# Tag Info

9

You have fallen victim to the renewal paradox, a.k.a. inspection paradox, a.k.a. length-biased sampling. $F_{\Delta}$ is the distribution of service time for the kth customer, but it is NOT the distribution of service time for the customer being served at a preselected time $T$. The very manner of selecting the customer based on observing at a preselected ...

9

Your calculations (factoring and simplification) are incorrect. $L$ is neither convex nor concave as a function of $\lambda$ and $\mu$. This can be concluded by examining the eigenvalues of the Hessian of $L$ with respect to $\lambda$ and $\mu$. I used MAPLE to compute the Hessian, and then evaluate its eigenvalues at the point $\lambda = 0.5, \mu = 1$. ...

7

This is an attempt at an answer, based on my current understanding: Background: Operations Research is an interdisciplinary field. You go from business administration and economics to theoretical computer science and mathematics. In practice, you observe a real-world situation, model it, solve the model (or develop a method for solving general instances), ...

7

Certainly, and to take the problem's structure into account, one could model the queue knowing - or guessing - prior information about the queue's structure and/or parameter distribution and use Bayesian inference. See for example the following sources and their references: Armero, C., & Bayarri, M. J. (1994). Bayesian prediction in M/M/1 queues. ...

7

Not directly answering your question of how to code it manually but for discrete simulation of queues in R I would strongly recommend the simmer package. The minimal code for your example would look like this (adapted from the tutorial). library(simmer) library(simmer.plot) lambda <- 2 queue <- trajectory() %>% seize("server", amount=1) %>% ...

6

Unfortunately, much of the performance analysis and transient approximations for time-varying systems with non-Markovian (non-exponential) properties are not easily obtained in book form (see note at bottom). This answer lists some books that don't require measure theory. Some Queueing & Renewal theory books: (non-measure theoretic) Probability, ...

4

Since you are unfamiliar with OR, I would recommend using discrete event simulation, which I think is the easiest approach for a newcomer (although it may require some programming chops, depending on what software you use). You will need a bit more than just average service completion times -- you will want a distribution of service times. (In the absence of ...

4

On the solving side, some hot topics include: presolving techniques GPU-powered algorithms algorithms designed for problems that consume a lot of memory algorithms for distributed architectures decompositions quantum computing optimisation algorithms large-scale factorisation algorithms/implementation domain reduction techniques automatic differentiation ...

4

I enjoyed Performance Modeling and Design of Computer Systems: Queueing Theory in Action (Amazon link) by Mor Harchol-Balter, which sounds like it fits your bill pretty well. I have it on my desk. ISBN-13: 978-1107027503

4

I have used Stochastic Modeling: Analysis and Simulation by Barry Nelson and found it to be a pretty gentle introduction. It covers stochastic processes, queuing, and simulation.

4

The following proof approach for Bruke's Theorem was given in this Lecture by Richard Clegg: Definition: A chain is called time-reversible if $p_{ij} = p^*_{ij}$ for all $i$ and $j$. This occurs if and only if: $p_{ij}\pi_i = p_{ji}\pi_j \ \ \forall i, j$. So the Birth-Death processes are time-reversible. Therefore all the queues which can be modelled as ...

4

After having read Chapter 5.3 of Decision Making Under Uncertainty by Mykel J. Kochenderfer, I have come to some conclusions. We are dealing with model uncertainty, in which case we can formulate a Bayes Adaptive Model. In the book that I read, the term model uncertainty refers more to not knowing what the transition probabilities nor the structure of the ...

4

Note: This answer is intended to show what I have learned from the valuable answer provided by @Mark L.Stone. His post answered my question of why the simulation is biased. Hence, this post provides only additional insight. I chose to post it as an answer and not an edit due to the original question already being lengthy. What has been learned comes from ...

3

I am not an expert in elevator traffic system simulation, but by googling you could find lots of related papers and literature. Some of them are as follows: Modelling of Elevator Traffic Systems Using Queuing Theory The modeling and simulation of elevator group control systems for public service buildings Also, there are some useful simulation software ...

3

Dyer and Proll (1977)1 showed that for an M/M/c queue, the mean waiting time is a strictly decreasing and convex function of c. Reference  Dyer, M. E., Proll, L. G. (1977). On the Validity of Marginal Analysis for Allocating Servers in M/M/c Queues. Management Science. 23(9):1019-1022.

3

This is a $M/M/1$ queue with Poisson arrival distribution with $\lambda =1/10$ and Exponential service distribution with $\mu=1/3$. The proportion of the time that the system is busy can be calculated by using the following equation: $$\rho=\frac{\lambda}{\mu} = \frac{1/10}{1/3}=0.3$$ which is exactly the proportion of time that a person arriving at the ...

2

I learned from Quantitative System Performance Computer System Analysis Using Queueing Network Models by Lazowska, et.al. Unfortunately, it is no longer published, but it is available for free online. It may seem a bit out of date today but it is considered the classic for queueing network analysis of computer performance. It does not really cover the ...

2

Introduction to queueing theory and stochastic teletraffic models$^1$. The aim of this textbook is to provide students with basic knowledge of stochastic models that may apply to telecommunications research areas, such as traffic modeling, performance evaluation, resource provisioning, and traffic management. These research areas are included in a ...

2

Since you are assuming infinite capacity, this sounds like an $M/M/\infty$ queueing system.

2

The steady-state probability of being served for an M/M/1 queue with exponential reneging times and no balking is $$p_s=\frac{1+z}{1+r(1+z)}$$ where $r=\lambda/\mu$ is the service intensity, and $$z=\exp(r\mu\tau)\cdot(r\mu\tau)^{-\mu\tau}\cdot\gamma(\mu\tau+1,r\mu\tau).$$ Here, $\gamma(x,a)$ is the unscaled lower incomplete gamma function:  \gamma(s,x)...

1

You can derive them from the balance equations. If you check Taha's or Lieberman's Introduction to OR books, you can find the proofs.

1

There are a couple of ways to derive the steady state probabilities for a $M/M/1/k$ queuing system with Markovian* arrivals (the first $M$), exponential service time distribution (the second $M$), a single server (the 1), and a finite total system capacity of $k$. Note this implies the queue can be at most $k-1$. *Recall a system with $M$ arrivals has ...

1

So far this is what I have come up with lambda <- 2 interarrivals <- rexp(5000,lambda) ## (2 items per minute) Provided the $\mu$ we expect that the interarrivals is about half a minute mean(interarrivals) <- 0.516 service.times <- rnorm(5000, mean=8,sd=1) where the service distribution is $N(8,1)$ arrival.times <- cumsum(interarrivals) ...

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