Questions tagged [queueing-theory]
Queueing theory is the mathematical study of waiting lines, or queues.
698 questions
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Moments of waiting time in a M/G/1 queue
I want to determine on how many moments of the service distribution does the moments of waiting time distribution depend on in a M/G/1 queue depend. I believe the $n^{th}$ Moment of Waiting Time ...
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How to calculate the expectation of the limit of this random process rigorously
Assume that the starting point continuously and stably transmits information to the end point, and the time intervals between any two adjacent messages from the starting point are independent and ...
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Queueing Theory System: Single Server & Forcing No Queue
I have an IT scenario at work where we have a single server that is dividing its time against all queries that arrive. Therefore, no queue forms as this single server divides its time against all ...
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Prove that the expected number of customers in an M/M/s queue is convex
I have an M/M/s queue with arrival rate $\lambda$ and service rate $\mu$, then the expected number of customers in the system with s servers is equal to $E[N_s] = \frac{\rho_s}{1-\rho_s}C(s,\rho) + \...
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Long run average holding cost
I have a single server queue with arrival rate $\lambda$. I have calculated that the steady state waiting time of a customer is $W$. Suppose it is given that the per unit time cost of having a ...
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Stationary distribution of $M/M/\infty$ queue
Looking in Wikipedia and other sources, I've found that the stationary distribution of an M/M/$\infty$ queue is given by
$$
\pi(x) = \left(\frac{\lambda}{\mu}\right)^x \frac{e^{-\lambda/\mu}}{x!} \...
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Queuing theory - Probability of servers being busy when packets are discarded if all servers are busy
Here are some simple twists on a queuing question that I cannot seem to get my head around.
a) Suppose that a server $S$ receives packets at rate $\lambda$. Call this arrival process $A$. The time ...
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Distribution of the longest queue's length in parallel queues
Considering $n$ people line up at $q$ queues.
Let's say all people choose which queue to line up randomly, so each people has probability $1/q$ to choose a particular queue. Then the length of any ...
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Calculating Variance in Waiting Time for a Queueing Network
I'm working on a queueing network model that incorporates blocking and features two states. After defining the global balance equations, I solved them for my parameters arrival rates (λ), service ...
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Queueing theory - connected queues stability condition
hey everyone, given a system with poissonian process split with probabilities p and q to queues with exponential serving times, notice you can move from being assigned to queue 1 to instead being ...
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M/M/1 Queues : Exclusive Queue Length is not Markov
For a M/M/1 queue let $N_q(t) = (Q(t)-1)^{+}$ be the number of customers in the queue except the one being served. We have to show that $N_q(t)$ is not a continuous-time Markov chain. [src: Sidney ...
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Cycle of queues
Consider a closed queueing network where there are two queueing nodes,
each with one server, FCFS queueing disciplines, and independent exponential
service times. The server at node $i$ has mean ...
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Burkes theorem and M/M/1 queue
Burke's theorem says that the output process of an $M/M/1$ queue with arrival rate $\lambda$ and service rate $\mu$ follows a Poisson with parameter $\lambda$. Suppose after service completion the ...
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How to apply queuing theory to find the long run proportion of customers who leave the system?
I am trying to apply queuing theory / birth and death process to the following.
Suppose customers arrive in a restaurant according to a Poisson process with rate $\lambda = 1$.
Suppose there are $2$ ...
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$M/G/\infty$: application of marking and transformation, finding the mean measure
Consider a queue $M/G/\infty$, starting with arrival time of calls as a PPP$(\Lambda)$ and lengths of calls as $iid$ random variables with common distribution $G$. The times when the calls terminate ...
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Understanding Sojourn times of M/D/1 queue
I am trying to understand how to approach a problem involving a Poisson Process queue with a deterministic service time. We have that the mean rate of arrival time is your standard $\lambda$ customers ...
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Finite queue transition probabilities with geometric exits
A server system with a finite queue is the following Markovian system: the state is defined as the length of the queue. At each time unit with probability $p$ regardless of the system’s state and ...
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Sanity check for Laplace-Stiletjes transform of compound process
I have a counting process $Y(t)=\sum_{i=1}^{N(t)}X_i$, where $X_1,X_2,\ldots$ are iid, and $N(t)$ is a non-negative integer random variable independent of $X_i$. I want to calculate the Laplace-...
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D/M/1 Queue Sojourn Time Variance
I am trying to model a queuing system, where the arrival process is deterministic and the service process is exponential, thus resulting in a D/M/1 queue. In that case the main factor for the rate at ...
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Response times of simulated M/M/1 queue are not exponentially distributed
I have created a simulation for an M/M/1 queuing system in Python using Simpy.
Simulations
The code of the simulation is just one class triggering two processes (in the Simpy sense of process):
One ...
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Generator of waiting time process in a queue
The text I am reading says that the infinitesimal generator of the waiting time process in a M/G/1 is given by
$$Gf(x)=\lambda \int_0^\infty [f(y+s)-f(y)]dF(s)-f'(y)1(y>0)$$
where $F$ is the ...
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residual waiting time and spent time
Does the expected value of the residual time (the amount of time one has to wait) equal the expected value of the spent time (the amount of time since the last arrival)? I know that the expected value ...
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Foster-Laypunov stability criterion for Continuous state spaces
I have been trying to prove the stability of a queuing system (stochastic), and the state-space I have obtained is uncountable.
I am aware of the Foster-Laypunov criterion, but as far as I know, it ...
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Distribution of number of customers in waiting queue in a M/M/1/S queueing system
What we deal with when talking about performance measures of this systems is mostly average values.
But how can I get the distributions of this values, i.e. the distribution of the number of customers ...
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M/M/1 queueing system with random dropping customers
I have an M/M/1 queueing system in which customers arrives at rate $\lambda$ and are served at rate $\mu$, but, upon entry, each customer can be randomly dropped with probability $1-p$ or enter the ...
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