Questions tagged [queueing-theory]
Queueing theory is the mathematical study of waiting lines, or queues.
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Non-standard Queuing Theory problem (Modified M/M/1)
Haven't see anything like it in queueing theory, the problem appears to be M/M/1 but with a twist?
Description:
The server starts serving calls only when the number of customers in the system becomes ...
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Intersections of line segment with discs distributed as blue noise?
My question
Suppose that we have discs of radius $r$ in the plane, where the discs' centres are distributed as blue noise with a minimum centre-to-centre distance $d$ with $2r < d$; equivalently (...
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Queue system with 2 parallel servers that works one at a time. Mean waiting time?
Consider a queueing system where customers arrive according to a Poisson process with rate $\lambda$, but the service facility consists of two parallel servers. A customer upon entry into the service ...
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Expected idle time of a server in an M/M/N queue
Consider a standard M/M/N queue where the jobs' arrival rate is $N\lambda$, the service rate of each of the $N$ servers is $1$, and $\lambda < 1$.
When a new job arrives, assign it to an idle ...
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Percentage Jobs drop in $M/M/1 K$ Queue with Finite Queue Length
In my simulation of the $M/M/1 K$ Queue, the arrival rate $\lambda$ is $2.7\ \mathrm{ jobs/s}$ while the service rate $\mu $ is $3 \ \mathrm{ jobs/s}$. The capacity $K$ of the buffer is $5$. I am ...
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Can you explain the queueing question/answer in this MBA exam?
In the paper that describes how ChatGPT could pass an MBA exam there is a question and associated answer relating to queueing that seems intuitively wrong. I copy the question and the answer the ...
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High water mark probability of an M/M/1 queue
I'm familiar with the steady-state behavior of an M/M/1 queue. For example, the arrival rate of $\lambda = 2$ per minute and servicing rate of $\mu = 3$ per minute means that utilization $\rho = \...
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Conditional average number of tasks in M/M/$\infty$ queue
I'm trying to solve the following problem:
Consider an M/M/$\infty$ queue in which the time intervals between task arrivals are i.i.d. exponential with parameter $\lambda$ and task durations are also ...
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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 ...
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Fraction of time that on-ramp holds $N$ or more vehicles
Given a M/M/1 queueing system where vehicles are admitted an on-ramp at a rate of $2$ vehicles/s and serviced by highway admission rate of $2.5$ vehicles/s. What is the fraction of time that the on-...
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Number of departures from n servers
Suppose There are n severs each of which has a service time exponentially distributed with mean m. What is the number of servers which complete their task till time t, nm m$\gg$t.
I am expecting it to ...
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How many people arrived in T/4
The number of people arrived during a time interval follows Poisson distribution. Suppose there is only one customer arriving in the shop during time $T$, how many people arrive during time $T/4$?
I ...
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How to solve a matrix functional equation containing matrix exponential?
For a given complex value $s$, I wonder how to solve a matrix functional equation for matrix $\hat{G}(s)$ in the following form
$$\hat{G}(s)=\int_0^\infty e^{[D_0+D_1\hat{G}(s)-sI]x}dB(x),$$
where $...
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A strategy for photographer using limited camera and films to take photos for all houses
I was asked a question about developing strategy for a photographer. This guy has a camera which can take maximum $n$ photos. And he hangs on the street consisting $p$ houses. He wants to take photos ...
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Number of departures in queues as an integral
In a queue with unit mean Poisson service times the number of departures are given by
$$D(t)=\int_0^t \mathcal N(ds)$$
where $\mathcal N$ is the unit mean Poisson process. How should I think of this ...
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Erlang B formula to solve a probability problem
I am trying to solve this problem where I need to express a probability problem as a queueing problem. I'm having troubles getting the distributions right.
Here's the problem:
There are $n$ keys, only ...
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Simple process equivalent definitions
How do I prove that the two definitions of a simple process are equivalent?
Definition 1.
Let arrival process $\xi(t)$ satisfy the Markov property.
Let the probability of $k$ arrivals within the time ...
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M/M/c queue is it time sliced or discrete?
I am using a queuing model based on an M/M/C queuing calculation. I am interested to know if it is generally a time slice model and not discrete? I'm not able to provide the calculation being used, ...
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Question about Markov Chain's - Queueing Theory
Consider a Markov chain $X_n$, for $ n=0, 1, 2, \ldots$, with states 0, 1, 2, whose transition probability
matrix is
P = \begin{bmatrix}
0 & 1/2 & 1/2\\
1/2 & 1/2 & 0\\
1 & 0 & ...
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Algorithm to create histogram of Queue
Say we have a standard queue, where entities are waiting to be processed.
A histogram can be generated which represents the "average length of the queue".
This should be somewhat ...
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M/M/1/1 queue with reentering
$\mathbf{Task}$: M/M/1/1 queue where user who get serviced reenters the queue with probability $\alpha$(going to the beginning of the queue). Find limiting probabilities.
$\mathbf{My}$ $\mathbf{...
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MMPP/M/1/K model
I am trying to understand a research for the call admission control optimization. so according to that research we have a below statement:
We take into consideration that packets arrive in bursts and ...
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Queue models with arrival rate equal to the service rate?
What happens in a queue system with arrival rate $\lambda$ equal to the service rate $\nu$?
It is possible to find its equilibrium distribution?
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Pair Matching Algorithm Ideas?
I am trying to work out a small task where I need to match up a sequence of events that have offsetting scores and see if there is a better theoretical framework to solve this in a more efficient way
...
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Why exponential distribution to model a queue?
I am studying the queing models and I have a doubt.
Why to model intrerarrival times between two clients is used the exponential distribution?
I know that this distribution has the memory less ...
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Why we need a probabilistic model for queues
Good evening.
I am thinking about the probabilistic queuing model M/M/n with n servers, arrival rate $\lambda$ and service rate $\mu$.
The model allows to estimate the waiting time in the queue and ...
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How do you compute the result of an equation with discrete values
A bit of background - I had a phonecall appointment with the drs and it wasn't until a few hours after the appointment time that I received the call. I decided to try and model how much extra time I ...
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Finding solution of an exponential function divided by factorial
I need to find the solution of $\dfrac{4^n}{n!}=51.8682$. Though I need $n$ to be a natural number, if $n\in\mathbb{R}$, I can take the celling of $n$ for problem purpose.
It can be done by trial and ...
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Determination of whether additional attendant is required or not
Question:
Workers come to tool store room to receive special tools for accomplishing a particular project assigned to them. The average time between two arrivals is $60$ seconds and the arrivals are ...
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Queueing Theory.Poisson distribution.
$\mathbf{Task}$: The machine's failure stream is the simplest, with an average failure interval of 300 hours of continuous operation. every day the machine is turned on for 8 hours. How long on ...
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How to calculate Markov Chain balance equations when number of servers change dynamically
I'm trying to understand how balance equations work when the number of servers in a system can change on demand. For example, say the number of servers in a system is 1 when the number of customers is ...
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Relationship between average of exponentially distributed arrival times and expected value of minimum of arrival times
I saw on a reply in the forum (https://math.stackexchange.com/a/4298641/1100748) that the average time of three exponentially distributed arrival times is equal to the expected value of the minimum of ...
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Stacking boxes into a limited height box, all with same width and length, only heights change
My problem is as follows:
I have ordered boxes to stack into a huge box.
All the boxes including the huge box have the same width and length.
The huge box has a constant height $H$.
All the other ...
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How to model this stochastic process?
I was thinking of a birth-death style stochastic process, but I'm not quite sure how to model it or classify it.
It is defined as follows. At each step in the process:
An individual is born/...
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occupancy rate of an M/M/$\infty$ queue
Derivation of occupancy rate of an M/M/$\infty$ queue
https://en.wikipedia.org/wiki/M/M/%E2%88%9E_queue
I am presuming occupancy rate $\rho = \dfrac{E[S]}{E[A]}$ where $S$ is the exponential service ...
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waiting time for D/M/1-LCFS queue
The stationary distribution of waiting time for a D/M/1 queue is well known if the first-come first-serve (FCFS) discipline is adopted.
If, however, the last-come first-serve (LCFS) discipline is ...
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Understanding the Markov Chain based expression
I am newbee in the field of stochastic process and I am reading a research paper wherein the following expression is given which I am not getting clearly.
Consider a discrete time Markov chain having ...
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M/M/1/10 queueing process with two different classes
I'm looking at a problem where we have calls queueing under two different classes, new calls and handovers. The number of calls arriving follow a Poisson process with $\lambda_{1} = 125$ per hour ...
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$M/M/2/3$ Queuing Theory Word-Problem
A service center consists of two servers, each working at an exponential rate of two services per hour. If customers arrive at a Poisson rate of three per hour, then, assuming a system capacity
of at ...
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Queueing multiple variables beyound Markov Blanket in Bayesian network
Afai understood, the variables beyound Markov Blanket does not influence on the node in Baesian Network.
Howewer, if i give some compound query, where 2 or multiple variables are given, and those ...
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Is Little's Law applicable to all Continous Time Markov Chain Models?
I was reading about Little's Law which is in general(infinite capacity system) form L = R*W ( R: throughput rate ,W : expected waiting time of a customer). I know it is applicable on M/M/S type ...
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FIFO (M/D/1/N queue) Overflow probability
I'm an engineer, trying to figure out the optimal FIFO depth for a particular application. The events I usually work with follow a Poisson distribution, and a fixed readout rate (thus the M/D/1/N ...
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How to determine $M(t) =E[X(t)]$ in an $M/M/1$ queue system with a differential equation?
Suppose that customers arrive at
a single-server service station in accordance with a Poisson process having rate
λ. That is, the times between successive arrivals are independent exponential
random ...
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Exsitence of stationary distribution for M/M/1 with non-homogeneous poisson arrival rate
Consider an M/M/1 queue where arrivals occur at rate $\lambda(t)$ according to a Poisson process at time $t$ and move the process from state $i$ to $i+1$, and service times have an exponential ...
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If task start time is chosen from a uniform random distribution, is inter-arrival time exponentially distributed
Question: If the start time is generated from uniform random distribution, is the inter arrival time exponentially distributed?
Context:
We have a number of tasks to be run every day (and we have the ...
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Poisson Process as Exponential Interarrivals
I was trying to understand the derivation of the poisson process as a counting process with exponential interarrival times, and came upon this source. They've defined $S_n$ as the time of the $n$th ...
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Numerically solving a PDE in Matlab: Eikonal in an exponential formulation and initial/boundary conditions
I am currently working on a small research project as a part of my degree. The project is centered around Continuous-time Markov Chains, and based on this paper :
https://hal.archives-ouvertes.fr/hal-...
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M($\lambda$)/M($\mu$)/n queue, recurrence and busy periods
Question
A surfing company has a very high number of surfboards to rent. The owner makes the following estimates. A new individual customer enters the shop on average every 5
minutes. Then, each ...
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Loss networks and Poisson processes
Consider a shop with capacity C, where customers spend an i.i.d exponential(1) amount of time before leaving through the back door. Customers arrive through the front door as a Poisson($\lambda$) ...
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On expected waiting time and distribution of departure times in M/M/1 queue, condition on the number of departures
Suppose we have an $M/M/1$ queue with arrival rate $\lambda$ and service rate $\mu$. Let $S_i$ and $D_i$ denote the arrival time and departure time of the $i$-th customer, and fix $S_1$ as $0$.
Given ...
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