Question

Consider an $\mathrm{M} / \mathrm{M} / 1$ queueing system in the steady state. (a) Show that the probability that there are $n$ or more customers in the system is $\rho^n$. (b) Use the result of part (a) to find the value of $\mu$, such that, for given values of $\lambda, n$, and $\alpha$, with $0<\alpha<1$, the probability of $n$ or more customers in the system is $\alpha$. This value of $\mu$ must be given explicitly by a formula in terms of $\lambda, n$, and $\alpha$. (c) Use the formula developed in part (b) to find $\mu$ if $\lambda=10, n=3$, and $\alpha=0.05$.

   Consider an $\mathrm{M} / \mathrm{M} / 1$ queueing system in the steady state.
(a) Show that the probability that there are $n$ or more customers in the system is $\rho^n$.
(b) Use the result of part (a) to find the value of $\mu$, such that, for given values of $\lambda, n$, and $\alpha$, with $0<\alpha<1$, the probability of $n$ or more customers in the system is $\alpha$. This value of $\mu$ must be given explicitly by a formula in terms of $\lambda, n$, and $\alpha$.
(c) Use the formula developed in part (b) to find $\mu$ if $\lambda=10, n=3$, and $\alpha=0.05$.

Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)

Arnold O. Allen 2nd Edition

Chapter 5, Problem 2

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Step 1

In an M/M/1 queue, arrivals follow a Poisson process with rate $\lambda$ and service times are exponentially distributed with rate $\mu$. The system is in steady state, and $\rho = \frac{\lambda}{\mu}$ is the traffic intensity, which must be less than 1 for Show more…

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Consider an $\mathrm{M} / \mathrm{M} / 1$ queueing system in the steady state. (a) Show that the probability that there are $n$ or more customers in the system is $\rho^n$. (b) Use the result of part (a) to find the value of $\mu$, such that, for given values of $\lambda, n$, and $\alpha$, with $0<\alpha<1$, the probability of $n$ or more customers in the system is $\alpha$. This value of $\mu$ must be given explicitly by a formula in terms of $\lambda, n$, and $\alpha$. (c) Use the formula developed in part (b) to find $\mu$ if $\lambda=10, n=3$, and $\alpha=0.05$.

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