# Questions tagged [queueing]

The mathematical study of waiting lines.

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28 views

### 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 ...
47 views

### 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 ...
10 views

### Simulate Erlang A with simpy [closed]

I am trying to simulate using simpy a set of calls that arrive in a 30min time interval, and with a given length. So far I have used the following hypothesis: Exponentially distributed arrival times ...
124 views

### 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 ...
36 views

### 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 ...
24 views

### 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 ...
5 views

### Is it better to rely on the average latency or on the percentile to apply Little's Law?

I was reading this post (Little’s Law- An insight on the relation between latency and throughput) about the relation of Latency vs. Throughput and using Little's Law to have an insight about them. ...
2 views

### Event source model for a queueing system and queue length estimation

I am facing a situation that I would like to analyzing using queueing theory but I am struggling to find a suitable model for my source. The system is incredibly simple so I'm quite sure there is ...
21 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 ...
10 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 ...
27 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 ...
69 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 ...
120 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 $L = \lambda$ ...
39 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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### 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. ...
22 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 ...
47 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 ...
137 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 ...
423 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. ...
201 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 ...
408 views

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### 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: ...
153 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 ...
522 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%; ...
124 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 ...
121 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 ...
284 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 $\lambda(t_i)/\int_{A}\lambda\left(t\right)dt$ ...
474 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 ...
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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### 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 ...
35 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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### 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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### 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 ...
609 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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### 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 ...
79 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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### 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 ...
143 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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### 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 ?
310 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 > ...
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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 ...