EEL6507sp09L20

From BoykinWiki

Jump to: navigation, search

Contents

EEL6507 Spring 2009, Lecture 20, Monday 2009/02/25 (Notes created by Gustavo Vejarano)

Erlangian Service

In the Erlangian service model, each customer brings r\,\! stages with it, and the customer leaves the system only when its r\,\! stages have been served. The server serves only one stage at a time within an exponentially-distributed service time.

Notation:

c\,\! is the number of customers

v\,\! is the service stage

r\,\! is the number of stages per customer

Service time distribution

The total service time for a costumer is the summation of r\,\! exponentially distributed random variables.

B_r^*(s) = \left ( \frac{r\mu}{s + r\mu} \right )^r\,\!

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

\left \langle E_r \right \rangle = \frac{1}{\mu}\,\!

\sigma_r^2 = \frac{1}{r \mu^2}\,\!

Example 1: Approximating a general distribution with the Erlangian process

If the system to be modeled has a mean service rate of 1 customer per second with a variation of 0.1 custumers, then

\left \langle E_r \right \rangle \simeq 1\,\!

\sigma_r \simeq \frac{1}{10} \rightarrow r = 100\,\!

Example 2: Two-stage (r = 2) case

The state of the system is determined by the total number of customers and the remaining (i.e., not served) number of stages of the customer in service.

State-transition-rate diagram for a 2-stage erlangian markovian process
State-transition-rate diagram for a 2-stage erlangian markovian process

Example 3: Three-stage (r = 3) case

State-transition-rate diagram for a 3-stage erlangian markovian process
State-transition-rate diagram for a 3-stage erlangian markovian process


Birth-death Erlangian model

The state of the system is determined by the total number of stages in the system, and it is denoted by s\,\!. Therefore,

s = r \left ( c - 1 \right ) + \left ( r - v \right)\,\!

\left \lfloor \frac{s}{r} \right \rfloor = c - 1\,\!

s\, \bmod\,r = v - 1\,\!

State-transition-rate diagram for an r-stage erlangian markovian process
State-transition-rate diagram for an r-stage erlangian markovian process

The difference equations are

\lambda p_{-1} = r \mu p_0\,\!

\left ( \lambda + r \mu \right ) p_k = r \mu p_{k+1}\,\! k < r - 1 \,\!

\left ( \lambda + r \mu \right ) p_k = \lambda p_{k-r} + r \mu p_{k+1} \,\! k \geq r - 1 \,\!

\sum_{k = -1}^{\infty} p_k = 1 \,\!


The probability of the system being empty is

\text{Prob} \left [ \text{empty} \right ] = p_{-1} \,\!


The probabilities of having 1, 2, and m customers in the system are

\text{Prob} \left [ c = 1 \right ] = \sum_{k = 0}^{r - 1} p_k \,\!

\text{Prob} \left [ c = 2 \right ] = \sum_{k = r}^{2r - 1} p_k \,\!

\text{Prob} \left [ c = m \right ] = \sum_{k = (m-1)r}^{mr - 1} p_k \,\!

Lecture Voice Recording

lecture_recording

Retrieved from "http://boykin.acis.ufl.edu/wiki/index.php/EEL6507sp09L20"