Queueing theory-steady state probability.

Question

Consider a queueing system comprising a single queue. Let $n$ be the system state, characterizing the number of customers in the queue, $n=0, 1, \ldots$. Let $P_n$ be the steady state probability of state $n$. While trying to find the steady-state probability of a system in Queueing theory, I arrived a solution ${P_{n}}$=constant. What properties of the system can I infer from this solution?

Answer 1

One system this could happen is an M/M/1/K queue when $\rho = 1$. In this case $P_n = \frac{1}{K+1}$.

The conclusion is that every state is equiprobable.

Observe that the mean queue length is

$$E[N] = \frac{K}{2}$$

Answer 2

If you have a finite queue, you can conclude that $E[N]=K/2$. If you have an infinite system, you can conclude that your system is such that $P_n=0$ for all $n$, i.e., it is either transient or null recurrent http://www.columbia.edu/~ks20/stochastic-I/stochastic-I-MCII.pdf

• Transient, e.g., M/M/1 with $\lambda > \mu$
• Null recurrent, e.g., M/M/1/K with $\lambda/\mu=1$ and $K \rightarrow \infty$: this means that the queueing delays grow unboundedly even though all states are reachable and you return to every state with probability 1 -- you return to every state with probability 1, but the time to return grows to infinite. In this case, your system is equivalent to a random walk with the same probability of moving to the left and to the right -- this kind of random walk is known to be null recurrent.