Questions tagged [probability-theory]

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

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3
votes
1answer
42 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 ...
1
vote
0answers
87 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 ...
11
votes
2answers
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 ...
3
votes
1answer
82 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 ...
5
votes
0answers
62 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 \...
9
votes
3answers
709 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 ...
3
votes
1answer
97 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, ...
7
votes
1answer
124 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 $\...
20
votes
2answers
695 views

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 ...
12
votes
4answers
977 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 ...