Questions tagged [queueing-theory]
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38
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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 ...
6
votes
1
answer
333
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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. ...
5
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1
answer
548
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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, ...
5
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1
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323
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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 ...
4
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2
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203
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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 ...
4
votes
1
answer
217
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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 ...
4
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1
answer
216
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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 "...
4
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1
answer
643
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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) ...
4
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0
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131
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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 ...
4
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2
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162
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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 $\...
3
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3
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413
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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 ...
3
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1
answer
1k
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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 ...
2
votes
1
answer
167
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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 ...
2
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0
answers
68
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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 $...
2
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0
answers
51
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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 ...
2
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0
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59
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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<...
2
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0
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89
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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
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1
answer
272
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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 $[0,\infty]$
...
1
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1
answer
59
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Queues wait for other queues- A communication problem
I am working on a problem which involves a single server that requires multiple inputs to do a computation. Each of these inputs arrive as a Poisson process with rate $\lambda$. Hence, a situation ...
1
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1
answer
1k
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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 &...
1
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1
answer
145
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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 ...
1
vote
1
answer
136
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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
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0
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100
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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/...
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0
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73
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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 ...
1
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0
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74
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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
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0
answers
113
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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
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0
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32
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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 ...
1
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0
answers
66
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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 $Y(t)$ ...
1
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0
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263
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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 ...
1
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0
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484
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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 ...
1
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0
answers
502
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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 ...
0
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1
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121
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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, ...
0
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1
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126
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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 ...
0
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1
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262
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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 ...
0
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0
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13
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Position dependent service time in queue
Is there any literature for queuing analysis (waiting time, capacity etc.) of a queue with service time that depends on the position of the customer in the queue?
I have encountered a problem where a ...
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0
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139
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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 ...
0
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0
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184
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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 ...
0
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1
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693
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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 ...