Question

A local Walmart is adopting its new Corporate customer drive-through order pick-up program.

Based on its serving region and demographics (households with smart phones, internet access, on-line shopping habits, etc.), it estimates that once launched, customers will arrive randomly according to a Poisson process with a mean hourly arrival rate of 100 cars per hour.

This randomness will occur despite the fact that customers schedule a time window for their arrival.

During their order pick up, some customers trust that their order is complete and quickly drive off, while others check every line item on their order for quality and accuracy, may require an order correction (rework), and thus may take a longer amount of time before driving off. As such, an expontentially distributed service time for each customer with an average of 10 minutes is appropriate.

It has been decided to begin construction and allocate staffing to support 17 drive-through lanes to support the Customer Pick Up (CPU) program at this Walmart location.

What is the average amount of time a customer will be in the system (waiting time plus service time)?

	a.	48 minutes
	b.	26 minutes
	c.	17 minutes
	d.	37 minutes

Answer 1

Given:

Arrival rate ( l)                  100         per Hour

Service rate (m)                                6              per Hour

# of servers (c)                  17

Calculations:

Probability that system is empty is given as and the formula is,

for each m from 0 to 16 we get the values as shown in the table:

n	p(n)	p(n)
0	1.07106E-08	1.1E-08
1	1.78511E-07	1.8E-07
2	1.48759E-06	1.5E-06
3	8.26439E-06	8.3E-06
4	3.44349E-05	3.4E-05
5	0.000114783	0.00011
6	0.000318842	0.00032
7	0.000759148	0.00076
8	0.001581558	0.00158
9	0.002928811	0.00293
10	0.004881351	0.00488
11	0.007395987	0.0074
12	0.010272204	0.01027
13	0.013169492	0.01317
14	0.015677966	0.01568
15	0.017419963	0.01742
16	0.018145795	0.01815

By p(n) we mean the probability that there are n customers in the system.

Hence the value of : = 0.092 from the table above

The value of = 84709165.7038

Therefore: = 1.07 E -08

Now we have:

Average number of customers waiting in queue =

On substituting the obtained values we have: = 45.3644

Hence the average customer will be in system = W, given as

:

Upon calculation we get  W = 0.6203 hrs = 37.216 minutes = 37 minutes

Hence the required answer is 37 minutes.