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# Questions tagged [queueing]

The mathematical study of waiting lines.

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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 $P(H>C)$} \...
83 views

49 views

### 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 ...
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 ...
1 vote
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 ...
299 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 ...
548 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 ...
1 vote
46 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 some ...
35 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 ...
1 vote
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 ...
25 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 ...
1 vote
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 ...
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 ...
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 $L = \lambda$ ...
1 vote
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 ...
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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 ...
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 ...
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 ...
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. ...
1 vote
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 ...
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{...
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 ...
1 vote
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 ...