The queueing-theory tag has no wiki summary.
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Queuing/statistics help appreciated
Would anyone be able to please help me with the following question?
A roadside rescue service in a small town relies on a tow truck that costs 40 dollars per hour to run, plus 4 dollars per rescue in ...
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1answer
32 views
Mean sojourn time for a non-Markovian chain? e.g., an M/G/C/C queueing system?
For an M/M/c/c queueing system (when it's at equilibrium) the mean sojourn time of each state can be calculated using the diagonal entries of its transition rate matrix (or the infinitesimal generator ...
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2answers
47 views
Martians and Jovians
In how many ways can five distinct Martians and eight distinct Jovians wait in line if no two Martians stand together?
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38 views
Formal Theory Regarding M/M/s Queue
I have some difficulties with formally deducing the Q-matrix or infinitesimal generator for M/M/s Queues. Although I undestrand the intuitive idea I would like to know the real formal definition of ...
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1answer
39 views
How to model a client / server interaction with queuing theory
I'm interested in modeling a server application where the normal flow of data is as follows:
Server A -> Server B -> Server C -> Server B -> Server A
That is to say, a job originating from A makes a ...
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0answers
41 views
Product form Solution of Jackson Network
Consider a Jackson network with nodes $\{i:1\leq i\leq n\}$ which have the arrival rates $\{\lambda_i\}_{i=1}^n$ from outside and service rates at each node $\{\mu\}_{i=1}^n$. Define ...
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1answer
39 views
Birth-death process: Calculate number of clients that can really enter the system for each unit of time
For a birth-death process that is M/M/1/4/$\infty$/G (as a queue system), how can I calculate the number of clients that can enter the system for each unit of time. Is this solving for $\lambda$ in $4 ...
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28 views
Stability condition for queueing model with 2 servers, non-identical hypoexponetial-2 and FCFS
We consider a queueing model with 2 servers and a shared queue, let us label the servers as $A$ and $B$. Jobs arrive in batches of size $N$ with rate $\lambda$. Each server has a hypoexponential ...
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1answer
169 views
Utilization difference between a multiple server, single queue and a multiple server, multiple queue system
I've been studying Queue theory for a while and I'm interested in finding out the solution to the following problem.
Suppose that we have a system with 4 processors where the arrival rate(λ) is 0.2 ...
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1answer
52 views
Infinite average queueing delay for M/M/1 queues
According to queueing theory, the average queueing delay for an M/M/1 queue can be calculated as
$\frac{1}{\mu-\lambda}$,
where $\mu$ is the average service rate, and $\lambda$ is the average ...
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1answer
116 views
utilization of m/m/c queue
What is the utilization of m/m/c queue ? in some texts such as http://www.amazon.com/Queueing-Networks-Markov-Chains-Applications/dp/0471565253
said: individual server utilization $\rho ...
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1answer
77 views
m/m/2 question in queueing theory
Customers arrive at a serving-system according to a Poisson process with rate 1. In the system there are two serving stations, A and B, which only take care of one customer at a time. If a customer ...
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1answer
97 views
markov chain application
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 exponentially distributed with expectation value 1/2 and independent of ...
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1answer
111 views
Question on M/M/s queue
costumers arrive to a service station according to a poisson prossees and on average 2 during
an hour.the service times and independent of the arrivals and internally independent with mean 45 minuts ...
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0answers
31 views
improving fluid analysis on queuing problems
Discrete-queuing models are hard to solve computationally and can become easily intractable with an increased number of state-action pairs. Markov decision processes can be employed to come up with ...
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2answers
54 views
Understanding this Poisson Queueing Process
Consider $N$ busses, each which breaks down independently at a rate of $\mu$. Once a bus breaks down it is sent to a repair shop which can only repair one bus at a time and takes an exponentially ...
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2answers
30 views
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
31 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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0answers
125 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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2answers
51 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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1answer
131 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
74 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
33 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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2answers
156 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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67 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
15 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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0answers
135 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
131 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. ...
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1answer
69 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 ...
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1answer
65 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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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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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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1answer
56 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 ...
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1answer
224 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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1answer
99 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
123 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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2answers
163 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 ...
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1answer
99 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
98 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 = ...
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1answer
1k 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
199 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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1answer
95 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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2answers
271 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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1answer
114 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 ...
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1answer
201 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 ...
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1answer
321 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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2answers
194 views
What math do I need to understand Markov Process and queueing theory?
The objective of the course is to learn how to model and compare elementary queues like the M/M/1 queue.
I'd like to know what are the minimal math knowledge I need to catch up with to understand ...
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2answers
149 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 ...
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1answer
145 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
388 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 ...
