# Questions tagged [queueing-theory]

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

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### What is the fraction of customers lost in a finite queue with one server, M/M/1/k? k = four places and s = 1 server

What is the fraction of customers lost in a finite queue with one server, M/M/1/k? $k =$ four places and $s = 1$ server $k=4, \lambda=\dfrac 1 {30}$, $\mu=\dfrac 1 {25}$ The steady-state probs are p0 ...
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### Proving recurrence of a random walk with unbalanced step sizes

This question arose from a queuing theory question. Part of the problem is to prove when the queue is positive recurrent. The jump chain of the queue is a reflecting random walk with the following ...
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### How to calculate resource utilization

I have learned some probability and statistics, and would like to apply it to a problem where I need to understand the resource utilization of a given set of servers in a time unit, say, a day. I ...
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### What are the sample points in sample space? What is the random variable?

As we know that interarrival time in queueing theory follows exponential distribution. I want to know what are the sample points and what is random variable corresponding to interarrival time? As ...
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### Customers arrive at a service center according to a Poisson process with a mean interarrival time of 15 minutes.

Customers arrive at a service center according to a Poisson process with a mean inter-arrival time of 15 minutes. What is the probability that no arrivals occur in the first half hour? What is the ...
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### Why is M/G/1 queue viewed only at departure times?

It is said in the book that for M/G/l queue, viewed only at departure times, leads to an imbedded discrete-time Markov chain. Viewing the queue only at arrival times does not yield a Markov chain. ...
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### Rest Service Time of a M/M/1-FIFO queue

How to derive the value for rest service time from arrival rate λ and mean service rate μ in a M/M/1-FIFO queue? Is this correct? ...
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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 ...
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### Can I use the model of an M/M/1 queue system as an aggregation of items?

I am studying M/M/1 queue systems and try to apply it to the behavior of my system. The thing is that my system works in a similar way as an ...
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### Regenerative Process Confusion…

I am struggling with the following problem and am hoping someone can give me some insights. Consider a regenerative process like the busy period ($BP$) in an M/G/1 queue. As an example, it is well ...
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### Solving another non-trivial recurrence relation

This is a generalization of question Yet another non-trivial recurrence relation to solve. . Let $E \in {\mathbb R}$ and $\lambda^{C} \ge 0$, $\lambda^{M} \ge 0$ and $\Lambda \ge 0$.In addition to ...