Suppose that you are the executive manager for establishing a distribution system for Home-OfficeDelivery (HOD) Demijohn water product your company sells and delivers in Rolla, MO (HOD-Demijohn water products are 10gallons of bottled water delivered to homes/offices upon receiving a call for delivery). Your customer base will consist of 2,000 units (a unit is either a home or an office). Specifically, a unit will give a call to your company's call-center, and then, your company will deliver the bottle to the unit. Assume that each unit will request one bottle at a time. Based on past data from other cities similar to Rolla, where your company operates, you are estimating that each unit in Rolla will consume one HODDemijohn bottle in 10 work-days on average, where a work-day is 8-hours.

For deliveries, you are planning to purchase scooters and hire delivery personnel. Each delivery person will have his/her own scooter. That is, the number of scooters is equal to the number of delivery personnel. Again, based on past data from other cities similar to Rolla, you are estimating that a delivery person can make 30 deliveries per work-day on average.

Given the above information, you are considering two-options for establishing the distribution system:

- Option 1: Rent a single (large) store in central Rolla, where you make all deliveries from. In this case, all of the customer calls will be directed to this store, and the deliveries will be made in a first-come first-served fashion.

- Option 2: Rent two (relatively smaller) stores: one in east Rolla and one in west Rolla. In this case, east-Rolla store will serve the 1000 units located in the east of Rolla; and, west-Rolla store will serve the other 1000 units located in the west of Rolla. Here, each store will have its own callcenter and the customer calls will be directed to the associated store based on their location. For instance, a customer calling from east-Rolla will speak to the east-Rolla store.

Assume that both the inter-arrival and service time distributions are exponential; the stores have sufficient inventory of HOD-Demijohn water bottles (i.e., there will not any stock-out); and there is no queuing capacity in any case.

a) What is the minimum number of delivery personnel/scooters to keep within the single store to be rented in central Rolla if Option 1 is selected so that your system can be stable?

b) Briefly explain the queuing system or systems if Option 1 is selected as the distribution system Suppose that 10 delivery personnel are hired, which means that 10 scooters are purchased, for the single store. Define the arrival rate, service rate, and use Kendall's notation to define the queueing system or systems if Option 1 is selected.

c) What are the minimum number of delivery personnel/scooters to keep in the east-Rolla and west-Rolla stores to be rented in east of Rolla and west of Rolla, respectively, if Option 2 is selected so that your system can be stable?

d) Briefly explain the queuing system or systems if Option 2 is selected as the distribution system Suppose that 5 delivery personnel are hired, which means that 5 scooters are purchased, for each store. Define the arrival rate, service rate, and use Kendall's notation to define the queueing system or systems if Option 2 is selected.

While establishing your distribution network, you want to satisfy the following standards:

- Standard 1: On average, there should not be more than 2 units waiting for a delivery personnel/scooter to leave a store to deliver to the unit.
- Standard 2: At least 80 percent of the units should have their orders delivered with 30 minutes.
- Standard 3: For at least 70 percent of the units, delivery of their order should start as soon as the order is placed.

e) Suppose that Option 1 is selected and there are 10 delivery personnel with 10 scooters (one for each delivery person). Determine whether the standards 1-3 are satisfied or not. Show your work by showing the inequalities that you are checking for each standard.

f) Suppose that Option 2 is selected and there are 5 delivery personnel with 5 scooters (one for each delivery person) at each store. That is, east-Rolla store has 5 delivery personnel and 5 scooters and west-Rolla store has 5 delivery personnel and 5 scooters. Determine whether the standards 1-3 are satisfied or not. Show your work by showing the inequalities that you are checking for each standard.

g) Now suppose that Option 1 is selected and there are 10 delivery personnel with 10 scooters (one for each delivery person). What is probability of all scooters being parked in front of the single store located in central Rolla?