I'm looking at a problem where we have calls queueing under two different classes, new calls and handovers. The number of calls arriving follow a Poisson process with $\lambda_{1} = 125$ per hour corresponding to new calls, and $\lambda_2 = 50$ per hour corresponding to handovers, and each call time is Exponential with $\mu = 30$ minutes.
Each call occupies one channel, and there are 10 channels overall. One channel is reserved purely for handover calls, and the remaining nine can take both handover and new calls. Now, I want to calculate the blocking probability for the remaining new calls, so is the following logic correct?
Proposed Answer:
New calls can only be serviced via the remaining nine channels. Let $C_1 = 0, 1, ..., 9$ denote the number of channels out of the remaining nine that are occupied, and $C_2 = 0, 1$ denote whether the handover channel is occupied.
The blocking probability is the probability that all channels are busy in $C_1$ across all possibilities of $C_2$, and by the law of total probability this should just be:
\begin{aligned} P(C_1 = 9) = \frac{(1-\rho_1)}{1-{\rho_1}^{10}}{\rho_1}^9; \space \rho_1 = \lambda/\mu \end{aligned}
Noting $\rho_1 = (\lambda_1 + \lambda_2)/\mu = 175/30 = 35/6$, we can substitute, yielding:
\begin{aligned} P(C_1 = 9) = 0.828 \end{aligned}