On two-queue Markovian polling systems with exhaustive service

Abstract

We consider a class of two-queue polling systems with exhaustive service, where the order in which the server visits the queues is governed by a discrete-time Markov chain. For this model, we derive an expression for the probability generating function of the joint queue length distribution at polling epochs. Based on these results, we obtain explicit expressions for the Laplace–Stieltjes transforms of the waiting-time distributions and the probability generating function of the joint queue length distribution at an arbitrary point in time. We also study the heavy-traffic behaviour of properly scaled versions of these distributions, which results in compact and closed-form expressions for the distribution functions themselves. The heavy-traffic behaviour turns out to be similar to that of cyclic polling models, provides insights into the main effects of the model parameters when the system is heavily loaded, and can be used to derive closed-form approximations for the waiting-time distribution or the queue length distribution.

Appendices

Appendix 1: Proof of Lemma 3.1

Proof

We first focus on the value of $\left| 1-f_1^{(\infty )}(z_2)\right| = \lim _{j \rightarrow \infty } \left| 1-f_1^{(j)}(z_2)\right|$. For arbitrary $j>0$, we have for any $z_2$ in the unit circle that

$$\begin{aligned} \left| 1-f_1^{(j)}(z_2)\right|&= \left| 1-f_1(f_1^{j-1}(z_2))\right| \\&= \left| \int \limits _{t=0}^\infty (1-e^{-\lambda _1(1-\widetilde{K}_{1,2}(f_1^{(j-1)}(z_2)))t}) \hbox {d}\mathbb {P}(P_2 < t)\right| \\&\le \int \limits _{t=0}^\infty \left| 1-e^{-\lambda _1(1-\widetilde{K}_{1,2}(f_1^{(j-1)}(z_2)))t}\right| \hbox {d}\mathbb {P}(P_2 < t), \end{aligned}$$

where the inequality constitutes the triangle inequality. Note that $\left| 1-e^{-x}\right| \le \left| x\right|$ for any $x \in \{z\in \mathbb {C}: \mathfrak {R}(z) > 0\}$, so that

$$\begin{aligned} \left| 1-f_1^{(j)}(z_2)\right|&\le \int \limits _{t=0}^\infty \lambda _1 t \left| 1-\widetilde{K}_{1,2}(f_1^{(j-1)}(z_2))\right| \hbox {d}\mathbb {P}(P_2 < t) \nonumber \\&= \lambda _1\mathbb {E}\left[ P_2\right] \left| 1-\widetilde{K}_{1,2}(f_1^{(j-1)}(z_2))\right| \nonumber \\&\le \lambda _1\mathbb {E}\left[ P_2\right] \left| \int \limits _{t=0}^\infty (1-e^{-\lambda _2(1-f_1^{(j-1)}(z_2))t}) \hbox {d}\mathbb {P}(P_1<t)\right| \nonumber \\&\le \lambda _1\mathbb {E}\left[ P_2\right] \lambda _2\mathbb {E}\left[ P_1\right] \left| 1-f_1^{(j-1)}(z_2)\right| . \end{aligned}$$

(32)

Iteration of (32) leads to

$$\begin{aligned} \left| 1-f_1^{(j)}(z_2)\right| \le \left( \lambda _1\mathbb {E}\left[ P_2\right] \lambda _2\mathbb {E}\left[ P_1\right] \right) ^j\left| 1-z_2\right| . \end{aligned}$$

(33)

By (1) we have that $\mathbb {E}[P_i] = \mathbb {E}[B_i](1-\rho _i)^{-1}$, so that

$$\begin{aligned} \lambda _1\mathbb {E}\left[ P_2\right] \lambda _2\mathbb {E}\left[ P_1\right] = \frac{\rho _1}{1-\rho _2}\frac{\rho _2}{1-\rho _1} < 1. \end{aligned}$$

(34)

The inequality follows since the queues are assumed to be stable, i.e., $0 \le \rho < 1$. Therefore, $\rho _1 = \rho -\rho _2 < 1-\rho _2$, and similarly $\rho _2 < 1-\rho _1$. A combination of (32) and (34) now leads to

$$\begin{aligned} 0 \le \lim _{j \rightarrow \infty } \left| 1-f_1^{(j)}(z_2)\right| \le \lim _{j \rightarrow \infty } \left( \lambda _1\mathbb {E}\left[ P_2\right] \lambda _2\mathbb {E}\left[ P_1\right] \right) ^j\left| 1-z_2\right| = 0. \end{aligned}$$

Since $\lim _{j \rightarrow \infty } \left| 1-f_1^{(j)}(z_2)\right| = 0$, we must have that $f_1^{(\infty )}(z_2) = \lim _{j \rightarrow \infty } f_1^{(j)}(z_2) = 1$.

By similar arguments, it can be shown that $f_2^{(\infty )}(z_1)=1$ for any $z_1$ in the unit circle. Finally, it is evident that $\widetilde{K}_{1,2}(1) = \widetilde{K}_{2,1}(1) = \widetilde{F}_1(1,1) = \widetilde{F}_2(1,1) = 1$. The lemma now follows. $\square$

Appendix 2: Proof of Lemma 3.2

Proof

We initially focus on the product $\prod _{j=0}^\infty a_1(f_1^{(j)}(z_2))$. By the theory of infinite products (see e.g., [26, Chapter 1]), we have that $\prod _{j=0}^\infty a_1(f_1^{(j)}(z_2))$ converges iff $\sum _{j=0}^\infty (1-a_1(f_1^{(j)}(z_2)))$ converges. To establish the latter, it is enough to prove that $\sum _{j=0}^\infty \left| 1-a_1(f_1^{(j)}(z_2))\right|$ converges. We observe that

$$\begin{aligned}&\left| 1-a_1(f_1^{(j)}(z_2))\right| \nonumber \\&\quad = \left| 1-\frac{r_{2,1}\widetilde{M}_{2,1}(\widetilde{K}_{1,2}(f_1^{(j)}(z_2)), f_1^{(j)}(z_2))}{1-r_{1,1}\widetilde{M}_{1,1}(\widetilde{K}_{1,2}(f_1^{(j)}(z_2)), f_1^{(j)}(z_2))} \frac{r_{1,2}\widetilde{M}_{1,2}(\widetilde{K}_{1,2}(f_1^{(j)}(z_2)), f_1^{(j)}(z_2))}{1-r_{2,2}\widetilde{M}_{2,2}(\widetilde{K}_{1,2}(f_1^{(j)}(z_2)), f_1^{(j)}(z_2))}\right| \nonumber \\&\quad = \left| \frac{\sum _{i=1}^2 A_{1,i}(f_1^{(j)}(z_2))(1-\widetilde{M}_{i,1}(\widetilde{K}_{1,2}(f_1^{(j)}(z_2)), f_1^{(j)}(z_2)))}{D(z_2)}\right. \nonumber \\&\qquad + \left. \frac{\sum _{i=1}^2 A_{2,i}(f_1^{(j)}(z_2))(1-\widetilde{M}_{i,2}(\widetilde{K}_{1,2}(f_1^{(j)}(z_2)), f_1^{(j)}(z_2)))}{D(z_2)}\right| , \end{aligned}$$

(35)

where

$$\begin{aligned} A_{1,1}(z_2)&= r_{1,1}(1-r_{2,2}), \\ A_{1,2}(z_2)&= (1-r_{1,1})(1-r_{2,2}), \\ A_{2,1}(z_2)&= (1-r_{1,1})(1-r_{2,2})\widetilde{M}_{1,2}(\widetilde{K}_{1,2}(z_2), z_2),\\ A_{2,2}(z_2)&= r_{2,2}(1-r_{1,1}\widetilde{M}_{1,1}(\widetilde{K}_{1,2}(z_2), z_2)) \hbox { and } \\ D(z_2)&= (1-r_{1,1}\widetilde{M}_{1,1}(\widetilde{K}_{1,2}(f_1^{(j)}(z_2)), f_1^{(j)}(z_2)))\\&\times (1-r_{2,2}\widetilde{M}_{2,2}(\widetilde{K}_{1,2}(f_1^{(j)}(z_2)), f_1^{(j)}(z_2))). \end{aligned}$$

Using the triangle inequality and similar arguments as those in the proof of Lemma 3.1, we note that for $1 \le i,k \le 2$ and $j>0$,

$$\begin{aligned}&\left| 1-\right. \left. \widetilde{M}_{i,k}(\widetilde{K}_{1,2}(f_1^{(j)}(z_2)), f_1^{(j)}(z_2))\right| \\&\quad \le \int \limits _{t=0}^\infty \left| 1-e^{-(\lambda _1(1-\widetilde{K}_{1,2}(f_1^{(j)}(z_2)))+\lambda _2(1-f_1^{(j)}(z_2)))t}\right| \hbox {d}\mathbb {P}(S_{i,k} <t) \\&\quad \le \mathbb {E}\left[ S_{i,k}\right] \left( \lambda _1\left| 1-\widetilde{K}_{1,2}(f_1^{(j)}(z_2))\right| + \lambda _2\left| 1-f_1^{(j)}(z_2)\right| \right) \\&\quad \le \mathbb {E}\left[ S_{i,k}\right] \lambda _2(\lambda _1\mathbb {E}\left[ P_1\right] +1)\left| 1-f_1^{(j)}(z_2)\right| . \end{aligned}$$

Moreover, it is trivially seen that $\left| A_{i,k}(z_2)\right| \le 1$ for $1 \le i,k \le 2$ and any $z_2$ in the unit circle. Furthermore, since $\left| \widetilde{M}_{i,k}(\widetilde{K}_{1,2}(z_2), z_2)\right| \le 1$, we have that $|D(z_2)| \ge (1-r_{1,1})(1-r_{2,2})$. Therefore, a combination of (33) and (35) with the triangle inequality leads to

$$\begin{aligned} \left| 1-a_1(f_1^{(j)}(z_2))\right|&\le \frac{\mathbb {E}\left[ S_{1,1}\right] +\mathbb {E}\left[ S_{1,2}\right] +\mathbb {E}\left[ S_{2,1}\right] +\mathbb {E}\left[ S_{2,2}\right] }{(1-r_{1,1})(1-r_{2,2})}\\&\times \,\lambda _2(\lambda _1\mathbb {E}\left[ P_1\right] +1)\left( \lambda _1\mathbb {E}\left[ P_2\right] \lambda _2\mathbb {E}\left[ P_1\right] \right) ^j\left| 1-z_2\right| \end{aligned}$$

This result obviously shows, in combination with (34), that $\sum _{j=0}^\infty \left| 1-a_1(f_1^{(j)}(z_2))\right|$ is bounded from above by a converging geometric sum. As such, $\sum _{j=0}^\infty \left| 1-a_1(f_1^{(j)}(z_2))\right|$ converges, so that $\prod _{j=0}^\infty a_1(f_1^{(j)}(z_2))$ converges. The convergence of the product $\prod _{j=0}^\infty a_2(f_2^{(j)}(z_1))$ can be established similarly. $\square$