Finite-buffer polling systems with threshold-based switching policy

Abstract

We consider a system of two separate finite-buffer M / M / 1 queues served by a single server, where the switching mechanism between the queues is threshold-based, determined by the queue which is not being served. Applications may be found in data centers, smart traffic-light control and human behavior. Specifically, whenever the server attends queue i ($Q_i$) and the number of customers in the other queue, $Q_j$ ($i,j=1,2$; $j\ne i$), reaches its threshold level, the server immediately switches to $Q_j$ whenever $Q_i$ is below its threshold. When a served $Q_i$ becomes empty we consider two scenarios: (i) non-work-conserving; and (ii) work-conserving. We present occasions where the non-work-conserving policy is more economical than the work-conserving policy when high switching costs are involved. An intrinsic feature of the process is an oscillation phenomenon: when the occupancy of $Q_i$ decreases the occupancy of the other queue increases. This fact is illustrated and discussed. By formulating the system as a three-dimensional continuous-time Markov chain we provide a probabilistic analysis of the system and investigate the effects of buffer sizes and arrival rates, as well as service rates, on the system’s performance. Numerical examples are presented and extreme cases are investigated.

Appendix

1.1 Proof of Theorem 2.3

Proof

By induction over k.

For $k=1$,

$$\begin{aligned} q_1(z)=\alpha _0(z)=(-\lambda _1\mu _1)^0\left( \left( {\begin{array}{c}0\\ 0\end{array}}\right) \alpha ^{0}(z)\alpha _0(z)+\left( {\begin{array}{c}0\\ -1\end{array}}\right) \alpha ^{1}(z)\right) . \end{aligned}$$

For $k=2$,

$$\begin{aligned} q_2(z)= & {} \alpha (z)\alpha _0(z)-\lambda _1\mu _1\\= & {} \sum _{l=0}^{1}{\left( -\lambda _1\mu _1\right) ^l \left( \left( {\begin{array}{c}1-l\\ l\end{array}}\right) \alpha ^{1-2l}(z)\alpha _0(z)+\left( {\begin{array}{c}1-l\\ l-1\end{array}}\right) \alpha ^{2-2l}(z)\right) }. \end{aligned}$$

We now show that the proposition is valid for any k. Suppose $k=2i$ (the case where $k=2i+1$ is similar and hence omitted from the presentation), and notice that for all $k\ge 0$, $\left( {\begin{array}{c}k\\ 0\end{array}}\right) =\left( {\begin{array}{c}k\\ k\end{array}}\right) =1$, and $\left( {\begin{array}{c}k\\ l\end{array}}\right) =\left( {\begin{array}{c}k-1\\ l\end{array}}\right) +\left( {\begin{array}{c}k-1\\ l-1\end{array}}\right)$, for every $0\le l\le k$,

$$\begin{aligned} q_k(z)&=\alpha (z)q_{k-1}(z)-\lambda _1\mu _1q_{k-2}(z)\\&=\alpha (z)\sum _{l=0}^{i-1}\left( -\lambda _1\mu _1\right) ^l \left( \left( {\begin{array}{c}2i-1-l-1\\ l\end{array}}\right) \alpha ^{2i-1-2l-1}(z)\alpha _0(z)\right. \\&\left. \quad +\left( {\begin{array}{c}2i-1-l-1\\ l-1\end{array}}\right) \alpha ^{2i-1-2l}(z)\right) \\&\quad -\lambda _1\mu _1\sum _{l=0}^{i-1}\left( -\lambda _1\mu _1\right) ^l \left( \left( {\begin{array}{c}2i-2-l-1\\ l\end{array}}\right) \alpha ^{2i-2-2l-1}(z)\alpha _0(z)\right. \\&\left. \quad +\left( {\begin{array}{c}2i-2-l-1\\ l-1\end{array}}\right) \alpha ^{2i-2-2l}(z)\right) \\ \end{aligned}$$

$$\begin{aligned}&=\sum _{l=0}^{i-1}\left( -\lambda _1\mu _1\right) ^l \left( \left( {\begin{array}{c}2i-1-l-1\\ l\end{array}}\right) \alpha ^{2i-1-2l}(z)\alpha _0(z)\right. \\&\left. \quad +\left( {\begin{array}{c}2i-1-l-1\\ l-1\end{array}}\right) \alpha ^{2i-2l}(z)\right) \\&\quad +\sum _{l=0}^{i-1}\left( -\lambda _1\mu _1\right) ^{l+1} \left( \left( {\begin{array}{c}2i-2-l-1\\ l\end{array}}\right) \alpha ^{2i-2-2l-1}(z)\alpha _0(z)\right. \\&\left. \quad +\left( {\begin{array}{c}2i-2-l-1\\ l-1\end{array}}\right) \alpha ^{2i-2-2l}(z)\right) \\&=\sum _{l=0}^{i-1}\left( -\lambda _1\mu _1\right) ^l \left( \left( {\begin{array}{c}2i-1-l-1\\ l\end{array}}\right) \alpha ^{2i-1-2l}(z)\alpha _0(z)\right. \\ \end{aligned}$$

$$\begin{aligned}&\left. \quad +\left( {\begin{array}{c}2i-1-l-1\\ l-1\end{array}}\right) \alpha ^{2i-2l}(z)\right) \\&\quad +\sum _{l=1}^{i}{\left( -\lambda _1\mu _1\right) ^{l} \left( \left( {\begin{array}{c}2i-2-l\\ l-1\end{array}}\right) \alpha ^{2i-2l-1}(z)\alpha _0(z)+\left( {\begin{array}{c}2i-2-l\\ l-2\end{array}}\right) \alpha ^{2i-2l}(z)\right) }\\&=(-\lambda _1\mu _1)^0\left( \left( {\begin{array}{c}2i-2\\ 0\end{array}}\right) \alpha ^{2i-1}(z)\alpha _0(z)+\left( {\begin{array}{c}2i-2\\ -1\end{array}}\right) \alpha ^{2i}(z)\right) \\&\quad +\sum _{l=1}^{i-1}{\left( -\lambda _1\mu _1\right) ^l \left( \left( {\begin{array}{c}2i-2-l\\ l\end{array}}\right) +\left( {\begin{array}{c}2i-2-l\\ l-1\end{array}}\right) \right) \alpha ^{2i-2l-1}(z)\alpha _0(z)}\\ \end{aligned}$$

$$\begin{aligned}&\quad +\sum _{l=1}^{i-1}{\left( -\lambda _1\mu _1\right) ^l\left( \left( {\begin{array}{c}2i-2-l\\ l-1\end{array}}\right) +\left( {\begin{array}{c}2i-2-l\\ l-2\end{array}}\right) \right) \alpha ^{2i-2l}(z)}\\&\quad +\left( -\lambda _1\mu _1\right) ^i\left( \left( {\begin{array}{c}2i-2-i\\ i-1\end{array}}\right) \alpha ^{2i-2i-1}(z)\alpha _0(z)+\left( {\begin{array}{c}2i-2-i\\ i-2\end{array}}\right) \alpha ^{2i-2i}(z)\right) \\&=\sum _{l=0}^{i}{\left( -\lambda _1\mu _1\right) ^l \left( \left( {\begin{array}{c}2i-l-1\\ l\end{array}}\right) \alpha ^{2i-2l-1}(z)\alpha _0(z)+\left( {\begin{array}{c}2i-l-1\\ l-1\end{array}}\right) \alpha ^{2i-2l}(z)\right) }\\&=\sum _{l=0}^{\lfloor {\frac{k}{2}}\rfloor }{\left( -\lambda _1\mu _1\right) ^l \left( \left( {\begin{array}{c}k-l-1\\ l\end{array}}\right) \alpha ^{k-2l-1}(z)\alpha _0(z)+\left( {\begin{array}{c}k-l-1\\ l-1\end{array}}\right) \alpha ^{k-2l}(z)\right) }. \end{aligned}$$

This completes the proof. $\square$

1.2 Proof of Theorem 2.4

Proof

We will proceed by induction over k. First we note that $A_2=\mathrm{diag}\left( \mu _1\right)$, so that, $A_2^{-1}=\mathrm{diag}\left( \frac{1}{\mu _1}\right)$. In addition, $A_0=\mathrm{diag}\left( \lambda _1\right) =\lambda _1I_{K_2}$. Now, from (2.24) we have

$$\begin{aligned} \vec {P}_1^1=-\frac{1}{\mu _1}\vec {P}_0^1A_1^0=\vec {P}_0^1C_1. \end{aligned}$$

Suppose that the proposition holds for all values up to some $k-1$, where $1\le k-1\le K_1-2$. We will show that it holds for $k\le K_1-1$. From (2.24) we have

$$\begin{aligned} \vec {P}_k^1=-\left( \vec {P}_{k-2}^1A_0+\vec {P}_{k-1}^1A_1\right) A_2^{-1}. \end{aligned}$$

Using the induction assumption with regard to the values of $\vec {P}_{k-2}^1$ and $\vec {P}_{k-1}^1$ we get

$$\begin{aligned} \vec {P}_k^1&=-\left( \vec {P}_{0}^1C_{k-2}A_0+\vec {P}_{0}^1C_{k-1}A_1\right) A_2^{-1}\\&=-\vec {P}_{0}^1\left( C_{k-2}A_0+C_{k-1}A_1\right) A_2^{-1}\\&=-\frac{1}{\mu _1}\vec {P}_{0}^1\left( \lambda _1C_{k-2}+C_{k-1}A_1\right) . \end{aligned}$$

Therefore $\vec {P}_k^1=\vec {P}_{0}^1C_k$, where $C_k=-\frac{1}{\mu _1}\left( \lambda _1C_{k-2}+C_{k-1}A_1\right)$.

This completes the proof. $\square$

1.3 Proof of Theorem 2.6

Proof

Similarly to the proof of Theorem 2.3, we proceed by induction over k.

For $k=1$,

$$\begin{aligned} C_1=\frac{-1}{\mu _1}A_1^0=\left( \frac{-1}{\mu _1}\right) ^1(-\lambda _1\mu _1)^0\left( \left( {\begin{array}{c}0\\ 0\end{array}}\right) A_1^0\left( A_1\right) ^0+\left( {\begin{array}{c}0\\ -1\end{array}}\right) \left( A_1\right) ^1\right) . \end{aligned}$$

For $k=2$,

$$\begin{aligned} C_2= & {} \frac{-1}{\mu _1}\left( \lambda _1I_{K_2}+\left( \frac{-1}{\mu _1}\right) A_1^0A_1\right) \\= & {} \left( \frac{-1}{\mu _1}\right) ^2 \sum _{l=0}^{1}{\left( -\lambda _1\mu _1\right) ^{l} \left( \left( {\begin{array}{c}1-l\\ l\end{array}}\right) A_1^0\left( A_1\right) ^{1-2l}+\left( {\begin{array}{c}1-l\\ l-1\end{array}}\right) \left( A_1\right) ^{2-2l}\right) }. \end{aligned}$$

We now prove that the proposition holds for any k. Suppose $k=2i+1$ (the case for even values, $k=2i$, is presented in the proof of Theorem 2.3),

$$\begin{aligned} C_k&=\frac{-1}{\mu _1}\left( C_{k-1}A_1+\lambda _1C_{k-2}\right) \\&=\frac{-1}{\mu _1}\Bigg (\left( \frac{-1}{\mu _1}\right) ^{2i}\sum _{l=0}^{i}\left( -\lambda _1\mu _1\right) ^l \left( \left( {\begin{array}{c}2i-1-l\\ l\end{array}}\right) A_1^0\left( A_1\right) ^{2i-2l-1}\right. \nonumber \\&\left. \quad +\left( {\begin{array}{c}2i-1-l\\ l-1\end{array}}\right) \left( A_1\right) ^{2i-2l}\right) A_1\\&\quad +\lambda _1\left( \frac{-1}{\mu _1}\right) ^{2i-1}\sum _{l=0}^{i-1}\left( -\lambda _1\mu _1\right) ^l \left( \left( {\begin{array}{c}2i-1-1-l\\ l\end{array}}\right) A_1^0\left( A_1\right) ^{2i-1-2l-1}\right. \nonumber \\ \end{aligned}$$

$$\begin{aligned}&\left. \quad +\left( {\begin{array}{c}2i-1-1-l\\ l-1\end{array}}\right) \left( A_1\right) ^{2i-1-2l}\right) \Bigg )\\&=\left( \frac{-1}{\mu _1}\right) ^{2i+1}\Bigg (\sum _{l=0}^{i}\left( -\lambda _1\mu _1\right) ^l \left( \left( {\begin{array}{c}2i-1-l\\ l\end{array}}\right) A_1^0\left( A_1\right) ^{2i-2l}\right. \nonumber \\&\left. \quad +\left( {\begin{array}{c}2i-1-l\\ l-1\end{array}}\right) \left( A_1\right) ^{2i+1-2l}\right) \\&\quad +\sum _{l=0}^{i-1}{\left( -\lambda _1\mu _1\right) ^{l+1} \left( \left( {\begin{array}{c}2i-2-l\\ l\end{array}}\right) A_1^0\left( A_1\right) ^{2i-2-2l}+\left( {\begin{array}{c}2i-2-l\\ l-1\end{array}}\right) \left( A_1\right) ^{2i-1-2l}\right) }\Bigg )\\ \end{aligned}$$

$$\begin{aligned}&=\left( \frac{-1}{\mu _1}\right) ^{2i+1}\Bigg (\sum _{l=0}^{k}\left( -\lambda _1\mu _1\right) ^l \left( \left( {\begin{array}{c}2i-1-l\\ l\end{array}}\right) A_1^0\left( A_1\right) ^{2i-2l}\right. \nonumber \\&\left. \quad +\left( {\begin{array}{c}2i-1-l\\ l-1\end{array}}\right) \left( A_1\right) ^{2i+1-2l}\right) \\&\quad +\sum _{l=1}^{i}{\left( -\lambda _1\mu _1\right) ^{l} \left( \left( {\begin{array}{c}2i-1-l\\ l-1\end{array}}\right) A_1^0\left( A_1\right) ^{2i-2l}+\left( {\begin{array}{c}2i-1-l\\ l-2\end{array}}\right) \left( A_1\right) ^{2i+1-2l}\right) }\Bigg )\\&=\left( \frac{-1}{\mu _1}\right) ^{2i+1}\Bigg ((-\lambda _1\mu _1)^0\left( \left( {\begin{array}{c}2i-1\\ 0\end{array}}\right) A_1^0\left( A_1\right) ^{2i}+\left( {\begin{array}{c}2i-1\\ -1\end{array}}\right) \left( A_1\right) ^{2i+1}\right) \\ \end{aligned}$$

$$\begin{aligned}&\quad +\sum _{l=1}^{i}{\left( -\lambda _1\mu _1\right) ^l \left( \left( {\begin{array}{c}2i-1-l\\ l\end{array}}\right) +\left( {\begin{array}{c}2i-1-l\\ l-1\end{array}}\right) \right) A_1^0\left( A_1\right) ^{2i-2l}}\\&+\sum _{l=1}^{i}{\left( -\lambda _1\mu _1\right) ^l\left( \left( {\begin{array}{c}2i-1-l\\ l-1\end{array}}\right) +\left( {\begin{array}{c}2i-1-l\\ l-2\end{array}}\right) \right) \left( A_1\right) ^{2i+1-2l}}\Bigg )\\&=\left( \frac{-1}{\mu _1}\right) ^{2i+1}\sum _{l=0}^{i}{\left( -\lambda _1\mu _1\right) ^l \left( \left( {\begin{array}{c}2i-l\\ l\end{array}}\right) A_1^0\left( A_1\right) ^{2i-2l}+\left( {\begin{array}{c}2i-l\\ l-1\end{array}}\right) \left( A_1\right) ^{2i+1-2l}\right) }\\&=\left( \frac{-1}{\mu _1}\right) ^{k}\sum _{l=0}^{\lfloor {\frac{k}{2}}\rfloor }{\left( -\lambda _1\mu _1\right) ^l \left( \left( {\begin{array}{c}k-l-1\\ l\end{array}}\right) A_1^0\left( A_1\right) ^{k-2l-1}+\left( {\begin{array}{c}k-l-1\\ l-1\end{array}}\right) \left( A_1\right) ^{k-2l}\right) }. \end{aligned}$$

This completes the proof. $\square$