EEL6507sp09L26

From BoykinWiki

Jump to: navigation, search

Contents

EEL6507 Spring 2009, Lecture 26, Wednesday 2009/03/18 (Notes created by Ajay Jain)

  • Series Parallel Servers
  • Departure Distribution

Series Parallel Servers

Series parallel servers system with same service rate
Series parallel servers system with same service rate

what is b(x) i.e service pdf for entire facility?

Same service rate in each stage

The coefficient of vartiation (C) is defined as:

b(x) = \sum_{i=1}^k\alpha_i b_i(x)

b_r(x) = \frac{r_i\mu_i \left(r_i\mu_i x\right)^{r_i-1}e^{-r_i\mu_i x}}{(r_i-1)!}\,\!

B^{*}(s) = (\frac{r_i\mu_i}{(s + r_i \mu_i)})^{r_i}

B^{*}(s) = \sum_{i=1}^k \alpha_i(\frac{r_i\mu_i}{(s + r_i \mu_i)})^{r_i}

Different service rate in each stage

Series parallel servers system with different service rate
Series parallel servers system with different service rate

B^{*}(s) = \sum_{i=1}^k \alpha_i B_i(s)

B_i^{*}(s) = \prod_{j=1}^{r_i} \frac{\mu_{ij}}{(s + \mu_{ij})}

B_i^{*}(s) = \sum_{i=1}^k \alpha_i \prod_{j=1}^{r_i} \frac{\mu_{ij}}{(s + \mu_{ij})}

Basically it gives more flexibility for PDF realisation.

Example Problem

An example
An example

B_i^{*}(s) = Laplace transform before i branch

B_i^{*}(s) = \alpha_i + \beta_i \frac{\mu_i}{(s + \mu_i)} B_{i+1}^{*}(s)

B_{r+1}^{*}(s) =1

B_r^{*}(s) = \alpha_r + \beta_r \frac{\mu_r}{(s + \mu_r)}

B_{r-1}^{*}(s) = \alpha_{r-1} + \beta_{r-1} \frac{\mu_{r-1}}{(s + \mu_{r-1})}(\alpha_r + \beta_r \frac{\mu_r}{(s + \mu_r)})


B_i^{*}(s) = \alpha_1 + \sum_{i=1}^r \alpha_{i+1} \prod_{j=1}^{i} \beta_j \frac{\mu_{j}}{(s + \mu_{j})}

Multiple queue network

Multiple Queue
Multiple Queue
  • To solve this first we will require Departure distribution of customer

Departure Distribution

We find this for M/M/m queues.


F_{k}(t) = Prob[N = k \ And\ T > t] \!

Where,

  • T = Time till next departure.
  • t= Time since previous departure.
  • Goal : To find time interval in which there was no departure which will give the distribution of departure time.

We proceed by talking small time interval in which there was no departure ,That can happpen in two ways

  • (1) No arrival and No departure
  • (2) One arrival and One departure

So

F_k(t+\Delta t)= \! No departure in interval (t + t+\delta t) \!

F_k(t+\Delta t)= F_k(t)(1-\Delta t\lambda)(1-\mu_k\Delta t)+ F_{k-1}(t)(1-\mu_{k-1}\Delta t)\Delta t \lambda \!

lecture voice recording

lecture_voice_recording

Retrieved from "http://boykin.acis.ufl.edu/wiki/index.php/EEL6507sp09L26"