QUESTION THREE

1. Auto vehicles arrive at a petrol pump, having one petrol unit, in Poisson fashion with an average of 10 units per hour. The service time is distributed exponentially with a mean of 3 minutes. Find the following:
   1. Average number of units in the system
   2. Average waiting time
   3. Average length of queue
   4. Probability that a customer arriving at the pump will have to wait The utilization factor for the pump unit
   5. Probability that the number of customers in the system is 2.

1. A factory is to produce two products P₁ and P₂. The products require machining on two critical machines M₁ and M_2. Product P₁ requires 5 hours on machine M₁ and 3 hours on machine M_2. Product P₂ requires 4 hours on machine M₁ and 6 hours on machine M_2. Machine M₁ is available for 120 hours per week during regular working hours and 50 hours on overtime. Weekly machine hours on M₂ are limited to 150 hours on regular working hours and 40 hours on overtime. Product P₁ earns a unit of profit of K8 if produced on regular time, K6 if produced on overtime on one machine and K4 if produced on overtime on both machines. Product P₂ earns a unit of profit K10 if produced on regular time, K8 if produced on overtime on both machines.
   1. Formulate an L.P model for designing an optimum production schedule for maximizing the profit.
   2. Solve the problem by simplex method