A harbor has a single dock to unload the containers from the incoming ships. The arrival and service rate at the harbor follow a Poisson distribution. The arrival rate and service rate are 8 ships per week and 14 ships per week respectively. Determine: - 1. Utilization of dock - 2. Average number of ships waiting in the queue and system - 3. Average waiting time per ship in the queue and system -

Mean arrival rate: $λ = 8$/week; Mean service rate: $μ = 14$/week

1. Traffic intensity (or utilization factor): $ρ = \dfracλ μ =\dfrac{8}{14} = 0.5714$

2. Average number of ships waiting in the system:

   $Ls = \dfrac{λ}{μ - λ} = \dfrac{8}{14 - 8} = 1.33$

   Average number of ships waiting in the queue:

   $Lq = \dfrac{ρ^2}{1-ρ} = \dfrac{(8/14)^2}{1-8/14} = 0.7619$

3. Average waiting time per ship in the system:

   $Ws = \dfrac{1}{μ - λ} = \dfrac{1}{14-8} = 0.1667 \ \ weeks$

   Average waiting time per ship in the queue:

   $Wq = \dfrac{ρ}{(μ - λ)} = \dfrac{8/14}{14-8} = 0.0952 \ \ weeks$