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Questions tagged [probability-theory]

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

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Supremum of a probabilistic function with ambiguity distribution set using Wasserstein metric

There is a proof of how to derive distributionally robust chance constraints with ellipsoid bound. $$\inf_{\mathbb{P}\in\mathcal{D}^{WD}} \mathbb{P}\{\|\mathbf{A\zeta-b}\|_2 \leq 1\} \geq 1-\epsilon$$ ...
Kaiming Zhang's user avatar
2 votes
0 answers
25 views

Service probability for M/M/c queue for reneging where reneging time is a function of service time

There is an M/M/1 (getting a result for M/M/c is ideal) queue with arrival rate ๐œ† and service rate ๐œ‡ and participants can renege. However, the difference between normal reneging settings (constant ...
Misha Lavsky's user avatar
3 votes
1 answer
483 views

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 ...
David M.'s user avatar
  • 2,107
1 vote
0 answers
114 views

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 ...
user.mokho's user avatar
12 votes
2 answers
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 ...
Amin's user avatar
  • 2,160
3 votes
1 answer
135 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 ...
ephemeral's user avatar
  • 917
5 votes
0 answers
87 views

What's the expected number of subsets of iid random variables with sum in given range? [closed]

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 \...
ydubey7's user avatar
  • 579
11 votes
3 answers
866 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 ...
Josh Allen's user avatar
3 votes
1 answer
842 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, ...
techbyte's user avatar
10 votes
1 answer
700 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 $\...
Pedro Gerum's user avatar
20 votes
2 answers
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Reference for "expectation preserves convexity"

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 ...
LarrySnyder610's user avatar
12 votes
4 answers
2k views

Normal demand and normal lead time; is lead-time demand normal?

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 ...
LarrySnyder610's user avatar