Questions tagged [queueing-theory]
Queueing theory is the mathematical study of waiting lines, or queues.
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What is the fraction of customers lost in a finite queue with one server, M/M/1/k? k = four places and s = 1 server
What is the fraction of customers lost in a finite queue with one server, M/M/1/k? $k =$ four places and $s = 1$ server
$k=4, \lambda=\dfrac 1 {30}$, $\mu=\dfrac 1 {25}$
The steady-state probs are p0 ...
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Proving recurrence of a random walk with unbalanced step sizes
This question arose from a queuing theory question. Part of the problem is to prove when the queue is positive recurrent. The jump chain of the queue is a reflecting random walk with the following ...
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How to calculate resource utilization
I have learned some probability and statistics, and would like to apply it to a problem where I need to understand the resource utilization of a given set of servers in a time unit, say, a day.
I ...
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What are the sample points in sample space? What is the random variable?
As we know that interarrival time in queueing theory follows exponential distribution. I want to know what are the sample points and what is random variable corresponding to interarrival time? As ...
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Customers arrive at a service center according to a Poisson process with a mean interarrival time of 15 minutes.
Customers arrive at a service center according to a Poisson process with a mean inter-arrival time of 15 minutes.
What is the probability that no arrivals occur in the first half hour?
What is the ...
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Why is M/G/1 queue viewed only at departure times?
It is said in the book that for M/G/l queue, viewed only at departure times, leads to an imbedded discrete-time Markov chain. Viewing the queue only at arrival times does not yield a Markov chain.
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Equation solution in queueing theory problem
I have been trying to solve a problem in queueing theory connected with queuing system with two separate servers. During my solution, i am needed to find $x$ which minimizes the following function:
${{...
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How to calculate mean of slowdown
Consider the system that jobs arrive at a server that services them in FCFS order. Job sizes (service time) are independently and identically distributed according to a random variable $S$. We define ...
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The number of customers in a store at certain time?
The question:
Customers arrive in a coffee shop one-by-one according to a Poisson process with intensity $\lambda = 6$ per hour. They stay at the coffee shop during a random duration $V$ which is ...
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M/M/1 processor sharing system service times
Altman and Shimkin (1998) discuss the case of a processor sharing system with Poisson arrivals with rate $\lambda$ and constant total service rate $\mu$. Let $x$ be the number of customers present in ...
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Find distribution of $Z$ using Laplace-Stiltjes transform
Let $X$, $Y$ be exponentially distributed with means $\lambda$ and $\mu$ ($\lambda < \mu$), respectively (Note that $X$ and $Y$ do not need to be independent). Let $Z$ be a random variable, which ...
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Continous time Markov Process M/M/2 queue - simple and finite
Time for arrival is Poisson$(\lambda)$ and serving is Exp$(µ)$. The Markov process is $Z_{t}$ = number of individuals at any time 't'.
When there are no restrictions:
The holding times are $a_{0}$ = ...
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G/M/1 queuing system stationary distribution.
I'm considering the following G/M/1 system: The interarrival times are Erlang(2, $\lambda$) distributed and the services times are exponentially distributed with mean $\mu$. Now I call L(t) the number ...
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1answer
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Generalization of pure birth process
The pure birth process is the generalization of the Poisson process where instead of the transition rate being $\lambda$ we write $\lambda_n$ to account for dependence of the rate on the state. What ...
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1answer
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What is the relation of the Erlang B and C formulas to the Erlang distribution?
I am confused as to whether the Erlang formulas in telecommunications are related to the Erlang distribution or whether they share the same name simply because they are coined after the same ...
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Queuing Analysis
Question
I am working on an exercise about queuing analysis. Below please find the picture of the question and my workout
I have done part a, and I am struggling in find the answer of part b. Is it ...
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Average queue length for M/M/c queue with time-dependent arrival rates
I'm trying to model the queueing behavior of a system with time-dependent arrival rates. For each interval $ t $, I have different values for the arrival rate $ \lambda(t) $ and for the number of ...
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Markov Process for a Shared Channel with Two Users
Question: Consider a communication channel where two users, A and B are sharing. Their arrival rates are $\lambda_A$ and $\lambda_B$ and service rates are $\mu_A$ and $\mu_B$. Only one user can use ...
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Queueing theory-steady state probability.
Consider a queueing system comprising a single queue. Let $n$ be the system state, characterizing the number of customers in the queue, $n=0, 1, \ldots$.
Let $P_n$ be the steady state probability of ...
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1answer
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Expected Waiting Time in a Queuing System $(M | M | 2 | 5)$
In the queuing system $(M | M | 2 | 5)$, the input flow rate is $240$ requests per hour, the average service time for one request is $30$ seconds. Find the average waiting time for an application in ...
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Arrival distribution of M/M/1 queue
Show that the arrivals $A_n$ of an M/M/1 queue $X$ with initial distribution $\eta_i := \rho^{i-1}(1-\rho)$ ($i \ge 1$), where $\rho$ is the traffic intensity, satisfy $X_{A_n} \sim \ \eta$.
I ...
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Limit order book
This question is related to Steady state of a non-trivial Markov chain. .
In a financial exchange customers submit orders which can be roughly divided into three types.Firstly, there are limit ...
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How to handle a queueing system with multiple servers with different service rate?
What model suits a real system, where there are:
multiple queues with different jobs
multiple single phase server with different service rate depending on the job
I am making up this example, but I ...
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26 views
Hitting time distributions for $M_t/M/\infty$ queues
Consider an $M_t/M/\infty$ queue where customers arrive as a NHPP with time-varying rate $\lambda(t)$ and the service time is exponentially distributed with parameter $\mu$ for each of the infinitely ...
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1answer
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M M 1 Queuing Model where $\lambda_n=0$ after some point
I have an m/m/1 queue with arrival rates $\lambda_0=3, \lambda_1=2,\lambda_2=1$ and for all $n>2$ we have $\lambda_n=0$. With constant service rates of $\mu=2$
I need here to calculate the ...
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PASTA using stationary distributions
Consider a M/M/1 system with arrival rate $\lambda$ and departure rate $\mu$ and $L^a$ is the process prior to an arrival.
Let $p^a(i, j) = P(L^a_{n+1} = j | L^a_{n} = i) $ and let $p_j = (1-\rho)\...
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Rest Service Time of a M/M/1-FIFO queue
How to derive the value for rest service time from arrival rate λ and mean service rate μ in a M/M/1-FIFO queue?
Is this correct? ...
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What is the stationary distribution? Job service times are Gamma$(k, \lambda_s)$ with $k>1$. You have $N$ servers.
Problem
Suppose that you have $N$ servers, $\{s_1, s_2, \ldots, s_N\}$. Let $T_i$ represent the random variable for service time on server $s_i$. $T_i \sim \text{Gamma}(k_i, \lambda_i)$. Suppose ...
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Can I use the model of an M/M/1 queue system as an aggregation of items?
I am studying M/M/1 queue systems and try to apply it to the behavior of my system. The thing is that my system works in a similar way as an ...
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Regenerative Process Confusion…
I am struggling with the following problem and am hoping someone can give me some insights.
Consider a regenerative process like the busy period ($BP$) in an M/G/1 queue. As an example, it is well ...
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Solving another non-trivial recurrence relation
This is a generalization of question Yet another non-trivial recurrence relation to solve. . Let $E \in {\mathbb R}$ and $\lambda^{C} \ge 0$, $\lambda^{M} \ge 0$ and $\Lambda \ge 0$.In addition to ...
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How to model feedback loop system in Simulink? [closed]
I am trying to simulate in Simulink a model of a control system with queue MM1K from the book Feedback Control of Computing Systems. The book says that this is just a block diagram, so I understand ...
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J.R. Norris Markov Chains 3.4.1
Continued questions for this one:
Markov chains and queues
Explain why, for some $\theta$ in $(0, 1]$, and $k = 0, 1, 2, ..., $
$$P(X \ \text{hits} \ A_0 \mid X_0 = A_k) = \theta ^k $$
Show that $...
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Is linearization with derivatives exact?
I wonder if my following calculation yields an approximate or an exact result. I would like to linearize a performance metric of the M/D/1 queue. As given here, the average number of entities in the ...
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Applicability of M/D/1 queue for multi-class demand arrivals
Assume a queue system working as follows. Let $\lambda_i$ represent the Poisson demand rate for a customer class $i$ and $\mu$ represent the constant (deterministic) service rate. We have multiple ...
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Calculation of the first time when a certain number of customers received service.
Problem
Let us consider M/M/infinity queueing system. I would like to know the time distribution when $b$ customers received service. For this, I first showed that the distribution of the number of ...
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Independence of the length of successive busy periods in the M/M/1 Queue
I am trying to figure out if the length of successive busy periods and an M/M/1 queue are independent.
For example, assume busy period $B_1$ occurs at $t_1$ and ends at $t_2$ with length $l_1$. After ...
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1answer
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Unclear about Sakasegawa formula
I'm referring to Sakasegawa's forumla for calculating average line length in a queuing system.
I don't understand the result intuitively. For example, our server is able to serve 10 customers per ...
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Deriving the Pollaczek-Khintchine formula for the mean number in system in a $M/G/1$ queue.
Let $\{X_n:n\geqslant0\}$ be an irreducible aperiodic Markov chain. Let the arrival process be governed by a Poisson process $\{N(t):t\geqslant 0\}$ of intensity $\lambda$ and the service times i.i.d. ...
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Modelling a fast-food drive-thru as a tandem queueing network.
For motivation, I thought of this problem as I was waiting in a drive-thru at Burger King, and wondered if it could be modeled mathematically given reasonable assumptions.
Suppose customers arrive to ...
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1answer
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Finding covariance for an M/G/$\infty$ queuing system
I am studying for an exam, and one of the recommended example problems is as follows:
Consider an M/G/$\infty$ queuing system. The arrivals form a Poisson Process $\mathbf{N} = \{N(t); t \ge 0\}$ ...
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1answer
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Distribution of patient's waiting time
The doctor has two visits scheduled: first at 11 AM and second at 11:30 AM. Time of each visit is exponential with expected value of 30 minutes and both times are independent.
Assuming, that every ...
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queuing theory - how to find the distribution of number of clients in G/D/1/n queue?
I'm having a trouble with analyzing G/D/1/n queue.
Queue definition:
Customers arrivals - the time between arrivals is uniformly distributed ([y,y+1]) and the service time is constant - x.
How can ...
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1answer
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What is the probability of finding someone when a packet arrives in a queue
In an $M/M/1/2$ queue what is the probability that a packet arrives and finds one packet in the queue, given that we have an arrival rate of $\lambda$ and departure rate $\mu$.
I have thought of a ...
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Exponential Random Variable and Conditional Probability
Person 1 enters a queue, Person 1 will eventually abandon this queue, where his impatience is an exponential random variable with rate $θ$. In $s$ minutes later Person 2 will enter the system and will ...
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1answer
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Average time spent in an M/M/1 queue when incoming packets have a rejection probability.
The mean time spent in an $M/M/1/\infty$ queue would be $\frac{1/\mu}{1-\rho}$, where $\rho = \lambda/\mu$ if I am not mistaken.
If the queue throws away incoming packets with probability $p=0.5$ how ...
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1answer
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Coefficient of Variation in service time of M/G/1 queue
The Coefficient of Variation (CV) $c$ is modelled as $c^2=c_0^2+A(1-A)m_r/t_0$, where the $M/G/1$ server is partially available with a mean service time $t_0$. Availability of the server is measured ...
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How do I define a cost function to trade latency and throughput?
I have a system that is a pipeline of three functions. Function P is a message producer. Function PA is a pre aggregate of data. ...
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cellular communications queuing theory
Not sure if this is the right place to ask but this is a homework question.
...
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Computing an expectation in an $M/M/c$ loss system with two channels
Consider an $M/M/c$ loss system (called station 1) with arrival rate
$\lambda$, where customers who find all channels busy (and are lost)
are served at a secondary infinite channel facility (...
