At a catalog order center for gardening supplies, customers browse catalogs, loo

Question

At a catalog order center for gardening supplies, customers browse catalogs, look at display items, and fill out an order card for items that they wish to purchase. They then join a single queue and wait to be served by the next of the six clerks who becomes available (arrivals follow a Poisson process with an average arrival rate of 49 customers per hour.) Clerks will then take the customer’s order card, enter the order in the computer, retrieve the items from a small warehouse, and process payment. A process specialist has recently proposed that the six clerks work in pairs, with each pair serving an individual customer at once, primarily as a means of reducing errors in orders and improving customer contact. One of the two clerks in a pair will thus handle customer administration (processing payment, etc.) while the other clerk retrieves items from the warehouse. It has been found in a trial run that the time to serve a customer averages 2.84 minutes/customer (assume negative exponential distribution). What is the average amount of time from when a customer joins the queue until they are finished being served (in minutes, rounded to two decimal places)?

Explanation / Answer

Let X be number of customer in the queue. X follows a poisson distribution with mean = 49. So Average amount of time from when a customer joins the queue until they are finished being served = average number of customers in the queue * average processing time per customer = 49*2.4 = 117.6 minutes = 1 hr 57 min 36 seconds

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