Each morning during rush hour, 10,000 people want to travel from New Jersey to New York City. If a person takes the commuter train, the trip lasts 40 minutes. If x thousand people per morning drive to New York, it takes 20 + 5x minutes to make the trip. This problem illustrates a basic fact of life: If people make their decisions individu- ally, they will cause more congestion than is actu- ally necessary. a. Show that if people make their decisions indi- vidually, an average of 4000 people will travel by road from New Jersey to New York. Here you should assume that people will divide up between the trains and roads in a way that makes the aver- age travel time by road equal to the travel time by train. When this “equilibrium" occurs, nobody has an incentive to switch from the road to the train or vice versa. b. Show that the average travel time per person is minimized if 2000 people travel by road.
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- Exercise 1 A supermarket is trying to decide how many cash registers to keep open. Suppose an average of 18 customers arrive each hour, and the average checkout time for a customers is 4 minutes. Interarrival times and service times are exponential, and the system may be modeled as an M/M/S/GD//0 queuing system. It costs $20 per hour to operate a cash register, and a cost of 25 cents is assessed for each minute the customer spends in the cash register area. How many registers should the store open?Chapter 5. (M/M/s/K Model). Leia is responsible for 8 computers in the IT offices in a small business called DataWish. Each computer 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 computer 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 (p) % Performance Measures PO (probability that sytem is empty) L (average units in units system) La (average units waiting in queue) units W (average time in the weeks system) Wq (average time in the queue) weeks c. If we assign a cost of waiting as $500 per week per computer and $800 per week of hiring Leia, what is the Total Time Cost per week cost of servicing the…Chapter 5. (M/M/s/K Model). Leia is responsible for 8 computers in the IT offices in a small business called DataWish. Each computer 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 computer for service and an exponential distribution to the actual service provided, answer the following questions: a. The arrival rate would bel 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 (p) Performance Measures c. If we assign a cost of waiting as $500 per week per computer and $800 per week of hiring Leia, what is the Total Time Cost per week cost of servicing the machines? PO (probability that sytem is empty) % dollars L (average units in system) units d. If DataWish decides to hire another individual to help Leia, what would the new Total Time Cost per…
- Chapter 5. (M/M/s/K Model). Leia is responsible for 12 computers in the IT offices in a small business called DataWish. Each computer runs on an average for 4 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 computer 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 $750 per week per computer and $800 per week of…DEMU 2103, va assignmen QUESTION 3 Metropolitan General Hospital is a city-owned and operated public hospital. Its emergency. room is the largest and most prominent in the city. Approximately 70% of emergency cases in the city come or are sent to Metro General's emergency room. As a result, the emergency room is often crowded and the staff is overworked, causing concern among hospital administrators and city officials about the quality of service and health care the emergency room is able to provide. One of the key quality attributes administrators focus on is patient waiting time that is, the time between when a patient checks in and registers and when the patient first sees an appropriate medical staff member. Hospital administration wants to monitor patient waiting time using statistical process control charts. At different times of the day over a period of several days, patient waiting times were recorded at random with the following results: Sample 1 2 3 4 5 6 7 8 9 10 1 27 22 16…A manufacturer wants to design an open box with a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume? Picture (suggested, not required) Primary equation: Constraint (secondary) equation: Maximization formula Interval of optimization (domain) Use the conclusion of the extreme value theorem to answer question posed
- 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…A small grocery store has a single cashier lane where groceries line up to pay for their purchases. Interarrival time of customers is exponentially distributed with mean of 3 mins. Due to the variety in type and number of items purchased, transaction time is approximately exponentially distributed with mean 2.5 mins. a. Compute for the average number of customers in the cashier lane. b. Find the probability that there are at most 4 customers in the lane. c. What is the proportion of time that the cashier is idle? d. Find the average time a customer spends in the cashier, including queuing time and transaction time. e. Buying a POS machine will help reduce average transaction time to only 2 mins. (20% reduction) with the aid of a barcode reader - eliminating the need to type item codes manually. If implemented, find the percent reduction in the average time a customer spends in the cashier.To finance a vacation in 4 years, Elsie saves $410 at the beginning of every six months in an account paying interest at 13% compounded semi dash annually. (a) What will be the balance in her account when she takes the vacation? (b) How much of the balance will be interest? (c) If she waits an additional year to start her vacation, and continues to save the same amount of money, how much more money does she have to spend? Question content area bottom Part 1 a) The balance in her account will be $ enter your response here. (Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.) Part 2 b) The amount of interest will be $ enter your response here. (Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.) Part 3 c) She will have an extra $ enter your response here to spend on her vacation. (Round the final answer to the nearest cent as needed. Round all…
- Problem 11-19 (Algorithmic) All airplane passengers at the Lake City Regional Airport must pass through a security screening area before proceeding to the boarding area. The airport has three screening stations available, and the facility manager must decide how many to have open at any particular time. The service rate for processing passengers at each screening station is 3 passengers per minute. On Monday morning the arrival rate is 5.2 passengers per minute. Assume that processing times at each screening station follow an exponential distribution and that arrivals follow a Poisson distribution. When the security level is raised to high, the service rate for processing passengers is reduced to 2 passengers per minute at each screening station. Suppose the security level is raised to high on Monday morning. Note: Use P0 values from Table 11.4 to answer the questions below. The facility manager's goal is to limit the average number of passengers waiting in line to 10 or fewer. How…Please solve 2 questions True or falseChapter 5. (M/M/s/K Model). Leia is responsible for 12 dataunits in the IT offices in a small business called DataWish. Each dataunit 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 dataunit 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 (p) % Performance Measures PO (probability that sytem is empty) % L (average units in system) units Lq (average units waiting in queue) units W (average time in the weeks system) Wq (average time in the queue) weeks c. If we assign a cost of waiting as $400 per week per dataunit and $800 per week of hiring Leia, what is the Total Time Cost per week cost of servicing the…