# Questions tagged [queueing-theory]

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### Is the departure process of an infinite server queue independent of the arrival process?

Assume we have a $M/M/\infty$ queue with arrival rate $\lambda$ and a service rate $\mu$. From Burke's theorem, the departure process of the queue is a Poisson process with rate $\lambda$. However, ...
1 vote
126 views

### The input and output processes in a single-server queue

Consider an $M/M/1$ queue with the arrival rate $\lambda>0$ and the service rate $\mu>\lambda$ (so that it is stable), in the stationary regime. Let $A_t$ be the number of arrivals in the time ...
1 vote
85 views

### Birth and death process $M/M/\infty$

I was reading about continuous time Markov chains, when I met for the first time the theory of queue processes. In particular, I considered the following situation which I found on Wikipedia, called M/...
87 views

### M/G/1 queue as a Markov renewal process: one-step transition probabilities

Seeking help on this interesting problem! any input is welcome and appreciated. I've posted on other places and decided to seek any possible help here! Background From many texts, we know that for an ...
91 views

### Epidemics: distribution of interarrival times

In models of disease transmission, after an individual is getting infected, he can generate a number of secondary infections. The number of secondary infections depends on the infectiousness of the ...
1 vote
61 views

### infinitesimal generators for G/G/1 queue

I read the infinitesimal generator for the M/M/1 queue and thought to generalize to the G/G/1 queue. More specifically, though the queue length process is not Markovian anymore, we could consider an ...
112 views

### A random walk/ruin theory problem with steps whose distribution has infinite mean

In what follows, I will make liberal use of the notations and terminology from ruin theory, just because I think it makes matters more intuitive. However, the problem I'm posing does not depend on its ...
229 views

### Limiting distribution in $M_t/M_t/1$ queue

Consider a $M/M/1$ queue with a constant arrival rate $\lambda$ and service rate $\mu$ with $\lambda < \mu$. We know that in this case the limiting distribution exists and it is a geometric ...
1 vote
73 views

### Stationary distribution of a Memoryless 2-type priority queue

I have come across the following priority queue, which seems quite natural to me. A single queue with 2 types of costumers, independent Poisson arrivals and Poisson services. First class costumers ...
1 vote
217 views

### Uniqueness of deconvolution after convolution?

I have the following question and I'd greatly appreciate any help! Basically, I have an arbitrary probability distribution with pdf $f(x)$, we can assume it's continuous with support on $[0,\infty]$ ...
114 views

### Laplace transform of sum of random variables in first hitting time problem

Let me refer to the example here. Suppose $X$ is a birth-death (BD) process (represents population size) that evolves by: $X \to X+1$ if a birth occurs with rate $\mu$, $X \to X-1$ if a death occurs ...
283 views

### Finding minimum operations to move ants through connected graph

I am working on a project that requires to find the minimum number of steps to move ants from source to sink in a graph; one step is the movement of all ants from one node to the next of the graph. ...
151 views

1 vote
65 views

### steady state distribution of a dynamical equation?

Given the following dynamical equation for $X(t)$ as follows: $X(t+1) = X(t) - \min\{X(t), M\} + Y(t)$, or can write it as follows: $X(t+1) = \max\{X(t) - M, 0\} + Y(t)$, Assume the PDF of $Y(t)$ ...
396 views

### A question about intuition of fluid limit in queuing system

This is a question about intuition in understanding the fluid limit queuing system. Assume we have a sequence of queuing systems $\{S^N\}_{N=1}^{\infty}$ with N servers and each server has unit ...
941 views

### Analyzing a multiple-queue single-server model

Consider the following multiple-queue single-server model of a packet network problem. At each discrete time $t=0,1,\ldots,n$, a packet may arrive at the server R with probability $1-\epsilon_1$. The ...
314 views

### Problem of random scheduling of queues of tasks

Consider $L$ queues in a discrete time system. At each time $n=0,1,2,\ldots$, one task would arrive at one of the queues with equal probability $\frac{1}{L}$. Immediately after that, a task scheduler ...
87 views

### Customers and Anti-Customer Queueing Problem: What is the Customer delete probability

Hello may I ask for your help? First the setting: I have got a problem with some queueing theory. The whole problem would be a grid of nodes, all nodes have an operation intensity $\mu_{i,j}$. ...
1 vote
260 views

### Average queue-length optimal queuing system

Consider a time-slotted queuing system which has two servers and two users. At each time slot, a packet for user $1$ arrives with probability $\lambda _1$, while a packet arrives for user $2$ with ...
204 views

### Continuity of the stationary distribution of $M/G/1$ queue w.r.t. the input rate

Let $(\lambda_n)_{n\geq0}$ be a sequence of positive numbers such that $\lambda_n\rightarrow \lambda$ as $n\rightarrow +\infty$. These $\lambda_n$ are the parameters of a sequence of Poisson Processes ...
I'm not sure if this question is suited to MO. I will happily delete if not. Situation Consider a general queueing system $\mathscr{S}$, whose customer arrival times are independent, and whose ...