I am currently studying about (non)preemptive priority queue, and below are some problems that I am getting stuck on. Hope somebody could give me some help on any of these problems.
Customers arrive to a single customer service agent according to a Poisson process with arrival rate $\lambda = 9$ per hour. $80\%$ of customers complete a standard transaction that takes a fixed amount of time, $\frac{1}{12}$ hour. $20\%$ of customers require special handling and take an exponential amount of time with mean $\frac{1}{6}$ hours. Determine the average wait in queue for standard-transaction customers when customers are first-come, first served.
A computer processer handles incoming jobs according to a non-preemptive priority queue discipline. Jobs arrive according to a Poisson process with rate 8 per minute. $80\%$ of jobs take $1/60$ min to process, $10\%$ take $1/12$ min to process, and $10\%$ take 1 minute to process. Determine the average time in queue for each class if the rule applied is Longest job first? How about the average time in queue if first-come, first-served is applied?
My thought: For $(1)$, what caused me a lot of difficulty is the fact that the two service distributions are not both exponential. I know that if both services are exponential, then we only need to compute $L_q = L_{q_1}+L_{q_2} = \lambda (\frac{\lambda_1}{u_1} + \frac{\lambda_2}{u_2})/(1- \frac{\lambda_1}{u_1} + \frac{\lambda_2}{u_2})$. Then dividing this by $\lambda_1$ (is this correct?) to get the result?
For $(2)$, I am completely stuck since I simply don't know the formula for the rule Longest job first. For shortest job first, the formula is: Average waiting in queue of class-x customer = $W_q^{x} = \frac{\lambda E[S^2]}{2(1-\sigma_x)^2}$ where $\sigma_x = \sum_{i=1}^{x} \rho_i$ = total load rate of all jobs less than $x$.