A Packages arrive at a warehouse that has a single reception point with a Poisson distribution mean of once every 12 minutes. It takes on the exponentially distributed average of 9-minutes to process each package. (a) What is the average wait time of a package in the queue? (b) On the average, how many packages are in the queue at any given time? (c) On the average, how many packages are in the system at any given tim
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A Packages arrive at a warehouse that has a single reception point with a Poisson
distribution mean of once every 12 minutes. It takes on the exponentially distributed
average of 9-minutes to process each package.
(a) What is the average wait time of a package in the queue?
(b) On the average, how many packages are in the queue at any given time?
(c) On the average, how many packages are in the system at any given time?
(d) On the average, what is the wait time of packages in the system?
(e) What percent of the time is the server idle?
(f) What is the probability that there are exactly 7 packages in the system at any given
point in time>
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- A small bank has two tellers, one for deposis and one for withdrawals. The service time for each teller is exponentially distributed, with a mean of 1 min. Customers arriving at the bank according to a poisson process, with mean rate 40 per hour; it is assumed that depositors and withdrawers constitute separate poisson processes, each with mean rate 20 per hour , and that no customer is both a depositor and a withdrawer. The bank is thinking of changing the current arrangement to allow each teller to handle both deposits and withdrawals. The bank would expect that each teller's mean service time would increase to 1.2 min, but it hopes that the new arrangement would prevent long lines in front of one teller while the other teller is idle, a situation that occurs from time to time under the current setup. Analyze the two arrangements with respect to the average idle time of teller and the expected number of customers in the bank at any given time.Ali Baba's Car Wash Service Centre is open 6 days a week, but its busiest day is always on Sunday. From the previous data, Ali Baba estimates that dirty cars arrive at the rate of one every two minutes, One car at a time is cleaned in this example of a single-channel waiting line. Assuming Poisson arrivals and exponential service times, find the following: iii) Compute the average time that a car spends in the system.A computer system queues up batch jobs and processes them on an FCFS basis. Between 2 and 5 P.M., jobs arrive at an average rate of 30 per hour and require an average of 1.2 minutes of computer time. Assume the arrival process is Poisson and the processing times are exponentially distributed.a. What is the expected number of jobs in the system and in the queue in the steady state?b. What are the expected flow time and the time in the queue in the steady state?c. What is the probability that the system is empty?d. What is the probability that the queue is empty?e. What is the probability that the flow time of a job exceeds 10 minutes?
- Give the resulting rate when two independent Poisson processes with rates λ1=2.546 and λ2=3.326 are merged. What is the exact rate?The manager of a market can hire either Mary or Alice. Mary, who gives you service at an exponential rate 20 customers per hour, can be hired at a rate of $3 per hour. Alice, who gives service at an exponential rate of 30 customers per hour, can hired at a rate of $C per hour. The manager estimates that, on the average, each customer’s time is worth $1 per hour and should be accounted for in the model. Assume customers arrive at a Poisson rate of 10 per hour. a) What is the average cost per hour if Mary is hired? If Alice is hired? b) Find C if the average cost per hour is the same for Mary and Alice.Suppose the customers arrive at a Poisson rate of on eper every 12 minutes, and that the service time is exponential at a rate of one service per 8 minutes. What are the average number of customers in the system(L) and the average time a customer spends in the system(W)?
- Suppose the customers arrive at a Poisson rate of on eper every 12 minutes, and that the service time is exponential at a rate of one service per 8 minutes. What are the average number of customers in the system(L) and the average time a customer spends in the system(W)? Now suppose that the arrival rate increases 20 percent.What is the corresponding change in L and W?Ali Baba‘s Car Wash Service Centre is open 6 days a week, but its busiest day is always on Sunday. From the previous data, Ali Baba estimates that dirty cars arrive at the rate of one every two minutes. One car at a time is cleaned in this example of a single-channel waiting line. Assuming Poisson arrivals and exponential service times, find the following: a) Compute the average time that a car spends in the system.The arrival of customers to a small local food restaurant may be modeled by a Poisson process with rate of 1 per 15 min period. Assume customer will only arrive if there is at least one customer present. Each customer on average stays for a time exponentially distributed with mean 20 minutes. This is modeled as a birth-death process. (i) Compute the probability that the capacity of 20 will be reached if we start with 5 customers.
- A production plant has three machines and the two products move about among these three machines. Upon completion of the process at machine i, it moves to the next available machine out of the two remaining machines. Therefore, there are always two busy machines. If the service times of these machines follow an exponential distribution with a rate of μ1=2/min, μ2=1 /min, μ3=4 /min, respectively, what proportion of time machine 1 will be idle?The lifetime of a bulb is modeled as a Poisson variable. You have two bulbs types A and B with expected lifetime 0.25 years and 0.5 years, respectively. When a bulb’s life ends, it stops working. You start with new bulb of type A at the start of the year. When it stops working, you replace it with a bulb of type B. When it breaks, you replace with a type A bulb, then a type B bulb, and so on. 1. Find the expected total illumination time (in years), given you do exactly 3 bulb replacements. 2. Your replacements are now probabilistic. If your current bulb breaks, you replace it with a bulb of type A with probability p, and with type B with probability (1 – p). Find the expected total illumination time (in years), given you do exactly nn bulb replacements, and start with bulb of type A. Answer for part 2 exists in closed form in terms of n and p.The average demand on a factory store for a certain electric motor is 8 per week. When the storeman places an order for these motors, delivery takes one week. If the demand for motors has a Poisson distribution, how low can the storeman allow his stock to fall before ordering a new supply if he wants to be at least 95% sure of meeting all requirements while waiting for his new supply to arrive?