6. The Security & Trust Bank employs 4 tellers to serve its customers. Customers arrive according to a Poisson process at a mean rate of 2 per minute. However, business is growing and management projects that the mean arrival rate will be 3 per minute a year from now. The transaction time between the teller and customer has an exponential distribution with a mean of 1 minute. Management has established the following guidelines for a satisfactory level of service to customers. The average number of customers waiting in line to begin service should not exceed 1. At least 90 percent of the time, the number of customers waiting in line should not exceed 3. a)Use the Excel sheet to determine how many tellers will be needed a year from now to completely satisfy these guidelines.

new shopping mall is considering setting up an information desk manned by one employee. Based upon information obtained from similar information desks, it is believed that people will arrive at the desk at a rate of 20 per hour. It takes an average of 2 minutes to answer a question. It is assumed that the arrivals follow a Poisson distribution and answer times are exponentially distributed. Find the probability that the employee is idle. Find the proportion of the time that the employee is busy. Find the average number of people receiving and waiting to receive some information. Find the average number of people waiting in line to get some information. Find the average time a person seeking information spends in the system. Find the expected time a person spends just waiting in line to have a question answered (time in the queue).

A cafeteria serving line has a coffee urn from which customers serve themselves. Arrivals at the urn 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 average, at the coffee urn?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?d. What is the probability that three or more people are in the cafeteria?e. If the cafeteria installs an automatic vendor that dispenses a cup of coffee at a constanttime of 15 seconds, how does this change your answers to (a) and (b)?

Customers arrive at Best-Bank-in-Town’s (BBT) sole ATM location at a rate of 21 customers per hour. It is assumed that the arrival process is random. BBT recently hired an intern who estimated that the average service time is 2 minutes per customer with a standard deviation of 1.2 minutes. [Assume that the arrival rate is the same throughout the day without peak and off-peak considerations and that there is only one ATM machine at that location] 1. Calculate average waiting time for ATM user at the BBT location 2. Calculate average length of queue at the ATM location (i.e. average numbers of customers waiting for service)? 3. What is the probability that an arriving customer has no waiting time to use the ATM? 4. BBT wants to cut the average waiting time in half without necessarily adding another ATM machine. They are in discussions with their ATM software provider who has told them that their new software is very consistent with information retrieval and can greatly minimize the…

A tax consulting firm has 5 counters in its office to receive people who have problems concerning their income, wealth and sales taxes. On the average 75 persons arrive in an 8hour day. Each tax adviser spends 25 minutes on an average on an arrival. If the arrivals are Poissonly distributed and service times are according to exponential distribution, find The probability of there being no customer in the system              The average number of customers in the system                                       iii.          Average number of customers waiting to be served                                  iv.   Average time a customer spends in the system                                        v.         Average waiting time for a customer

Suppose that arrivals to a hairdresser follow a Poisson process with mean 12 customers per hour. Calculate the probabilities of (a) no customers in 20 minutes; (b) exactlyone customer in 5 minutes; (c) exactly 12 customers in a hour; and (d) fewer than threecustomers in 10 minutes.

At a one man barber shop, customers arrive according to poison distribution with a mean arrival rate of 5 per hour and hair cutting time was exponentially distributed with an average hair cutting time was exponentially distributed with an average hair cut taking 19 minutes. It is assumed that because of excellent reputation, customers were always willing to wait. Calculate the following a. Average number of customers in the shop and average numbers waiting for a haircut b .Percentage of time arrival can walk in right without having to wait c. The percentage of customers who have to wait before getting into the barber’s chair

What is the answer in this number 2?

Doña Blanca plans to open a small car wash and must decide how much space to allocate to waiting cars. She estimates that customers will arrive randomly (Poisson process) at an average rate of 1 every 4 minutes, unless the waiting area is full, in which case the arriving customers will take their car elsewhere. The total time attributable to washing a car has exponential distribution with mean 3 minutes. Compare the fraction of potential customers lost due to lack of waiting space if a) 2 and b) 4 spaces (in addition to the wash location) are provided.