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Finding the infinitesimal generator of a M/M/2 queue [closed]

I have a M/M/2 queue with a total population of 5. The arrival times are independent exponential random variables with mean of $\lambda$ and the service times have a mean of $\mu$. The initial number ...
0
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0answers
11 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 ...
-1
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0answers
10 views

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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0answers
9 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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0answers
9 views

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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0answers
8 views

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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0answers
12 views

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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0answers
21 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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21 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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1answer
13 views

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 ...
2
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1answer
53 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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0answers
18 views

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 ...
0
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1answer
25 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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1answer
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. ...
2
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1answer
79 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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1answer
34 views

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 ...
2
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0answers
134 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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1answer
65 views

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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3answers
146 views

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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2answers
45 views

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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1answer
44 views

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 ...
1
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1answer
57 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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0answers
51 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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1answer
38 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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1answer
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 ...
2
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0answers
59 views

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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1answer
44 views

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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1answer
61 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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0answers
40 views

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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1answer
43 views

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 ...
0
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2answers
56 views

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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0answers
12 views

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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0answers
102 views

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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0answers
18 views

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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1answer
58 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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1answer
54 views

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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0answers
26 views

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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0answers
39 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 ...
2
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1answer
84 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 ...
4
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1answer
49 views

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 ...
1
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1answer
23 views

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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0answers
28 views

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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1answer
86 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 ...
1
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1answer
43 views

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 ...
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0answers
41 views

Probabilistic model of parallel web servers

Note: The following probabilistic model of parallel web servers is abstracted from an engineering project. I am not good at probability theory and I am seeking some evaluations and suggestions. ...
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1answer
68 views

M/M/1 queue with probability of new client leaving

I'm looking at a M/M/1 queue system and trying to show that $\{M_t\}_{t\geq}0$, the number of clients in the system, is a birth-death process. In the simplest of cases this is true if $\lambda_i = ...
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1answer
20 views

Closed queueing networks

When applying Norton's theorem to a closed queueing network, most documentation I found assumes same service rates for all queues. How do you calculate the throughput and delay if the service times ...
0
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1answer
153 views

M/D/c queue with different service times

In formulas on M/D/c queue it is assumed that service time is the same for all servers. Are there formulas for the case when service times differ between servers (in general case there would be as ...
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0answers
108 views

Poisson distribution-Queueing theory

Vehicles arrive at a junction, in order to swing left, create a line queue ( tail) . The number of vehicle follow Poisson distribution. The length of cycle for the traffic light (for left turns ) is 1 ...
2
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1answer
254 views

Conditional expectation of conditional expectation

I have a question about conditional expectation. I have always problem with that... It is a step of a proof that I just don't get... I appreciate any help! I have the random variable $$B=S+ ...