text: # A $M/M/\infty$ queue with fixed probability going back to the end of the queue
  
  Suppose for a $$M/M/\infty$$ queue with arrival rate $$\lambda$$ and service rate $$\mu$$, with probability $$p$$ the service is successful and the person will leave the queue, and otherwise she will go back to the end of the queue. I am wondering if it is equivalent to another $$M/M/\infty$$ queue? If this is the case, what is this queue's arrival and service rates and how to prove it?
  
  Let $$S$$ be the (total) service time of a customer and $$N$$ the number of times a customer is served. Then $$S\mid N=n\sim\mathsf{Erlang}(n,\mu)$$ and $$N\sim\mathsf{Geo}(p)$$, so by the law of total probability \begin{align} f_S(t) &= \sum_{n=1}^\infty f_{S\mid N=n} (t)\mathbb P(N=n)\\ &= \sum_{n=1}^\infty \frac{\mu(\mu t)^{n-1}}{(n-1)!} e^{-\mu t}(1-p)^{n-1}p\\ &= pe^{-\mu t}\sum_{n=0}^\infty \frac{((1-p)(\mu t))^n}{n!}\\ &= p\mu e^{-p\mu t}, \end{align} and hence $$S\sim\mathsf{Expo}(p\mu)$$.
  The resulting detailed balance equations $$\pi_n = \frac{\lambda}{np\mu}\pi_{n-1}$$, $$n\geqslant 1$$ differ from those of a regular $$M/M/\infty$$ queue only by the factor of $$\frac 1p$$ on the right-hand side and accordingly the stationary distribution has Poisson distribution with mean $$\frac\lambda{p\mu}$$.
explanation: The text extract describes a mathematical problem involving a queueing theory scenario. Specifically, it discusses an \(M/M/\infty\) queue with a fixed probability \(p\) that a customer will either leave the queue or go back to the end of the queue after service. The extract poses a question about whether this modified queue is equivalent to another \(M/M/\infty\) queue and, if so, what the arrival and service rates of this equivalent queue would be and how to prove it. The text then provides a detailed mathematical derivation to show that the service time \(S\) follows an exponential distribution with rate \(p\mu\), and it concludes with the detailed balance equations and the stationary distribution of the modified queue.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # A $M/M/\infty$ queue with fixed probability going back to the end of the queue
  
  Suppose for a $$M/M/\infty$$ queue with arrival rate $$\lambda$$ and service rate $$\mu$$, with probability $$p$$ the service is successful and the person will leave the queue, and otherwise she will go back to the end of the queue. I am wondering if it is equivalent to another $$M/M/\infty$$ queue? If this is the case, what is this queue's arrival and service rates and how to prove it?
  
  Let $$S$$ be the (total) service time of a customer and $$N$$ the number of times a customer is served. Then $$S\mid N=n\sim\mathsf{Erlang}(n,\mu)$$ and $$N\sim\mathsf{Geo}(p)$$, so by the law of total probability \begin{align} f_S(t) &= \sum_{n=1}^\infty f_{S\mid N=n} (t)\mathbb P(N=n)\\ &= \sum_{n=1}^\infty \frac{\mu(\mu t)^{n-1}}{(n-1)!} e^{-\mu t}(1-p)^{n-1}p\\ &= pe^{-\mu t}\sum_{n=0}^\infty \frac{((1-p)(\mu t))^n}{n!}\\ &= p\mu e^{-p\mu t}, \end{align} and hence $$S\sim\mathsf{Expo}(p\mu)$$.
  The resulting detailed balance equations $$\pi_n = \frac{\lambda}{np\mu}\pi_{n-1}$$, $$n\geqslant 1$$ differ from those of a regular $$M/M/\infty$$ queue only by the factor of $$\frac 1p$$ on the right-hand side and accordingly the stationary distribution has Poisson distribution with mean $$\frac\lambda{p\mu}$$.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract describes a mathematical problem involving a queueing theory scenario. Specifically, it discusses an \(M/M/\infty\) queue with a fixed probability \(p\) that a customer will either leave the queue or go back to the end of the queue after service. The extract poses a question about whether this modified queue is equivalent to another \(M/M/\infty\) queue and, if so, what the arrival and service rates of this equivalent queue would be and how to prove it. The text then provides a detailed mathematical derivation to show that the service time \(S\) follows an exponential distribution with rate \(p\mu\), and it concludes with the detailed balance equations and the stationary distribution of the modified queue.
  
  Label: question-and-answer