Modem standards for wireless telecommunication, such as IEEE 802. 16s (WiMAX"), foresee in several power-saving mechanisms. One of the main mechanisms is sleep-mode operation, which allows the mobile device to switch off the antenna during a negotiated time (it goes to sleep), thus reducing energy consumption. However, 1гаШс that arrives al the base station (BS) incurs an extra delay, because its delivery can only start after the sleep period has ended. The sleep-mode mechanism in WiMAX has provisions to vary the lengths of subsequent sleep periods, which allows to exploit correlation in the arriving traffic. We depart from our earlier analyses and study the mode! of the sleep-mode mechanists specifically under light-traffic conditions. This results in relatively short and
simple formulas that give a lot of insight

Bruneel, Herwig

De Turck, Koen

Fiems, Dieter

Wittevrongel, Sabine

English

De Turck, K.

Fiems, D.

Wittevrongel, S.

Bruneel, H.

BSU Digital Library

Performance of sleep-mode mechanisms under light-traffic conditions

Ghent University Academic Bibliography

PERFORMANCE OF SLEEP-MODEMECHANISMS UNDERLIGHT-TRAFFIC CONDITIONSK. De Turck1, D. Fiems1, S. Wittevrongel1,H. Bruneel11 SMACS Research Group, Department TELIN1 Ghent University, Belgiumkdeturck@telin.ugent.beModern standards for wireless telecommunication, such as IEEE 802.16e (WiMAX),foresee in several power-saving mechanisms. One of the main mechanisms is sleep-modeoperation, which allows the mobile device to switch off the antenna during a negotiatedtime (it goes to sleep), thus reducing energy consumption. However, traffic that arrivesat the base station (BS) incurs an extra delay, because its delivery can only start afterthe sleep period has ended. The sleep-mode mechanism in WiMAX has provisions tovary the lengths of subsequent sleep periods, which allows to exploit correlation in thearriving traffic. We depart from our earlier analyses and study the model of the sleep-mode mechanism specifically under light-traffic conditions. This results in relativelyshort and simple formulas that give a lot of insight.Keywords: light-traffic approximation, power-saving mechanisms, wireless commu-nications, IEEE WiMAX1. INTRODUCTIONWe study the performance of the energy saving mechanism defined in the IEEE802.16e [1] standard for BroadbandWireless Access networks. Consider the downstreamcommunication between a Mobile Station (MS) and its serving Base Station (BS) overa wireless link. To allow the MS to extend its battery life, the standard provides asleep-mode mechanism that allows the MS to turn off its radio interface for a certaintime whenever the BS has no packets in its queue. Specifically, if the BS’s queue isempty, the MS starts a sleep period during which it remains powered down and thuscannot be reached by the BS. After this period, the MS is briefly reactivated to checkwhether there are packets waiting for it at the BS. If not, the MS initiates a secondsleep period, a third, and so on. However, if any packets arrived at the BS during thelast sleep period, the MS remains powered and enters awake mode. This allows the BSto transmit all the packets in its buffer exhaustively, after which the whole procedure isrepeated. In case of IEEE 802.16e, an exponential update strategy is used, where thesleep periods double in length every time until a certain maximum is reached. However,in this paper, we allow more general sequences of sleep period lengths.1Quite a few authors have already investigated the performance of the IEEE sleep-mode mechanism. We refer to [2] for an overview. Our paper was the first to take intoaccount traffic correlation.The structure of the paper is as follows. We elaborate on the model in section 2. Thelight-traffic technique is explained in section 3. We give the main results in section 4.Finally, we numerically verify the exact results we obtained in our previous paper [2]with the light-traffic results we derived here in section 5.2. MODEL AND PROBLEM STATEMENTWe model the BS as a discrete-time single-server queue with infinite capacity and afirst-come-first-served discipline. Time is divided into fixed-length intervals called slotssuch that changes in the system can occur at slot boundaries only. The service (i.e.transmission) times of the packets, expressed as a number of slots, are independentand have common probability generating function (pgf) S(z) and mean E[S]. Packetsarrive in the queue according to a D-BMAP with M phases. This arrival process ischaracterised by the values a(k, j|i), (k ≥ 0, i, j ∈ {1, · · · ,M}), denoting the proba-bility that, given phase i in a slot, there are k packet arrivals and the phase switchesto j in the next slot. These probabilities can conveniently be arranged in the M ×Mprobability generating matrix (pgm) A(z) with entries [A(z)]ij =∑∞k=0 a(k, j|i)zk. Thestationary probabilities of the phases are given by the entries of a row vector a thatsatisfies a = aA(1) and a1 = 1, where 1 is a column vector of ones of appropriatelength. The mean number of arrivals per slot is given by λ = aA′(1)1.The sleep-period update strategy is denoted by a sequence (t1, t2, . . .) where theintegers tn (n ≥ 1) are the lengths of the subsequent sleep periods of the same idleperiod. Let us also define τn =∑ni=1 ti (n ≥ 1) and τ0 = 0 indicating the starting slotsof the sleep periods relative to the start of the idle period. We assume that the lengthsof the sleep periods become equal above a certain value J , i.e. tn = tJ , for n ≥ J .We cite here the results we obtained in [2] regarding the virtual waiting time andthe energy consumption. We also derived the distributions of the queue content andthe packet delay, but in this paper we focus on the two aforementioned quantities. Thevector generating function W(z) is a row vector ofM generating functions with entriesdefined as follows:[W(z)]i =∞∑k=0Pr[w = k, b = i]zk, (1)where b and w are random variables denoting the phase and the virtual waiting timeat the beginning of a random slot in stationary regime. The virtual waiting time at atime instant is defined as the time until the server is ready to serve a virtual packetthat arrives exactly at that time instant. We found in [2] thatW(z) = w∞∑n=1A(0)τn−1(ztn − 1)(zI−A(S(z)))−1, (2)where w is a 1×M vector that can be determined by a variety of techniques.2We define the energy consumption E of the base station as the probability thatduring a random slot the antenna is switched on. Note that the antenna is switched onduring every busy slot and also during every last slot of a sleep period. We found thatE = ρ+w∞∑n=0A(0)τn1. (3)Although the above sketched solution technique is very powerful, there are a coupleof drawbacks to it. Firstly, computing the vector w is a numerically complex process,especially if the number of background states M is large. Secondly, the exact influenceof arrival correlation is not evident from the equations. We try to circumvent theseproblems by considering a Taylor expansion in the region that –for this application– ismost interesting from a practical point of view, namely where the traffic is low.3. LIGHT-TRAFFIC TECHNIQUEAlthough perhaps less well-known than their heavy-traffic counterparts, light-trafficapproximations have a long history, and very different approaches have been developedto tackle a wide variety of systems. For an excellent overview, see e.g. [4].It is very important how light-traffic conditions are reached: different ways of reach-ing low traffic can give different results. Here we take the so-called pi-thinning approach:we start from a ‘normal’ arrival process, and then consider the family of related pro-cesses where each arrival is retained with a probability pi, independent from anythingelse. Light-traffic conditions are then reached in the limit pi → 0.We want to obtain the mean virtual waiting time and the mean power consumptionunder light traffic. Generally, these quantities are studied by relating the distributionsat different time instants and then, by invoking stationarity, we get an equation for thedesired quantity. Under light traffic, we can take the more direct approach of averagingover all sample paths.For this technique to work, we assume that the operation of the system has startedat time t = −∞, and hence, assuming that the system is stable, has reached stationarityin slot 0. Let S be the set of sample paths. A sample path s for this model recordsthe arrival times and the service durations of each packet, resulting in two sequencesi1, i2, · · · and j1, j2, · · · , as well as a variable g signifying the number of slots until theservice of the very first packet starts. Note that g is uniformly distributed between 0and tJ − 1. Let ψ be the quantity of interest, for example the virtual waiting time Wat the beginning of slot 0 or the energy consumption E during slot 0. Then we cancompute the expectation of this quantity under pi-thinning as follows:Epi[ψ] =∑s∈SPpi(s)f(s), (4)where Ppi(s) measures the probability of a sample path under pi-thinning, and functionf(.) computes the quantity ψ for given sample path s. As the variable g and the servicetimes are independent of the arrival process, and moreover not subject to pi-thinning,3average over them separately. From henceforth we only include the arrival times in thesample path.Such probabilities are functions of pi, of which we can make a Taylor expansion.Under light traffic, we consider only coefficients in, say, the n lowest powers of pi. Aswe will see, this leads to an enormous reduction in the amount of sample paths we needto consider. Note that the probability matrix generating function of the arrival processunder pi-thinning is given by:Api(z) = A(1− pi + pi z), (5)which gives rise to the following Taylor-expansion in pi = 0:Api(z) =∞∑n=01n!A(n)(1)pin(z − 1)n=∞∑k=0zk∞∑n=k(−1)n−k(nk)1n!A(n)(1)pin=∞∑k=0zkCk(pi). (6)Note that the probability of a sample path can written as an infinite product oftransition matrices Ck(pi), premultiplied by vector a and postmultiplied by vector 1,where each k signifies the number of arrivals during a certain slot. From this we observethat the probability of a sample path with n arrivals is a power series in pi where thelowest non-zero coefficient is in pin.We state, without proof the following theorem, which shows the formula for a secondorder light-traffic approximation of a quantity ψ. The proof follows the lines of theexpositions in [3] and [5], but must be adapted to D-BMAP arrivals. It is quite technicaldue to a tricky interchange of limits.Theorem 1. The expectation of a quantity ψ with associated function f(.) underpi-thinning has the following second-order approximation:Epi[ψ] = f({}) + piλ+∞∑i=−∞(f({i})− f({})) (7)+ pi2+∞∑i1,i2=−∞R(|i1 − i2|) (f({i1, i2})− f({i1})− f({i2}) + f({}) +O(pi3), (8)where {i1, i2, · · · } denotes a sample path with arrivals in slots i1, i2 and so on. Thefunction R(.) is defined asR(0) =12aA′′(1)1,R(n) =12aA′(1)A(1)n−1A′(1)1, where n ≥ 1. (9)We can view R(.) as a sort of autocorrelation function.44. MAIN RESULTSThe functions fE(s) and fW (s) that determine respectively the energy consumptionand the unfinished work during slot 0 for a given sample path s follow quite straight-forwardly from the system equations. For two arrivals, the complete definitions arealready quite lengthy, and hence we omit them in favor of the results themselves.After some tedious manipulations, we find from formula (8) that the energy con-sumption is approximated byEpi[E] =1tJ+ piλ{E[S] + J − 1−E[S] + τJ−1tJ}++2pi2J−1∑j=0Xj(τJ−1 − τjtJ− J + 1 + j)− R(0)(τJ−1 − τjtJ− J + 1)+O(pi3),(10)where the Xjs are defined asX0 =1tJtJ−1∑i=0∞∑k=1Pr[S = k]k+i∑n=0R(n) (11)Xj =1tJtJ−1∑i=0∞∑k=1Pr[S = k]k−1∑m=0R(k +m+ i+ τj−1), if 0 < j < J. (12)This expression for P [E] has an intuitive basis. The constant term tells what happenswhen there are no arrivals at all: the system will forever be in the sleep period withlength tJ , and hence have an energy consumption1tJ. In the first order derivative wesee the amount of extra energy consumption induced by one arrival. The second orderderivative shows the influence of the overlap of two packet ‘life times’ (service time plussubsequent sleep periods). The terms Xj are such that pi2Xj gives the probability ofhaving a sample path with two arrivals that have exactly j sleep periods in between.For the mean virtual waiting time we find:Epi[W ] =tJ − 12+ piλ{12E[S(S − 1)] + TJ−1 − τJ−1tJ − 12}++2pi2{J−1∑j=0Xj(tJ − 12τJ−1 − TJ−1 + E[S])+ Y E[S]−X0 E[S]2++J−1∑j=1Xj(tJ − 12(E[S]− τj)− Tj − E[S]tj)}+O(pi3), (13)whereY =1tJtJ−1∑i=0∞∑k=1Pr[S = k]τJ−1+k−1∑n=0nR(n) (14)Tj =j∑i=1ti(ti − 1)2(15)55. NUMERICAL EXAMPLE0 0.005 0.01 0.015 0.0220222426283032λ′Epi[W]0 0.005 0.01 0.015 0.020.020.040.060.08λ′Epi[E]Fig. 1. Plots showing the exact curve of Epi[E] and Epi[W ] vs. λ′ := λpi as well as their firstand second order approximations (in straight lines, dashes and long dashes respectively).We verified the obtained light-traffic limits with our earlier results for an on/offarrival source and for sleep strategy (4, 8, 16, 32, 64, · · · ) and found indeed an agreement.6. CONCLUSIONLight-traffic approximations are a very interesting tool for the analysis of power-saving protocols. We can not only reconstruct our previous work under light traffic,but all sorts of generalizations seem possible as well. It is remarkable that the firstorder approximation is insensitive to traffic correlation. It would be interesting to findout whether this is due to either the applied thinning scheme or the Markovian trafficcorrelation, or whether this can be established more generally.REFERENCES1. IEEE 802.16e-2005 Part 16: Air interface for fixed and mobile broadband wirelessaccess systems // 2006.2. De Turck K., De Vuyst S., Fiems D., Wittevrongel S. Performance analysis of theIEEE 802.16e sleep mode for correlated downlink traffic // Telecommunicationsystems 2008. V. 39. No 2. P. 145–156.3. Reiman M., Simon B. Open queueing systems in light traffic // Mathematics ofoperations research, 1989. V. 14. No. 1. P. 26–59.4. Kovalenko I. Rare events in queueing theory. A survey // Queueing systems 1994.V. 16. No 1. P. 1–49.5. Tsoukatos. K. Heavy and light traffic regimes for M |G|∞ traffic models // PhDthesis. 1999.6

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Performance of sleep-mode mechanisms under light-traffic conditions

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