Recent content by TaPaKaH

Suppose we have two series of independent Bernoulli experiments with unknown probabilities ##p_1## and ##p_2##. The first series registers ##x_1## successes in ##n_1## trials while the second series registers ##x_2## successes in ##n_2## trials. Is there a way that we can compute probability...

Suppose we have a random service time ##T## with residual service time ##R## observed at some point along the way. What is the correct way to call ##T## (in 1-2 words, without having to introduce ##R##) if: 1) For any observation time, ##\mathbb{E}R\leq\mathbb{E}T##? 2) For any observation...

So far I did case-by-case analysis branching on terms, at which maxima for ##f(x+2e_i)##, ##f(x+e_i)## and ##f(x)## are achieved. Six of eight cases went rather smooth, but two cases are giving me problems. Case 1: Case 2: I guess the proof lies in rearranging the terms in a smart way and...

Suppose we have a function ##f:\mathbb{N}\times\mathbb{N}\to\mathbb{R}_+## that isincreasing: ##f(x+e_i)\geq f(x)## for any ##x\in\mathbb{N}^2## and ##i\in\{1,2\}##;convex: ##f(x+2e_i)-f(x+e_i)\geq f(x+e_i)-f(x)## for any ##x\in\mathbb{N}^2## and ##i\in\{1,2\}##.How could one show that a...

Having one exponential is perfectly acceptable, it's the ##\exp(...\exp(...))## bit that's causing the integration problems. It seems that ##g(x)=(1-e^{-\frac{\lambda}{\beta}})\cdot\min(\frac{1}{e},1-e^{-\beta x})## is a good enough lower bound. Thank you. EDIT: Just found an even better...

A function ##f:\mathbb{R}^3_+\to[0,1]## defined as ##f(\lambda,\beta,x)=1-e^{-\frac{\lambda}{\beta}\left(1-e^{-\beta x}\right)}## serves a lot of pain under integration. As this function is used to describe a lower bound, could anyone suggest another non-zero function that would be smaller than...

Your example makes perfect sense. But what if we assume that ##F(x)## is increasing in ##x## and ##F(0)=0##?

Suppose we have a function ##F:\mathbb{R}_+\to\mathbb{R}_+## such that ##\frac{F(y)}{y}## is decreasing. Let ##x## and ##y## be some ##\mathbb{R}_+##-valued random variables. Would ##\mathbb{E}x\leq\mathbb{E}y## imply that ##\mathbb{E}F(x)\leq\mathbb{E}F(y)##?

I tried the second approach: the busy period ##BP## for type-1 customers has the following Laplace-Stieltjes transform: ##\tilde{BP}(s)=\frac{\lambda_1+\mu_1+s-\sqrt{(\lambda_1+\mu_1+s)^2-4\lambda_1\mu_1}}{2\lambda}## so we can indeed model the type-2 queue as a queue, in which server uptimes...

Well, the global balance equations system is as...

Two-dimensional diagram is what I initially started with but it eventually reduced to this, which I thought might be "easier" to solve.

The unconditioned number of type 1 customers in the system is a Markov chain with the following transition rates: This yields balance equations ##p_n=\left(\frac{\lambda_1}{\mu_1}\right)p_{n-1}## that are easy to solve. But, when we condition on there being exactly one type 2 customer, the...

Homework Statement Suppose we have two types of customers, whose interarrival times are exponentially distributed with parameters ##\lambda_1## and ##\lambda_2##. These customers have to share one server that gives preemptive priority to customers of type 1. The service times are...

Suppose we have a Markov chain with stationary distributions ##p_n=\frac{a}{nb+c}p_{n-1}## for ##n\in\mathbb{N}## where ##a,b## and ##c## are some positive constants. It follows that ##p_n=p_0\prod_{i=1}^n\frac{a}{ib+c}##. Normalisation yields...

Assume we have a number ##S_0##. For ##i=1..n## define$$S_i=\begin{cases}(1+b)S_{i-1}\text{ with probability }p\\(1+a)S_{i-1}\text{ with probability }1-p\end{cases}$$. What is the expected value of ##S_n##?