The Lets-Do-A-Deal new car sales dealership needs to review and plan its services. It's a very busy dealership with an easy to access location. A new car sale is made on average to every second walk in customer. Sales staff work mostly on a commission basis (there's also a small retainer). Sitting in the glass panelled office, the sales team keep an eye on the lot and take the walk-up customers in turn. But good sales staff need a guarantee of regular opportunities and don't wish to be waiting around too long for new targets (i.e., potential new car buyers). Helen, the dealership proprietor, suspects that some of the better salespeople have left in the last few years due to this issue. She's tried various sales team numbers (from two to seven) but has just three now as the team resists employing new team members arguing this reduces their own chance to make a sale. And anyway, the sales staff get edgy if they have to sit around for more that 15 minutes waiting for their next target. Helen feels conflicted by taking on more sales staff but her specified (and advertised) service standard is that all customers will be greeted on arrival - she feels that a failure here reduces customer trust and so results in a lost sale about half the time. Helen's wondering whether there may be a way to get a better handle on the pattern of potential new car buyers entering the lot. There's no obvious pattern of walk-ups over the day (Monday to Friday 9:00 a.m. to 6:00 p.m.) - it seems pretty random. But she needs to find the right balance between customer entrypatterns and sales staff numbers, this currently being three. The problem is that having too many salespersons means them waiting around too long for customers and too few means that potential unserved customers may be lost. She's paying Shirley, an acknowledged expert in services management, $5000 to advise her about this issue. Shirley documents the following data:

Sales persons spend an average of 20 minutes with each new customer (whether this leads to a sale or no sale).

For any given hour (based on a random sample), an average of 14 targets are available (potential customers that typically enter the new car lot and then wander around the cars). The target arrival pattern follows a Poisson distribution.

1. Following a customer walk-in, specify how long the next-in-line sales person should expect to wait (on average) for the next customer.

2. What is the probability (to two decimal places) of a sales person waiting 20 minutes for the next customer (show working)?

3. How many customers likely to be unmet on arrival each hour at the current sales staff level? Explain or justify your answer.

4. What advice should Shirley provide to Helen on the suitable number of persons to have in the sales team based on the data? State the implications of this advice given the current situation and also whether you consider that Shirley's' fee was justified (i.e., a good investment).

5. Write a statement to communicate the staff level situation (and any changes) to the current sales team. What alternative strategies that may be used to address the issues identified by Shirley?