EEL6507sp09L37

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EEL6507 Spring 2009, Lecture 37, Friday 2009/04/17 (Notes created by Kyungyong Lee)

Topic : G/M/m

Embedded Markov Chain Approach

\begin{alignat}{2}q_n'=\text{number in the system at arrial of }C_n\end{alignat}

\begin{alignat}{2}v_{n+1}'=\text{the number of customers served between }C_n\text{ and }c_{n+1}\end{alignat}

\begin{alignat}{2}q_{n+1}=q_n'+1-v_{n+1}'\end{alignat}

Markov points = Arrival points

Chain Diagram

State transition probability diagram from Kleinrock
State transition probability diagram from Kleinrock
\begin{alignat}{2} p_{i,j}& =\text{Probability when }C_{n+1}\text{ arrives, there are j customers, given that when }C_n\text{ arrived, there were } i\text{ customers} \\ & = Prob[q_{n+1}'=j|q_n'=i] \\ & = Prob[i-v_{n+1}'=j-1|q_n'=i] \\ & = Prob[v_{n+1}'=i-j+1|q_n'=i] \end{alignat}

Regions of pij

Range of validity for pijequation
Range of validity for pijequation

\text{In Region 1 : }i<m,j \le i+1, j \le m\text{(No waiting in this region)}

Prob[v_{n+1}'=i-j+1|q_n'=i] = \int_{0}^{\infty}a(t)Prob[i-j+1\text{ get service }| \text{ interval }t, q_n'=i]dt

\begin{alignat}{2} Prob[\text{any given customer will depart within t sec after the arrival of }C_n]=1-e^{-\mu t} \end{alignat}

\begin{alignat}{2} Prob[\text{a given customer will not depart by time t}]=e^{-\mu t} \end{alignat}

Prob[v_{n+1}'=i-j+1|q_n'=i] = \binom{i+1}{i+1-j}(1-e^{-\mu t})^{i+1-j}e^{-\mu tj}

Lecture Vocie Recording

lecture_recording

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