Expert: Lynn Houston - 8/13/2008

Question
Juliet and Damien are equal partners in a fruit and vegetable business and have noticed that their busiest time is between 12 noon and 1pm each day. They want to investigate the number of staff needed to serve on an additional counter that is only open for the one-hour period to cater for the overflow of customers. This counter can have 1,2 or 3 different service points each with its own server, and customers would be served at the next available point.

At 12:00, there are already 5 customers who need to be served at this counter(5 customers are queueing). On average, a customer arrices every 2 minutes during this busy hour. The extra counter closes at 1:00, but anyone still in the queue at that time will be served. Server 1 serves continuously if there are people waiting. A new server (server 2 or 3) is added to the counter whenever the queue has 4 or more people waiting.

On average, during this one hour, most customers require 3 minutes service time, but every fourth customer requires 5 minutes of service time.

Formulate a table showing number of customers, customer arrical time, server (1, 2 or 3), start service time, finish service time, customer waiting time and server idle time.

Thanks for your help and i think this is a really hard question so any help would be appriciated.

Answer
I can't accurately post a table in this space, so I'll try and explain how to do the problem.  First, start with a spreadsheet and number 1-35 down the first column (with the header, customer).  Then next column is the arrival time.  At 12 noon, there were already 5 ppl waiting, so the arrival time for the first 5 ppl is 12.  From there, add 2 minutes for each person below.  The next column will be the server.  Since there are already 5 ppl in line, 2 servers will be working.  Alternate 1 + 2 for at least the 1st 5.  Next column is the start time, for customer 1 and 2, that will be 12 noon, server 1 will wait on customer 3 when she finishes with cust 1 at 12:03, server 2 will take cust 4 at the same time and server 1 will take customer 5 at 12:06.  Server 2 will take cust 6 at 12:06.  Now, because the 5th person takes 5 minutes, server one will finish with cust 5 at 12:11 while server 2 finished with cust 6 at 12:09, so the next cust, 7, will be served by server 2.  The start time of a customer, will be the start time of the servers previous cust + 3 or 5 depending if it's a 5th cust or not.  

The thing you need to look at as you go along with end times, is, how many ppl are now waiting in line while the others are being served.  At some pt, you will have to bring in server 3.  And, at some pt, the start time won't be the end time of the last customer, because a new customer hasn't arrived yet.  The start time will be when the next customer arrives.  once you have the times for all 35, you can use the table to determine cust wait time (time they were served, minus time they arrived) and server idle time (any time between the end time of one customer and the start time of the next one).

Hope this makes sense.