Questions tagged [queueing]

The mathematical study of waiting lines.

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In queueing delay, why can there be $+1$ added?

Here in p.4. a typical treatment on computing of queue delay is given. However, I've seen the equation $$W_n=W_{n-1} + Y_{n-1} - X_n$$ where $W_n$ is queueing delay, $Y_n$ is the service time of $n$...
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23 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 ...
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320 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. ...
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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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182 views

Time series tracking queue optimization problem

In order to track prices of many different products from different sources, I must optimally schedule a group of trackers dedicated to price collection (ie. collect one price at a time for each ...
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1answer
47 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 ...
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1answer
55 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 $L = \lambda$ ...
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Why is the Poisson distribution chosen to model arrival processes in Queueing theory problems?

When we consider Queueing theory scenarios where individuals arrive to a serving node and queue up, usually a Poisson process is used to model the arrival times. These scenarios come up in network ...
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28 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 ...
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163 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 ...
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28 views

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 $\...
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118 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 $...
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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. ...
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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 ...
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15 views

How to determine the rate of input in a queue M/M/c?

I know the exit rate ($\mu$) and the average waiting time in the queue ($W_q$). I need solve to rate of input ($\lambda$) in a queue. $\rho = \frac{\lambda}{c\mu} < 1$ $\pi_0 = \left[\left(\sum_{...
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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 ...
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76 views

Modeling worker performance parameters for optimum allocation of tasks to workers

Lets say we have an English to French translation task in a company and there are 100s of workers who are proficient in doing this task but each worker has its own unique attributes which allow them ...
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Dynamics of birth-death process with discouraged arrivals (alternatively, M/M/1 queue with balking customers)

Take a continuous-time birth-death process, where $k \in \{0,1,\ldots\}$ is the count and the arrival rate of death is $\mu \geq 0$ for $k = 1, 2, \ldots$ the arrival rate of births is $\alpha_k > ...
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135 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 ...
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1answer
384 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{...
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471 views

Poisson distribution problem - traffic problem

Hi So I have this question below. I know I need to model each lane as a separate Poisson distribution. The possible answers are: a) 11.4%; 22.4%; 33.4%; 44.4%; 55.4% b) 2.74%; 4.74%; 12.74%; 34.74%; ...
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37 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 ...
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2k views

Assumption for an M/M/1 queue

When a queueing system is modeled as an M/M/1 queue, it is assumed that the arrival time of jobs has Poisson distribution and the service rate has exponential distribution. I am wondering what ...
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72 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 ...
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153 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 ...
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Total hourly profit for a single-server food stand

Customers arrive to a single-server food stand according to a Poisson process with rate $20$ per hour. The time to serve a customer is exponentially distributed with a mean of $2$ minutes. (a) The ...
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244 views

Probability of collisions in queues of Poisson process

I have a process whereby objects of width $W$ land on a gene at rate $F$ (per second, poisson process, lets assume), and then start to move along at constant speed $V$. I'm trying to work out the ...
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208 views

How to determine 'burstiness' of data?

This is both a Math- and an R-question. I have a vector of POSIXct dates and a I want to determine the characteristics of the data. Example with numbers: ...
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141 views

What is the average number of students waiting in line

For this Question, am having a problem determining the average number of students in the queue. My problem arises since by using the formula, am getting a decimal answer less than 1. The formula I ...
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1answer
91 views

Length of a queue

Let we have a queue of independent events. The events are processed in the order of arrival, processing one event takes $t$ time. Events arrive in average $N$ events per identity time. What it the ...
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110 views

Burke's Theorem for rejection from Erlang-B loss queue

I have some general uncertainties regarding the rejection process from an Erlang-B loss queue ($M/M/c/c$), where the total capacity of the queue is equal to the number $c$ of servers. Consider the ...
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117 views

What is the average expectation value of two specific objects in a queue?

I need some hand to solve this problem. Suppose a queue can be filled by three type objects (M, F and A). Depending on the number of each objects the line length can be different. We are looking ...
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99 views

Showing a queueing system is a Markov Chain

I generally understand how to do this but I'm having trouble with a formal proof. "Consider an $M/M/1/m+1$ queue with exponential arrivals rate $\lambda$, exponential service rate $\mu$, and finite ...
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89 views

Mean length of time spent queueing in $M/E_2/1$ system?

Context: Consider a $M/E_2/1$ queueing system, where the customer arrival rate is $\lambda$ and the service time distribution has a gamma distribution with parameters $2$ and $\mu$, i.e. with p.d.f. $\...
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227 views

Conditional distribution (on N) of arrival times in a nonhomogenous poisson process

Conditional on $N(t)$, given some $\lambda(t)$ characterizing some Nonhomogenous poisson point process, the distribution of an arrival time $t_i$ is $\lambda(t_i)/\int_{A}\lambda\left(t\right)dt$ ...
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412 views

The Number of Exponential Summands in a Fixed Interval is Poisson

What is the most clever way to prove that the number of independent exponential summands in a fixed interval is distributed as a Poisson random variable? I can do it one way, but I would like to know ...
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Which methods do people use to understand queueing networks?

Queueing networks can be analyzed through analytic results (in some cases), approximation methods or simulation (discrete-event simulation, system dynamics). Analytic solutions do not exist in general....
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94 views

Revenue Forecasting using Markov Chain or Queuing Theory?

I am trying to forecast revenue for a HealthTech giant that sales HealthTech Hospital Equipment like Ultrasound, Magnetic resonance, CT AMI etc. The nature of business is Build to Order, which means ...
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State-dependent tandem network of queues

I've been studying network of queues and I would like to model a system by a tandem state-dependent network of queues. I know Jackson network works for $M/M/$ networks, but not for queues where the ...
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111 views

How many random selections needed to be 90%, 95%, 99% confident that a specific selection occurred?

For example, let's say that I have a bag of 100 marbles labeled 1 to 100. A "selection" is randomly picking a marble from the bag and then placing it back into the bag. The marble selected is not ...
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568 views

How to visualise queue wait time

Suppose I have a process that works as follows: collect all outstanding requests process all of them, which takes time commit all results in one transaction repeat I get a data point (start, ...
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30 views

Treatment of waiting time data with repeated observations for a finite time

A call center is open for 9 hours every day. My data are the times of occurrence of calls, divided in days. Some days are busier than the others and some days no calls at all occur. I have to estimate ...
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147 views

Simple service time in M/M/1

I'm trying to understand queueing systems and I found some notes online. The define $\lambda$ as the mean arrival rate, and $\mu$ as the mean service rate (the average number of customers who can be ...
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136 views

Model design for queue position prediction

The goal of the model is to predict progression in position in a FIFO queue, given our current position and other predictor variables. This is progression in the sense that the number ahead of us in ...
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7k views

Exponential Service Times When a Minimum Service Time is Reasonable

In many queuing models it is assumed that the service time follows an exponential distribution with parameter $\mu=1/\lambda$, where $\lambda$ is the average rate of service. An example might be a ...
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627 views

Why is the exponential distribution chosen to model service time in Queuing theory?

Why is the exponential distribution chosen to model service time in Queuing theory ?
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Distribution of “priority” emails among agents with different speeds (strange question)

but I figure if someone knows how to answer it, it may be someone here. Basically I have this weird distribution where the customer service agent speed (in terms of contacts per hour) is very ...
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3k views

Monte carlo simulation in R

I am trying to solve the following exercise but I actually have no clue on how to start doing this. I've found some code in my book that looks like it but it's a completely different exercise and I ...
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mean queue delay ( nonpreemtive priority)

I'm trying to solve a problem where all arriving items (arrival exponential $\lambda = 1/5$) are divided into into groups, those who are served within 5 units of time and those who have their service ...
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185 views

Stability of a GI/G/1 queue with $\rho=1$?

The final theorem in Chapter 19 of Meyn and Tweedie's Markov Chains and Stochastic Stability tells us that if the mean inter-arrival time $\lambda$ of a GI/G/1 queue is greater than its mean service ...