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

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Does this resilience/resource scheduling analogy make sense?

A firend has recently presented an analogy for the rescheduling of Doctors (big topic in the UK atm) across a 7 day week as opposed to a 5 day week with a skelton staff at weekends - I'm ignoring ...
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Service Systems Problem with Repair

Can anyone help me with this problem? A data centre is equipped with $M = 5$ servers, completely interchangeable. Each of them can fail independently of the others. The time to failure of a ...
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Waiting time in Queue with $N$ policy

Suppose that a queue has $N$-policy, that is, only when there are $N$ customers in the queue the service starts. In such a queue with deterministic arrival rate $\lambda$ and deterministic service ...
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Queuing systems, what is the fraction of customers lost?

In a car repair shop with only one birth to repair the cars, cars arrive for repair in a poison process at a mean rate of four cars per day. The manager decides not to accept any car for repair if he ...
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Queuing systems with different arrival and departure job multiplicities

Consider the following. A queue has an arrival rate of $\lambda$, where a single job enters the queue. Next, the job is processed by the service at rate $\mu$, and it emits three jobs in response to ...
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23 views

Load balance N customers over K servers with different capacities

Let's say we have N customers that supply a stream of requests, but each customer i supplies different number of requests per minute - $R_i$. All requests are identical in terms of the amount of ...
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39 views

Distribution of the random time for queuing system to change from full to empty.

Question: Find the distribution for the (random) time it takes an $M/M/1/2$ queuing system with $\lambda = \mu = 1$ to change its state from being full to being empty. ($\lambda, \mu$, arrival rate ...
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Simulation of the variance of a typical waiting time W(q) in a queue

Write a computer programme that by means of stochastic simulation finds an approximation of the variance of a typical waiting time W(q) (in the queue) before service for a typical customer arriving to ...
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45 views

M/M/1 or M/M/n?

In a queuing systems with a single queue that receives $n$ poisson arrival streams with arrival rates $\lambda_1, \lambda_2, ...,\lambda_n$, and exponential service rates of $\mu_1,..., \mu_n$, we can ...
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Waiting Time or Response Time Distribution for M/G/1/K Queue

I can't find any formulas for the LST of the waiting time distribution or the response time distribution of the M/G/1/K queue. Does anybody know any references and sources that they could point me to? ...
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Poisson $ (\lambda) $ process: Relation between $ n^{th} $ arrival time $ S_n $ and maximum of $ n $ arrival times

Poisson $ (\lambda) $ process: Relation between $ n^{th} $ arrival time $ S_n $ and maximum of $ n $ arrival times Sheldon Ross's Stochastic Processes, second ed: Consider a Poisson ($\lambda$) ...
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$M/M/2/4$ simulation of the probability that the queue gets full during first $10$ time units.

Let $X(t)$ denote the total number of customers at time $t \geq 0$ in an $M/M/2/4$ queuing system in steady-state (/started according to its stationary distribution) with Poisson arrival process with ...
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Where does the properties of queue systems come from?

I just started a course on queue theory, yet equations are given for granted without any demonstrations, which is very frustrating... Thus Why is the mean number of people in a queue system ...
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Fork-Join queue stability

According to the wikipedia article: There are $N$ queues, the index of a single queue is $i \in N$. For the fork–join queue to be stable the input rate $\lambda$ must be strictly less than sum ...
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How to find the service rate $\rho$?

The engineer of a small atelier observes. $6$ workers employed in this workshop are versatile, so that any order can be done by any of them. Nevertheless, the engineer is stressed because he ...
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Birth-Death process with shifted exponential distribution

In the general framework of $M/M/1$ queue we have rate $\lambda$ and an exponential service time $\mu$, we can set up the transition rate matrix intuitively. However, if the service times satisfy ...
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46 views

What distribution model should be used to model customer arrival times?

I have this multiple choice problem that is testing my understanding of distribution models. I cannot come up with the correct one to solve the problem. Any help would be greatly appreciated! ...
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57 views

Queueing system: M/M/2 vs 2*M/M/1

I want to examine the difference between two systems: Single queue with arrival rate $2\lambda$ and 2 servers with serving rate $\mu$ A systems with 2 queues, each with arrival rate of $\lambda$ and ...
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Proof that Poisson process interarrival time $T(N+1)-T(N)$ with $T(N)<t<T(N+1)$ is Gamma$(2,\lambda)$

Suppose a Poisson process $N(t)\sim\text{Poisson}(\lambda t)$. Let $T(N)$ be the time of the last arrival before time $t$ and $T(N+1)$ be the time of the first arrival after time $t$. From ...
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What is the mean number of cars waiting to pay at a toll?

Minute is the chosen time scale. A highway ramp has only one tollbooth. Cars introduce themselves according to a Poisson process of $\lambda = 0.3$. The at the toolboth follows an exponential law with ...
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66 views

Verify that $L = \lambda W$, $M/M/2$

I have a $M/M/2$ system, with traffic intensity $\rho = \frac{\lambda}{2\mu}$. I will call $\boldsymbol \pi$ the stationary distribution. Then, I have formulas for $\pi_0$ and $\pi_k$, and I also have ...
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True or false: The process $\{ X(t), t \geq 0 \}$ at a $M/M/s$ queue is a reversible Markov process.

Let $X(t)$ denote the number of customers in a system at time $t$. The process $\{ X(t), t \geq 0 \}$ at a $M/M/s$ queue is a reversible Markov process. Is this statement true or false for: (a) ...
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37 views

Probability that two or more workers are being served or waiting to be served?

The problem is Customers arrive at a tool crib according to a Poisson process of rate $\lambda = 5$ per hour. There is a single tool crib employee, and the individual service times are ...
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32 views

Simple $M/M/1$ service time

I'm trying to understand queueing systems and I found some notes online. They define $\lambda$ as the mean arrival rate, and $\mu$ as the mean service rate (the average number of customers who can be ...
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Balance equations looks the same in different cases for a problem

In a supermarket customers arrive at the cash desk with a Poisson process with an average of 30 customers per hour. There is one cash desk and the service time is exponential with an average of 2 ...
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How to find $E(L^q)$ in queue processes?

Let say we have a Coast Guard center wich have a emergency center hold only one rescuer. accidents comes according to a Poisson process. There is usually 16 casualties on 8 hours. Casualties are ...
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Prove theorem that concerns to sums of exponentials.

I try to solve this theorem: Theorem: Let $x_1, x_2,...$ iid with exponential distribution with rate $\lambda$. The density function of $S_n$ is given by: $$ P(S_n \le ...
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M/M/m Queue birth and death parameters

I would like to determine $\theta_k$ for a $M/M/2$ system. I understand that $\theta_0 = 1$. I also believe that $\lambda_1 = \lambda_2 = \lambda$, and $\mu_1 = \mu$, and $\mu_2 = 2\mu$. Therefore, I ...
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21 views

assign customers to queue to minimize time

I have a problem where I have a fixed number of N customers, and I have a number Q of queues and each queue serves the customers at different rates, so my question is, is there an algorithm, ...
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19 views

Why multiple Poisson streams taken as sum of IID rather than joint PDF?

I have a very basic doubt regarding queueing theory. We argue that merging of more than one IID Poisson streams (say $i$ streams) with respective parameters $\lambda_i$ is also Poisson with parameter ...
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Wait time in queue for 2 server system (exponential process)

A customer has to be server 1 before being served by server 2. Service times are exponential with rates $\mu_1$ and $\mu_2$ respectively. After being done at server 1, you wait at that station till ...
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Transition Probability of M/M/1 Queue given any constant observing period T

I am trying to find the transition probability for $M$/$M$/$1$ queue given any constant observing period $T$ (if $T$ go to infinity the transition matrix will degenerate to a matrix with identical ...
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model with markov chain

Suppose to have the following situation: At a bar at each time unit arrives a certain number of customer with probabilities $p_1,p_2,...,p_n$. In the bar there are 3 bartenders so 3 customer can be ...
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Average customers/hour equivalent to arrival rate?

For a Basic Single Server problem, M/M/1, I'm given the average customers/hour at clerks desk. The solution takes that figure as λ. That doesn't make sense, that figure is essentially how many ...
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Inversion of Laplace transforms - simplifying the Bromwich integral

I have trouble following the derivation of equation $(2)$ in this paper. The authors define the Laplace transform of a real-valued function $f(t)$ of a positive real variable $t$ as \begin{equation} ...
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Understanding the Fluid limit model in Queueing Theory

I am self-studying the fluid limit model in queuing theory. The basic setting is that: let $Q(t)$ be the total number of customers in the system. Consider the simplest case of M/M/1 queuing system. ...
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Poisson arrivals, poisson departures with rate proportional to queue length

Suppose arrivals to a queue (/collection, more accurately) follow $Poisson(\lambda_a)$. In other words, the arrival rate is constant $\lambda_a$. Departures, however, at time $t$ go at rate ...
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Pay off of adding new service station

I have to solve the following problem: In a factory they do reperations of a certain type of machine. The machines that need reparations arrives at the factory with a frequency of 4 every hour, such ...
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interarrival time

I am trying to model the arrival rate (delay between the arrivals) of the cars in my city. I have some real data with the resolution of one minute. For example, Number of counted cars at position x at ...
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What is the probability of never reaching a state in Markov chain?

The markov chain consists of states $a_1$,...$a_k$. The initial state is $a_1$. For the state $a_{i}$ the next state can either be $a_{i+1}$ with probability p, or the initial state $a_1$ with ...
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What is the name of this problem about queuing processes?

The problem is: suppose we have $p$ processes labeled $1, \ldots, p$, with runtimes $(t_1, \ldots, t_p) \in \mathbb{R}_{++}^p$. In what order should the processes be set up to minimize overall runtime ...
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44 views

Combined arrival rate

Let us suppose a scenario with two clients, $a$ and $b$, each one generating load at rate $\lambda_a$ and $\lambda_b$, respectively. The server receives the requests from both clients. What will be ...
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Is there a Little law for a network of two connected queues?

From Patterson et al' Computer Organization and Design: Throughput and Response Time Do the following changes to a computer system increase throughput, decrease response time, or both? ...
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Survey on large deviation bounds of queuing delay in CSMA scheduling

I am trying to do some literature survey on the theoretical guarantees in uplink scheduling algorithms. I found there exist a series of papers from UIUC and UC Berkeley by L.Jiang, J. Walrand, R. ...
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What is the capacity of Earth? Is the Earth stable?

I'm learning queueing theory and just finished Little's Law and Utilization. If the Earth is interpreted as a system that provides a service for its customers, is it unstable? Let the customers be ...
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139 views

Probability distribution of number of waiting customers in front of a counter [closed]

The number of customers arriving at a bank counter is in accordance with a Poisson distribution with mean rate of 5 customers in 3 minutes. Service time at the counter follows exponential distribution ...
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Confusion regarding Burke's theorem

Arrivals occur at rate $\lambda$ according to a Poisson process the service time have an exponential distribution with parameter $1/\mu$ in an M/M/1 queue, where $\mu$ is the mean service rate where ...
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Effective inter-arrival time converge to mean

I am fairly new to statistics and just recently encountered queueing theory. I have programmed a simulation for a $M/M/1$ queue in which I specify the inter-arrival times and service times. I input ...
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From one-dimensional to two-dimensional Markov chains

I have a $M/M/1$ queueing system that is described below: There are two types of customers in the system with different arrival rates, $\lambda_{sg}$ and $\lambda_{sb}$. Service rate is $\mu$. Type ...
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Final rank, given initial rank and a probability distribution

I got an interesting problem today, though I could not find a closed form solution to it. Imagine a setting where we have people submitting solutions, which are ranked. Given some initial rank of an ...