Question

Henry Philly needs to decide where to get a haircut. He has narrowed the choice down to two local hair salons - Large Hair Salon (LHS) and Small Hair Cutters (SHC). During busy periods, a new customer walks into LHS every 15 minutes (with a standard deviation of 15 minutes). At SHC, a customer walks in every hour (with a standard deviation of 1 hour). LHS has a staff of 4 barbers, while SHC has 1 barber. Typical service time at either salon lasts 30 minutes (with a standard deviation of 30 minutes). a. If Henry walks into LHS during a busy period, how long must he wait in line before he can see a barber? (Note: Only include the waiting time, not any service time.)

Answer 1

Henry's expected wait time at LHS during busy periods is theoretically minimal since the arrival rate seems to match the combined service rate of the barbers. However, the high variability in arrival and service times could result in actual wait times being longer under certain circumstances. The concept of diminishing marginal productivity also suggests that having more barbers doesn't always equate to shorter waits due to potential inefficiencies.

Step-by-step explanation:

To calculate Henry Philly's expected wait time at Large Hair Salon (LHS), we will consider the arrival rate of customers, the number of barbers, and the service time. The average customer arrival rate at LHS during busy periods is every 15 minutes, which means there are 4 customers arriving per hour. With 4 barbers available and a service time of 30 minutes, each barber can serve 2 customers per hour. In theory, this matches the arrival rate, indicating that Henry should not have to wait, assuming customers are evenly distributed.

However, the variability indicated by a standard deviation equal to the mean suggests great variability in arrival times, so Henry might experience a wait if many customers happen to arrive in quick succession. Without a queueing theory model that takes variability into account, we can't give a precise expected wait time. Therefore, the best we can do with the given information is to state that Henry could potentially walk right in if he arrives when a barber is free, or he might have to wait if multiple customers have already arrived.

In reality, variability in arrival and service times means there is no fixed wait time, but queueing theory can help estimate it if more detailed information is available. This explanation illustrates the significance of having multiple barbers to reduce wait times, similar to how a main waiting line at a post office or supermarkets A and B can reduce customer waiting time variation. Furthermore, understanding the principle of diminishing marginal productivity in businesses like a barber shop can indicate why merely increasing staff may not always lead to efficiency gains.