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# Arrival and Service Rates for check out lanes

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The problem is this:

In planning for a major redesign, Beth collected data at her store on several consecutive Saturday mornings. She noticed that customers arrived at the checkout at a rate of approximately 100 per hour. Fully 20 percent of the customers had 10 items or less. Those people took about 2 minutes to serve on average, while customers with more than 10 items took about 4 minutes to process. Beth expects service time to improve when universal price code readers are installed in the new design. Help Beth with her design for the system.
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I need verification or correction of my service rate adn arrival solutions. I came up with the solution below, but would like to know if I have the correct arrival and service rate for all three:

Therefore, if arrival rate is calculated per hour we need to convert the service rate to hours:
a. 20% of time, service rate = 60/2 = 30 customers/hr (for users with less than 10 items it takes 2 mins), thus per hour = 60/2= 30 / hr)

b. 80% of time, service rate = 60/4 = 15 customers/hr (for users with more than 10 items it takes 4 mins)

Avg Service Rate for both scenarios:
20% x 30 + 80% x 15 = 6+12 = 18/hr
Avg Service Rate for customers with less than 10 items
20% x 30hrs = 6/hr

Avg Service Rate for customers with items (more than) < 10
80% x 15hrs = 12/hr
I now need to:

Based on my calculations, what is the arrival and service rate for the express lane with 10 items or less? What is the arrival rate and service rate for the other customers with more than 10 items and what is the average arrival rate and service rate for all 100 arrivals if we do not calculate in the express lane?

Once I have this correct, I can then go ahead and use the Excel QM to calcuclate the answers below.I do not necessarily need you to do this part, I am more concerned with finding out if I have the correct arrival rate and service rates:

Average service time
&#955; calculation
Âµ calculation
Seven Servers
Eight Servers
1 Express Server
2 Express Servers
6 regular servers
7 regular servers

https://brainmass.com/math/probability/arrival-and-service-rates-for-check-out-lanes-152041

#### Solution Preview

You did calculate arrival and service rates correctly (see below for my work on the problem).

If you had one Express Lane (10 items of less) and one traditional lane (more than 10 items):
Express Lane: arrival rate = 20 customers/hr and service rate = 30 customers/hr.
Traditional Lane : arrival rate = 80 customers/hr and service rate = 15 customers/hr.

If you had only one traditional lane and no express lane (i.e. one lane for all customers):
arrival rate = 100 customers/hr and service rate = 18 customers/hr

My ...

#### Solution Summary

A step by step mathematical answer to the following question: In planning for a major redesign, Beth collected data at her store on several consecutive Saturday mornings. She noticed that customers arrived at the checkout at a rate of approximately 100 per hour. Fully 20 percent of the customers had 10 items or less. Those people took about 2 minutes to serve on average, while customers with more than 10 items took about 4 minutes to process. Beth expects service time to improve when universal price code readers are installed in the new design. Help Beth with her design for the system.

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