Transition Probability of M/M/1 Queue given any constant observing period T

Question

I am trying to find the transition probability for $M$/$M$/$1$ queue given any constant observing period $T$ (if $T$ go to infinity the transition matrix will degenerate to a matrix with identical rows, each of them is the stable state distribution).

For any continuous Markov process, in general, the transition probability can be solved with Kolmogorov equations. But in $M$/$M$/$1$ queue, Kolmogorov equations will lead to a infinite dimension matrix, which I don't know how to solve.

I found some results specify to $M/M/1$ from books (such as Geoffery Grimmett's book "Probability and Random Processes", chapter 11), but only the probability that starts from state $0$ and transits to any state $j$ after constant time $T$ is given as follows. (There is another form with infinite summation can been found in the book.)

$p_{0j}(T) = K_{n}(T)-K_{n+1}(T)$,

where $K_{n}(T)=\int_{0}^{T}{(\lambda/\mu)^{n/2} n s^{-1} e^{-s(\lambda+\mu)} I_n(2s\sqrt{\lambda \mu} )ds }$, and $I_{n}(x)$ is modified Bessel function.

However what I want is the more general case, i.e. starts from state $i$ and transits to any state $j$. What would that be? Hope anyone can help me.

Answer 1

Let $X(t)$ be the number of customers in the $M/M/1$ system at time $t$. The transition functions that you are interested in are

\begin{equation} p_{i,j}(t) := \mathbb{P}(X(t) = j \mid X(0) = i), \quad i,j = 0,1,2,\ldots, ~ t \ge 0. \end{equation}

Many classic books on queueing theory cover the topic of transition functions in the $M/M/1$ queue. For example, you can find expressions for $p_{i,j}(t)$ here:

1. Takács, L, 1962. Introduction to the Theory of Queues. (Chapter 2, Theorems 1 and 2)
2. Cohen, J.W, 1969. The single server queue. (Chapter II.2.1)

Expressions for $p_{i,j}(t)$ are typically quite difficult though.