# Tagged Questions

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

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### Intuition of the Mean wait time in queuing system

In queuing theory, (with a single queue and a single server) , given A is service rate (of customers) and B is arrival rate(of customers) We know that, the average time a customer waits in the ...
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### Showing a queueing system is a Markov Chain

I generally understand how to do this but I'm having trouble with a formal proof. "Consider an $M/M/1/m+1$ queue with exponential arrivals rate $\lambda$, exponential service rate $\mu$, and finite ...
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### Tauberian theorems in queing theory

I'm trying to use Tauber's theorem below (Feller 1971, chapter XIII.5) "Let U be a measure with a Laplace transform $\omega(\lambda)$ defined $\forall \lambda >0$ and $t,\tau>0$ s.t. $t\tau=1$, ...
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### Queueing theory M/M/k - probability of number of busy servers seen by next arrival process

Consider a $n$ server parallel queueing system, need to calculate the probability of $1$ busy server as seen by next arrival process. $\lambda$$=$$arrival$ $rate$ $of$ $processes$ ; $\mu$$=$$service$ ...
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### Concerning an infinite server queue with Poisson arrivals

Here's the statement of the problem (from Ross's Introduction to Probability Models): For those unfamiliar with "infinite server queues," they are described here. In this case, however, the service ...
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### Queueing Theory Probability of Customers Arriving

This is a question on my midterm practice exam, but for some reason we weren't given solutions so it's not very helpful. There's a single-server queueing system which arrivals follow a Poisson ...
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### Random Early Discard Markov Chain

I'm trying to sketch the Markov chain for a Random Early Discard queueing policy where customers arrive to the queue of infinite size according to a Poisson process with rate $\lambda$. Customers that ...
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### Balance Equations for M/M/1/m Queue

I think I found the solution for this problem, but they don't show all the steps. I was wondering if someone could explain to me how they get from the three balance equations to the solution.
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### Does this resilience/resource scheduling analogy make sense?

A firend has recently presented an analogy for the rescheduling of Doctors (big topic in the UK atm) across a 7 day week as opposed to a 5 day week with a skelton staff at weekends - I'm ignoring A&...
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### Service Systems Problem with Repair

Can anyone help me with this problem? A data centre is equipped with $M = 5$ servers, completely interchangeable. Each of them can fail independently of the others. The time to failure of a ...
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### Waiting time in Queue with $N$ policy

Suppose that a queue has $N$-policy, that is, only when there are $N$ customers in the queue the service starts. In such a queue with deterministic arrival rate $\lambda$ and deterministic service ...
Consider the following. A queue has an arrival rate of $\lambda$, where a single job enters the queue. Next, the job is processed by the service at rate $\mu$, and it emits three jobs in response to ...