# Questions tagged [queueing-theory]

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

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### Minimising average holding cost per unit time is equivalent to minimising average delay per customer in queue

This question has now been asked on Operations Research Stack Exchange as it is probably better suited there. Introduction When one thinks about a queue, it is natural to want to find a policy that ...
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### Simple Poisson process question [closed]

A small post office, with a single server, has space for a maximum of 3 customers. New customers arrive according to a Poisson process at a rate of 4 per hour. The service times are independent and ...
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### Average delay in a Processor Sharing M/GI/1 queue

I'm following the derivation of the average delay in a Processor Sharing M/GI/1 queue from https://www.netlab.tkk.fi/opetus/s383141/kalvot/E_psjono.pdf (slide 6). They end up with the following ...
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### Stochastic simulation in R with large matrices

I am running a simulation in $R$ of a queueing system. One of the problems I have is that the running time is heavily impacted by a read operation on a matrix that has to be performed multiple times (...
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### Queuing theory, practical exercise.

The problem: There are 4 sets of printers in the printing service. On average there are 25 people / hour that would like to use the service. On average one client prints their necessary documents in 4 ...
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### Calculate when a queue will run full given it's input / output rate and capacity

I just came across an interesting question: let's say we have a security waiting line at an airport which can physically provide space for 3500 passengers at the same time we know that the personnel ...
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### Limiting distribution for a periodic single-server queue

Consider a $M/M/1$ queue with a constant arrival rate $\lambda$ and service rate $\mu$ with $\lambda < \mu$. We know that in this case the limiting distribution exists and it is a geometric ...
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### Fictitious Queue

Suppose I have one big system composed of two subsystems which have independent Poisson arrivals with rates $\lambda_1$ and $\lambda_2$. Let $N_1(t)$ and $N_2(t)$ denote the number of customers at ...
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### Thinking users interactive system

I'm trying to solve this exercise but I am not sure about the correctness. Exercise: Consider the following measurement data for an interactive system: measurement interval: 5 min number of users: 50 ...
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### Average production of an item and proportion of destroyed items

An item is produced on a machine that sometimes breaks down. Without breakdown the time of production of a batch of 100 items is 20h (deterministic) and the machine works in an uninterrupted way by ...
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### “no atom at the origin” of a service time distribution?

Let $S$ be the service time of a queue, and $G(s)$ be the cumulative distribution function for $S$. What does it mean for $G(s)$ to have "no atom at the origin"? Or might an atom at the ...
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### Queuing theory: practical exercise; three workers, 9 spots in the queue, mean service time - 30 minutes.

There are three workers that can service the clients at the same time. If one of the workers doesn't have a client to service, he goes to help the other workers. Mean service time is 30 minutes. ...
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### Determining $\lambda$ and $\mu$ for a queuing system

I have just started studying the topic of queuing systems and I am having trouble with grasping the intuition that I need to start solving questions. Consider a M/M/1 queue where arrivals occur as a ...
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### Deriving Poisson arrival process from operation times

I have a problem from my Stochastic Processes class I have no idea how to solve. We have been studying Poisson processes including models of queues, but this question involves a small amount of sample ...
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### M/G/1 queue with group arrivals

This question is about exercise 70 of the book 'Queueing Systems' (https://www.win.tue.nl/~iadan/queueing.pdf) from Ivo Adan and Jacques Resing. The task is the following: Consider a queueing system ...
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### Processing time in 2-phase vs 1-phase queueing systems

I'm trying to analyse a simple system there requests come in, processed, and sent back. Sending request back takes some time, so there is an option to do it async (i.e. offload sending to another ...
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### Modified M/M/1 Queue

I have queueing system in which the arrival rate is Poisson with rate $\lambda$ and the service times are exponentially distributed with rate $\mu > \lambda$. The twist in this problem that a user ...
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### Queuing question

Can you construct a queueing questions whose test point is the pure death model and make sure the necessary information and data are provided. Can you give me one simple example about pure death model
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### Distribution of packets leaving a server delaying packets with exponential distribution

From Burke's theorem, we know that for the M/M/$\infty$ queue in the steady-state with arrivals a Poisson process with rate parameter $\lambda$ and service time exponentially distributed with ...
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### How can I calculate metrics of 2 independent M/M/1/K systems?

I know how to calculate those metrics if there is a single M/M/1/K queueing system. but what if the entire system is a splitting Poisson process(has 2 independent M/M/1/K queueing systems in parallel)?...
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### Is the period of an irreducible Markov Chain the same for all states?

I'm a bit confused about this theorem about irreducible Markov Chains: If a Markov chain is irreducible, then all its states have the same period $d(i) := g.c.d.\{n > 0|P^n(i, i) > 0\}$.[source]...
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### How to calculate probability that I will encounter full barbershop when I go for a cut?

Let's say for example that you want to get a fresh cut. I am eeger to know how could one calculate probability that barbershop will be full? What are the parameters that should be looked. I don't need ...
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### Is there any mathematic tools could help to get the maximum queue capacity required to keep it non-empty when the arrival is not Poisson?

Assume there is a queue with input flow formulized as a function $f_{in}(X, t)$ with random variable $X$ and the output speed is a constant $K$, $f_{out}(t) = K*t$. Can it be defined as a G/D/1 queue? ...
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### Is there any theorem like Burke's theorem in the case of discrete-time queues?

Suppose that we have a queueing system with slotted time-line, single server and Bernoulli distribution for the arrival and service processes. The probability of a packet arrival into the queue is $p$ ...
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### Find probability that a gap greater than 7 seconds occurs between two consecutive arrivals

So I have a question that basically says: Customers arrive at a facility at a mean rate of $12$ per minute. Find the Probability that a gap of greater than $7$ seconds occurs between two consecutive ...
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### Why isn't the throughput = (1 / response time) by their definition?

I found that all say that throughput is inversely proportional to response time. But I found it's confusing that if I see their relationship by their definitions, isn't each of them a reciprocal of ...
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### Key Renewal and Renewal Type Equations: Finding $E[X_{N(t)+1}^k]$

We're studying renewal processes in my probability class, and I'm a unsure as to where to start with the following practice problem: $N(t)$ is a renewal process and where $X_{N(t)+1}$ is the length of ...
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### Probability of Poisson arrival in a vanishingly small interval

Consider a homogeneous Poisson arrival process, and we look at the time interval $(t,t+\varepsilon]$ for a vanishingly small $\varepsilon>0$. Why is it the case that the probability of more than 1 ...
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### M/M/c non-preemptive priority queue with batch service

I am looking for work on bulk service queueing models with non-preemptive priority. It seems to me that there is a lot done in terms of bulk arrivals but I cannot find a paper that focuses on bulk ...
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### Intuition behind intensity parameter in Poisson point process

Consider a homogeneous Poisson point process as function of time, $N(t)$ with the parameter $\lambda$, where $N(t)$ represents the total number of occurrences (e.g. phone calls) up to and including ...
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### Queueing System - Probability

Consider a supermarket with three cash settling boxes and only one queue created from buyers. Four people, A, B, C and D enter the supermarket at almost the same time and in that order. A, B and C go ...
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### If we don't know the transition probability, then how to prove a markov chain is recurrent or irreducible?

For example, we only have the next state information: $q(n+1) = q(n) + a - c$, where $a$ can be regarded as some arrival and $c$ can be regarded as some action. Also, $a,c \in \{0,1,\cdots\}$, which ...
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### How to find the best Queue in a non normal distribution

I've been thinking about this problem for a little bit but did not find a satisfactory answer, maybe you guys can help me! I have N queues processing data all day, and I have to find the best or a few ...
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### Stability (equilibrium) condition in M/M/1 queue with both open and closed classes

Mixed queueing network models are those in which some classes are open and some are closed. In the example network illustrated above, classes A and B are open and depart from the network after CPU ...
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### For M/M/1 queue, why is the number in the queue exclusive of the customer being served not a Markov chain?

For a M/M/1 queue, we let $N_q(t)=(Q(t)-1)_+$ be the the number in the queue exclusive of the customer being served, where $Q_{ij}(t)$ is the probability of going from state $i$ to $j$ in time $t$. ...
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