An Analytical Model Driven by Fluid Queue for Battery Life Time of a User Equipment in LTE-A Networks ∗ Anupam Gautam A,S

Physical Communication 30 (2018) 213–219

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Physical Communication

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Full length article An analytical model driven by fluid queue for battery life time of a user equipment in LTE-A networks ∗ Anupam Gautam a,S. Dharmaraja b, a Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India b Bharti School of Telecommunication Technology and Management, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India article info a b s t r a c t

Article history: During the Discontinuous Reception (DRX) mechanism, a User Equipment (UE) turns off most of its Received 13 October 2017 components for a prolonged period. The battery charge of a UE, which is a continuous quantity, arises Received in revised form 28 May 2018 as a fluid model. In this scenario, we represent the system as the reservoir where battery charge gets Accepted 13 August 2018 accumulated or is depleted gradually over the time, subject to a random environment constructed by the Available online 6 September 2018 DRX mechanism. Therefore, the DRX mechanism acts as the background process for the fluid queue model MSC: of the battery charge. In a UE, battery charge available at any time t is considered as the fluid available 00-01 in the reservoir. Different discharging rates are considered for the various states of the DRX mechanism. 99-00 Using the fluid queue model, cumulative distribution function for the battery life of a DRX enabled UE is Keywords: derived. Further, the expression for the mean battery life is obtained and analyzed numerically. Battery life © 2018 Elsevier B.V. All rights reserved. Fluid queue Discontinuous reception mechanism Long term evolution-advanced networks

1. Introduction the time. Also while using UEs, most of the activities involved are web browsing, gaming and multimedia apps which again leads to In recent years, smartphones have emerged as the basic com- battery consumption at a high rate [7]. modity for the young generation. However, some of the major In this paper, battery content of a DRX enabled UE is investi- constraints for the smartphone users are, short battery life due to gated. Further, an analysis is performed to improve the battery life various applications, large display screen and high data rates. For of a UE. Battery content of a UE is called charge which is a real the current generation Long Term Evolution/Long Term Evolution- quantity whose discretized approximation would not be as good. Advanced (LTE/LTE-A) introduced by Third Generation Partnership Considering that, we have used fluid queue modeling technique Project(3GPP), a power saving mechanism known as Discontinu- to model the battery charge of the UE. In general, fluid queues ous Reception (DRX) mechanism has been launched [1,2]. are used to represent a model where some amount is depleting or LTE/LTE-A standard supports two types of Radio Resource Con- accumulating gradually over a time, subject to a random environ- trol (RRC) modes, RRC_ CONNECTED and RRC_IDLE [3]. These ment [8,9]. modes are categorized based on the handling of RRC connection. In past, various studies have been performed to reduce the The fundamental difference between LTE/LTE-A DRX and other energy consumption in smartphones. For example, a risk-theoretic existing DRX mechanism is that LTE/LTE-A DRX mechanism allows Markov fluid queue model is proposed for the computation of the a UE to enter a sleep period while the UE is in RRC_CONNECTED first battery outage probabilities in a given finite time horizon [10]. mode, i.e., still registered with evolved NodeB (eNB), and therefore Further, this model is used in obtaining the optimum operational leads to more power saving in UEs [4]. parameters for the threshold policies. In [11], the outage probabil- In previous years, various surveys have shown that as the bat- ity performance of a new relay selection scheme for the energy har- tery capacity of the UEs is improved, the screen size and other data vesting relays based on the wireless power transfer is investigated. packet processing also have been improved in that time span [5, A finite-state Markov chain is developed to capture the evolution 6]. This is one of the reason that why has the increased battery of relay batteries under the proposed relay selection scheme. But capacity not improved the battery life of UEs significantly over this system may become complex with the increase in the number of relay nodes and further, Markovian-based analysis turns even ∗ Corresponding author. more difficult. An efficient transmission scheme is proposed for E-mail addresses: [email protected] (A. Gautam), the downlink of an OFDMA multicell multiuser system with mul- [email protected] (S. Dharmaraja). tiuser decoding capable receivers. This scheme works optimally https://doi.org/10.1016/j.phycom.2018.08.004 1874-4907/© 2018 Elsevier B.V. All rights reserved. 214 A. Gautam, S. Dharmaraja / Physical Communication 30 (2018) 213–219 in small network scenario but fails to do well in large network etc. [18]. A two level fluid queue model is formed for investigating scenarios [12]. the battery life with different rates of discharging in the respective An approach to automatically access to the Power Line Commu- levels. Further, in [28], it is discussed that steady-state analysis nication (PLC) network by identifying the modulation of signals is only provides some information on the battery life, which is not proposed. Further, an MIMO based cooperative modulation iden- sufficient viz., for studying the dynamic behavior of the battery dis- tification scheme is developed for the power saving [13]. An en- charge. In light of this, transient analysis conducted in this article hanced intercell interference coordination by considering central- will be of critical value, in understanding the dynamic behavior of ized processing controller with almost blank subframe (eICIC-CPC- the battery life, in controlling the discharge of the battery, and in ABS) is proposed for the heterogeneous networks (HetNets). This studies relating to the rate of convergence to the steady-state. Thus, method provides significant savings in transmitting power [14]. one of the main contribution of this paper lies in obtaining the In [15], authors demonstrated that by controlling both transmit distribution of battery life of a DRX enabled UE in LTE-A networks. power and stored energy usage of Base Stations (BSs), energy The rest of the paper is organized as follows. The DRX mecha- costs can be effectively reduced. Specifically, they propose a two- nism and its configuration are introduced in Section2. Proposed stage BS operation scheme where an optimization and control analytical fluid queue model is discussed in detail in Section3. subproblem is solved at each stage, respectively. In [16], downlink- In Section4, an analytical model driven by fluid queue model is power-control scheme is presented for femtocell with no overhead developed and its performance analysis is performed. Results are signaling exchange with the macrocell. The transmit power can be obtained and compared numerically in Section5. Pointers to the reduced by 52.6% which leads to a power effective solution in UEs. conclusion and some future work directions are given in Section6. To study the effect of the battery charge level on the energy consumption of the battery operated devices, a set of measure- 2. DRX mechanism and its configuration ments have been framed [17]. On the basis of these measurements, the authors have found a clear correlation between battery voltage DRX allows a UE to monitor Physical Downlink Control Channel level and the power consumption. In [18], a battery life extension (PDCCH) discontinuously, when the UE is not receiving the data method has been proposed. A technique of selective data reception packets from the related eNB [27]. When a UE is not monitoring on a UE according to the state of a battery in the UE is used. In this the PDCCH, the UE enters in a power saving state during which work, the authors have proposed that the battery state information most of its components are turned off. This exercise reduces the of a UE is transmitted to the corresponding application server with power consumption appreciably [29]. During DRX mechanism, the keep-alive signaling or initial signaling messages. An analytical UE wakes up for a while from sleep to monitor the PDCCH and model fitting has been proposed for the specific DRX mechanism by keeps on returning to the power saving state if packet’s arrival using a semi-Markov process [19]. Two key performance indicators is not detected. Moreover, if there is arrival of a packet, the UE affected by the DRX mechanism, the power saving factor and the resumes to provide service to the arriving packets in the power average buffering delay of the radio-off periods, are derived. Infor- active state, i.e., a state during which the UE remains on, [30]. mation about the battery life is discussed in [20,21]. The battery life Following are the DRX parameters considered in this paper, [29, of mobile devices have been improved by focusing on improving 31]. the energy efficiency of the hardware and the software. Some of • Inactivity timer (T ): It is the time duration during which a the ways of improving energy efficiency are like, by reducing the IN UE waits before initiating the DRX. It works as a timer which amount of data being transmitted, increasing the capacity of the re-initiates itself after a successful reception of the data on internal battery, or restricting the resources allocated for the idle the PDCCH. On the expiration of the inactivity timer, the UE (background) applications [22]. enters into a short DRX cycle. One more approach used widely is code offloading. In this • Short DRX cycle (T ): It is the length of the first DRX cycle after method migrating mobile code is executed remotely in the cloud S the initiation of DRX mechanism in a UE. This short DRX cycle or on the dedicated servers which lead to the energy saving on is repeated for a predetermined number of times if no data the UEs [23]. Further, in [24] the optimal active power flow model packet arrives and ends otherwise. In this duration, the UE for UEs is discussed. In [25], the authors have investigated battery turns off most of its components. life of a UE subject to the stochastically determined charging and • Short DRX cycle timer (N ): It is a timer which regulates the discharging periods. A multi-regime fluid queue model, imposing S number of short DRX cycles before starting the first long DRX a threshold at some value is proposed. Performance analysis of cycle. batteries of the multi-hop wireless network under energy harvest- • Long DRX cycle (T ): It is the length of the first long DRX cycle ing is conducted, where the network is dependent on the renew- L which starts on the expiration of N short DRX cycles if no able energy for the power consumption [26]. Further, the energy S data packet arrives (T ≥ T ). Further, this cycle keeps on levels are modeled in the battery using a stochastic fluid-flow L S repeating itself for a predetermined number of times. model to determine the availability and time-averaged latency. • On duration timer (T ): It is the length of a very small time In recent times, a lot of work has been carried out in evaluating ON interval inside a DRX cycle (short or long) at its end. During the performance of the DRX mechanism under various limitations T , the UE monitors the PDCCH for the arrival of a new and conditions. Out of these studies, most of the work define ON packet. A packet arrival indication at the PDCCH during T analytical model, to capture the behavior of the DRX enabled UEs ON wakes up the UE, and it begins to serve the data packet. for different types of traffic [27]. To the best of our knowledge, a • Busy period (T ): It is a length of time during which the UE fluid queue modeling of the battery life of a UE, equipped with the B provides service to the packets after waking up from DRX. DRX mechanism has not been performed so far. This motivated us to propose a fluid queue model for examining the battery life of a Hence, as depicted in Fig.1, DRX mechanism is composed of the UE during DRX mechanism in LTE-A networks. power saving state and the power active state. Further, the power In this paper, a novel approach to find the analytical results saving state comprised of the short DRX cycle and the long DRX for the battery life of the UEs is considered. We have derived cycle, whereas the power active state comprised of the inactiv- the battery life distribution for the Lithium polymer battery of a ity timer period and the busy period. Thus, the DRX mechanism UE. Advantages of this type of batteries includes its light weight, consists of the four states, the busy period, inactivity timer period, reasonable size at low price, more resistance to the overcharging, short DRX cycle and the long DRX cycle. A. Gautam, S. Dharmaraja / Physical Communication 30 (2018) 213–219 215

Fig. 1. DRX mechanism (see, [29]).

3. Model description is defined as the first passage time, starting with charge level C, given that the process X(t) hits 0. In this section, we model the battery charge level in a UE subject In most of the current generation UEs, a notification is to random time spent in the power saving and the power active shown when the battery charge goes below a particular states caused by the DRX mechanism. Now, before proceeding level and user makes the device to go in a battery saving further, we discuss the battery charge of a UE. mode. During Case (i), user is cut-off from some of the In a Lithium-ion battery, the charging pattern is divided into applications which consume maximum energy viz., video two phases from the fully depleted 0% state-of-charge (SOC) to calling, online gaming, etc. Hence, the UE goes to a mode 100% SOC. In the first phase, the UE’s charge controller circuit where it is forced to decrease UE’s discharging rate which outputs a high constant current (CC) which delivers high power are considered as µil, i ∈ S in aforementioned Case (i). to the battery. As the battery voltage reaches a certain threshold, Further as discussed in Section2, the DRX mechanism typically 4.2 V, the charge management controller circuit tran- consists of four major states, i.e., the busy period, inactiv- sitions to the constant voltage (CV) phase, which maintains the ity timer period, short DRX cycle and the long DRX cycle. voltage threshold and allows current drawn by the battery to fall Let they be denoted by state 1, 2, 3, 4, respectively. Now, off gradually. Also, the power delivered to the battery is typically since Y (t) denotes the state of UE during DRX mechanism highest in the CC phase, and drops off during the CV phase. at time t. Then, {Y (t), t ≥ 0} is a stochastic process with a Let the battery of the UE has finite capacity of charge, say C and finite state space, say S = {1, 2, 3, 4}. Also, depending on the UE can possesses all four states of DRX mechanism, if it is not the pattern of the packet arrival, transition from a given discharged completely. Further, define state to another state can take place at any instant of time. X(t) := the level of remaining battery charge content Assume that time spent by the UE in the aforementioned states of DRX mechanism (i.e., inactivity timer, short DRX of the UE at time t, cycle, long DRX cycle and busy period) follow Markov Y (t) := the state of the UE during DRX mechanism at time t. property. For mathematical analysis, we assume that the Since, X(t) denotes the level of the remaining battery charge con- time spent by the UE in these four states follows expo- tent of the UE at time t. Then {X(t), t ≥ 0} is a stochastic process nential distribution. Also, every state can be reached from every other state. Then, the stochastic process {Y (t), t ≥ with state space {x | 0 ≤ x ≤ C} where C ∈ R. Also, there exists 0} can be nicely modeled as an irreducible Continuous B ∈ R such that 0 ≤ B < C is the minimum level of charge of the UE’s battery after which the charge dissipation rate changes as Time Markov Chain (CTMC) with a state space S, and shown in Fig.2. the state transition diagram for this CTMC is shown in Therefore, battery charge level of the UE is divided into two Fig.3. Let α1, α2 and α3 be the rates of transition from levels according to the CV and the CC phases. Let l and h be the state 1 to state 2, state 2 to state 3 and state 3 to state charge levels when 0 ≤ X(t) ≤ B and B < X(t) ≤ C denoting CC 4, respectively. Further, consider β1, β2 and β3 as rates and CV phases, respectively. Further, for the sake of mathematical of transition from state 4 to state 1, state 4 to state 2, analysis, aforementioned two levels are considered as Case (i) and and state 3 to state 1, respectively. These transition rates Case (ii), respectively. are independent of charging, and discharging rates of the Now, we discuss the Case (i) and Case (ii) in detail as follows: UE as transition in {Y (t), t ≥ 0} depends on the packet transmission. Case (i) When 0 ≤ X(t) ≤ B In this case, let the charging rate be λ and the discharging Let Q = [qij]i,j∈S be the infinitesimal generator matrix of the CTMC { ≥ } = − ∈ rate be µil, i ∈ S. Further, the battery life is defined as the Y (t), t 0 and qi qii where qij, i, j S are transition rates first passage time, starting with charge level B, given that from state i to state j. Let ı = (πi)i∈S be the stationary distribution the process X(t) hits 0. of {Y (t), t ≥ 0}. Case (ii) When B < X(t) ≤ C Now, {(X(t), Y (t)), t ≥ 0} forms a two-dimensional Markov In this case, the charging rate remains λ but the discharg- process such that the first component of this Markov process, ing rate is considered to be µih, i ∈ S such that µih > {X(t), t ≥ 0} varies according to the second component {Y (t), t ≥ µil. This change in discharging rate is assumed to keep 0}. That is, the battery of the UE experiences distinct rates of in account the real life scenario. Further, the battery life charging and discharging depending upon the state possessed by 216 A. Gautam, S. Dharmaraja / Physical Communication 30 (2018) 213–219

Fig. 2. Change in battery charge dissipation rate with its respective levels.

the random variables W l = inf{t > 0 : X(t) = 0 | 0 ≤ X(u) ≤ B, u > 0}, W h = inf{t > 0 : X(t) = 0 | B < X(u) ≤ C, u > 0}, (3) and their respective conditional Cumulative Distributive Functions (CDFs) are given by l = [ l ≤ l = Hi,j(t, x) P W t, Y (W ) j, 0 ≤ X(t) ≤ B |Y (0) = i, X(0) = x], h = [ h ≤ h = Hi,j(t, x) P W t, Y (W ) j, Fig. 3. Proposed CTMC model of the DRX mechanism. B < X(t) ≤ C |Y (0) = i, X(0) = x]. (4) k k Table 1 By the definition of W , for k = l, h, we have W = 0 if X(0) = 0 Charge buffer rates. k = = ∑4 and rik < 0, otherwise P(W 0) 0. We assume i=1 rikπi < k Battery charge content level r1k r2k r3k r4k 0, k = l, h, to make the random time W as a finite quantity.

B < x ≤ C, (k = h) λ − µ1h λ − µ2h λ − µ3h λ − µ4h The governing system of partial differential equations (PDEs) for ≤ ≤ = − − − − 0 x B, (k l) λ µ1l λ µ2l λ µ3l λ µ4l the fluid queue model, for i, j ∈ S, k = l, h, is given by ∂Hk (t, x) ∂Hk (t, x) 1,j 1,j k k + (λ − µ1k) = −α1H (t, x) + β4H (t, x) ∂t ∂x 1,j 2,j the UE during DRX mechanism mode. Thus, {Y (t), t ≥ 0} forms a k k + β3H (t, x) + β1H (t, x), (5) random environment for the first component, i.e., {X(t), t ≥ 0}. 3,j 4,j ∂Hk (t, x) ∂Hk (t, x) Further, consider 2,j 2,j k k + (λ − µ2k) = −(α2 + β2)H (t, x) + α1H (t, x) ∂t ∂x 2,j 1,j rik: the rate at which the UE’s battery is discharged, + β Hk (t, x), (6) i ∈ S, k = l, h. 2 4,j ∂Hk (t, x) ∂Hk (t, x) 3,j 3,j k k For i ∈ S and k = l, h, we have + (λ − µ3k) = α2H (t, x) − (α3 + β3)H (t, x), (7) ∂t ∂x 2,j 3,j dX(t) ∂Hk (t, x) ∂Hk (t, x) = 4,j 4,j k k rik, when X(t) > 0, (1) + (λ − µ4k) = α3H (t, x) − (β1 + β2)H (t, x), (8) dt ∂t ∂x 3,j 4,j dX(t) with initial conditions as follows = max{rik, 0}, when X(t) = 0. (2) dt k = ≥ ≤ ̸= Hi,j(t, 0) 1 if t 0 and rik 0, i j, Corresponding values for r , k = l, h are given in Table1. Also, ik l H (t, B) = 0 if t ≥ 0 and rik ≥ 0, i ̸= j, when rik ≤ 0, it implies that the level of charge content is i,j decreasing in the UE [32]. h = ≥ ≥ ̸= Hi,j(t, C) 0 if t 0 and rik 0, i j, k = ≤ Hi,j(0, x) 0 if 0 < x C, 4. Performance analysis k = ̸= Hi,j(0, 0) 0 if rik > 0, i j. For practical design and performance evaluation, it is essential On writing the above system of PDEs in the matrix form, we to obtain information about the buffer occupancy distribution of have aforementioned fluid queue model. Let the battery discharge of the k k ∂˜H (t, x) ∂˜H (t, x) k UE device starting at t = 0 be x, i.e., X(0) = x. When x > 0, define − Dk = Q˜H (t, x), k = l, h, (9) ∂t ∂x A. Gautam, S. Dharmaraja / Physical Communication 30 (2018) 213–219 217

∈ ∗k where where bi,j, i, j S are the constants to be evaluated and ˜Hj (s, x) is a column vector which is given by ⎡ k k k k ⎤ H1,1(t, x) H1,2(t, x) H1,3(x, t) H1,4(t, x) ⎡ ∗ ⎤ ⎢ k k k k ⎥ ˜H k(s, x) k H2,1(t, x) H2,2(t, x) H2,3(t, x) H2,4(t, x) 1,j H (t, x) = ⎢ ⎥ ∗ ˜ ⎢ k k k k ⎥ ⎢ k ⎥ H (t x) H (t x) H (t x) H (t x) ∗k ⎢˜H2 j(s, x)⎥ ⎣ 3,1 , 3,2 , 3,3 , 3,4 , ⎦ H (s x) = , j ∈ S k = l h ˜j , ⎢ ∗ ⎥ , , , . k k k k ⎢˜H k(s, x)⎥ H4,1(t, x) H4,2(t, x) H4,3(t, x) H4,4(t, x) ⎣ 3,j ⎦ ∗k ˜H4,j(s, x) and Dk = diag[rik]i∈S,k=l,h. Also, initial conditions are given by Now, we have 16 equations which can be solved to obtain the b , i, j ∈ S, using the LST of boundary conditions which are given Hk (t, 0) =1 if t ≥ 0 and r ≤ 0, i ̸= j, i,j ˜i,j ik by l H (t, B) =0 if t ≥ 0 and rik ≥ 0, i ̸= j, ˜i,j ∗ 1 H k(s 0) = if r ≤ 0 i ̸= j h = ≥ ≥ ̸= ˜i,j , ik , , ˜Hi,j(t, C) 0 if t 0 and rik 0, i j, s k = ≤ Hl (s, B) =0 if r ≥ 0, i ̸= j, ˜Hi,j(0, x) 0 if 0 < x C, i,j ik h k = ̸= ∈ = H (s, C) =0 if rik ≥ 0, i ̸= j, ˜Hi,j(0, 0) 0 if rik > 0, i j, i, j S, k l, h. (10) i,j ∗k = ≤ ˜Hi,j (0, x) 0 if 0 < x C, ∗k = ̸= ∈ = 4.1. Distribution of the battery life Hi,j (0, 0) 0 if rik > 0, i j, where i, j S, k l, h. (15) Now, on using the conditions given in Eq.(15) for solving Eq.(14), For solving system of PDEs(9), we take Laplace Stilijes Trans- we obtain equations corresponding to each j ∈ S, respectively, form(LST) of PDE with respect to t on both sides of(9). which can be used further in simplifying the solution. k Define LST of Hi,j (t, x) as ∫ ∞ 4.2. The expected battery life ∗k −st ∂Hi,j(x, t) Hi,j (s, x) = e dt, i, j ∈ S, k = l, h, 0 ∂t Let X(0) = x0 and Y (0) = y0 for some given finite and allowable and a row (or column) vector of LSTs of functions is the LST of x0 and y0. row (or column) vector of functions, respectively. Also, it exists if Now, to obtain the expected battery life, we have ∂H (x,t) i,j is piecewise continuous in [0, ∞), and is of exponential or- ∂t 4 der. Thus,(9) reduces to following system of Ordinary Differential −sW k ∑ ∗k E[e ] = H (s, x0), k = l, h, (16) Equations (ODEs): ˜y0,j j=1 ∗k ∗k k dH (s, x) ∗k k s˜H (s, x) − ˜H (0, x) − Dk = ˜H (s, x)Q, k = l, h. where W , k = l, h are defined in Eq.(3). Hence, further moments dx can be obtained as follows: Now, on using the initial condition given by Eq.(10), we obtain k m [ −sW ] m k m d E e ⏐ ∗ E [W ] = (−1) ⏐ , k = l, h, m = 1, 2,.... (17) k m ∂˜H (s, x) ∗k ds ⏐s=0 Dk = (sI − Q)˜H (s, x), k = l, h, (11) ∂x For m = 1, Eq.(17) gives the expected battery life. ∗ Also, the probability that the battery life ends in state j ∈ S is given where ˜H k(s, x) is the LST of ˜Hk(t, x). Now, the ODE obtained in the system(11) is easier to solve, and by the LST of the CDF for the battery life W k k = l h can be found. To k , , P{Y (W ) = j|X(0) = x0, Y (0) = y0} obtain it, first, we solve the ODE. ∗ = Hy (∞, x0) = H (0, x0), k = l, h. (18) Let Z1(s), Z2(s), Z3(s), and Z4(s) be four scalar solutions of the ˜ 0,j ˜y0,j characteristic equation obtained for the matrix system in Eq.(11), k = { | = = } ∈ = Now, define Wi inf t > 0 X(t) 0, Y (0) i , i S, k l, h. which is given by k Then, Wi is the battery life of the UE initiating in state i. ∈ = det(DkZj(s) − sI + Q) = 0, j ∈ S, k = l, h. (12) For i, j S, k l, h, consider U : the residual time in state i, Corresponding to each Z (s), j ∈ S, we can find a column vector i j L : the state of the CTMC obtained for DRX mechanism ψ (s) which satisfies j which is entered after state i.

Zj(s)Dkψj(s) = (sI − Q)ψj(s), j ∈ S, k = l, h. (13) k Random variables Wj and Ui are independent. Further, let Hence, for a given s, Z1(s), Z2(s), Z3(s), and Z4(s) are the eigen- values, and ψ (s), ψ (s), ψ (s), and ψ (s) are the respective right { qij 1 2 3 4 , for i ̸= j, qi eigenvectors. Since for a given real matrix A with a real eigenvalue pij := (19) λ, then there exists a real non-zero column vector v such that qi = −qii for i = j.

Av = λv. For i, j ∈ S, assuming that background process {Y (t), t ≥ 0} is in state i at time t = 0, we have Thus, the solutions to a system of ODE(11) are given by ∫ ∞ ∗ −sW k ∗ H k(s, B) = e j H k(t, B)dt, ∗k Z1(s)x Z2(s)x ˜i,j i,j ˜Hj (s, x) = b1,j(s)e ψ1(s) + b2,j(s)e ψ2(s) 0 k Z3(s)x Z4(s)x −sW + b3,j(s)e ψ3(s) + b4,j(s)e ψ4(s), =E[e j ], k ∈ = −s(Ui+W ) j S, k l, h (14) =E[E[e j |L = j]], 218 A. Gautam, S. Dharmaraja / Physical Communication 30 (2018) 213–219

Table 2 Parameters values. λ = 3 s−1 C = 1QB = 0.2Q −1 −1 −1 q12 = α1 = 2 s q23 = α2 = 1 s q34 = α3 = 1 s −1 −1 −1 −1 q41 = β1 = 1 s q42 = β2 = 1 s q31 = β3 = 1 s q21 = β4 = 1 s −1 −1 −1 −1 µ1h = 2 s µ2h = 1.8 s µ3h = 1.6 s µ4h = 1.4 s −1 −1 −1 −1 µ1l = 1 s µ2l = 0.8 s µ3l = 0.6 s µ4l = 0.4 s

− k ∑ −sUi sWj = pijE[e ]E[e ], k = l, h, (20) j̸=i where pi,js are the transition probabilities which are obtained in Eq.(19). Since Ui is the residual time in the state i ∈ S of the stochastic process {Y (t), t ≥ 0}, which is an irreducible CTMC then, for i ∈ S, we have ∫ ∞ − − qij [ sUi ] = sUi = ̸= ∈ E e e fUi (u)du, , i j, j S, (21) 0 qij + s where fUi (u) is the probability density function of Ui and qijs are the transition rates from state i to state j corresponding to the Fig. 4. CDF of the battery life. CTMC {Y (t), t ≥ 0}. Hence, on substituting Eqs.(16),(19) and(21) in Eq.(20), we get q q ∗k = ∑ ∗k iv iv ˜Hi,j (s, B) ˜Hv,j(s, B) −qii qiv + s v∈S,v̸=i,rvj≥0 q q + ∑ ∗k iv iv = ˜Hv,j(s, B) , k l, h. (22) −qii qiv + s v∈S,v̸=i,rvj<0 Authors have verified the result obtained in Eq.(22) with that of Ref. [20], which validates the proposed analytical model in theory. Proposed analytical model is a notably generalizable model, and produces results for which a comparison with the state-of-the-art technology is not achievable. It is due to the fact that the purpose of this work is, not to give experimental results and discussion, but to offer the guidelines for providing an advanced theoretical Fig. 5. Expected battery life of the UE. understanding of the battery life.

5. Numerical illustration It is an easy and accurate analytical model for the performance Since the authors did not find any existing literature having analysis of the charge content in a UE in LTE-A networks. This such implementation in LTE-A networks then, the authors have work can be used in providing the important guidelines for the taken these values in consideration of the analysis performed improvement in Quality of Service of the user. The battery life in [25], and are mentioned in Table2. Further, one can use the dif- distribution, and the expected battery life are the performance ferent values and can get the results from the proposed analytical measures which are obtained and analyzed numerically for the model. considered model. CDF obtained for Eq.(22) is depicted graphically in Fig.4. We In future, we plan to explore battery life distribution for a non- observe, as the first level of charge, i.e., as the value of B increases, Markovian DRX mechanism. Further, we propose to investigate the CDF decreases. It means when the level of charge takes values, the battery life of a UE for the other power saving techniques. In B = 0.2Q 1 and B = 0.4Q , the discharging rate of the battery is the manuscript, the analytical model and the numerical results are provided. In future, for the application purpose, various constraints decreased sharply, i.e., µih to µil, then the battery life finishes faster. Fig.5 presents the expected battery life which is obtained can be implemented to obtain a better regularization to the solu- using Eq.(17). The battery life of the UE with and without DRX tion. Also, when the constraints/priors are modified in the above mechanism, is presented graphically. We observe that with the mentioned analytical model, it is possible to update the model inclusion of DRX mechanism after a point, the battery life increases accordingly as a future research work. sharply.

6. Conclusion and future work Acknowledgments

Most of the work conducted in this area do not involve the Initially, this research was supported by National Board of analysis of the battery life using fluid queue. In this work, a fluid Higher Mathematics (NBHM), India under project grant Transient queue model dealing with the battery life of a UE during a power Analysis of Fluid Queues. Anupam is supported through a se- saving mode, called DRX mechanism, is proposed and investigated. nior research fellowship from Council of Scientific and Indus- trial Research (CSIR), India. Authors are thankful to the editor 1 Reader should not get confused for Q mentioned in Page 13 (here as unit of and two anonymous reviewers for their valuable suggestions and charge) and Q (otherwise, as infinitesimal rate matrix) mentioned before. comments which helped improve the paper to great extent and A. Gautam, S. Dharmaraja / Physical Communication 30 (2018) 213–219 219 gratefully acknowledges for the financial support received from [22] R. Bolla, R. 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He appointed as tion, in: Proceedings of 15th IEEE Real-Time and Embedded Technology and ‘Jaswinder & Tarvinder Chadha Chair Professor’ for teaching and research in the area Applications Symposium (RTAS), 2009, pp. 215–220. of Operations Research from May 2010 to July 2015. He has held visiting positions [18] M.W. Kim, D.G. Yun, J.M. Lee, S.G. Choi, Battery life time extension method at the Duke University, USA, Emory University, USA, University of Calgary, Canada, using selective data reception on smartphone, in: Proceedings of The IEEE University of Los Andes, Bogota, Colombia, National University of Colombia, Bogota, International Conference on Information Network, 2012, pp. 468–471. Colombia, University of Verona, Verona, Italy, Sungkyunkwan University, Suwon, [19] P.J. Hsieh, G.Y. Lin, C.Y. Chen, H.Y. Wei, Accurate modeling of the DRX mecha- Korea and Universita degli Studi di Salerno, Fisciano, Italy. nism with predetermined DRX cycles based on the 3GPP LTE standard, Mobile His research interests include applied probability, queueing theory, stochastic Netw. Appl. 21 (2) (2012) 259–271. modeling, performance analysis of computer and communication systems and [20] A. Rahmati, A. Qian, L. Zhong, Understanding human-battery interaction on financial mathematics. He has published over 40 papers in refereed international mobile phones, in: Proceedings of the 9th International Conference on Human journals and over 20 papers in refereed international conferences in these areas. He Computer Interaction with Mobile Devices and Services, 2007, pp. 265–272. is an Associate Editor of International Journal of Communication Systems and an [21] S. Jin, W. Yue, C. Meng, Z. Saffer, A novel active DRX mechanism in LTE Associate Editor of Opsearch. He is co-author of a text book entitled ‘‘Introduction technology and its performance evaluation, J. Ind. Manag. Optim. 11 (3) (2015) to Probability and Stochastic Processes with Applications’’ in John Wiley (US Edition, 849–866. New Jersey, June 2012) and (Asian Edition, New Delhi, Jan. 2016) and co-author of a text book entitled ‘‘Financial Mathematics: An Introduction’’ in Narosa, Nov. 2012.