In an M/MA queueing system, the arrival rate is 9 customers per hour and the service rate is 12 customers per hour. (Round youranswers to 3 decimal places.)a. What is the probability that the server will be idle?Probabilityb. What is the probability of having exactly 4 customers in the system?Probabilityc. What is the probability of having 3 or fewer customers in the system?Prubabtyd. What is the expected number of customers in the queue(La)? (Round your answer to 3 decimal pleces.)Expected number of customerse. What is the expected waiting time in the queue(Ng)? (Express the waiting time in hours, round your answer to 3 decimal pleces)Expected wasting time

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Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter12: Queueing Models

Section12.4: Important Queueing Relationships

Problem 6P

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In an M/M/1 queueing system, the arrival rate is 8 customers per hour and the service rate is 11 customers per hour. (Round your answers to 3 decimal places.) What is the probability that the server will be idle? What is the probability of having exactly 4 customers in the system? What is the probability of having 3 or fewer customers in the system?
In an M/M/1 queueing system, the arrival rate is 7 customers per hour and the service rate is 12 customers per hour. Note: Round your answers to 3 decimal places. What is the probability that the server will be idle? What is the probability of having exactly 4 customers in the system? What is the probability of having 4 or fewer customers in the system?
Customers arrive at a one window drive according to a poison distribution with mean of 10 minutes and service time per customer is exponential with mean of 6 minutes. The space in front of the window can accommodate only three vehicles including the serviced ones. Other vehicles are have to wait outside this space. Calculate:a. A. probability that an arriving customer can drive directly to the space in front of the windowb. Probability that an arriving customer will have to wait outside the directed space c. How long an arriving customer is expected to wait before getting the service?
In an M/M/1 queueing system, the arrival rate is 7 customers per hour and the service rate is 12 customers per hour. (Round your answers to 3 decimal places) a. What is the probability that the server will be idle? b. What is the probability of having exactly 4 customers in the system? c. What is the probability of having 4 or fewer customers in the system?
4. The problem: In a queueing problem modelled as (M/M/2), if L (number of units in the system) 0.75 units and service rate for each s (server) is 3 units per hour, what is the arrival rate? To answer the problem or question, you were given the following data/statements: = (1) The average time a unit spends in the queue is 2.5 mins. (2) The average time a unit spends in the system is 22.5 hours. Which of the following appropriately describes the data/statements provided? Select the corresponding letter and support your answer. A. Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient. B. Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.
c. The probability that more than five cars arrive in any given hour
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Suppose the waiting time at a certain checkout counter is bi-modal. With probability 0.85, the waiting time follows an exponential distribution with a mean waiting time of four minutes. With probability 0.15, the waiting time equals 20 minutes. a) Compute the mean and median waiting time at the checkout counter. b) Compute the variance of the waiting time at the checkout counter. c) Compute the probability that an individual customer waits longer than 5 minutes at the checkout counter.
Users are connected to a database server through a network. Users request files from the Database server. The database server takes a period of time that is exponentially distributed with mean 5 seconds to process a request. Find the probability that 10 requests are processed by the server during the first 1 minutes. Determine the probability that no request is processed by the server during the first 2 minutes Determine the average number of requests processed in 2 minutes
11. Automobiles arrive at a tollbooth on a highway completely at random according to a Poisson process with rate À = 4 cars per hour. Determine the following: a. The probability that the time between any two successive arrivals exceeds 20 minutes. b. The probability that exactly four cars arrive in any given hour. c. The probability that more than five cars arrive in any given hour. d. The probability that more than 10 cars arrive in any given two-hour period. e. A person working for the transportation department begins counting arrivals at the tollbooth at 8 A.M. What is the probability that he counts 20 arrivals before 12 noon? (Use a normal approximation for your calculations.)
please dont answer in image format. Do type Thnks
Consider a Poisson queue with the following steady-state probabilities: po = 1/3, p1 = 1/2, p2 = 1/6, and pn = for n = 3,4,5, ... . What is the minimum number of stools that would accommodate all customers in the system at least 75% of the time in steady state? 3 stools O 2 stools none of the other choices O 1 stool

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a. What is the probability that the server will be idle? Probability b. What is the probability of having exactly 4 customers in the system? Pobabty What is the probablity of having 3 or fewer customers in the system?