Questions tagged [queueing-theory]

Queueing theory is the mathematical study of waiting lines, or queues.

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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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Choosing between M/M/1 vs D/M/1 queuing model

I'm designing a system where I could choose the arrival times. I'm trying to understand which distribution will provide a better system performance and SLA. Context: Users have some task run once a ...
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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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Bounds for the first time of the workload hitting zero in a G/G/1 queue

Consider a G/G/1 queue, with sequence of i.i.d inter-arrival time $T_1, T_2, T_3,\cdots$ and i.i.d service time $S_1, S_2, S_3, \cdots$. Suppose system is initialized with workload $w>0$ and ...
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Joint Waiting Times distribution in M/M/1 queue

Consider an M/M/1 queue, where arrivals to the queue occur according to a Poisson process of rate $\lambda$. We assume FCFS (First Come First Serve) service discipline with i.i.d. exponential service ...
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maximal increment of renewal process and partial sums when only second moment exists

I am working on a problem that needs to bound the increment of the absolute value of centered partial sum process and its associated renewal process. the iid partial sum process $S(t)=\sum_{i=1}^{\...
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Expected queue length of a Poisson process while service time is based on an unknown distribution with known expected value $\beta$

Suppose we have a service provider with an in-process task queue in which tasks arrive based on a Poisson distribution with parameter $\lambda$. Moreover, the service time of each task is based on an ...
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Estimation of many jobs running concurrently given distribution?

In our system, we are running jobs which are queued with a distribution determined by analyzing empirical data, sample below: Time between consecutive jobs (s) How many times this happened 0 257 1 ...
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Convergence rate of Erlang loss b formula

Consider the Erlang loss B formula $E(N, x) = \frac{\frac{x^N}{x!}}{\sum\limits_{i=0}^N \frac{x^i}{i!}}$, where $0<x<\infty$ and $N$ is a positive integer. As $N \to \infty$, $E(N, x) \to 0$ and ...
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How is the Markov chain transition matrix of an M/M/1 queueing process derived (or interpreted)?

I have a basic understanding of Markov chains as transition probabilities between states, but I just started reading a bit on queueing processes, and found this matrix on Wikipedia and several ...
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Ergodicity of a queue

I am simulating a queueing network (I assume the details are irrelevant, because the question is theoretical, this is just the context) with exponential inter-arrival and service times. Let's assume ...
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Could you define a rate of emptiness in a M/M/1 system?

Suppose to have one server and two queues, say A and B. In both queues arrive jobs with exponential inter-arrival time with rate λ, but only A is connected to the server. The server has an exponential ...
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Maximum input throughput, i.e., arrival rate [at steady state] that a queue can exhibit, while guaranteeing a maximum waiting time threshold.

Given a certain unbounded queue where customers wait before being served by a single worker. Suppose the worker service time follows a distribution G. It is required to guarantee that: NO customer ...
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departure process of M/G/1 queue with hyper-exponential service times

I am working on two queues in tandem; the first queue is M/G/1 with hyper-exponential service times and the second queue has exponential service times. I want to know if the second queue can be ...
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Why is an M/M/1 queueing system unstable when $\lambda=\mu$?

It is common knowledge that an M/M/1 queueing system is stable if $\lambda<\mu$, where $\lambda$ is the arrival rate and $\mu$ is the service rate. When $\lambda>\mu$, it is quite obvious that ...
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Average queue length in short-run at early stage

Consider a simple M/M/1 queue (for example), with arrival rate $\lambda$ and service rate $\mu$. Under the stability condition ($\lambda<\mu$), Queueing Theory says that the average queue length in ...
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M/M infinity queueing model.

Customer arrivals at a 7-Eleven is Poisson at the rate of 20 per hour. They can be assumed to spend an average of 12 minutes picking up merchandise, with the length of time having an exponential ...
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Upper bound the probability that the maximum of i.i.d. r.v.'s (e.g. busy periods) exceeding a threshold

Suppose that $B_1, B_2,\ldots, B_n$ are a series of positive independent and identically distributed random variables. The moment generating function (MGF) of $B_i$'s is known, denoted as $M_{B}(\...
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Finding Arrival rate and Service rate of a Queuing Systems.

I'm student and I want to analyze performance of my client-server application, the response time can go from 0.6s to 0.8s depending the task and can handle 5 request at a time (M/M/5). Now I define ...
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M/G/1 Queue as a Markov Renewal Process: One step transition probabilities

Seeking help on this interesting problem! any input is welcome and appreciated Background From many texts, we know that for an M/G/1 queue, denote $I_n$ as the number of customers in the system ...
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Show $\gamma(t)\leq 0$ for almost all $t$ with $\max_{u\leq t} \int_u^t \gamma \,\mathrm d\lambda = 0$

Given a locally integrable function $\gamma: \mathbb R_{\geq0}\rightarrow \mathbb R$, we define the absolutely continuous function $\Gamma(t) := \max_{u\leq t} \int_u^t \gamma \,\mathrm d\lambda$. I ...
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Derivative of $\Gamma(t):=\max_{u\leq t} \int_u^t \gamma \,\mathrm d\lambda$

Given a locally integrable function $\gamma:\mathbb R_{\geq0}\rightarrow\mathbb R$ we define the continuuous, non-negative function $\Gamma(t):=\max_{u\leq t}\int_u^t \gamma \,\mathrm d\lambda$. $\...
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Wrong expected value definition in book? [duplicate]

I am currently hospitalised and reading a queueing theory book. I encountered in a proof this, and I fail to understand how this is true: $$E[R_j]=\int_0^\infty{P(R_j>u)du}$$ Other than the fact ...
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Existence of Locally (Lebesgue-)Integrable Function

Given a locally integrable function $f: \mathbb R_{\geq0} \rightarrow \mathbb R_{\geq0}$, I wonder whether there exists an equivalent function that operates at a certain capacity $\nu\in\mathbb R_{>...
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Expected Number of Batches of Size $K$

I think this is a question for the compound Poisson process which I am unable to solve. The question is as follows: Batches of customers arrive at a shop at instants $S: Ω → {\Bbb{R}_+}^{\Bbb{Z}_+}$ ...
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Adding feedback to an M/M/c queue

I am trying to model a queue with the following properties: A multi-server/channel queuing model. Arrivals occur via a Poisson process and the service times are exponentially distributed. After ...
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M/M/4 queue distribution and expected wait time

I have M/M/4 queue with an arrival rate of λ= 1.5 per minute and an expected service time of 1/γ= 2 minutes. At arrival four persons are being served and 10 are waiting in line. Q: What distribution ...
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Deriving the state transition diagram for Markov Process

I'm preparing for an exam and one of the preparation questions for the exam is the following: Consider a computer system with TWO processors and NO waiting queue. Out of the two processors, one is ...
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Balance Equations Intuition and Non Markovian Queues

I have a question regarding the intuition under the balance equation in a birth death process, wich can be used to solve simple Markovian queues. The idea is in the long term, the times u go from ...
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Queueing average time with 3 servers

Suppose the bank has three service lines. Customers arrive to the first, second, and third lines according to an exponential distribution with rates $\lambda = 3$, $\lambda = 4$, and $\lambda = 8$. ...
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Queueing Theory - Differences between Utilization factor and 1-P0?

Whilst studying Queueing Theory, I noticed that when calculating the utilization factor and 1-P0, the two values would be different. As the utilization factor is the expected fraction of time the ...
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Infinitesimal Generator for the G/G/1 queue

I read the infinitesimal generator for the M/M/1 queue and thought to generalize to the G/G/1 queue. More specifically, though the queue length process is not Markovian anymore, we could consider an ...
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Model car queueing with two gates

Say a school has two gates at which parents can pick up their kids. One gate is only for primary school students and the other is only for secondary school students. The school has a primary school to ...
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Stationary distribution of M/M/∞ queue with customer types

I'm new to queueing theory but think I have a problem sufficiently well-described by a M/M/∞ queue. I am seeking to determine the proportion of time that is spent servicing $n$ customers (...
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Can two consecutive Brownian queues be empty at the same time?

For a function $f:\mathbb R \rightarrow \mathbb R$, define $f(s,t) = f(t) - f(s)$. A two-sided Brownian motion $B$ is defined by $$ B(t) = \begin{cases} B^{(1)}(t), &t \ge 0 \\ B^{(2)}(-t) &t &...
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What is the stationary distribution for this continuous-time variation of the M/D/1 queue process

Consider the following process: $X$ is a nonnegative real function of a continuous time variable $t$, so long as $X\geq 0$, it decreases deterministically at rate $\mu$ (i.e., $X(t+dt) = X(t) - \mu\, ...
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How to calculate average waiting time in a M/D/c queue?

I consider an M/D/C queueing model where arrivals are Poisson distributed, processing time is fixed at D, and the server is C. However, we could not find a valid way to calculate the average waiting ...
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$M/Erlang-2/1$ number of customers distribution

Is there a known closed-form distribution of the number of customers in a $M/Erlang-2/1$ system with service rate $2/\mu$ and $\lambda$ as the arrival rate to the system? I wonder if such a general ...
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Find the average waiting time in M/M/1 Queue

Suppose that two types of customers arrive to a queue with a single server. Type $A$ customers arrive according to a Poisson process with on average $3$ minutes between customers, while type $B$ ...
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