For my project, I had to simulate a hospital waiting queue, and ended up stuck with this equation.
$$ 1=\sum_{i=0}^\infty \left(\frac{\lambda}{\mu+i\gamma}\right)^iP_0 $$
Could any kind soul help me with the RHS of the equation?
Also, I have found out that this expression $\sum_{i=0}^\infty \left(\frac{\lambda}{\mu}\right)^i$ can be simplified to get $\frac{1}{1-\frac{\lambda}{\mu}}$ but I don't understand why. Please enlighten me on this. Thank you!
Edit:
Thanks Almagest and Did for helping and clarifying. I realised that the equation is wrong; what I was trying to come up with was this:
$$ P_1 = \frac{\lambda P_0}{\mu} $$ $$ P_2 = \frac{\lambda P_1}{\mu + \gamma} = \frac{\lambda^2 P_0}{\mu(\mu + \gamma)} $$ $$ P_3 = \frac{\lambda P_2}{\mu + 2\gamma} = \frac{\lambda^3 P_0}{\mu(\mu + \gamma)(\mu + 2\gamma)} $$ $$ P_4 = \frac{\lambda P_3}{\mu + 3\gamma} = \frac{\lambda^4 P_0}{\mu(\mu + \gamma)(\mu + 2\gamma)(\mu + 3\gamma)} $$ and so on..
Where:
- $P_n$ is the probability that there are n patients in the system.
- $\lambda$ is the arrival rate.
- $\mu$ is the service rate.
- $\gamma$ is the death rate of the patient.
Therefore, I want to sum up all the terms from 0 to infinity to be 1, and express $\gamma$ in terms of $\lambda, \mu,$ and $P_0$.