A security firm proposed that two-way cameras be installed at some room openings. Each camera has the ability to monitor the two rooms between which the camera is located. For example, if a camera were located at opening number 4, rooms 1 and 4 would be covered; if a camera were located at opening 11, rooms 7 and 8 would be covered; and so on. Management decided not to locate a camera system at the entrance to the display rooms. The objective is to provide security coverage for all eight rooms using the minimum number of two-way cameras. (a) Formulate a 0-1 integer linear programming model that will enable Bayside's management to determine the locations for the camera systems. (Let x, be the 0-1 which is 1 if a camera is installed at opening i, and 0 otherwise, for i = 1, 2, ..., 13.) Min s.t. Room 1 Room 2 Room 3 Room 4 Room 5 Room 6 Room 7 Room 8 x = 0, 1, for i = 1, 2, .... 13 (b) Solve the model formulated in part (a) to determine how many two-way cameras to purchase and where they should be located. The gallery should install cameras with (X1, X2 X3 X4 X5 X6 X, X, X, X10' x 11' X 12'x13): The Bayside Art Gallery is considering installing a video camera security system to reduce its insurance premiums. A diagram of the eight display rooms that Bayside uses for exhibitions is shown in the figure below; the openings between the rooms are numbered 1 through 13. Entrance Room 1 Room 2 Room 3 Room Room 4 4 7 7 Room 8 5 10 11 12 Room 13 Room 6 8

The Rockwell Electronics Corporation retains a service crew to repair machine breakdown that occur on an average of λ = 3 per day (approximately Poisson in nature). The crew service an average of μ = 8 machines per day, with a repair time distribution that resemble the exponential distribution. a. What is the probability that more than one machine is in the system? Probability that more than two are broken and waiting to be repaired or being serviced? More than three? More than four?

NewTel is a telephone company with a policy of filling positions internally through promotions, rather than hiring from outside. Until recently, the company had a strong engineering focus and tended to promote people to senior executive positions from the engineering areas. Consequently, almost all of the company's 14 senior executives joined the company over 20 years ago as junior engineers. There is increasing pressure on NewTel to become more market and service oriented. As a result, four people were hired last year from firms to fill new senior executive positions in marketing and service departments. The external hires were necessary because current employees were not sufficiently qualified. Now, there are signs of tension among senior executives, particularly during budget deliberations where there is limited discretionary spending on new corporate activities. The four new hires have been thwarted in their attempts to have the company invest more in marketing and customer…

Chapter 5. (M/M/s/K Model). Leia is responsible for 13 data units in the IT offices in a small business called DataWish. Each data unit runs on an average for 3 weeks and requires an average of ½ or 0.5 weeks to get reset by Leia before it can start up again. If we assign the Poisson distribution to the “arrival” of each data unit for service and an exponential distribution to the actual service provided, answer the following questions: a. The arrival rate would be units per week and the service rate would be units per week. b. Calculate the following operating characteristics for the M/M/s/K system: Utilization Factor (ρ)   % Performance Measures     P0 (probability that sytem is empty)   % L (average units in system)   units Lq (average units waiting in queue)   units W (average time in the system)   weeks Wq (average time in the queue)   weeks c. If we assign a cost of waiting as $400 per week per data unit and $800 per…