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Questions tagged [queueing-theory]

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

Queueing theory is the mathematical study of waiting lines, or queues. In queueing theory a model is constructed so that queue lengths and waiting times can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service (Wikipedia).

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Little's Law, Queueing Theory, and the Universal Scalability Law

I'm trying to use Little's Law to translate Neil J. Gunther's Universal Scalability Law into the latency-versus-utilization domain, and I'm having a little bit of trouble with both the concepts (which are confusing) and the algebra (which is never…
user381310
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Can you explain the queueing question/answer in this MBA exam?

In the paper that describes how ChatGPT could pass an MBA exam there is a question and associated answer relating to queueing that seems intuitively wrong. I copy the question and the answer the author of the paper gives (NOT ChatGPT) in full…
User65535
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For a M/M/1/K System, why is utilization = $\rho=\lambda T_s$

Before I ask my question these are the symbols I used: $w$ = Customers waiting in queue $\mu$ = Service rate $T_s = 1/\mu$ = Average service time $\lambda$ = Average arrival time $\rho$ = Server utilization How come $\rho=\lambda T_s$ for a M/M/1/K…
Krimson
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If a fun park wants to reduce ride waiting time, should the length of the rides be increased or decreased?

At a recent trip to a fun park I was struck by the long waiting time for rides (eg roller coasters). So I was wondering: if you changed the duration of a ride (eg let a merry-go-round go around ten times instead of five times or let a roller coaster…
Reinstate Monica - Goodbye SE
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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 type of service. Each facility contains a single…
Kamini
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Does a single-line result in shorter average queue time?

Set-up: Travelers are waiting in line to be processed by immigration officers. There are N officers, each with their own counter. No assumptions about line-switching, constant processing speed, homogeneous processing speed, or homogeneous entrance…
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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 tasks per time unit and each processor has a…
ksm001
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Average number of customers in M/M/1/K queueing system

For an M/M/1/4 queue with $\lambda=7,\mu=8,K=4$ and $\rho=7/8$, $$L=\frac{\rho}{1-\rho}-\frac{(K+1)\rho^{K+1}}{1-\rho^{K+1}}$$ I don't understand how the expression for $L$ is derived here. I know the first term in there $(\frac{\rho}{1-\rho})$…
pihyvev
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Little's Formula for M/G/1/c queue

Suppose we have an M/G/1/c loss system, with equilibrium distribution $\;\pi,\;$ service times $\;S_i\;$ and arrival rate $\;\lambda.\;$ I'm trying to show that $\;(1-\pi_0)=(1-\pi_c)\lambda \mathbb{E}S_1.\;$ The question suggests use of Little's…
Ben Derrett
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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 jobs arrive at a rate $\lambda_a$. Suppose that…
Logan Schelly
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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: $$p_n=Β_mF(n)\prod\limits_ {i=1} ^{N}\lambda_i^{n_i}$$ Here $B_m$ is constant,…
user52561
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Distribution of waiting time in queueing system

Suppose I'm in line at a bank. There are 4 bank tellers, and 12 people in line in front of me. All the tellers are idle in the beginning so the first 4 customers get processed immediately. Each customer takes a random amount of time to be processed,…
Bai Li
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Removing some arrivals from a Poisson Process

The inter-arrival time of a Poisson Process, $t$, conforms to the exponential distribution, so the probability density function for $t$ is $f(t)=\lambda e^{-\lambda t}$, $t>0$. ($\lambda$ is the arrival rate of the Poisson Process.) For two adjacent…
Bloodmoon
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M/M/s Queue - Gas Station

I have been trying to wrap my head around an M/M/s queue problem but I can't seem to understand what's going on. The problem is queuing at a gas station and I have done some research about the M/M/s queue model but can't seem to figure out how it…
user130256
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A problem on queues.

"Consider a queueing process with two servers in which the inter-arrival times and service times are exponential with parameters $\lambda$ and $\mu$ respectively. Let $X_n$ be the queue length at the time of arrival of the $n^{\text{th}}$…
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