Topics:
Little’s Law:
\[ E(L) = \lambda E(T) \]
Utilization:
\[ \rho = \frac{\lambda}{\mu} \]
Number of total customers \(L\). This is only dependent on \(\rho\).
\[ P(L = n) = p_{n} = ( 1 - \rho ) \rho^{n} \]
Expected value of \(T\)
\[ E(T) = \frac{1}{\mu ( 1 - \frac{\lambda}{\mu} )} = \frac{1}{\mu ( 1 - \rho )} = \frac{1}{ \mu - \lambda } \]
Expected value of \(L\)
\[ E(L) = \frac{ \lambda }{\mu ( 1 - \rho )} = \frac{\rho}{( 1 - \rho )} \]
\[ E(T_{q}) \mu^{2} - E(L_{q}) \mu - \lambda = 0 \]
Relationship between expected value of time to service customer to service rate
\[ E(T_{s}) = \frac{1}{\mu} \]
Relationship between expected value of time to service customer to service rate
\[ E(T_{q}) = E(T) - E(T_{s}) = \frac{ \rho }{ \mu (1 - \rho )} \]