Consider two queuing systems. The first has one server and no limit on the length of the queue. Customers arrive according to a Poisson process with rate λ. The service time is exponentially distributed with rate μk. μk is proportional to the number of people in the system. That is, where k is the number of people in the system and μ is a constant. μk = kμ λ = 1.5 μ = 1.6 A) Determine the steady-state probability π1 (Please round to 3 decimal places): B) Calculate the average number of people in the system (Please round to 3 decimal places): C) Calculate their average time in the system: The second is an M/M/∞ queue. It has an infinite number of servers and no limit on the number of customers. Customers arrive according to a Poisson process with rate λ. The service time of each server is exponentially distributed with rate μ. A) Determine the steady-state probability π1 (Please round to 3 decimal places): B) Calculate the average number of people in the system (Please round to 3 decimal places): C) Calculate their average time in the system:
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We can use the following formula for the steady-state probabilities in this system: πk = (λ/μ)^k * π0, for k = 1, 2, 3, ... Show more…
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Adi S.
1. Consider a queuing system with two servers whose service times are independent and are exponentially distributed with a mean of 1 hour. The servers do not assist each other. Assume customers arrive into the system according to a Poisson rate of 20 per hour. Assume that if a prospective customer enters the system and sees two people waiting in line, the prospective customer immediately balks. (That is, the prospective customer immediately leaves; it is as if the prospective customer did not enter the system.) (a) Analyze the queue using the state transition diagrams discussed during lectures. For each possible number k of customers in the system, determine the steady-state probability that k customers are in the system. (b) What is the steady-state probability that a prospective customer will balk? (c) What is the steady-state average number of customers in the system?
Areen D.
Consider a 2 server system where customers arrive according to a Poisson process with rate ̀λ, and where each arrival is sent to the server currently having the shortest queue. (If they have the same length queue then the choice is made at random.) The service time at either server is exponential with rate μ, where λ < 2μ. For n ≥ 0, say that the state is (n, n) if both servers currently have n customers, and say that the state is (n, m), n < m, if one of the servers has n customers and the other has m. a) Write down the balance equation equating the rate at which the process enters and leaves a state for state (0, 0). b) Write down the balance equations equating the rate at which the process enters and leaves states of the form (0, m), m > 0. c) Write down the balance equations for the states (n, n), n > 0. d) Write down the balance equations for the states (n, m), 0 < n < m. e) In terms of the solution of the balance equations, find the average time a customer spends in the system.
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