# Questions tagged [probability-theory]

For questions on the mathematical modeling of processes with randomized outcomes.

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### Service probability for M/M/1 queue with reneging

Consider an M/M/1 queue with arrival rate $\lambda$ and service rate $\mu$, where participants renege after a random amount of time which follows an exponential distribution with mean $\tau$. We ...
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### Markov Chain - Stationary or Borderline distribution [closed]

I have a Markov chain problem below, where : The problem An urn initially contains 3 black balls and 1 red ball. The balls are indistinguishable to the touch. One ball is randomly drawn. If this ball ...
3k views

### Is it necessary to study rigorous math courses in OR?

I am a business student with engineering background and I am studying papers published in some journals like Management Science, Operations Research, Math of OR and they use some notations and ...
85 views

### Queuing Theoretic Model with Memory

Consider a telephone company which receives call request at some arrival rate and serves each request with some service rate. This can be modeled using a Poisson Process. However I wish to model the ...
64 views

Suppose $X_1,\ldots,X_n$ are drawn i.i.d from a uniform distribution on $[0,1]$ and let $x$ be the random vector $(X_1,\ldots,X_n)$. Then consider the random variable $Y_v = v^\top x$ for all $v \in \... 3answers 727 views ### Modeling the Choose function In statistics, one often encounters the choose function${x \choose y}$which encodes the number of ways of choosing$y$items from a set of$x$items. How would one go about modeling a choose ... 1answer 152 views ### Question about a queueing problem Arrivals at a telephone booth are considered to be Poisson with an average time of 10 minutes between one arrival and the next. The length of phone calls is assumed to be distributed exponentially, ... 1answer 138 views ### The departure process of an$\rm M/M/\infty$queue Burke's theorem says that the output process of an$\rm M/M/1$, an$\rm M/M/C$, and a$\rm M/M/\infty$queue with arrival rate$\lambda$and service rate$\mu$follows a Poisson with parameter$\...
It is well known that expectation preserves convexity: If $f(x)$ is convex and $Y$ is a random variable, then $\mathbb E[f(x-Y)]$ is convex. This property arises in, for example, inventory theory. I ...
In a continuous-review $(r,Q)$ inventory system under a type-1 service level constraint, if the demand per unit time is distributed as $N(\mu,\sigma^2)$ and the lead time, $L$, is a constant, then the ...