Questions tagged [queueing-theory]

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How to modeling continuous batching in large-scale inference with queuing theory approach?

I want to model continuous batching in large model inference problems, but my knowledge in data theory is insufficient, and I haven't found an appropriate queuing theory model to use for modeling. So, ...
2 votes
0 answers
134 views

Sum of arrival times of Chinese Restaurant Process (CRP)

Suppose that a random sample $X_1, X_2, \ldots$ is drawn from a continuous spectrum of colors, or species, following a Chinese Restaurant Process distribution with parameter $|\alpha|$ (or ...
6 votes
1 answer
331 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. ...
2 votes
0 answers
66 views

Distribution of waiting time conditioned on a fixed time length

FYI, this question is a duplicate from math stack exchange I ask here again because I got no response. Suppose, I work in a factory production line. The time for me to finish wrapping product $A$ (or $...
0 votes
1 answer
115 views

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, ...
0 votes
1 answer
122 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 ...
1 vote
1 answer
135 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
0 answers
98 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/...
0 votes
1 answer
257 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
0 answers
71 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 ...
4 votes
0 answers
130 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 ...
1 vote
0 answers
74 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
1 answer
266 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 $[0,\infty]$ ...
4 votes
2 answers
203 views

Reference on a markov chain / Queue

Im looking for a reference that treats the Markov Chain defined by $$W_i=(W_{i-1}-1)\vee X_i$$ where $X_i\geq 0$ are i.i.d discrete variables. In particular im interested in a reference that treats ...
0 votes
0 answers
138 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 ...
4 votes
2 answers
160 views

Poisson counting process subinterval distribution

Suppose $N(\omega,t)$ is a homogeneous Poisson counting process with a constant parameter $\lambda,\,\forall\omega \in\Omega$ where $\Omega$ is the sample space. Given positive real numbers $T$ and $\...
1 vote
0 answers
111 views

Showing existence of a solution to an underdetermined system of equations with non-negativity constraints

Let $K$ be a positive integer, let $p\in (0,1)$, and let $\{W(k,i),W^B(k,i), \varphi_k(i)\}_{1\leq i\leq k\leq K}$ be variables. I need to prove that there exists a solution to the following system ...
1 vote
0 answers
32 views

Practical statistics for queueing networks

There is a theory for queueing networks where we postulate some nicely behaving base distributions of arrival processes and service processes and then calculate the behaviour of the system. Now, in ...
2 votes
0 answers
51 views

Finding a queuing model for waste accumulation

I've been tasked with modeling the accumulation of solid waste in an urban setting. In particular, the objective is to find the steady state distribution describing the amount of waste in a given ...
0 votes
1 answer
686 views

M/G/1 queue - probability that waiting time is zero

so: I have a M/G/1-queue with Poisson arrivals with rate lambda=1 and the service time being the sum of two exp-distributed variables vith rates u1=1 and u2=2. If we let Wq be the time an average ...
5 votes
1 answer
320 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 ...
2 votes
0 answers
59 views

Trying to show expected wait is convex -- need to show an expression is positive

I need to show that the following expression is positive $$ (B+1) (2 B+1) z_0^B-(B+2) (\rho +1) z_0-2 (B+1) (B-1) ((\rho +1) z_0-\rho )+(B-1) (\rho +1) > 0 $$ where $B\geq 1$ is an integer, $0<...
3 votes
1 answer
1k 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 ...
1 vote
0 answers
66 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 $Y(t)$ ...
3 votes
3 answers
413 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 ...
1 vote
0 answers
262 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 ...
2 votes
0 answers
89 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
1 answer
1k views

Matrix Generator for M/M/1 Queue Waiting Time Distribution

I "believe" that generator, $\bf W$, of the waiting time distribution for the M/M/1 queue is given by the following (I'm not sure if this is even correct): ${\bf W} =\left( \begin{array}{ccccc} 0 &...
4 votes
1 answer
215 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 ...
0 votes
0 answers
182 views

Repeatedly changing queue behavior

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 ...
4 votes
1 answer
640 views

Concurrency related problems in $n$ independent, parallel $M/M/1$ queues

Queueing Model: Consider $n$ independent, parallel $M/M/1$ queues with identical arrival rate $\lambda$ and service rate $\mu$. For each $M/M/1$ queue, we use the FCFS (First Come First Served) ...
1 vote
1 answer
143 views

Minimal variance for phase-type distributions?

Let $\mathcal{D}(m)$ be the set of phase-type distributions constructed from $m+1$-state Markov chains. Recall that the coefficient of variation of a distribution $D$ is the ratio of the standard ...
4 votes
1 answer
214 views

A queuing process where customers must be detected

Imagine a scenario where customers arrive in some queue according to a Poisson process with rate parameter $\lambda_{arr}$, and where the process of responding to the customers has a kind of "...
1 vote
0 answers
484 views

Wiener-Hopf Integral/Lindley's Equation

Lindley's equation is well known within queueing theory and is as follows $F(y) = - \int_0^\infty F(x)dH(y-x)$ However, many textbooks only consider the case where 0 $\le$ y $\le \infty$ (which ...
6 votes
2 answers
389 views

If Mean Residual Lifetime is approximately constant, Residual Lifetime is Approximately Exponential in a Strong Sense

Suppose the "mean residual lifetime," $\mathbb{E}[X-x|X≥x]$ is approximately constant for large $x$. Then, I believe that the conditional tail distribution is approximately exponential, in the sense ...
1 vote
0 answers
502 views

Comparing two Markov chains

I thought that this question is more appropriate for math.stackexchange, where I asked it, but seeing how I got no response, here it goes: I am interested in the question of the positive recurrence ...
5 votes
1 answer
548 views

Are there interesting problems involving arbitrarily long time series of small matrices?

Are there well-known or interesting applied problems (especially of the real-time signal processing sort) where arbitrarily long time series of small (say $d \equiv \dim \le 30$ for a nominal bound, ...