/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 Customers arrive at a service st... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 25

Customers arrive at a service station, manned by a single server who serves at an exponential rate \(\mu_{1}\), at a Poisson rate \(\lambda\). After completion of service the customer then joins a second system where the server serves at an exponential rate \(\mu_{2}\). Such a system is called a tandem or sequential queueing system. Assuming that \(\lambda<\mu_{i}, i=1,2\), determine the limiting probabilities. Hint: Try a solution of the form \(P_{n, m}=C \alpha^{n} \beta^{m}\), and determine \(C, \alpha, \beta\).

Short Answer

Expert verified
The limiting probabilities of the tandem queueing system are given by \(P_{n, m} = C \alpha^n \beta^m\), where \(n\) and \(m\) represent the number of customers in the first and second systems, respectively, and \(C, \alpha, \beta\) are constants found as: \[ \alpha = \frac{\lambda + \mu_1 - \mu_2 \beta}{\lambda \beta} \] \[ \beta = \frac{\lambda \beta^{m - 1} + \mu_1 \left(\frac{\lambda + \mu_1 - \mu_2 \beta}{\lambda \beta}\right) \beta^{m - 1}}{\mu_2 \beta^m + \lambda \left(\frac{\lambda + \mu_1 - \mu_2 \beta}{\lambda \beta}\right) \beta^m} \] \[ C = \frac{1}{\sum_{n = 0}^{+\infty} \sum_{m = 0}^{+\infty} \alpha^n \beta^m} \] These limiting probabilities represent the long-term behavior of the tandem queueing system and can be used to analyze and optimize its performance.

Step by step solution

01

In a tandem queueing system, a customer arrives at service station 1 at a Poisson arrival rate \(\lambda\). After being served, they proceed to service station 2, which serves at an exponential rate \(\mu_2\). There is only one server at each station, and it is given that \(\lambda < \mu_i\) for \(i = 1, 2\). #Step 2: Establish limiting probabilities solution form#

We are asked to determine the limiting probabilities of this system using the solution form \(P_{n,m} = C \alpha^n \beta^m\), where \(P_{n, m}\) represents the probability of having \(n\) customers in the first system and \(m\) customers in the second system, and \(C, \alpha, \beta\) are constants that we need to determine. #Step 3: Define the Poisson processes and exponential rates#
02

A Poisson process with rate \(\lambda\) represents the arrival of customers, meaning that the probability of \(n\) arrivals in an interval of length \(t\) is given by: \[ p(n, t) = e^{-\lambda t} \frac{(\lambda t)^{n}}{n!} \] The exponential rates \(\mu_1\) and \(\mu_2\) represent the service times at each station. The probability density function of an exponential distribution with rate \(\mu\) is: \[ f(t) = \mu e^{-\mu t} \] #Step 4: Set up the balance equations for the system#

To find the limiting probabilities, we need to set up balance equations for the system. The balance equations represent the relationship between the rates of customers entering and leaving the queues. For our tandem queueing system, the balance equations are: \[ \lambda P_{0,0} + \mu_1 P_{1,0} = \mu_2 P_{0,1} + \lambda P_{1,1} \] \[ \lambda P_{0, m - 1} + \mu_1 P_{1, m - 1} = \mu_2 P_{0, m} + \lambda P_{1, m}, \quad m > 1 \] These equations describe how customers flow through the two stations in the tandem queueing system. #Step 5: Solve for C, 伪, and 尾#
03

Substituting the solution form \(P_{n,m} = C \alpha^n \beta^m\), we find: \[ \lambda C + \mu_1 C \alpha = \mu_2 C \beta + \lambda C \alpha \beta \] \[ \lambda C \beta^{m - 1} + \mu_1 C \alpha \beta^{m - 1} = \mu_2 C \beta^m + \lambda C \alpha \beta^m, \quad m > 1 \] Divide these equations by \(C\): \[ \lambda + \mu_1 \alpha = \mu_2 \beta + \lambda \alpha \beta \] \[ \lambda \beta^{m - 1} + \mu_1 \alpha \beta^{m - 1} = \mu_2 \beta^m + \lambda \alpha \beta^m, \quad m > 1 \] Solve the first equation for \(\alpha\): \[ \alpha = \frac{\lambda + \mu_1 - \mu_2 \beta}{\lambda \beta} \] Substitute this expression into the second equation, and solve for \(\beta\): \[ \beta = \frac{\lambda \beta^{m - 1} + \mu_1 \left(\frac{\lambda + \mu_1 - \mu_2 \beta}{\lambda \beta}\right) \beta^{m - 1}}{\mu_2 \beta^m + \lambda \left(\frac{\lambda + \mu_1 - \mu_2 \beta}{\lambda \beta}\right) \beta^m} \] With the obtained values for \(\alpha\) and \(\beta\), we can determine the constant \(C\): \[ C = \frac{1}{\sum_{n = 0}^{+\infty} \sum_{m = 0}^{+\infty} \alpha^n \beta^m} \] #Step 6: Obtain the limiting probabilities#

Finally, having determined \(C, \alpha, \beta\), we obtain the limiting probabilities: \[ P_{n, m} = C \alpha^n \beta^m \] These probabilities represent the system's long-term behavior, which can be used to better understand and optimize the performance of the tandem queueing system.

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Most popular questions from this chapter

In Example \(6.20\), we computed \(m(t)=E[O(t)]\), the expected occupation time in state 0 by time \(t\) for the two-state continuous-time Markov chain starting in state \(0 .\) Another way of obtaining this quantity is by deriving a differential equation for it. (a) Show that $$ m(t+h)=m(t)+P_{00}(t) h+o(h) $$ (b) Show that $$ m^{\prime}(t)=\frac{\mu}{\lambda+\mu}+\frac{\lambda}{\lambda+\mu} e^{-(\lambda+\mu) t} $$ (c) Solve for \(m(t)\).

Let \(Y\) denote an exponential random variable with rate \(\lambda\) that is independent of the continuous-time Markov chain \(\\{X(t)\\}\) and let $$ \bar{P}_{i j}=P\\{X(Y)=j \mid X(0)=i\\} $$ (a) Show that $$ \bar{P}_{i j}=\frac{1}{v_{i}+\lambda} \sum_{k} q_{i k} \bar{P}_{k j}+\frac{\lambda}{v_{i}+\lambda} \delta_{i j} $$ where \(\delta_{i j}\) is 1 when \(i=j\) and 0 when \(i \neq j\). (b) Show that the solution of the preceding set of equations is given by $$ \overline{\mathbf{P}}=(\mathbf{I}-\mathbf{R} / \lambda)^{-1} $$ where \(\overline{\mathbf{P}}\) is the matrix of elements \(\bar{P}_{i j}, \mathbf{I}\) is the identity matrix, and \(\mathbf{R}\) the matrix specified in Section \(6.8\). (c) Suppose now that \(Y_{1}, \ldots, Y_{n}\) are independent exponentials with rate \(\lambda\) that are independent of \(\\{X(t)\\}\). Show that $$ P\left\\{X\left(Y_{1}+\cdots+Y_{n}\right)=j \mid X(0)=i\right\\} $$ is equal to the element in row \(i\), column \(j\) of the matrix \(\overline{\mathbf{p}}^{n}\). (d) Explain the relationship of the preceding to Approximation 2 of Section \(6.8\).

Consider a system of \(n\) components such that the working times of component \(i, i=1, \ldots, n\), are exponentially distributed with rate \(\lambda_{i} .\) When failed, however, the repair rate of component \(i\) depends on how many other components are down. Specifically, suppose that the instantaneous repair rate of component \(i, i=1, \ldots, n\), when there are a total of \(k\) failed components, is \(\alpha^{k} \mu_{i}\) (a) Explain how we can analyze the preceding as a continuous-time Markov chain. Define the states and give the parameters of the chain. (b) Show that, in steady state, the chain is time reversible and compute the limiting probabilities.

In the \(M / M / s\) queue if you allow the service rate to depend on the number in the system (but in such a way so that it is ergodic), what can you say about the output process? What can you say when the service rate \(\mu\) remains unchanged but \(\lambda>s \mu ?\)

The following problem arises in molecular biology. The surface of a bacterium consists of several sites at which foreign molecules - some acceptable and some not - become attached. We consider a particular site and assume that molecules arrive at the site according to a Poisson process with parameter \(\lambda\). Among these molecules a proportion \(\alpha\) is acceptable. Unacceptable molecules stay at the site for a length of time which is exponentially distributed with parameter \(\mu_{1}\), whereas an acceptable molecule remains at the site for an exponential time with rate \(\mu_{2}\). An arriving molecule will become attached only if the site is free of other molecules. What percentage of time is the site occupied with an acceptable (unacceptable) molecule?
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