# 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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1 vote
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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 ...
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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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1 vote
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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 ...
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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 ...
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### Laplace-Stieltjes Transforms and distribution

I was going through a paper, I came across below relation, T=\begin{cases} C, & \text{with probability $P(H<C)$}\\ 0, & \text{with probability $P(H>C)$} \...
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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 ...
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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 ...
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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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1 vote
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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 ...
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1 vote
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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 ...
1 vote
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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 ...
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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 ...
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1 vote
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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 ...
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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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### 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 $L = \lambda$ ...
1 vote
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 ...