Past record indicate that each of the five speed page printers at the US Department of Commerce in Washington DC needs repair after about 20 hours of use. Breakdown have been determined to be Poisson distributed. The one technician on duty can service a printer in an average of 2 hours, following an exponential distribution. If the printer downtime cost \( \$ 120 \) per hour and technician is paid \( \$ 25 \) per hour. Determine the a. Probability that the system is empty b. Average length of the queue c. Average number of customers (units)in the system d. Average waiting time in the queue e. Average time in the system f. Probability of 3 units in the system g. Determine the total waiting and service cost
Added by Retchin B.
Close
Step 1
The printers need repair after about 20 hours of use. Since there are 5 printers, the arrival rate is 5/20 = 0.25 repairs per hour. The technician can service a printer in an average of 2 hours, so the service rate is 1/2 = 0.5 repairs per hour. a. The Show more…
Show all steps
Your feedback will help us improve your experience
Rukhmani Jain and 64 other Discrete Mathematics educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Recommended Videos
Customers arrive at a two-server system at a Poisson rate $\lambda$. An arrival finding the system empty is equally likely to enter service with either server. An arrival finding one customer in the system will enter service with the idle server. An arrival finding two others in the system will wait in line for the first free server. An arrival finding three in the system will not enter. All service times are exponential with rate $\mu_{\text {, }}$ and once a customer is served (by either server), he departs the system. (a) Define the states. (b) Find the long-run probabilities. (c) Suppose a customer arrives and finds two others in the system. What is the expected time he spends in the system? (d) What proportion of customers enter the system? (c) What is the average time an entering customer spends in the system?
Croup C.
A computer center is equipped with four identical mainframe computers. The number of users at any time is 25. Each user is capable of submitting a job from a terminal every 15 minutes, on the average, but the actual time between submissions is exponential. Arriving jobs will automatically go to the first available computer. The execution time per submission is exponential with mean 2 minutes. Compute the following: (a) The probability that a job is not executed immediately upon submission. (b) The average time until the output of a job is returned to the user. (c) The average number of jobs awaiting execution. (d) The percentage of time the entire computer center is idle. (e) The average number of idle computers.
Aarti K.
3. Customers arrive at a bank according to a Poisson process with rate of one customer every two minutes. Find the probability that the fifth customer arrives at the bank within 10 minutes after the bank opens. (A) .560 (B) .440 (C) .500 (D) .616 (E) .384 4. Consider the situation described in problem #3. If no customers have arrived within the first two minutes after the bank has opened, what is the probability that it will be at least two more minutes before the first arrival? (A) e^-2 (B) e^-4 (C) 1 - e^-4 (D) e^-1 (E) 1 - e^-1 5. The length of time that I wait for my bus in the morning is exponentially distributed with mean θ = 10 minutes. What is the probability that the minimum of my waiting times over the next 5 days is more than 6 minutes? Assume that the waiting times on different days are independent. (A) e^-0.6 (B) e^-3.0 (C) 1 - e^-0.6 (D) 1 - e^-3.0 (E) (1 - e^-0.6)^5
Chai S.
Recommended Textbooks
Discrete Mathematics and its Applications
Higher Level Mathematics
Discrete Mathematics
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD