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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? "...
romesh prasad's user avatar
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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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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 ...
RookieScientist's user avatar
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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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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 ...
Frontier_Setter's user avatar
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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 ...
insipidintegrator's user avatar
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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, ...
Yaddle's user avatar
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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 $2$ ...
MilesToGo's user avatar
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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 ...
IPreferAlgebra's user avatar
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the queue length X(t) in birth and death process

I am learning about the birth and death process, and I know an infinite M/M/1 queue with impatient customers can be described as a birth and death process, with arrival rate $\lambda$, service rate $\...
Ellen1230's user avatar
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Probability a customer will not be served in an M/G/k queue

Consider an M/G/k queue. The arrivals are exponential with rate $\lambda$. There are $k$ servers, but no queue - meaning that if a customer arrives and find no servers free, he simply goes home. How ...
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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 ...
kal_elk122's user avatar
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Lower bound for probability of birth-death process at 0

Consider a birth-death process with birth transition rate of 1 and death transition rate of $r + \gamma$ at every state $r \in \mathbb{N}$. Can we come up with an lower bound efor the steady-state ...
Alireza Amani's user avatar
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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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Generator for total time spent in the system

I know that the generator of the waiting time of a M/G/1 queue, that is, the time spent by a customer before starting service is given by $$Gf(y)=\lambda \int_0^\infty [f(y+s)-f(y)]dF(s)-f'(y)1(y>0)...
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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 ...
Genwolf's user avatar
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A $M/M/\infty$ queue with fixed probability going back to the end of the queue

Suppose for a $M/M/\infty$ queue with arrival rate $\lambda$ and service rate $\mu$, with probability $p$ the service is successful and the person will leave the queue, and otherwise she will go back ...
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Is the formula for residence time equivalent to Little's Law?

Is the equation T=M/F where T is residence time, M is quantity and F is flow rate equivalent to W=L/lambda where W is waiting time, L is queue length and lambda is flow rate? It seems to me that the ...
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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 ...
Andry's user avatar
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Equilibrium condition of queue theory with arrival depending on probability

Consider a queue (1 server only) where arrivals occurs based on a Poisson process with $\lambda > 0$ and the dropout $\mu > 0$. The thing is each arrival can correspond to 1 client with ...
Davi's user avatar
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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 ...
abc's user avatar
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Probability Analysis of Arrival Times in a Poisson Process within a Single-Server Queueing Model at a Theme Park

I'm working on a problem regarding a Poisson process and I could use some assistance. The question is as follows: A theme park ride follows a Poisson process, where people arrive at a rate of 3 people ...
knight5478's user avatar
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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 ...
APerson's user avatar
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Waiting time in a merged queue described by a random variable with exponential distribution.

I have a problem with merging 2 exponential distributions. Let's say that: people need to wait in a queue $1$ and then wait in queue $2$ or queue $3$ (with probability $\frac{1}{2}$ for each). ...
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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 ...
Siddharth Ambekar's user avatar
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Queue Kendall Notation

How to understand that queue model in Kendall notation? M/G/S/Q I don't understand letter G. Can someone give good resource or simple explanation what G stands for?
DDDDDD's user avatar
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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 ...
Alex_DeLarge's user avatar
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Help solving a partial differential equation in queuing theory

I have been working on a problem in queuing theory, and in order to understand the steady state behaviour, I have obtained a PDE. The equation is $(\mu + 2\lambda)f(x, y) + \displaystyle\frac{\partial ...
Siddharth Ambekar's user avatar
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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 ...
Alex_DeLarge's user avatar
2 votes
1 answer
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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 ...
Alex_DeLarge's user avatar
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How to calculate the M/G/1 Queue Variance of the mean queue size L?

I have been working so long to figure out the variance of the queue size L for M/GI/1 queue. From many resources, I can find the Expected Queue Size L, but for some reason I cannot find any textbook/...
kirin's user avatar
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how much is the value of $P_K$ in a M/M/1/K queue when the value of $\lambda$/$\mu$ is greater than 1?

I know that $P_K$ can be obtained as follows: Assume that $\rho$ is $\lambda/\mu$ and that the probability that there are $m$ customer in the system is $P_m$. $P_0+P_1+...+P_K=1$ ⇔ $P_0+\rho P_0+...+\...
mkim's user avatar
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1 answer
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Simplification of Summation in Queuing theory for queue with no after effects

I am looking at a queuing problem with no aftereffects. We have the following convention $v_{k}(t) = P({k \text{events happen in the time interval} (0,t)})$. We have the following equation we are ...
Ramesh Kadambi's user avatar
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Service probability for M/M/c queue where reneging time depends on service time

There is an M/M/1 (getting a result for M/M/c is ideal) queue with arrival rate 𝜆 and service rate 𝜇 and participants can renege. However, the difference between normal reneging settings (constant ...
Misha Lavsky's user avatar
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Non-standard Queuing Theory problem (Modified M/M/1)

Haven't see anything like it in queueing theory, the problem appears to be M/M/1 but with a twist? Description: The server starts serving calls only when the number of customers in the system becomes ...
ChonChon's user avatar
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1 answer
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Queue system with 2 parallel servers that works one at a time. Mean waiting time?

Consider a queueing system where customers arrive according to a Poisson process with rate $\lambda$, but the service facility consists of two parallel servers. A customer upon entry into the service ...
Alex_DeLarge's user avatar
1 vote
0 answers
135 views

Expected idle time of a server in an M/M/N queue

Consider a standard M/M/N queue where the jobs' arrival rate is $N\lambda$, the service rate of each of the $N$ servers is $1$, and $\lambda < 1$. When a new job arrives, assign it to an idle ...
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