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Figuring Waiting Times

A-On average 2 customers arrive per hour at a Foto-Mat to process film. There is one clerk in attendance that on average spends 15 minutes per customer.

1. What is the average queue length and average number of customers in the system?
2. What is the average waiting time in queue and average time spent in the system?
3. What is the probability of having 2 or more customers waiting in queue?
4. If the clerk is paid $4 per hour and a customer's waiting cost in queue is considered $6 per hour. What is the total system cost per hour?
5. What would be the total system cost per hour, if a second clerk were added and a single queue were used?

B-White & Sons wholesale fruit distributions employ a single crew whose job is to unload fruit from farmer's trucks. Trucks arrive at the unloading dock at an average rate of 5 per hour Poisson distributed. The crew is able to unload a truck in approximately 10 minutes with exponential distribution.
1. On the average, how many trucks are waiting in the queue to be unloaded?
2. The management has received complaints that waiting trucks have blocked the alley to the business next door. If there is room for 2 trucks at the loading dock before the alley is blocked, how often will this problem arise?
3. What is the probability that an arriving truck will find space available at the unloading dock and not block the alley?

Solution Preview

A1) There is an average of 0.5 customers in the queue and 2 customers in the system per hour. A customer arrives and it takes 15 minutes to care for them. Since on average one customer arrives every 30 minutes - 15/30 minutes = .5 customers. If you define the system as standing in line and being cared for, then it is two an hour. If you define the system as only being cared for, then it would be the same as the first part, 0.5 customers per hour.
A2) There are none in queue, actually. If a customer arrives at the top of the hour and another arrives at the bottom of the hour, ...

Solution Summary

The author uses statistics to solve queue time problems.