<s>
Task	O
systems	O
are	O
mathematical	O
objects	O
used	O
to	O
model	O
the	O
set	O
of	O
possible	O
configuration	O
of	O
online	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
The	O
objective	O
of	O
an	O
online	B-Algorithm
algorithm	I-Algorithm
for	O
task	O
systems	O
is	O
to	O
create	O
a	O
schedule	O
that	O
minimizes	O
the	O
overall	O
cost	O
incurred	O
due	O
to	O
processing	O
the	O
tasks	O
with	O
respect	O
to	O
the	O
states	O
and	O
due	O
to	O
the	O
cost	O
to	O
change	O
states	O
.	O
</s>
<s>
If	O
the	O
cost	O
function	O
to	O
change	O
states	O
is	O
a	O
metric	O
,	O
the	O
task	O
system	O
is	O
a	O
metrical	B-Algorithm
task	I-Algorithm
system	I-Algorithm
(	O
MTS	O
)	O
.	O
</s>
<s>
Metrical	B-Algorithm
task	I-Algorithm
systems	I-Algorithm
generalize	O
online	O
problems	O
such	O
as	O
paging	B-General_Concept
,	O
list	O
accessing	O
,	O
and	O
the	O
k-server	B-Algorithm
problem	I-Algorithm
(	O
in	O
finite	O
spaces	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
metric	O
,	O
is	O
a	O
metrical	B-Algorithm
task	I-Algorithm
system	I-Algorithm
.	O
</s>
<s>
As	O
usual	O
for	O
online	O
problems	O
,	O
the	O
most	O
common	O
measure	O
to	O
analyze	O
algorithms	O
for	O
metrical	B-Algorithm
task	I-Algorithm
systems	I-Algorithm
is	O
the	O
competitive	B-Algorithm
analysis	I-Algorithm
,	O
where	O
the	O
performance	O
of	O
an	O
online	B-Algorithm
algorithm	I-Algorithm
is	O
compared	O
to	O
the	O
performance	O
of	O
an	O
optimal	O
offline	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
For	O
deterministic	O
online	B-Algorithm
algorithms	I-Algorithm
,	O
there	O
is	O
a	O
tight	O
bound	O
on	O
the	O
competitive	B-Algorithm
ratio	I-Algorithm
due	O
to	O
Borodin	O
et	O
al	O
.	O
</s>
<s>
For	O
randomized	O
online	B-Algorithm
algorithms	I-Algorithm
,	O
the	O
competitive	B-Algorithm
ratio	I-Algorithm
is	O
lower	O
bounded	O
by	O
and	O
upper	O
bounded	O
by	O
.	O
</s>
