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

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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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Queuing Model Finite Population Question

Manufacturing Company has a group of six identical machines; each machine operates an average of 20 hours between breakdowns on average. One maintenance operative looks after the machines and takes an ...
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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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Delay dependency single server

I have a question regarding dependencies between consecutively served customers at a server. Assume I have two customers arriving at the same server with a fixed gap of $Y$. I'm interested in the ...
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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 ...
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Probability that a birth--death process crosses level $n$ in $(0,T)$

This question is inspired by this question. Jobs arriving according to a Poisson process with rate $\lambda$. Jobs stay in the system for a fixed amount of time $d$ and depart thereafter. Let $X(t)$ ...
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How to compute the arrival rate

I have a data set of interarrivals . I need to compute the arrival rate. Should I compute the mean interarrival and then inverse it to get the arrival rate ? or should I inverse all the ...
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Queue theory - M/D/k - Probability of never having a queue before a time T

This is probably a known result, but I couldn't find any resource pointing directly to the issue I'm trying to solve. Suppose you start a logistic mission that needs that during its time $T_m$ a ...
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37 views

M/M/1 vs G/G/1 vs G/M/1

I am using queuing theory to model a router. I have a model that assumes Poisson traffic and I need to modify it as my actual traffic is not Poisson I want to ask what's the main difference between ...
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Customers and Anti-Customer Queueing Problem: What is the Customer delete probability

Hello may ask for your help? First the setting: I have got a problem with some queueing theory. The whole problem would be a grid of nodes, all nodes have an operation intensity $\mu_{i,j}$. ...
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M/M/1 Queueing theory with change in rates

Consider a single-server exponential system in which customers arrive at a rate $\lambda$ and have a regular service rate $\mu$. When a customers arrives and the system is busy, the customer joins the ...
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Probability of a message being processed in a given interval

Let's say I have a queue and a server that connects to the queue in an interval basis and process all the available messages. The server connects, tries to get a message, if there is one then server ...
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Server farm model and non-explosiveness

Consider the following server farm model. Customers arrive at a server farm according to a Poisson process at rate $\lambda$, each requesting a machine for an amount of time that has an exponential ...
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Expectation of the time difference between starting times in queueing theory

Consider 2 independent, parallel $M/M/1$ queues $Q_1, Q_2$ with identical arrival rate $\lambda$ (corresponding to an exponential random variable $A \sim \text{Exp}(\lambda)$) and service rate $\mu$ ...
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A type of continuous time Markov process

I am looking for a stochastic process model with the following features. It is a continuous time Markov process---modelling, if you like, the evolution of a population. New arrivals are added to ...
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116 views

Jackson's theorem to optimize mean queue length of a traffic model

I am working on traffic signals for a city transport system. I modeled the city transport using a queuing network as shown in the following image Arrival rate of "A" cars from outside is S1 and ...
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Laplace-Stieltjes transform M/G/1-queue

Two types of jobs arrive at a machine. Type $1$ jobs arrive according to a Poisson process with rate 45 per hour and need an exponential service time with mean $1/2$ minute. Type $2$ jobs arrive ...
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Distribution of the number of customers

We have the following situation Passengers are brought with small vans from the airport to hotels nearby. At one of those hotels on average $6$ vans per hour arrive according to a Poisson process. ...
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Distribution of server utilisations in an M/M/c queuing model with an unusual dispatching discipline

I'm studying an M/M/c queuing model with an unusual (?) dispatching discipline: Servers are numbered 1...c The servers have an identical mean service time, exponentially distributed (as usual), ...
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2 User Queuing Model Probability Problem

Consider two users who arrive to a system with exponential arrival times with parameters $\lambda_a$ and $\lambda_b$. Once they arrive, the users stay in the system for an exponentially distributed ...
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Problem with Little's law

I am estimating number of threads required by my server to execute clients requests efficiently and initially I starts 4 threads on the server.Request arrival rate on my server is 4 request/sec and ...
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Time between arrivals Distribution

I am simulating a hair parlor queue with m number of queues and 3 different types of services (queues). I was doing the time between arrival with a uniform distribution with a min value and a max ...
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Why Little law is not fit for web request arrival rate scenario?

I am estimating number of threads required by my Server to execute clients requests efficiently and initially i starts 4 threads on the server.Request arrival rate on my server is 4 request/sec and ...
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Question about $M/GI/ \infty $ queue

Consider an $M/GI/ \infty $ queue with the following service time distribution: the service time is $1/\mu_i$ with probabbility $p_i$, and $\sum_{i=1}^kp_i=1$ and $\sum_{i=1}^kp_i/\mu_i=1/\mu$. In ...
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Can You Help Me With This Continuous Markov Chain Question?

Consider 2 machines, both of which have an exponential lifetime with mean $\frac{1}{\lambda}$. There is a single repairman that can service machines at an exponential rate $\mu$. Set up the ...
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Interarrival time for Counting process?

Messages arrive at an interactive message center according to a counting process with the average inter-arrival time of 15 seconds. Choosing a frame size of 5 seconds, compute the probability that ...
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M/G/1 queueing problem

I need to prove that in the M/G/1 queueing system with Poisson arrivals with parameter lambda and exponential service time with parameter mu, that q_k = (lambda/(lambda+mu))^k (mu/(lambda+mu)).
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Compute the standard deviation of the monthly cost due to blackouts

Network blackouts occur at an average rate of 5 blackouts per month. Assuming a suitable continuous-time counting process, a. Compute the probability of more than 3 blackouts during a given month. ...
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Determine the probability that the time interval between successive job arrivals is :

Consider a server with Poisson job-arrival stream at an average rate of 60 per hour. Determine the probability that the time interval between successive job arrivals is a. longer than 4 min b. ...
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queuing problem, related to marriage algorithm

Say we have an nxn matrix and for every entry a_{ij}, it equals 1 if flight j starts after flight i ends. Otherwise it is 0. Suppose the largest matching contains M marriages (i.e. 1's in nxn matrix ...
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Markov Process - formulate a Markov chain model for this system ( what is q(i,j)?)

Potential customers arrive at a full-service, two-pump gas station according to a Poisson process at a rate of 40 cars per hour. There are two service attendants to help customers, one for each pump. ...
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Modelling a continious-time queue which behaves differently when there are more or less people being served.

For my research I am trying to model a continuous-time queue which behaves differently when there are more or less people being served. The arrival rate in the queue is constant, however the departure ...
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Optimal scheduling policy for two server model

Suppose there are two servers with exponential arrival rates $\mu_{1}$ and $\mu_2$ such that $\mu_1 > \mu_2$. These two servers have a shared infinite buffer, where there is independent Poisson ...
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75 views

m/m/1 Queuing Theory - Average Delay/Utilization Graph confusion

I'm currently coding a single server single queue m/m/1 simulation with Lambda as arrival events rate (Poisson) Omega as service rate (Exponential) and I'm having problem understanding how to ...
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parameters of queuing theory

Consider a queuing system that follows the model M / M / 1 wherein the average time between consecutive arrivals is 12 secons and the average service time is 3 seconds. ...
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Can I think of both arrival times and service times in a Markov chain as Poisson processes?

According to the Wikipedia article about M/M/1 queues, the rate at which new jobs arrive is a Poisson process with parameter $\lambda$, and the rate at which the jobs are finished is an exponential ...
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Applying Generating Function Approach to a $M/E_r/1$ queue

(This question is about Exercise 27 on page 55 from these lecture notes.) We consider a $M/E_r/1$ queue with arrival rate $\lambda$ and mean service time $r/\mu$. We let ...
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Formula for Pipelining

Can someone help me come up with a formula to describe the time it takes for a process to take when using pipelining? For example, suppose I have to send a message from ...
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114 views

For a M/M/1/K System, why is utilization = $\rho=\lambda T_s$

Before I ask my question these are the symbols I used: $w$ = Customers waiting in queue $\mu$ = Service rate $T_s = 1/\mu$ = Average service time $\lambda$ = Average arrival time $\rho$ = Server ...
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Multiclass Markov process

There are two car M/M/1 queues Q1 and Q2. Arrival rate of Red car and Green car in Q1 is $\lambda_{1R}$ and $\lambda_{1G}$ respectively. Similarly arrival rate of red car and green car in Q2 is ...
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Funny queuing model probability distribution question

I like toilet jokes, so I am interested in mathematical Problems like follows: A Boy has a bladder with volume $V$. In his bladder it is inflowing the amount of urine (in cm³/sec) denoted by $\mu$; ...
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Survival function of birth-death process

There is a linear birth-death process with $N$ states + an absorbing state $0$, with $$\Pr[X_{t+1}=0|X_{t}=0]=1, \\ \Pr[X_{t+1}=i+1|X_{t}=i]=\Pr[X_{t+1}=i-1|X_{t}=i]=q_i, i\in [1..N-1],$$ and ...
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One queue vs multiple queues: how to calculate mean and variance?

I am having trouble understanding the solution to a queueing problem. You are at an amusement park. There are two ticketing areas. The first one has 1 ticketing machine and a 10-person line. The ...
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Calculating optimal priorities in M/M/c queueing system

This isn't a "solvable" question per-se, it's more a question of whether this can be solved, or modeled, and how I would do it. It's basically a more complex M/M/c system. The scenario: Customer ...