Math and small business?

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 arrival time, server (1, 2 or 3), start service time, finish service time, customer waiting time and server idle time.

I had made a detailed table for you, but cannot find a way to make it presentable in this answer box.

Here is some summary data for you.

You will have 35 customers in the hour. The last one arrives at 1:00 om, and serving her will take three minutes into overtime. Also customer number 34 arrives at 12:58, and you will need one minute overtime to serve him.

You have a tradeoff between best possible customer service (minimum wait) and slightly slowe customer service, you can save the expense of one server. The starting queue of five takes very little time to work off with three servers, who rhen take turns being idle. It takes somewhat longer for two servers to catch up, but near the end of the hour, they have an occasional idle minute, but a customer may still have to wait a minute.

Using three servers,

Server 1 -- 16 custs, work 50 minutes, idle 13 minutes
Server 2 -- 10 custs, work 42 minutes, idle 18 minutes
Server 3 -- 9 custs, work 29 minutes, idle 32 minutes

2 custs (c4 and c5) wait three minutes for service
1 cust (c7) wait two minutes for service
1 cust (c6) wait one minute for service
31 customers had no wait.

Using two servers,

Server 1 -- 18 custs, work 60 minutes, idle 2 minutes
Server 2 -- 17 custs, work 61 minutes, idle 2 minutes

2 custs (c5 and c6) wait six minutes for service
3 custs (c7, c8 and c10) wait five minutes for service
4 custs (c9, c11, c12 and c14) wait four minutes for service
6 custs (c3, c4, c13,c15, c16 and c18) wait three minutes for service
4 custs (c17, c19, c20 and c22) wait two minutes for service
6 custs (c21, c23, c24, c26,c30 and c34) wait one minute for service
10 custs had no wait (c1, c2, c25, c27, c28, c29, c31, c32, c33, c35)