The wheat harvesting season in the American Midwest is short

The wheat harvesting season in the American Midwest is short, and most farmers deliver their truckloads of wheat to a giant central storage bin within a two-week span. The central bin is owned cooperatively, and it is to every farmer’s advantage to make the unloading/storage process as efficient as possible. The cost of grain deterioration caused by unloading delays, the cost of truck rental, and idle driver time are significant concerns to the cooperative members. Although farmers have difficulty quantifying crop damage, it is easy to assign a waiting and unloading cost of $18 per hour per truck. During the harvest season the storage bin is open and operated 16 hours per day, 7 days per week, and is capable of uploading 35 trucks per hour according to an exponential distribution. During the hours the bin is open, full trucks arrive at an average rate of 30 per hour, following a Poisson pattern.

a) Find the average number of trucks in the unloading system.

b) Find the average time per truck in the system.

c) Find the utilization rate for the bin area.

d) Find the probability that there are more than three trucks in the system at any given time. e) Find the total daily cost to the farmers of having their trucks tied up in the unloading process.

Solution