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Discrete-time renewal input state dependent queue with working vacations and change over time

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Abstract

This paper considers a discrete-time renewal input state dependent queue with multiple working vacations and change over time. Using the supplementary variable and recursive techniques, the steady state system length distributions at different time epochs have been derived. Based on the system length distributions, some performance measures of the model have been discussed. The optimum values of service rates have been found through quadratic fit search method under a given cost structure.

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Acknowledgments

The authors are thankful to the referees for their valuable comments and suggestions which have helped in improving the quality of the presentation of this paper.

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Authors and Affiliations

  1. Department of Applied Mathematics, Andhra University, Visakhapatnam, 530003, India

    P. Vijaya Laxmi & V. Suchitra

Authors
  1. P. Vijaya Laxmi
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  2. V. Suchitra
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Correspondence to P. Vijaya Laxmi.

Appendix :

Appendix :

Some part of the computational algorithm is presented below. Appendix A1 presents the derivation of ζ n,z and \(\zeta _{n,z}^{(l)}\). The computation of e n,z , f n,z and \(e_{n,z}^{(l)},~f_{n,z}^{(l)}\) is presented in A2.

1.1 A1 Calculation of ζ n,z and \(\zeta _{n,z}^{(l)}\)

For n=N,

$$\begin{array}{@{}rcl@{}} \zeta_{N,z}& =& \left\{ \begin{array}{l} \bar\phi A^{*(1)}(z)\left(\bar\gamma_{N-1}\psi_{N-1}+\bar\gamma_{N}\psi_{N}\right), ~z=\delta_{N},\\ \frac{\bar\phi A^{*}(z)(\bar\gamma_{N-1}\psi_{N-1}+\bar\gamma_{N}\psi_{N})-\bar\phi\bar\gamma_{N}\psi_{N}}{z-\delta_{N}}, ~z\neq\delta_{N}, \end{array} \right. \\\\ \zeta_{N,z}^{(l)}&= &\left\{ \begin{array}{l} \frac{\bar\phi A^{*(l+1)}(z)\left(\bar\gamma_{N-1}\psi_{N-1}+\bar\gamma_{N}\psi_{N}\right)}{l+1}, ~z=\delta_{N},\\ \frac{\bar\phi A^{*(l)}(z)(\bar\gamma_{N-1}\psi_{N-1}+\bar\gamma_{N}\psi_{N})-l\zeta_{N,z}^{(l-1)}}{z-\delta_{N}}, ~z\neq\delta_{N}. \end{array} \right. \end{array} $$

For n=N−1,

$$\begin{array}{@{}rcl@{}} \zeta_{N-1,z}& =& \left\{ \begin{array}{l} \bar\phi A^{*(1)}(z)\left(\gamma_{N-1}\psi_{N-1}+\bar\gamma_{N-2}\psi_{N-2}+\gamma_{N}\psi_{N}\right) \\+\bar\phi\gamma_{N}\zeta_{N,z}^{(1)} ,~z=\delta_{N-1},\\ \frac{\bar\phi\psi_{N-1} \left(\gamma_{N-1}A^{*}(z)-\bar\gamma_{N-1}\right)+\bar\phi\gamma_{N}\zeta_{N,z} }{z-\delta_{N-1}}\\ +\frac{\bar\phi A^{*}(z)\bar\gamma_{N-2}\psi_{N-2}+\gamma_{N}\bar\phi\psi_{N}\left( A^{*}(z)-1\right)}{z-\delta_{N-1}}, z\neq\delta_{N-1}, \end{array} \right. \\\\ \zeta_{N-1,z}^{(l)}& =& \left\{ \begin{array}{l} \bar\phi A^{*(l+1)}(z)\left(\gamma_{N-1}\psi_{N-1}+\bar\gamma_{N-2}\psi_{N-2}+\gamma_{N}\psi_{N}\right) \\+\bar\phi\gamma_{N}\zeta_{N,z}^{(l+1)} , ~z=\delta_{N-1},\\ \frac{\bar\phi A^{*(l)}(z) \left(\bar\gamma_{N-2}\psi_{N-2}+\gamma_{N-1}\psi_{N-1}+\gamma_{N}\psi_{N}\right)}{z-\delta_{N-1}}\\ +\frac{\bar\phi\gamma_{N}\zeta_{N,z}^{(l)}-l\zeta_{N-1,z}^{(l-1)}}{z-\delta_{N-1}}, ~~z\neq\delta_{N-1}. \end{array} \right. \end{array} $$

For 1≤n≤N−2,

$$\begin{array}{@{}rcl@{}} \zeta_{n,z}& =& \left\{ \begin{array}{l} \bar\phi A^{*(1)}(z)\left(\gamma_{n}\psi_{n}+\bar\gamma_{n-1}\psi_{n-1}\right)+\bar\phi\gamma_{n+1}\zeta_{n+1,z}^{(1)} , ~z=\delta_{n},\\ \frac{\bar\phi\psi_{n}\left(\gamma_{n}A^{*}(z)-\bar\gamma_{n}\right) +\bar\phi\gamma_{n+1}\left(\zeta_{n+1,z}-\psi_{n+1}\right)}{z-\delta_{n}} \\+\frac{\bar\phi A^{*}(z)\bar\gamma_{n-1}\psi_{n-1}}{z-\delta_{n}}, ,~z\neq\delta_{n}, \end{array} \right. \\\\ \zeta_{n,z}^{(l)}& =& \left\{ \begin{array}{l} \frac{\bar\phi A^{*(l+1)}(z)\left(\gamma_{n}\psi_{n}+\bar\gamma_{n-1}\psi_{n-1}\right)+\bar\phi\gamma_{n+1}\zeta_{n+1,z}^{(l+1)}}{l+1} , ~z=\delta_{n},\\ \frac{\bar\phi A^{*(l)}(z)\left(\gamma_{n}\psi_{n}+\bar\gamma_{n-1}\psi_{n-1}\right) +\bar\phi\gamma_{n+1}\zeta_{n+1,z}^{(l)}-l\zeta_{n,z}^{(l-1)}}{z-\delta_{n}} , z\neq\delta_{n}, \end{array} \right. \end{array} $$

1.2 A2 Calculation of e n,z ,f n,z and \(e_{n,z}^{(l)}, f_{n,z}^{(l)}\)

For n=N,

$$\begin{array}{@{}rcl@{}} e_{N,z}& =& \left\{ \begin{array}{l} \phi A^{*(1)}(z)(\bar\gamma_{N-1}\psi_{N-1} +\bar\gamma_{N}\psi_{N}) +\phi\bar\gamma_{N}\zeta_{N,z}^{(1)}\\ +A^{*(1)}(z)\bar\mu_{N-1}t_{N-1}, ~z=\bar\mu_{N},\\ \frac{\phi A^{*}(z)(\bar\gamma_{N-1}\psi_{N-1} +\bar\gamma_{N}\psi_{N}) +\phi\bar\gamma_{N}\zeta_{N,z}+A^{*}(z)\bar\mu_{N-1}t_{N-1}}{z-\bar \mu_{N}}\\ -\frac{\phi\bar\gamma_{N}\psi_{N}}{z-\bar\mu_{N}} ,~z\neq\bar\mu_{N}, \end{array} \right. \\\\\\f_{N,z}& =& \left\{ \begin{array}{l} A^{*(1)}(z)(\bar\mu_{N-1}d_{N-1}+\bar\mu_{N}d_{N}), ~z=\bar\mu_{N},\\ \frac{A^{*}(z)(\bar\mu_{N-1}d_{N-1}+\bar\mu_{N}d_{N})-\bar\mu_{N}d_{N}}{z-\bar \mu_{N}} ,~z\neq\bar\mu_{N}, \end{array} \right. \\\\\\e_{N,z}^{(l)}& =& \left\{ \begin{array}{l} \frac{\phi A^{*(l+1)}(z)(\bar\gamma_{N-1}\psi_{N-1} +\bar\gamma_{N}\psi_{N}) +\phi\bar\gamma_{N}\zeta_{N,z}^{(l+1)}}{(l+1)}\\+\frac{A^{*(l+1)}(z)\bar\mu_{N-1}t_{N-1}}{(l+1)}, ~z=\bar\mu_{N},\\ \frac{\phi A^{*(l)}(z)(\bar\gamma_{N-1}\psi_{N-1} +\bar\gamma_{N}\psi_{N}) +A^{*(l)}(z)\bar\mu_{N-1}t_{N-1}}{z-\bar \mu_{N}}\\ +\frac{\phi\bar\gamma_{N}\zeta_{N,z}^{(l)}-l e_{N,z}^{(l-1)}}{z-\bar\mu_{N}} ,~z\neq\bar\mu_{N}, \end{array} \right. \\\\\\f_{N,z}^{(l)}& =& \left\{ \begin{array}{l} \frac{A^{*(l+1)}(z)(\bar\mu_{N-1}d_{N-1}+\bar\mu_{N}d_{N})}{(l+1)}, ~z=\bar\mu_{N},\\ \frac{A^{*(l)}(z)(\bar\mu_{N-1}d_{N-1}+\bar\mu_{N}d_{N})-lf_{N,z}^{(l-1)}}{z-\bar \mu_{N}} ,~z\neq\bar\mu_{N}. \end{array} \right. \end{array} $$

For n=N−1,

$$\begin{array}{@{}rcl@{}} e_{N-1,z}& =& \left\{ \begin{array}{l} \phi A^{*(1)}(z)(\bar\gamma_{N-2}\psi_{N-2}+\gamma_{N-1}\psi_{N-1} +\gamma_{N}\psi_{N}) +\phi\bar\gamma_{N-1}\zeta_{N-1,z}^{(1)} \\+\phi\gamma_{N}\zeta_{N,z}^{(1)}+\mu_{N}e_{N,z}^{(1)} +A^{*(1)}(z)(\bar\mu_{N-2}t_{N-2}+\mu_{N-1}t_{N-1}),\!z=\bar\mu_{N-1},\\ \frac{\phi A^{*}(z)(\bar\gamma_{N-2}\psi_{N-2}+\gamma_{N-1}\psi_{N-1} +\gamma_{N}\psi_{N})+\phi\bar\gamma_{N-1}\zeta_{N-1,z}}{z-\bar \mu_{N}}\\ -\frac{\phi(\bar\gamma_{N-1}\psi_{N-1} +\gamma_{N}\psi_{N})+A^{*}(z)(\bar\mu_{N-2}t_{N-2}+\mu_{N-1}t_{N-1} )}{z-\bar \mu_{N-1}} \\+\frac{\phi\gamma_{N}\zeta_{N,z}+\mu_{N}e_{N,z}-\bar\mu_{N-1}t_{N-1}}{z-\bar \mu_{N-1}} ,~z\neq\bar\mu_{N-1}, \end{array} \right. \\\\\\f_{N-1,z}& =& \left\{ \begin{array}{l} \mu_{N}f_{N,z}^{(1)}+A^{*(1)}(z)\left(\bar\mu_{N-2}d_{N-2} +\mu_{N-1}d_{N-1}+\mu_{N}d_{N}\right), ~z=\bar\mu_{N-1},\\ \frac{\mu_{N}f_{N,z}+A^{*}(z)\left(\bar\mu_{N-2}d_{N-2} +\mu_{N-1}d_{N-1}+\mu_{N}d_{N}\right)}{~z-\bar\mu_{N-1}}\\ -\frac{\mu_{N}d_{N}-\bar \mu_{N-1}d_{N-1}}{~z-\bar\mu_{N-1}} ,~z\neq\bar\mu_{N-1}, \end{array} \right. \end{array} $$
$$\begin{array}{@{}rcl@{}} e_{N-1,z}^{(l)}& =& \left\{ \begin{array}{l} \frac{\phi A^{*(l+1)}(z)(\bar\gamma_{N-2}\psi_{N-2}+\gamma_{N-1}\psi_{N-1} +\gamma_{N}\psi_{N}) }{l+1}\\+\frac{\mu_{N}e_{N,z}^{(l+1)}+A^{*(l+1)}(z)(\bar\mu_{N-2}t_{N-2}+\mu_{N-1}t_{N-1})}{l+1} \\+\frac{\phi(\bar\gamma_{N-1}\zeta_{N-1,z}^{(l+1)}+\gamma_{N}\zeta_{N,z}^{(l+1)})}{(l+1)}, ~z=\bar\mu_{N-1},\\ \frac{\phi A^{*(l)}(z)(\bar\gamma_{N-2}\psi_{N-2}+\gamma_{N-1}\psi_{N-1} +\gamma_{N}\psi_{N}) +\mu_{N}e_{N,z}^{(l)}}{z-\bar \mu_{N-1}}\\ +\frac{\phi(\bar\gamma_{N-1}\zeta_{N-1,z}^{(l)}+\gamma_{N}\zeta_{N,z}^{(l)})+A^{*(l)}(z)(\bar\mu_{N-2}t_{N-2}+\mu_{N-1}t_{N-1})}{z-\bar \mu_{N-1}}\\ -\frac{le_{N-1,z}^{(l-1)}}{z-\bar \mu_{N-1}} ,~z\neq\bar\mu_{N-1}, \end{array} \right. \\\\f_{N-1,z}^{(l)}& =& \left\{ \begin{array}{l} \frac{A^{*(l+1)}(z)\left(\bar\mu_{N-2}d_{N-2} +\mu_{N-1}d_{N-1}+\mu_{N}d_{N}\right)}{(l+1)}\\+\frac{\mu_{N}f_{N,z}^{(l+1)}}{(l+1)} ,~z=\bar\mu_{N-1},\\ \frac{\mu_{N}f_{N,z}^{(l)}+A^{*(l)}(z)\left(\bar\mu_{N-2}d_{N-2} +\mu_{N-1}d_{N-1}+\mu_{N}d_{N}\right)}{~z-\bar\mu_{N-1}}\\ - \frac{le_{N-1,z}^{(l-1)}}{~z-\bar\mu_{N-1}} ,~z\neq\bar\mu_{N-1}. \end{array} \right. \end{array} $$

For 1≤n≤N−2,

$$\begin{array}{@{}rcl@{}} e_{n,z}& =& \left\{ \begin{array}{l} \phi\left\{\gamma_{n+1} \zeta_{n+1,z}^{(1)}+\bar\gamma_{n} \zeta_{n,z}^{(1)} +A^{*(1)}(z)(\gamma_{n}\psi_{n}+\bar\gamma_{n-1}\psi_{n-1})\right\}\\+ \mu_{n+1}e_{n+1,z}^{(1)}+A^{*(1)}(z)(\bar\mu_{n-1}t_{n-1}+~\mu_{n}t_{n}), ~z=\bar\mu_{n},\\ \frac{\phi\left\{\gamma_{n+1} \zeta_{n+1,z}+\bar\gamma_{n} \zeta_{n,z}-(\gamma_{n+1}\psi_{n+1}+\bar\gamma_{n}\psi_{n}) \right\}+\mu_{n+1}e_{n+1,z}}{z-\bar\mu_{N-1}}\\ +\frac{A^{*}(z)(\gamma_{n}\psi_{n}+\bar\gamma_{n-1}\psi_{n-1})+A^{*}(z)(\bar\mu_{n-1}t_{n-1}+\mu_{n}t_{n})}{z-\bar\mu_{n}} \\ -\frac{(\mu_{n+1}t_{n+1}+\bar\mu_{n}t_{n})}{z-\bar\mu_{N-1}},~z\neq\bar\mu_{n}, \end{array} \right.\\ \\\\f_{n,z}& =& \left\{ \begin{array}{l} \mu_{n+1}f_{n+1,z}^{(1)}+A^{*(1)}(z)\left(\bar\mu_{n-1}d_{n-1} +\mu_{n}d_{n}\right), ~z=\bar\mu_{n},\\ \frac{\mu_{n+1}f_{n+1,z}+A^{*}(z)\left(\bar\mu_{n-1}d_{n-1} +\mu_{n}d_{n}\right)}{~z-\bar\mu_{n}}\\-\frac{(\bar \mu_{n}d_{n}+ \mu_{n+1}d_{n+1})}{~z-\bar\mu_{n}} ,~z\neq\bar\mu_{n}, \end{array} \right. \end{array} $$
$$\begin{array}{@{}rcl@{}} e_{n,z}^{(l)}& =& \left\{ \begin{array}{l} \phi\left\{\gamma_{n+1} \zeta_{n+1,z}^{(l+1)}+\bar\gamma_{n} \zeta_{n,z}^{(l+1)} +A^{*(l+1)}(z)(\gamma_{n}\psi_{n}+\bar\gamma_{n-1}\psi_{n-1})\right\}\\+ \mu_{n+1}e_{n+1,z}^{(l+1)}+A^{*(l+1)}(z)(\bar\mu_{n-1}t_{n-1}+~\mu_{n}t_{n}), ~z=\bar\mu_{n},\\ \frac{\phi\left\{\gamma_{n+1} \zeta_{n+1,z}^{(l)}+\bar\gamma_{n} \zeta_{n,z}^{(l)} +A^{*(l)}(z)(\gamma_{n}\psi_{n}+\bar\gamma_{n-1}\psi_{n-1})\right\}}{z-\bar\mu_{n}}\\+\frac{ \mu_{n+1}e_{n+1,z}^{(l)}+A^{*(l)}(z)(\bar\mu_{n-1}t_{n-1}+\mu_{n}t_{n})-le_{n,z}^{(l-1)}}{z-\bar\mu_{n}} ,~z\neq\bar\mu_{n}, \end{array} \right. \end{array} $$
$$\begin{array}{@{}rcl@{}} f_{n,z}^{(l)}& =& \left\{ \begin{array}{l} \mu_{n+1}f_{n+1,z}^{(l+1)}+A^{*(l+1)}(z)\left(\bar\mu_{n-1}d_{n-1} +\mu_{n}d_{n}\right), ~z=\bar\mu_{n},\\ \frac{\mu_{n+1}f_{n+1,z}^{(l)}+A^{*(l)}(z)\left(\bar\mu_{n-1}d_{n-1} +\mu_{n}d_{n}\right)-le_{n,z}^{(l-1)}}{~z-\bar\mu_{n}} ,~z\neq\bar\mu_{n}. \end{array} \right. \end{array} $$

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Laxmi, P.V., Suchitra, V. Discrete-time renewal input state dependent queue with working vacations and change over time. OPSEARCH 52, 285–306 (2015). https://doi.org/10.1007/s12597-014-0185-2

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