Neve Commercial Bank is the only bank in the town of York, Pennsylvania. On a typical Friday, an average of 10 customers per hour arrive at the bank to transact business. There is currently one teller at the bank, and the average time required to transact business is 4 minutes. It is assumed that service times may be described by the negative exponential distribution. If a single teller is used, find:

The average time in the system.

Neve Commercial Bank is the only bank in the town of York, Pennsylvania. On a typical Friday, an average of 10 customers per hour arrive at the bank to transact business. There is currently one teller at the bank, and the average time required to transact business is 4 minutes. It is assumed that service times may be described by the negative exponential distribution. If a single teller is used, find:

The average time in the line.

Neve Commercial Bank is the only bank in the town of York, Pennsylvania. On a typical Friday, an average of 10 customers per hour arrive at the bank to transact business. There is currently one teller at the bank, and the average time required to transact business is 4 minutes. It is assumed that service times may be described by the negative exponential distribution. If a single teller is used, find:

The average number in the line.

Janson’s Department Store in Stark, Ohio, maintains a successful catalog sales department in which a clerk takes orders by telephone. If the clerk is occupied on one line, incoming phone calls to the catalog department are answered automatically by a recording machine and asked to wait. As soon as the clerk is free, the party who has waited the longest is transferred and serviced first. Calls come in at a rate of about 12 per hour. The clerk can take an order in an average of 4 minutes. Calls tend to follow a Poisson distribution, and service times tend to be negative exponential.

The cost of the clerk is $\$ 10$ per hour, but because of lost goodwill and sales, Janson's loses about $\$ 25$ per hour of customer time spent waiting for the clerk to take an order.

Pamela Janson is considering adding a second clerk to take calls. The store's cost would be the same $\$10$ per hour. Should she hire another clerk? Explain your decision.

Janson’s Department Store in Stark, Ohio, maintains a successful catalog sales department in which a clerk takes orders by telephone. If the clerk is occupied on one line, incoming phone calls to the catalog department are answered automatically by a recording machine and asked to wait. As soon as the clerk is free, the party who has waited the longest is transferred and serviced first. Calls come in at a rate of about 12 per hour. The clerk can take an order in an average of 4 minutes. Calls tend to follow a Poisson distribution, and service times tend to be negative exponential.

The cost of the clerk is $\$ 10$ per hour, but because of lost goodwill and sales, Janson's loses about $\$ 25$ per hour of customer time spent waiting for the clerk to take an order.

What is the average number of customers waiting to place an order?

Customers arrive at Rich Dunn's Styling Shop at a rate of 3 per hour, distributed in a Poisson fashion. Rich's service times follow a negative exponential distribution, and Rich can complete an average of 5 haircuts per hour.

Find the average number of customers waiting for haircuts.

Customers arrive at Rich Dunn's Styling Shop at a rate of 3 per hour, distributed in a Poisson fashion. Rich's service times follow a negative exponential distribution, and Rich can complete an average of 5 haircuts per hour.

Find the percentage of time that Rich is busy.

An important application of regression analysis in accounting is in the estimation of cost. By collecting data on volume and cost and using the least squares method to develop an estimated regression equation relating volume and cost, an accountant can estimate the cost associated with a particular manufacturing volume. Consider the following sample of production volumes and total cost data for a manufacturing operation.

$$\begin{matrix} \text{Production Volume (units)} & \text{Total Cost (\$)}\\ \text{400} & \text{4000}\\ \text{450} & \text{5000}\\ \text{550} & \text{5400}\\ \text{600} & \text{5900}\\ \text{700} & \text{6400}\\ \text{750} & \text{7000}\\ \end{matrix}$$

The company’s production schedule shows 500 units must be produced next month. What is the estimated total cost for this operation?

Gainesville Cigar stocks Cuban cigars that have variable lead times because of the difficulty in importing the product: lead time is normally distributed with an average of 6 weeks and a standard deviation of 2 weeks. Demand is also a variable and normally distributed with a mean of 200 cigars per week and a standard deviation of 25 cigars. Explain what these two service levels mean. Which is preferable.

Gainesville Cigar stocks Cuban cigars that have variable lead times because of the difficulty in importing the product: lead time is normally distributed with an average of 6 weeks and a standard deviation of 2 weeks. Demand is also a variable and normally distributed with a mean of 200 cigars per week and a standard deviation of 25 cigars. For a 90% service level, what is the ROP?

fₙ is the nth Fibonacci number. Show that

$$f_{n+1}f_{n−1} − f_n^2 = (−1)^n$$

when n is a positive. integer.

verify that(0, 0)is a critical point, show that the system is locally linear, and discuss the type and stability of the critical point(0, 0)by examining the corresponding linear system. dx/dt=x−y2,dy/dt=x−2y+x2

Three departments milling (M), drilling (D), and sawing (S) are assigned to three work areas in Victor Berardis's machine shop in Vent, Ohio. The number of workpieces moved per day and the distances between the centers of the work areas, in feet, follow.

Pieces Moved Between Work Areas Each Day

Distances Between Centers of Work Areas (Departments) in Feet

It costs $\$2$ to move 1 work piece 1 foot. What is the cost?

AlwaysRain Irrigation, Inc. would like to determine capacity requirements for the next four years. Currently, two production lines are in place for making bronze and plastic sprinklers. Three types of sprinklers are available in both bronze and plastic: $90$ -degree nozzle sprinklers, $180$-degree nozzle sprinklers, and $360$-degree nozzle sprinklers.

$$\begin{array}{lcccc} &&\text { Yearly Demand }\\ \hline & \mathbf{1} \text { (in 000s) } & \mathbf{2} \text { (in 000s) } & \mathbf{3} \text { (in 000s) } & \mathbf{4} \text { (in 000s) } \\ \hline \text { Plastic 90 } & 32 & 44 & 55 & 56 \\ \text { Plastic } 180 & 15 & 16 & 17 & 18 \\ \text { Plastic 360 } & 50 & 55 & 64 & 67 \\ \text { Bronze } 90 & 7 & 8 & 9 & 10 \\ \text { Bronze } 180 & 3 & 4 & 5 & 6 \\ \text { Bronze 360 } & 11 & 12 & 15 & 18 \\ \hline \end{array}$$

Both production lines can produce all the different types of nozzles. The bronze machines needed for the bronze sprinklers require two operators and can produce up to $12,000$ sprinklers. The plastic injection molding machine needed for the plastic sprinklers requires four operators and can produce up to $200,000$ sprinklers. Three bronze machines and only one injection molding machine are available. What are the capacity requirements for the next four years? (Assume that there is no learning.)