Questions tagged [queueing]

The mathematical study of waiting lines.

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14
votes
2answers
143 views

Distribution to reflect situation where some waiting leads us to expect more waiting

In reading Blake Master's notes on Peter Thiel's lecture on start ups, I came across this metaphor of the technology frontier: Picture the world as being covered by ponds, lakes, and oceans. You’...
14
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2answers
19k views

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 ...
11
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1answer
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 ...
11
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1answer
808 views

A hair dresser's conundrum

My hairdresser Stacey always puts on a happy face, but is often stressed about managing her time. Today Stacey was overdue for my appointment and very apologetic. While getting my haircut I wondered: ...
8
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2answers
974 views

General approaches to model car traffic in a parking garage

a friend of mine has asked me to help him with predictive modelling of car traffic in a medium sized parking garage. The garage has its busy and easy days, its peak hours, dead hours opening hours (it ...
7
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1answer
406 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 ...
7
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1answer
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 ...
6
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1answer
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 ...
6
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3answers
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 ...
5
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2answers
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%; ...
5
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0answers
297 views

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 > ...
4
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1answer
1k views

Distribution of arrival times to server for an M/M/1 queue (what the server experiences)

In an M/M/1 queue, we know that inter-arrival times are exponentially distributed, and that service times are the same. What is the distribution of to-server inter-arrival times (aka service start ...
4
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1answer
152 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 ...
4
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1answer
421 views

Mean service time of a $M/E_2/1$ queueing system?

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. $\mu^2te^{-...
4
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1answer
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 ?
4
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1answer
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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0answers
243 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 ...
3
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1answer
129 views

How can I even out a random distribution while minimising how far each data point is moved?

I have a software application which uses a queue and multiple processors to process those jobs. Jobs get re-run on a daily basis for customers, but we also have new customers signing up regularly. ...
3
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0answers
168 views

Continuous time Markov chain forward equation

This is a homework question and I need suggestion how to approach it. We have given the transitions $\ i\rightarrow i+1$ with rate $\lambda(i)$ where $\ i \ge 1$ $\ i\rightarrow i-1$ with rate $\...
2
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3answers
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 ...
2
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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 ...
2
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2answers
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: ...
2
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1answer
312 views

How to predict and optimize a queue?

I am trying to determine if I can model my system as a M/M/1 queue and if so do the numbers I get from it help me at all. I can model my system like this: System Description A. I want to spec out ...
2
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1answer
54 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 $L = \lambda$ ...
2
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1answer
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 $...
2
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1answer
85 views

Modelling worst, average and best case capacity of a system

A system has an incoming request rate of N requests per second. The system hosts a pool of x workers where x >= N. Each worker can complete a single request in approximately t seconds. The maximum ...
2
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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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0answers
6 views

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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0answers
37 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. ...
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0answers
16 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 ...
2
votes
1answer
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. ...
2
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0answers
140 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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0answers
567 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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0answers
61 views

How to predict how long a worker will take to perform a task when performance on the task is only known for other workers?

I am trying to model a queue system (via simulation) to see if I can make different work assignments to improve the overall performance (based on a number of metrics). I have historical records of ...
2
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0answers
520 views

Queueing Theory: How to estimate steady-state queue length for single queue, N servers?

I have a real-life situation that can be solved using Queueing Theory. This should be easy for someone in the field. Any pointers would be appreciated. Scenario: There is a single Queue and N ...
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1answer
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 ...
1
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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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2answers
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 ...
1
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1answer
45 views

Is our queue processing speeding up?

We have been collecting statistics on a queueing and processing system that powers part of our business. We have multiple servers processing items in a queue. We recently added new servers, and we ...
1
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1answer
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 ...
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1answer
280 views

Transition matrix to a compact set of rules

I would like to ask you about methods of converting a discrete time Markov chain, represented by a fully known transition matrix, into a relatively small set of transition rules. For example, let ...
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1answer
22 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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0answers
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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0answers
26 views

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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0answers
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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0answers
34 views

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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0answers
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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0answers
33 views

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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0answers
29 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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0answers
39 views

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 ...