Analysis of energy efficiency of small cell base station in 4G/5G networks

Abstract

Base Stations (BSs) sleeping strategy is an efficient way to obtain the energy efficiency of cellular networks. To meet the increasing demand of high-data-rate for wireless applications, small cell BSs provide a promising and feasible approach but that consumes more power. Hence, energy efficiency in small cell BSs is a major issue to be concerned. To get the energy efficiency, in this research work, we have addressed the total power consumption and delay of User Requests (URs) in the small cell as well as 5G small cell BSs with sleeping strategy and N limited scheme. One of the effective ways to reduce the power consumption is introduce BSs sleeping strategy. Here, the small cell BSs are modeled as a M/G/1 queueing system with two different types of sleep modes namely, short sleep mode and long sleep mode. Short sleep mode is consists of maximum M number of short sleep. Here, the steady state probabilities of the small cell BSs in active, short sleep and long sleep modes are derived using supplementary variable technique. The expressions for expected power consumption and expected delay are also obtained. Finally, to get the energy consumption from the small cell BSs, the optimization of expected power consumption and expected delay is presented. The numerical results displays the impact of various parameters on power consumption and delay in small cell BSs. Also, the comparative analysis for the proposed model with the existing model has been presented.

Appendices

Appendix A

Proof of Theorem 1

Since the system does not change over time in steady state, the assumptions $P_{1,n}(x)=\lim _{t\rightarrow \infty } P_{1,n}(x,t),\quad Q^1_{j,n}(x)=\lim _{t\rightarrow \infty }Q^1_{j,n}(x,t), Q^2_{n}(x)= \lim _{t\rightarrow \infty }Q^2_{n}(x,t),$ are taken to obtain the steady state probabilities. Using these assumptions in Eqs. (1) to (11), the steady state equations are obtained as follows:

$$\begin{aligned} -\displaystyle \frac{{d}}{{d}x}P_{1,0}(x)= & {} -\lambda P_{1,0}(x)+P_{1,1}(0)s(x) \end{aligned}$$

(A.1)

$$\begin{aligned} -\displaystyle \frac{{d}}{{d}x}P_{1,n}(x)= & {} -\lambda P_{1,n}(x)+P_{1,{n+1}}(0)s(x)+\lambda P_{1,{n-1}}(x), \nonumber \\&\quad 1\le n\le N-2 \end{aligned}$$

(A.2)

$$\begin{aligned} -\displaystyle \frac{{d}}{{d}x}P_{1,n}(x)= & {} -\lambda P_{1,n}(x)+P_{1,{n+1}}(0)s(x)\nonumber \\&+\lambda P_{1,{n-1}}(x) +\sum _{j=1}^{M}Q^{1}_{j,n+1}(0)s(x)\nonumber \\&+\lambda s(x)\int _{0}^\infty Q^{2}_{n}(y)dy , n= N-1 \end{aligned}$$

(A.3)

$$\begin{aligned} -\displaystyle \frac{{d}}{{d}x}P_{1,n}(x)= & {} -\lambda P_{1,n}(x)+P_{1,{n+1}}(0)s(x)+\lambda P_{1,{n-1}}(x)\nonumber \\&+\sum _{j=1}^{M}Q^{1}_{j,n+1}(0)s(x),\qquad n \ge N. \end{aligned}$$

(A.4)

$$\begin{aligned} -\displaystyle \frac{{d}}{{d}x}Q^{1}_{1,0}(x)= & {} -\lambda Q^{1}_{1,0}(x)+P_{1,0}(0)v_{1}(x) \end{aligned}$$

(A.5)

$$\begin{aligned} -\displaystyle \frac{{d}}{{d}x}Q^{1}_{1,n}(x)= & {} -\lambda Q^{1}_{1,n}(x)+\lambda Q^{1}_{1,n-1}(x), \qquad n \ge 1 \end{aligned}$$

(A.6)

$$\begin{aligned} -\displaystyle \frac{{d}}{{d}x}Q^{1}_{j,0}(x)= & {} -\lambda Q^{1}_{j,0}(x)+Q^{1}_{j-1,0}(0)v_{1}(x), 2 \le j \le M \nonumber \\ \end{aligned}$$

(A.7)

$$\begin{aligned} -\displaystyle \frac{{d}}{{d}x}Q^{1}_{j,n}(x)= & {} -\lambda Q^{1}_{j,n}(x)+Q^{1}_{j-1,n}(0)v_{1}(x)+\lambda Q^{1}_{j,n-1}(x),\nonumber \\&\quad 2 \le j \le M ; 1 \le n \le N-1 \end{aligned}$$

(A.8)

$$\begin{aligned} -\displaystyle \frac{{d}}{{d}x}Q^{1}_{j,n}(x)= & {} -\lambda Q^{1}_{j,n}(x)+\lambda Q^{1}_{j,n-1}(x),\nonumber \\&\quad 2 \le j \le M ; n \ge N \end{aligned}$$

(A.9)

$$\begin{aligned} -\displaystyle \frac{{d}}{{d}x}Q^{2}_{0}(x)= & {} -\lambda Q^{2}_{0}(x)+Q^{1}_{M,0}(0)v_{2}(x) \end{aligned}$$

(A.10)

$$\begin{aligned} -\displaystyle \frac{{d}}{{d}x}Q^{2}_{n}(x)= & {} -\lambda Q^{2}_{n}(x)+Q^{1}_{M,n}(0)v_{2}(x)+\lambda Q^{2}_{n-1}(x),\nonumber \\&\quad 1 \le n \le N-1 \end{aligned}$$

(A.11)

The Laplace-Stieltjes transform of $P_{1,n}(x)$, $Q^{1}_{j,n}(x)$ and $Q^{2}_{n}(x)$ are defined as

$$\begin{aligned}&\widetilde{P}_{1,n}(\theta )= \int _{0}^{\infty }e^{-\theta x} P_{1,n}(x)dx ; \\&\widetilde{Q}^{1}_{j,n}(\theta )= \int _{0}^{\infty }e^{-\theta x}Q^{1}_{j,n}(x)dx \end{aligned}$$

$$\begin{aligned} \widetilde{Q}^{2}_{n}(\theta )= & {} \int _{0}^{\infty }e^{-\theta x}Q^{2}_{n}(x)dx. \end{aligned}$$

Taking Laplace-Stieltjes transform on both sides of the Eqs. (A.1) to (A.11),

$$\begin{aligned}&\theta \widetilde{P}_{1,0}(\theta )-{P}_{1,0}(0)=\lambda \widetilde{P}_{1,0}(\theta )-P_{1,1}(0)\widetilde{S}(\theta ). \end{aligned}$$

(A.12)

$$\begin{aligned}&\theta \widetilde{P}_{1,n}(\theta )-{P}_{1,n}(0)=\lambda \widetilde{P}_{1,n}(\theta )-P_{1,n+1}(0)\widetilde{S}(\theta )\nonumber \\&\qquad -\lambda \widetilde{P}_{1,n-1}(\theta ), 1 \le n \le N-2. \end{aligned}$$

(A.13)

$$\begin{aligned}&\theta \widetilde{P}_{1,n}(\theta )-{P}_{1,n}(0)=\lambda \widetilde{P}_{1,n}(\theta )-P_{1,n+1}(0)\widetilde{S}(\theta )-\lambda \widetilde{P}_{1,n-1}(\theta )\nonumber \\&\quad -\sum _{j=1}^{M}Q^{1}_{j,n+1}(0)\widetilde{S}(\theta ) -\lambda \widetilde{S}(\theta ) \int _{0}^{\infty }Q^{2}_{n}(y)dy. \end{aligned}$$

(A.14)

$$\begin{aligned} \theta \widetilde{P}_{1,n}(\theta )-{P}_{1,n}(0)= & {} \lambda \widetilde{P}_{1,n}(\theta )-P_{1,n+1}(0)\widetilde{S}(\theta )-\lambda \widetilde{P}_{1,n-1}(\theta )\nonumber \\&-\sum _{j=1}^{M}Q^{1}_{j,n+1}(0)\widetilde{S}(\theta ) , \qquad n \ge N. \end{aligned}$$

(A.15)

$$\begin{aligned}&\theta \widetilde{Q}^{1}_{1,0}(\theta )-Q^{1}_{1,0}(0) =\lambda \widetilde{Q}^{1}_{1,0}(\theta )-P_{1,0}(0)\widetilde{V}_{1}(\theta ). \end{aligned}$$

(A.16)

$$\begin{aligned}&\theta \widetilde{Q}^{1}_{1,n}(\theta )-Q^{1}_{1,n}(0) =\lambda \widetilde{Q}^{1}_{1,n}(\theta )-\lambda \widetilde{Q}^{1}_{1,n-1}(\theta ) , n \ge 1. \end{aligned}$$

(A.17)

$$\begin{aligned}&\theta \widetilde{Q}^{1}_{j,0}(\theta )-Q^{1}_{j,0}(0) =\lambda \widetilde{Q}^{1}_{j,0}(\theta )-Q^{1}_{j-1,0}(0)\widetilde{V}_{1}(\theta ),\nonumber \\&\quad 2 \le j \le M. \end{aligned}$$

(A.18)

$$\begin{aligned}&\theta \widetilde{Q}^{1}_{j,n}(\theta )-Q^{1}_{j,n}(0) =\lambda \widetilde{Q}^{1}_{j,n}(\theta )-Q^{1}_{j-1,n}(0)\widetilde{V}_{1}(\theta )\nonumber \\&\quad -\lambda \widetilde{Q}^{1}_{j,n-1}(\theta ), 2 \le j \le M ; 1 \le n \le N-1. \end{aligned}$$

(A.19)

$$\begin{aligned}&\theta \widetilde{Q}^{1}_{j,n}(\theta )-Q^{1}_{j,n}(0) =\lambda \widetilde{Q}^{1}_{j,n}(\theta )\nonumber \\&-\lambda \widetilde{Q}^{1}_{j,n-1}(\theta ), 2 \le j \le M ; n \ge N. \end{aligned}$$

(A.20)

$$\begin{aligned}&\theta \widetilde{Q}^{2}_{0}(\theta )-Q^{2}_{0}(0)=\lambda \widetilde{Q}^{2}_{0}(\theta )- Q^{1}_{M,0}(0)\widetilde{V}_{2}(\theta ). \end{aligned}$$

(A.21)

$$\begin{aligned} \theta \widetilde{Q}^{2}_{n}(\theta )-Q^{2}_{n}(0)= & {} \lambda \widetilde{Q}^{2}_{n}(\theta )- Q^{1}_{M,n}(0)\widetilde{V}_{2}(\theta )- \lambda \widetilde{Q}^{2}_{n-1}(\theta ),\nonumber \\&\quad 1 \le n \le N-1. \end{aligned}$$

(A.22)

To find the steady state probability generating function (PGF) of the number of URs in the queue at an arbitrary time epoch, the following PGFs are defined as:

$$\begin{aligned} \widetilde{P}_1(z,\theta )&=\sum _{n=0}^{\infty }\widetilde{P}_{1,n}(\theta )z^n;\qquad P_1(z,0)&=\sum _{n=0}^{\infty }P_{1,n}(0)z^n \nonumber \\ \widetilde{Q}^{1}_{j}(z,\theta )&=\sum _{n=0}^{\infty } \widetilde{Q}^{1}_{j,n}(\theta ) z^n;\qquad Q^{1}_{j}(z,0)&=\sum _{n=0}^{\infty } Q^{1}_{j,n}(0)z^n \nonumber \\ \widetilde{Q}^{2}_{n}(z,\theta )&=\sum _{n=0}^{N-1}\widetilde{Q}^{2}_{n}(\theta )z^n;\qquad Q^{2}_{n}(z,0)&=\sum _{n=0}^{N-1} Q^{2}_{n}(0)z^n. \nonumber \\ \end{aligned}$$

(A.23)

Multiplying Eq. (A.16) by $z^0$, (A.17) by $z^n,(n=1,2,\ldots )$ and taking summation from 0 to $\infty$ and using Eq. (A.23)

$$\begin{aligned}&\theta \sum _{n=0}^{\infty }\widetilde{Q}^{1}_{1,n}(\theta )z^n-\sum _{n=0}^{\infty }Q^{1}_{1,n}(0)z^n \nonumber \\&\quad =\lambda \sum _{n=0}^{\infty }\widetilde{Q}^{1}_{1,n}(\theta )z^n-\lambda \sum _{n=1}^{\infty }\widetilde{Q}^{1}_{1,{n-1}}(\theta )z^n \nonumber \\&\qquad - P_{1,0}(0)\widetilde{V}_{1}(\theta )\nonumber \\&{[}\theta -\lambda +\lambda z]\widetilde{Q}^{1}_{1}(z,\theta )=Q^{1}_1(z,0)-P_{1,0}(0)\widetilde{V}_{1}(\theta ).\nonumber \\ \end{aligned}$$

(A.24)

Multiplying Eq. (A.18) by $z^0$, (A.19) by $z^n, (n=1,2,\ldots , N-1)$, (A.20) by $z^n (n=N,N+1,\ldots )$ taking summation from 0 to $\infty$ and using Eq. (A.23)

$$\begin{aligned}&\theta \sum _{n=0}^{\infty } \widetilde{Q}^{1}_{j,n}(\theta )z^n-\sum _{n=0}^{\infty }Q^{1}_{j,n}(0)z^n \nonumber \\&\quad = \lambda \sum _{n=0}^{\infty }\widetilde{Q}^{1}_{j,n}(\theta )z^n-\lambda \sum _{n=1}^{\infty }\widetilde{Q}^{1}_{j,n-1}(\theta )z^n\nonumber \\&\qquad +\widetilde{V}_{1}(\theta )\sum _{n=0}^{N-1}\widetilde{Q}^{1}_{j-1,n}(0)z^n.\nonumber \\&\quad {[}\theta -\lambda +\lambda z]\widetilde{Q}^{1}_{j}(z,\theta )=Q^{1}_{j}(z,0)-\widetilde{V}_{1}(\theta )\nonumber \\&\quad \sum _{n=0}^{N-1}\widetilde{Q}^{1}_{j-1,n}(0)z^n, 2 \le j \le M. \end{aligned}$$

(A.25)

Multiplying Eq. (A.21) by $z^0$, (A.22) by $z^n (n=1,2,\ldots , N-1)$ and taking summation from 0 to $N-1$ and using Eq. (A.23)

$$\begin{aligned}&\theta \sum _{n=0}^{N-1}\widetilde{Q}^{2}_{n}(\theta )z^n-\sum _{n=0}^{N-1}Q^{2}_{n}(0)z^n\nonumber \\&\quad =\lambda \sum _{n=0}^{N-1}\widetilde{Q}^{2}_n(\theta )z^n-\lambda \sum _{n=1}^{N-1}\widetilde{Q}^{2}_{n-1}(\theta )z^n\nonumber \\&\quad -\widetilde{V}_{2}(\theta )\sum _{n=0}^{N-1}\widetilde{Q}^{1}_{M,n}(0)z^n.\nonumber \\&\quad {[}\theta -\lambda +\lambda z]\widetilde{Q}^{2}(z,\theta )= Q^{2}(z,0)\nonumber \\&\quad -\widetilde{V}_{2}(\theta )\sum _{n=0}^{N-1}Q^{1}_{M,n}(0)z^n+\lambda z^N \widetilde{Q}^{2}_{N-1}(\theta ). \end{aligned}$$

(A.26)

Multiplying Eq. (A.12) by $z^0$, (A.13) by $z^n (n=1,2,\ldots , N-2)$, (A.14) by $z^{N-1}$, (A.15) by $z^n,(n=N,N+1,\ldots )$ and taking summation from 0 to $\infty$ and using Eq. (A.23)

$$\begin{aligned}&\theta \sum _{n=0}^{\infty }\widetilde{P}_{1,n}(\theta )z^n-\sum _{n=0}^{\infty }P_{1,n}(0)z^n \nonumber \\&\quad = \lambda \sum _{n=0}^{\infty }\widetilde{P}_{1,n}(\theta )z^n-\lambda \sum _{n=1}^{\infty }\widetilde{P}_{1,{n-1}}(\theta )z^n\nonumber \\&\quad -\widetilde{S}(\theta ) (\sum _{n=0}^{\infty }P_{1,{n+1}}(0)z^n \nonumber \\&\quad +\sum _{n=N-1}^{\infty } \sum _{j=1}^{M}Q^{1}_{j,n+1}(0)z^n\nonumber \\&\quad +\lambda z^{N-1}\int _{0}^\infty Q^{2}_{N-1}(y)dy).\nonumber \\&z[\theta -\lambda +\lambda z]\widetilde{P}_{1}(z,\theta ) \nonumber \\&\quad = [z-\widetilde{S}(\theta )] P_{1}(z,0) -\widetilde{S}(\theta )\Big ({-P_{1,{0}}(0)}+ \sum _{j=1}^{M} Q^{1}_{j}(z,0)\nonumber \\&\quad -\sum _{j=1}^{M}\sum _{n=0}^{N-1}Q^{1}_{j,n}(0)z^n+\lambda z^N \widetilde{Q}^{2}_{N-1}(0)\Big ). \end{aligned}$$

(A.27)

Substituting $\theta =\lambda -\lambda z$ in Eqs. (A.24),(A.25),(A.26), and (A.27), we obtain the following

$$\begin{aligned} Q^{1}_{1}(z,0)= & {} P_{1,0}(0)\widetilde{V}_{1}(\lambda -\lambda z). \end{aligned}$$

(A.28)

$$\begin{aligned} Q^{1}_{j}(z,0)= & {} \widetilde{V}_{1}(\lambda -\lambda z)\sum _{n=0}^{N-1} Q^{1}_{j-1,n}(0)z^n, \nonumber \\&2\le j\le M. \end{aligned}$$

(A.29)

$$\begin{aligned} Q^{2}(z,0)= & {} \widetilde{V}_{2}(\lambda -\lambda z)\sum _{n=0}^{N-1}Q^{1}_{M,n}(0) z^n\nonumber \\&-\lambda z^N \widetilde{Q}^{2}_{N-1}(\lambda -\lambda z). \end{aligned}$$

(A.30)

$$\begin{aligned} P_{1}(z,0)= & {} \frac{\widetilde{S}(\lambda -\lambda z)}{[z-\widetilde{S}(\lambda -\lambda z)]}\Big (-P_{1,0}(0) +\sum _{j=1}^{M}(Q^{1}_{j}(z,0)\nonumber \\&\quad -\sum _{n=0}^{N-1}Q^{1}_{j,n}(0)z^n)+\lambda z^N \widetilde{Q}^{2}(0)\Big ). \end{aligned}$$

(A.31)

Substituting Eq. (A.28) in (A.24), we have

$$\begin{aligned} \widetilde{Q}^{1}_{1}(z,\theta ) = \frac{(\widetilde{V}_{1}(\lambda -\lambda z)-\widetilde{V}_{1}(\theta ))}{[\theta -\lambda +\lambda z] }P_{1,0}(0). \end{aligned}$$

(A.32)

Substituting Eq. (A.29) in (A.25), we obtain the following

$$\begin{aligned} \widetilde{Q}^{1}_{j}(z,\theta )= & {} \frac{(\widetilde{V}_{1}(\lambda -\lambda z)-\widetilde{V}_{1}(\theta ))}{[\theta -\lambda +\lambda z]}\sum _{j=2}^{M} \sum _{n=0}^{N-1}\widetilde{Q}^{1}_{j-1,n}(0)z^n,\nonumber \\&2 \le j\le M. \end{aligned}$$

(A.33)

Substituting Eq. (A.30) in (A.26), we get

$$\begin{aligned} \widetilde{Q}^{2}(z,\theta )= & {} \frac{(\widetilde{V}_{2}(\lambda -\lambda z)-\widetilde{V}_{2}(\theta ))}{[\theta -\lambda +\lambda z]}\sum _{n=0}^{N-1}Q^{1}_{M,n}(0)z^n\nonumber \\&+\frac{\lambda z^N (\widetilde{Q}^{2}_{N-1}(\lambda - \lambda z)-Q^{2}_{N-1}(\theta )}{[\theta -\lambda +\lambda z]}. \end{aligned}$$

(A.34)

Substituting Eq. (A.31) in (A.27), we obtain

$$\begin{aligned} \widetilde{P}_{1}(z,\theta )= & {} \frac{[\widetilde{S}(\lambda -\lambda z)-\widetilde{S}(\theta )]}{[\theta -\lambda +\lambda z][z-\widetilde{S}(\lambda -\lambda z)]}\nonumber \\&\Big [\Big (-P_{1,0}(0)+\sum _{j=1}^{M}(Q^{1}_{j}(z,0)\nonumber \\&-\sum _{n=0}^{N-1}Q^{1}_{j,n}(0)z^n)+\lambda z^N \widetilde{Q}^{2}(0)\Big )\Big ]. \end{aligned}$$

(A.35)

$\square$

1.1 The probability generating Function of number of user requests in the queue P(z) at an arbitrary time epoch

$$\begin{aligned} P(z)= & {} \widetilde{P}_1(z,0)+\sum _{j=1}^{\infty } \widetilde{Q}^{1}_{j}(z,0)+ \widetilde{Q}^{2}(z,0). \end{aligned}$$

Substituting Eqs. (A.32), (A.33), (A.34) and (A.35) in the above equation, we get

$$\begin{aligned} P(z)= & {} \frac{[\widetilde{V}_{1}(\lambda -\lambda z)]}{\lambda [z-\widetilde{S}(\lambda -\lambda z)]}P_{1,0}(0)+\frac{[\widetilde{V}_{1}(\lambda -\lambda z)]}{\lambda [z-\widetilde{S}(\lambda -\lambda z)]}\nonumber \\&\sum _{n=0}^{N-1}q^1_n z^n+\frac{1-z+\widetilde{V}_{2}(\lambda - \lambda z)(z-\widetilde{S}(\lambda - \lambda z)}{(z-1)\lambda (z-\widetilde{S}(\lambda - \lambda z)}\nonumber \\&\sum _{n=0}^{N-1}Q^{1}_{M,n}(0)z^n \nonumber \\&+\frac{z^2 \lambda -z\lambda -1+(1+\lambda - z \lambda )\widetilde{S}(\lambda -\lambda z)}{\lambda (z-1)(z-\widetilde{S}(\lambda - \lambda z))}\nonumber \\&z^N \widetilde{Q}^{2}_{N-1}(0)-\lambda z^N \widetilde{Q}^{2}_{N-1}(\lambda - \lambda z). \end{aligned}$$

(A.36)

where $q^{1}_n=\sum _{j=1}^{M-1}Q^1_{j,n}(0)$. The probability generating function P(z) has to satisfy $P(1) = 1$. In order to satisfy the condition, applying L’Hospital’s rule and evaluating $\lim _{z\rightarrow 1} P(z)$ and equating the expression to 1, $( 1-\lambda E(S)) > 0$ is obtained. Let $\rho = \lambda E[S]$. Thus $\rho <1$ is the stability condition for the proposed model.

Appendix B

Proof of Theorem 2

Since $\beta _i$ is the probability of i URs arrive during the short sleep period.

$$\begin{aligned} \beta _i=\int _{0}^{\infty } \frac{e^{-\lambda t}{(\lambda t)}^i}{i!} \,dV(t). \end{aligned}$$

Multiplying both sides of the above equation by $z^i$ and taking the summation from $i=0$ to $\infty$, we obtain the following

$$\begin{aligned} \sum _{i=0}^\infty \beta _i z^i= & {} \sum _{i=0}^\infty \int _{0}^{\infty } \frac{e^{-\lambda t}{(\lambda t)}^i}{i!}z^i \,dV(t).\\= & {} \int _{0}^{\infty } \sum _{i=0}^\infty \frac{e^{-\lambda t}{(\lambda t)}^i}{i!}z^i \,dV(t).\\= & {} \int _{0}^{\infty } \sum _{i=0}^\infty e^{-\lambda t}e^{\lambda tz}\,dV(t).\\= & {} \widetilde{V}(\lambda -\lambda z).\\ \end{aligned}$$

$\square$

Appendix C

Proof of Theorem 3

From Eqs. (A.28), (A.29) and using (A.23), we get

$$\begin{aligned}&\sum _{j=1}^{M}Q^{1}_{j}(z,0)= \widetilde{V}_1 (\lambda - \lambda z )\left[ P_{1,0}(0)+ \sum _{j=2}^{M}\sum _{n=0}^{N-1}Q^{1}_{j-1,n}(0)z^n\right] .\\&\sum _{j=1}^{M}\sum _{n=0}^{\infty }Q^{1}_{j,n}(0)z^n = \widetilde{V}_1 (\lambda - \lambda z )\left[ P_{1,0}(0)+ \sum _{j=1}^{M-1}\sum _{n=0}^{N-1}Q^{1}_{j,n}(0)z^n\right] .\\&\sum _{n=0}^{\infty }(q^{1}_{n}+Q^{1}_{M,n}(0))z^n = \widetilde{V}_1 (\lambda - \lambda z )\left[ P_{1,0}(0)+\sum _{n=0}^{N-1} q^{1}_n z^n\right] . \end{aligned}$$

Using Theorem 1, we get

$$\begin{aligned}&\sum _{n=0}^{\infty }(q^1_n+Q^1_{M,n}(0))z^n\\&\quad = \left( \sum _{m=0}^{\infty }\beta _m z^m\right) \left( P_{1,0}(0)+ \sum _{n=0}^{N-1}(q^{1}_{n}z^n\right) \end{aligned}$$

where $\beta _i$ is the probability that ‘i’ URs arrive during the short sleep period. $\square$

Equating the coefficients of $z^n , n= 0,1,\ldots N-1$ on both sides of equation, we have

$$\begin{aligned} q^1_n+Q^1_{M,n}(0)= & {} \beta _n P_{1,0}(0)+ \sum _{i=0}^{n}\beta _i q^{1}_{n-i}.\nonumber \\ q^1_n+Q^1_{M,n}(0)= & {} \beta _n P_{1,0}(0)+\beta _0 q^1_n+\sum _{i=1}^{n}\beta _i q^{1}_{n-i}.\nonumber \\ (1-\beta _0)q^1_n= & {} \beta _n P_{1,0}(0)+\sum _{i=1}^{n}\beta _i q^{1}_{n-i}-Q^1_{M,n}(0).\nonumber \\ q^1_n= & {} \frac{[\beta _n P_{1,0}(0)+\sum _{i=1}^n\beta _iq^1_{n-i}-Q^1_{M,n}(0)]}{1-\beta _0},\nonumber \\&n=,1,2,\ldots , N-1. \end{aligned}$$

(C.1)

and

$$\begin{aligned} q^1_0= & {} \frac{1}{1-\beta _0}[\beta _0 P_{1,0}(0)-Q^1_{M,0}(0). \end{aligned}$$

(C.2)

By substituting $b_0=\frac{\beta _0}{1-\beta _{0}}$ and $c_{0}= \frac{Q^1_{M,0}(0)}{1-\beta _{0}}$, the above equation reduces to $q^{1}_{0} = b_{0}P_{1,0}-c_{0}$.

In Eq. (C.1), when $n=1$ we get

$$\begin{aligned} q_1^{1}= & {} \frac{\beta _{1}P_{1,0}+\beta _{1}q^{1}_{0}-Q^{1}_{M,1}(0)}{1-\beta _{0}} \end{aligned}$$

Substituting Eq. (C.2) in the above equation, we get

$$\begin{aligned}&q^1_1= b_{1}P_{1,0}-c_{1}\quad \text {where}\quad b_{1}\\&\quad =\frac{\beta _{1}+\beta _{1}b_{0}}{1-\beta _0} \text {and} \quad c_1=\frac{\beta _1 c_0 +Q^{1}_{M,1}(0)}{1-\beta _0} \end{aligned}$$

In Eq. (C.1), when $n=2$, we get

$$\begin{aligned} q^{1}_{2}= & {} \frac{\beta _{2}P_{1,0}+\beta _{1}q^{1}_{1}+\beta _{2}q^{1}_{0}-Q^{1}_{M,2}(0)}{1-\beta _{0}}\\= & {} b_{2}P_{1,0}-c_{2} \end{aligned}$$

$$\begin{aligned}&\text {where}\quad b_{2} = \frac{\beta _{2}+\beta _{1}b_{1}+\beta _{2}b_{0}}{1-\beta _{0}} \qquad \text {and}\quad \\&c_{2} = \frac{\beta _{1}c_{1}+\beta _{2}c_{0}+Q^{1}_{M,2}(0)}{1-\beta _{0}} . \end{aligned}$$

$$\begin{aligned} \text {In general}, \quad Q^{1}_{n}= & {} b_{n}P_{1,0}-c_{n} \end{aligned}$$

$$\begin{aligned}&\text {where}\quad b_{n}=\frac{\beta _{n}+\sum _{i=1}^{n}\beta _{i}b_{n-i}}{1-\beta _{0}} \quad \text {and}\quad \\&c_{n}=\frac{Q^{1}_{M,n}(0)+\sum _{i=1}^{n}\beta _{i}c_{n-i}}{1-\beta _{0}} . \end{aligned}$$

From Eq. (A.28), we have

$$\begin{aligned} Q^1_1(z,0)= & {} P_{1,0}(0)\widetilde{V}_1(\lambda -\lambda z).\\ \sum _{n=0}^\infty Q^1_{1,n}(0)z^n= & {} \Big (\sum _{n=0}^\infty \beta _n z^n \Big )P_{1,0}(0). \end{aligned}$$

Coefficient of $z^n$ in the above equation is $Q^1_{1,n}$ =$\beta _n P_{1,0}(0)$.

From Eq. (A.29), we have

$$\begin{aligned} Q^1_2 (z,0)= & {} \widetilde{V}_1(\lambda -\lambda z)\sum _{n=0}^{N-1} Q^1_{1,n}(0)z^n.\\ \sum _{n=0}^\infty Q^1_{2,n}(0)z^n= & {} \Big (\sum _{m=0}^\infty \beta _m z^m\Big )\Big (\sum _{n=0}^{N-1}Q^1_{1,n}(0)z^n \Big ) \\ \sum _{n=0}^\infty Q^1_{2,n}(0)z^n= & {} \Big (\sum _{m=0}^\infty \beta _m z^m\Big )\Big (\sum _{n=0}^{N-1}(\beta _n P_{1,0}(0)z^n) \Big ) \\ \sum _{n=0}^\infty Q^1_{2,n}(0)z^n= & {} P_{1,0}(0)\Big (\beta _0\beta _0+(\beta _0\beta _1+\beta _1 \beta _0)z\\&+(\beta _0\beta _2+\beta _1\beta _1+\beta _2\beta _0)z^2\\&+(\beta _0\beta _3+\beta _1\beta _2+\beta _2\beta _1+\beta _3 \beta _0)z^3+ \ldots . \end{aligned}$$

Coefficient of $z^n$ in the above equation is

$$\begin{aligned} Q^1_{2,n}(0) =P_{1,0}(0)\sum _{k_{1}=0}^j \beta _{k_{1}}\beta _{j-k_{1}}. \end{aligned}$$

From Eq. (A.29), we have

$$\begin{aligned} Q^1_3 (z,0)= & {} \widetilde{V}_1(\lambda -\lambda z)\sum _{n=0}^{N-1} Q^1_{2,n}(0)z^n\\ \sum _{n=0}^\infty Q^1_{3,n}(0)z^n= & {} \Big (\sum _{m=0}^\infty \beta _m z^m\Big )\Big (\sum _{n=0}^{N-1}Q^1_{2,n}(0)z^n\Big ) \\ \sum _{n=0}^\infty Q^1_{3,n}(0)z^n= & {} P_{1,0}(0)\Big (\beta _0\beta _0\beta _0 +(\beta _0(\beta _0\beta _1+\beta _1 \beta _0)\\&+\beta _1\beta _0\beta _0)z+(\beta _0(\beta _0\beta _2 +\beta _1\beta _1+\beta _2\beta _0)\\&+\beta _1(\beta _0\beta _1+\beta _1\beta _0)+\beta _2\beta _0\beta _0)z^2 + \ldots . \end{aligned}$$

Coefficient of $z^n$ in the above equation is

$$\begin{aligned} Q^1_{3,n}(0)= & {} P_{1,0}(0)\Big (\sum _{k_{2}=0}^j \beta _{j-{k_{2}}}\Big )\Big (\sum _{k_{1}=0}^{k_2} \beta _{k_{1}}\beta _{k_{2}}-{k_{1}}\Big ). \end{aligned}$$

By generalizing this, we get the coefficient of $z^n$ in $Q_{M} (z,0)$ is

$$\begin{aligned} Q_{M,j}(0)= & {} P_{1,0}(0)\sum _{k_{M-1}=0}^{j}\beta _{j-{k_{M-1}}}\sum _{k_{{M-2}=0}}^{k_{M-1}}\beta _{k_{M-1}-k_{M-2}}\\&\quad \ldots \sum _{k_{{1}=0}}^{k_{2}}\beta _{k_{2}-k_{2}}\beta _{k_{1}},\\&\quad 0 \le j \le N-1. \end{aligned}$$

Appendix D

Proof of Theorem 4

From Eq. (A.21), we have

$$\begin{aligned} (\theta -\lambda )\widetilde{Q}^2_0(\theta )= & {} Q^2_0(0)-Q^1_{M,0}(0)\widetilde{V}_2(\theta ). \end{aligned}$$

(D.1)

Substituting $\theta = \lambda$ in (D.1), we get

$$\begin{aligned} Q^2_0(0)= & {} Q^1_{M,0}(0)\widetilde{V}_2(\lambda ). \end{aligned}$$

Therefore,

$$\begin{aligned} \widetilde{Q}^2_0(\theta )= & {} \frac{\widetilde{V}_2(\lambda )-\widetilde{V}_2(\theta )}{\theta - \lambda }Q^1_{M,0}(0)\\= & {} k_0 Q^1_{M,0}(0) \quad \text {where} \qquad k_0 = \frac{\widetilde{V}_{2}(\lambda )-\widetilde{V}_{2}(\theta )}{\theta - \lambda }. \end{aligned}$$

Substituting $n = 1$ in Eq. (A.22), we get

$$\begin{aligned} (\theta -\lambda )\widetilde{Q}^2_1(\theta )= & {} Q^2_1(0)-Q^1_{M,1}(0)\widetilde{V}_2(\theta )-\lambda \widetilde{Q}^2_0(\theta ). \end{aligned}$$

(D.2)

when $\theta =\lambda$, the above equation reduces to

$$\begin{aligned} Q^2_1(0)= & {} Q^1_{M,1}(0)\widetilde{V}_2(\theta )+\lambda \widetilde{Q}^2_0(\theta ). \end{aligned}$$

Substituting the above in Eq. (D.2), we get

$$\begin{aligned} \widetilde{Q}^2_1(\theta )= & {} \frac{1}{\theta - \lambda }\Big [(\widetilde{V}_2(\lambda ) - \widetilde{V}_2(\theta ))Q^1_{M,1}(0)\\&+\lambda \widetilde{(}Q^2_0(\lambda ) - \widetilde{Q}^2_0(\theta )\Big ]\\= & {} k_0 Q^1_{M,1}(0)+\lambda (\frac{\widetilde{Q}^2_0(\lambda ) - \widetilde{Q}^2_0(\theta )}{\theta -\lambda })\\= & {} k_0 Q^1_{M,1}(0)-\lambda (\frac{\widetilde{V}^1_2(\lambda ) +k_0)}{\theta - \lambda })Q^1_{M,0}(0)\\= & {} k_0 Q^1_{M,1}(0)-\lambda k_1 Q^1_{M,0}(0)\quad \text {where}\\&\quad k_1=\Big (\frac{\widetilde{V}^1_2(\lambda ) +k_0}{\theta - \lambda }\Big ). \end{aligned}$$

Substituting $n = 2$ in Eq. (A.22), we get

$$\begin{aligned} (\theta -\lambda ) \widetilde{Q}^{2}_{2}(\theta )= & {} Q^{2}_{2}(0)- Q^{1}_{M,2}(0)\widetilde{V}_{2}(\theta )-\lambda \widetilde{Q}^{2}_{1}(\theta ). \end{aligned}$$

(D.3)

When $\theta = \lambda$, the above equation reduced to

$$\begin{aligned} Q^2_{2}(0)= & {} Q^{1}_{M,2}(0)\widetilde{V}_{2}(\lambda )+\lambda \widetilde{Q}^{2}_{1}(\lambda ). \end{aligned}$$

Substituting the above in Eq. (D.3), we get

$$\begin{aligned} \widetilde{Q}^{2}_{2}(\theta )= & {} k_{0} Q^{1}_{M,2}(0)-\lambda Q^{1}_{M,1}(0)\Big (\frac{\widetilde{V}^{1}_{2}(\lambda )+k_{0}}{\theta - \lambda }\Big )\\&+\lambda ^2 Q^{1}_{M,0}(0)\Big (\frac{\widetilde{V}^{2}_{2}(\lambda )+k_{1}}{\theta - \lambda }\Big ) \\= & {} k_{0}Q^{1}_{M,2}(0)+\lambda k_{1} Q^{1}_{M,1}(0)+\lambda ^2 k_{2}Q^{1}_{M,0}(0) \end{aligned}$$

where $k_{2} = \frac{\widetilde{V}^{2}_{2}({\lambda })+k_{1}}{(\theta - \lambda )}$.

Substituting $n = 3$ in Eq. (A.22), we get

$$\begin{aligned} (\theta -\lambda ) \widetilde{Q}^{2}_{3}(\theta )= & {} Q^{2}_{3}(0)- Q^{1}_{M,3}(0)\widetilde{V}_{2}(\theta )-\lambda \widetilde{Q}^{2}_{1}(\theta ). \end{aligned}$$

(D.4)

When $\theta = \lambda$, the above equation reduced to

$$\begin{aligned} Q^{2}_{3}(0)= & {} Q^{1}_{M,3}(0)\widetilde{V}_{2}(\lambda )+\lambda \widetilde{Q}^{2}_{1}(\lambda ). \end{aligned}$$

Substituting the above in Eq. (D.4), we get

$$\begin{aligned} \widetilde{Q}^{2}_{3}(\theta )= & {} \Big (\frac{\widetilde{V}_{2}(\lambda )-\widetilde{V}_{2}(\theta )}{\theta -\lambda }\Big )Q^{1}_{M,3}(0)+\lambda \Big (\frac{\widetilde{Q}^{1}_{2}(\lambda )-\widetilde{Q}^{1}_{2}(\theta )}{\theta -\lambda }\Big )\\= & {} k_{0}Q^{1}_{M,3}(0)-\frac{\lambda }{\theta -\lambda }(\widetilde{V}^{1}_{2}(\lambda )+k_{0})Q^{1}_{M,2}(0) \\&-\lambda (\widetilde{V}^{2}_{2}(\lambda )+k_{1})Q^{1}_{M,1}(0) + \lambda ^2 (V^{3}_{2}(\lambda )+k_{2})Q^{1}_{M,0}(0)\\= & {} k_{0}Q^{1}_{M,3}(0)-\lambda k_{1} Q^{1}_{M,2}(0)+ \lambda ^2 k_{2}Q^{1}_{M,1}(0)\\&-\lambda ^3 k_{3}Q^{1}_{M,0}(0) \end{aligned}$$

where $k_{3} = \frac{V^{3}_{2}(\lambda )+k_{2}}{\theta -\lambda }$.

Generalizing this, we have

$$\begin{aligned} \widetilde{Q}^{2}_{i}(\theta )= \sum _{j=0}^{i}(-\lambda )^{j}k_{j}Q^{1}_{M,i-j}(0),\quad 0\le i\le N-1 \end{aligned}$$

where $k_{j}=\frac{\widetilde{V}_{2}^{j}(\lambda )+k_{j-1}}{\theta -\lambda }\quad \text {and} \qquad k_{0}=\frac{\widetilde{V}_{2}(\lambda )-\widetilde{V}_{2}(\theta )}{\theta -\lambda }$. $\square$