The queueing-theory tag has no wiki summary.
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How does the rewriting of the following two equations work?
I am failing to understand the proof of coming to the steady-state formula in queueing theory. This is probably due to the fact that I may have forgotten (and cannot find it back) some of the algebra ...
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1answer
21 views
Mean number of particle present in the system: birth-death process, $E(X_t|X_0=i)$, $b_i=\frac{b}{i+1}$, $d_i=d$
Let $\{X_t\}$ be a birth–and–death process with birth rate
$$
b_i = \frac{b}{i+1},
$$
when $i$ particle are in the system, and a constant death rate
$$
d_i=d.
$$
Find the expected number of particle ...
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68 views
Modified M/M/1/2 with 2 possible arrival rates and M/M/1/5 queue
I've been stuck on this question for hours, and could use some help :)
"An M/M/1/2 queue has service rate $\mu$ and arrival rate of either $\lambda_1$ or $\lambda_2$. The rate can change only when ...
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1answer
14 views
Tandem queue - response time distribution
In tandem queue with two queuing system, each server has exp(mu0) and exp(mu1) service time distribution and arrival rate is poisson(lambda). Scheduling policy is FCFS.
What would be the response time ...
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52 views
Question on M/M/2 queue variation
I have the following question:
Two workers handle three machines(i.e. we can at most repair two machines at a time). The time until the machine breaks down is exponential distributed with expectation ...
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1answer
46 views
Expected value of minimum of exponentials
I am not sure of the following. I have $(i-1)$ exponential random variables with rates $\theta$ and $\mu$ and I want the expected value that the particular $\mu$ random variable is the minimum. Think ...
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2answers
28 views
Waiting after finishing a single queue
I'm a little confused by a conditional expectation question:
You have two exponentially distributed random variables, and you need to compute an expectation that looks like
$$
E[T_{1}|T_{2} > ...
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0answers
53 views
Question about queueing theory and queueing systems [closed]
i have presentation about queuing theory and i need some latest journal and article about it please introduce or give latest journal and article (my subject its around markov chain in Queuing theory ...
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2answers
130 views
When does the next bus come?
People arrive at a bus stop according to a Poisson process at rate $\lambda$ per minute. The bus leaves every $n$ minutes, but you have no idea when the last bus left. You observe that there are $k$ ...
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0answers
39 views
Multiple-server Queuing problem
Full disclosure: This is homework and I'm behind with the course, but I'm not here to ask for the answer, instead I'm hoping someone can point me in the right direction for self-study. I've been ...
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0answers
13 views
References on discrete $M/M/\infty$ queue
Do you know any paper/book/web site talking about discrete $M/M/\infty$ queue?
By discrete I mean at each step some clients come and some clients leave, probability of arrival/leave are not defined ...
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121 views
Boundedness of expected reward Markov chain (may be related to discret $M/M/\infty$ queue)
[EDIT]:
I read a bit on $M/M/\infty$ queue and it may not be the right comparison and my notation may be confusing (I'm in discrete time and $\lambda,\mu$ look likes rates when they are probability). ...
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1answer
83 views
M/M/1 queues And finding equilibrium probability that the shop is empty
Customers arrive at a barbers shop at the incidents of a Poisson process of rate λ. Each person is served in order of arrival (by the single barber), and takes an exponential, rate μ service time. ...
0
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1answer
51 views
Birth processes with immigration and catastrophe
On the volcanic island of Montserrat the number of species increases(by immigration from neighbouring islands) at rate α. However, at rate η the volcano explodes, and all life is wiped out, although ...
2
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1answer
58 views
How to get transition rates in a $M/M/\infty$ queue
I am told for an $ M/M/\infty$ queue the transition rates $q$ are as follows.
$q(n,n+1) = \lambda$
$q(n,n-1) =n\mu$
Can anybody explain the intuition behind $q(n,n-1)$?
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0answers
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Is Queueing Theory dead? [closed]
I was studying queueing theory for my class and noticed that we are now able to either solve with certainity most queiening problems or simulate them.
is queueing a dead research area?
I read this ...
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0answers
34 views
Stationary process of a generalized closed Jackson network
We have a generalized closed Jackson network where $\mu_i(n)=\frac{G(n-e_i)}{F(n)}$
How can I prove that the stationary process is given by the following formula: $$p_n=Β_mF(n)\prod\limits_ {i=1} ...
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0answers
43 views
Proof of $ \text{Poisson}(\lambda p) $-arrivals.
We have a queue where people pass out of it with $ \text{Poisson}(\lambda) $ and they come in with probability $ p $.
I understand that the arrivals follow $ \text{Poisson}(\lambda p) $, but how can ...
1
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1answer
113 views
Markov Chain Transition Intensity Conversion
I have a question about converting a 3-state discrete state, continuous-time, markov chain to a 2-state.
My 3-state model has states: Well (state 1), Ill (state 2) and Dead (state 3).
...
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0answers
62 views
Queueing model with two servers
I have a two-server queue with Poisson arrival rate and $\lambda$ exponential services with $\mu$ ( first server service rate) and 2$\mu$ ( 2nd server service rate). Capacity is infinite.
Then why is ...
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0answers
87 views
Problem with two server Queue
I am practicing for an exam on queuing theory and I found this question somewhat confusing. Appreciate if somebody can shed some light on it.
There are 2 facilities, A and B, which provide the same ...
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1answer
101 views
M/GI/1 service time distribution
I want to compute the distribution of the waiting time and the number of jobs for M/GI/1 where the service time is Heavy-Tailed or more specifically Pareto. I found this paper ...
2
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1answer
93 views
$M/M/1$ Running job distribution
Consider an $M/M/1$ queue with $\mu=1$. Suppose that a job comes in and see only one job is running in the server. I want to know the distribution of duration of the running job . Simulation shows ...
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1answer
83 views
Chernoff bound for Geometric RVs compared to exact tail bound
I keep getting a result I can't interpret.
X is a Geometric RV with distribution ($0<\rho<1$)
$$
\pi_k = \rho^k(1- \rho)
$$
so directly applying Geometric series the tail bound is
$$
B_1 = ...
1
vote
1answer
823 views
What is the correct inter-arrival time distribution in a Poisson process?
Given a Poisson process (e.g. radioactive decay) with rate $\lambda$, then the expression $\exp(-\lambda t)$ is the probability of observing no counts in time interval $t$. This can be interpreted ...
2
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1answer
147 views
Find the probability that the second customer to arrive has to wait to be served if arrival time is exponential and serving time is uniform
Customers line up to be serviced according to a Poisson process at an average rate of five per hour. If the time it takes to serve one customer is a continuous uniform random variable on $[0,4]$, ...
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0answers
68 views
How to model this system?
I am not a mathematician, but I need your help please, to resolve my problem:
I would like to study a stability of a network, for this end I have to modelise it as a queue system with the following ...
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0answers
46 views
Probability that a presentation will be on time
I'm visiting a grand conference soon which has 30 lectures in a single day.
Unfortunately they don't say when each of the lectures will be held. Only that each will be held in the order written ...
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0answers
60 views
Mathematically optimizing size of a FIFO buffer
Suppose one has a data buffer in software, where the data from the buffer is being consumed at some constant rate $F_c = \frac{1}{T_c}$, and the data is being produced by some internal process. The ...
0
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1answer
66 views
Calculating expected residual service times
This is a queueing theory-related question. Suppose we have two types of arrivals, call them A and B, who arrive according to a Poison proces with rates $\lambda_A = 1/20$ per second and $\lambda_B = ...
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Scheduling variables sized work items efficiently
(I have also posted this question at stackoverflow.com because I'm not sure where it should belong.)
I have a system with the following inputs:
Set of work items to be completed. These are variable ...
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2answers
220 views
How to find the Laplace transform?
Jobs arrive to a computer facility according to a Poisson process with rate $\lambda$ jobs / hour. Each job requires a service time $X$ which is uniformly distributed between $0$ and $T$ hours ...
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0answers
102 views
Two stage cyclic queue
Given a cyclic queue of two servers of exponential service rates, if there are N customers at one server at time t, how do i start about showing that N can be modeled as a birth and death process? and ...
2
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1answer
184 views
M/G/1/K - evalutate birth and death rates
Within a queue with capacity = K and exponential interarrival times, death rate is μ and birth rate λ.
A packet is discarded when the queue is full.
When the source is active there's a probability ...
0
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1answer
296 views
variable death / birth rate in stochastic process
Within a queue with capacity = K death rate is μ and birth rate λ.
A packet is discarded when the queue is full with probability Pk=P(K elements in the queue)
Moreover there's a probability $p1 > ...
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1answer
142 views
What math do I need to understand Markov Process and queueing theory?
I'm overwhelmed by this course that goes into queueing theory, talks about stationary process, simple point process, markov process, little's law, etc. All of which I've never heard of and my math ...
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2answers
133 views
Queueing Theory: Why does this hold for a M/M/1 queue?
For a M/M/1 queue, calculating the estimated number of jobs $n$ in the queue is given by:
$$E[n] = \sum_{i=1}^{\infty} p_i i = \sum_{i=1}^{\infty} \rho^i (1-\rho) i .$$
The final result for a M/M/1 ...
0
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1answer
111 views
common queue and two servers
in a checkout system, customers arrive according to Poisson rate λ. The system consists of two parallel boxes, in Box 1 time is exponential of rate μ1 and box 2 exponential of rate μ2.
There is only ...
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2answers
259 views
Good Queuing Theory Introductory Textbook
I am an undergraduate student who is going to be taking a queuing theory introductory course next semester, I am wondering what's a good introductory book out there? (my math background is probability ...
2
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2answers
517 views
One vs multiple servers - problem
Consider the following problem:
We have a simple queueing system with $\lambda%$ - probabilistic intensity of queries per some predefined time interval.
Now, we can arrange the system as a single ...
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2answers
119 views
Serving customers algorithm
Well I have a problem with a Christmas assignment and my teacher is not responding(maybe he is skiing somewhere now) so I will need some help.
The algorithm is about an office and the waiting time of ...
2
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1answer
165 views
Comparing two Markov chains
I am interested in the question of the positive recurrence of a Markov chain that (in some sense) converges to another Markov chain known to be positive recurrent. The following is a concrete example ...
2
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1answer
560 views
Little's Formula for M/G/1/c queue
Suppose we have an M/G/1/c loss system, with equilibrium distribution $\;\pi,\;$ service times $\;S_i\;$ and arrival rate $\;\lambda.\;$ I'm trying to show that $\;(1-\pi_0)=(1-\pi_c)\lambda ...
0
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1answer
396 views
Queueing Theory: Difference between Time in Queue and Waiting Time?
I'm looking at the formulas here: http://www.cs.auckland.ac.nz/courses/compsci742s2c/lectures/p-q.pdf
It has one for r-th percentile of waiting time ($r$% of customers wait less than said time), and ...
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1answer
100 views
Queueing Theory - Probability that all jobs have been served?
Suppose I have M/M/1 system with $\lambda = 4$ per hour and $\mu = 5$ per hour. How can I find out if all jobs have been served after, say, 8 hours? At first I thought about doing $P(n > 40)$ since ...
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2answers
259 views
The processing time of an M/M/1 queue
Suppose I have a queue with $\lambda$ and $\mu$. I can calculate the probability that there are 2 objects in the queue trivially, but how can I compute, for example, the probability that it takes an ...
4
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1answer
234 views
Crowded and quiet periods in a $M/M/1$ queue
I'm trying to solve the following exercise (not homework):
Consider a $M/M/1$ queue with an arrival rate of 60 customers per hour and a mean service time of 45 seconds. A period during which there ...
3
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2answers
333 views
Birth and Death Process Question (Queuing)
A small shop has two people who can each serve one customer at a time. There is also space for two customers to wait. Anyone who arrives and sees that the shop is full will go to another store. ...
2
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1answer
271 views
Queueing Theory: How to estimate steady-state queue length for single queue, N servers?
I have a real-life situation that can be solved using Queueing Theory.
This should be easy for someone in the field. Any pointers would be appreciated.
Scenario:
There is a single Queue and N ...
