Calls arrive at Lynn Ann Fish's hotel switchboard at a rate of 2 per minute. The average time to handle each is 20 seconds. There is only one switchboard operator at the current time. The Poisson and negative exponential distributions appear to be relevant in this situation.

What is the probability that the operator is busy?

There is only one copying machine in the student lounge of the business school. Students arrive at the rate of $\lambda=$ 40 per hour (according to a Poisson distribution). Copying takes an average of 40 seconds, or $\mu=90$ per hour (according to a negative exponential distribution). Compute the following:

The average number of students in the system.

There is only one copying machine in the student lounge of the business school. Students arrive at the rate of $\lambda=$ 40 per hour (according to a Poisson distribution). Copying takes an average of 40 seconds, or $\mu=90$ per hour (according to a negative exponential distribution). Compute the following:

The average length of the queue.

There is only one copying machine in the student lounge of the business school. Students arrive at the rate of $\lambda=$ 40 per hour (according to a Poisson distribution). Copying takes an average of 40 seconds, or $\mu=90$ per hour (according to a negative exponential distribution). Compute the following:

The average time in the system.

Virginia's Ron McPherson Electronics Corporation retains a service crew to repair machine breakdowns that occur on average $\lambda=3$ per 8-hour workday (approximately Poisson in nature). The crew can service an average of $\mu=8$ machines per workday, with a repair time distribution that resembles the negative exponential distribution.

How many machines are waiting to be serviced at any given time?

Calls arriving at a switchboard follow a Poisson process with parameter $\lambda = 5.2$ per minute. $\mathbf{(a)}$ What is the expected waiting time between the arrivals of two calls? $\mathbf{(b)}$ What is the probability that the waiting time between the arrivals of two calls is less than 10 seconds? $\mathbf{(c)}$ What is the distribution of the time taken for ten calls to arrive at the switchboard? $\mathbf{(d)}$ What is the expectation of the time taken for ten calls to arrive at the switchboard? $\mathbf{(e)}$ What is the probability that more than five calls arrive at the switchboard during a 1-minute period?

An average of 120 students arrive each hour (interarrival times are exponential) at State College’s Registrar’s Office to change their course registrations. To complete this process, a person must pass through three stations. Each station consists of a single server. Service times at each station are exponential, with the following mean times: station 1, 20 seconds; station 2, 15 seconds; station 3, 12 seconds. On the average, how many students will be present in the registrar’s office for changing courses?

Suppose the number of phone calls arriving at a switchboard per hour is Poisson distributed with mean 7 calls per hour. Find the probability that no phone calls arrive during a certain hour.

The time between the arrivals of vehicles at a particular intersection follows an exponential probability distribution with a mean of 12 seconds.

Sketch this exponential probability distribution.

Show how the average time a customer must wait for a teller depends on the number of servers.

What should companies do to develop talented women to take top management positions? Provide support for your recommend

A telephone call arrived at a switchboard at random within a one-minute interval. The switch board was fully busy for 15 seconds into this one-minute period. What is the probability that the call arrived when the switchboard was not fully busy?

The time between arrivals of vehicles at a particular intersection follows an exponential probability distribution with a mean of 12 seconds.

What is the probability that the arrival time between vehicles is 12 seconds or less?

Renuka Jain's Car Wash takes a constant time of $4.5$ minutes in its automated car wash cycle. Autos arrive following a Poisson distribution at the rate of 10 per hour. Renuka wants to know:

The average waiting time in line.