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Stability of a two-queue cyclic polling system with BMAPs under gated service and state-dependent time-limited service disciplines

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Abstract

The stability of a cyclic polling system, with a single server and two infinite-buffer queues, is considered. Customers arrive at the two queues according to independent batch Markovian arrival processes. The first queue is served according to the gated service discipline, and the second queue is served according to a state-dependent time-limited service discipline with the preemptive repeat-different property. The state dependence is that, during each cycle, the predetermined limited time of the server’s visit to the second queue depends on the queue length of the first queue at the instant when the server last departed from the first queue. The mean of the predetermined limited time for the second queue either decreases or remains the same as the queue length of the first queue increases. Due to the two service disciplines, the customers in the first queue have higher service priority than the ones in the second queue, and the service fairness of the customers with different service priority levels is also considered. In addition, the switchover times for the server traveling between the two queues are considered, and their means are both positive as well as finite. First, based on two embedded Markov chains at the cycle beginning instants, the sufficient and necessary condition for the stability of the cyclic polling system is obtained. Then, the calculation methods for the variables related to the stability condition are given. Finally, the influence of some parameters on the stability condition of the cyclic polling system is analyzed. The results are useful for engineers not only checking whether the given cyclic polling system is stable, but also adjusting some parameters to make the system satisfy some requirements under the condition that the system is stable.

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Acknowledgments

The authors thank the associate editor and reviewers for their invaluable comments and suggestions which lead to the considerable improvement of this paper. The research is supported by the National Natural Science Foundations of China (Grant Nos. 61271107 and 61401286).

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Correspondence to Jianyu Cao.

Appendices

Appendix 1: The proof of Theorem 1

Proof

Without loss of generality, suppose \({S_n} = \left( {{i^{(1)}},{v^{(1)}},{i^{(2)}},{v^{(2)}}} \right) \in S\), and consider the state \({S_{n + 1}} = \left( {N_{n + 1}^{(1)},J_{n + 1}^{(1)},N_{n + 1}^{(2)},J_{n + 1}^{(2)}} \right) \). There are the relations \(N_{n + 1}^{(1)} = A_n^{(1)}\) and \(N_{n + 1}^{(2)} = {i^{(2)}} + A_n^{(2)} - C_n^{(2)}\), where \(A_n^{(k)}\) is a random variable which denotes the number of customers arriving at queue k during cycle n, and \(C_n^{(2)}\) is a random variable which denotes the number of 2-customers served (excluding the 2-customer whose service is interrupted) during the 2-service period in cycle n. \(C_n^{(2)}\) is determined by the state of the cyclic polling system at time \(TD_n^{(1)}\) in the stochastic sense, and the state of the cyclic polling system at time \(TD_n^{(1)}\) is determined by the state \({S_n} = \left( {{i^{(1)}},{v^{(1)}},{i^{(2)}},{v^{(2)}}} \right) \) in the stochastic sense independently of n. \(A_n^{(k)}\) and \(J_{n + 1}^{(k)}\) are determined by the cycle time of cycle n in the stochastic sense. The cycle time of cycle n is determined by the state \({S_n} = \left( {{i^{(1)}},{v^{(1)}},{i^{(2)}},{v^{(2)}}} \right) \) in the stochastic sense independently of n. Thus \(\left\{ {S_n}\right\} \) is a homogeneous Markov chain.

Under assumption A.2, given \(v_1 \in M_{1}\), \(v_2 \in M_{2}\), any state \(\left( {{i^{(1)}},{v^{(1)}},{i^{(2)}},{v^{(2)}}} \right) \in S\) can reach some state \(\left( i^{(1)}_1,v_1,i^{(2)}_1,v_2\right) \), since \({{\mathbf {D}}_1}\) and \({{\mathbf {D}}_2}\) are irreducible. Moreover, the state \(\left( {i_1^{(1)},v_1,i_1^{(2)},v_2} \right) \) can reach \(\left( 0,v_1,0,v_2\right) \), since \({{\mathbf {D}}_{1,0}}\) and \({{\mathbf {D}}_{2,0}}\) are stable. So the state \(\left( 0,v_1,0,v_2\right) \) can be reached from any state \(\left( {{i^{(1)}},{v^{(1)}},{i^{(2)}},{v^{(2)}}} \right) \in S\). From the above, there is a set C consisting of \(\left( 0,v_1,0,v_2\right) \) and the states which can be reached from the state \(\left( 0,v_1,0,v_2\right) \), such that C is absorbing and irreducible. Because \({{\mathbf {D}}_{1,0}}\) and \({{\mathbf {D}}_{2,0}}\) are stable, the state \(\left( 0,v_1,0,v_2\right) \) can be reached from itself by one-step transition. Thus all the states in C are aperiodic. Finally let \(D = S\backslash C\). Since any state in S can reach the state \(\left( 0,v_1,0,v_2\right) \in C\), if \(D \ne \emptyset \), then each state in D can reach C. The proof is complete. \(\square \)

Appendix 2: The proof of Theorem 2

Proof

Define the vector \({{\mathbf {X}}_i}\) by \({{\mathbf {X}}_i} = i{\mathbf {e}}\) \((i \in \mathrm{N})\). From the relation (8), if \({i_0} \in \mathrm{N}^+\) and \(i \in \left\{ {{i_0} + 1,{i_0} + 2, \ldots } \right\} \), then the left-hand side of the inequality (9) is equal to

$$\begin{aligned}&\sum _{j = 0}^\infty {{{\mathbf {P}}_{ij}}{{\mathbf {X}}_j} - {{\mathbf {X}}_i}}\nonumber \\&= \sum _{k = 0}^\infty {\left[ {\int _{0}^\infty {{{\mathbf {P}}^{(1)}}(k,\tau )dB_1^{ * (i)}(\tau )} \int _{0}^\infty {\sum _{n = 0}^\infty {n{{\mathbf {P}}^{(1)}}(n,x)d{R_1} * {H_k} * {R_2}(x)} } } \right] {\mathbf {e}}}\nonumber \\&\phantom {=} + \sum _{k = 0}^\infty {\left[ {\int _{0}^\infty {k{{\mathbf {P}}^{(1)}}(k,\tau )dB_1^{ * (i)}(\tau )} \int _{0}^\infty {\sum _{n = 0}^\infty {{{\mathbf {P}}^{(1)}}(n,x)d{R_1} * {H_k} * {R_2}(x)} } } \right] {\mathbf {e}}} - i{\mathbf {e}}. \end{aligned}$$
(42)

Based on the relation (15) in [22], the following relation is obtained:

$$\begin{aligned} \sum _{n = 0}^\infty {n{{\mathbf {P}}^{(1)}}(n,x)} {\mathbf {e}} = {\mathbf {e}}{\lambda _1}x + \left( {{e^{{{\mathbf {D}}_1}x}} - {\mathbf {I}}} \right) {({\mathbf {e}}{{\varvec{\theta }}_1} + {{\mathbf {D}}_1})^{ - 1}}{{\mathbf {d}}_1}, \end{aligned}$$
(43)

where \({{\mathbf {d}}_1} = \sum _{k = 1}^\infty {k{{\mathbf {D}}_{1,k}}{\mathbf {e}}}\). Using the relation (43), the relation (42) is transformed to,

$$\begin{aligned} \sum _{j = 0}^\infty {{{\mathbf {P}}_{ij}}{{\mathbf {X}}_j} - {{\mathbf {X}}_i}}=&\int _{0}^\infty {{e^{{{\mathbf {D}}_1}\tau }}dB_1^{ * (i)}(\tau )} \times \left[ {\left( {{\mathbf {e}}{\lambda _1}({r_1} + {r_2})} \right) - {{({\mathbf {e}}{{\varvec{\theta }}_1} + {{\mathbf {D}}_1})}^{ - 1}}{{\mathbf {d}}_1}} \right] \nonumber \\&+ \int _{0}^\infty {\left[ {\sum _{k = 0}^\infty {{h_k}{{\mathbf {P}}^{(1)}}(k,\tau )} } \right] dB_1^{ * (i)}(\tau ){\lambda _1}{\mathbf {e}}}\nonumber \\&+ \int _{0}^\infty {\left[ {\sum _{k = 0}^\infty {{{\mathbf {P}}^{(1)}}(k,\tau )\int _{0}^\infty {{e^{{{\mathbf {D}}_1}x}}d{H_k}(x)}} } \right] dB_1^{ * (i)}(\tau )}\nonumber \\&\times \int _{0}^\infty {{e^{{{\mathbf {D}}_1}x}}d{R_1}(x)} \times \int _{0}^\infty {{e^{{{\mathbf {D}}_1}x}}d{R_2}(x)} {({\mathbf {e}}{{\varvec{\theta }}_1} + {{\mathbf {D}}_1})^{ - 1}}{{\mathbf {d}}_1}\nonumber \\&+ \left( {\int _{0}^\infty {{e^{{{\mathbf {D}}_1}\tau }}dB_1^{ * (i)}(\tau )} - {\mathbf {I}}} \right) {({\mathbf {e}}{{\varvec{\theta }}_1} + {{\mathbf {D}}_1})^{ - 1}}{{\mathbf {d}}_1} + i({\rho _1} - 1){\mathbf {e}}. \end{aligned}$$
(44)

Since \({{e}^{{{\mathbf {D}}_{1}}t}}=\sum \limits _{k=0}^{\infty }{\frac{{{\left( {{\mathbf {D}}_{1}}t \right) }^{k}}}{k!}}\), there is the following relation:

$$\begin{aligned} {e^{{{\mathbf {D}}_1}t}}{\mathbf {e}} = {\mathbf {e}}. \end{aligned}$$
(45)

From the relation (3), it is shown that each element of \({e^{{{\mathbf {D}}_1}t}}\) is nonnegative. So from the relation (45), \({e^{{{\mathbf {D}}_1}t}}\) is a stochastic matrix for arbitrary \(t\ge 0\). Therefore, the supremum norm of \({e^{{{\mathbf {D}}_1}t}}\) is equal to 1, namely, \({\left\| {{e^{{{\mathbf {D}}_1}t}}} \right\| _\infty } = 1\). \({\left\| {\sum _{k = 0}^\infty {{h_k}{{\mathbf {P}}^{(1)}}(k,\tau )} } \right\| _\infty } \le {h_{\max }} < \infty \), since \(\sum _{k = 0}^\infty {{h_k}{{\mathbf {P}}^{(1)}}(k,\tau )}\le {h_{\max }}{e^{{{\mathbf {D}}_1}\tau }}\). \({\left\| {\sum _{k = 0}^\infty {{{\mathbf {P}}^{(1)}}(k,\tau )\int _{0}^\infty {{e^{{{\mathbf {D}}_1}x}}d{H_k}(x)}} } \right\| _\infty } = 1\), since \(\int _{0}^\infty {{e^{{{\mathbf {D}}_1}x}}d{H_k}(x)}{\mathbf {e}}={\mathbf {e}}\) and \(\sum _{k = 0}^\infty {{{\mathbf {P}}^{(1)}}(k,\tau )}{\mathbf {e}} = {\mathbf {e}}\). Thus, if i tends to infinity, then the sign of the expression (44) is identical to the sign of the expression \(\left( {{\rho _1} - 1} \right) \). If \({\rho _1} < 1\), then the left-hand side of (9) is negative for a sufficiently large i. From the relation (44), the inequality (10) is true for a finite i. The proof is complete. \(\square \)

Appendix 3: The proof of Lemma 1

Proof

Suppose that the initial state of \(\left\{ S'_n\right\} \) is \({S'_1} = ({i_1},{v_1}) \in D'\) \(({i_1} > 0)\) and \({S'_2} = ({i_2},{v_2})\) is reached from \(({i_1},{v_1})\) by one-step transition. From the relation (8), the transition probability from \(({i_1},{v_1})\) to \(({i_2},{v_2})\) is the \(({v_1},{v_2})\)-th element of the matrix \({{\mathbf {P}}_{{i_1}{i_2}}}\),

$$\begin{aligned} {{\mathbf {P}}_{{i_1}{i_2}}}= \sum _{k = 0}^{{i_2}} {\int _{0}^\infty {{{\mathbf {P}}^{(1)}}(k,\tau )dB_1^{ * ({i_1})}(\tau )} \int _{0}^\infty {{{\mathbf {P}}^{(1)}}({i_2} - k,x)d{R_1} * {H_k} * {R_2}(x)} }. \end{aligned}$$
(46)

The transition probability from \((0,{v_1})\) to \(({i_2},{v_2})\) is the \(({v_1},{v_2})\)-th element of the matrix \({{\mathbf {P}}_{{0}{i_2}}}\),

$$\begin{aligned} {{\mathbf {P}}_{0{i_2}}}= \sum _{k = 0}^{{i_2}} {\int _{0}^\infty {{{\mathbf {P}}^{(1)}}(k,\tau )d{R_1}(\tau )} \int _{0}^\infty {{{\mathbf {P}}^{(1)}}({i_2} - k,x)d{H_0} * {R_2}(x)} }. \end{aligned}$$
(47)

It is shown that the relations (46) and (47) have similar structure. From (4), (5), (6), and (7), given a probability distribution G(t) on \([0, + \infty )\) with a positive and finite mean, the sign of each element of the matrix \(\int _{0}^\infty {{{\mathbf {P}}^{(1)}}(k,t)dG(t)}\), \(k \in \mathrm{N}\), depends only on \({{\mathbf {P}}^{(1)}}(k,t)\). Thus for \({{\mathbf {P}}_{{i_1}{i_2}}}\) and \({{\mathbf {P}}_{{0}{i_2}}}\), their elements at the same position have the same sign. So \({S'_2} = ({i_2},{v_2})\) can be reached from \((0,{v_1})\) by one-step transition with a positive probability. \({S'_2} = ({i_2},{v_2}) \in C'\), since \((0,v_1)\in C'\). The proof is complete. \(\square \)

Appendix 4: The proof of Corollary 1

Proof

From Lemma 5 of [30] and Corollary 3 of [35],

$$\begin{aligned} {{C_{\text {s}}} = \lambda _1^{-1}{\varvec{\pi '}}(1){\mathbf {e}}.} \end{aligned}$$
(48)

Substituting (48) into the right-hand side of the relation (17) yields

$$\begin{aligned} {{C_{\text {s}}} = {{\left( {{r_1} + {r_2} + {{\varvec{\pi }}_0}{h_0}{\mathbf {e}} + \sum _{k = 1}^\infty {{{\varvec{\pi }}_k}\sum _{l = 0}^\infty {{\mathbf {B}}_{1,l}^{(k)}{h_l}} } {\mathbf {e}}} \right) } \bigg /{(1 - {\rho _1})}}.} \end{aligned}$$
(49)

Based on the relations (16), (18), and (49), the formula for \(r_{\text {s}}^{(2)}\) can be written as the following:

$$\begin{aligned} {r_{\text {s}}^{(2)} = \frac{{(1 - {\rho _1})\left( {{\varvec{\pi }}_0}{d_0}{\mathbf {e}} + \sum _{k = 1}^\infty {{{\varvec{\pi }}_k}\sum _{l = 0}^\infty {{\mathbf {B}}_{1,l}^{(k)}{d_l}} } {\mathbf {e}} \right) }}{{{{r_1} + {r_2} + {{\varvec{\pi }}_0}{h_0}{\mathbf {e}} + \sum _{k = 1}^\infty {{{\varvec{\pi }}_k}\sum _{l = 0}^\infty {{\mathbf {B}}_{1,l}^{(k)}{h_l}} } {\mathbf {e}}}}}.} \end{aligned}$$
(50)

By substituting the relation (50) into the relation \(\lambda _2<r_{\text {s}}^{(2)}\), the relation (24) can be obtained and it implies that \(\rho _1<1\) is true. The proof is complete. \(\square \)

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Cao, J., Xie, W. Stability of a two-queue cyclic polling system with BMAPs under gated service and state-dependent time-limited service disciplines. Queueing Syst 85, 117–147 (2017). https://doi.org/10.1007/s11134-016-9504-z

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