Exact tail asymptotics for fluid models driven by an M/M/c queue

Abstract

In this paper, we investigate exact tail asymptotics for the stationary distribution of a fluid model driven by the M / M / c queue, which is a two-dimensional queueing system with a discrete phase and a continuous level. We extend the kernel method to study tail asymptotics of its stationary distribution, and a total of three types of exact tail asymptotics are identified from our study and reported in the paper.

Appendix: Tauberian-like theorem

Denote

$$\begin{aligned} \varDelta _{1}(\phi , \varepsilon )=\left\{ x: |x|\le |x_{0}|+\varepsilon , |\arg (x-x_{0})|> \phi , \varepsilon >0, 0<\phi <\frac{\pi }{2}\right\} . \end{aligned}$$

Let $f_{n}$ be a sequence of numbers with the generating function

$$\begin{aligned} f(x)=\sum _{n\ge 1} f_{n} x^{n}. \end{aligned}$$

Lemma 16

(Flajolet and Odlyzko 1990) Assume that f(x) is analytic in $\varDelta _{1}(\phi , \varepsilon )$ except at $x=x_{0}$ and

$$\begin{aligned} f(x)\sim K(x_{0}-x)^{s} \ \ \text{ as }\ \ x\rightarrow x_{0} \ \ \text{ in } \ \ \varDelta _{1}(\phi , \varepsilon ). \end{aligned}$$

Then, as $n\rightarrow \infty$, (i) If $s \not \in \{0, 1, 2, \ldots \}$,

$$\begin{aligned} f_{n}=\frac{K }{\varGamma (-s)}n^{-s-1}x_{0}^{-n}, \end{aligned}$$

where $\varGamma (\cdot )$ is the Gamma function.

(ii) If s is a non-negative integer, then

$$\begin{aligned} f_{n}= o(n^{-s-1}x_{0}^{-n}). \end{aligned}$$

For the continuous case, let

$$\begin{aligned} g(x)=\int _{0}^{\infty }e^{xt}f(t)\mathrm{d}t. \end{aligned}$$

Denote

$$\begin{aligned} \varDelta _{2}(\phi , \varepsilon )=\{x: \mathfrak {R}(x)\le |x_{0}|+\varepsilon , x\ne x_{0}, \varepsilon>0, |\arg (x-x_{0})|> \phi \}. \end{aligned}$$

The following lemma is shown in Theorem 2 in [4].

Lemma 17

Assume that g(x) satisfies the following conditions:

• (i) The leftmost singularity of g(x) is $x_{0}$, with $x_{0} > 0$. Furthermore, we assume that as $x\rightarrow x_{0}$,

  $$\begin{aligned} g(x)\sim (x_{0}-x)^{-s} \end{aligned}$$

  for some $s\in \mathbb {C}\backslash Z_{-}$.

• (ii) g(x) is analytic on $\varDelta _{2}(\phi _{0}, \varepsilon )$ for some $\phi _{0}\in (0, \frac{\pi }{2}]$.

• (iii) g(x) is bounded on $\varDelta _{2}(\phi _{1}, \varepsilon )$ for some $\phi _{1}> 0$.

Then, as $t\rightarrow \infty$,

$$\begin{aligned} f(t)\sim e^{-x_{0} t} \frac{t^{s-1}}{\varGamma (s)}, \end{aligned}$$

where $\varGamma (\cdot )$ is the Gamma function.