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Cost scrutiny of discrete-time priority queue with cluster arrival and Bernoulli feedback

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Abstract

This work describes the economic feasibility of a single server discrete-time queueing model, (Geo/G/1) where interarrival times have a geometric distribution, and service times have a general distribution. This work is motivated by the case of discrete-time queueing models under priority scheme for solving many congestion issues of the telecommunication system wherein few calls are treated as prioritized calls and system manager may handle it properly. Herein a state-dependent arrival policy is used. It is assumed that the clients arrive in groups of varying sizes, and incorporates only one server queueing system with unlimited capacity. Under a discrete-time system with Markovian service practice, clients are serviced one at a time. If a client is dissatisfied with his service, he will most likely be directed back to the front of the queue. This concept is commonly referred to as Bernoulli feedback (BF) in queueing scenario. Just after every service, it is presumed that the server either starts to identify the next client to be serviced with some probability, or the server starts a solo vacation procedure with its complementary probability and this process is referred as Bernoulli vacation (BV). In addition, preferred and impatient clients are examined too. We investigate the Markov chain that underpins the queueing system in question, and its normalizing condition. The average number of consumers in the queue and the system are found using a generating function method. The numeral expositions are ascertained to delve the impact of different parameters on various performance metrics which can give information to system management in order to monitor the system's functioning condition and decrease congestion. We then used direct search method (DSM) and Particle Swarm Optimization (PSO) approaches to present a comparative study to assist system administrators or decision-makers by economically regulating the system. Furthermore, the results of the provided model are contrasted to those of a soft computing approach termed as ANFIS (Adaptive Neuro-Fuzzy Inference System).

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Data availability

The real data taken in Sect. 7 to support the findings of this study are openly available in [27]. And for rest of the data, the authors declare that this research is an observational study, does not involve any surveys or participants in any capacity and thus, no ethical approval is required.

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Acknowledgements

The authors are grateful to anonymous reviewers for their insightful comments and suggestions. The expertise of all has improved this article in innumerable ways and saved us from many errors.

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No funding was received while conducting this study.

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Authors and Affiliations

  1. Amity Institute of Applied Sciences, Amity University, Noida, 201301, U. P., India

    Vaishnawi Shree & Shweta Upadhyaya

  2. Department of Mathematics, BITS Pilani, Jhunjhunu, 333031, Rajasthan, India

    Rakhee Kulshrestha

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  1. Vaishnawi Shree
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  2. Shweta Upadhyaya
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  3. Rakhee Kulshrestha
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Appendix: Notations used in above equations

Appendix: Notations used in above equations

$$F_{\mathcalligra{i}} (1) = 1 - \mathcalligra{p}_{\mathcalligra{i}} \Theta \alpha ;\quad \mathcalligra{i} = 1,2;\quad F^{\prime}_{\mathcalligra{i}} (1) = \mathcalligra{p}_{\mathcalligra{i}} \Theta \overline{\alpha } E[{\rm X}];\quad \mathcalligra{i} = 1,2;\quad F^{\prime\prime}_{\mathcalligra{i}} (1) = \mathcalligra{p}_{\mathcalligra{i}} \Theta \overline{\alpha } E[{\rm X}^{2} ];\quad \mathcalligra{i} = 1,2$$
$$T_{1} (1) = F(1) = \overline{{\mathcalligra{p}_{1} }} + \mathcalligra{p}_{1} \overline{\Theta } + \Theta \overline{\alpha } \left( {\overline{\mathcalligra{w}} \mathcalligra{p}_{1} + \mathcalligra{w}\overline{{\mathcalligra{p}_{1} }} a_{0} } \right);\quad T^{\prime}_{1} (1) = \Theta \overline{\alpha } \left( {\overline{\mathcalligra{w}} \mathcalligra{p}_{1} + \mathcalligra{w}\overline{{\mathcalligra{p}_{1} }} a_{0} } \right)E[{\rm X}];$$
$$T^{\prime\prime}_{1} (1) = \Theta \overline{\alpha } \left( {\overline{\mathcalligra{w}} \mathcalligra{p}_{1} + \mathcalligra{w}\overline{{\mathcalligra{p}_{1} }} {\text{a}}_{0} } \right)E[{\rm X}^{2} ];\quad \pounds_{1} (1) = \overline{\mathcalligra{r}} + \mathcalligra{r}T_{1} (1);\quad \pounds_{2} (1) = \overline{{\mathcalligra{p}_{1} }} \mathcalligra{q} + \overline{{\mathcalligra{p}_{2} }} \overline{\mathcalligra{q}} \frac{{{\varvec{S}}_{3} (F_{2} (1))F(1)}}{{F_{2} (1)}};$$
$$\mathcal{H}^{\prime}(1) = \Theta \overline{\alpha } \left( {\overline{\mathcalligra{w}} \mathcalligra{p}_{1} + \mathcalligra{w}\overline{{\mathcalligra{p}_{1} }} {\text{a}}_{0} } \right)\left( {E[{\rm X}] - 1} \right);\quad F^{\prime\prime}(1) = \Theta \overline{\alpha } \left( {\overline{\mathcalligra{w}} \mathcalligra{p}_{1} + \mathcalligra{w}\overline{{\mathcalligra{p}_{1} }} {\text{a}}_{0} } \right)\left\{ {E[{\rm X}^{2} ] + 2\left( {1 - E[{\rm X}]} \right)} \right\};$$
$$\pounds(1) = \pounds_{1} (1)\pounds_{2} (1);\quad \pounds^{\prime}(1) = \pounds^{\prime}_{1} (1)\pounds_{2} (1) + \pounds_{1} (1)\pounds^{\prime}_{2} (1);\quad \pounds^{\prime\prime}(1) = \pounds^{\prime\prime}_{1} (1)\pounds_{2} (1) + 2\pounds^{\prime}_{1} (1)\pounds^{\prime}_{2} (1) + \pounds_{1} (1)\pounds^{\prime\prime}_{2} (1);$$
$$\pounds^{\prime}_{1} (1) = \frac{{\mathcalligra{p}_{1} \Theta \mathcalligra{r}}}{{\left[ {F_{2} (1)} \right]^{2} }}\left[ {F_{2} (1)\left\{ \begin{gathered} 2\alpha {\varvec{S}}_{2} (F_{1} (1))T_{1} (1) + \alpha \varvec{S}^{\prime}_{2} (F_{1} (1))F^{\prime}_{1} (1)T_{1} (1) \hfill \\ - \overline{\alpha } E[{\rm X}]{\varvec{S}}_{2} (F_{1} (1))T_{1} (1) + \alpha {\varvec{S}}_{2} (F_{1} (1))T^{\prime}_{1} (1) \hfill \\ \end{gathered} \right\} - \alpha {\varvec{S}}_{2} (F_{1} (1))T_{1} (1)F^{\prime}_{2} (1)} \right];$$
$$\pounds^{\prime}_{2} (1) = \overline{{\mathcalligra{p}_{1} }} \mathcalligra{q}\mathcalligra{w} + \frac{{\overline{{\mathcalligra{p}_{2} }} \overline{\mathcalligra{q}} }}{{\left[ {F_{2} (1)} \right]^{2} }}\left[ {F_{2} (1)\varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1) + {\varvec{S}}_{3} (F_{2} (1))F^{\prime}(1)F_{2} (1) - {\varvec{S}}_{3} (F_{2} (1))F(1)F^{\prime}_{2} (1)} \right];$$
$$\begin{aligned} \pounds^{\prime\prime}_{1} (1) & = \frac{{\mathcalligra{p}_{1} \Theta \mathcalligra{r}}}{{\left[ {F_{2} (1)} \right]^{3} }}\left[ \begin{aligned} & F_{2} (1)\left\{ {F_{2} (1)N^{\prime\prime}(1) - \, \alpha {\varvec{S}}_{2} (F_{1} (1))T_{1} (1)F^{\prime\prime}_{2} (1)} \right\} \\ & - \, 2F^{\prime}_{2} (1)\left\{ {F_{2} (1)N^{\prime}(1) - \, \alpha {\varvec{S}}_{2} (F_{1} (1))T_{1} (1)} \right\} \\ \end{aligned} \right]\quad {\text{where}} \\ N^{\prime}(1) & = 2\alpha {\varvec{S}}_{2} (F_{1} (1))T_{1} (1) - \overline{\alpha } E[{\rm X}]{\varvec{S}}_{2} (F_{1} (1))T_{1} (1) + \alpha \varvec{S}^{\prime}_{2} (F_{1} (1))F^{\prime}_{1} (1)T_{1} (1) + \alpha {\varvec{S}}_{2} (F_{1} (1))T^{\prime}_{1} (1) \\ N^{\prime\prime}(1) & = {\varvec{S}}_{2} (F_{1} (1))T_{1} (1)\left\{ {2\alpha - 4\overline{\alpha } E[{\rm X}] - \overline{\alpha } E[{\rm X}^{2} ]} \right\} + 2\left( {2\alpha - \overline{\alpha } E[{\rm X}]} \right)\left\{ {\varvec{S}^{\prime}_{2} (F_{1} (1))F^{\prime}_{1} (1)T_{1} (1) + {\varvec{S}}_{2} (F_{1} (1))T^{\prime}_{1} (1)} \right\} \\ & \quad + \alpha T_{1} (1)\left\{ {\varvec{S}^{\prime\prime}_{2} (F_{1} (1))\left( {F^{\prime}_{1} (1)} \right)^{2} + \varvec{S}^{\prime}_{2} (F_{1} (1))F^{\prime\prime}_{1} (1)} \right\} + 2\alpha \varvec{S}^{\prime}_{2} (F_{1} (1))F^{\prime}_{1} (1)T^{\prime}_{1} (1) + \alpha {\varvec{S}}_{2} (F_{1} (1))T^{\prime\prime}_{1} (1); \\ \end{aligned}$$
$$\pounds^{\prime\prime}_{2} (1) = \frac{{\overline{{\mathcalligra{p}_{2} }} \overline{\mathcalligra{q}} }}{{\left[ {F_{2} (1)} \right]^{3} }}\left[ \begin{aligned} & F_{2} (1)\left[ {F_{2} (1)\left\{ \begin{aligned} & 2\varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F^{\prime}(1) + {\varvec{S}}_{3} (F_{2} (1))F^{\prime\prime}(1) \\ & + \varvec{S}^{\prime\prime}_{3} (F_{2} (1))\left( {F^{\prime}_{2} (1)} \right)^{2} F(1) + \varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime\prime}_{2} (1)F(1) \\ \end{aligned} \right\} - {\varvec{S}}_{3} (F_{2} (1))F^{\prime\prime}_{2} (1)F(1)} \right] \\ & - 2F^{\prime}_{2} (1)\left[ {F_{2} (1)\left\{ {{\varvec{S}}_{3} (F_{2} (1))F^{\prime}(1) + \varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)} \right\} - {\varvec{S}}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)} \right] \\ \end{aligned} \right];$$
$$\begin{aligned} \pounds^{\prime}_{3} (1) & = \pounds^{\prime}_{1} (1)\left\{ {\mathcalligra{q}\left( {\overline{\mathcalligra{w}} + \overline{{\mathcalligra{p}_{1} }} \mathcalligra{w}} \right) + \frac{{\overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))F(1)}}{{F_{2} (1)}}} \right\} - \, \mathcalligra{p}_{1} \Theta \alpha E[{\rm X}] - \, F_{1} (1) - F^{\prime}_{1} (1) \\ & \quad + \pounds_{1} (1)\left[ \begin{aligned} & \mathcalligra{q}\left\{ {\overline{\mathcalligra{w}} \left( {1 - \overline{{\mathcalligra{p}_{1} }} {\text{a}}_{0} } \right)E[{\rm X}] + \left( {1 - {\text{a}}_{0} } \right)\overline{{\mathcalligra{p}_{1} }} \mathcalligra{w} + \overline{{\mathcalligra{p}_{1} }} \mathcalligra{w}E[{\rm X}]} \right\} + \frac{{\overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))F(1)\left( {1 - \overline{{\mathcalligra{p}_{2} }} {\text{a}}_{0} } \right)E[{\rm X}]}}{{F_{2} (1)}} \\ & + \, \frac{{\overline{\mathcalligra{q}} \left[ {F_{2} (1)\left\{ {{\varvec{S}}_{3} (F_{2} (1))F^{\prime}(1) + \varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)} \right\} - {\varvec{S}}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)} \right]}}{{\left[ {F_{2} (1)} \right]^{2} }} \\ \end{aligned} \right]; \\ \end{aligned}$$
$$NR^{\prime}(1) = \alpha {\varvec{S}}_{1} (F_{1} (1))\left[ {\left( {1 - A(\overline{{\mathcalligra{p}_{0} }} )} \right)E[{\rm X}] - 1} \right];$$
$$\begin{aligned} NR^{\prime\prime}(1) & = 2\left\{ {\alpha \varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime}_{1} (1) - \overline{\alpha } E[{\rm X}]{\varvec{S}}_{1} (F_{1} (1))} \right\}\left[ {\left( {1 - A(\overline{{\mathcalligra{p}_{0} }} )} \right)E[{\rm X}] - 1} \right] \\ & \quad + \, \alpha {\varvec{S}}_{1} (F_{1} (1))\left( {1 - A(\overline{{\mathcalligra{p}_{0} }} )} \right)E[{\rm X}^{2} ]; \\ \end{aligned}$$
$$\begin{aligned} \pounds^{\prime\prime}_{3} (1) \, & = \, \pounds^{\prime\prime}_{1} (1)\left\{ {\mathcalligra{q}\left( {\overline{\mathcalligra{w}} + \overline{{\mathcalligra{p}_{1} }} \mathcalligra{w}} \right) + \frac{{\overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))F(1)}}{{F_{2} (1)}}} \right\} \\ & \quad + \, 2\pounds^{\prime}_{1} (1)\left[ \begin{gathered} \mathcalligra{q}\left\{ {\overline{\mathcalligra{w}} \left( {1 - \overline{{\mathcalligra{p}_{1} }} {\text{a}}_{0} } \right)E[{\rm X}] + \left( {1 - {\text{a}}_{0} } \right)\overline{{\mathcalligra{p}_{1} }} \mathcalligra{w} + \overline{{\mathcalligra{p}_{1} }} \mathcalligra{w}E[{\rm X}]} \right\} + \frac{{\overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))F(1)\left( {1 - \overline{{\mathcalligra{p}_{2} }} {\text{a}}_{0} } \right)E[{\rm X}]}}{{F_{2} (1)}} \hfill \\ + \, \frac{{\overline{\mathcalligra{q}} \left[ {F_{2} (1)\left\{ {{\varvec{S}}_{3} (F_{2} (1))F^{\prime}(1) + \varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)} \right\} - {\varvec{S}}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)} \right]}}{{\left[ {F_{2} (1)} \right]^{2} }} \, \hfill \\ \end{gathered} \right] \\ & \quad - \, \mathcalligra{p}_{1} \Theta \alpha E[{\rm X}^{2} ] - \, 2F^{\prime}_{1} (1) - F^{\prime\prime}_{1} (1) \\ & \quad + \, \pounds_{1} (1)\left[ \begin{aligned} & \mathcalligra{q}\left\{ {\overline{\mathcalligra{w}} \left( {1 - \overline{{\mathcalligra{p}_{1} }} {\text{a}}_{0} } \right)E[{\rm X}^{2} ] + 2\left( {1 - {\text{a}}_{0} } \right)\overline{{\mathcalligra{p}_{1} }} \mathcalligra{w}E[{\rm X}] + \overline{{\mathcalligra{p}_{1} }} \mathcalligra{w}E[{\rm X}^{2} ]} \right\} \\ & + \frac{{2\overline{\mathcalligra{q}} \left( {1 - \overline{{\mathcalligra{p}_{2} }} {\text{a}}_{0} } \right)E[{\rm X}]\left[ {F_{2} (1)\left\{ {{\varvec{S}}_{3} (F_{2} (1))F^{\prime}(1) + \varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)} \right\} - {\varvec{S}}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)} \right]}}{{\left[ {F_{2} (1)} \right]^{2} }} \\ \end{aligned} \right] \\ & \quad + \, \frac{{\overline{\mathcalligra{q}} \pounds_{1} (1)\left[ \begin{aligned} & F_{2} (1)\left[ \begin{aligned} & F_{2} (1)\left\{ \begin{aligned} & 2\varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F^{\prime}(1) + {\varvec{S}}_{3} (F_{2} (1))F^{\prime\prime}(1) \\ & + \varvec{S}^{\prime\prime}_{3} (F_{2} (1))\left( {F^{\prime}_{2} (1)} \right)^{2} F(1) + \varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime\prime}_{2} (1)F(1) \\ \end{aligned} \right\} \\ & - {\varvec{S}}_{3} (F_{2} (1))F^{\prime\prime}_{2} (1)F(1) \\ \end{aligned} \right] \\ & - 2F^{\prime}_{2} (1)\left[ {F_{2} (1)\left\{ {{\varvec{S}}_{3} (F_{2} (1))F^{\prime}(1) + \varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)} \right\} - {\varvec{S}}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)} \right] \\ \end{aligned} \right]}}{{\left[ {F_{2} (1)} \right]^{3} }}; \\ \end{aligned}$$
$$\xi (1) = \mathcalligra{p}_{1} \Theta \alpha {\varvec{S}}_{2} (F_{1} (1));\quad \xi_{1} (1) = 1 - \mathcalligra{p}_{1} \Theta \alpha \left( {1 - {\varvec{S}}_{2} (F_{1} (1))} \right);$$
$$\xi^{\prime}_{1} (1) = \, 1 - \mathcalligra{p}_{1} \Theta \alpha + \mathcalligra{p}_{1} \overline{\alpha } \Theta E[{\rm X}] + \mathcalligra{p}_{1} \Theta \alpha \left( {E[{\rm X}]{\varvec{S}}_{2} (F_{1} (1)) \, + \varvec{S}^{\prime}_{2} (F_{1} (1))F^{\prime}_{1} (1)} \right);$$
$$\xi^{\prime}_{2} (1) = \mathcalligra{p}_{1} \Theta \left( {\alpha - E[{\rm X}]} \right);\quad \xi^{\prime\prime}_{2} (1) = - \mathcalligra{p}_{1} \Theta \left( {2\overline{\alpha } E[{\rm X}] + E[{\rm X}^{2} ]} \right);$$
$$\xi^{\prime}(1) = \, \xi_{1} (1)\xi^{\prime}_{2} (1) + \, \mathcalligra{p}_{1} \Theta \alpha \left[ {\left( {E[{\rm X}] + 1 - \mathcalligra{p}_{1} \Theta \alpha + \mathcalligra{p}_{1} \Theta E[{\rm X}]} \right){\varvec{S}}_{2} (F_{1} (1)) + \varvec{S}^{\prime}_{2} (F_{1} (1))F^{\prime}_{1} (1)} \right];$$
$$\begin{aligned} \xi^{\prime\prime}(1) & = \, 2\xi^{\prime}_{1} (1)\xi^{\prime}_{2} (1) + \, \xi^{\prime}_{1} (1)\xi^{\prime\prime}_{2} (1) \, \\ & + \, \mathcalligra{p}_{1} \Theta \alpha \left[ \begin{aligned} & \left\{ {E[X^{2} ]\left( {1 + \mathcalligra{p}_{1} \Theta } \right) + 2E[{\rm X}]\left( {1 - \mathcalligra{p}_{1} \left( {\alpha - \overline{\alpha } } \right)\Theta + \mathcalligra{p}_{1} \Theta E[{\rm X}]} \right)} \right\}{\varvec{S}}_{2} (F_{1} (1)) \, + \, \varvec{S}^{\prime}_{2} (F_{1} (1))F^{\prime\prime}_{1} (1) \\ & + \, 2\left( {E[{\rm X}] + 1 - \mathcalligra{p}_{1} \Theta \alpha + \mathcalligra{p}_{1} \Theta E[{\rm X}]} \right)\varvec{S}^{\prime}_{2} (F_{1} (1))F^{\prime}_{1} (1) \, + \varvec{S}^{\prime\prime}_{2} (F_{1} (1))\left( {F^{\prime}_{1} (1)} \right)^{2} \, \\ \end{aligned} \right]; \\ \end{aligned}$$
$$NR^{\prime}_{3} (1) = \xi^{\prime}(1) - \, \mathcalligra{p}_{1} \Theta \alpha \varvec{S}^{\prime}_{2} (F_{1} (1))F^{\prime}_{1} (1) - \, \mathcalligra{p}_{1} \Theta {\varvec{S}}_{2} (F_{1} (1))\left[ {\alpha - \overline{\alpha } E[{\rm X}] + \alpha \left\{ {\mathcalligra{p}_{1} \Theta E[{\rm X}] + F_{1} (1)} \right\}} \right];$$
$$\begin{aligned} NR^{\prime\prime}_{3} (1) & = \xi^{\prime\prime}(1) - \, \mathcalligra{p}_{1} \Theta \alpha \left[ {\varvec{S}^{\prime}_{2} (F_{1} (1))F^{\prime\prime}_{1} (1) + \varvec{S}^{\prime\prime}_{2} (F_{1} (1))\left( {F^{\prime}_{1} (1)} \right)^{2} } \right] \\ & \quad - \, 2\mathcalligra{p}_{1} \Theta \varvec{S}^{\prime}_{2} (F_{1} (1))F^{\prime}_{1} (1)\left[ {\alpha - \overline{\alpha } E[{\rm X}] + \alpha \left\{ {\mathcalligra{p}_{1} \Theta E[{\rm X}] + F_{1} (1)} \right\}} \right] \\ & \quad - \, \mathcalligra{p}_{1} \Theta {\varvec{S}}_{2} (F_{1} (1))\left[ \begin{aligned} & - 2\overline{\alpha } E[{\rm X}] - \overline{\alpha } E[{\rm X}^{2} ] + \alpha \left( {\mathcalligra{p}_{1} \Theta E[{\rm X}^{2} ] + 2F^{\prime}_{1} (1)} \right) \\ & + \, 2\left( {\mathcalligra{p}_{1} \Theta E[{\rm X}] + F_{1} (1)} \right)\left( {\alpha - \overline{\alpha } E[{\rm X}]} \right) \\ \end{aligned} \right]; \\ \end{aligned}$$
$$\begin{aligned} DR^{\prime}(1) & = \left\{ {\alpha \varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime}_{1} (1) - \overline{\alpha } E[{\rm X}]{\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right)} \right\}\left[ {\pounds(1) + \pounds_{1} \left( 1 \right)\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))\frac{F(1)}{{F_{2} (1)}}} \right)} \right] \\ & \quad + \, \alpha {\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right)\left[ \begin{aligned} & \pounds^{\prime}(1) + \pounds(1)E[{\rm X}]\left( {1 - A(\overline{{\mathcalligra{p}_{0} }} )} \right) + \pounds^{\prime}_{1} \left( 1 \right)\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))\frac{F(1)}{{F_{2} (1)}}} \right) \\ & + \, \pounds_{1} \left( 1 \right)\left\{ \begin{aligned} & E[{\rm X}]\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))\frac{F(1)}{{F_{2} (1)}}} \right) + \overline{{\mathcalligra{p}_{1} }} {\text{a}}_{0} \mathcalligra{q}\mathcalligra{w}\left( {E[{\rm X}] - 1} \right) \\ & + \mathcalligra{p}_{2} \overline{\mathcalligra{q}} \left\{ {\frac{{F_{2} (1)\left( {\varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1) + {\varvec{S}}_{3} (F_{2} (1))F^{\prime}(1)} \right) - {\varvec{S}}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)}}{{\left( {F_{2} (1)} \right)^{2} }}} \right\} \\ \end{aligned} \right\} \\ \end{aligned} \right] \\ & \quad - \, \mathcalligra{p}_{1} \Theta \alpha \left[ {\alpha E[{\rm X}]{\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right) + \alpha \varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime}_{1} (1) - \overline{\alpha } E[{\rm X}]{\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right)} \right] - \alpha F^{\prime}_{1} (1){\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right) \\ & \quad - \, F_{1} \left( 1 \right)\left[ {\alpha - E[{\rm X}] + \alpha E[{\rm X}]{\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right) + \alpha \varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime}_{1} (1)} \right]; \\ \end{aligned}$$
$$\begin{aligned} NR^{\prime\prime}_{2} (1) & = \left[ {2\alpha \left( {1 - E[{\rm X}]} \right) - {\text{a}}_{0} \alpha \left( {1 - 2E[{\rm X}]} \right) - \overline{\alpha } E[{\rm X}]\left( {1 - {\text{a}}_{0} } \right)} \right]DR^{\prime}(1) \\ & \quad + \left[ {\left( {1 - {\text{a}}_{0} } \right)\left( {\alpha - E[{\rm X}]} \right) - \alpha E[{\rm X}]\left( {A(\overline{{\mathcalligra{p}_{0} }} ) - {\text{a}}_{0} } \right)} \right]NR^{\prime}_{1} (1) \\ & \quad + \alpha {\varvec{S}}_{1} (F_{1} (1))\left\{ {\alpha \pounds^{\prime}_{3} (1) - \left( {1 - {\text{a}}_{0} } \right)\left( {\alpha - E[{\rm X}]} \right)\pounds(1)} \right\}\left[ {\left( {1 - A(\overline{{\mathcalligra{p}_{0} }} )} \right)E[{\rm X}] - 1} \right]; \\ \end{aligned}$$
$$\begin{aligned} NR^{\prime\prime\prime}_{2} (1) & = \left[ {2\left( {\alpha - \overline{\alpha } E[{\rm X}]} \right) - \left( {1 - {\text{a}}_{0} } \right)\left\{ {2E[{\rm X}]\left( {\overline{\alpha } + \alpha E[{\rm X}]} \right) + E[{\rm X}^{2} ]\left( {\overline{\alpha } + 2\alpha } \right)} \right\}} \right]DR^{\prime}(1) \\ & \quad + \left[ {2\alpha \left( {1 - E[{\rm X}]} \right) - {\text{a}}_{0} \alpha \left( {1 - 2E[{\rm X}]} \right) - \overline{\alpha } E[{\rm X}]\left( {1 - {\text{a}}_{0} } \right)} \right]DR^{\prime\prime}(1) \\ & \quad + \left[ \begin{aligned} & \left( {A(\overline{{\mathcalligra{p}_{0} }} ) - {\text{a}}_{0} } \right)\left\{ { - 2E[{\rm X}] - \left( {1 + \alpha } \right)E[{\rm X}^{2} ] + 2\alpha E[{\rm X}]\left( {1 - E[{\rm X}]} \right)} \right\} \\ & + \left( {1 - A(\overline{{\mathcalligra{p}_{0} }} )} \right)\left\{ {2E[{\rm X}]\left( {\alpha - E[{\rm X}]} \right) - E[{\rm X}^{2} ] + \overline{\alpha } E[{\rm X}]} \right\} \\ \end{aligned} \right]NR^{\prime}_{1} (1) \\ & \quad + \left[ {\left( {1 - {\text{a}}_{0} } \right)\left( {\alpha - E[{\rm X}]} \right) - \alpha E[{\rm X}]\left( {A(\overline{{\mathcalligra{p}_{0} }} ) - {\text{a}}_{0} } \right)} \right]NR^{\prime\prime}_{1} (1) \\ & \quad + \, \left[ {\alpha \pounds^{\prime\prime}_{3} (1) - 2\overline{\alpha } E[{\rm X}]\pounds^{\prime}_{3} (1) - \left( {1 - {\text{a}}_{0} } \right)\left\{ \begin{aligned} & \left( {\alpha - E[{\rm X}]} \right)\left( {3\pounds(1)E[{\rm X}] + 2\pounds^{\prime}(1)} \right) \\ & + \pounds(1)\left( {E[{\rm X}^{2} ] + 2\overline{\alpha } E[{\rm X}]} \right) \\ \end{aligned} \right\}} \right]NR^{\prime}(1) \\ & \quad + \, \left[ {\alpha \pounds^{\prime}_{3} (1) - \left( {1 - {\text{a}}_{0} } \right)\pounds(1)\left( {\alpha - E[{\rm X}]} \right)} \right]NR^{\prime\prime}(1); \\ \end{aligned}$$
$$\begin{aligned} NR^{\prime}_{1} (1) & = \mathcalligra{p}_{1} \Theta \alpha \left[ {\alpha E[{\rm X}]{\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right) + \alpha \varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime}_{1} (1) - \overline{\alpha } E[{\rm X}]{\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right)} \right] + \, \alpha F^{\prime}_{1} (1){\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right) \\ & \quad + \, F_{1} \left( 1 \right)\left[ {\alpha - E[{\rm X}] + \alpha E[{\rm X}]{\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right) + \alpha \varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime}_{1} (1)} \right] \\ & \quad - \, \left\{ {\alpha \varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime}_{1} (1) - \overline{\alpha } E[{\rm X}]{\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right)} \right\}\left[ {\pounds(1) + \pounds_{1} \left( 1 \right)\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))\frac{F(1)}{{F_{2} (1)}}} \right)} \right] \\ & \quad - \, \alpha {\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right)\left[ \begin{aligned} & \pounds^{\prime}(1) + \pounds(1) + \pounds^{\prime}_{1} \left( 1 \right)\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))\frac{F(1)}{{F_{2} (1)}}} \right) \\ & + \, \pounds_{1} \left( 1 \right)\left\{ \begin{aligned} & E[{\rm X}]\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))\frac{F(1)}{{F_{2} (1)}}} \right) + \overline{{\mathcalligra{p}_{1} }} {\text{a}}_{0} \mathcalligra{q}\mathcalligra{w}\left( {E[{\rm X}] - 1} \right) \\ & + \mathcalligra{p}_{2} \overline{\mathcalligra{q}} \left\{ {\frac{{F_{2} (1)\left( {\varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1) + {\varvec{S}}_{3} (F_{2} (1))F^{\prime}(1)} \right) - {\varvec{S}}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)}}{{\left( {F_{2} (1)} \right)^{2} }}} \right\} \\ \end{aligned} \right\} \\ \end{aligned} \right]; \\ \end{aligned}$$
$$\begin{aligned} DR^{\prime\prime}(1) & = \, \left\{ \begin{aligned} & \alpha \varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime\prime}_{1} (1) + \alpha \varvec{S}^{\prime\prime}_{1} (F_{1} (1))\left( {F^{\prime}_{1} (1)} \right)^{2} \\ & - \, 2\overline{\alpha } E[{\rm X}]\varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime}_{1} (1) - \overline{\alpha } E[{\rm X}^{2} ]{\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right) \\ \end{aligned} \right\}\left[ {\pounds(1) + \pounds_{1} \left( 1 \right)\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \frac{{\mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))F(1)}}{{F_{2} (1)}}} \right)} \right] \\ & \quad + \, 2\left\{ \begin{aligned} & \alpha \varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime}_{1} (1) \\ & - \, \overline{\alpha } E[{\rm X}]{\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right) \\ \end{aligned} \right\}\left[ \begin{aligned} & \pounds^{\prime}(1) + \pounds(1)E[{\rm X}]\left( {1 - A(\overline{{\mathcalligra{p}_{0} }} )} \right) + \pounds_{1} ^{\prime}\left( 1 \right)\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \frac{{\mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))F(1)}}{{F_{2} (1)}}} \right) \\ & + \, \pounds_{1} \left( 1 \right)\left\{ \begin{aligned} & E[{\rm X}]\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \frac{{\mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))F(1)}}{{F_{2} (1)}}} \right) + \overline{{\mathcalligra{p}_{1} }} {\text{a}}_{0} \mathcalligra{q}\mathcalligra{w}\left( {E[{\rm X}] - 1} \right) \\ & + \mathcalligra{p}_{2} \overline{\mathcalligra{q}} \left\{ {\frac{{F_{2} (1)\left( {\varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1) + {\varvec{S}}_{3} (F_{2} (1))F^{\prime}(1)} \right) - {\varvec{S}}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)}}{{\left( {F_{2} (1)} \right)^{2} }}} \right\} \\ \end{aligned} \right\} \\ \end{aligned} \right] \\ \end{aligned}$$
$$\begin{aligned} & \quad + \, \alpha {\varvec{S}}_{3} (F_{2} (1))\left[ \begin{aligned} & \pounds^{\prime\prime}(1) + \pounds(1)E[{\rm X}^{2} ]\left( {1 - A(\overline{{\mathcalligra{p}_{0} }} )} \right) + \left\{ {\pounds^{\prime}_{1} \left( 1 \right) + \pounds^{\prime\prime}_{1} \left( 1 \right)} \right\}\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \frac{{\mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))F(1)}}{{F_{2} (1)}}} \right) \\ & + \, 2\pounds^{\prime}_{1} \left( 1 \right)\left\{ \begin{aligned} & E[{\rm X}]\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \frac{{\mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))F(1)}}{{F_{2} (1)}}} \right) + \overline{{\mathcalligra{p}_{1} }} {\text{a}}_{0} \mathcalligra{q}\mathcalligra{w}\left( {E[{\rm X}] - 1} \right) \\ & + \mathcalligra{p}_{2} \overline{\mathcalligra{q}} \left\{ {\frac{{F_{2} (1)\left( {\varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1) + {\varvec{S}}_{3} (F_{2} (1))F^{\prime}(1)} \right) - {\varvec{S}}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)}}{{\left( {F_{2} (1)} \right)^{2} }}} \right\} \\ \end{aligned} \right\} + 2\pounds^{\prime}(1)\left( {1 - A(\overline{{\mathcalligra{p}_{0} }} )} \right) \\ & + \, \pounds_{1} (1)\left\{ \begin{aligned} & E[{\rm X}^{2} ]\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \frac{{\mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))F(1)}}{{F_{2} (1)}}} \right) + \overline{{\mathcalligra{p}_{1} }} {\text{a}}_{0} \mathcalligra{q}\mathcalligra{w}E[{\rm X}^{2} ] \\ & + \, 2E[{\rm X}]\mathcalligra{p}_{2} \overline{\mathcalligra{q}} \left\{ {\frac{{F_{2} (1)\left( {\varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1) + {\varvec{S}}_{3} (F_{2} (1))F^{\prime}(1)} \right) - {\varvec{S}}_{3} (F_{2} (1))F_{2} ^{\prime}(1)F(1)}}{{\left( {F_{2} (1)} \right)^{2} }}} \right\} \\ \end{aligned} \right\} \\ & + \, \frac{{\mathcalligra{p}_{2} \overline{\mathcalligra{q}} \pounds_{1} (1)}}{{\left( {F_{2} (1)} \right)^{3} }}\left\{ \begin{aligned} & F_{2} (1)\left[ \begin{aligned} & F_{2} (1)\left\{ \begin{aligned} & 2\varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F^{\prime}(1) + {\varvec{S}}_{3} (F_{2} (1))F^{\prime\prime}(1) \\ & + \varvec{S}^{\prime\prime}_{3} (F_{2} (1))\left( {F^{\prime}_{2} (1)} \right)^{2} F(1) + \varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime\prime}_{2} (1)F(1) \\ \end{aligned} \right\} \\ & - {\varvec{S}}_{3} (F_{2} (1))F^{\prime\prime}_{2} (1)F(1) \\ \end{aligned} \right] \\ & - 2F^{\prime}_{2} (1)\left[ {F_{2} (1)\left( {\varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1) + {\varvec{S}}_{3} (F_{2} (1))F^{\prime}(1)} \right) - {\varvec{S}}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)} \right] \\ \end{aligned} \right\} \\ \end{aligned} \right] \\ & \quad - \, \mathcalligra{p}_{1} \Theta \alpha \left[ \begin{aligned} & \left( {\alpha - \overline{\alpha } } \right)E[{\rm X}^{2} ]{\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right) + 2\left( {\alpha - \overline{\alpha } } \right)E[{\rm X}]\varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime}_{1} (1) \\ & - 2\overline{\alpha } \left( {E[{\rm X}]} \right)^{2} {\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right) + \alpha \varvec{S}^{\prime\prime}_{1} (F_{1} (1))\left( {F^{\prime}_{1} (1)} \right)^{2} + \alpha \varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime\prime}_{1} (1) \\ \end{aligned} \right] \, - \, \alpha F^{\prime\prime}_{1} (1){\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right) \, \\ & \quad - \, 2F^{\prime}_{1} (1)\left[ {\alpha \left( {1 - E[{\rm X}]} \right) - E[{\rm X}]\left( {\overline{\alpha } - \alpha {\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right)} \right) + \alpha \varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime}_{1} (1)} \right] \\ & \quad - \, F\left( 1 \right)\left[ \begin{aligned} & \alpha \varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime\prime}_{1} (1) + \alpha \varvec{S}^{\prime\prime}_{1} (F_{1} (1))\left( {F^{\prime}_{1} (1)} \right)^{2} - 2\overline{\alpha } E[{\rm X}] \\ & + 2\alpha E[{\rm X}]\varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime}_{1} (1) - E[{\rm X}^{2} ]\left\{ {1 + \alpha {\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right)} \right\} \\ \end{aligned} \right]; \\ \end{aligned}$$
$$\begin{aligned} NR^{\prime\prime}_{1} (1) & = \, \mathcalligra{p}_{1} \Theta \alpha \left[ \begin{aligned} & \left( {\alpha - \overline{\alpha } } \right)E[{\rm X}^{2} ]{\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right) + 2\left( {\alpha - \overline{\alpha } } \right)E[{\rm X}]\varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime}_{1} (1) \\ & - 2\overline{\alpha } \left( {E[{\rm X}]} \right)^{2} {\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right) + \alpha \varvec{S}^{\prime\prime}_{1} (F_{1} (1))\left( {F^{\prime}_{1} (1)} \right)^{2} + \alpha \varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime\prime}_{1} (1) \\ \end{aligned} \right] \\ & \quad + \, \alpha F^{\prime\prime}_{1} (1){\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right) \, + \, 2F^{\prime}_{1} (1)\left[ {\alpha \left( {1 - E[{\rm X}]} \right) - E[{\rm X}]\left( {\overline{\alpha } - \alpha {\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right)} \right) + \alpha \varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime}_{1} (1)} \right] \\ & \quad + \, F_{1} \left( 1 \right)\left[ \begin{aligned} & \alpha \varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime\prime}_{1} (1) + \alpha \varvec{S}^{\prime\prime}_{1} (F_{1} (1))\left( {F^{\prime}_{1} (1)} \right)^{2} - 2\overline{\alpha } E[{\rm X}] \\ & + 2\alpha E[{\rm X}]\varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime}_{1} (1) - E[{\rm X}^{2} ]\left\{ {1 + \alpha {\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right)} \right\} \\ \end{aligned} \right] \\ & \quad - \, \left\{ \begin{aligned} & \alpha \varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime\prime}_{1} (1) + \alpha \varvec{S}^{\prime\prime}_{1} (F_{1} (1))\left( {F^{\prime}_{1} (1)} \right)^{2} \\ & - 2\overline{\alpha } E[{\rm X}]\varvec{S}^{\prime}_{1} (F_{1} (1))F^{\prime}_{1} (1) - \overline{\alpha } E[{\rm X}^{2} ]{\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right) \\ \end{aligned} \right\}\left[ {\pounds(1) + \pounds_{1} \left( 1 \right)\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \frac{{\mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))F(1)}}{{F_{2} (1)}}} \right)} \right] \\ \end{aligned}$$
$$\begin{aligned} & \quad - \, 2\left\{ \begin{gathered} \alpha \varvec{S}^{\prime}_{1} (F_{1} (1))F_{1} ^{\prime}(1) \hfill \\ - \overline{\alpha } E[{\rm X}]{\varvec{S}}_{1} \left( {F_{1} \left( 1 \right)} \right) \hfill \\ \end{gathered} \right\}\left[ \begin{aligned} & \pounds^{\prime}(1) + \pounds(1) + \pounds^{\prime}_{1} \left( 1 \right)\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \frac{{\mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))F(1)}}{{F_{2} (1)}}} \right) \\ & + \, \pounds_{1} \left( 1 \right)\left\{ \begin{gathered} E[{\rm X}]\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \frac{{\mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))F(1)}}{{F_{2} (1)}}} \right) + \overline{{\mathcalligra{p}_{1} }} {\text{a}}_{0} \mathcalligra{q}\mathcalligra{w}\left( {E[{\rm X}] - 1} \right) \hfill \\ + \mathcalligra{p}_{2} \overline{\mathcalligra{q}} \left\{ {\frac{{F_{2} (1)\left( {\varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1) + {\varvec{S}}_{3} (F_{2} (1))F^{\prime}(1)} \right) - {\varvec{S}}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)}}{{\left( {F_{2} (1)} \right)^{2} }}} \right\} \hfill \\ \end{gathered} \right\} \\ \end{aligned} \right] \\ & \quad - \, \alpha {\varvec{S}}_{3} (F_{2} (1))\left[ \begin{aligned} & 2\pounds^{\prime}(1) + \pounds^{\prime\prime}(1) + \pounds^{\prime}_{1} \left( 1 \right)\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \frac{{\mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))F(1)}}{{F_{2} (1)}}} \right) + \pounds^{\prime\prime}_{1} \left( 1 \right)\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \frac{{\mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))F(1)}}{{F_{2} (1)}}} \right) \\ & + \, 2\pounds_{1} ^{\prime}\left( 1 \right)\left\{ \begin{gathered} E[{\rm X}]\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \frac{{\mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))F(1)}}{{F_{2} (1)}}} \right) + \overline{{\mathcalligra{p}_{1} }} {\text{a}}_{0} \mathcalligra{q}\mathcalligra{w}\left( {E[{\rm X}] - 1} \right) \hfill \\ + \mathcalligra{p}_{2} \overline{\mathcalligra{q}} \left\{ {\frac{{F_{2} (1)\left( {\varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1) + {\varvec{S}}_{3} (F_{2} (1))F^{\prime}(1)} \right) - {\varvec{S}}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)}}{{\left( {F_{2} (1)} \right)^{2} }}} \right\} \hfill \\ \end{gathered} \right\} \\ & + \, \pounds_{1} (1)\left\{ \begin{gathered} E[{\rm X}^{2} ]\left( {\mathcalligra{p}_{1} \overline{\mathcalligra{w}} \mathcalligra{q} + \frac{{\mathcalligra{p}_{2} \overline{\mathcalligra{q}} {\varvec{S}}_{3} (F_{2} (1))F(1)}}{{F_{2} (1)}}} \right) + \overline{{\mathcalligra{p}_{1} }} {\text{a}}_{0} \mathcalligra{q}\mathcalligra{w}E[{\rm X}^{2} ] \hfill \\ + \, 2E[{\rm X}]\mathcalligra{p}_{2} \overline{\mathcalligra{q}} \left\{ {\frac{{F_{2} (1)\left( {\varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1) + {\varvec{S}}_{3} (F_{2} (1))F^{\prime}(1)} \right) - {\varvec{S}}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)}}{{\left( {F_{2} (1)} \right)^{2} }}} \right\} \hfill \\ \end{gathered} \right\} \\ & + \, \frac{{\mathcalligra{p}_{2} \overline{\mathcalligra{q}} \pounds_{1} (1)}}{{\left( {F_{2} (1)} \right)^{3} }}\left\{ \begin{gathered} F_{2} (1)\left[ \begin{aligned} & F_{2} (1)\left\{ \begin{aligned} & 2\varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F^{\prime}(1) + {\varvec{S}}_{3} (F_{2} (1))F^{\prime\prime}(1) \\ & + \varvec{S}^{\prime\prime}_{3} (F_{2} (1))\left( {F^{\prime}_{2} (1)} \right)^{2} F(1) + \varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime\prime}_{2} (1)F(1) \\ \end{aligned} \right\} \\ & - {\varvec{S}}_{3} (F_{2} (1))F^{\prime\prime}_{2} (1)F(1) \\ \end{aligned} \right] \hfill \\ - 2F^{\prime}_{2} (1)\left[ {F_{2} (1)\left( {\varvec{S}^{\prime}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1) + {\varvec{S}}_{3} (F_{2} (1))F^{\prime}(1)} \right) - {\varvec{S}}_{3} (F_{2} (1))F^{\prime}_{2} (1)F(1)} \right] \hfill \\ \end{gathered} \right\} \\ \end{aligned} \right]; \\ \end{aligned}$$

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Shree, V., Upadhyaya, S. & Kulshrestha, R. Cost scrutiny of discrete-time priority queue with cluster arrival and Bernoulli feedback. OPSEARCH (2024). https://doi.org/10.1007/s12597-024-00742-8

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