# Tagged Questions

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### M/M/3 queue - reducing wait time by adding servers

Full question below: You are the manager of the customer support division in your company. Your division uses 3 telephone lines operated by 3 separate customer service representatives. A customer is ...
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### Queuing model $M/M/\infty$

I am considering a queuing model of the form $M/M/\infty$, you find properties of this queue here: http://en.wikipedia.org/wiki/M/M/%E2%88%9E_queue I am interested in the average busy period of this ...
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### Queuing Theory with Poisson Distribution

Suppose customers arrive in a one-server queue according to a Poisson distribution with rate lambda=1 (in hours). Suppose that the service times equal 1/4 hour, 1/2 hour, or one hour each with ...
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### Finding the inverse of Laplaceâ€“Stieltjes transformation and Convolution related to Probability

I would like to ask you something I do not understand from my book. If I have G following exponential distribution with $G(t)=1-e^{-\lambda t},t\geq 0$ then ...
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### M/M/c Queue determine $\Pi_0$

I'm trying to determine how to calculate the probabilities of going from one state to another. In a M/M/4 queue. The information I initially have is the following: $\lambda$ and ...
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### Probability of a tie

One hundred students are divided into two equal groups. Both groups vote (yes or no). Find the probability that the groups 1 and 2 both reach a majority. Students vote independently and the ...
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### Formal Theory Regarding M/M/s Queue

I have some difficulties with formally deducing the Q-matrix or infinitesimal generator for M/M/s Queues. Although I undestrand the intuitive idea I would like to know the real formal definition of ...
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### Understanding this Poisson Queueing Process

Consider $N$ busses, each which breaks down independently at a rate of $\mu$. Once a bus breaks down it is sent to a repair shop which can only repair one bus at a time and takes an exponentially ...
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### When does the next bus come?

People arrive at a bus stop according to a Poisson process at rate $\lambda$ per minute. The bus leaves every $n$ minutes, but you have no idea when the last bus left. You observe that there are $k$ ...
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### Boundedness of expected reward Markov chain (may be related to discret $M/M/\infty$ queue)

[EDIT]: I read a bit on $M/M/\infty$ queue and it may not be the right comparison and my notation may be confusing (I'm in discrete time and $\lambda,\mu$ look likes rates when they are probability). ...
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### Is Queueing Theory dead? [closed]

I was studying queueing theory for my class and noticed that we are now able to either solve with certainity most queiening problems or simulate them. is queueing a dead research area? I read this ...
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### Proof of $\text{Poisson}(\lambda p)$-arrivals.

We have a queue where people pass out of it with $\text{Poisson}(\lambda)$ and they come in with probability $p$. I understand that the arrivals follow $\text{Poisson}(\lambda p)$, but how can ...
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### Markov Chain Transition Intensity Conversion

I have a question about converting a 3-state discrete state, continuous-time, markov chain to a 2-state. My 3-state model has states: Well (state 1), Ill (state 2) and Dead (state 3). ...
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### Find the probability that the second customer to arrive has to wait to be served if arrival time is exponential and serving time is uniform

Customers line up to be serviced according to a Poisson process at an average rate of five per hour. If the time it takes to serve one customer is a continuous uniform random variable on $[0,4]$, ...
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### One vs multiple servers - problem

Consider the following problem: We have a simple queueing system with $\lambda%$ - probabilistic intensity of queries per some predefined time interval. Now, we can arrange the system as a single ...
Suppose I have M/M/1 system with $\lambda = 4$ per hour and $\mu = 5$ per hour. How can I find out if all jobs have been served after, say, 8 hours? At first I thought about doing $P(n > 40)$ since ...
Suppose I have a queue with $\lambda$ and $\mu$. I can calculate the probability that there are 2 objects in the queue trivially, but how can I compute, for example, the probability that it takes an ...