Dynamical modelling and cost optimization of a 5G base station for energy conservation using feedback retrial queue with sleeping strategy

Abstract

Dense network deployment is now being evaluated as one of the viable solutions to meet the capacity and connectivity requirements of the fifth-generation (5G) cellular system. The goal of 5G cellular networks is to offer clients with faster download speeds, lower latency, more dependability, broader network capacities, more accessibility, and a seamless client experience. However, one of the many obstacles that will need to be overcome in the 5G era is the issue of energy usage. For energy efficiency in 5G cellular networks, researchers have been studying at the sleeping strategy of base stations. In this regard, this study models a 5G BS as an $M^{[X]}/G/1$ feedback retrial queue with a sleeping strategy to reduce average power consumption and conserve power in 5G mobile networks. The probability-generating functions and steady-state probabilities for various BS states were computed employing the supplementary variable approach. In addition, an extensive palette of performance metrics have been determined. Then, with the aid of graphs and tables, the resulting metrics are conceptualized and verified. Further, this research is accelerated in order to bring about the best possible (optimal) cost for the system by adopting a range of optimization approaches namely particle swarm optimization, artificial bee colony and genetic algorithm.

Appendices

Appendix A

In the case where $\rho < 1$, the embedded Markov chain (MC) $\{F_n;n\in N\}$ can be said to be ergodic, where $\rho = {\mathcal {E}}(X_1)[1-{\mathcal {Q}}^*(\nu )]+r-\nu {\mathcal {E}}(X_1)[1+\eta {\mathcal {E}}(S^1)]{\mathcal {E}}(L^1)$

Proof

It is straightforward to verify that ergodicity is a sufficient condition by applying Foster’s criteria [29], which points out that the chain might be irreducible and aperiodic. If a non-negative fn. c(d),  $d\in N$ and $\epsilon >0,$ exists that guarantees that the mean drift $\varsigma _d=H[c(f_ {n+1})-c(f_n)/f_n=d]$ is limited for all $d\in N$ and $d\in N$ and $\varsigma _d \le -\epsilon$, excluding possibly for a finite no. of $d'$s, then the MC is ergodic. When considering the function $c(d)=d$ in our scenario, we get

$$\begin{aligned} \varsigma _d= {\left\{ \begin{array}{ll} r-\nu {\mathcal {E}}(X_1)[1+\eta {\mathcal {E}}(S^1)]{\mathcal {E}}(L^1), ~~\text {if }c=0 \\ {\mathcal {E}}(X_1)[1-{\mathcal {Q}}^*(\nu )]+r-\nu {\mathcal {E}}(X_1)[1+\eta {\mathcal {E}}(S^1)]\\ \quad {\mathcal {E}}(L^1)-1, ~~~~\text {if }c=1,2,\dots \\ \end{array}\right. } \end{aligned}$$

It is clear that ergodicity must exist in order for the inequality below to exist.

$$\begin{aligned} {\mathcal {E}}(X_1)[1-{\mathcal {Q}}^*(\nu )]+r-\nu {\mathcal {E}}(X_1)[1+\eta {\mathcal {E}}(S^1)]{\mathcal {E}}(L^1) <1 \end{aligned}$$

If the MC $\{F_ n;n\epsilon N\}$ meets Kaplan’s criterion, we may easily ensure non-ergodicity in accordance with Sennott et al. [31] particularly $\varsigma _d<\infty$ for all $d\ge 0$ and $\exists$ $d_0\in N$ s.t $\varsigma _d \ge 0$ for $d\ge d_ 0$. The fact that “Kaplan’s condition”is met in our scenario is underscored by the existence of a w such that $g_{ld}= 0$ for $d<l-m$ and $l>0$, in which $g_{ld}$ is the one-step transition matrix of $\{F_n;n\in N\}$. Consequently, it is indicated that the MC is non-ergodic by

$$\begin{aligned} {\mathcal {E}}(X_1)[1-{\mathcal {Q}}^*(\nu )]+r-\nu {\mathcal {E}}(X_1)[1+\eta {\mathcal {E}}(S^1)]{\mathcal {E}}(L^1) \ge 1 \end{aligned}$$

$\square$

Appendix B

By substituting the eqns. (35), (36) and (33) in (26) and performing some calculations, we eventually arrive at,

(57)

Here, by substituting the eqns. (35) to (39) into the eqns. (27) to (31), we get,

(58)

(59)

(60)

(61)

(62)

Appendix C

$$\begin{aligned} Nr_q^{'''}(1)&=-6 \nu {\mathcal {E}}(X_1)(1+s{\mathcal {E}}(S^1))[r+{\mathcal {E}}(X_1)(1-{\mathcal {Q}}^*(\nu ))\\&\quad -\nu {\mathcal {E}}(X_1)(1+\eta {\mathcal {E}}(S^1)){\mathcal {E}}(L^1)\\&\quad -1] [M^{'}-\nu {\mathcal {E}}(X_1)+\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}[\nu {\mathcal {E}}(X_1)(1+s){\mathcal {E}}(B^1)\\&\quad +\nu s{\mathcal {E}}(X_1){\mathcal {E}}(G_2^1)]]\\&\quad -9M^{'} \nu {\mathcal {E}}(X_1)(1+s{\mathcal {E}}(S^1))\nu {\mathcal {E}}(X_1)(1-{\mathcal {Q}}^*(\nu ))\\&\quad +2\nu {\mathcal {E}}(X_1)(1+\eta {\mathcal {E}}(S^1)){\mathcal {E}}(L^1)[-\nu {\mathcal {E}}(X_1)\\&\quad +\eta {\mathcal {E}}(X_1){\mathcal {E}}(S^1)]]-9[2{\mathcal {E}}(X_1)+U^{'}(1+s)\\&\quad +\frac{(1+s)}{{\mathcal {G}}_{1,0}^*(\nu )}-\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}[\nu {\mathcal {E}}(X_1) (1+s){\mathcal {E}}(B^1)\\&\quad +\nu s{\mathcal {E}}(X_1){\mathcal {E}}(G_2^1)]][2(1-{\mathcal {Q}}^*(\nu )) \nu {\mathcal {E}}(X_1)(1\\&\quad +s{\mathcal {E}}(S^1))\nu {\mathcal {E}}(X_1)+\nu [2{\mathcal {E}}(X_1)(1\\&\quad +s{\mathcal {E}}(S^1))\nu {\mathcal {E}}(L^1)[-\nu {\mathcal {E}}(X_1)+\eta {\mathcal {E}}(X_1){\mathcal {E}}(S^1)]]]\\ Nr_q^{''''}(1)&=-6[(\nu {\mathcal {E}}(X_1))^2(1+s{\mathcal {E}}(S^1))][(1-{\mathcal {Q}}^*(\nu ))[M^{''}\\&\quad -4M^{'}+4r[2{\mathcal {E}}(X_1)+U^{'} (1+s)+\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}(1+s)\\&\quad -\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}[\nu (1+s){\mathcal {E}}(X_1){\mathcal {E}}(B^1)\\&\quad +\nu s {\mathcal {E}}(X_1){\mathcal {E}}(G_2^1)]]] -12[2{\mathcal {E}}(X_2)\\&\quad +2{\mathcal {E}}(X_1)+U^{''}(1+s) -2[\nu (1+s){\mathcal {E}}(X_1){\mathcal {E}}(B^1)\\&\quad +\nu s {\mathcal {E}}(X_1) {\mathcal {E}}(G_2^1)][U^{'}+\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}]+2U^{'}(1+s)\\&\quad -\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}[(1+s){\mathcal {B}}^{*''}(1)+s{\mathcal {G}}_2^{*''}(1)]\\&\quad +2\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}\nu (1+s){\mathcal {E}}(X_1){\mathcal {E}}(B^1)\nu s {\mathcal {E}}(X_1){\mathcal {E}}(G_2^1)]] \\&\quad -12\nu [M^{'}+[2{\mathcal {E}}(X_1)+U^{'} (1+s) +\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}(1+s)\\&\quad -\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}[\nu (1+s){\mathcal {E}}(X_1){\mathcal {E}}(B^1) +\nu s {\mathcal {E}}(X_1){\mathcal {E}}(G_2^1)]]]\\ \end{aligned}$$

$$\begin{aligned}&\quad [(1-{\mathcal {Q}}^*(\nu )){\mathcal {E}}(X_1){\mathcal {T}}^{''}(1)\\&\quad +{\mathcal {L}}^{*''}(1)[-\nu {\mathcal {E}}(X_1)+\eta \nu {\mathcal {E}}(X_1){\mathcal {E}}(S^1)]]\\&\quad -12[M^{'} +\nu [2{\mathcal {E}}(X_1)+U^{'}(1+s)(U^{'} +\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )})\\&\quad -\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}[\nu (1+s){\mathcal {E}}(X_1){\mathcal {E}}(B^1)\\&\quad +\nu s {\mathcal {E}}(X_1){\mathcal {E}}(G_2^1)]]]\nu {\mathcal {E}}(X_1)(1+\eta {\mathcal {E}}(S^1))({\mathcal {E}}(L^1)\\&\quad +(1-{\mathcal {Q}}^*(\nu ))\nu {\mathcal {E}}(X_2))\\&\quad -12\nu {\mathcal {E}}(X_1) (1+\eta {\mathcal {E}}(S^1)){\mathcal {E}}(L^1)[[-\nu {\mathcal {E}}(X_1)\\&\quad +\eta \nu {\mathcal {E}}(X_1){\mathcal {E}}(S^1)][\nu [2{\mathcal {E}}(X_2)\\&\quad +2{\mathcal {E}}(X_1)+U^{''} (1+s)-2[\nu (1+s){\mathcal {E}}(X_1){\mathcal {E}}(B^1)\\&\quad +\nu s {\mathcal {E}}(X_1){\mathcal {E}}(G_2^1)][U^{'} +\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}]\\&\quad +2U^{'}(1+s) -\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}[(1+s){\mathcal {B}}^{*''}(1) +s{\mathcal {G}}_2^{*''}(1)]\\&\quad +2\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}\nu (1+s){\mathcal {E}}(X_1){\mathcal {E}}(B^1)\nu s \\&\quad {\mathcal {E}}(X_1){\mathcal {E}}(G_2^1)]\\&\quad +(1-{\mathcal {Q}}^*(\nu ))[M^{''}\nu -12M^{'}\nu {\mathcal {E}}(X_1)]]\\&\quad +12(1-{\mathcal {Q}}^*(\nu ))(1+\eta {\mathcal {E}}(S^1))({\mathcal {E}}(X_1))^2[2{\mathcal {E}}(X_1)\\&\quad +[U^{'}+\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}](1+s)\\&\quad -\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}[\nu (1+s){\mathcal {E}}(X_1) {\mathcal {E}}(B^1) +\nu s{\mathcal {E}}(X_1){\mathcal {E}}(G_2^1)]]]\\&\quad -12[r+{\mathcal {E}}(X_1)(1-{\mathcal {Q}}^*(\nu ))\\&\quad -\nu {\mathcal {E}}(X_1)(1+\eta {\mathcal {E}}(S^1)) {\mathcal {E}}(L^1)-1][ -\nu {\mathcal {E}}(X_1)(1+\eta {\mathcal {E}}(S^1))][M^{''}\\&\quad -\nu {\mathcal {E}}(X_2)+\nu [2U^{'}[\nu (1+s){\mathcal {E}}(X_1){\mathcal {E}}(B^1) +\nu s {\mathcal {E}}(X_1){\mathcal {E}}(G_2^1)]\\&\quad +\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}[(1+s){\mathcal {B}}^{*''}(1)\\&\quad +s {\mathcal {S}}^{*''}(1)]]]+12[M^{'} -\nu {\mathcal {E}}(X_1)\\ \end{aligned}$$

$$\begin{aligned}&\quad +\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}[\nu (1+s){\mathcal {E}}(X_1){\mathcal {E}}(B^1)\\&\quad +\nu s {\mathcal {E}}(X_1){\mathcal {E}}(G_2^1)]][{\mathcal {T}}^{''}(1)(r-1)\\&\quad +\nu {\mathcal {E}}(X_1) (1+\eta {\mathcal {E}}(S^1))[(1-{\mathcal {Q}}^*(\nu )){\mathcal {E}}(X_2) +{\mathcal {L}}^{*''}(1)]\\&\quad -\nu {\mathcal {E}}(X_1)(1+\eta {\mathcal {E}}(S^1)){\mathcal {E}}(L^1)[{\mathcal {T}}^{''}(1)\\&\quad -2\nu {\mathcal {E}}(X_1)(1+\eta {\mathcal {E}}(S^1))[r+(1-{\mathcal {Q}}^*(\nu )){\mathcal {E}}(X_1)]]\\&\quad +(1-{\mathcal {Q}}^*(\nu )){\mathcal {E}}(X_1)[2r\nu {\mathcal {E}}(X_1)(1+\eta {\mathcal {E}}(S^1))\\&\quad +{\mathcal {T}}^{''}(1)]]\\ Dr_q^{'''}(1)&=6\nu {\mathcal {E}}(X_1)(1+\eta {\mathcal {E}}(S^1))\nu {\mathcal {E}}(X_1)[r+{\mathcal {E}}(X_1)(1\\&\quad -{\mathcal {Q}}^*(\nu ))-\nu {\mathcal {E}}(X_1)(1+\eta {\mathcal {E}}(S^1))\\&{\mathcal {E}}(L^1)-1]\\ Dr_q^{''''}(1)&= 12[[r+{\mathcal {E}}(X_1)(1-{\mathcal {Q}}^*(\nu ))\\&\quad -\nu {\mathcal {E}}(X_1)(1+\eta {\mathcal {E}}(S^1)){\mathcal {E}}(L^1)-1][{\mathcal {T}}^{''}(1){\mathcal {E}}(X_1)\\&\quad +{\mathcal {E}}(X_2)\nu {\mathcal {E}}(X_1)(1+\eta {\mathcal {E}}(S^1))]\\&\quad +\nu {\mathcal {E}}(X_1)(1+\eta {\mathcal {E}}(S^1)){\mathcal {E}}(X_1)[2[r{\mathcal {E}}(X_1) (1-{\mathcal {Q}}^*(\nu ))\\&\quad -\nu {\mathcal {E}}(X_1)(1+\eta {\mathcal {E}}(S^1)){\mathcal {E}}(L^1)[r\\&\quad +{\mathcal {E}}(X_1)(1-{\mathcal {Q}}^*(\nu ))]]\\&\quad +{\mathcal {E}}(X_2) (1-{\mathcal {Q}}^*(\nu )) -{\mathcal {T}}^{''}(1)]]\\ Nr_s^{''''}(1)&= Nr_q^{''''}(1)-6 \nu {\mathcal {E}}(X_1)(1+\eta {\mathcal {E}}(S^1)){\mathcal {E}}(L^1)[\nu {\mathcal {E}}(X_1)\\&\quad +\nu {\mathcal {E}}(X_1)\eta {\mathcal {E}}(S^1)][1-4M^{'}] +24M^{'}(\nu {\mathcal {E}}(X_1))^2(1\\&\quad +\eta {\mathcal {E}}(S^1))(1-{\mathcal {Q}}^*(\nu ))\\ {\mathcal {T}}^{''}(1)&= \nu {\mathcal {E}}(X_2)(1+\eta {\mathcal {E}}(S^1)) +(\nu {\mathcal {E}}(X_1))^2\eta {\mathcal {E}}(S^2)\\ {\mathcal {L}}^{*''}(1)&= \nu ({\mathcal {E}}(X_2){\mathcal {E}}(L^1) +{\mathcal {E}}(X_1){\mathcal {E}}(L^2));~ {\mathcal {B}}^{*''}(1) \\&= \nu ({\mathcal {E}}(X_2){\mathcal {E}}(B^1) +{\mathcal {E}}(X_1){\mathcal {E}}(B^2)) \\ {\mathcal {S}}^{*''}(1)&= \nu ({\mathcal {E}}(X_2){\mathcal {E}}(S^1)+{\mathcal {E}}(X_1){\mathcal {E}}(S^2)); ~ {\mathcal {G}}_2^{*''}(1) \\&= \nu ({\mathcal {E}}(X_2){\mathcal {E}}(G_2^1) +{\mathcal {E}}(X_1){\mathcal {E}}(G_2^2)) \\ U^{''}&=\frac{(\nu )^2}{{\mathcal {G}}_{1,0}^*(\nu )}[{\mathcal {E}}(G_{1,0}^2)\\&\quad +2\sum _{m=1}^{M}p_m{\mathcal {E}}(G_{1,m}^1){\mathcal {E}}(G_{1,0}^1) +\sum _{m=1}^{M}p_m{\mathcal {E}}(G_{1,m}^2)]\\ M^{''}&=\frac{\nu }{{\mathcal {G}}_{1,0}^*(\nu )}[{\mathcal {E}}(G_{1,0}^2) +2\sum _{m=1}^{M}p_m{\mathcal {E}}(G_{1,m}^1){\mathcal {E}}(G_{1,0}^1)\\&\quad +\sum _{m=1}^{M}p_m{\mathcal {E}}(G_{1,m}^2)] \end{aligned}$$