数理解析研究所講究録1114 巻 1999 年 46-57

46

The Optimal Auto Sleep Scheduling for a Computer Systemwith Batch Arrival of$\mathrm{n}\mathrm{a}\mathrm{n}\mathrm{S}\mathrm{a}\mathrm{C}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}\mathrm{S}}$岡村寛之 \dag er , 土肥正 \dag er , 尾崎俊治 \dag er
and SHUNJI, TADASHI

HIROYUKI\dagger Department of Industrial and Systems Engineering, Hiroshima University
4-1 Kagamiyama 1 Chome, Higashi-Hiroshima 739-8527, Japan.
$\mathrm{D}()\mathrm{H}\mathrm{I}^{\uparrow}$$\mathrm{o}\mathrm{K}\mathrm{A}\mathrm{M}\mathrm{U}\mathrm{R}\mathrm{A}\dagger$$\mathrm{O}\mathrm{S}\mathrm{A}\mathrm{K}\mathrm{I}^{1}\perp$

Abstractof how to determine the optilnal sleep tinzing
This paper addresses awhen the computer user should turn the hard disk or the display off in order to savethe electric power after the computer has not been accessed. We consider a stochastic
lnodel to obtain the optimal sleep timing strategy which lnininlizes the expected power
consumed per unit time in the steady-state, where access requirements arrive at theaccording to a renewal process alld contain sonle lightweight processes. Then.
the expected
form for the idle period is proposed toapproximation
theprocess. We also derive the condition
per unit time based on theto exist the unique optimal sleep timing.
$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{e}\ln$$\mathrm{s})^{r}\mathrm{S}\mathrm{t}\mathrm{e}\ln$$\mathrm{r}\mathrm{e}_{\mathrm{P}^{\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{t}}}$$\mathrm{p}\mathrm{o}\mathrm{v}\iota r\mathrm{e}\mathrm{l}$$\mathrm{w}\mathrm{o}\mathrm{l}\cdot \mathrm{k}\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{d}$1Introduction)
the automatic sleep function of the hard disk or the display in a conlputer
. the
. In
to be important in terms of poweris rapidlv
Then.
computer systems as a standard function.
auto-sleep function is equipped inproblenl. ill particular.
is the mostauto-sleep
functionthefordesignoptimaltheexanlple.
on the hard disk ofwith lilnited capacity of battery. For
for notebook
up from sleep mode is larger than thata computer, the electronic power .onsumed tonotalways effective to designitisoperation.
Thust
consumed by the normalsuch tllat moves its state to the sleep mode whenever there is no access requirement.
First. the optimal design problem for tlle auto-sleep function was collsidered by Sandoh.
. Kaio and Osaki [5] also
Hirakoshi and Sandoh [4].
Hirakoshi and Kawai [3]for the samethetoa nollp(
)
(
above
.
it is
.
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of
and wassaki .
.
.
valid
auto-sleepIII( .
two killtl,h of models\v{c}rt
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of
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(
.
job
at the[.
(thIItlllif.
desktopacilc
areill which the other accessto a buffer{ ask
as network
thewllell one job has beell procebsed. alld collsi(
are thebothforstrategies
printers. They proved that the optimal sleep timing
always the system off after the process for ajob is colllpleted
switching strategies,
if the access requirelnents arrive according to the homogeneous Poissonor not do at$\mathrm{s}\mathrm{y}\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{n}$$\mathrm{R}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{l}\mathrm{y}\backslash$$\mathrm{f}\not\subset \mathrm{i}L\mathrm{t}$$1\overline{1}1\mathrm{a}\mathrm{n}\mathrm{a}_{\mathrm{e})}^{\sigma}\mathrm{e}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{t}[1.2]$$\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{o}^{\sigma}\mathrm{n}\mathrm{i}\mathrm{Z}\mathrm{e}\mathrm{d}\circ$$\mathrm{a}1_{1}\mathrm{n}\mathrm{o}\mathrm{s}\mathrm{t}$$\mathrm{i}_{111}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$$\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{p}\iota \mathrm{l}\mathrm{t}\mathrm{e}\mathrm{l}\cdot \mathrm{S}$$\mathrm{w}\mathrm{a}\mathrm{r}\ln$$\mathrm{c}$$\mathrm{t}1_{1\mathrm{e}\mathrm{S}}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\ln$$\mathrm{D}\mathrm{o}1_{1}\mathrm{i}$$\mathrm{a}\mathrm{l}\overline{\perp}\mathrm{d}$$\mathfrak{i}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{C}^{\cdot}11\perp \mathrm{c}\mathrm{f}\mathrm{l}\iota \mathrm{o}\mathrm{c}\mathrm{l}$$\mathrm{e}\downarrow \mathrm{s}\mathrm{t}\mathrm{i}_{1}11\mathrm{a}\mathrm{t}\mathrm{e}$$\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}1\overline{1}1\mathrm{c}‘\iota 1,\mathrm{s}\mathrm{l}\mathrm{e}\mathrm{e}_{1^{)}}\mathrm{t}\mathrm{i}_{11}1\mathrm{i}_{1\overline{1}}\mathrm{g}$$\mathrm{I}^{\mathrm{J}1\mathrm{O}}.\mathrm{P}^{\mathrm{o}\mathrm{s}}\mathrm{e}\mathrm{d}$$11\mathrm{o}\mathrm{t}_{\mathrm{t}^{\mathrm{Y}}(}1\mathrm{t}11i\backslash \mathrm{t}\mathrm{t}11$$\mathrm{H}\mathrm{o}\backslash \iota^{\tau}(\backslash 1\tau(^{3}1^{\cdot}$$|*\mathrm{c}\mathrm{m}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{h}^{7}\mathrm{O}1^{\cdot}\mathrm{k}‘ \mathrm{b}$$\mathrm{s}_{\llcorner}\mathrm{i}\ln_{1}$$1\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{e}(1\mathrm{t}11(^{\backslash }1111(\iota_{\mathrm{Q}\mathrm{l}1}\backslash r\mathrm{i}11_{\mathrm{o}}()$$\mathrm{I})1^{\cdot}0])1_{\mathrm{C}\mathrm{m}\iota}\mathrm{b}$,$\mathrm{b}\mathrm{t}()(1_{1}\partial \mathrm{s}\mathrm{t}\mathrm{i}(\rceil_{)(^{\mathrm{Y}}}1_{1\zeta}’\backslash \backslash \cdot \mathrm{i}()1^{\cdot}$$\mathrm{f}\mathrm{o}1^{\cdot}\mathrm{n}1111\mathrm{c}‘ \mathrm{t}\mathrm{t}\mathrm{i}_{0}1\perp \mathrm{h}\backslash 1^{\tau}(^{1}1(^{1}111_{\zeta}\mathrm{t}(1(\backslash \rceil)\backslash ()1\sigma\zeta\prime \mathfrak{j}11\perp\rceil 11^{\cdot}C\mathrm{t}$$\beta_{\backslash }\mathrm{I}\mathrm{o}\Gamma(^{\Delta}$$\mathrm{h}\lambda^{}.*\uparrow(1111$

$-$

$]$

$\mathrm{t}11(^{\backslash }$$\mathrm{i}_{11((1}11\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{t}()\mathrm{f}\mathrm{o}11(^{\mathrm{y}}1^{)1\mathrm{C}\backslash \mathfrak{c}^{\backslash }},111\mathrm{a}\mathrm{f}\mathrm{i}\mathrm{o}11$$1)101)1(^{\backslash }111\mathrm{t}^{}\mathrm{X}\mathrm{t}1^{\cdot}(^{\backslash }111\mathrm{e}1_{\lambda}$$(^{r}1_{\lambda^{r}}’ 1)(^{1}$$l’ 11\mathrm{c}\backslash 1^{r}1^{)1\mathrm{O}}1^{)}(\mathrm{s}(^{\mathrm{y}}(1$$111((\iota(^{\tau}1(‘\backslash 11(1r1^{\backslash }\backslash _{1^{)(}}^{\tau}\backslash$$\mathrm{t}11(^{\backslash }.rightarrow 1^{r}\mathrm{h}\mathrm{t}(^{\backslash }11\perp$$\backslash \backslash ^{\tau}\mathrm{i}\mathrm{t}\mathrm{h}()1\iota \mathrm{T}(\dot{\mathrm{r}}\backslash 11\Gamma..(\Delta 11\mathrm{c}\mathfrak{i}\mathrm{t}\mathrm{i}()11$$\mathrm{c}\backslash \mathrm{C}((\backslash ,\mathrm{b},*1(^{1}(1^{1}1\mathrm{i}\mathrm{l}\mathrm{e}^{\mathrm{Y}}111(1\mathrm{i}\iota \mathrm{t}_{h}\mathrm{d}1^{\cdot}1\mathrm{i}1^{-}\epsilon)(\mathfrak{j}$$\mathrm{D}$(($\rangle 1_{1}\mathrm{i}c11\}$

$1$

$()$

$[(\rangle$$1\mathrm{t}\mathrm{l}\mathrm{O}(\iota(^{\tau}1)\backslash \backslash ’ \mathrm{i}\mathrm{t}\mathrm{l}\mathrm{l}(\mathrm{t}11(\iota$$1^{\cdot}\mathrm{e}^{\mathrm{s}}{}^{\mathrm{t}}1^{)((}1t\mathrm{i}\backslash ()1\backslash \cdot$$\mathrm{h}1_{01(^{\mathrm{t}}}$$\rceil 11()(1_{\mathrm{C}^{\mathrm{l}1}}\tau\backslash ’ \mathrm{i}\mathrm{r}1_{1(}\cdot\dot{\subset}\mathrm{t}\mathrm{l}\mathrm{l}((111c\mathrm{t}\mathrm{t}\mathrm{i}()11\mathrm{c}‘|\mathrm{h}\mathrm{h}\mathrm{l}\mathrm{t}111(^{)},\mathrm{b}\mathrm{t}1_{1}\subset‘ 1\mathrm{f}()\mathrm{t}1_{1(1}\tau \mathrm{c}\backslash (.((^{\backslash }\mathrm{h}*1(\backslash (11\iota \mathrm{i}1(^{\backslash }111(’ 1)1‘\backslash \mathrm{t}\backslash 1\mathrm{l}\mathrm{i}\backslash \cdot \mathrm{G}(\iota$$\nwarrow 1)\mathrm{e}(\mathrm{i}\mathrm{f}\mathrm{i}(.(.\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{Y}.1’\backslash \backslash r1)\mathrm{e}$$111\backslash \backslash ^{\mathrm{T}}1_{1(^{3}\mathrm{n}}01\perp \mathrm{c}$$\mathrm{s}_{1}.\backslash ^{\gamma}.’$11\mathrm{c}‘ \mathrm{t},\backslash \rceil$$\mathrm{c}‘ \mathrm{i}11(\iota \mathrm{f}()(1\iota\nwarrow \mathfrak{t}\backslash ‘ \mathrm{s}()11\mathrm{t}1_{1}(^{)}111\backslash 1\mathrm{l}\mathrm{t}\mathrm{i}-\mathrm{t}\mathrm{l}‘ \mathrm{s}(^{\backslash }$$\mathrm{C}^{1}\mathrm{e}11\mathrm{I})1^{\cdot}(((^{\backslash }\mathrm{b}_{\mathrm{s}}*(^{\backslash }\mathrm{C}\mathrm{l}\mathrm{r}\mathrm{t}\mathrm{l}\mathrm{e}\backslash \mathrm{c}_{\mathrm{f}}‘ \mathrm{i}11\mathrm{e}\cdot(^{\backslash }1(^{1}(1$\mathrm{s}\mathrm{t}$$\mathrm{O}\mathrm{J}\mathrm{l}\mathrm{t}1_{1}\mathrm{e}$$\iota 1111‘ \mathrm{b}\mathrm{t}_{\mathrm{d}}11(\mathrm{e}\mathrm{f}\mathrm{o}1$$‘11_{1_{C111(}^{\mathfrak{k}}}$$\mathrm{T}\backslash _{1^{)(^{\backslash }}}’$$111\mathrm{O}(1(^{\backslash }1(\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{C}_{\backslash }\mathrm{b}\mathrm{I})\mathrm{O}1\perp \mathrm{c}1‘ \mathrm{h}$$\mathrm{c}\mathrm{o}111\mathrm{P}\iota 1\mathrm{t}_{\mathrm{C}}1^{\cdot}$$\mathrm{a}(\mathrm{c}\iota \mathrm{t}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{t}\iota \mathrm{l}\mathrm{a}\mathrm{t}\epsilon\backslash \mathrm{C}\mathrm{l}$$\mathrm{r}\mathrm{e}(1^{\mathrm{t}\mathrm{i}\mathrm{t}\iota}\mathrm{I}\xi^{\mathrm{Y}}1\mathrm{n}\mathrm{C}^{1}\mathrm{n}\mathrm{q}\mathrm{C}‘\backslash 11^{\cdot}\mathrm{i}_{\mathrm{Y}\epsilon}- 1\mathrm{C}\mathrm{l}1’\subset\backslash \mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}$$\mathrm{b}.1^{r}.‘\backslash \mathrm{r}_{\mathrm{e}\ln}$($\mathrm{l}\mathrm{e}\mathrm{r}\mathrm{S}$$11\overline{1}\mathrm{t}\mathrm{l}\mathrm{l}\mathrm{t}\mathrm{i}-$$‘ \mathrm{h}\lambda^{r}\mathrm{b}\mathrm{t}\mathrm{c}^{\mathrm{Y}}\mathrm{n}\mathrm{l}\mathrm{s}_{\wedge}^{\backslash }n\mathrm{c}‘ \mathrm{h}$$\mathrm{n}\hat{\perp}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{s}$$i.\mathrm{e}.,$$\mathrm{a}1]_{\backslash }$process.

$\mathrm{t}\mathrm{u}\Gamma 11$

47However, notice that the actual multi-task system is required to process a huge number of transactions during a limited period. This implies that the auto-sleep schedulingproblem should be considered for the buffer system with batch arrival of transactions, suchthat a single arrival is composed of some lightweight transactions called threads. Threads
are parts of a single arrived process which are independently scheduled each other in thesystem. In this paper, we design the optimal auto-sleep schedule for a computer systemwith batch arrival of transactions. If a single arrival has only one thread, the model underconsideration can be reduce to Type II model in kamura, Dohi and Osaki. In that,sense, the model in this paper is an extension of Type II model.
The paper is planned as follows. Section 2 describes the auto-sleep model with batch
arrival of transactions. In Section 3, we give an implicit orm of the expected power
consumed per unit time in the steady-state under the assumption that the access requirements arrive at the system following a compound renewal process. Section 4 concerns theapproximation problem for the expected power consumed per unit time. Then we propose
an approximation form of the idle period distribution. The condition to exist the uniqueoptimal sleep timing is derived approxilnately. In Section 5, we give solne nulnerical examples, and investigate the asymptotic property of the idle period derived in Section 4.
Also. we examine the dependence on lnodel paralneters for the optimal auto-sleep timing.
Finally, the paper is concluded with some remarks.
$O$$\mathrm{L}^{7]}\lceil$$\mathrm{f}l$2Auto-Sleep Model for a Computer System

We assume that the stochastic system under consideration can take the following fourstates:busy state: The process is started at least for an access. wheretillleunits are needed for making ready the process. The process is continued until thebuffer is empty. and the system is turned imlnediately an idle state after conlpletingthe final process. In the buby state. we suppose
the electronic power consumed
per unit time is $P_{1}(>0)$ .
$rightarrow(>0)$$\mathrm{r}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\Gamma \mathrm{e}\mathrm{n}\overline{\perp}\mathrm{e}\coprod \mathrm{t}$$\mathrm{t}_{}\mathrm{h}\mathrm{a}\mathrm{t}$idle state: In this state. the system is waiting at least for an access. If theaccess requirelnent arrives at the system up to the time lilnit $t_{0}(0\leq t_{0}<\infty)$ . theprocess is started at that time. otherwise. the system is moved to a sleep ode. Ill
the idle state. the systenl also consumes tlle electrollic powel $P_{1}(>0)$ per unit tinle.
$\mathrm{r}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{m}e1\overline{1}\mathrm{t}$$11\overline{\perp}$$\cdot$sleep state:that thecolne instate. (To
the analvsis. welnode is.
$\mathrm{S}_{11}\mathrm{p}\mathrm{p}o.\mathrm{h}\mathrm{e}$$‘ \mathrm{s}\backslash ’ \mathrm{S}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{l}\backslash$$111(^{\backslash },\mathrm{s}1\epsilon^{\backslash \in_{1^{)}}}\mathrm{Y}$$‘ \mathrm{s}\mathrm{i}\mathrm{n}11^{)}1\mathrm{i}\mathrm{f}\backslash ^{r}$$|\backslash 1\mathrm{e}\mathrm{e}_{1^{)}}$

If
thall olle
state moves 10 astatetllat tlle.

$\mathrm{d}_{\mathrm{o}\mathrm{r}\coprod\overline{\perp}\mathrm{a}}\mathrm{D}\mathrm{t}$$.\mathrm{b}\lambda^{\mathcal{T}},\mathrm{h}\mathrm{t}(^{\backslash }111$$111\mathrm{O}\mathrm{l}\mathrm{e}$$\mathrm{a}\mathrm{c}\cdot \mathrm{c}\cdot \mathrm{e}\mathrm{b}\mathrm{s}\mathrm{l}\mathrm{c}(1^{11\mathrm{i}}1^{\cdot}\mathrm{C}\mathrm{n}1(\backslash \mathrm{l}\mathrm{l}\mathrm{t}$$1\nwarrow_{\mathrm{c}11}^{f}’ 111^{-}111)$$‘*|\iota$]) )$\mathrm{I}$$0‘*\mathrm{e}$$\mathrm{i}\mathrm{l}$)$11\mathrm{t}1(^{\backslash }(1\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{C}^{\backslash }1\lambda^{r}$

,

tlle

$(^{11}(^{1}(\mathrm{t}\mathrm{l}\mathrm{o}\mathrm{I}\mathrm{l}\mathrm{i}(1)(\backslash \backslash ^{\mathrm{Y}}\mathrm{C}1()(.(\iota\iota \mathrm{i}\Gamma\zeta^{)}\mathrm{C}1(1_{11\mathrm{l}}\cdot \mathrm{i}_{1}1^{\circ}0$$/_{\text{・}^{}\prime}(1\mathrm{r}o$warm-up state: Aft$\mathrm{h}\mathrm{l}\mathrm{C}^{\backslash }(^{\rangle}1^{)}111()(1(\backslash$$\mathrm{b}\lambda^{r}‘ \mathrm{S}\mathfrak{s}\mathrm{G}111$

in the$\mathrm{t}11$is.

$(^{\backslash }1\mathrm{a}1^{\cdot}1^{\cdot}\mathrm{i}\nwarrow*\mathrm{i}\mathrm{n}_{0}\circ \mathrm{d}11C\iota’(((^{\backslash _{\mathrm{h}}}\mathrm{h}1\iota^{\backslash }(111\mathrm{i}1(\backslash 111(111\mathrm{t}$.

$\mathrm{t}11(^{1}‘ \mathrm{h}\backslash \mathrm{b}\mathrm{t}(^{\backslash }111$i,s.

$\backslash \backslash \cdot 1_{1}(\backslash 1()\mathrm{s}(>())\mathrm{t}\mathrm{i}111(^{\mathrm{Y}}111\mathrm{l}\mathrm{i}\mathrm{r}\mathrm{s}_{C\backslash \Gamma}()11(^{\backslash }\mathrm{t}^{1}(1(\backslash (1\mathrm{f}()1\mathrm{t}1_{1(^{1}\mathrm{t}}\backslash r1^{\cdot}111-(\mathrm{t}11])$. the state is Dlove(1 to a$\dagger$)lllode is knowll to be. we denote
warnl-uptime.
$\mathrm{s}\mathrm{l}\mathrm{e}\mathrm{e}_{1)}$$\mathrm{p}\mathrm{e}1^{\cdot}\mathrm{i}_{0}\mathrm{d}$$\iota 1.\mathrm{s}\backslash ’$$\backslash \backslash ^{_{C}}\backslash 1111\mathrm{c}(1111)\mathrm{f}\mathrm{i}\cdot()111\subset 1$$\mathrm{A}\mathrm{f}\uparrow(^{\mathit{1}}1\backslash \tau_{\mathrm{c}}^{\tau}|1111\mathrm{i}_{1}1_{0}^{(\mathrm{J}}\mathrm{t}11(’$(. Sincebtafe
hnlall relative to tllat collsullled
the electronic$1^{\cdot}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}^{\mathrm{i}}1^{\cdot}$$P_{2}(P_{2}>P_{1})\mathrm{a}_{\mathrm{L}}\mathrm{s}$$\mathrm{i}_{\ln}111(\backslash$$\iota \mathrm{i}‘ \mathrm{d}\mathrm{t}_{\mathrm{G}}1_{\mathrm{t}}r$$\mathrm{t}1_{1}\mathrm{e}$).

$(^{31\mathrm{e}\mathrm{C}\mathrm{t}}\mathrm{i}_{01}1\mathrm{i}(1)\mathrm{O}\backslash \nwarrow \mathcal{T}(^{\backslash }1$$\mathfrak{c}1\mathrm{t}^{\sim}\iota 1^{\cdot}\mathrm{i}\mathrm{n}\circ$$\mathrm{I}^{)\mathrm{o}\mathrm{w}\mathrm{G}\mathrm{r}}\mathrm{C}‘ \mathrm{o}\mathrm{D}\mathrm{s}\mathrm{u}\mathrm{n}\mathrm{l}\mathrm{C}\Delta \mathrm{e}\mathrm{l}_{\mathrm{P}}(^{1}1^{\cdot}$

the
ullit

$\mathrm{w}\mathrm{a}\mathrm{r}\mathrm{n}_{\overline{1}^{-}}\mathrm{u}\mathrm{p}$Suppose that the access requiremellts arrive at the system according to a compoundrenewal process. Denote $\{X_{k} : k=1,2. \cdots\}$ as a sequence of inter-arrival times between$(k-1)$ -th and k-th arrivals. Then,are non-negative i.i.d. random variables, having$X_{k}$

48

the probability distribution $F(t)$ with mean $1/\lambda(>0)$ and variance $\sigma_{A}(>0)$ . There aresome threads contained in an access requirement. The number of threads contained inwith finite mean $\mu_{B}(>0)$ and variancek-th arrival is given by the random variable$\sigma_{B}(>0)$ .
The other threads arrived during the processing time are accumulated in the,buffer. The k-th thread contained in n-th arrival is processed for a service time
$G(t)$
mean
finitewith
distribution
probability
thehavingwhich is the random variable$1/\mu_{S}(>0)$ and variance $\sigma_{S}(>0)$ . The system processes threads exhaustively until thebuffer becomes empty. Since the system under consideration is very similar to the $GI/GI/l$queueing system with a server vacation (see Takagi [8]), we may analyze the stochastic
properties for the auto-sleep system in the framework of queueing analysis.
$N_{k}$$S_{n,k}$Expected Power Formulation3

To formulate the expected power consumed per unit time in the steady-state, we introduce
the following two random variables (see Fig. 1);$\zeta_{x}$(time length of busy period) : time period for an exhaustive processing, provided thatthe server vacation period is$x$.

when thcre is no(time length of idle period) : timeis .
vacationtheserverbuffer. provided that$\mathrm{p}\mathrm{e}1^{\cdot}\mathrm{i}_{0}\mathrm{d}$$\eta_{x}$$\mathrm{a}\mathrm{c}\cdot \mathrm{c}\mathrm{e}\mathrm{s}\mathrm{S}\mathrm{r}\mathrm{e}\mathrm{q}\iota \mathrm{i}\mathrm{r}\mathrm{Q}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{t}$in the$x$$\mathrm{p}\mathrm{e}1^{\cdot}\mathrm{i}_{0}\mathrm{d}$, where in
fullction of . by
Let us define the probabilitv
is the survivor filnction. Fronl the wcll-known result for queueillg
general(sce Takagi [8]). we obtain the followillg eqllilibriulll equatioll:
analysis with server va($I(t|x)= \mathrm{P}\mathrm{r}\{r\int x\leq t\}$$\mathrm{d}\mathrm{i}_{\iota}\mathrm{q}\mathrm{t}\Gamma \mathrm{i}\mathrm{b}11\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$? \int_{l’}$$\overline{\phi}(\cdot)=1-\phi(\cdot)$$.\mathrm{a}\mathrm{t}\mathrm{i}_{0}11$$\mathrm{E}[(_{x}]=\rho(_{T+}\mathrm{E}[\zeta_{x}]+\mathrm{E}[\eta x])$(1),

is thc traffic intensity. Since Eq. (1) holds only if $\rho<1$ . we assulne in
where
the rest part of this paper that the condition $\rho<1$ holds.
The systeln is resumed at the poillt of time wllen tlle state nioves fronl the sleep mode
pointsone. Then. we define the tinle period between the successive
to the$T(t_{0})$
sleep
the
that
providedcycle.
length
time
of
one
the
mean
denoteas one cycle. Let
exceeds the sleep tilning .
timing is fixed as . Noting whether the first idle period
the mean time length of one cycle is written by$\rho=\lambda\mu_{B}/\mu_{S}$$1^{\cdot}\mathrm{e}\mathrm{s}\iota \mathrm{t}\mathrm{n}_{\overline{1}}\mathrm{i}\mathrm{n}\mathrm{g}$$\}_{)\mathrm{u}}\mathrm{s}\mathrm{y}$$t_{0}$$t_{0}$$\eta_{s+\tau}$$\cdot$$\mathit{1}^{1}(t_{0})$$=$$./()t_{0}${$s+\tau+\mathrm{E}[(_{s+\mathcal{T}}]+t+\prime \mathit{1}_{\Gamma}^{1}$(to)} $dI(t|s+\tau)$$+/t_{\mathrm{t}}.)\mathrm{x}\{_{\mathrm{S}}\mathrm{s}+\mathcal{T}+\mathrm{E}[_{\mathrm{t}_{\mathrm{b}}]t}.+\mathcal{T}+\}(lI(\dagger|\backslash +\mathcal{T})$$\backslash \backslash ^{\tau}1_{1}\mathrm{t}\backslash 1^{\cdot}(^{)l()}’\int\prime f_{\mathrm{t}})$)isof tlle llext$\mathrm{t}1_{1\langle \mathrm{C}_{\lrcorner}\backslash _{1}}.$$\mathrm{t}11(’|)^{1}‘ \mathrm{t}\circ’ \mathrm{i}1111\mathrm{i}_{1}1^{(})0$1,ofidle )$(^{1}(\uparrow \mathrm{e}\backslash (\iota \mathrm{r}\mathrm{i}111(11(111_{\circ}(’ \mathrm{t}11\mathrm{f}_{1()1}.11\mathrm{t}]_{1\mathfrak{c}})\mathrm{r}_{(111}\mathrm{y}\cdot 1\mathrm{i}11C\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{t})11$$\mathrm{c}\cdot\backslash (1\mathrm{t}^{\tau}$. It is$(^{\mathrm{Y}}\backslash \cdot \mathrm{i}$($1\mathrm{t}^{\backslash }\mathrm{l}\mathrm{l}\mathrm{t}$tlldt the$\mathrm{h}_{1_{\backslash }}\cdot*\dagger$.

(2)$\mathrm{t}1_{1}\mathfrak{c}^{1}\mathrm{h}_{1\wedge}.\backslash \mathrm{t}\mathrm{i}(11\mathrm{f}^{1}]^{)(^{\backslash }}1\mathrm{i}()(1_{1}111\mathrm{t}\mathrm{i}1$$(^{\backslash }1^{\cdot}\mathrm{i}\mathrm{o}(\iota$

$1$

is

$\backslash \backslash ^{\tau}()1_{1_{c1T}’}()$$\cdot$$\mathrm{H}e\mathrm{D}\mathfrak{c}$

(.

$=$

$(t_{\{)})$

froni$\mathrm{E}$$/()t_{()}\{’-+\mathrm{E}[(_{\mathcal{T}}]+\dagger+\cdot \mathit{1}^{\cdot}(\prime f\mathrm{t}))\}(fI(/|_{\mathcal{T})+}/t_{\mathrm{t})}\mathrm{x}\{\mathcal{T}+\mathrm{E}[(\overline{l}]+\dagger\}$( $1^{\mathrm{t}}‘’\cdot(2)$$T(\dagger 0)=_{\backslash }\mathrm{s}$.

alld (3). tllc lllca11 time

$1\mathrm{e}11_{6}^{\circ}\mathrm{f}\mathrm{h}$of olle$\mathfrak{c}_{\mathrm{L}}’\backslash r\mathrm{C}^{\cdot}1\mathrm{e}\}$)$\mathrm{t}$($]J(f|_{\overline{\mathit{1}}}).$(.3)) $\mathrm{C}^{\cdot}0111e.\mathrm{b}$$+ \mathcal{T}+\mathrm{E}[(_{s+}\tau]+\mathrm{E}[_{7}\mathit{1}s+\tau]+\frac{I(t_{0}|s+\tau)}{\overline{I}(t\mathrm{o}|_{\mathcal{T}})}\{\mathcal{T}+\mathrm{E}[(\mathcal{T}]+\mathrm{E}[\eta_{\mathcal{T}}]\}.$(4)

Furthermore, substituting Eq. (1) into Eq. (4), we have$\mathcal{I}^{1}(t_{0})=\frac{1}{1-\rho}\{s+\tau+\mathrm{E}[\eta s+\tau]+(\tau+\mathrm{E}[\eta \mathcal{T}])\frac{I(t_{0}|_{S}+\tau)}{\overline{I}(t0|_{\mathcal{T}})}\}$

.

(5)

49

$——\mathrm{L}--$$-\mathcal{L}------$: idle period: busy period–: sleep mode
. lenewal point$\eta_{\tau}$.

$\zeta_{\backslash }$Figure 1: Possible realization of the stochastic system.

Let fhe random variable $N$ denote the total nunlber of transition from the idle t,o the buby
during one cycle. Then. it is verified that $N$ has the following probability mass function:$\overline{I}(t_{\mathit{0}^{1S}}+\tau)$

for $n=0$ .

$I(t_{0}|S+\tau)\overline{I}(t0|\mathcal{T})I(\mathrm{f}0|\tau)^{n-1}$

for $n=1_{i}2$ .

(6)$\mathrm{P}\mathrm{r}\{N=n\}=\{$$\cdots$.

Froln the property of the geonletric distribution. the expected value of $N$ is$\mathrm{E}[N]=\frac{I(t_{0}|s+\tau)}{\overline{I}(t_{0}|\tau)}$

Using Eq. (7). the.

(7)time length of one cycle has the following sinlple form:$\mathrm{n}\overline{\perp}\mathrm{e}\mathrm{a}\mathrm{n}$$\ulcorner l^{\urcorner}(\mathrm{f}\mathrm{o})=\frac{1}{1-\rho}\{\beta+\tau+\mathrm{E}[_{l}\overline{l}\mathrm{s}+\mathcal{T}]+\mathrm{E}[N](\tau+\mathrm{E}[_{\mathit{1}]}\gamma)\mathcal{T}\}\cdot$

Eq. (8) (1(11Ill a$\backslash \backslash ^{\tau}11\mathrm{c}\backslash 1^{\cdot}\mathrm{e}$iblrtio]l $I(t|\cdot)$ .

$11(1,\mathrm{h}\backslash 01\downarrow 1_{\lambda^{r}0}11\mathrm{t}1_{1(}1\mathrm{i}\mathrm{e}\mathrm{i}1\zeta^{\rangle}\mathrm{c}\mathrm{l}\mathrm{i}1*\mathrm{t}_{1}$$\mathrm{f}\mathrm{a}\mathrm{s}11\mathrm{i}_{0}11.\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{i}1i\mathrm{d}1^{\cdot}$

to tllc

onc$1)(\mathfrak{n}\cdot \mathrm{e}\mathrm{r}\mathrm{e}\mathrm{o}11_{\mathrm{v}}^{\mathrm{Q}},\mathrm{t}1111\mathrm{c}\mathrm{c}1\subset 1\iota 1\mathrm{r}\mathrm{i}11()0$$C(\mathrm{f}_{0})$$(\aleph)$$=$$\mathrm{n}1(_{C}^{s\prime}\mathrm{t}11\mathrm{f}\mathrm{i}\mathrm{n}1(^{\}1\iota 1_{\circ}^{0\uparrow 11}}()$$\mathrm{c}‘ \mathrm{t}^{r}(1\xi^{1}‘ \mathrm{d}$.Sof$(\mathrm{i}1e(1^{\tau}(1(^{1}$.

$\backslash \backslash ^{-}()(1()\mathrm{h}_{11}\mathrm{t}^{\mathrm{t}}\mathrm{t}1_{1}(^{)}\mathrm{t}\circ \mathrm{f}_{\dot{C}}\mathrm{i}\mathrm{l}\mathrm{c}\mathrm{x}_{\mathrm{P}}\mathrm{t}\Delta(\mathrm{t}(’ \mathrm{t}\iota$$J_{()}^{l}(\}\{F_{\mathit{2}^{S}}+P_{1}(\tau+\mathrm{E}[(_{s+\mathcal{T}}]+t)+C, (t_{\mathrm{U}})\}dI(t|s+\tau)$$+./t_{()}\mathrm{x}\{P_{2}s+P_{1}(\tau+\mathrm{E}[\eta_{S+}\mathcal{T}]+t_{0})\}dI(f|s+\tau)$

where$C_{r}(t_{0})$$=$$\int_{0}^{t_{0}}\{P_{1}(\tau+\mathrm{E}[\zeta_{\mathcal{T}}]+t)+C_{r}(t_{0})\}dI(t|\tau)$.

$(^{(}\mathrm{J})$

50

(10)

$+ \int_{t_{0}}^{\infty}\{P_{1}(\tau+\mathrm{E}[\eta_{s+\mathcal{T}}]+t_{0})\}dI(t|\tau)$. From a few algebraic manipulations,is the expected power consumed for the period
by
given
cycle
is
one
the total expected power consumed for$T_{r}(t_{0})$$C(t_{0})$$=$$P_{2}s+P_{1}\{\tau+\mathrm{E}[\zeta_{s+\tau}]+\mathrm{E}[\eta_{s+\tau}\wedge t_{0}]\}+P_{1}\mathrm{E}[N]\{\mathcal{T}+\mathrm{E}[\zeta_{\tau}]+\mathrm{E}$$=$$\{\frac{\rho}{1-\rho}P_{1}+P_{2}\}S+\frac{P_{1^{\mathcal{T}}}}{1-\rho}+P1\{\frac{\rho}{1-\rho}\mathrm{E}[\eta s+\tau]+\mathrm{E}[\eta s+\tau\wedge t0]\}$[$\eta_{\tau}$A,$t_{0}$

]

$\}$

(11)

$+ \{\frac{P_{1}\tau}{1-\rho}+P_{1}(\frac{\rho}{1-\rho}\mathrm{E}[\eta \mathcal{T}]+\mathrm{E}[\eta\tau\wedge t\mathrm{o}])\}\frac{I(t_{0}|_{S+\mathcal{T})}}{\overline{I}(t0|_{\mathcal{T}})}$

where$\mathrm{E}$[$\eta_{x}$A$t_{0}$(12)] $= \mathrm{E}[\min(\eta x’ t0)]=\int_{0}^{t0}udI(u|x1+t0\overline{I}\text{ノ}(t_{0}|x)$Finally. the expected power consumed per unit time in the steady-state is, fronl the wellknown renewal reward argulnent,$V(t_{()})= \lim_{tarrow\infty}\frac{\mathrm{E}[\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}1_{\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{S}}\mathrm{u}\mathrm{l}\mathrm{n}\mathrm{e}\mathrm{d}_{\mathrm{P}^{\mathrm{O}}}\mathrm{w}\mathrm{e}\mathrm{r}\mathrm{i}_{1}1(0.t]]}{t}=\frac{C(\mathrm{f}\mathrm{o})}{T(t_{0})}$Then. the problem is to seek the optimal sleep timing4$t_{0}^{*}\mathrm{w}\mathrm{h}\mathrm{i}_{\mathrm{C}}\mathrm{h}$

(13)

.

minimizes $V(t_{0})$ .

The Approximation of the Idle Period

To represent the expected power consumed per unit tillle $V(t_{0})$ explicitly, we have to derive
the probability distribution $I(t|x)$ for the idle period. It is. however. difficult to obtain
the explicit form in the ordinary renewal arrival case. Thus. we propose an approximation. Let us denote the arrival tinle
method for the idle period based on the workload$\{T_{n}:n=0.1,2.
\cdots\}$, whereof the n-th access requirement by$\mathrm{p}\mathrm{r}\mathrm{o}\mathfrak{c}\cdot \mathrm{e}\mathrm{s}\mathrm{s}$$T_{n}= \sum_{k=0}nx_{k}$Then. for an arbitrarv. whele$\mathrm{t}\mathrm{i}\mathrm{n}\overline{\perp}\mathrm{e}t\in(T_{r?}<t<T_{n+1})$

.

(14)

. we define the workload process$\{\mathrm{T}\mathrm{T}^{\tau}(t):\gamma\geq$$0\}$$\mathrm{T}\mathrm{T}^{\mathrm{Y}}(\{)=\mathrm{T}\mathrm{T}’(’l_{l1}\gamma)+-(t-l_{\prime},)$

TU

(15)

.

(1 )
$()$

$(’l_{1},+)= \mathrm{T}\mathrm{T}^{-}(\mathit{1},, -)+\sum_{/_{\backslash =}1}^{(1}6\backslash ‘"/_{\backslash }\cdot$

.
alld
where
ib busv at tilllc . otllerwise. the syslelll is idle.
Notice that if $W(f)>0$ then theand that theof access requirenlents occurs atthat the firstserver vacation period is $s+\tau$ . If the number of access requirements which experience thefirst empty buffer in the system is . then the idle period can be represented as$\mathrm{T}\iota^{r}(t-)=1\mathrm{i}111_{(}-\mathrm{t})\mathfrak{s}\mathrm{T}^{\tau}(t-\xi \mathrm{I}(‘ \mathrm{s}\mathrm{e}\mathrm{e}\mathrm{G}^{1}\Gamma\circ‘ \mathrm{b}\mathrm{s}\prime C\backslash 11(1\mathrm{H}‘ C\iota \mathrm{r}\mathrm{l}\cdot \mathrm{i}‘ \mathrm{b}[^{(})])$$\mathrm{T}\mathrm{T}^{\tau}(i+)=1\mathrm{i}\mathrm{n}1_{\epsilon-\cup}\mathfrak{s}\mathrm{T}^{\mathrm{v}}(t+\epsilon)$$t$$\mathrm{s}\backslash ’ \mathrm{b}\mathrm{t}\mathrm{e}\ln$$\mathrm{t}\mathrm{i}\mathrm{l}\overline{\mathrm{l}}\mathrm{l}\mathrm{e}f=$$\mathrm{a}\mathrm{r}\mathrm{l}\cdot \mathrm{i}\backslash \cdot \mathrm{a}\mathrm{l}$$()$$\mathrm{S}\mathrm{u}_{\mathrm{I}^{)}\mathrm{P}^{\mathrm{O}\mathrm{S}\mathrm{e}}}$$n^{*}$$\eta_{s+\tau}=-W(\tau_{n^{*-}})$

.

(17)

51

Figure 2: Behavior of the workload process.

Figure 2 illustrates the configuration of workload process. For the idle period with theserver vacation. we obtain the following renewal-type equation for an
meastlrable
function$\mathrm{a}\mathrm{r}\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{r}\mathrm{a}\Gamma v\vee$$f_{\backslash }$$\mathrm{E}[f(\eta s+\tau)]=\int_{0}^{S}\mathrm{E}[f(\eta S-x)]dI(x|\mathcal{T})+\int_{s}^{\infty}f(x-s)dI(_{X|\mathcal{T})}$

Lemma 1 For an arbitrary measurable function
given by$f\cdot$. the asymptotic

.

(18)

of the idle

$propert_{j}\mathrm{t}$$per’/od?\backslash s$$s arrow\infty 1\mathrm{i}\mathrm{n}\overline{1}E[f(\eta_{\tau+s})]=\frac{E[\int_{0}^{\eta_{\mathcal{T}}}f(_{Z})d_{\tilde{p}}]}{E[\eta_{\mathcal{T}}]}$

.

(19)

Proof. Taking the Laplace transform to both sides of Eq. (18). we have
(20)$/0 \propto\exp(-\zeta\iota S)\mathrm{E}[f(\eta s+\mathcal{T})]ds=\frac{\mathrm{E}[/_{0^{\eta_{\mathcal{T}}}}\exp\{-\mathfrak{a}(\uparrow\overline{/}\mathcal{T}^{-_{\tilde{\mathcal{L}}}})\}f(z)d\hat{\mathcal{L}}]}{[1-\mathrm{e}_{\backslash }\mathrm{x}\mathrm{p}(-\mathit{0}\eta_{\tau})]},\cdot$$\backslash \backslash ^{\tau}1_{1}(^{\backslash }1^{\cdot}0^{\backslash }(\{\mathrm{i},\mathrm{s}\mathrm{a}111^{\tau}$(Olnplex)$111\mathrm{l}\mathrm{n}11-$.

$\backslash \backslash ^{\mathrm{Y}}\mathrm{C}\supset 11-\mathrm{k}11()\backslash \backslash ^{\tau}11\mathrm{A}\dagger)\mathrm{e}1_{\mathrm{h}}^{\cdot}\backslash 1\in\backslash \mathrm{l}\mathrm{n}111\mathrm{a}\lfloor\lceil 1()]$If$f(_{\sim})=(^{\backslash }\mathrm{X}1)(-(\{arrow)$ill$\mathrm{e}1^{\cdot}$.

$\mathrm{L}‘\backslash 111111_{C}’\mathrm{t}1$.

. $(2())$ .

($\mathrm{L}\mathfrak{c}^{\Delta}\mathrm{t}\mathrm{t}\mathrm{i}11_{6}^{\mathrm{U}}c\backslash arrow()\mathrm{i}_{11}\mathrm{E}_{1}$$\prime \mathrm{r}1_{1(^{\mathrm{Y}}}1^{)1()()}.\mathrm{f}$is$(()1\mathrm{l}\mathrm{l}1)1\mathfrak{c})\mathfrak{f}\mathrm{G}(1$$\mathrm{t}1_{1\mathrm{G}}1^{\cdot}(^{1}.\mathrm{s}\mathrm{t}11\mathrm{t}$

i.s (

.

$1\mathrm{t}\mathrm{l}\mathrm{C}$fotlle right-1lan( side of Eq. (21) is the salncof . we can approxinlate the idle
$1$

(21)

is:

tlle
for sllfficiently larger , than . that$\mathrm{L}\mathrm{a}_{\mathrm{I}}\mathrm{J}\mathrm{l}\mathrm{a}\mathrm{C}^{\cdot}\mathrm{c}$$\mathrm{d}\mathrm{i}_{5}\mathrm{t}_{\Gamma \mathrm{i}\mathrm{b}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}}\iota 1$$7)_{\mathcal{T}}$.

$\mathrm{t}11(^{\backslash }11\mathrm{i}\uparrow \mathrm{i}\mathrm{s}‘*(\}(^{s}11\mathrm{t}1_{1c1}($$\sigma-1\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{t}\infty \mathrm{E}[\mathrm{e}\mathrm{Y}\mathrm{X}1^{)(-}(|/_{\mathit{1})]}\tau+\mathrm{s}=\frac{1-\mathrm{E}_{\lfloor^{(^{\backslash }}1^{)}}^{\mathrm{r}}\lrcorner\backslash (-c1//_{\mathcal{T}})]}{(\iota \mathrm{E}[1-]/\mathcal{T}}.$$\mathrm{S}\mathrm{i}\mathrm{l}\mathrm{l}(\mathrm{c}$$\mathrm{t}11\mathrm{C}^{3}$$(\}.\mathrm{E}.\mathrm{D}$$\mathrm{p}\mathrm{e}1^{\cdot}\mathrm{i}_{0}\mathrm{d}$$I(t|_{S}+ \tau)\approx\frac{1}{\mathrm{E}[\eta_{\mathcal{T}}]}.\int_{0}^{t}\overline{I}(s|\mathcal{T})d_{S}$fran,bforlll$\zeta^{\backslash }(1\iota\iota \mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t})1\mathrm{i}_{1}\ln$$\mathrm{a}\mathrm{i}\supset$$\mathrm{s}$.

$\tau$(22)

52

Substituting Eq. 22 into Eqs. (4) and (11), we derive approximately the expected power
consumed per unit time in the steady-state as$V(t \mathrm{o})\approx V_{a}(t_{0})=\frac{C_{a}(t_{0})}{T_{a}(t_{0})}$

(23)

,

where
(24)

$T_{a}(t \mathrm{o})=\frac{1}{1-\rho}\{s+\tau+\frac{\mathrm{E}[\eta^{2}\tau]}{2\mathrm{E}[\eta_{\tau}]}+(\tau+\mathrm{E}[\eta_{\tau}])0^{0}\int_{\overline{\overline{I}(}}t\overline{I}(t|_{\mathcal{T})}dt\}t0|\mathcal{T})\mathrm{E}[\eta_{\tau}]$and$C_{a}(t_{0})$$- \rho\rho P_{1}+P_{2}\}s+\frac{P_{1^{\mathcal{T}}}}{1-\rho}+P1\{\frac{\rho}{1-\rho}\mathrm{E}[\eta_{s}+\tau]+\int_{0}^{t_{0}}\int_{t}^{\infty}\overline{I}(u|\tau)dudt\}$$=$$\{_{\overline{1}}$$+\{$$\frac{P_{1^{\mathcal{T}}}}{1-\rho}+P_{1}(\frac{\rho}{1-\rho}\mathrm{E}[\eta_{\tau}]+\int^{t_{0}}0\overline{I}(t|\tau)dt)\}\frac{\int_{0}^{i_{0}}\overline{I}(t|_{\mathcal{T}})dt}{\overline{I}(t_{0}|\mathcal{T})\mathrm{E}[\eta\tau]}$

To obtain the optimal sleep timing, we define the mean residual life
;
rate

.

$R(t_{0}|\tau)$(25)

and hazard$r(t_{0}|\mathcal{T})$(26)$R(t0|_{\mathcal{T}})= \frac{\int_{t_{0}}^{\infty}\overline{I}(t|\mathcal{T})dt}{\overline{I}(t0|_{\mathcal{T}})}$

and$r(t_{0}| \mathcal{T})=\overline{\overline{I}(t_{0}|\mathcal{T})}1(\frac{dI(t_{0}|\mathcal{T})}{dt_{0}})$

(27)

.

Further. define$\hat{t}_{0}=\inf\{t_{0}>0;\frac{d\log R(t0|\mathcal{T})}{d\log r(t0|\mathcal{T})}+_{\overline{\overline{I}(t_{0}|\mathcal{T})R(t|}}\mathrm{E}[\eta_{\mathcal{T}}]0T)>1\}$

.

(28)

The following result gives the optimal auto sleep timing which minimizes approximately
the expected power consumed per unit time in the steady-state.
). the idle $period$ , has decreasing hazard
Theorem 1 Suppose that, for all. ). then there exist.
rate. If an auto-sleep $ti7ningt0$ satisfies $P_{1}>V_{o}(t_{0})$ for all
(to)
)
izes
sleep
an
$t_{0}\in[t^{\wedge}0\cdot\infty$$\cdot$$t_{0}\in[\mathrm{f}_{0}^{\wedge}$

Proof.

$\mathrm{I}_{11()1(}1$($1^{\cdot}$

to$wh’/(h?17irl_{\text{ノ}}irr’$$t\dot{\uparrow}m,\dot{?,}\eta gf_{()}^{\star}\in[f_{()}^{\wedge}x$$un\dot{\uparrow}queopti_{7}r\prime al$$\mathrm{I}$)$1^{\cdot}(\mathrm{V}(^{\backslash }\mathrm{t}1_{1\mathrm{C}}((11\iota\cdot(^{)}\mathrm{X}\mathrm{i}\mathrm{t}\nwarrow’$of$V_{c;}(t_{()})$.

$\propto$$\mathrm{s}$$V_{\mathit{0}}$$\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{i}_{1}1\mathrm{c}$$\overline{C},r(t_{(}))\equiv’(1-(J)\overline{I}(f_{()}|\mathcal{T})c,,(r\mathrm{r}))$

(29)

.

(30)

and
$\overline{T}_{o}(t_{0})\equiv(1-\rho)\overline{I}(\gamma_{01)}\mathcal{T}T_{o}(t_{0)}$wherewith respect to. Further. define the numerator of the derivative of$i.e.$ ,asdivided by$V_{(?}(t_{0)}=\overline{C}_{G}(t0)/\tilde{T}_{o}(t_{0})$$q(t_{0})$$=$$t_{0}$.

$\overline{I}(t0|\tau)r(t0|\mathcal{T})$$q(t_{0}),$$P_{1}(1- \rho)\tilde{T}_{a}(t\mathrm{o})\{\frac{1}{r(t0|_{\mathcal{T}})}\int_{0}^{t\mathrm{o}_{\overline{I}(}}t|\mathcal{T})dt-\int_{0}^{t_{0}}\int_{t}^{\infty}\overline{I}(u|\tau)dudt\}$$V_{c\iota}(t_{0})$

53

(to).

(31)

$+ \{\frac{\tilde{C}_{a}(\infty)}{\mathrm{E}[\eta_{\tau}]r(t_{0}|_{\mathcal{T})}}-\tilde{c}_{a}(0)\mathrm{I}^{\tilde{T}_{a}}(t\mathrm{o})-\{\frac{\tilde{T}_{a}(\infty)}{\mathrm{E}[\eta_{\tau}]r(t0|\mathcal{T})}-\overline{\tau}_{a}(\mathrm{o})\}\tilde{c}_{a}$Differentiating$q(t_{0})$with respect to$\frac{d}{dt_{0}}q(t_{0})$$=$$t_{0}$

yields$\frac{P_{1}(1-\rho)\tilde{\tau}_{a}(t_{0})\overline{I}(t0|\mathcal{T})}{r(t0|_{\mathcal{T}})}\{1-r(t_{0}|\mathcal{T})R(t_{0}|\mathcal{T})$$- \frac{dr(t0|\mathcal{T})/dt0}{r(t0|_{\mathcal{T}})}(_{\overline{\overline{I}(t_{0}|)}}^{\mathrm{E}[}\eta_{\tau}]t_{0}-R(|_{\mathcal{T}))}\tau\}${$- \frac{dr(t_{0}|\tau \mathrm{I}/dt0}{\mathrm{E}[\eta_{\mathcal{T}}]r(t0|\tau)2\tilde{T}a(\infty)\tilde{\tau}(ot0)}$$P_{1}-V_{a}$(to)}.

(32)

If $dr(t0|\mathcal{T})/dt_{0}<0$ holds for all), the first term of Eq. (32) is strictly positive
for all. Therefore, for the range satisfying $P_{1}>V(t_{0})$ , it can be seen that$q(t_{0})/dt_{0}>0$ . Since this implies $d^{2}V_{a}(t_{0})/dt_{0}^{2}>0$ , the convexityofis proved for.
Q.E.D.
$to\in[\hat{t}_{0,\infty}$$t_{0}\in[\hat{t}_{0}, \infty)$$V_{(\mathrm{J}}(t_{0})$$t_{0}\in[\hat{t}0, \infty)$

Remark 1 Theorem 1 shows that there is a local
sleep tinling for strong conditions $P_{1}>V(t_{0})$ and). Hence the local optimal sleep timing can be calculatedusing any numerical optimization lnethod such as the deepest
method. NewtonRaphson method. etc. On the other hand, to fin. the global optimal sleep tilning. wehave to check the global behavior of the function
for all . Since $V_{a}(\infty)=P_{1}$ . thesolution space can be limited to the range at which $P_{1}>V(r_{0})$ holds. This result leads tothe fact that the nulnerical optimization lnethod should be applied for all.
$\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}_{11}\overline{1}\mathrm{a}1$$t_{0}\in[\hat{t}_{0\cdot\infty}$$\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{l}\overline{\mathrm{l}}\mathrm{t}$$\mathrm{d}$$V_{o}(t_{0})$$t_{0}$$t_{0}\in[\hat{t}_{0\cdot\infty})$5

Numerical Examples

In this section. we illvestigate the asymptotic properties of the approximated idle period,and examine the dependence on model parameters for the optimal sleep timing. Supposethat the time interval between the successive access requirelnents obeys the phase-type
distribution. The phase-type distribution represents the tilne until the absorption in thecolltinuous time AIarkov chain. with the following distribution function;$F(f)=1-\alpha\exp(Tt)e$ .

is a ?7 $l\cross m$ lnatrix
vector and is a colunlll

negative diagoll$\backslash \mathrm{v}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{r}\mathrm{e}\prime l^{\mathrm{Y}}$$e$$1_{1\mathrm{f}\mathrm{i}\backslash }r\mathrm{i}_{1}$$1^{\tau}(^{\backslash }(\mathrm{t}\mathrm{o}1^{\cdot}$of$1\mathrm{s}$. Also. we(33)$c\mathfrak{i}\prime 1(\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{I})011e\mathrm{n}\mathrm{t}_{\mathrm{S}}$$‘ \mathrm{s}\mathrm{t}\mathrm{l}\mathrm{P}1^{)\mathrm{o}\mathrm{s}}‘ \mathrm{e}$tllat ille.

$\alpha$is a$\backslash \backslash L(\Gamma \mathrm{k}1(^{J}C\backslash (\iota$$1^{\cdot}(^{\mathit{1}}(1^{\backslash 1\mathrm{i}\cdot 11\mathrm{t}}\mathrm{i}(\supset 111\mathfrak{k})()\})\mathrm{C}1\tau‘\backslash \uparrow 1\perp \mathrm{e}\mathrm{n}\mathrm{f}\mathrm{o}\mathrm{l}1()\backslash \backslash ^{\vee}\mathrm{i}_{1\mathrm{l}}\mathrm{g}\sigma\backslash -\mathrm{x}1^{y()1}1\circ 11\mathrm{t}\mathrm{i}\mathrm{a}1(1\mathrm{i}‘\wedge \mathrm{t}_{\mathrm{F}}\mathrm{i}\}_{)}\iota 1\mathrm{t}\mathrm{i}()111\backslash \prime \mathrm{i}(1_{1}i\backslash _{\mathrm{I}^{)_{(}\mathfrak{i}1c}\mathit{1}}\downarrow 111(^{1}\mathrm{t}_{\mathrm{C}^{1}}1^{\cdot}/$$G(t)=1-\mathrm{e}\mathrm{x}_{1(\ell\dagger)})-_{l}$the condition tllat thebution (see Neuts [11]):$\mathrm{U}\mathrm{n}(\mathrm{l}\mathrm{e}\mathrm{l}\cdot$$\mathrm{s}\mathrm{e}1^{\cdot}\backslash ’\mathrm{e}\mathrm{r}\backslash _{C}’‘$}($\mathrm{C}‘\{\mathrm{t}\mathrm{i}_{01}1$$\mathrm{I}$of)$1^{\cdot}\mathrm{o}\mathrm{t}3^{\tau}c1|_{)}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\}^{\tau}$$\dot{c}111\mathrm{r}‘ \mathrm{t}(’(.(^{)}$$\mathrm{s}*$

“

:

.

$(_{)}^{J}.‘\lrcorner)$is . tlle idle )$j\cdot$$\mathrm{C}^{\backslash }1^{\cdot}\mathrm{i}\mathrm{o}\mathrm{d}\mathrm{h}^{I}\epsilon 1.\mathrm{b}$$\mathrm{I}$$I(t|x)=1- \frac{\alpha\exp\{(T+\tau^{0}\alpha G)x\}c_{\mathrm{C}}\mathrm{x}_{1^{\mathrm{J}}}(Tt)e}{\alpha e\mathrm{x}\mathrm{p}\{(T+T^{0}\alpha G)x\}Ge}$thc$\mathrm{f}\mathrm{o}110\backslash \backslash ’ \mathrm{i}_{1\mathrm{l}}\mathrm{g}_{\mathrm{C}}1\mathrm{i}_{h}\mathrm{t}1^{\cdot}\mathrm{i}-$.

where $T^{0}=-Te$ , andis a transition probability matrix with components
is the probability that the phase makes a transition from to .
$G$

$i$

$j$

(35)
$[G]_{ij}$

which

54

Figure 3: Asymptotic behavior of the mean time length of the idle period with varying
service rate.

In this section, we assume the following model paralneters;$T=\lambda$ .
$\lambda=1.0.0.3,0.5$

or 0.7.

$\alpha=(1.0,0.0)$.

$\mu=1.0.1/0.1.1/0.5$ or

1/0.9.

Figures 3 and 4 depict the behavior of the mean tillle length of the idle period for the.

$\rceil)()11\zeta \mathfrak{i}\nwarrow\cdot \mathrm{i}\mathrm{o}1^{\cdot}$$\mathrm{F}\mathrm{i}_{\mathfrak{Q}}(’ 111(\backslash 4:\mathrm{A}_{\mathrm{h}\iota^{r}1},111^{)}\mathrm{r}_{()}(\mathrm{i}($of$\mathrm{t}11\mathrm{e}^{\mathrm{Y}}$lllC‘d11$\mathrm{t}\mathrm{i}111(^{\backslash }1\langle^{)}11_{\mathrm{b}}^{\mathrm{t}\mathrm{J}}\mathrm{t}11$

of 4 he$\mathrm{i}(f1()1^{)(})1^{\cdot}\mathrm{i}\mathrm{o}$($1\backslash \backslash \cdot \mathrm{i}\mathrm{t}\mathrm{l}_{1}$Vcu)$1^{\tau}\mathrm{i}11_{\circ}^{(}$.

$i\{11^{\cdot}\mathrm{i}_{\lambda_{C}\mathrm{t}}.1_{1\subset\backslash \mathrm{t}}(\backslash$)(nleallthat. Tlles.
atioll witll
tllalldepends on the arrival of tlle
tinle length of the idle
werequilements.
accessofintel-arrival
lnean
the
tlle workload. Also. 1 aking accormt ofobserve that the meall time length of the idle period collverges to a value asylnptoticallywhen the server vacation is about 4\sim 5 times as long as the mean inter-arrival of accessrequirements.
Next, we investigate the dependence on model parameters for the optimal sleep timing,

servel

$1\mathrm{i}\nwarrow^{}(\rangle 11^{r}$$\mathrm{v}\mathrm{a}\mathrm{c}$$\backslash ^{-}(‘\iota \mathrm{r}\backslash ^{r}\mathrm{i}1l^{/}c\iota\prime \mathrm{n}(1\lambda$$\mathrm{i}\mathrm{n}$$\{)1^{\cdot}\mathfrak{c}^{1}.\mathrm{b}111\mathrm{f}_{\backslash }\mathrm{s}\epsilon^{1}\mathrm{x}\mathrm{p}1\mathfrak{c}$$\mathrm{t}11(^{1}$$1\mathrm{e}_{\backslash }\mathrm{s}\mathrm{p}e$$\mathrm{a}\mathfrak{c}(\mathrm{C}_{1}\mathrm{b}_{9}\mathrm{q}\mathrm{l}\mathrm{e}(1^{1}1\mathrm{i}1^{\cdot}\mathrm{e}\mathrm{n}\mathrm{l}\mathrm{C})\mathrm{n}\mathrm{t}\mathrm{l}\mathrm{b}1\prime c\mathrm{t}\mathrm{t}11\mathrm{C}\mathrm{J}1^{\cdot}$$1\supset \mathrm{e}\Gamma \mathrm{i}_{\mathrm{o}\mathrm{d}}$55Table 1: Dependence on the arrival rate for the optimal sleep timing.

Figure 5: Dependence on the electronic power
unit tinle in steady-state.

$P_{2}$

for the expected power consullled per

where the following parameters are assulned:$P_{1}=1$ ,$P_{2}=10$.

$s=10.0$ .

$\tau=0.5$

.

(36)

and where the arrival and service paralneters are the same as theexalllple. In
Table 1 we give the nunierical value of the optinlal sleep tinlillg and the associated lnininiulll expectedforvalues of.
. it\daggerll\v{c}lt
foulldthe lllinilllllnl
ease.s 11lollotol\downarrow i(
for
.
of
is
of
(
)
.
tliat
.
thrlt
is a 111(
Also. ill
. ).
() tinlingtlle bellaviol of tlle
for thepowel
witll $P_{2}=(.).()$ .
. ]. $12.()$ . $14.0$ . In Fig. 5. it seen tllat lheofpower
sellsitivelv as the power consunled inincreases. On the otller hand. it ibfrom Fig. 6 that the optimal sleep tinling is insensitive to warnl-llI) tinle.
$\mathrm{p}_{1}\cdot \mathrm{e}1^{\mathrm{v}}\mathrm{i}\mathrm{o}\mathrm{t}\mathrm{l}\mathrm{S}$$\mathrm{s}\langle^{\mathrm{Y}}\backslash ^{\tau}\mathrm{G}1\prime c\iota 1$$\mathrm{t}11()\mathrm{d}’1^{\cdot}\mathrm{r}\mathrm{i}\lambda 7\mathrm{a}\mathrm{l}1)_{(\downarrow}\mathrm{r}\mathrm{a}111\mathrm{e}^{\supset}\mathrm{t}\mathrm{C}1^{\cdot}\lambda$$1^{\mathrm{J}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}$$\mathrm{F}_{1()11}.1\ulcorner 1_{c\backslash }^{i}’]_{)}1(11$$\mathrm{i}‘ \mathrm{s}$$\epsilon^{\mathrm{J}}\mathrm{x}_{1^{)}}(^{)}(\mathrm{t}\mathrm{C}(1\mathrm{I}^{\mathrm{J}\mathrm{O}}\backslash \backslash ’ e1\mathrm{i}\mathrm{l}1(1^{\cdot}$$\lambda$$\zeta‘\iota]1.\backslash \cdot$$\mathrm{t}11(^{\mathrm{Y}}\mathrm{i}\mathrm{n}\mathfrak{c}\cdot 1^{\cdot}(^{\backslash }\dot{c}\iota \mathrm{h}\mathrm{i}11(\mathrm{b})$$\mathrm{d}\mathrm{D}(1\mathrm{t}\mathrm{l}\mathrm{l}\mathrm{d}(\mathrm{t}1_{1(^{\backslash }}\backslash ‘ C11^{\cdot}\mathrm{i}_{\dot{c}}\iota \mathrm{t}\mathrm{i}()11$$\mathrm{t}1_{1\zeta^{\mathrm{y}}}()1^{)}\mathrm{t}\mathrm{i}111\dot{c}\mathfrak{i}1‘ \mathrm{b}1\mathfrak{c}^{1}(\mathrm{Y}1^{)}\mathrm{r}\mathrm{i}_{111}\mathrm{i}_{11^{(}})\mathrm{Q}$$\mathrm{i}11\mathrm{r}\zeta^{)}1\zeta\{11^{\cdot}\mathrm{i}_{\mathrm{Y}^{i}}\cdot\zeta \mathrm{t}1$$r\perp 11(^{\backslash }‘ \mathrm{h}(^{\backslash }1(^{)}‘*1\iota 1\mathrm{t}\mathrm{t}\mathrm{i}11(\iota \mathrm{i}$$\mathrm{t}11\mathrm{t}^{\backslash }\mathrm{d}\mathrm{l}\cdot 1^{\cdot}\mathrm{i}\backslash \mathrm{d}\mathrm{l}1)\mathrm{d}1_{C}‘\backslash 111(1\mathrm{t}(^{\mathrm{Y}}1’$$\mathrm{t}11()()1$$\dot{c}\iota \mathrm{t}\mathrm{C}^{\backslash }$$\mathrm{g}_{1\xi^{1}\mathrm{d}\mathrm{t}}(\backslash 1\mathfrak{f}1\mathrm{l}_{\subset}\backslash \mathrm{l}\mathrm{l}\mathrm{t}1_{1\mathrm{d}}\mathrm{t}$$\mathrm{t}11(^{\backslash }111(^{\backslash }C111$$[\mathrm{i}_{111\mathrm{a}}1‘ \mathrm{s}1(^{\supset}(^{\supset}1)\mathrm{r}\mathrm{i}_{111\mathrm{i}_{11^{\mathrm{t})}}}\wedge(1\mathfrak{c}\backslash 1^{)(})11(1‘*,*\mathrm{t}\mathrm{i}()11()1\mathfrak{n}\backslash ^{r}()11$$c\mathrm{t}\mathrm{n}(\iota$$\mathrm{t}11(^{\backslash }\mathrm{d}11\mathrm{t}\mathrm{o}^{- \mathrm{b}}1(\mathrm{Y}(^{\backslash }1)\mathrm{f}\iota_{\iota 11(}‘ \mathrm{t}\mathrm{i}_{011}$$()\mathrm{l}(^{i}()\mathrm{H}\mathrm{t}^{1}(\mathrm{t}\mathrm{i}\backslash (^{1}1)()\tau \mathrm{t}’\mathrm{t}\mathrm{Y}1^{\cdot}-\Leftrightarrow \mathrm{d}\tau\cdot \mathrm{i}_{1}1^{(J}0$$\ulcorner$$\mathrm{f}_{111}1\mathfrak{c}\mathrm{t}\mathrm{i}(y11\mathrm{w}11(’\iota 1\mathrm{d}(((^{\backslash }\mathrm{S}_{\iota}\mathrm{h}1\mathfrak{c}^{s}(\mathrm{i}11\mathrm{i}_{1(1}\backslash 11(i11\mathrm{f},\mathrm{b}\mathrm{l}\mathrm{l}\mathrm{d}\lambda(^{1}\zeta\iota 1\mathrm{i}_{\mathrm{o}}^{\zeta)}1_{1}\mathrm{t}\mathrm{t}_{1\zeta}’\{\mathrm{f}\mathrm{f}\mathrm{i}$$1$$\mathrm{c}^{1}\mathrm{x}_{\mathrm{P}^{\mathrm{t}^{\backslash }\mathrm{c}\mathrm{f}}}\cdot(3$$\mathrm{a}\mathrm{n}(1(\mathfrak{s}=10.\mathrm{t})$$V(t_{()})$$\mathrm{F}^{\neg}\mathrm{i}_{p_{7}^{\mathrm{t})}}‘ \mathrm{s}$$‘\backslash 1\mathrm{e}\epsilon_{\mathrm{I}}^{\mathrm{Y}}$$(\backslash 11(1()$$\backslash \backslash (^{1}\circ()\mathrm{i}1^{\cdot}(\backslash$$1(\mathrm{I}.()$$t_{\mathrm{r}\rfloor}$$\lfloor$$.()$$\mathrm{i}_{\mathrm{L}}\mathrm{s}$$.\mathrm{s}1_{1\mathrm{a}_{1}}\supset \mathrm{e}$$\mathrm{w}\mathrm{a}\mathrm{r}\mathrm{n}\mathrm{l}-\iota \mathrm{l}\mathrm{I}yP_{2}$$\mathrm{e}\mathrm{x}_{1^{)\mathrm{e}\mathrm{t}\cdot \mathrm{t}}}\mathrm{C}\mathrm{c}1$$(.1_{1_{C}\mathrm{i}\mathrm{n}\mathrm{g}}t(^{\backslash }h$$\mathfrak{c}\cdot \mathrm{l}\mathrm{e}\mathrm{a}\mathrm{r}$56

Figure 6: Dependence on the warm-up tilne for the expected power consumed per unit
time in steady-state.

6

Concluding Remarks

In this paper. we have proposed the stochastic model to generate the auto sleep schedule
for a computer system with batch arrival of transactions. In the case of the renewal accessrequirements, the expected power consumed per unit tinle in steady-state has been formulated. Also. we have derived the approximation form of the idle period from its asylnptotic
property. With the proposed approximation method. we llave derived a sufficient condition
to exist the unique optimal sleep timing which minilnizes the expected power consumed
per unit time in steady-state. In numerical examples. we have investigated asylnptotic
properties of the idle period. and have exanlined the sensitivity of model parameters forthe optimal sleep timing.

References[1] Shimotono. S. (1997). Power managelnent for notebook PC (in Japanese). Joumal of

IPSJ. 38. 135-142.

[2] Aihara. T.. Sekiva. K. and Kuroda. A. (1997). Low-power svstcm technologie.h
).
(in
not
of IEICE. 80. 370 37().
$G\iota_{)(\mathrm{o}}\mathrm{k}((\ln_{\mathrm{P}^{\iota}}1\mathrm{t}\mathrm{e}\mathrm{r}.*$[:;] Sandoh. H..

$\mathrm{J}_{8}\mathrm{P}^{\mathrm{a}11}\mathrm{C}\mathrm{s}\mathrm{c}$slli. H. alld$\mathrm{H}\mathrm{i}_{\mathrm{l}\mathrm{a}}\mathrm{k}o$$\mathrm{d}\mathrm{l}\mathrm{l}\mathrm{t}\circ-‘*1\in 1(11)‘ \mathrm{b}\backslash ^{\tau}‘ \mathrm{s}\mathrm{t}\xi^{1}\mathrm{m}$.

$\zeta_{()},’\gamma/’ l$)$J_{o\mathrm{t}r\gamma}7al$$\mathrm{K}_{\mathrm{f}1\backslash \mathrm{v}c}\urcorner \mathrm{i}$. H. (1996). All (\rangle ptillldl. $23$ . 221-227.

$\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{l}()$to$\mathrm{h}1(^{\tau}\mathrm{C})1$)$\mathrm{f}()1^{\cdot}$all$(\mathit{1}\dagger\theta^{\prime r\cdot,\mathrm{b}}\beta d\zeta)_{\mathit{1}^{J\theta}}\mathcal{T}\zeta \mathrm{t}t/\mathit{0}n.6R_{C_{}\mathrm{q}(^{}(\prime},,Ct/$[4] Hirakoshi. H. alld Salldoll. H. (1996). All optinlal
consideringubage stateb. Iatf, ematical and$\mathrm{t}\mathrm{w}’ 0$$\mathrm{f}\mathrm{o}1^{\cdot}$$l\mathrm{I}$$\mathrm{r}\mathrm{i}111(^{\backslash }$to$h1(^{)}\mathrm{C}^{1}1)$

fol clll$\mathrm{a}1\iota \mathrm{t}\mathrm{o}-\mathrm{h}\mathrm{l}\mathrm{c}\mathrm{c}^{1}1^{)}$$Comput_{\mathrm{C}^{\lrcorner}}\Gamma^{\cdot}\mathrm{J}/I(de[l\dot{l}ng$svstelll. to appear.

[5] Dohi. T.. Kaio. N. and Osaki. S. (1997). Nonparanletric approach to power-savillg
strategies for a portable personal computer. Electronics and Communications in Japan.
Part 3. 80, 80-90.
[6] Okamura, H., Dohi, T. and Osaki, S. (1998), On the effect of power saving by auto-sleep(in Japallese),functions on a computer system I–modeling by a renewal
39,
IPSJ,
on
1858-1869.
Transactions$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}-$57[7] Okamura, H., Dohi, T. and Osaki, S. (1999), On the effects of power saving by autosleep functions on a computer system –queueing(in Japanese), Transactions on IPSJ, 40, 1027-1040.
$\mathrm{I}\mathrm{I}$$\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{s}-$[8] Takagi, H. (1991), Queueing Analysqs, Vol. 1, Elsevier Science Publisher, Netherlands.

[9] Gross, D. and Harris, C. M. (1998), Fundamentals of Queueing Theory, John Wiley&
Sons, New York.
[10] Karlin, S. (1969), A First Course in Stochastic Processes, Academic Press, New York.

[11] Neuts, M. F. (1981), Matrzx-Geometnc Solutions in Stochastic Models: An Algorithmic Approach, The Johns Hopkins University Press, Baltimore.