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EEL6507 Spring 2009, Lecture 22, Monday 2009/03/02 (Notes created by Gustavo Vejarano)

M/Er/1 Review

The M/Er/1 system turns into an M/D/1 system when $r \,\!$ (i.e., the number of stages per customer) goes to infinity.

$\begin{align} \lim_{r \to \infty} B^*(s) & = \lim_{r \to \infty} \left ( \frac{r\mu}{s + r\mu} \right )^r \\ & = \lim_{r \to \infty} \left ( 1 + \frac{s}{r\mu} \right )^{-r} \\ & = \lim_{r \to \infty} \left ( \left ( 1 + \frac{s}{r\mu} \right )^{-r \frac{\mu}{s}} \right )^{\frac{s}{\mu}} \\ & = \left ( \lim_{x \to \infty} \left ( 1 + \frac{1}{x} \right )^{-x} \right )^{\frac{s}{\mu}} \\ & = \left ( e^{-1} \right)^{\frac{s}{\mu}} \\ & = e^{- \frac{s}{\mu}} \end{align}\,\!$

$\lim_{r \to \infty} b_r(x) = \delta \left ( x - \frac{1}{\mu} \right ) \,\!$

Therefore,

$\underset{(r = 1)}{M/M/1} \to \underset{(1 < r < \infty)}{M/E_r/1} \to \underset{(r \to \infty)}{M/D/1}\,\!$

Bulk Arrival

Simplest Bulk Arrival

Each arrival brings $r \,\!$ customers. This is the same markov chain as M/Er/1.

State-transition-rate diagram for the simplest-bulk-arrival markovian process

Variable Bulk Arrival

Each arrival brings an i.i.d. random number of customers.

$\text{Prob} \left [ k \text{ customers in an arrival event} \right ] = g_k \,\!$

State-transition-rate diagram for a variable-bulk-arrival markovian process

The difference equations are

$\lambda p_0 = \mu p_1 \,\!$

$\left ( \lambda + \mu \right ) p_k = \mu p_{k+1} + \lambda g_1 p_{k-1} + \lambda g_2 p_{k-2} + \ldots \,\!$

Solving for $P(z) \,\!$

$\left ( \lambda + \mu \right ) \sum_{k = 1}^{\infty} z^k p_k = \mu \sum_{k = 1}^{\infty} z^k p_{k + 1} + \lambda \sum_{k = 1}^{\infty} \sum_{i = 1}^k g_i p_{k - i} z^k \,\!$

$\left ( \lambda + \mu \right ) (P(z) - p_0) = \mu z^{-1} (P(z) - p_0 - z p_1) + \lambda \sum_{i = 1}^{\infty} \sum_{k = i}^{\infty} g_i p_{k - i} z^{k - i} z^i \,\!$

$\left ( \lambda + \mu \right ) (P(z) - p_0) = \mu z^{-1} (P(z) - p_0 - z \frac{\lambda}{\mu} p_0) + \lambda \sum_{i = 1}^{\infty} g_i z^i \sum_{k = i}^{\infty} p_{k - i} z^{k - i} \,\!$

$\left ( \lambda + \mu \right ) (P(z) - p_0) = \mu z^{-1} (P(z) - p_0 - z \frac{\lambda}{\mu} p_0) + \lambda \sum_{i = 0}^{\infty} g_i z^i \sum_{k = 0}^{\infty} p_k z^k \,\!$

$\left ( \lambda + \mu \right ) (P(z) - p_0) = \mu z^{-1} \left (P(z) - p_0 \left ( 1 + z \frac{\lambda}{\mu} \right ) \right ) + \lambda G(z) P(z) \,\!$

$\left ( 1 + \frac{\lambda}{\mu} \right ) P(z) - z^{-1} P(z) - \frac{\lambda}{\mu} G(z) P(z) = \left (1 + \frac{\lambda}{\mu} \right ) p_0 - \left (z^{-1} + \frac{\lambda}{\mu} \right ) p_0 \,\!$

$P(z) \left (z + \frac{\lambda}{\mu} z - 1 - \frac{\lambda}{\mu} z G(z) \right ) = z p_0 + z \frac{\lambda}{\mu} p_0 - p_0 - z \frac{\lambda}{\mu} p_0 \,\!$