Question on M/M/s queue

Question

customers arrive to a service station according to a poisson prossees and on average 2 during an hour. The service times and independent of the arrivals and internally independent with mean 45 minutes. At must 3 can be in system (and while one is being served the other wait)put xt=number of customer in the system at time a) Determine the birth and death frequencies in the birth and death chain (Xt)t≥0. b) Determine lim→∞Pi(Xt=0) for i=0,1,2,3. can you explain for me solve problem like this with markov chain

Answer 1

Your model is an M/M/c queue with arrival rate $\lambda=2$ jobs per hour and service rate of each server $\mu=\frac{1}{0.75}=\frac{4}{3}$ jobs per hour. You have $c=3$ servers. Transition rates are given by the $Q$ matrix in the linked article.

"Births" in this model are customer arrivals, so the birth rate is a constant $\lambda$ in all states. "Deaths" are when jobs are serviced. When $X_t \geq 3$ all the servers are busy so the death rate is $3\mu$, when $X_t=2$ the death rate is $2\mu$ and when $X_t=1$ the death rate is $\mu$. There are no deaths when $X_t=0$ because there are no individuals to die.

The probabilities you seek to compute are all the same because for large values of $t$ the starting state doesn't matter, $$\lim_{t\rightarrow \infty}\mathbb P_i(X_t=0) = \pi_0$$ for $i=0,1,2,3$. Perhaps the probabilities you are looking for are $$\lim_{t\rightarrow \infty}\mathbb P(X_t = i) = \pi_i.$$ Formulas for the $\pi_i$ are given in the linked article.