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

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Intuition of the Mean wait time in queuing system

In queuing theory, (with a single queue and a single server) , given A is service rate (of customers) and B is arrival rate(of customers) We know that, the average time a customer waits in the ...
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Property of renewal point processes

For a renewal process where $f(t)$ is the number of arrivals in time $t$ and $S_k$ is the $k^{th}$ time of arrival, how can we show: $$f(\alpha S_k)/k \xrightarrow{\text{a.s.}}\alpha $$ as $k \...
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Distance between two point processes

Is there a distance metric that we can use to see how close are two point processes? If instead of point process, we deal with random variables, there are a bunch of distance metrics that we can use ...
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M/G/1 queuing system with two arrivals

I have a queuing system with two independent Poisson arrivals with rates $\lambda_1$ and $\lambda_2$. But, the service time for each arrival is different. Suppose f_1(s) and f_2(s) are the pdf of ...
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infinite server queue general arrival/process

Are there any known expressions for the variance of number of customers in a $G/G/\infty$ system? For $M/G/\infty$ systems, we know the average number of customers in the system follows a Poisson ...
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References request: two-queue, one-server model with pre-emptive queue priority and finite buffers

Sorry of the title is a mouthful. I'm developing a queue model with the following characteristics: Two queues: One contains an infinite number of people (Queue A) while the other (Queue B) is ...
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Optimal average utility of the processing network needed

In "Utility Optimal Scheduling in Processing Networks" by Michael J. Neely et al an example of processing network is provided. There are three queues ($q_1,q_2,q_3$) in the network and two processors (...
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Steady state distribution needed

I have a chain $C_t$. At every instant $t$ an exponential random variable $X_t$ with parameter $\lambda$ is added to the chain or if the chain has a value greater than $Q$ then a value $Q$ is ...
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Distribution of waiting times in a M/G/1 queue for fixed number of jobs.

Is there a formula to calculate the distribution of the mean waiting time for a server which processes a workload of $N$ jobs, which behaves like a M/G/1 queue? I would like to be able to answer the ...
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Asymptotic behavior of queueing system with a server that takes breaks

I'm working on the following problem: A single server works on an infinite supply of jobs. The amount of time it takes the server to work on a single job is exponential with rate $\mu$, ...
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Is there a general method for solving this type of recurrence?

Edit: Here is the original problem; it is possible that my recurrence for the stationary distribution $\pi$ is incorrect. Consider a single server queue where customers arrive according to a ...
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in a M/M/1/K Queue, the ratio of losses when the incoming rate is combination of two rates

Generally, In a M/M/1/K system, the incoming rate is $\lambda$, effective incoming rate $\lambda_e$ is equal to $\lambda(1-P_k)$, where $P_k$ is the probability that queue waiting space is full. This ...
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Why Response time goes to infinite when Utilization approaches 1?

I've started to learn queueing theory & model. When we take a look at the graph (Utilization vs. Response time), it shows Response time goes to infinite when Utilization approaches 1, which is ...
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How to calculate the residual service time of a customer in the following queueing theory question?

Consider a machine where jobs arrive according to a Poisson stream with a rate of 4 jobs per hour. Half of the jobs have a processing time of exactly 10 minutes, a quarter have a processing time of ...
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In queueing-theory, M/M/1/K, if in steady state, is the mean arrival rate equal to mean departure rate(not service rate)?

I am trying to calculate the Waiting time of a box that contains two queueing systems(qs) in serial. The arrivals are on left, they enter qs1, exit , then enter qs2, then exit qs2 i.e exits the whole ...
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Why do customers in the following queue have a residual service time of 5?

Consider an $M/D/1$ queue with three types of customers. The customers arrive according to a Poisson process. Each hour, 1 type 1 customer, 2 type 2 customers and 2 type 3 customers arrive on average. ...
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Calculating infinite series for a hospital waiting queue

For my project, I had to simulate a hospital waiting queue, and ended up stuck with this equation. $$ 1=\sum_{i=0}^\infty \left(\frac{\lambda}{\mu+i\gamma}\right)^iP_0 $$ Could any kind soul help ...
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How to solve the following queuing theory question using the P-K formula?

This was a midterm question which I did not get correct. Customers arrive at a grocery store's checkout counter according to a Poisson process with a rate 1 per minute. Each customer carries a number ...
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Poisson arrivals to Jackson network

I'm really confused as to what can be considered Poisson arrivals or departures in an open Jackson network. Say we have a Jackson network of with $K$ servers, with exogenous Poisson arrivals but with ...
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Are queues CTMC?

The $M/M/1$ queue have all the properties of the countable state continuous time markov chain. Is any general queue also a countable state CTMC?
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Shape of distribution between arrivals in a poisson process

Note that both diagrams are refferring to the same problem. The difference is that I'm not sure if graph I'm supposed to visualize is the PDF or the CDF, so I drew them both and hope someone will ...
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Help with the poisson process and jackson networks.

I'm posting the following question from my notes as an image because it has a diagram within it. It's from my lecture notes. I start to get confused when $\delta $ is choses to be much smaller ...
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Find the expectation of the arrival times of a queuing system?

Suppose ${X_1, X_2, \dots}$ are independent identically distributed random variables defined by the density $f(x)=\lambda e^{-\lambda x}$. The renewal process $N={N(t): t>=0}$ is defined by the ...
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Find the waiting time at a copy machine with 2 classes and non-preemptive priority

People arrive at a copy machine according to a Poisson process with rate one per minute. The number of copies to be made by each person is uniformly distributed between 1 and 10. Each copy requires 3 ...
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Prove a $M/D/1$ has the smallest waiting time among $M/G/1$

Fix $\lambda, \rho = \lambda E[S], \rho < 1$ Use the Pollaczek-Khinchine formula for an $M/G/1$ system $$ w^Q = \frac{\lambda E[S^2]}{2(1 - \rho)} $$ To show that if we take the ...
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M/M/1 queue derivation: how to “recursively solve in dependence on $p_0$”

I want to sketch out the derivation of the equations for an M/M/1 queue for a presentation I'm giving. I can understand most of the derivation from Willig but I don't understand this section from p10 ...
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Find $E[S]$ in an interrupted $M/G/1$ queue

Let arrivals in a $M/G/1$ queue be $\lambda$, and service rate exponential with rate $\mu$ but may be interrupted by an independent crisis event with independent duration uniformly distributed between ...
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How to interpret 'arbitrary customer'

My question is about how to interpret 'arbitrary customer' in the following scenario (see question 2. listed below): "At a single server service station two types of arrivals occur. According to a ...
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Questions about queue theory

A service center consists of one server, working at an exponential rate of four 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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Making money off the length of a queue

Take the M/M/1 queue (exponential inter-arrival times, exponential service times, one server). Consider the queue to have initially n(0) customers. The queue runs for a finite amount of time $T$. ...
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Theory of Queueing

There's in a banc two identical queues and totally separated : these are two queues of type $M/M/1$. For each of them, the arrivals are separated by exponential times of parameter $\nu$, the time ...
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Queues with general arrival time and service time distributions

In a G/G/1 queue, the interarrival and service time are IID random variables with mean 1/lambda y 1/mu respectively (lambda is the rate for arrivals and mu is the rate for the service). My question es:...
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Distribution that describes # of memoryless events in an interval but the mean is not constant.

Context: I'm taking a stochastic processes class right now and we got a bit into queueing theory. In all the queue's we've considered, arrivals follow a poisson process. This seems unrealistic in some ...
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Markov chains and queues

I do not understand how may I use the Markov Chain $Y$ and and describe the system $X$ using the states that the exercise suggest. I was searching queue's examples and -i understand this is a M/M/1 ...
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Queue serving packets in chunks: waiting time and output process?

Consider a queue with a Poisson-distributed arrival rate of packets (with the mean $\lambda$). Now the packets are not served immediately, but only in chunks of size $N$ (i.e., once $N$ packets are ...
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Stationnary distribution - Time of service [closed]

There's in a banc two identical queues and totally separated : these are two queues of type $M/M/1$. For each of them, the arrivals are separated by exponential times of parameter $\nu$, the time ...
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How to model continuous arrivals given a changing mean?

If items are known to arrive at a certain fixed rate, then the probability of any particular number of arrivals during an interval fits a Poisson distribution. However, this assumes that the mean ...
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What is the transient behavior of an M/M/1 queue

Suppose I initialise an M/M/1 queue with a queue length of $q_0$. How do I describe $\mathrm{E}[q(t)]$ as a function of time? I know that as $t\to\infty$ , the expression goes to $\frac{\lambda ^2}{\...
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Showing a queueing system is a Markov Chain

I generally understand how to do this but I'm having trouble with a formal proof. "Consider an $M/M/1/m+1$ queue with exponential arrivals rate $\lambda$, exponential service rate $\mu$, and finite ...
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Tauberian theorems in queing theory

I'm trying to use Tauber's theorem below (Feller 1971, chapter XIII.5) "Let U be a measure with a Laplace transform $\omega(\lambda)$ defined $\forall \lambda >0$ and $t,\tau>0$ s.t. $t\tau=1$, ...
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Queueing theory M/M/k - probability of number of busy servers seen by next arrival process

Consider a $n$ server parallel queueing system, need to calculate the probability of $1$ busy server as seen by next arrival process. $\lambda$$=$$arrival$ $rate$ $of$ $processes$ ; $\mu$$=$$service$ ...
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Concerning an infinite server queue with Poisson arrivals

Here's the statement of the problem (from Ross's Introduction to Probability Models): For those unfamiliar with "infinite server queues," they are described here. In this case, however, the service ...
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Queueing Theory Probability of Customers Arriving

This is a question on my midterm practice exam, but for some reason we weren't given solutions so it's not very helpful. There's a single-server queueing system which arrivals follow a Poisson ...
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Random Early Discard Markov Chain

I'm trying to sketch the Markov chain for a Random Early Discard queueing policy where customers arrive to the queue of infinite size according to a Poisson process with rate $\lambda$. Customers that ...
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Balance Equations for M/M/1/m Queue

I think I found the solution for this problem, but they don't show all the steps. I was wondering if someone could explain to me how they get from the three balance equations to the solution.
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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 A&...
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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 ...