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Questions tagged [queueing]

The mathematical study of waiting lines.

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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($\alpha$, $\...
Ellen1230's user avatar
4 votes
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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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Web Service Markov Chain - state probability calculation

I have the following problem: Let us consider a Web server software that fails at the failure rate gp, running on a machine (node) that fails independently at the failure rate gm. An automatic failure ...
user380232's user avatar
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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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1 answer
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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 ...
David's user avatar
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Tandem system in queuing theory

We have a queue networking comprising of two queues $A$ and $B$ such that customers after finishing the service in $A$ go directly to $B$ then from $B$ go out of the network after service in $B$. ...
endeavor's user avatar
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Modeling Scenario as M/M/1 Queue

I have a rather cursory knowledge of queueing processes (am currently an undergraduate taking a course that discusses them) but had a question about a scenario for which I am to prove it may be ...
Carter Hall's user avatar
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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. ...
Spätzle's user avatar
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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 ...
324's user avatar
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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 ...
PerfectNeglect's user avatar
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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. ...
coar's user avatar
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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
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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 ...
stats_noob's user avatar
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1 answer
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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 $\Delta_i \sim F_{\Delta}$...
Dylan Solms's user avatar
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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 ...
ghostDs's user avatar
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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 ...
Parseval's user avatar
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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, $A$ ...
docdev's user avatar
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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
2 votes
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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 ...
Konstantin's user avatar
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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
1 vote
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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 $P(H>C)$} \...
Pramod_achar's user avatar
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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
1 vote
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32 views

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
71 views

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 ...
A.M.'s user avatar
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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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8 votes
1 answer
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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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1 vote
1 answer
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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 ...
Coddy's user avatar
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3 votes
1 answer
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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
1 vote
0 answers
161 views

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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2 votes
0 answers
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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
88 views

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
  • 136
2 votes
1 answer
829 views

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 ...
linofex's user avatar
  • 155
2 votes
1 answer
951 views

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
0 answers
199 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
567 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
2 votes
0 answers
180 views

Distribution of shifted and censored random variable

Suppose I have a non-negative random variable $X$ with finite mean $\mu$. Let's for example assume it denotes demand, and we denote by $X(t)$ demand in period $t$, demand in different periods are iid. ...
chris12's user avatar
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2 votes
0 answers
34 views

optimal scheduling allocation between online and walk-in queues

I have a scheduling queue that can be split between walk-ins and online booking appointments. In order to serve these queues I have a limited number of resources available. What is the optimal way to ...
phoenix373's user avatar
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0 answers
299 views

simulation in R: reading arrival times and service times from a set and have priority queue

I want to simulate an outpatient clinic that does not have walk-in. based on a scheduling model, I have already calculated arrival times and examination times and whether the patient will be a show or ...
Maryam's user avatar
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7 votes
1 answer
280 views

Finite state machine with gamma distributed waiting times

I have a state machine with positive and negative inputs. The time between positive inputs follows a gamma distribution ($X_+ \sim \Gamma(k_+, \theta_+)$), and the time between negative inputs follows ...
mwoods's user avatar
  • 73
3 votes
2 answers
2k views

Queuing Theory: Customers arrive at a fast-food restaurant at the rate of 120 customers per hour

During lunch hour, customers arrive at a fast-food restaurant at the rate of 120 customers per hour. The restaurant has one line, with three workers taking food orders at independent service stations. ...
Nicklovn's user avatar
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1 vote
1 answer
481 views

M/G/1 queue and Pollaczek-Khintchine formula

My question is about the interpretation of symbols used in a description of the derivation of the Pollaczek-Khintchine formula, as outlined on pp 240 - 242 of Cox and Miller's "The theory of ...
adrianmcmenamin's user avatar
2 votes
1 answer
594 views

Probability Generating Functions and Discrete Time Queuing Chains

I've been learning about Markov Chains for a reading course I'm doing and I've become stuck on some of the notes I was given. It defines a discrete time queuing chain for a server by $$X_{t+1}=\begin{...
Jonathan.Lidcombe's user avatar
0 votes
0 answers
44 views

Utilization vs Percentile

My objective is to have high utilization of the resources by keeping only the required number of resources alive. Low utilization means the resources were not used effectively, while high ...
pauljeba's user avatar
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1 vote
2 answers
80 views

Maximizing car park usage efficiency

I'm trying to create a predictive model for a closed car park. This model is expected to estimate available parking spaces given date and time. Brief information: the car park has a limited capacity ...
LiquidSpirit's user avatar
4 votes
1 answer
345 views

CDF of multiple exponential random variables

Assume we are are serviced by core $I$, where $I=i$ and $i=[0, n]$, with probability $p_i$. Also assume that the time needed by each $i$ in order to complete a job is an exponential random variable ...
user154281's user avatar