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

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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. $$ 1=\sum_{i=0}^\infty (\frac{\lambda}{\mu+i\gamma})^iP_0 $$ Could any kind soul help me with the ...
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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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24 views

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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38 views

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 ...
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25 views

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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35 views

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 ...
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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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33 views

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 ...
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22 views

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 ...
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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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40 views

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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37 views

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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65 views

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 ...
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1answer
29 views

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 ...
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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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26 views

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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27 views

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 ...
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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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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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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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56 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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$M/M/2/4$ 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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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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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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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 ...