Online Server Allocation in a Server Farm
via Benefit Task Systems
[Extended Abstract]
T.S. Jayram
Tracy Kimbrely
Robert Krauthgamerz
Baruch Schiebery
Maxim Sviridenkoy
ABSTRACT
A web ontent hosting servie provider needs to dynamially alloate servers in a server farm to its ustomers' web
sites. Ideally, the alloation to a site should always suÆe
to handle its load. However, due to a limited number of
servers and the overhead inurred in hanging the alloation of a server from one site to another, the system may
beome overloaded. The problem faed by the web hosting
servie provider is how to alloate the available servers in
the most protable way. Adding to the omplexity of this
problem is the fat that future loads of the sites are either
unknown or known only for the very near future.
In this paper we model this server alloation problem, and
onsider both its oine and online versions. We give a polynomial time algorithm for omputing the optimal oine alloation. In the online setting, we show almost optimal algorithms (both deterministi and randomized) for any positive lookahead. The quality of the solution improves as the
lookahead inreases. We also onsider several speial ases
of pratial interest. Finally, we present some experimental
results using atual trae data that show that one of our
online algorithm performs very lose to optimal.
Interestingly, the online server alloation problem an be
ast as a more general benet task system that we dene.
Our results extend to this task system, whih aptures also
the benet maximization variants of the k-server problem
and the metrial task system problem. It follows that the
benet maximization variants of these problems are more
tratable than their ost minimization variants.
IBM Almaden Researh Center, 650 Harry Road, San Jose,
CA
95120. E-mail: jayramalmaden.ibm.om
y IBM T.J. Watson Researh Center, P.O. Box 218,
Yorktown Heights, NY 10598.
E-mail:
fkimbrel,sbar,svirigwatson.ibm.om
z Department of Computer Siene and Applied Mathematis, Weizmann Institute of Siene, Rehovot 76100, Israel.
E-mail: robiwisdom.weizmann.a.il . Part of this work
was done while this author was visiting IBM T.J. Watson
Researh Center.
Permission to make digital or hard copies of all or part of this work for
personal or classroom use is granted without fee provided that copies are
not made or distributed for profit or commercial advantage and that copies
bear this notice and the full citation on the first page. To copy otherwise, to
republish, to post on servers or to redistribute to lists, requires prior specific
permission and/or a fee.
STOC’01, July 6-8, 2001, Hersonissos, Crete, Greece.
Copyright 2001 ACM 1-58113-349-9/01/0007 ... 5.00.
$
1. INTRODUCTION
Web ontent hosting is an important emerging market.
Data enters and web server \farms" are proliferating (see,
e.g., [19, 20℄). The rationale for using suh enters is that
servie providers an benet from eonomies of sale and
sharing of resoures among multiple ustomers. This benet translates, in turn, to lower ost of maintenane for the
ustomers who purhase these hosting servies. Web ontent hosting servies are strutured in many ways. One of
the most prevailing ways is outsouring: the ustomers deliver their web site ontent to the web ontent hosting servie
provider, and the servie provider is responsible for serving
this ontent in response to HTTP requests. Most of the servie providers use \farms" of ommodity servers to ahieve
this goal.
One of the omponents in the payment for suh a servie is \pay per served request". Thus, one of the main
objetives of the servie provider is to maximize the revenue
from served requests while keeping a tab on the amount of
resoures used. Ideally, the alloation to a web site should
always suÆe to serve its requests. However, due to a limited number of servers and the overhead inurred in hanging the alloation of a server from one site to another, the
system may beome overloaded, and requests may be left
unserved. We make the realisti assumption that requests
are not queued, and a request is lost if it is not served at
the time it is reeived. The problem faed by the web hosting servie provider is how to utilize the available servers in
the most protable way. Adding to the omplexity of this
problem is the fat that future loads of the sites are either
unknown or known only for the very near future. In this
paper we model this problem, and onsider both its oine
and online versions.
In our model time is divided into intervals of xed length.
We assume that eah site's demand is uniformly spread
throughout eah suh interval. Server alloations remain
xed for the duration of an interval. We assume that servers
are realloated only at the beginning of an interval, and that
a realloated server is unavailable for the length of the interval during whih it is realloated. This represents the time
to \srub" the old site (ustomer data) to whih the server
was alloated, to reboot the server and to load the new site
to whih the server has been alloated. The length of the
time interval is set to be equal to the nonnegligible amount
of time required for a server to prepare to serve a new ustomer. In urrent tehnology, this time is in the order of 5
to 10 minutes. In the sequel, we normalize time so that the
length of a time interval is one unit.
Algorithm
Lookahead Lower Bound
Upper Bound
Deterministi L = 1
4 4
o
n
p
general L 1 + L + L2
min 1 + L ; 1 + L L L +1
Randomized general L
1+ L 1+ L
1
1
1
+1
4
7
1
1
Table 1: Results
Eah server has a rate of requests it an serve in a time
interval. For simpliity, we assume that all rates are idential. Due to pratial onerns (seurity onstraints plaed
by ustomers), we do not allow sharing of servers at the same
time. Customers share servers only in the sense of using the
same servers at dierent times, but do not use the same
servers at the same time. Thus, even in ase of overload,
some of the servers may be underutilized if they are alloated to sites with rates of requests lower than the servers'
rate. In the sequel, we normalize rates so that the servers'
rate is 1, and allow the demand of sites to be frational.
We assume that eah ustomer's demand is assoiated
with a benet gained by the servie provider in ase a unit
demand is satised. Given the number of available servers,
the objetive of the servie provider is to nd a time-varying
server alloation that maximizes the benet gained by satisfying sites' demand. We all this server alloation problem
the web server farm problem. In the oine version of this
problem, future demands of the sites are known. In the
fully online version, only the demand of the urrent interval
is known at the beginning of the interval. It makes sense to
assume that some amount of future demand is known to the
servie provider, and we model it by lookahead. Lookahead
L means that the demand of the sites is known for the next
L time intervals (following the urrent interval). In reality,
lookahead will require that some foreasting mehanism be
used to predit future demands. Fortunately, web traÆ exhibits properties of preditability over the short term whih
render this pratial as demonstrated in [13, 18℄. We measure the performane of an online alloation algorithm using
its ompetitive ratio; that is, the maximum over all instanes
of the ratio of the benet gained by the optimal oine alloation to the benet gained by the online alloation on the
same instane. Sine we are dealing with a maximization
problem, this ratio is always greater than 1, and the goal is
to make it as small as possible.
Interestingly, our model an be ast as a more general
benet task system. In this task system we are given a set
of states for eah time t and a benet funtion. The system
an be in a single state at eah time, and the benet gained
between times t and t +1 is a funtion of the system states at
times t and t +1. The goal is to nd a time-varying sequene
of states that maximizes the total benet gained; that is,
at eah time t we need to determine to whih state should
the system move (and this will be the state of the system at
time t +1), and we gain the benet that is determined by the
benet funtion. Similar to the web server farm problem, in
the oine version, the benet funtion is known in advane.
In the fully online version, only the possible benets of the
urrent step are known at the time a move is determined,
and in ase of lookahead L, the benets of the next L steps
(in addition to the urrent) are known. Note that number
of states in a benet task system might be super-polynomial
if the states are not given expliitly.
It an be shown that benet task systems apture also
benet maximization variants of well studied problems, suh
as the k-server problem [16, 9℄ and metrial task systems [10,
9℄ (see Setion 2). Thus, our results hold for these variants
as well, and show that the benet variants of these problems
may be more tratable than their ost minimization variants.
Our results
We observe that the oine version of the web server farm
problem an be solved in polynomial time. In the online
setting, we rst note that the ompetitive ratio in the fully
online version is unbounded; we then onsider the ase of
a positive lookahead for both deterministi algorithms and
randomized ones. We obtain tight bounds on the ompetitive ratio for the randomized ase and almost tight bounds
for the deterministi ase. In the deterministi ase we also
onsider the speial ase L = 1. The spei results are
summarized in Table 1. Our bounds on the ompetitive
ratio extend to the benet task system problem, assuming
that an optimal sequene of moves for the period of the
lookahead (i.e. for the next L + 1 time steps) is eÆiently
omputable. Therefore, our upper bounds (i.e. algorithms)
are presented in the more general framework of the benet
task system problem, and the lower bounds are given for the
more restrited web server farm problem.
In addition, we onsider two speial ases of the web server
farm problem with lookahead 1. One is the basi ase of one
server and two sites. For the
p ompetitive ratio in this ase we
obtain a lower bound of 3 3=2 p 2:598 and an upper bound
is the golden ratio. The
of 1 + 2:618, where =
seond ase is that of k servers and arbitrary number of sites,
for whih we give an algorithm that ahieves a ompetitive
ratio 2 if k is even and 2 + 2=(2k 1) if k is odd.
Several of our algorithms use the natural approah of \disounting the future". Speially, when the algorithm faes
the hoie between a guaranteed benet immediately and
a potential benet in the future, the deision is made by
omparing the guaranteed benet value with a disounted
value of the potential future benet. This disount fator is
exponential in the number of time units that it would take
the potential benet to be materialized. The reason for disounting future benets is straightforward. By the time a
benet will be materialized, things might hange and the algorithm might deide to make another hoie for a potential
(even larger) benet.
Another approah used by our algorithms in the ase of
lookahead is an \intermittent reset". Some number of future
steps are determined and then a single time step is \wasted"
to realloate the servers in preparation for another sequene
of future steps. The time step for realloation is hosen so
that the amount of benet unrealized due to the realloation
is guaranteed to be small with respet to the benet that is
gained.
1+ 5
2
We use atual web traÆ data of ommerial sites and
randomly generated data to evaluate our algorithms. Our
experiments show that the disount algorithm performs better, with less lookahead, than the intermittent reset algorithm, and is within 0:2% of optimal on all the ommerial
trae instanes, even with lookahead one.
Related work
Our web server farm problem an be viewed as a sheduling
problem. The requests are represented by unit length jobs,
and the k servers are mahines. Eah job has a deadline
whih is exatly one time unit after its release, and a value.
The goal is to maximize the throughput; that is, to maximize
the value of the sheduled jobs. An additional fator is setup:
it takes one time unit to set up a mahine to handle a job
for one web site if it handled a job in the previous time unit
for a dierent site.
Several papers have onsidered the problem of throughput
maximization in sheduling jobs on parallel mahines with
release times and deadlines, both in the oine and online
settings [5, 3, 7, 8, 14℄. In thePstandard notation, this type of
problem is denoted by P jri j wi (1 Ui ). The type of setup
onsidered here is related to \bath setup" sheduling, for
whih several oine problems have been studied (see [17℄ for
a survey). The online problems onsidered here also relate
to previous work on online all admission and bandwidth
alloation [12, 15, 4, 6℄. Again, the issue of setups does not
appear in any of these papers.
The benet task system an also be viewed as a prot
maximization version of the request-answer game of [9℄. To
the best of our knowledge this is the rst attempt to onsider a task system for prot maximization rather than ost
minimization problems. Awerbuh et al. [2℄ onsidered a
similar problem in whih a deision maker that has no way
to aurately predit future performane of several options
has to hoose one option that will have the best future performane. This problem has an unbounded deterministi
ompetitive ratio, and the authors propose a randomized
algorithm. Lookahead is not onsidered in [2℄.
In the next setion we give more aurate
denitions of our models and show their relations to the
k-server model and metrial task systems. In Setion 3 we
desribe an oine algorithm for the server farm problem. In
Setion 4 we devise two deterministi online algorithms for
the benet task system, based on disounting the future and
on an intermittent reset, and we give upper bounds on their
ompetitive ratio. In Setion 5 we prove lower bounds on the
ompetitive ratio of any deterministi online algorithm for
the more restrited web server farm problem. In Setion 6
we onsider some speial ases of the online deterministi
web server farm problem. In Setion 7 we give tight bounds
for the randomized ase. We onlude in Setion 8 with
some experimental results based on traes of ommerial
retail sites.
Organization.
server has a \servie rate" whih is the number of requests
to a web site eah server an serve in a time unit. Without
loss of generality, we normalize the demands by the servie
rate so that a server an serve unit demand per time unit
and demands of a site may be frational. A web server an
be alloated to no more than one site at eah time unit and
it takes a time unit to hange the alloation of a server.
A problem instane onsists of the number of servers k,
the number of sites s, a nonnegative benet matrix fbi;t g
denoting the benet gained by serving a request of site i 2
[1::s℄ for time step t 2 [1:: ℄, and a nonnegative demand
matrix fdi;t g denoting the number of requests at site i for
time step t. The goal is to nd for eah site i 2 [1::s℄, a
time-varying alloation fai;t g of servers, so as to maximize
the total benet,
as follows. The alloation must satisfy that
for eah t, Psi ai;t k. Only a0i;t = minfai;t ; ai;t g of the
servers alloated to site i for time step t are \produtive",
i.e., atually serve requests.
that the total benet
P WePget
s b minfd ; a0 g =
of
alloation
f
ai;t g is t
i;t
i;t i;t
i
Pan P
s b minfd ; a ; a
i;t
i;t i;t i;t g. The initial alloation
i
t
is xed to be, say, ai; = 0 for all i.
In the oine version of this problem we are given the
omplete demand matrix fdi;t g and we need to ompute
the omplete alloation fai;t g. In the fully online version
we need to ompute the alloation fai;t g at time
t given
the demands so far, i.e., fdi;t g for all i and t0 t. It is
not diÆult to show that any fully online algorithm may
perform arbitrarily badly with respet to the oine optimal
solution. This is true beause of the one time unit lag in
a server's availability. Thus we onsider online algorithms
with lookahead. In the online version with lookahead L,
at eah time t we are additionally given the demands and
benets for times t0+1; : : : ; t+L, i.e., the entries di;t and bi;t
for i 2 [1::s℄ and t 2 [t +1::t + L℄, and we need to ompute
the alloation at time t, i.e., ai;t for i 2 [1::s℄.
The benet task system problem: The web server
farm problem is a speial ase of the generalized task system benet problem. In this problem we are given (i) a set
of possible states Ut , for eah time t 2 [0:: ℄, Sand (ii) a nonnegative benet funtion B whose domain is t (Ut Ut ),
that denotes, for eah time t, the benet that is arued (at
time t + 1) by the transition from a state in Ut to a state in
Ut . The goal is to hoose a P
state st for eah time t so as
to maximize the total benet t B (st; st ). The initial
state is xed by letting that U = fs g.
In the oine version of the problem all the state sets and
the benet funtion are known in advane. In the online
version with lookahead L, the state st 20 Ut should be omputed based on knowing only Ut for t t + L, and the
restrition of the funtion B to pairs of these sets of states.
Observe that the web server farm problem an be ast in
this setting by identifying eah possible alloation of servers
to sites at time t with a state Si;t, and dening the benet
funtion B (Si;t ; Sj;t ) to be the benet gained by (serving
some number of requests while) hanging the alloation at
time t from the one represented by Si;t to the alloation at
time t +1 represented by Sj;t . (In a sense, the set of states
for all times are the same.) The number of states is exponential in the number of servers k, so the states and the benet
funtion are impliit and follow from the more suint representation of the web server farm problem. For example,
the values B (Si;t; Sj;t ) are not listed expliitly, and any
single value an be eÆiently omputed when neessary.
1
=1
=1
=1
0
0
THE PROBLEMS
The web server farm problem: Suppose that we are
given s web sites that are to be served by k web servers. (For
simpliity, we assume that all servers are idential.) Time
is divided into units (a.k.a. steps). It is assumed that the
demand of a web site is uniform in eah time unit. Eah
0
0
+1
+1
1
=0
0
+1
+1
+1
0
0
+1
2.
=1
1
=1
Benet maximization variants of well known problems an
also be ast in this setting of a benet task system. Consider
the benet version of the k-server problem. This version of
the k-server is similar to the lassial k-server problem [16℄
with one dierene. Instead of minimizing the ost of satisfying all requests, we dene for eah time t and any possible
pair of server ongurations, one at time t 1 and one at
time t, a net benet gained by satisfying the request starting
from the onguration at time t 1 and ending at the onguration at time t. (Note that the onguration at time
t must inlude at least one server at the point of the request.) This benet has to be nonnegative and is omposed
of a xed positive omponent that is redued by the ost to
move from the onguration at time t 1 to the onguration at time t. The goal in this ase is to maximize the
total benet. It is not diÆult to see that this problem an
also be modeled by the benet task system dened above
and thus all our results apply to the benet version of the
k-server problem.
Similarly, onsider the benet version of a metrial task
system. This version is similar to the lassial metrial task
system [10℄, with the dierene that eah task is assoiated
with a vetor of benets, one for eah state, suh that the net
benet after subtrating the transition ost is nonnegative.
This model as well an be ast as a benet task system and
our results apply.
3.
OFFLINE ALGORITHM FOR THE WEB
SERVER FARM PROBLEM
In this setion we show that the oine web server farm
problem an be solved in polynomial-time. Note that an
algorithm that solves this oine problem is also a omponent in the deterministi online algorithms of Setion 4, as
these online algorithms rely on an ability to nd an optimal
sequene of alloations in the limited future given by the
lookahead (i.e., the next L + 1 time units).
We redue the web server farm problem to the well-known
minimum-ost network ow problem, whih an be solved in
polynomial time; see for example [1, 11℄. We remark that
when sk is polynomial in the input size, the web server farm
problem an be solved in polynomial time also using dynami programming. However, in general k might be exponential in the input size.
1. The oine web server farm problem an be
Theorem
redued in polynomial time to a minimum-ost network ow
problem. Hene, it an be solved in polynomial time.
Reall that the input for the minimum ost network ow problem is a direted network, two speial verties
alled the soure vertex and the sink vertex, an amount of
ow to be injeted into the soure vertex, and a nonnegative
apaity and a ost for eah edge. The goal is to nd from
all the ows from s to t that respet the edge apaities and
are of size k, one that has a minimal ost, where the ost of
a ow is the sum, over all edges, of the produt of the ow
on this edge and its ost.
In fat, we desribe below a redution from the web server
farm problem to the analogous maximum-ost network ow
problem. The latter problem is essentially the same as the
minimum-ost network ow problem (and thus an be solved
in polynomial time), sine the edge osts are not restrited
Proof.
in sign, i.e., are allowed to be negative. We remark that in
our network, all the paths from s to t are of equal length,
and therefore another way to guarantee that all osts are
nonnegative is to inrease the osts of all the edges by the
same suÆiently large number.
The outline of the redution is as follows. Given an instane of the web server farm problem, we onstrut a (direted) network with s \rails", one per site. Eah rail is a
hain of edges, eah representing one time step. (Atually,
we will split these edges into three parallel edges, for reasons
whih will beome lear shortly.) Flow along a rail represents the alloation of servers to the orresponding site. In
addition, we onstrut a set of \free pool" nodes, one per
time step, through whih ow will pass when servers are
realloated from one site to another.
Let fdi;t g with i 2 [1::s℄ and t 2 [1:: ℄ be the demand
matrix. We onstrut nodes ni;t for i 2 [1::s℄ and t 2 [1:: ℄,
along with nodes ft for t 2 [1:: ℄. For eah site s and eah
time step t, we onstrut three edges from ni;t to ni;t .
The rst has apaity bdi;t and ost bi;t . The seond has
apaity one and ost bi;t (di;t bdi;t ). The last has innite
apaity and ost 0. Flow along the rst edge represents the
benet of alloating servers to site i during time step t, up
to the integer part of di;t . Flow along the seond represents
the remaining benet, bi;t times the frational part of di;t, to
be olleted by one more server (that will be only partially
produtive). Flow along the third represents the fat that
extra servers an remain alloated to i, but do not ollet
any benet. Clearly the rst edge will saturate before ow is
present on the seond, and similarly the seond will saturate
before ow is present on the third.
We also onstrut edges of innite apaity and ost 0
from ni;t to ft and from ft to ni;t , for eah t 2 [1:: ℄ and
i 2 [1::s℄. These represent the movement of servers from one
site to another. (We use an intermediate node ft to keep the
number of edges linear.)
Finally, we onstrut a soure into whih we injet ow
k, with innite apaity zero ost edges to eah ni; , and a
sink with innite apaity zero ost edges from eah ni; .
It is not hard to see that an integral ow of ost C in
this network orresponds to an alloation fai;t g with benet equal to C , and vie versa. It is well known that sine
all the edge apaities are integral, there is minimum-ost
(or maximum-ost) ow in the network that is integral, and
that, furthermore, suh a ow an be eÆiently found. This
implies an eÆient algorithm that nds an optimal alloation fai;t g. Note that the size of the network (number of
nodes and edges) is linear in the size of the demand matrix fdi;t g, and thus the oine web server farm problem an
be solved within roughly the the same time as the best algorithm for minimum-ost network ow on a network with
O(s ) nodes and edges.
1
1
1
4. DETERMINISTIC ONLINE ALGORITHMS
FOR BENEFIT TASK SYSTEMS
In this setion we desribe two deterministi algorithms
for benet task systems with lookahead L. The rst algorithm is a future disountingp
algorithm that ahieves a
ompetitive ratio of (1 + 1=L) L L + 1. Asymptotially in
L this ratio is 1 + (log L)=L. The seond algorithm is an
intermittent reset algorithm that ahieves a ompetitive ratio of 1 + 4=(L 7), whih is better than the rst algorithm
when L is suÆiently large. Eah algorithm requires that
one an eÆiently nd a sequene of moves that is optimal
(in a ertain sense) for the period of the lookahead (i.e. for
the next L + 1 time steps). In the partiular ase of the
web server farm problem, we an ahieve this requirement
by using the oine algorithm of Setion 3.
4.1 The future discounting algorithm
We present a deterministi
p algorithm that ahieves ompetitive ratio (1 + 1=L) L L + 1 for the benet task system
problem. The algorithm is based on disounting future benets in an exponential way, as follows. Consider a sequene
of L +1 moves that ollets the benets b ; b ; : : : ; bL (in this
order) in the next L +1 time steps. We dene the antiipated
L (for a
benet of this sequene to be b + b = + : : : + bL =
p
L
parameter > 1 that we later hoose to be = L + 1).
The disounting algorithm greedily follows at eah time
step the largest antiipated benet possible at that time.
Namely, at every time step the algorithm nds a sequene
of L + 1 moves (for the next L + 1 time steps) whose antiipated benet is maximal, and performs the rst move in
this sequene. We stress that a new sequene is alulated
at every time step, and that the sequene found in the urrent time step does not have to agree with the one found in
the previous time step.
Note that this disounting algorithm requires eÆient omputation of a sequene of L + 1 moves with the maximum
antiipated benet. The following lemma provides this requirement when the number of states at every time t is polynomial in the input size, e.g. when the states of the benet
task system are given expliitly. In the web server farm problem, the number of (impliit) states is super-polynomial, but
we an use the oine algorithm of Setion 3, with straightforward modiations for handling the initial state and the
disount.
2. Suppose that the number of states possible for
any one time is at most M . Then a sequene of L +1 moves
0
0
1
1
Lemma
with the largest antiipated benet an be omputed in time
that is polynomial in M (and L).
Use dynami programming that goes iteratively
over the time steps and aumulates disounted benets. As
the additional benet of the next time step depends only on
the urrent state, it suÆes to have for eah time step t a
table of size M , and ompute the table for time step t + 1
from that of time step t.
3. The ompetitive ratio
p of the above disounting algorithm is at most (1 + 1=L) L L + 1.
Proof.
Theorem
Consider among the sequenes of L + 1 moves
that are available for the online algorithm at time t, the sequene with the largest antiipated benet, and let b ; b ; : : : ; bL
be the benets
olleted by this sequene. Dene ONt = b
and ONtL = b = + : : : + bL =L , and then ONt + ONtL
is the largest antiipated benet over all sequenes of L + 1
moves that are available to the online algorithm at time
t. The online algorithm performs the rst move in this sequene, and thus the benet it ollets at time t is ONt .
Fixing an oine algorithm arbitrarily, let OF Ft denote
the benet
that this oine algorithm ollets at time t, and
let OF FtL be a shorthand for OF Ft =+: : :+OF Ft L =L .
Proof.
0
1
0
+1
1
+1
+1
+
One sequene of moves that the online algorithm onsiders
at time t is to rst join the oine algorithm, (i.e., move
to the same state as the oine algorithm at time t), and
then follow the oine algorithm for the next L steps.
The
antiipated benet of this sequene is at least OF FtL . Sine
the online algorithm follows at time t the sequene of moves
with the largest antiipated benet, we get that
ONt + ONtL OF FtL
(1)
Another sequene of moves that is onsidered by the online algorithm at time t is to follow the sequene that had
the largest antiipated benet at the previous time step t 1
(without the rst step of that sequene, whih was performed at time t 1, and with an arbitrary move added
at the end of the sequene.) The ontribution of these benets to the antiipated benet at time t is larger by a fator
of than their ontribution at time t 1, and so the antiipated benet of this sequene of moves is at least ONtL .
Sine the online algorithm follows at time t the sequene of
moves with the largest antiipated benet, we get that
ONt + ONtL ONtL
(2)
Adding (1 ) times inequality (1) and times inequality (2), we get that
+1
+1
+1
1
1
1
1 )OF F L : (3)
t
Adding up inequality (3) over all time steps t we get that
(it is straightforward that we an assume, without loss of
generality, that all the benets in the rst and last L + 1
time steps are zero regardless of the state):
X
X
ONt (1 1 ) OF FtL
L ON L + (1
ONt + ONt+1
t
t
+1
+1
t
X
= (1 1 ) 1 + : : : + 1L
OF Ft :
t
The last equality follows
sine every benet that the oine
ollets ours in Pt OF FtL with eah of the disounts
1=; : : : ; 1=L . We onlude that
the ompetitive ratio of
L L + 1):
the algorithm is (hoosing = p
+1
P
OF Ft
Pt
t ONt
1
1
L
p
= L 1 = (1 + 1=L) L L + 1 :
+1
1
L+1
4.2 The intermittent reset algorithm
We present a deterministi intermittent reset online algorithm that has ompetitive ratio 1 + O(1=L) for the benet
task system problem. This algorithm is motivated by the
randomized algorithm presented in Setion 7 that saries
one out of every L +1 steps in order to gain the optimal benet between saried steps. Here, we sarie one out of
every (L) steps, but the hoie of steps to sarie is done
arefully in a deterministi fashion and the overall benet
that we gain is lose to the expeted benet of the randomized algorithm.
The algorithm works in iterations of length at least L=2
and at most L. Let si denote the start time of iteration i.
Eah iteration i onsists of four steps, as follows.
In a sense, this is a potential funtion analysis. See for
example [9℄.
1
Consider all times t = si + L=2 + 1 : : : ; si + L. For
eah suh time t, let xt denote the maximum benet
of any transition from a state at time t to a state at
time t + 1. Let T be the time t that minimizes xt, i.e.,
xT = minfxt : si + L=2 + 1 t si + Lg.
Compute the optimal sequene of transitions that
starts at any state at time si + 1 and ends at any
state at time T . This an be done either by dynami
programming similar to Lemma 2 or by applying the
oine algorithm of Setion 3. (In the rst iteration
the initial state is given and thus we may ompute the
optimal sequene of transitions from the initial state to
any state at time T . The next step has to be modied
aordingly.)
Move from the urrent state at time si to the state
at time si + 1 that is the starting state of the optimal sequene omputed above, and ontinue with the
optimal sequene of moves to the state at time T .
Set the starting point si = T for the next iteration.
4. The ompetitive ratio of the above intermittent reset algorithm is at most 1 + L .
Step 1:
Step 2:
Step 3:
Step 4:
+1
Theorem
4
7
We assume that L is a multiple of 4, and otherwise round it downwards to the nearest multiple of 4 (at
the expense of inreasing the ompetitive ratio as explained
later). We will all the starting times of the iterations
\gaps." Denote the gaps by s ; s ; : : :. We divide the transitions taken by the algorithm into two omponents, the
moves taken in the gaps and the steps taken in the rest of
the times. Consider the path taken by the online algorithm.
It may not ollet any benet in the gaps, but sine it omputes the optimal paths between gaps (allowing arbitrary
initial and terminal states adjaent to the gaps), learly the
benet of the optimal oine between gaps is no larger than
the benet of the online algorithm between gaps.
Now we bound the benet of the optimal oine algorithm
in the gaps. Consider iteration i starting at si and omputing T = si . Consider the portion of the online path from
si + L=2 + 1 to si . Denote the length of this portion by
a, i.e., a = si
(si + L=2 + 1). Consider the portion
of the path omputed in the next iteration from si + 1
to si + L + 1. Denote the length of this portion by b, i.e.,
b = si + L + 1 (si + 1). By our hoie of L, L=2 is even
and a + b = L=2 1. Thus one of a and b is even and the
other is odd. Assume a is odd and b is even; the proof is
similar in the opposite ase. (We note that ideally, we would
like both a and b to be even, but hoosing a + b to be even
may lead to a worse ase when both are odd.)
We laim that the benet ahieved by the online algorithm
in these portions of the path is at least xT (a + b 1)=2.
This is true sine in every time step in the range there is a
transition of benet xT and thus the online algorithm an
ahieve benet at least xT on every other transition, even
if it ollets no benet on the other half of the transitions.
Sine a is odd, it an ollet at least xT (a 1)=2 in the rst
portion, and sine b is even, it an ollet at least xT b=2
in the seond.
Putting these together along with a + b = L=2 1 we have
that the online benet is at least
xT ( a 1 + b ) = x T L 4 :
2
2
4
Proof.
1
2
+1
+1
+1
+1
+1
Sine si + L + 1 is always less than si + L=2 + 1, these
portions are disjoint among iterations.
Summing over all
iterations we have ON ( L ) Pi xTi , where ON denotes the total benet olleted by the online algorithm
and xTi denotes the maximum benet possible in the ith
gap. AountingPfor the portions outside the gaps we have
ON OF F
i xTi , where OF F denotes the benet of
the optimal oine algorithm. Putting these together and
allowing for values of L that are not multiples of 4 yields
the desired bound of 1 + L .
+1
4
4
4
7
5. LOWER BOUNDS FOR DETERMINISTIC ONLINE ALGORITHMS
In this setion we prove three lower bounds on the ompetitive ratio of deterministi online algorithms for the web
server farm problem with one server. The rst lower bound
is for general lookahead, the seond is for lookahead 1 (and
arbitrary number of sites), and the third is for lookahead 1
and two sites.
5.1 A lower bound for general lookahead
We show a lower bound of 1+ L + L2 on the ompetitive
ratio of a deterministi online algorithm for the web server
farm problem with one server. The same lower bound then
follows for the benet task system problem. Formally, we
have the following theorem.
5. The ompetitive ratio of any deterministi
1
1
+1
Theorem
online algorithm for the web server farm problem with one
server (and hene for the benet task system problem) is
larger than 1 + L1 + L21+1 .
Consider a deterministi online algorithm whose
input is a 3(L+3) demand table fdi;t g. Sine the algorithm
has lookahead L, it is given at time t = 0 only the rst L +1
olumns, as illustrated below. Sine the rst olumn is all
zeros, we may assume without loss of generality that the
server is initially alloated to the rst site. We denote the
sequene of loations of the server by triangles, and use a
parameter a that will be determined later.
3
2
.0 1 a : : : a ? ?
fdi;t g = 4 0 1 a : : : a ? ? 5
0 1 a ::: a ? ?
Sine at time t = 0 all three rows look the same, we may
assume without loss of generality that the algorithm deides
to alloate the server at this time to the rst site, and it
ollets a benet of 0. Next, at time t = 1 another olumn
is revealed to the algorithm, and the situation is as follows.
3
2
.0 .1 a : : : a 0 ?
fdi;t g = 4 0 1 a : : : a a ? 5
0 1 a ::: a a ?
At time t = 1 the algorithm an either alloate the server
to the rst site again or to another site. Consider rst the
ase that the algorithm alloates the server to the rst site
again, so the algorithm ollets a benet of 1. In this ase
the last olumn is revealed (at time t = 2) to be all zeros,
and the situation is the following.
3
2
.0 .1 .a : : : a 0 0
fdi;t g = 4 0 1 a : : : a a 0 5
0 1 a ::: a a 0
Proof.
The total benet that the online an ollet in this ase is at
most 1 + (L 1)a, while the oine ould have olleted the
whole seond row, i.e., 1 + La. Therefore, the ompetitive
ratio of the algorithm would be at least
a
r0 = 1 + La = 1 +
1 + (L 1)a
1 + (L 1)a :
Consider now the ase that the algorithm alloates the
server at time t = 2 to a site other than the rst one, so it
ollets a benet of 0. Without loss of generality we may
assume that the server is alloated to the seond site. At
time t = 2, the last olumn is revealed to the algorithm, and
the situation is the following.
3
2
.0 .1 a : : : a 0 0
fdi;t g = 4 0 1 .a : : : a a 0 5
0 1 a ::: a a a
The total benet that the online an ollet in this ase is
at most La, while the oine ould have olleted the whole
third row, i.e., 1+(L +1)a. Therefore, the ompetitive ratio
of the algorithm would be at least
1 + (L + 1)a = 1 + a + 1 :
r00 =
La
La
We onlude that the ompetitive ratio of any deterministi
p algorithm is at least minfr0 ; r00 g. Choosing a = (L +
L + 4)=2, we get that a = 1 + La and hene
r0 = r00 = 1 + a + 1 = 1 + 1 + 1 :
2
2
La
L aL
0
Sine a < L +1=L, we get that minfr ; r00 g > 1+ L1 + L21+1 ,
as laimed.
This lower bound (slightly) improves over the 1+ L straightforward lower bound. As the latter bound is the ompetitive
ratio threshold for randomized algorithms (see Setion 7),
this proves that for any xed lookahead L, the thresholds of
deterministi and randomized algorithms are dierent.
1
5.2 Lower bounds for lookahead one
We prove two lower bound results for deterministi algorithms for the web server farm problem with one server and
lookahead 1. The rst result shows that no algorithm an
ahieve a ompetitive ratio better than 4 for this problem.
This mathes our algorithm from Setion 4.1 that ahieves a
ompetitive ratio 4. The seond result shows that in the ase
where the number of sites is two,pno algorithm an ahieve
a ompetitive ratio better than 3 3=2 2:598. This is almost tight with our algorithm from Setion 6.1 that ahieves
a ompetitive ratio roughly 2.618. The proofs of the following two theorems are omitted due to spae onstraints.
Theorem 6. Consider the web server farm problem with
one server and lookahead 1. For every < 4, there exists an
s > 0 suh that the ompetitive ratio of any deterministi
online algorithm for the problem with s sites is at least .
Theorem 7. Consider the web server farm problem with
p
two sites, one server and lookahead 1. For every < 3 3=2,
the ompetitive ratio of any deterministi online algorithm
for the problem is at least .
6. SPECIAL CASES
In this setion we onsider two speial ases of the web
server farm problem. The rst is the ase of one server and
two sites with lookahead one. The seond is multiple servers
with lookahead one.
6.1 One server, two sites, and lookahead one
We devise a deterministi online algorithm for the web
server farm problem with one server, two sites and lookahead
L = 1. p
The algorithm has ompetitive ratio r = 1+ where
=
1:618 is the golden ratio. This ratio is nearly
the best possible pin this ase, as we present (in Setion 5) a
lower bound of 2:598 on the ompetitive ratio of any
deterministi algorithm for this problem. We remark that
the ompetitive ratio that we obtain in this ase is muh
better than in the ase of arbitrary number of sites, where
the best possible ompetitive ratio is 4.
The algorithm is based on disounting future benets. It
is similar to the algorithm presented in Setion 4.1 (speialized to the ase L = 1), exept that the disount fator is
taken to be . The analysis for this ase adds some ompliation to that of the general ase, in order to take advantage
of the limited number of sites. For example, a disount fator of 2 optimizes the analysis of Setion 4.1, and then the
ompetitive ratio is shown to be at most 4. The atual ompetitive ratio in this ase is 3, whih an be shown by an
analysis similar to the one that we give here.
The following theorem states the improved ompetitive
ratio for this ase.
p 8. The disounting algorithm with disount fator =
ahieves ompetitive ratio r = 1 + 2:618
1+ 5
2
3
2
3
Theorem
1+ 5
2
for the web server farm problem with one server and two
sites.
We rst desribe the algorithm in more detail.
We assoiate with eah possible alloation its antiipated
benet, whih ombines the immediate benet of this alloation and its future benet. To ompensate for the unertainty of a future benet (the algorithm might later deide
not to ollet these benets), we disount it (with respet to
an immediate benet) by a fator of . In other words, an
ation that auses the algorithm to ollet0 a benet b and
to have the potential to ollet a benet b in the future has
antiipated benet of b + b0 .
Suppose that the situation of the online algorithm at some
time t is as illustrated below. Assume, without loss of generality, that the alloation of the server in the previous time
unit is to the rst site, as denoted by the triangle. Sine the
lookahead is L = 1, the algorithm knows only the demands
at the urrent time t and at the next time t + 1.
0
fdi;t g = :: :: :: .xy xy0 ?? ?? :: :: ::
Proof.
The algorithm deides on
the alloation of the server at time
t as follows. If x + x0 y 0 then the previous alloation is
kept, i.e., the online algorithm alloates the server at time t
to the rst site. Otherwise, the algorithm hanges the alloation and the server is alloated to the seond site. Observe
that this deision ahieves the largest antiipated benet for
the algorithm. If it deides to stay with the alloation as
before, it ollets a benet of x and will have the option
to ollet at the next time t + 1 an additional
benet x0 ,
yielding an antiipated benet x + x0 . If the algorithm
deides to hange the alloation to the other site, then it
ollets no immediate benet (at this point there is no way
for the online algorithm to ollet the benet y), and in the0
next time t +1 it will have the option to0 ollet
a benet y ,
yielding an antiipated benet 0 + y = y0.
We will prove by indution on T that
X
X
OF Ft + OF FT
(4)
r ONt + qT ONT 2
tT
+1
+1
+1
tT
where ONt is the atual
benet olleted at time t by the
online algorithm; ONt is the potential benet that the online algorithm will ollet at time t if it alloates the
server
to the same site as at time t 1; OF Ft and OF Ft are dened similarly for an arbitrary oine algorithm; and qt = if at time t 1 the online and the oine algorithms have
alloate the server to the same site (i.e., if they have the
same \starting position" for time
t), and qt = 1 otherwise.
(Note that ONt is either ONt in ase the potential benet
is gained or 0.)
Proving (4) would omplete the proof, sine we an assume, without loss of generality, that all the demands in the
rst and last L + 1 time steps
are zero, and thus obtain for
the last time step T that r Pt ONt Pt OF Ft , as required.
The base ase of the indution follows from padding the
demand matrix with zeros from both sides. For the indutive
step, i.e., that the indution hypothesis (4) for time T follows
from that of time T 1, it suÆes to show that
r ONT + qT ONT
qT ONT
(5)
OF FT + OF FT
OF FT
We prove Equation (5) by verifying 0it over0 all ases. Denote
the benets at times T; T +1 by x; x ; y; y as above. Assume
without loss of generality that at time T 1 the server is
alloated to the rst site. The total number of ases is eight,
sine there are two possible deisions for the online, and four
possible deisions for the oine.
Consider rst the ases in whih the alloation of the ofine at times T and T + 1 are the same.
Then at time T
the oine
ollets
the
benet
OF FT , and hene OF FT =
OF FT . In addition ONT = x, so inequality (5) simplies to
r ONT + qT ONT
qT x OF FT :
1. qT = ; qT = . Then both oine and online ollet
x. We need to show that
r x + x0 x x0
i.e., that x + ( 1)x0 0, whih learly holds. p
2. qT = 1; qT = 1. Then oine ollets y and online
ollets x. We need to show that
r x + x0 x y 0
i.e., that x + x0 y0, whih holds sine online keeps
the previous alloation. p
+1
+1
+1
+1
+1
+1
+1
+1
3. qT = ; qT = 1. Then oine ollets x and online
does not ollet x. We need to show that
r 0 + y 0 x x0
whih holds sine online hanges alloation. p
In a sense, this is a potential funtion analysis. See [9℄.
+1
2
4. qT = 1; qT = . Then oine ollets y and online
does not ollet x. We need to show that
r 0 + y0 x y0
i.e., that y0 xp= x, whih holds sine online
hanges alloation.
Consider now the ases in whih the alloation of the ofine at times T and T + 1 are dierent. Then the benet
of
the oine at time T is OF FT = 0. In addition ONT = x,
so inequality (5) simplies to
r ONT + qT ONT
qT x OF FT
OF FT
5. qT = ; qT = . Then oine and online both do not
ollet x. We need to show that
r 0 + y0 x y0 x
i.e., that y0 p x, whih learly holds sine online hanges
alloation.
+1
1
+1
1
+1
+1
+1
6. qT = 1; qT = 1. Then oine does not ollet y and
online does not ollet x. We need to show that
r 0 + y 0 x x0 y
0
and sine0 online hanges
palloation we indeed get y x + x x + x0 y .
+1
7. qT = ; qT = 1. Then oine does not ollet x and
online ollets x. We need to show that
r x + x0 x y 0 x
i.e., that 2x + x0 y0 , whih holds sine online keeps
the previous alloation.
8. qT = 1; qT = . Then oine does not ollet y and
online ollets x. We need to show that
r x + x 0 x x0 y
i.e., that x + ( 1)x0 + y 0, whih learly holds.
+1
+1
6.2 Multiple servers with lookahead one
We present an online algorithm for the web server farm
problem with multiple servers and lookahead one, as stated
in the following theorem. We omit the proof of the lower
bound for lak of spae.
9. There is a deterministi algorithm for the
Theorem
web server farm problem that ahieves ompetitive ratio 2
when the number of servers k is even, and ompetitive ratio
k = 2 + 2=(2k 1) when k 3 is odd. Furthermore, no
online algorithm ahieves a lower ompetitive ratio.
We distinguish between even and odd numbers
of servers. For an even k, the algorithm splits the set of
servers into two sets of equal size. The rst set of servers
is alloated so as to always ollet the k=2 largest benets
on the odd time steps. These servers spend the even time
steps in realloation to those sites where they will be able
to ollet in the following time step, whih is odd, the k=2
largest benets. On even time steps, these servers may ollet zero benet in the worst ase. The seond set of servers
Proof.
Workload
ran-24
om1-15
om1-25
om2-15
om2-25
Avail.
9494
4634
4634
7265
7265
Opt.
7871
3904
4634
4227
6013
Dis. L = 1
7750 (98.5%)
3899 (99.9%)
4634 (100 %)
4217 (99.8%)
6007 (99.9%)
Dis. L = 4
7803 (99.1%)
3901 (99.9%)
4634 (100%)
4218 (99.8%)
6009 (99.9%)
Reset. L = 4
7484 (95.1%)
3875 (99.3%)
4629 (99.9%)
4163 (98.5%)
5966 (99.2%)
Reset. L = 8
7753 (98.5%)
3894 (99.8%)
4634 (100%)
4183 (99.0%)
5990 (99.6%)
Table 2: Experimental results
is sheduled in a symmetri way. It ollets the k=2 largest
benets on the even time steps and spends the odd time
steps on realloation to the best sites for the following time
step whih is even. We onlude that at eah time t, the
total benet that the k servers ollet is at least as large as
the k=2 largest benets at this time t, whih is learly at
least half the benet that an optimal oine algorithm an
ollet at this time t. Therefore, the ompetitive ratio of
this algorithm is at most 2.
For an odd k, the algorithm splits the servers into three
groups. Two groups are of size (k 1)=2 eah, and the third
group ontains one server. The rst group of servers ollets
the (k 1)=2 largest benets on the odd time steps. The
seond group ollets the (k 1)=2 largest benets on the
even time steps. Let Ft denote the (k + 1)=2 largest benet
in step t. The last server tries to ollet Ft on eah time
step, as follows. On the step t the server either ollets Ft
or it moves to the site q suh that dq;t = Ft . The server
ollets Ft if 2Ft > Ft ; otherwise the server moves to site
q . If the server ollets Ft on some step t, it saries the
step t + 1 on a move to the k th best site q for step t + 2,
i.e., dq;t = Ft .
For the sake of analysis we split time into phases: the
rst phase starts at time step 0 (by onvention we assume
that benets at this time step are equal to zero). A phase
ends at time step t + 1 if the server from the third group
ollets Ft in step t. The next phase starts at step t +2, and
so on. Notie that the server fromk the third group always
starts the phase alloated to the th best
site. Consider
the rst phase. We laim that Ft 1=4 Pti Fi . Indeed,
2Fi Fi for i = 1; : : : ; t 1 and 2Ft > Ft . Therefore
+1
+1
+1
+1
2
+2
+2
+1
2
+1
=1
+1
+1
t+1
X
i=1
Fi < F t
t+1
X
i=1
1=2t i 2Ft
1
X
j =0
1=2j = 4Ft :
For p P
k, if S (; p) is a sum
P of the p largest benets in step
, then S (; p) p=k OF F and
t
t
X
X
k
1
S ;
ON 2 + Ft
t
t
X
X
43 S ; k 2 1 + 14 S ; k +2 1
X
t
3(k8k 1) + (k 8+k 1)
OF F
= 2k4k 1 OF F = k OF F :
The same proof holds for any phase, and therefore we
have proved that our algorithm is k -ompetitive for odd
k 3.
+1
+1
=1
=1
+1
+1
=1
=1
+1
=1
1
7. RANDOMIZED ONLINE ALGORITHMS
In this setion we derive tight bounds on the ompetitive ratio of randomized online algorithms with lookahead
L. These bounds are stated in the following theorem. We remark that our randomized algorithm requires only O(log L)
random bits, independent of the input length.
10. There is a randomized online algorithm
for the benet task system problem with ompetitive ratio
1 + 1=L. Furthermore, no randomized algorithm ahieves a
Theorem
lower ompetitive ratio, even for the web server farm problem with one server.
The idea behind the algorithm is that with lookahead L, we an ompute an optimal path for L time steps.
However, we may have to make a poor move, in whih we
ollet little or no benet, in order to reah the rst state
on this path. After following this path, we an again hoose
the best state to move to during the next time step, possibly without olleting any benet at all, but ensuring the
best starting point for the next L steps. We refer to this as
the \resetting" algorithm, sine it operates in stages, after
eah of whih it resets to an optimal state for the upoming
stage. The online algorithm will ollet at least as muh
benet as the optimal oine algorithm on every step exept
the resetting steps. Notie that there are L + 1 possible
\phases": phase j means (potentially) giving up all benet
during steps i (L +1)+ j for eah i, where 0 j L. Sine
the sets of steps whose benet is given up is these phases
are mutually disjoint, the total benet lost in all phases is
at most the oine benet. Randomizing over these hoies,
the online algorithm loses in expetation only a 1=(L + 1)
fration of the oine benet, and the ompetitive ratio is at
most = L = 1 + 1=L.
For any > 0, we show a lower bound of (L + 1)=(L +
) for the one-server problem. Our randomized adversary
generates a demand matrix fdi;t g with d1=e rows and L +2
olumns (whih an be repeated indenitely), and a benet
matrix of all ones. The rst olumn of the demand matrix
ontains all zeroes, the next L olumns ontain all ones,
and exatly one randomly hosen row in the last olumn
ontains a one and the rest ontain zeroes. The optimal
hoie is for the server to move during the rst step to the
row ontaining the one in olumn L + 2, and to stay in
that row for L + 1 steps, olleting a total benet of L + 1.
Any randomized online strategy stands only an hane of
hoosing this row, and with probability at least 1 must
either miss the benet in this olumn, or forgo benet in
some earlier olumn in order to ollet it. Thus the expeted
benet olleted by the randomized online algorithm is at
most L + .
Proof.
1
1 1 ( +1)
8.
EXPERIMENTAL RESULTS
We have implemented the oine optimal, disount, and
intermittent reset algorithms for the web server farm problem and measured their performane. We used a randomly
generated input, and inputs generated from ommerial retail web server logs. The online algorithms perform very well
on these inputs, both in ases of overloaded and underloaded
servers. In fat, the disount algorithm with lookahead one
and the intermittent reset algorithm with lookahead eight
math the optimal oine performane in the ase of underloaded servers. (This is not surprising, sine it an be shown
that if the optimal oine algorithm is able to ollet all the
available benet, then the greedy algorithm with lookahead
one ahieves this optimal benet.) The disount algorithm
performs better, with less lookahead, than the intermittent
reset algorithm, and is within 0:2% of optimal on all the
ommerial trae instanes, even with lookahead one. We
use the (untuned) value of 2 as the disount fator.
The random problem instane has 3 ustomers and 400
time steps. Demands are generated uniformly between 0
and 16, and revenues uniform between 0 and 2.
The ommerial traes were olleted from two dierent
retail sites, whih will remain unnamed due to ondentiality onstraints. One of these is separated into \browse" and
\buy" transations. We treat these separately (i.e., so that
we have three dierent \ustomers"), and assign revenue
three times as great to the \buy" transations sine they
represent atual monetary transations. Eah trae represents the requests arriving at a web server over a 24-hour
period. We separate these into 288 ve-minute bukets, i.e.,
our time unit is ve minutes. We sale the inputs using realisti server rates, whih results in demands ranging in the
low tens of servers for two ustomers, and up to two servers
for the third (the one that handles the \buy" transations).
Table 2 shows the algorithms' performane. The olumn
labeled \Avail." denotes the total available revenue for the
problem instane, i.e., the revenue that would be olleted
with an innite number of servers. The next olumn, labeled
\Opt", shows the revenue olleted by the optimal oine algorithm. The next four olumns show the revenue olleted
by the disount algorithm and the intermittent reset algorithm with diering amounts of lookahead (one and four for
the disount algorithm, and four and eight for the intermittent reset algorithm). All values are rounded to the nearest
integer. The \Workload" olumn indiates the trae used
(random and the two ommerial traes, om1 and om2)
and the number of servers.
Acknowledgements.
We thank Maria Minko for helpful disussions and insights.
9.
REFERENCES
[1℄
R.K. Ahuja, T.L. Magnanti and J.B. Orlin
[2℄
B. Awerbuh, Y. Azar, A. Fiat and F.T. Leighton
Flows: Theory, Algorithms, and Appliations.
Hall, 1993.
, Network
Prentie
,
\Making Commitments in the Fae of Unertainty: How to
Pik a Winner Almost Every Time". Pro. 28th ACM
Symp. on Theory of Computing (STOC), pp. 519{530,
1996.
[3℄
A. Bar-Noy, R. Bar-Yehuda, A. Freund, J. Naor and
[4℄
A. Bar-Noy, R. Canetti, S. Kutten, Y. Mansour and
B. Shieber, \A unied approah to approximating
resoure alloation and sheduling". Pro. 32nd ACM
Symp. on Theory of Computing (STOC), pp. 735{744,
2000.
, \Bandwidth alloation with preemption".
28 (1999), pp. 1806{1828.
[5℄ A. Bar-Noy, S. Guha, J. Naor and B. Shieber,
\Approximating the throughput of multiple mahines
under real-time sheduling". Pro. 31st ACM Symp. on
Theory of Computing (STOC), pp. 622{631, 1999.
[6℄ A. Bar-Noy, Y. Mansour and B. Shieber,
\Competitive dynami bandwidth alloation". Pro. 17th
B. Shieber
SIAM Journal on Computing,
[7℄
ACM SIGACT-SIGOPS Symp. on Priniples of
Distributed Computing (PODC), pp. 31{39, 1998.
S. Baruah, G. Koren, D. Mao, B. Mishra,
A. Raghunathan, L. Rosier, D. Shasha and F. Wang
[8℄
\On the ompetitiveness of online real-time task
sheduling". Pro. 12th IEEE Symp. on Real Time
Systems, pp. 106{115, 1991.
,
S. Baruah, G. Koren, B. Mishra, A. Raghunathan,
, \Online sheduling in the
L. Rosier and D. Shasha
presene of overload".
Pro. 32nd IEEE Symp. on
Foundations of Computer Siene (FOCS), pp. 101{110,
[9℄
1991.
, On-Line Computation and
Cambridge University Press, 1998.
[10℄ A. Borodin, N. Linial and M.E. Saks, \An optimal
on-line algorithm for metrial task system". J. ACM,
39(1992), pp. 745{763.
[11℄ W. J. Cook, W. H. Cunningham, W. R. Pulleyblank,
and A. Shrijver, Combinatorial optimization. John
Wiley & Sons In., New York, 1998.
[12℄ J.A. Garay, I.S. Gopal, S. Kutten, Y. Mansour and
M. Yung, \EÆient online all ontrol algorithms". Pro.
2nd Israel Conf. on Theory of Computing and Systems,
pp. 285{293, 1993.
[13℄ A.K. Iyengar, Mark S. Squillante and L. Zhang,
\Analysis and Charaterization of Large-Sale Web Server
Aess Patterns and Performane". World Wide Web,
2(1999), pp. 85{100.
[14℄ G. Koren and D. Shasha, \D-over: An optimal on-line
sheduling algorithm for overloaded real-time systems".
SIAM Journal on Computing, 24 (1995), pp. 318{339.
[15℄ R.J. Lipton and A. Tomkins, \On-line interval
sheduling". Pro. 5th ACM-SIAM Symp. on Disrete
Algorithms (SODA), pp. 302{311, 1994.
[16℄ M.S. Manasse, L.A. MGeoh and D.D. Sleator,
\Competitive algorithms for server problems". J. of
Algorithms, 11 (1990), pp. 208{230.
[17℄ C.N. Potts and L.N.Van Wassenhove, \Integrating
sheduling with bathing and lot sizing: A review of
algorithms and omplexity". J. of the Operational Researh
Soiety, 43 (1992), pp. 395{406.
[18℄ M.S. Squillante, D.D. Yao and L. Zhang, \Web TraÆ
Modeling and Web Server Performane Analysis", Pro.
38th IEEE Conf. on Deision and Control, pp. 4432{4437,
1999.
[19℄ http://www.ibm.om/servies/webhosting/A. Borodin and R. El-Yaniv
Competitive Analysis.
full servies.html
[20℄ http://www.exodus.om/managed servies/ontent distribution/index.html