - λ} = \dfrac{1}{1-1/2} = 2 hours$

4. The average time a machine waits for service:

   $Wq = \dfrac{ρ}{(μ - λ)} = \dfrac12 ×2 = 1 hour$