What is the transient behavior of an M/M/1 queue

Question

Suppose I initialise an M/M/1 queue with a queue length of $q_0$. How do I describe $\mathrm{E}[q(t)]$ as a function of time? I know that as $t\to\infty$ , the expression goes to $\frac{\lambda ^2}{\mu (\mu - \lambda)}$ but I don't know how it reaches that steady state value.

Answer 1

By queue length I suppose you mean the number of customers waiting in the queue, so this does not include the customer in service (otherwise the $\frac{\lambda^2}{\mu(\mu - \lambda)}$ is incorrect). Note that $q_0 = 0$ is ambiguous, is there a customer in service or not?

Let $L(t)$ be the number of customers in the system at time $t$. I only know of an expression for the mean number of customers in the system at time $t$ given that we start with $i$ customers at time $0$, indicated by $\mathbb{E}[L(t) \mid L(0) = i]$. This expression appears in Takács (1962) - Introduction to the Theory of Queues on page 27. It does not look nice at all. So I think there is little hope of finding a nice expression for the mean number of customers in the queue at time $t$ given we start with $q_0$ customers in the queue at time $0$.

If you are only interested in the behavior for $t$ large, think of $t \to \infty$, then Section 2.1 of Cohen (1969) - The Single Server Queue, might be of much interest to you. Again, this describes $\mathbb{E}[L(t) \mid L(0) = i]$ as opposed to what you mention in your question.

Answer 2

Let $\{X_n\}$ be the embedded jump chain of $q(t)$ and $R_1=\inf\{t>0:X_{q(t)}=0\}$. Then the process $\{q(t+R_1):t\geqslant 0\}$ is independent of $\{q(t):t\leqslant R_1\}$ and $R_1$. Letting $\{R_n:n\geqslant1\}$ be a $\mathsf{iid}$ sequence and $S_n=\sum_{k=1}^n R_k$, be the corresponding (delayed) renewal process, it follows that the counting process $q(t)$ "regenerates" itself at the epochs $\{R_n\}$. That is, for each $n\geqslant 1$, $\{q(t+R_n):t\leqslant1\}$ is independent of $\{q(t):0\leqslant t\leqslant R_n\}$ and $R_n$, and further, $\{q(t+R_n):t\leqslant1\}$ is stochastically equivalent to $\{q(t+R_1):t\leqslant1\}$. This simplifies the analysis of the transient behavior of the system, as we need only study its behavior on the renewal cycles $(S_n,S_{n+1}]$.