Question

Question 2.

In this question you will be comparing two different queuing configurations to see which one is better in reducing delays.

NOTE: SHOW ALL YOUR WORK. USE 4 DECIMAL PLACES IN ALL CALCULATIONS.

Historical data at a retail store (e.g. Walmart) shows that the total average arrival rate of customers to checkout lanes (cashiers) at the store is 240 customers per hour during the peak hours. The arrivals can be modeled by a Poisson distribution. There are 20 cashiers working during peak hours. Each cashier can check out 14 customers/hr.

In Configuration 1, each cashier has a dedicated queue in front of him/her. This leads to 20 separate queues at the checkout area. In this configuration, assume that the arrivals of checkouts are equally distributed across the 20 queues: the average arrival rate to each queue = 240 customers per hour / 20 queues = 12 customers per hour per queue.

In Configuration 2, there is one serpentine queue which is served by all 20 cashiers. That is, all customers join the same queue for checkouts. Whenever a cashier becomes available, the customer in front of the queue gets served by that cashier. Note that the average arrival rate to the single checkout queue is 240 customers per hour in this configuration.

a) What is the average wait time in the system (in minutes) for a customer in Configuration 1[1] ? Use the Excel Queuing Models spreadsheet to determine the answer.

         

b) What is the average wait time in the system (in minutes) for a customer in Configuration 2? Use the Excel Queuing Models spreadsheet to determine the answer.

  

c) Why is the average time a customer spends in the system in Configuration 1 different than the average time a customer spends in the system in Configuration 2?

          

d)   ( What is the average time spent in the queue (in minutes) per customer in Configuration 1?

           Wq =

e) What is the average time spent in the queue (in minutes) per customer in Configuration 2?

           Wq =

f) What is the average number of customers waiting in the queue at any time in Configuration 1?

           Lq =

g) What is the average number of customers waiting in the queue at any time in Configuration 2?

           Lq =

h) Answer the remaining questions based on your work in parts (a) to (g) and based on the information provided in the following articles:

A Long Line for a Shorter Wait at the Supermarket: http://nyti.ms/xZ10Ae,

How to Pick the Fastest Lane at the Supermarket: goo.gl/1J1IjD .

What really drives you crazy about waiting in line (it actually isn’t the wait at all) http://wapo.st/1PSafCh?tid=ss_tw

i. Based on your answers to parts (a) to (g), which generally performs better based on queuing metrics: a serpentine line (one queue leading to several servers), or a dedicated queue in front of each server? Explain clearly why this configuration performs better.

                      ii. What are some potential reasons to use the other configuration, despite its poorer queuing system performance? Give a specific example of a company that has done so, along with their reasoning for it.

                    iii. State two psychological factors that influence customers’ experiences in queuing systems. For each psychological factor, provide a suggestion for what a company might do to address it and thereby improve customers’ experiences.

Answer 1

a) Config 1 is a M/M/1 queue model,

Arrival rate, ? = 12 per hr

Service rate, ? = 14 per hr

Average wait time in system (W) = 1/(?-?) = 1/(14-12) = 0.5 hr = 30 minutes

b) Config 2 is a M/M/20 (20 servers) queue model

Calculation of operating characteristics of queue is as under

Average wait time in system (W) = 0.0816 hr = 4.9 minutes

c) Wait time in system is different because in config, each server is dedicated to serve only 20 customers per hour, so there is likelihood that there are customers waiting in other queues while there are idle servers in other queues. This likelihood is eliminated in config 2, because there is single queue, so any of the idle server can serve a waiting customer.

d) Average time spent in queue in config 1 (Wq) = ?/(?*(?-?)) = 12/(14*(14-12)) = 0.429 hour = 25.7 minutes

e) Average time spent in queue in config 2 (Wq) = 0.0102 hr = 0.612 minutes (refer calculation table in part b