# BB207 Management Science Sample Assignment

**Question 1 **

Five administration assistants use an office copier. The average time between arrivals for each assistant is 40 minutes, which is equivalent to an arrival rate of 1/5=0.025 arrival per minute. The mean time each assistant spends at the copier is 5 minutes, which is equivalent to a service rate of 1/5=0.20 per minute. Use the model with a finite calling population to determine the following:

- The probability that the copier is idle. (Answer: 4790)
- The average number of administration assistants in the waiting line. (Answer: 3110)
- The average number of administration assistants at the copier. (Answer: 832)
- The average time an assistant spends waiting for the copier. (Answer: 9846 mins)
- The average time an assistant spends at the copier. (Answer: 9846 mins)
- During an 8-hour day, how many minutes does an assistant spend at the copier? How much of this time is waiting time? (Answer: 8152 mins; 35.8152 mins)

**Question 2**

Schips Department Store operates a fleet of 10 trucks. The trucks arrive at random times throughout the day at the store’s truck dock to be loaded with new deliveries or to have incoming shipments from the regional warehouse unloaded. Each truck returns to the truck dock for service two times per 8-hour day. Thus the arrival rate per truck is 0.25 trucks per hour. The service rate is 4 trucks per hour. Using the Poisson arrivals and exponential service times model with a finite calling population of 10 trucks, determine the following operating characteristics:

- The probability that no trucks are at the truck dock. (Answer: 44056)
- The average number of trucks waiting for loading/unloading. (Answer: 48952)
- The average number of trucks in the truck dock area. (Answer: 04896)
- The average waiting time before loading/unloading begins. (Answer: 0.21875)
- The average waiting time in the system. (Answer: 46875)
- What is the hourly cost of operation if the cost is $50 per hour for each truck and $30 per hour for the truck dock? (Answer: $82.45)

**Question 3**

A large insurance company maintains a central computing system that contains a variety of information about customer accounts. Insurance agents in a six-state area use telephone lines to access the customer information database. Currently, the company’s central computer system allows three users to access the central computer simultaneously. Agents who attempt to use the system when it is full are denied access; now waiting is allowed. Management realizes that with its expanding business, more requests will be made to the central information system. Being denied access to the system is inefficient as well as annoying for agents. Access requests follow a Poisson probability distribution with a mean of 42 calls per hour. The service rate per line is 20 calls per hour.

- What is the probability that 0, 1, 2 and 3 access lines will be in use? (Answer: P
_{0}=0.1460; P_{1}=0.3066; P_{2}=0.3220; P_{3}=0.2254) ) - What is the probability that an agent will be denied access to the system?(Answer: 0.2254)
- What is the average number of access lines in use? (Answer: 62666)
- In planning for the future, management wants to be able to handle calls per hour; in addition, the probability that an agent will be denied access to the system should be no greater than the value computed in part (b). How many access lines should this system have? (Answer: 4 lines)

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