Performance Analysis of Battery Power Management Schemes in Wireless Mobile. Devices

Transcription

1 Performance Analysis of Battery Power Management Schemes in Wireless Mobile Devices Balakrishna J Prabhu, A Chockalingam and Vinod Sharma Det of ECE, Indian Institute of Science, Bangalore, INDIA Abstract In this aer, we analyze the erformance of battery ower management schemes in wireless mobile devices using a queueing theory aroach We model the battery as a server with finite service caacity and data ackets as customers to be served With an intent to exloit the recharging caability of the battery when left idle, we allow the battery to go on intentional vacations during which the battery can recharge itself The recharge thus built u can effectively increase the number of customers served (in other words, battery life can be extended) Such imroved battery life erformance would, however, come at the exense of increased acket delay erformance We quantify the battery life gain versus delay erformance trade-off in this aroach through analysis and simulations By considering a continuous recharge model of the battery, we derive exressions for the number of customers served and the mean delay for an M=GI=1 queueing system without and with server vacations We show that allowing intentional vacations during busy eriods hels to increase battery life, and that this aroach can be beneficial when alied on traffic of delay-tolerant alications We also roose a acket delay constrained ower saving algorithm that will exloit the recharge henomenon when acket delay constraints are imosed I INTRODUCTION Portable user terminals for mobile communications must rely on limited battery ower for their oeration [1] It has been shown that energy savings in ortable devices can be sought at different layers of the wireless rotocol stack [2],[3], not necessarily at the devices/circuits level alone Also, several studies characterizing the battery discharge behavior have shown that ulsed discharge erforms better than continuous discharge [4]-[5] Particularly, the battery can recharge itself (ie, recover the otential) if left idle after discharge This recharge henomenon is illustrated in Fig 1 Figure 1 shows the battery otential (in volts) as a function of time with continuous discharge and ulsed discharge The arameter d reresents the duty cycle of the ulsed discharge attern (ie, d = :1 means 1% discharge time followed by 9% relaxation time, and so on) The lots in Fig 1 are generated using the battery simulation rogram develoed by the Chemical Engineering deartment at UC, Berkeley [6] A 2:76V Lithium ion battery with a cutoff voltage of 1:9V is considered It is assumed that the battery ceases to deliver ower once the voltage dros below the cutoff voltage In other words, the time This work was suorted in art by the Deartment of Science and Technology, Govt of India, New Delhi, under scheme III5(32)/99-ET taken for the battery to fall below the cutoff voltage is the battery life The sloe of battery discharge is determined by the discharge current density Larger the value of this current density steeer will be the discharge sloe, and hence lesser will be the time taken to reach the cutoff voltage In Fig 1 the discharge current density is taken to be 5A=m 2 From Fig 1 it can be seen that if the battery is discharged continuously it takes about 7:5 minutes to reach the cutoff voltage On the other hand, if the battery is discharged in ulsed mode, the battery recovers the voltage during the relaxation eriods and it takes more time to reach the cutoff voltage For examle, if the duty cycle of the ulsed discharge is 5% (ie, d =:5) it takes more than 4 minutes to reach the cutoff voltage Also, since the duty cycle is 5, the total on-time of the battery is more than 2 minutes which is about three times the on-time in continuous discharge mode As the discharge current density is taken to be the same in both continuous and ulsed discharge modes, this essentially means that the battery can deliver energy for a longer duration This recharge effect advantage in ulsed mode can be exloited for imroved energy efficiency in acket communications in wireless mobile devices The contribution in this aer is the roosal of methods to exloit the recharge henomenon in batteries and the analysis of battery life gain and acket delay erformance using a queueing theory aroach Recently, in [7] and [8] Chiasserini and Rao roosed a robabilistic recharge model for the battery Through simulations, they showed that the recharge henomenon can be exloited for battery life gain through suitable traffic shaing algorithms and battery level sensed routing strategies [8],[9] In this aer, we model the battery as a server with finite caacity and the arriving ackets at the mobile terminal as the customers to be served Each transmitted acket consumes energy roortional to the acket size, transmission bit rate, wireless link design, etc We roose to intentionally allow the server (battery) to go on vacation for calibrated amount of time, essentially allowing idle times for the battery to recharge itself By doing so, the number of customers served can be increased (in other words, battery life can be increased) We derive exressions for the number of customers served and the average delay for an M=GI=1 queueing system without and with server vacations We show that allowing forced vacations /2/$17 (c) 22 IEEE 825

2 during busy eriods hels to increase battery life However, since the customers (ackets) have to wait in the queue when the server goes on vacation, the battery life gain will come at the exense of increased delay erformance of the ackets It would be of interest to guarantee secified acket delay erformances in ractical systems Hence, in order to achieve bounded delay erformance, we further roose an algorithm that will exloit recharge henomenon when acket delay constraints are imosed Battery voltage (V) 28 d=1 27 d=5 d=9 26 continuous Time (min) Fig 1 Battery otential versus time for different duty cycles of discharge Current density =5A=m 2 Cutoff voltage =1:9V recharge robability has been assumed to follow an exonential function that decreases with increasing discharged caacity In other words, the recharge caability of the battery at any time is made roortional to the charge available in the battery at that time (ie, more the available charge, more is the ability to recharge) In addition to the nominal caacity N, the battery is assumed to be characterized by another arameter called the theoretical caacity, T charge units [7] The battery is considered to be fully drained if either the available charge goes to zero or T charge units of caacity has been delivered, which ever occurs first Tyically, the theoretical caacity of the battery is taken to be one or two orders higher than the nominal caacity [7] Battery Charge 1 2 l r1 l ll r1 e!!!!q s QQQQΦ e r2φh s HH e H H Hψ r3ψs bb e b bb b II SYSTEM MODEL Consider data acket transmission from a mobile terminal that draws ower from a battery of finite caacity We model the system as a queueing system with the battery as the server and ackets as customers, as shown in Fig 2 The acket interarrival time is assumed to be exonential (ie, Poisson arrivals with rate ) and service times are assumed to be iid with a general distribution We assume that the battery has a nominal caacity of N charge units That is, if the transmission of a unit length acket consumes one charge unit and if the battery is discharged continuously, then the battery can serve N such ackets till it gets comletely discharged Packets - Fig 2 Buffer ν» μ Battery Battery modeled as a server Battery Discharge/Recharge Model: In [7],[8], the battery discharge behavior is assumed to be discrete, ie, each acket occuies a fixed slot size, and each acket transmission consumes one charge unit (or integer number of charge units) Also, in [7],[8] the battery recharge behavior (when the battery remains idle) too is modeled as a discrete rocess, ie, the battery recharges by one charge unit with some robability if left idle for one slot duration The s: denotes the start of a busy eriod e: denotes the end of a busy eriod r s: recharge sloes (r i 1 >r 2 >r 3) Fig 3 Continuous discharge/recharge model Time In this aer, we consider both the discharge as well as the recharge behavior of the battery as continuous henomena, as illustrated in Fig 3 While serving ackets in busy eriods, the battery looses charge linearly at a constant sloe of unity During idle eriods, the battery recharges linearly with varying sloes deending on the battery level 1 at the beginning of the idle eriod The recharge model is more clearly exlained as follows We divide the range of charge from to N using P threshold values, 1; 2; :::; P The recharge sloe is taken to be r1;r2;:::;r P+1, resectively, when the battery level at the beginning of the idle eriod is in the range 1 to N, 2 to 1,, to P By choosing r1 >r2 >:::>r P+1, the model ensures that the ability to recharge reduces with decreasing battery level which is more realistic [6] Thus, the arameters P, i s, and r i s characterize the recharge behavior of the battery, which can be used in the mathematical analysis of the battery life gain due to recharge during idle/vacation eriods III ANALYSIS We are interested in analyzing the battery erformance in terms of mean number of ackets served and mean acket delay when the battery s recharge behavior is exloited by intentionally allowing the battery to go on vacations 1 The term battery level is used to denote the amount of charge resent in the battery /2/$17 (c) 22 IEEE 826

3 We first analyze and quantify the battery life gain due to idle eriods (ie, when buffer is emty) inherent in an M=GI=1 queue We then analyze a system where the server exhaustively serves all ackets in the queue and takes intentional vacations before the start of the next busy eriod We call this as a system with exhaustive service with vacations We also roose a non-exhaustive service system with vacations where the server takes vacations during the busy eriod First consider a simle M=GI=1 queue Define a cycle as the time from start of one idle eriod to the start of the next idle eriod A cycle consists of one idle and one busy eriod as shown in Fig 4 Note that the first cycle starts after the first busy eriod In order to obtain the exected number of customers served, we 1) obtain the distribution of the amount of charge left at the end of the first busy eriod, 2) obtain the exected number of cycles after the first busy eriod till the charge goes to zero, and 3) obtain the exected number of customers served in each cycle, as follows Remaining Work = P [» N z] + = 1+ Z N z z Z N z z E z1 [fi ] df Z1 (z 1 ) E z1 [fi ] df Z1 (z 1 ); (3) where F Z1 (z 1 ) is the cdf of To obtain the distribution of Z i, we need to obtain the distributions of Y i and B i The distribution of Y i is obtained as follows Let T i denote the duration of the idle eriod of the i th cycle and z denote the charge at the beginning of the i th cycle Then and F Yi (y) = Y i = min(r k T i ;N z ); (4) ρ FTi (y=r k ) y» N z 1 y>n z ; where k 2f1; 2;:::;P+1g is determined by the charge threshold values between which z lies Since T i is distributed as ex( ), the cdf of Y i is given by F Yi (y) = ( 1 e y r k y» N z 1 y>n z : (5) (6) First busy eriod Cycle Time Comuting the distribution of B i for a general M=GI=1 queue may not be so easy Here, we limit ourselves to exonential service times with rate μ Then, B i has density [1] Fig 4 Definition of a cycle in the system For the M=GI=1 queue the idle eriod is distributed as ex( ) Let Z i = Y i B i ; (1) where B i is the charge consumed during the busy eriod and Y i is the amount of recharge during the idle eriod in the i th cycle, resectively Then, Z i is the total charge lost (Y i <B i ) or gained (Y i > B i ) in the i th cycle Let fi be the random variable denoting the number of cycles till the charge goes to zero, and let z be a random variable which denotes the charge at the end of the first busy eriod We need to find fi = inf ( nx n : Z i» z i=1 ) : (2) Let E z [fi ] denote the exected number of cycles given that charge after the first busy eriod is z We can then write the integral equation E z [fi ] = P [» z] + N z Z z (1 + E z1 [fi ]) df Z1 (z 1 ) f I 1 2b μ e b( +μ) Bi (b) = b ; (7) ρ where I 1 ( ) is the modified Bessel function of the first order, and ρ = μ The cdf of, F Z1 (z 1 ) can then be written as = N z z F Z1 (z 1 )= Z1 max(; z 1) F Y1 (z 1 + b)f B1 (b) db + max(; z 1) F Y1 (z 1 + b)f B1 (b) db (8) N z z 1 f B1 (b) db: Eqn (3) can be numerically solved to obtain E z [fi ] To obtain E[fi ] we average E z [fi ] over z,as E[fi ]= Z N E N z [fi ] df B (z); (9) where F B is the distribution of the first busy eriod The exected number of customers served can be obtained as the /2/$17 (c) 22 IEEE 827

4 roduct of the exected number of cycles and the exected number of customers served in a busy eriod as follows 2 C =(1+E[fi ]) 1 1 ρ : (1) Note that the exected number of customers served is uer bounded by the theoretical caacity of the battery, μt The mean delay of ackets in the M=M=1 queue is given by [11] W = μ 2 (1 ρ) : (11) A Exhaustive Service System with Vacations Next, we consider a system where the server takes intentional vacations The server exhaustively serves all ackets in the queue and takes a vacation If after comletion of vacation, there is still no acket in the queue, it takes another vacation, and so on We assume the vacation time distribution to be ex(ffi), where ffi can be a system arameter We roceed in similar fashion as before to derive the exected number of customers served and mean delay In addition to idle times in a simle M=GI=1 queue, the vacation times too contribute to the idle time distribution in this case The idle time distribution of this system can be shown to be ex(ffi), where = The distribution of +ffi Y i in this system with vacation is the same as that given in Eqn (5) In the system without vacation, the busy eriod starts as soon as a customer arrives following an idle eriod However, in the system with vacations considered in this subsection, the busy eriod may start after queuing u a random number of customers Hence, to find the distribution of the busy eriod in a cycle we need to condition on the number of customers waiting before the start of the busy eriod The Lalace transform of the busy eriod in a cycle, considering the secial case of exonential service times, can be written as 1X B (s) = n=1 [B(s)] n (n); (12) where B(s) is the Lalace transform of the busy eriod of a simle M=M=1 queue, and (n) is the robability of n customers waiting at the start of a busy eriod The exression for B(s) can be written as [1] μ B(s) = + + s (μ + + s) 2 4 μ : (13) 2 The exression for (n) can be derived as follows P (at least one arrival occurs during a vacation) = + ffi ; ( v) n P (n arrivals occur in a vacation) =ffi e ( ( +ffi)v) dv: n! 2 Actually this is an aroximation (in fact, an uer bound) because the last cycle may not be comlete Hence, P (n arrivals occur given at least one arrival occurs) = ffi ( v) n e ( ( +ffi)v) dv: (14) From Eqns (12), (13), (14), B (s) can be obtained as, B (s) = n! ffi B(s) + ffi B(s) : (15) The df of the busy eriod, f B, can then be obtained by inverting the above equation as [12] e 2( +ffi)u I1 4( + ffi) f B (b) = 2 b +2u μ ffi e ( +μ) b μb(b +2u) u b du: (16) From the above distributions of idle eriod, recharge, and busy eriod, the exected number of cycles E[fi ] can be obtained from Eqns (3) and (9) The exected number of customers served in the first busy eriod is (1 ρ) 1 The exected number of customers served in the subsequent busy +ffi 1 eriods is given by ffi 1 ρ customers served can then be written as C = 1 1 ρ + E[fi ] + ffi ffi The exected number of 1 1 ρ : (17) Here again, the exected number of customers served is uer bounded by the theoretical caacity of the battery The mean delay for this system with vacations can be written as [11] W = μ 2 (1 ρ) + 1 ffi : (18) B Non-exhaustive system with vacations Another way to imrove battery life is to interrut service during busy eriod and allow vacations Secifically, we allow the server to go on vacation after serving K ackets in a row during a busy eriod We assume the duration of this vacation to be iid and distributed as ex(ffi) Both ffi and K are arameters which can be chosen to allow desired vacation times In order to analyze this scheme, consider an aroximate system which models the non-exhaustive system with vacations at high arrival rates Assume the arrival rate is high so that there are always ackets to send in the queue In this case, we can take the busy eriod to be the sum of K service times Thus, for the case of ex(μ) service times the df of the busy eriod can be written as f Bi (b) = μk b (K 1) e μb : (19) (K 1)! /2/$17 (c) 22 IEEE 828

5 The vacation eriod is distributed as ex(ffi) We then use Eqns (5), (3), and (9) to obtain the exected number of cycles to discharge comletely, E[fi ] The number of ackets served C is then given by, No of ackets served C KE[fi = ] : (2) μ Analysis Simulation Fig 5 Number of ackets served versus arrival rate in the system without vacations N =1 1 =75, 2 =5, 3 =25 r1 =:4, r2 =:3, r3 =:2, r4 =:1μ =1 IV RESULTS AND DISCUSSION We consider the following system arameters: nominal caacity N = 1 charge units, number of thresholds P = 3, threshold values 1 =75, 2 =5, 3 =25, recharge sloes r 1 = :4, r 2 = :3, r 3 = :2, r 4 = :1, and service time arameter μ = 1 First, from Eqn (1), we comute the number of ackets served in a simle M=M=1 queue with battery recharge due to inherent idle eriods Figure 5 shows the number of ackets served for different arrival rates, obtained through analysis and simulation In the numerical evaluation of Eqn (3), the integral is discretized in stes of 1 and converted into a set of linear equations Further, instead of evaluating all the entries of the coefficient matrix we comute the entries for one row and use the same for the other rows for a given threshold region This significantly reduces the comutation time while maintaining the accuracy of the results as observed from the close match between the analytical and simulation results It can be observed that the recharge due to inherent idle eriods can increase the number of customers served to about three times the battery s nominal caacity of 1 charge units, articularly at low arrival rates (eg, 35 ackets served at =:3) Next, in Fig 6, we lot the number of customers served in the system with intentional vacations and exhaustive service (as er Eqn (17)) as described in Section III-A The corresonding mean delay erformance is lotted in Fig 7 The erformance curves are arameterized by the vacation time arameter ffi The erformance of the system without vacation (as er Eqn 1) is also lotted for comarison It is observed that allowing additional vacations during idle eriods does not rovide significant increase in the number of customers served For examle, when ffi = 1 the number of customers served increases only by less than 1% When the vacation times are made larger (smaller values of ffi, ffi =:5; :1 ) the number of ackets served, in fact, reduces when comared to the system without vacations This behavior can be attributed to two reasons: a) even if large vacation times are given, the battery will not be able to recharge beyond its nominal caacity N, and b) increasing vacation times allows more ackets to queue u resulting in longer busy eriods which deletes the battery charge faster The mean delay increases with increasing mean vacation time as seen in Fig 7 The usefulness of the above observation is that leaving vacation times in an exhaustive service system is not effective and alternate schemes to exloit the recharge effect need to be looked into One such alternate aroach is to start service as soon as the battery recharges to its nominal caacity The erformance of this aroach has been evaluated using simulations and the resulting number of ackets served for different values of ffi are lotted in Fig 8 It is observed that this aroach results in moderate gains in number of ackets served at moderate arrival rates (in the range =:3 to 5) However, at high arrival rates there is no significant gain Consequently, we investigate yet another aroach which is the non-exhaustive system with vacations as described in Section III-B No of ackets served delta=5 delta=1 delta= Fig 6 Number of ackets served versus arrival rate in the exhaustive service system with vacations for different values of ffi N =1 1 =75, 2 =5, 3 =25 r1 =:4, r2 =:3, r3 =:2, r4 =:1 μ =1 Figures 9 and 1 illustrate the number of customers served and the mean delay erformance of the non-exhaustive service system with vacations for various arrival rates obtained through simulations The lots are arameterized with arameter K for ffi = :5 It is observed that for low arrival rates both the systems without and with vacations erform similar, reaching the theoretical caacity of the battery, T, which is taken to be 1N For examle, when the arrival rate is quite low ( = :2) the inherent idle eriods themselves will allow sufficient recharge to deliver the theoretical caacity, and therefore allowing additional vacation times has no /2/$17 (c) 22 IEEE 829

6 25 mobile devices Mean Delay delta=5 delta=1 delta=1 No of ackets served K=1 K=2 K=4 K=6 K=8 K=1 Fig 7 Mean delay versus arrival rate in the exhaustive service system with vacations for different values of ffi N =1 1 =75, 2 =5, 3 =25 r 1 =:4, r 2 =:3, r 3 =:2, r 4 =:1 μ =1 No of ackets served Arrival rate delta=1 delta=1 delta=5 Fig 8 Number of ackets served versus arrival rate in the exhaustive service system with vacations limited to time to recharge to N for different values of ffi N =1 1 =75, 2 =5, 3 =25 r 1 =:4, r 2 =:3, r 3 =:2, r 4 =:1 μ =1 effect However, at high arrival rates the system with vacations increasingly serves more ackets than the system without vacations for increasing values of K For examle, taking a vacation each time a customer from the queue is served (ie, K =1) the theoretical caacity can be reached even at high arrival rates For larger values of K the number of ackets served initially decreases with increasing arrival rate u to some oint after which the number of ackets served increases This can be exlained as follows At low arrival rates the number of ackets in the queue will be less and hence the robability of taking vacations will be less leading to lesser recharge However, for large arrival rates the buffer will be mostly occuied leading to more vacations and more recharge The mean delay erformance gets increasingly worse with lower values of K Though K = 1 and 2 result in close to theoretical caacity in terms of number of customers served, the delay for K =2is significantly lower than that of K =1, indicating the ossibility of choice of the arameters of K and ffi that can result increased battery life with less degradation in delay Thus the non-exhaustive service system with vacations can be alied on the traffic of delay-tolerant alications ( , file transfers, etc) to increase battery life in wireless Fig 9 Number of ackets served versus arrival rate in the non-exhaustive service system with vacations for different values of K N = 1 1 =75, 2 = 5, 3 = 25 r 1 = :4, r 2 = :3, r 3 = :2, r 4 = :1 μ = 1 ffi =:5 Mean Delay K=1 K=2 K=4 K=6 K=8 K= Fig 1 Mean delay versus arrival rate in the non-exhaustive service system with vacations for different values of K N = 1 1 = 75, 2 = 5, 3 =25 r 1 =:4, r 2 =:3, r 3 =:2, r 4 =:1 μ =1 ffi =:5 V DELAY CONSTRAINED POWER SAVING The roosed schemes in Section III resulted in increased battery life at the exense of increased acket delay erformance The ackets delays are found to increase monotonically with increasing intentional relaxation times It would be of interest to guarantee secified acket delay erformances in ractical systems Hence, in order to achieve bounded delay erformance we consider the following algorithm for serving ackets ffl Consider Poisson arrival of ackets with rate and exonential service times with mean μ 1, so that ρ = =μ ffl The device maintains a count of the total number of ackets, N s, served so far and the sum of the delays exerienced by each acket, D s ffl After serving K ackets the device comutes the number of ackets, N c, which when served continuously will discharge the battery to its cutoff voltage We can aroximate the battery otential to be a linearly decreasing function of time when discharged at a constant current density This discharge sloe can be obtained a riori /2/$17 (c) 22 IEEE 83

7 from the battery discharge behavior as lotted in Fig 1 Using the knowledge of the resent battery voltage and the discharge sloe, the device can calculate the time it will take to reach the cutoff voltage and hence the number of ackets it can serve continuously, N c, and the sum of the delays that will be exerienced by these ackets when served continuously, D c ffl The device then calculates the average delay D as, D = D s+d c N s+n c ffl Let D be the delay constraint imosed in the system If D D then the battery stos taking further vacations and serves the remaining ackets in a continuous fashion till the cutoff voltage is reached On the other hand, if D<Dthen the battery continues to take a vacation for every K ackets served The erformance of this delay constrained algorithm has been evaluated using the battery simulation rogram from Berkeley [6] Figures 11 and 12 show the number of ackets served and mean delay erformance of the system with vacations and delay constraint The system erformance without vacations is also lotted for comarison The acket arrival rocess is Poisson with rate and the service time is exonential with mean μ 1 =:2sec The vacation time is taken to be exonential with rate ffi =:5μ K is chosen to be equal to 4 The current density is taken as 1A=m 2 It is seen that the system with vacation results in increased number of ackets served comared to the system without vacation Also, as the delay constraint is relaxed (eg, from D = 5secs to D =1secs) the number of ackets served is increased For examle, when ρ = :5 the system without vacation serves about 135 ackets whereas the system with vacations serves about 16 and 24 ackets for D =5secs and D =1secs, resectively Also, from Fig 12 it can be observed that the roosed algorithm ensures that the mean delay erformance never exceeds the imosed delay constraint Number of ackets served vacations, D=5 secs vacations, D=1 secs Rho Fig 11 Number of ackets served vs ρ = =μ μ = :2sec ffi=μ = :5 K =4 Mean delay (secs) vacations, D=5 secs vacations, D=1 secs Rho Fig 12 Mean delay vs ρ = =μ μ =:2sec ffi=μ =:5 K =4 aroach We modeled the battery as a server with finite service caacity and roosed schemes to allow intentional server vacations to exloit the battery recharge effect for increased battery life We analyzed and quantified the increase in battery life due to idle and vacation eriods in a M=GI=1 queue Systems with vacations having exhaustive and non-exhaustive service were considered We showed that a non-exhaustive service system with vacations can result in significant gain in battery life, and that this aroach can be beneficial when alied on traffic of delay-tolerant alications in wireless mobile devices We also roosed an algorithm that exloited the recharge henomenon when acket delay constraints are imosed REFERENCES [1] Energy management in ersonal communications and mobile comuting, secial issue, IEEE Personal Commun, June 1998 [2] A Chockalingam and M Zorzi, Energy efficiency of media access rotocols for mobile data networks, IEEE Trans on Commun, vol 46, no11, , November 1998 [3] W Heinzelman, A Chandrakasan, and H Balakrishnan, Energy efficient routing rotocols for wireless microsensor networks, Proc 33 rd Hawaii Intl Conf on System Sciences (HICSS ), January 2 [4] R M Lafollette and D Bennion, Design fundamentals of high ower density, ulsed discharge, lead-acid batteries II: Modeling, Jl Electrochem Soc, vol 137, no 12, , December 199 [5] B Nelson, R Rinehart, and S Varley, Ultrafast ulsed discharge and recharge caabilities of thin-metal film battery technology, 11 th IEEE Intl Pulsed Power Conf, , June 1997 [6] J S Newman, FORTRAN rograms for simulation of electrochemical systems htt://wwwcchemberkeleyedu/jsngr/ [7] C F Chiasserini and R R Rao, Pulsed battery discharge in communication devices, Proc Mobicom 99, August 1999 [8] C F Chiasserini and R R Rao, A traffic control scheme to otimize the battery ulsed discharge, Proc IEEE MILCOM 99, November 1999 [9] C F Chiasserini and R R Rao, Energy efficient battery management, Proc INFOCOM 2, March 2 [1] L Kleinrock, Queueing Systems Vol 1:Theory John Wiley, 1975 [11] D Bertsekas and R Gallager, Data Networks Prentice-Hall, Englewood Cliffs, New Jersy, 1987 [12] G E Roberts and H Kaufman, Table of Lalace Transforms W B Saunders Comany, 1966 VI CONCLUSIONS We analyzed the erformance of battery ower management schemes in wireless mobile devices using a queueing theory /2/$17 (c) 22 IEEE 831