Questions tagged [queueing-theory]

Queueing theory is the mathematical study of waiting lines, or queues.

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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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M/M/k queue problem with with exponential rate distribution [closed]

A room has $L$ lamps. Suppose that the lamps have independent lifetimes, with exponential rate distribution $\lambda$ ($0 < \lambda < \infty$). Suppose, furthermore, that in the instants of a ...
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Independence assumption for interarrival time [closed]

I am new to Queuing systems. There is an independent assumption made for the interarrival time. Can someone please explain to me why this assumption is true, can you provide me with an example? "...
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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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1 vote
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Queues wait for other queues: A communication problem

I am working on a problem which involves a single server that requires multiple inputs to do a computation. Each of these inputs arrive as a Poisson process with rate $\lambda$. Hence, a situation ...
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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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Let $W_t$ be the standard Wiener process, find probability $P_t(u)=P(|W(s)|\leq u,0\leq s\leq t)$.

For an one-dimensional standard Wiener process $W_t$, find $P_t(u)=P(|W(s)|\leq u,0\leq s\leq t)$. This arises from the problem to find the first time a random walk has travelled a given distance, or ...
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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/c Queue Model Solutions for Average Waiting Time and Queue Length

I am seeking assistance with a queueing theory problem involving the M/M/c queue model from my textbook. I have attempted to solve the problem and would greatly appreciate it if someone could review ...
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1 vote
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Number of ways arrival and departure events happen in a FIFO queue.

Consider a FIFO queue with an upper bound of queue size $U$. $N$ people already in the queue when this person A arrives at time $t_0$. Now there are $N+1$ people in the queue ($N+1\leq U$). Suppose A ...
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How to calculate the response time with M/M/c/PS(Processor Sharing) models in queueing theory?

I'm trying to model the process scheduling mechanism in Linux using queueing theory models. Assuming that both the arrival and processing times of processes in the system follow a Poisson distribution ...
1 vote
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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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Interpreting $\displaystyle\frac{\lambda}{c\mu}<1$ in steady-state solution of M/M/c model

In this video, the professor embarks on finding the equilibrium solution to an M/M/c queueing model, with the condition that $\displaystyle\frac{\lambda}{c\mu}<1$, where: Customers arrive into the ...
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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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Proving a property related to $M/M/c$ queues - Queueing theory.

My goal is to show that in a $M/M/c$ queueing system it is satisfied that $$L_s = L_q + \frac{\lambda}{\mu},$$ where $L_s$ represents the average number of costumers in the system, $L_q$ represents ...
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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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Reference request: Queueing and reliability theory from a stochastic point of view

I am an analyst and looking for good references on the topic "Queueing and reliability theory" but from a probabilistic perspective. Ideally, I would like to find a monograph on this topic, ...
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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/M/c queue, but customers might leave the queue due to impatience

Given a M/M/c queue (Poisson arrival, exponentially distributed service time, c servers). The queue is unlimited in length and operates by first-in-first-out. Each customer who arrives and needs to ...
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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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Moments of waiting time in a G/G/1 queue

Are there any results for moments of waiting or sojourn time(total time spent by a job in the system including its own service) for a G/G/1 queue. I know that in the special case of M/G/1 queues the ...
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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 ...
1 vote
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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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Average number of customers at the service facility of an M/M/1/S queueing system

My textbook (pg 121, formula 4.39) says: The average number of customers at the service facility: $N_S = P[k=0]E[N_S|k=0] + P[k>0]E[N_S|N>0] = 1 - P_0 = \rho(1-P_S)$ But I can't understand why ...
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