Questions tagged [queueing-theory]
Queueing theory is the mathematical study of waiting lines, or queues.
698 questions
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Why do we need \rho (utilization)<1 in queuing theory?
I’m studying basic queueing theory, in particular a single–server queue with one arrival stream and one server (a G/G/1 type setup).
Let
A be the interarrival time with mean 𝔼(A),
B be the service ...
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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 confusion around when a customer can rejoin the queue
If I have an M/M/1 queue, then I have rates $$q_{i,i+1} = \lambda, \quad q_{i,i-1} = \mu$$
Since it is guaranteed that I never have the transition $i\to i$, I can comfortably let $q_{i,i} = -\sum_{j\...
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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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Distribution of busy servers in a modified Erlang Loss system
Consider an $M/M/k/0$ queueing system (we have $k$ servers and if our system is already full, any clients will immediately leave). The arrival rate is $\lambda$ and the work rate is $\mu$. Now, let's ...
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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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How can I express this expectation as something I'm able to calculate?
Consider a queue with one server and infinitely many customers already, and a time-varying service rate $\mu(t)$. I want to get a closed expression for the expected time it takes to finish serving $n$ ...
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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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Is it better to hire $1$ fast barista vs $2$ slower baristas?
This is another math puzzle I heard today.
Consider a M/M/K queue (https://en.wikipedia.org/wiki/M/M/c_queue) in a cafe. Lets say the cafe has a rule that each queue is FIFO (first in first out), each ...
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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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If a new line opens, what's the optimal strategy for $i$th person?
If there are $n$ people in a queue, and a new line opens, each person can choose either to stay in the original line or to switch to the new line. Each person should decide at the same time. If a ...
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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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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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