Customers arrive at a video rental desk at the rate of 7 per minute(Poisson).Each server can handle 2.334 customers per minute(Poisson). There are 7 servers in the system. What is the probability of 10 customers in the system? O a. 0.009 O b. 0.05 O c. 0.963 O d. 0.002 O e. 0.005
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- Customers arrive at a video rental desk at the rate of 14 per minute(Poisson).Each server can handle 4.668 customers per minute(Poisson). If there are 6 servers, determine the probability of 5 or fewer customers in the system. a. 0.03 b. 0.025 c. 0.049 d. 0.901The Regency Hotel has enough space at its entrance for six taxicabs to line up, wait for guests,and then load passengers. Cabs arrive at the hotel every 8 minutes; if a taxi drives by the hoteland the line is full, it must drive on. Hotel guests require a taxi every 5 minutes, on average. Ittakes a cab driver an average of 3.5 minutes to load passengers and luggage and leave the hotel(exponentially distributed). What is the average time a cab must wait for a fare?Customers arrive at a video rental desk at the rate of 14 per minute(Poisson).Each server can handle 6.55 customers per minute(Poisson). If there are 3 servers, determine the average time it takes to rent a video tape. a. 0.09 minutes b. 1.266 minutes c. 0.146 minutes d. 0.243 minutes
- Buses arrive at the downtown bus stop and leave for themall stop. Past experience indicates that 20% of the time,the interval between buses is 20 minutes; 40% of the time,the interval is 40 minutes; and 40% of the time, the intervalis 2 hours. If I have just arrived at the downtown bus stop,how long, on the average, should I expect to wait for a bus?Each airline passenger and his or her luggage must bechecked to determine whether he or she is carrying weaponsonto the airplane. Suppose that at Gotham City Airport, anaverage of 10 passengers per minute arrive (interarrivaltimes are exponential). To check passengers for weapons,the airport must have a checkpoint consisting of a metal detector and baggage X-ray machine. Whenever a check-point is in operation, two employees are required. A checkpoint can check an average of 12 passengers perminute (the time to check a passenger is exponential). Underthe assumption that the airport has only one checkpoint,answer the following questions:a What is the probability that a passenger will have towait before being checked for weapons?b On the average, how many passengers are waiting in line to enter the checkpoint? 1082 CHAPTER 2 0 Queuing Theoryc On the average, how long will a passenger spend atthe checkpoint?The Rockwell Electronics Corporation retains a service crew to repair machine breakdowns that occur on an average of l = 3 per day (approximately Poisson in nature). The crew can service an average of m = 8 machines per day, with a repair time distribution that resembles the exponential distribution. (a) What is the utilization rate of this service system? (b) What is the average downtime for a machine that is broken? (c) How many machines are waiting to be serviced at any given time? (d) 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?
- Currently, Commercial Banks have 1 unit of ATM machines to meet customer needs. It is known that there is an average of 12+x customers in the system. Then it is also known that the average time spent by someone in the system is 10 minutes. Assume that the system follows M/M/1. The Commercial Bank operational manager is considering whether it is necessary to add 1 unit of ATM machine. What are your considerations? To strengthen the consideration, compare the queue length, queue length, and utilities before and after adding an ATM machine? Note: x denoted by the last number of my ID Number.Example, if my ID number is 2602296545, then the number used are “5”Determining the Number of ServersIn the service department of the Glenn-Mark Auto Agency, mechanics requiring parts for auto repair or service present their request forms at the parts department counter. The parts clerk fills a request while the mechanic waits. Mechanics arrive in a random (Poisson) fashion at the rate of 40 per hour, and a clerk can fill requests at the rate of 20 per hour (exponential).If the cost for a parts clerk is $30 per hour and the cost for a mechanic is $60 per hour, determine the optimum number of clerks to staff the counter. (Because of the high arrival rate, an infinite source may be assumed.)The arrival rate of patients at a hospital counter follows Poisson distribution with a mean of 25 per hour. The service rate of the treatment to patients also follows Poisson distribution with a mean of 45 per hour. What is the probability of having 0 patients in the system (P 0 )? What is the probability of having 10 patients in the system (P 10 )? What is the probability of having 15 patients in the system (P 15 )? Find Ls, Lq, Ws and Wq
- During the early morning hours, customers arrive at a branch post office at an average rate of 45 per hour (Poisson), while clerks can handle transactions in an average time (exponential) of 4 minutes each. Find: (a) The average number of customers waiting for service if 5 clerks are used. (b) The minimum number of clerks needed to keep the average waiting time in the system to under 5 minutes. (c) If clerk cost is $40 per hour and customer waiting time represents a “cost” of $30 per hour, how many clerks can be justified on a cost basis? Please show work.A university cafeteria line in the student center is a self-serve facility in which students select the food items they want, then form a single line to pay the cashier. Students arrive at a rate of about four per minute according to a Poisson distribution. The single cashier ringing up sales takes about 12 seconds per customer, following an exponential distribution. a. What is the probability that there are more than two students in the system? More than three students? More than four? b. What is the probability that the system is empty? c. How long will the average student have to wait before reaching the cashier? d. What is the expected number of students in the queue? e. What is the average number in the system? f. If a second cashier is added (who works at the same pace), how will the operating chareristics computed in parts (b), (c), (d), and (e) change? Assume that customers wait in a single line and go to the first available cashier.A cafeteria serving line has a coffee urn from which customers serve themselves. Arrivals at the um follow a Poisson distribution at the rate of three per minute. In serving themselves, customers take about 15 seconds, exponentially distributed. a. How many customers would you expect to see on the average at the coffee um? b. How long would you expect it to take to get a cup of coffee? c. What percentage of time is the urn being used?