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

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Average wait time for a library book [on hold]

If a library has 20 copies of a book that circulates for 21 days and all of them are checked out and there are 40 reserves (i.e. people waiting ahead of me) what is the average wait time I can expect? ...
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Expected number of lines in use in call centre (markov process: queuing theory)

Suppose we have a call centre with infinitely many lines to be able to call to. Calls come in a rate of $\lambda$ and customers are served with rate $\mu$. It is easy to see that the $Q$-matris looks ...
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A queueing model issue.

I am very beginner in Queueing Theory and I am learning in my own. I am struggling in the following situation. Suppose in a service center if a job arrives it will immediately being processed if a ...
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38 views

M/M/1 and M/M/m queueing models for a client-server-database system

I have designed a system and have to do now analysis using queueing models with which I have problems. First, I will briefly explain my system. I have a client-server-database system. There are N ...
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Pollaczek–Khinchine formula for claims with expotential distribution - derivation

I am trying to understand ruin probability formula using Pollaczek–Khinchine formula described here: http://en.wikipedia.org/wiki/Ruin_theory $$\psi(x)=\left(1-\frac{\lambda ...
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Stationary probability in an M/M/$1$ queue with a lazy server

Customers arrive to a single server queue according to a Poisson process with rate $\lambda$. Each customer requires Exponential($\mu$) service time. In the beginning when there are $0$ ...
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What is the probability of a M/M/1 queue having less than 5 customers?

(Not homework) I'm doing some studying for an upcoming final, which includes material on queueing theory (the class is on design/analysis of computer networks, not queueing theory). I've worked ...
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25 views

Queue-length and waiting time of M/D/1-queue

I studied the M/G/1 queue by myself. Now as an application, I considered the M/D/1- queue. I know that the results can be find in the internet, but I haven't seen any calculations. Let $Q$ be an ...
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pseudocode about registers and clients

I have projects that requires to simulate a market with 3 registers. Every second an amount of clients come to the registers and we assume that each clients takes 4 seconds to the register before he ...
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13 views

How many staff should this employer hire given this set of customer queue and service speed?

Suppose James owns a restaurant and gets on average X customers per minute and each takes Y minutes to be served. What is the minimum number of staff to ensure that 99% of customers are served in ...
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Proportion of arrivals taking a particular path in a Routing Matrix

I have a routing matrix with Node-0 being the source/sink (outside world) and there are service Nodes 1,2..k in the system. The matrix has entries R_ij = Probability of an arrival at Node-i moving to ...
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M/M/s/s Queue properties

Let Q be an M/M/s/s queue with m servers and a total capacity of m slots, so that a new customer can enter system only when there is one server free. Arrival rate is λ, service rate μ. I have the ...
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Analysis of shared server queueing system

To be able to analyse the processes in an organizational department, I had the idea that the processes can be modelled as a queueing system. Now suppose the service times are exponentially ...
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27 views

From an M/G/1-queue to the M/M/1-queue

Suppose I have an $M/G/1$-queue $Q$. Then I have an embedded Markov chain $Q(D)$ and the following theorem: If $\rho=\lambda \mathbb{E}[S]<1$, then $Q(D)$ is ergodic with a unique stationary ...
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23 views

M/M/1 queue, probability of time spending in queue..

Let $W$ be the time $nth$ customer spends in the queue when $n$ go to $\infty$. How do we write down the formula for $P[W \le t | N = k]$ ? where $N$ is the number of customer in the system.
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time for n or more than n customers in a M/M/1 queue system

The question asks for how much time (given an 8hr working day) in a day are there 2 or more customers in the system. If I am to calculate probability for 2 customers in the system and probability for ...
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56 views

Expected Service Times for truncated exponential

I'm trying to solve a problem where all arriving items (arrival exponential $\lambda = 1/5$) are divided into into groups, those who are served within 5 units of time and those who have their service ...
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Need guidance on a Queuing problem

I can't really go into specifics, I'm more just looking for terms that I can research to get on the right track. Classes of model/processes etc. A close analogy to my problem: I need to optimally ...
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28 views

meaning of stationary distribution, null-recurrence and transience in queueing theory

I have 3 question about the meaning of the mathematical term in the reality. Let the queue $Q$ have the stationary distribution $\pi_n=(1-\rho)\rho^n$ for $n\geq0$. Does it mean, when $t$ is large, ...
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35 views

Explicit extinction probability of busy period

Let $Q$ be a M/G/1-queue. We denote by $B$ the busy period of the queue, that is defined as follows: $$B:=\inf \{ t>0: Q(t+T_1)=0 \},$$ where $T_1$ is the arriving time of the first costumer. ...
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81 views

Removing some arrivals from a Poisson Process

The inter-arrival time of a Poisson Process, $t$, conforms to the exponential distribution, so the probability density function for $t$ is $f(t)=\lambda e^{-\lambda t}$, $t>0$. ($\lambda$ is the ...
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Poisson Modeling/ Queue Theory - Reference Material

Can anyone reccomend some practical reference material related to building and implementing queueing theory models. using stochastic (prefferably Poisson) processes? We are looking to build out a few ...
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135 views

Changing a queueing processes

Situation Consider a general queueing system $\mathscr{S}$, whose customer arrival times are independent, and whose service times are independent; both of these are allowed to have general ...
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M/G/1 queue has embedded Markov chain

I tried to prove that the M/G/1 queue has an embedded discrete-time Markov chain. But I'm not sure if I have done it right and properly. Specially I'm not 100% sure if i calculated right the ...
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Transition Matrix of M/M/1 Queue

We know that for an M/M/1 queue the state space is $S=\{0,1,2,... \}$. Further the probability to go from state $i$ to $i+1$ is $\lambda$ for all $i$ in $S$. Moreover, to go from $i$ to $i-1$ is the ...
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Questions about the functions of the operations of a queue

I am looking at the following two functions of two operations of a queue: ...
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How would I find the PDF of one poisson variable in terms of another?

I am looking for the PDF of a Poisson variable. The setup is that pit crews are working on racecars on the track after the drivers reports trouble. The pit crew's expected work time is exponentially ...
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61 views

Determine wait time in queue without arrival rate

I'm interested in determining wait time in a queue. For example, I'm at the grocery store, there's a single line leading to a set of 5 cashiers, 10 people are in front of me. I know on average it ...
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53 views

Queueing Delay(W) for M/D/1 queue with different value of service times

I have a problem of calculate the queueing delay of M/D/1 queue. There are two different types of packets with different size. Such that, the arrival rate $\lambda$ and service times $\mu$ will not be ...
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45 views

Queuing theory-Multiple server (reducing simple recurrence formulas)

The equations given in 6.3 have been reduced which really eases the computation in further studies. But I tried to find the method of reducing these but I could not find a way at all. Any hints will ...
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43 views

How do you find the probability of a certain state in Markov Chain?

This question appears without answer in an old exam I found (not a homework question) Suppose messages that enter a system need to be processed by two servers. They arrive at the system at a ...
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Taking Laplace-Stieltjes transform to find virtual idle time in G/M/1 queue

I am reviewing some queueing problems from Gross and Harris, and had a question on problem 5.40 part b. The problem is stated as follows: Part B: Show that the stationary output of a $ G/M/1 $ queue ...
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Poisson processes and queues

I am trying to understand Poisson processes and queues. I have this exercise: Consider a fuel station with two fuel pumps and one park. Each car that comes to the fuel station when the pumps and the ...
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63 views

what is difference between open and mixed queue

please consider this image: In this picture that I got it from Here write said that Network C is Open,B is Mixed and A is Closed. I want to know why C isn't mixed? ...
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G/G/1 Queues - Book with Discrete Time Markov Chain examples

Need some book recommendation or links which have examples how to solve G/G/1 queues with detailed Discrete Time Markov Chain drawn and how to get the steady state distribution, the average number of ...
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Solution of a system of differential equations for a continuous time Markov chain.

The equations arise as the Laplace transforms of the forward equations of a continuous time Markov chain for a three-state system, with the following transition rates: Transition , rate $0 ...
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Continuous Markov chains, arriving pairs

I have been trying to sort out this exercise but really stuck on this. Preparing myself for exams and found many exercise on continuous Markov chains but I am always stuck when it comes to transition ...
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conditional mean of geometric RV

Say, there are three nodes: $S$, $R$, $D$. $S$ transmits to $R$, $R$ stores the packets, and later transmits to $D$. At any time, either $S$ or $R$ is selected to transmit according to some random ...
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Solving Probabilities for M/M/1 Queue Waiting Time Generating Function

I "believe" that generator, $\bf Q$, of the waiting time distribution for the $M/M/1$ queue is given by the following (I'm not 100% sure if this is even correct): $\bf Q$ = $\left( ...
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Estimating queue overflows

(This come up trying to debug a real system at work, and got me wondering...) We have a server that consumes a queue, one item at time. Processing the item is quite quick, but not instant. There is ...
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59 views

Which Queue to Join at the Super Market

Last night I started wonder about the fastest way to take a shopping trip with my university flat mates and was wonder about how we should queue for the check out. I have a feeling that queue theory ...
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What is the distribution of the service-starting time lag w.r.t. two concurrent customers from two parallel $M/M/1/1$ queues?

Consider two parallel, independent $M/M/1/1$ queues (denoted $Q_i, Q_j$) with identical arrival rate $\lambda$ and service rate $\mu$, using FCFS (First Come First Served) discipline. Note that the ...
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Conditional Poisson PMF ~ the joint PMF not independent?

Let X denote the number of customers who arrive during a service time and Y the first service time. Customers arrive in a Poisson process with rate $\lambda$. Then: $P(X=x|Y=y)= e^{-(\lambda y)} ...
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42 views

Steady state queue-length distribution of M/G/$1$ queue

The steady state system-length distribution of a FCFS $M/G/1$ queue is well known (see here for reference). I am trying to find it from a different approach, using Little's law, but I am not sure if I ...
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85 views

What does a customer see when it begins to be served in $M/M/1$ queue?

In queueing theory, the PASTA (Poisson Arrivals See Time Averages) principle [wiki] justifies $a_n = P_n$ where $$a_n = \text{proportion of customers that find } n \text{ customers in the system when ...
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Rouché theorem in queuing theory

I was looking for the uses of Rouché's theorem, and I came across queuing theory. An article stated that it is a workhorse theorem in this field, but as much as I tried to find some examples on the ...
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Poisson arrival and selective removal

Users arrive according to Poisson process with rate λ. If every third user is removed, then do the remaining users form a Poisson process with rate 2λ/3? If every other user is removed, then do the ...
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Stationary distribution of Waiting Time in a $GI/GI/1$ queue

I am trying to find if there is any literature where I can find formulas for the stationary distribution of a $GI/GI/1$ queue. Specifically, I need to find $P(W=0)$ where $W$ is the steady state ...
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91 views

M/M/1 Queuing Theory Question

Lets say I have packets arrive to a terminal at Poisson rate $\lambda$ per hour and my terminal has an exponential service rate $\mu$ per hour (so the mean service time is $\frac{1}{\mu}$). So this is ...
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Strict proof of markovity of queing system of type $M/M/n/\infty$

I have a queing system of type $M/M/n/\infty$. The service time is exponential, and the arrival process is poisson. I do understand that because of these two facts the future of the system in ...