We have a FCFS queueing system with two job types $a$ and $b$. We model the system as a discrete time markov chain where the state is a vector $(k,j_1, \ldots, j_k)$ where $k$ is the number of jobs and $j_1$ is the job type of the latest arival and $j_k$ is the type of the job that will be served next. If the current state is $(k,j_1, \ldots, j_k)$ the next state will be $(k+1,a,j_1, \ldots, j_{k})$ with probability $\lambda_a$ or $(k+1,b,j_1, \ldots, j_{k})$ with probability $\lambda_b$ or $(k-1,j_1, \ldots, j_{k-1})$ with probability $\mu$; otherwise the number of jobs does not change. What is the stationary distribution of the system?
From the analysis of the system where there are only one type of job I guess the total number of jobs in the stationary distribution will be $k$ with probability $\left(\frac{\lambda_a+\lambda_b}{\mu}\right)^k\left(1-\frac{\lambda_a+\lambda_b}{\mu}\right)$. And my naive guess will be the type of jobs will be follow a binomial distribution where out of the $k$ jobs the number of jobs of type $a$ will be binomial with parameter $\frac{\lambda_a}{\lambda_a+\lambda_b}$. Is this correct? And how do I go about proving this?
You’re right. There’s not all that much to prove. You can imagine the jobs originally all being of one type and then randomly assigning one of the two types to them at the time you sample their distribution; clearly the distribution can’t be affected by when you assign the types.