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Questions tagged [queueing]

The mathematical study of waiting lines.

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What is the distribution of needed hospital beds?

Suppose I am modelling a hospital service with $k$ number of beds. Initially there are $m$ number of beds being used, where $m \leq k$, each of which has a known amount of time that it has been ...
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What is the rate of waiting time increase for a M/M/1 queue where $\lambda = \mu$?

I am almost finished reading The Goal, which emphasizes the importance of managing bottlenecks in production. I'm more into service networks than production, but there is a lot to carry over from one ...
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Kendall's Notation for Kelly Networks?

Background Kendall's notation (also see Ciw docs') is a convenience when you work with queues a lot. It not only provides an abbreviation, but its wide use helps make literature searches more specific....
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What paper did Hall suggest the queuing rule of thumb $s \geq \max ( 1, \rho + \sqrt{\rho})$?

According to this site: Hall (1991) cited an argument of his previous paper that operation research profession could and should be more scientific and less mathematical. In fact, Hall also suggested ...
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Stochastic Task Processing Times in Queueing Theory

I'm struggling with an operations research problem which has 3 stations containing 3 different task processing times and different coefficients of variation (for example, station 1 has 3 tasks with ...
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Is the visit ratio vector non-negative in each of its components?

I am reading the wiki on mean value analysis. I am writing a Python implementation of it, and it occurred to me that I was getting some negative values. I want to validate whether that is expected ...
Galen's user avatar
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What is a static buffer priority (SBP) service discipline?

In Anton Braverman et al 2023 they mention their assumed service discipline: "Our results assume that the network operates under a static buffer priority (SBP) service discipline and require ...
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Little's Law + Kingman's Formula --> Approximation of Expected Length of G/G/1?

Little's Law gives us $$\mathbb{E}[L] = \lambda \mathbb{E}[W]$$ where $L$ is the number of customers in the queue + being served $\lambda$ is the arrival rate $\mu$ is the service rate $W$ is the ...
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How can I obtain the distribution of the number of customers in a non-starionary MM1 queue given an interval partition of stationary transitions?

I am considering this transient solution for the probability mass function over the number of "customers" in an MM1 queue: $$ p(k; t, i, \lambda, \mu) =\\ \exp \left( - (\lambda + \mu) t \...
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Is there a heavy traffic approximation for percent under benchmark?

Suppose I have an waitlist of patients waiting to be served. Each patient has a benchmark number of days that the service should be completed by as a goal (not a hard constraint of the modelling). ...
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Are there other types of blocking in queueing networks beyond Type I?

The Ciw documentation states In Ciw, Type I blocking (blocking after service) is implemented for restricted networks. After service, a customer’s next destination is sampled from the transition ...
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What are these interruptions in queueing networks?

I'm interested in modifications of Kendall queueing networks where agents in the network can be arbitrarily (according to a distribution or a deterministic function) moved from one part of the network ...
Galen's user avatar
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General insight from G/G/m queue approximations

I've been asked to add in some general queue theory insight into some simulation work i've done. Basically, just to include a few lines here and there about what the basic theory would suggest would ...
Controller's user avatar
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Which likelihood function is correct?

I have a confusion related to the likelihood function. I suppose that users waiting time $W$ follows an Exp distribution with the rate $\lambda$, and the prior of $\lambda$ follows Gamma($\alpha$, $\...
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Should I use the mean or median of my data for queueing models?

I am working on a project with a call center. Long story short, I am analzying the data revolved around the incoming calls to this call center in order to eventually use a queueing model. A queueing ...
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How to model distribution of arrivals in chunks

I want to model a queue system where customers tend to arrive in chunks. The queue represents vehicle arriving at a traffic light (consider just a single approach) and they tend to arrive in chunks ...
Federico Taschin's user avatar
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Queue M/M/2, proportion of unserved customers

In my work, I managed to arrive at the following scenario: Consider that I have a queue with two servers whose arrival rate is a poisson process $\lambda = 2 $ customers per hour and the service time ...
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Queue Management Algorithm With "Ages"

Imagine we are running a physiotherapy clinic, working 5 days a week. The overall number of treatments given in a day is $k$, and the daily treatments are randomly drawn among the registered patients. ...
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Setting up an M/M/1 Queueing Model in R with No Waiting Area

I am trying to set up a queueing model in R using the queueing library. I am trying to model a scenario with no waiting area. In other words, if the service ...
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Looking for general advice/guidelines on deriving a probability density function for waiting times for a queue

So I'm a student in my final year of high school, and the course that I take (IBDP) requires us to write a 12-20 page 'mathematical exploration' on a topic of our choosing. I am interested in deriving ...
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How to model this queueing process

I need help with the following problem. In my eyes, the description of it is a bit sloppy/unclear, so hopefully someone can help me figure out how the related questions can be answered satisfactorily. ...
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How is the Following Picture Possible?

Source: https://blog.wolfram.com/2013/03/21/the-mathematics-of-queues/ In this picture, it would appear that multiple customers can enter and exit the queue at the same time. In real life, this is ...
stats_noob's user avatar
2 votes
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Queues with non-constant arrival rates

https://cran.r-project.org/web/packages/simmer/index.html I am interested in simulating some basic queues in R. However, I want to try to simulate queues that have "non-constant" arrival ...
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Significant bias introduced into simple simulation

Introduction Service is allocated to an infinite source of customers i.e. there is always a service in progress. The duration of the $i^{th}$ service is generally distributed $\Delta_i \sim F_{\Delta}$...
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How Much Time Must the Shopkeeper Wait? -- Exponential Distribution

In a store, the distance between customer arrivals follows an exponential distribution with a parameter of 8 minutes. The second seller starts his shift at 10:30 while the last customer entered the ...
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Simulating a simple queueing problem to calculate $\Pr(B > 4).$ If-statement not working as I expect it

I have the following problem: Consider an M/M/3/4 queuing system with $\lambda=\mu=1$ that is the arrival time is exponentially distributed with parameter $\lambda = 1$ and the service times are ...
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Probability of being still in the system in a queue system

I have one queue with two servers $S_1$ and $S_2$.The serving times are modeled $\sim exp(\mu_1)$ and $\sim exp(\mu_2)$ respectively. The first server is free while the second has two clients, $A$ ...
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Probability of service in a queue theory problem with exponential random variable

I have one queue with two servers $S_1$ and $S_2$.The serving times are modeled $\sim exp(\mu_1)$ and $\sim exp(\mu_2)$ respectively. The first server is free while the second has two clients, $A$ ...
docdev's user avatar
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Why are departure rates of $M/M/c$ queues not equal to service rate $\mu$?

I think this issue should have an answer somewhere but I could not find in any materials. In every textbook I read about $M/M/c$ queueing systems, it is always acknowledged from the beginning that the ...
fermented_bean's user avatar
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$M/M^B/1$ Burke's theorem : what is the distribution of the output batch interarrival times?

Setup: Take an M/M/1 queue: the inputs arrive according to a Poisson process at rate $\lambda$, the service time per item is distributed exponentially with mean $1/\mu$, $\mu > \lambda$ the ...
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When the number occurences in a time interval are not Poisson distributed?

The lectures statistics I followed also presented the Poisson distribution. We were taught that the number of events occurring in a time interval, that this statistic follows a Poisson distribution. $ ...
Match Maker EE's user avatar
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Laplace-Stieltjes Transforms and distribution

I was going through a paper, I came across below relation, \begin{equation} T=\begin{cases} C, & \text{with probability $P(H<C)$}\\ 0, & \text{with probability $P(H>C)$} \...
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Simulate discrete state space CTMC from generator matrix

Consider a generator matrix $Q\in\mathbb{R}^{h\times h}$ for a discrete state space $\{1,...,h\}$. I want to determine the probability of a single transition of a stochastic process $X(t)$ with $X(0)=...
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How to Solve an Arrival Rate for a given Latency Percentile?

I've been cranking on this for some time and think some help would be nice. Say I've got an M/M/1 queue, and while I can't change my service time distribution, I can certainly adjust my arrival rate. ...
Lally Singh's user avatar
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Poisson counting process subinterval distribution

Suppose $N(t)$ is a homogeneous Poisson counting process with a constant parameter $\lambda$. Given positive real numbers $T$ and $\tau$, and non-negative integer $n$, what is the probability that $N(...
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Steady state properties M/M/Inf/1/N queue

Say I have a fixed population size of $N$ individuals, each with exponential arrival times $\lambda$ to an infinite number of queues with exponential service times $\mu$. I think the transition rate ...
robsmith11's user avatar
2 votes
1 answer
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Arrival distribution of an M/M/1 queue

Show that the arrivals $A_n$ of an M/M/1 queue $X$ with initial distribution $\eta_i := \rho^{i-1}(1-\rho)$ ($i \ge 1$), where $\rho$ is the traffic intensity, satisfy $X_{A_n} \sim \ \eta$. I ...
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users and licences

I have the following question: Suppose I have a product with $U$ users and $S$ seats (licences). Where $S < U$. A user may only use the given product if there are seats available, and during the ...
iliar's user avatar
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Queuing theory for elevators

It's been a while since I had my probability course based on Sheldon Ross' book "Probability Models", and while I never went into econometrics, I was very interested in the queuing theory section. I ...
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Finding Average Wait Time From Number of People Waiting

Problem: We are trying to estimate the average wait time for a process. Only data we have is how many people are in the system at a given time, how many have entered this period, how many exited this ...
Ofya's user avatar
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In the following queuing problem what assumption led the author to assume the probability of each task happening at 1/180

I am going through this algorithms and data structure course which implements a queue DS to simulate a printing queue. Following is the solution described: To model this situation we need to use some ...
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How to model pooling instead of queueing

I'm building a software system where constructing a new object takes a few minutes, say T minutes. I can create up to K objects in parallel. Users come randomly and request an object, but they are not ...
Edwin Young's user avatar
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Birth and death process, and calculating waiting time using Little's law

Assume that an individual only has two possible states: susceptible (S) and infected (I). Further, assume that the individuals in the population are independent, and that for each susceptible ...
Pame's user avatar
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calculating variance in a queueing model which looping flow

For the verification of a Discrete Event Model of a queueing model that I programmed, I would like to calculate the steady state values of the queueing times at different stations and their variance ...
LFSvenne's user avatar
1 vote
1 answer
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Birth Death process

An office has two employees that process incoming orders. these two are always busy and they process the orders at the rate of 100/day for each person. However they are smokers. On an average they ...
rgk's user avatar
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2 votes
1 answer
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Expected value of $e^{vS}$, where $S$ is an exponential

I am studying queueing theory and in particular I am dealing with priority queues with preemption. I found this very interesting paper that treats various topics of interest. The system is composed ...
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2 votes
1 answer
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Simulating a queue in R [closed]

It's noted that the number of folks in a stationary system will maintain an average equal to the rate of arrival multiplied by the mean of the service distribution. Where the formula $L = \lambda$ ...
user253521's user avatar
1 vote
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370 views

Distribution of the idle time in a queue

I have iid random variables $D_t \sim \mathcal{N}(\mu, \sigma^2)$ and a constant $Q$. Define a random variable $Z_t = (Z_{t-1} + Q - D_t)^+$. Essentially $Z$ is the waiting time distribution in a ...
chris12's user avatar
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probabilities related to a transient single-server queue

Consider an $N = 1$ server queue with arrival rate $\lambda > 0$ and service rate $\mu = 1$. If the process is transient, find $\rho{_{x0}}$ for $x ≥ 1$. My attempt: The process is transient if $\...
Jin Yu Li's user avatar
4 votes
1 answer
848 views

Finding the mean and variance of an infinite server queue

I am presented with the following homework problem: Let $X(t)$, $t > 0$, be the infinite server queue and suppose that initially there are $x$ customers present. Compute the mean and variance of $...
Jin Yu Li's user avatar