M/G/1 queue with increased arrival rate when empty

Question

It is well known how to analyze the classical M/G/1 queue. I am interested in an M/G/1 queue where the arrival rate is equal to $\lambda$ when the queue is nonempty, but it is equal to some $\tilde \lambda > \lambda$ when the queue is empty. One option is to go through the analysis of the classical M/G/1 queue and see how to adapt it, but I would expect these results are also well known. However I do not find any reference for this "adapted version" of the M/G/1 queue.

Answer 1

The problem you are interested in does not require separate study and can be solved using the known results for the M/G/1 queue. I assume that you ask about the steady state probabilities of the system size.

Here is how it is done. Let us refer to the system M/G/1 queue with arrival rate $\lambda$ and service time distribution $B(x)$ by as S1 and to the system M/G/1 queue with arrival rate ${\tilde \lambda}$ when empty and $\lambda$ otherwise, and service time distribution $B(x)$ as S2.

M/G/1 queues can be analysed in many ways. Let us use supplementary variable technique. As a supplementary variable take elapsed service time.

Let us use upper index to distinguish between the quantities related to S1 and S2. Denote by $p^1_0$ the steady-state probability of S1 to be empty and by $p^1_i(x)$, $i \ge 1$, the the steady-state probability density of $i$ customers in the system and elapsed service time equal to $x$. By $p^2_0$ and $p^2_i(x)$ let us denote the same quantities but to the S2.

Consider S1. The expressions for the computation of $p^1_0$ and $p^1_i(x)$ are well-known. For example, you can use the expressions (5.4.12) and (5.4.13) in P.P. Bocharov, C. D'Apice, A.V. Pechinkin and S. Salerno: Queueing Theory, Modern Probability and Statistics, VSP, The Netherlands, 2004. It is important to see how (5.4.12) looks like: $$ \lambda p^1_0 = p^1_1(0) \int_0^\infty e^{-\lambda u }dB(u). $$

Once $p^1_0$ and $p^1_i(x)$ are computed, notice the following. As long as there are customers in S1 and S2 (i.e. as long as they are not empty), their operation is identical from the probabilistic point of view. Mathematically it means that their state probabilities are the same up to a positive constant, which is not known yet. Denote this constant by $C$. Then what was just said means that $p^2_i(x)=Cp^1_i(x)$ for all $i\ge 1$ and $x\ge 0$.

Now consider S2. For S2 you can write out the balance equation for the empty state and it will look like the equation above i.e. $$ {\tilde \lambda} p^2_0 = p^2_1(0) \int_0^\infty e^{-\lambda u }dB(u). $$

But as it was mentioned above, $p^2_i(x)=Cp^1_i(x)$ for all $x\ge0$ and thus $p^2_i(0)=Cp^1_i(0)$. This gives $$ p^2_0 = C {p^1_1(0) \over {\tilde \lambda} } \int_0^\infty e^{-\lambda u }dB(u). $$

Now you can use the normalization condition to compute the unknown constant $C$. You have for the system S2: $$ p^2_0 + \sum_{i=1}^\infty \int_0^\infty p^2_i(u)du=1, $$ wherefrom $$ C = \left ( {p^1_1(0) \over {\tilde \lambda} } \int_0^\infty e^{-\lambda u }dB(u) + \sum_{i=1}^\infty \int_0^\infty p^1_i(u)du \right )^{-1}. $$

Now the steady state distribution of the system size in S2 is completely determined.