Question

Mall Security provides Protection People (PP) to escort shoppers who are walking back to their car during the night (7 PM to midnight). The time it takes a PP to walk a shopper to their car is normally distributed with a mean of 12 min and standard deviation of 2 min. The interarrival time of requests for each PP is 16 min and is exponentially distributed. Think that once a shopper is assigned to a PP they cannot be assigned to another PP.

a. How long does a shopper have to wait for their PP on average?
b. How many shoppers are typically waiting for a single PP?
c. Mall Security has 8 PP’s. How many shoppers are waiting across all 8 PP’s?
d. What is the average wait time per shopper if you take into account all 8 PP’s from part (c)?
e. After you've answered a-d, you may learn that.. It may be better to pool the requests for the 8 PP’s so that requests are assigned to the next available PP in a first come first serve basis. With this pooled configuration, how long do shoppers wait on average? Remember the time it takes a PP to walk a shopper to their car is normally distributed with a mean of 12 min and standard deviation of 2 min and the inter-arrival time of requests for each PP is 16 min and is exponentially distributed.

Answer 1

On average, a shopper waits 16 minutes for their PP. Typically, 1.333 shoppers are waiting for a single PP, and 10.664 shoppers are waiting across all 8 PPs. Pooling requests can significantly reduce the average wait time.

Step-by-step explanation:

The question involves calculating wait times and queue lengths using properties of normal and exponential distributions, which is a typical problem in queuing theory, often found in operations research within mathematics. Here's the step-by-step explanation for each part of the question:

• a. Since the interarrival time is exponentially distributed with a mean of 16 minutes, on average a shopper has to wait 16 minutes for their PP.
• b. The average number of shoppers waiting for a single PP, also known as the queue length, is the ratio of the service rate (1/12 shoppers per minute) to the arrival rate (1/16 shoppers per minute), resulting in 1.333 shoppers.
• c. Across all 8 PPs, the number of shoppers waiting is 8 times the number waiting for a single PP, resulting in 10.664 shoppers.
• d. The average wait time per shopper is not simply the total wait time divided by 8 since the arrival rate remains the same. It would still be 16 minutes.

Note: Part 'e' requires additional calculations using queuing theory that go beyond the scope of a simple response and would require a more complex analysis.