Tail asymptotics of the waiting time and the busy period for the ${{\varvec{M/G/1/K}}}$ queues with subexponential service times

Abstract

We study the asymptotic behavior of the tail probabilities of the waiting time and the busy period for the $M/G/1/K$ queues with subexponential service times under three different service disciplines: FCFS, LCFS, and ROS. Under the FCFS discipline, the result on the waiting time is proved for the more general $GI/G/1/K$ queue with subexponential service times and lighter interarrival times. Using the well-known Laplace–Stieltjes transform (LST) expressions for the probability distribution of the busy period of the $M/G/1/K$ queue, we decompose the busy period into a sum of a random number of independent random variables. The result is used to obtain the tail asymptotics for the waiting time distributions under the LCFS and ROS disciplines.

Appendix: Useful Literature Results

For convenience, we collected useful literature results for this paper in this appendix.

The following two basic properties of the subexponential distribution can be found in Embrechts et al. [17].

Lemma 6.1

(Lemma 1.3.5 in [17]) If $F \in \mathcal S$, let $X$ be a random variable having $F$ as its distribution. Then, (a)

$$\begin{aligned} \lim _{x\rightarrow \infty } P(X\!>\!x\!+\!y | X\!>\!x ) \!=\! \lim _{x\rightarrow \infty } \frac{\overline{F}(x+y)}{\overline{F}(x)}\!=\!1, \; \text{ uniformly} \text{ for} \text{ any} \text{ compact} \text{ set} \text{ of} y; \end{aligned}$$

and (b)

$$\begin{aligned} \lim _{x\rightarrow \infty } e^{\epsilon x} P(X>x)=\infty =E(e^{\epsilon X}), \quad \text{ for} \text{ all} \epsilon >0. \end{aligned}$$

The following lemma says that the class $\mathcal S$ is closed under tail-equivalence.

Lemma 6.2

(Lemma A 3.15 in [17]) Suppose that $F$ and $G$ are distributions on $(0,\infty )$. If $F \in \mathcal S$ and

$$\begin{aligned} \lim _{x \rightarrow \infty } \frac{\overline{G}(s)}{\overline{F}(x)} =c \in (0, \infty ), \end{aligned}$$

then, $G \in \mathcal S$.

The following result is useful for obtaining the asymptotic tail probability of the sum of independent random variables having subexponential distributions.

Lemma 6.3

(Lemma A 3.28 of in [17]) Let $Y_1, Y_2, \ldots , Y_m$ be independent r.v.s and $F \in \mathcal S$. Assume

$$\begin{aligned} \lim _{x\rightarrow \infty } \frac{P(Y_i>x)}{\overline{F}(x)} = a_i \in [0, \infty ], \quad i=1, 2,\ldots ,m. \end{aligned}$$

Then,

$$\begin{aligned} \lim _{x\rightarrow \infty } \frac{P\left( \sum _{i=1}^mY_i>x \right) }{\overline{F}(x)} = \sum _{i=1}^m a_i. \end{aligned}$$

Corollary 6.1

If $F_1$ and $F_2$ are two distributions such that $F_1\in \mathcal S$ and $\overline{F}_2(x)=o(\overline{F}_1(x))$, then $F_1*F_2\in \mathcal S$ and $\overline{F_1*F_2}(x) \sim \overline{F}_1(x)$ as $x\rightarrow \infty$.

Lemma 6.4

(Geometric Sum, p. 580 in [17]) Let $Y_1, Y_2, \ldots$ be a sequence of i.i.d. r.v.s with a common d.f. $F(x)$. Let $N$ be an integer-valued random variable, which is independent of the sequence $Y_i$ and has a geometric distribution $P(N=k) = (1-p)p^{k}$, $k=0, 1,2,\ldots$. Define $S_0\equiv 0$ and $S_n = Y_1+ Y_2+ \cdots + Y_n$ for $n \ge 1$. Then,

$$\begin{aligned} G(x)\overset{\triangle }{=}P(S_N \le x)=(1-p)\sum _{k=0}^{\infty } p^{k} F^{*k}(x), \end{aligned}$$

where $F^{*n}$ denotes the $n$-fold convolution of $F$ ($F^{*0}$ is defined as the Heaviside unit step function), and the LST of $G(x)$ is given by

$$\begin{aligned} \gamma (s)&= \frac{1-p}{1-p f(s)}, \end{aligned}$$

where $f(s)$ is the LST of $F(x)$.

Lemma 6.5

(p. 296 in Asmussen [4]) Let $Y_1, Y_2, \ldots$ be i.i.d. r.v.s with a common subexponential distribution $F$ and let $N$ be an integer-valued random variable, independent of the sequence $Y_i$, with $Ez^N < \infty$ for some $z>1$. Then,

$$\begin{aligned} P(Y_1+Y_2+\cdots +Y_N>x) \sim EN \overline{F}(x). \end{aligned}$$

Specifically, if $P(N=k) = (1-p)p^{k}$, $k=0, 1,2, \ldots$, then

$$\begin{aligned} P(Y_1+Y_2+\cdots +Y_N>x) \sim \frac{p}{1-p} \overline{F}(x)\quad \text{ as} x\rightarrow \infty . \end{aligned}$$

The following results can be easily verified.

Lemma 6.6

• (1) Let $Y_i$ be a r.v. whose d.f. has the LST $g_i(s)$, $i=1, 2,\ldots ,n$. Define

  $$\begin{aligned} X = Y_i \text{ with} \text{ probability} p_i,\ 1\le i \le n, \end{aligned}$$

  where $\sum _{i=1}^n p_i=1$. Then, the d.f. of $X$ has the LST $g_X(s)=\sum _{i=1}^n p_i f_i(s)$;

• (2) Let $Y_1, Y_2, \ldots , Y_n$ be independent r.v.s whose d.f.s have the LST $g_1(s),g_2(s),\ldots ,g_n(s)$, respectively. Then, $g(s)=\prod _{i=1}^n g_i(s)$ can be viewed as the LST of the d.f. of the r.v. $X\overset{d}{=}\sum _{i=1}^n Y_i$.