NewCo Design is London’s eco & socially responsible floral design studio, and widely recognized as London's Best Florist. They are known for innovative and elegant floral arrangements in a wide range of styles. Importantly, it is a floral studio and not a retail store. It is closed to the public and orders must be made either through their website or by phone. They offer same-day delivery for orders placed before 11am. The average price of an arrangement, including delivery, is $91 plus tax. The gross margin after cost of flowers sold is 60%. They pay more than the living wage for London ($25 per hour for their floral arrangers and delivery people). They also source their flowers ethically and limit use of plastics in their arrangements.

They own a fleet of eight electric vans that they use to deliver the flowers throughout the city. Starting at 8AM the van drivers depart with 10 to 20 orders, typically taking 2-3 hours to make their deliveries and return to reload for the next delivery tour. To ensure that the customers enjoy their flowers as much as possible, the last delivery van departs at 3pm.

Orders that arrive after 11am, if completed by before 3pm, may be sent out the same day, but no promise is made to do so. Orders that are completed after 3pm are held in their large walk-in refrigerator and sent out the next day.

A big challenge for NewCo is the variability of orders and staffing to meet the orders. A simple flower arrangement in a vase might take 10 minutes to put together, while an extensive piece might take 30 minutes or more. The time to process an order is 20 minutes on average and is given by a gamma distribution with parameters (a=4, b=5); here the parameterization implies the mean = ab and the std. dev. = b√a.

Currently there are 8 flower arrangers who work at NewCo from 7am until 3pm, and 5 other flower arrangers who work from 11 am until 7pm. (For simplicity, assume they can eat lunch while working.) There are on average 20 orders placed through the website each hour from 7am until 7pm. The total number of orders placed each night between 7pm and 7am is on average 50 with a standard deviation of 7.07. (Assume there is no weekly cyclical pattern, only the daily one.)

Question:

NewCo gets nervous about making their service promise if all orders received by 11am have not been started in production by 2pm so that they can be loaded and shipped by 3pm. Suppose at 11am on Thursday there are 100 unprocessed orders. What is the likelihood that NewCo will not be able to start processing all 100 orders by 2pm? That is, should NewCo be nervous Thursday at 11am?