From BoykinWiki

Jump to: navigation, search

EEL6507 Spring 2009, Lecture 03, Monday 2008/01/12 (Notes created by Ajay Jain)

Continue With Little Results.

Introduce Markov Process.

Little Result For Waiting Time

$γ q (t)$ . The total time spent by all customer in $(0, t)$ .

where, $\gamma(t) = \int_{0}^{t} N(\tau) d\tau$ .

$W t$ . The Waiting time per customer in $(0, t)$ .

$W_t = \frac{\gamma(t)}{t}$

$\bar{N (t)}$ is Average no. of customer in queue in $(0, t)$ .

$\frac{\int_{0}^{t}N_q(\tau)d\tau}{t}$ = $\frac{\gamma_q(t)}{t}$ = $\frac{\bar{\alpha(t)} W_t}{t}$ $\bar{N}_q (t) = \lambda \bar{W}$

Now Service Time

$\begin{align} \bar{N}_q (t) &= \lambda \bar{X}\\ \bar{N}_ {system}&= \bar{N}_{queue}+ \bar{N}_{server}\\ \lambda \bar{T} &=\lambda \bar{W}+\lambda \bar{X} \\ \bar{T} &= \bar{W}+\bar{X} \end{align}$

i.e average system time =average wait time +average service time

Traffic Intensity

It is defined as rate of work enters the system

$\frac{customer}{sec}$ $\frac{work}{customer}$ = $\lambda\bar{X}$

The Unit is Erlang

Utilization Factor

It is the ratio of the rate at which work enters the system to the maximum rate at which the system performs the this work

$\rho = \frac{Arrival rate}{service rate}=\frac{\lambda}{m/\bar{X}}= \frac{\lambda \bar{X}}{m}$

Where m is No. of Servers

Busy Probability

Let probability of being idle = $P o$

Let in period $τ$ there are on average $λτ$ customers arrives.

So Busy Time for server = $τ(1 - P o)$ .

Total customer served = $\frac{\tau (1- P_o)}{\bar{X}}$ .

We know that total customer served should be same as total customer arrived.

Hence, $\begin{align} \frac{\tau (1- P_o)}{\bar{X}}&=\lambda\tau\\ \ (1- P_o)&=\lambda \bar{X}\\ \ P_o&=1-\rho \end{align}$

Markov Process

A set of random variables $X n$ forms a markov chain if the probability that the next value(state) is $X n + 1$ depends only upon the current value(state) $X n$ and not upon any previous values.

Expressed analytically

$P [X (t n + 1 = X n + 1 | X (t n = X n, X (t n - 1 = X n - 1,....., X (t 1 = X 1)] = P [X (t n + 1 = X n + 1 | X (t n = X n]$

Where, $t 1$ < $t 2$ < $t 3$ < .....< $t n$

EEL6507sp09L03

From BoykinWiki

Contents

EEL6507 Spring 2009, Lecture 03, Monday 2008/01/12 (Notes created by Ajay Jain)

Little Result For Waiting Time

Traffic Intensity

Utilization Factor

Busy Probability

Markov Process

Views

Personal tools

Navigation

Search

Toolbox