Fdtxjr

What should be the ideal length of sentences in a blog post for ease of reading?

What is this high flying aircraft over Pennsylvania?

What is the meaning of the following sentence?

How do you justify more code being written by following clean code practices?

Is there a reason to prefer HFS+ over APFS for disk images in High Sierra and/or Mojave?

Why can't the Brexit deadlock in the UK parliament be solved with a plurality vote?

Visualizing the difference curve in a 2D plot?

The Digit Triangles

Why do Radio Buttons not fill the entire outer circle?

Why the "ls" command is showing the permissions of files in a FAT32 partition?

How to make a list of partial sums using forEach

How to leave product feedback on macOS?

What does "tick" mean in this sentence?

How do I Interface a PS/2 Keyboard without Modern Techniques?

How do I tell my boss that I'm quitting in 15 days (a colleague left this week)

"Oh no!" in Latin

Sound waves in different octaves

Is there anyway, I can have two passwords for my wi-fi

I'm just a whisper. Who am I?

How to make money from a browser who sees 5 seconds into the future of any web page?

Giving feedback to someone without sounding prejudiced

Do I have to know the General Relativity theory to understand the concept of inertial frame?

Can I run 125kHz RF circuit on a breadboard?

Check if object is null and return null

Steady-State Workload for a Compound-Poisson-Process (Lévy-Driven-Queue)

Lévy measure of a pure jump processDoes $mathbbE(|X_t|) = O(t)$ hold for a Lévy process $(X_t)_t geq 0$?bound on compound poisson processCharacteristic function of compound Poisson processDistribution of the increments of a Compound Poisson processCompound poisson process invariant measureNumber of arrivals of a compound Poisson process?Calculate $mathbbE[N(E) e^-rE]$ for Compound Poisson $(N(t))_t$ and exponential $E$.

1

$begingroup$

Suppose that $X in mathbbCP(r, lambda, b(cdot))$ with the jumps being distributed exponentially with mean $frac1vartheta$. (Hence we are in a special case of compound Poisson input: an $M/G/1$ queue.) The distribution of the jobs is denoted with $B$. Let $Q$ be the corresponding stationary workload for which holds
$$
beginalign
mathbbP=(Q leq x)= mathbbP bigg( sum_n=1^N B_n^res leq x bigg) qquad (1)
endalign
$$
where $mathbbP(N=n)= big( 1-fraclambda mathbbEr big) big( fraclambda mathbbEr big)^n$ and $B_1^res, B_2^res,...$ are i.i.d. samples from the residual lifetime distribution of $B$ defined by
$$
mathbbP big(B^res leq xbig) = frac1mathbbE[B] int_0^x mathbbP (b > y) dy
$$
with $mathbbE[B] = int_0^infty mathbbP (b > y) dy$.

The Questions:

1. What is the explizit expression for $mathbbE[Q]$ and $mathbbV[B]$?




2. Find the distribution of $N$ and $B_1^res$ in (1).

I am thankful for any answer or idea!

calculus stochastic-processes stochastic-calculus levy-processes

share|cite|improve this question

asked Mar 14 at 8:29

ChristinaChristina

234

New contributor

Christina is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  1. Have you tried using the GPK formula? I.e., if you have $kappa(alpha)=log E[e^-alpha Q]=fracalphavarphi'(0)varphi(alpha)$ then $E[Q]=-kappa'(0)$? 2. You already wrote down the distributions of the residual service time and the the Geometric distribution of $N$.
  $endgroup$
  – QQQ
  Mar 14 at 11:05
  
  

  
  
  
  

add a comment | 

1

$begingroup$

Suppose that $X in mathbbCP(r, lambda, b(cdot))$ with the jumps being distributed exponentially with mean $frac1vartheta$. (Hence we are in a special case of compound Poisson input: an $M/G/1$ queue.) The distribution of the jobs is denoted with $B$. Let $Q$ be the corresponding stationary workload for which holds
$$
beginalign
mathbbP=(Q leq x)= mathbbP bigg( sum_n=1^N B_n^res leq x bigg) qquad (1)
endalign
$$
where $mathbbP(N=n)= big( 1-fraclambda mathbbEr big) big( fraclambda mathbbEr big)^n$ and $B_1^res, B_2^res,...$ are i.i.d. samples from the residual lifetime distribution of $B$ defined by
$$
mathbbP big(B^res leq xbig) = frac1mathbbE[B] int_0^x mathbbP (b > y) dy
$$
with $mathbbE[B] = int_0^infty mathbbP (b > y) dy$.

The Questions:

1. What is the explizit expression for $mathbbE[Q]$ and $mathbbV[B]$?




2. Find the distribution of $N$ and $B_1^res$ in (1).

I am thankful for any answer or idea!

calculus stochastic-processes stochastic-calculus levy-processes

share|cite|improve this question

asked Mar 14 at 8:29

ChristinaChristina

234

New contributor

Christina is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  1. Have you tried using the GPK formula? I.e., if you have $kappa(alpha)=log E[e^-alpha Q]=fracalphavarphi'(0)varphi(alpha)$ then $E[Q]=-kappa'(0)$? 2. You already wrote down the distributions of the residual service time and the the Geometric distribution of $N$.
  $endgroup$
  – QQQ
  Mar 14 at 11:05
  
  

  
  
  
  

add a comment | 

$begingroup$

Suppose that $X in mathbbCP(r, lambda, b(cdot))$ with the jumps being distributed exponentially with mean $frac1vartheta$. (Hence we are in a special case of compound Poisson input: an $M/G/1$ queue.) The distribution of the jobs is denoted with $B$. Let $Q$ be the corresponding stationary workload for which holds
$$
beginalign
mathbbP=(Q leq x)= mathbbP bigg( sum_n=1^N B_n^res leq x bigg) qquad (1)
endalign
$$
where $mathbbP(N=n)= big( 1-fraclambda mathbbEr big) big( fraclambda mathbbEr big)^n$ and $B_1^res, B_2^res,...$ are i.i.d. samples from the residual lifetime distribution of $B$ defined by
$$
mathbbP big(B^res leq xbig) = frac1mathbbE[B] int_0^x mathbbP (b > y) dy
$$
with $mathbbE[B] = int_0^infty mathbbP (b > y) dy$.

The Questions:

1. What is the explizit expression for $mathbbE[Q]$ and $mathbbV[B]$?




2. Find the distribution of $N$ and $B_1^res$ in (1).

I am thankful for any answer or idea!

calculus stochastic-processes stochastic-calculus levy-processes

share|cite|improve this question

asked Mar 14 at 8:29

ChristinaChristina

234

New contributor

Christina is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

asked Mar 14 at 8:29

ChristinaChristina

234

New contributor

Christina is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

asked Mar 14 at 8:29

ChristinaChristina

234

New contributor

Christina is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 14 at 8:29

ChristinaChristina

234

New contributor

Christina is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 14 at 8:29

ChristinaChristina

234