<s>
In	O
queueing	O
theory	O
,	O
a	O
discipline	O
within	O
the	O
mathematical	O
theory	O
of	O
probability	O
,	O
the	O
backpressure	B-Protocol
routing	I-Protocol
algorithm	O
is	O
a	O
method	O
for	O
directing	O
traffic	O
around	O
a	O
queueing	O
network	O
that	O
achieves	O
maximum	O
network	O
throughput	O
,	O
which	O
is	O
established	O
using	O
concepts	O
of	O
Lyapunov	B-Algorithm
drift	I-Algorithm
.	O
</s>
<s>
Backpressure	B-Protocol
routing	I-Protocol
considers	O
the	O
situation	O
where	O
each	O
job	O
can	O
visit	O
multiple	O
service	O
nodes	O
in	O
the	O
network	O
.	O
</s>
<s>
Backpressure	B-Protocol
routing	I-Protocol
is	O
an	O
algorithm	O
for	O
dynamically	O
routing	O
traffic	O
over	O
a	O
multi-hop	B-General_Concept
network	I-General_Concept
by	O
using	O
congestion	O
gradients	O
.	O
</s>
<s>
The	O
algorithm	O
can	O
be	O
applied	O
to	O
wireless	O
communication	O
networks	O
,	O
including	O
sensor	B-Architecture
networks	I-Architecture
,	O
mobile	B-Protocol
ad	I-Protocol
hoc	I-Protocol
networks	I-Protocol
(	O
MANETS	B-Protocol
)	O
,	O
and	O
heterogeneous	O
networks	O
with	O
wireless	O
and	O
wireline	O
components	O
.	O
</s>
<s>
Ad	B-Device
Hoc	I-Device
on-Demand	I-Device
Distance	I-Device
Vector	I-Device
Routing	I-Device
(	O
AODV	B-Device
)	O
,	O
</s>
<s>
geographic	B-Protocol
routing	I-Protocol
,	O
and	O
extremely	B-Protocol
opportunistic	I-Protocol
routing	I-Protocol
(	O
ExOR	B-Protocol
)	O
.	O
</s>
<s>
They	O
considered	O
a	O
multi-hop	B-General_Concept
packet	O
radio	O
network	O
with	O
random	O
packet	O
arrivals	O
and	O
a	O
fixed	O
set	O
of	O
link	O
selection	O
options	O
.	O
</s>
<s>
Backpressure	O
is	O
mathematically	O
analyzed	O
via	O
the	O
theory	O
of	O
Lyapunov	B-Algorithm
drift	I-Algorithm
,	O
and	O
can	O
be	O
used	O
jointly	O
with	O
flow	O
control	O
mechanisms	O
to	O
provide	O
network	O
utility	O
maximization	O
.	O
</s>
<s>
Consider	O
a	O
multi-hop	B-General_Concept
network	I-General_Concept
with	O
N	O
nodes	O
(	O
see	O
Fig	O
.	O
</s>
<s>
Extended	O
formulations	O
of	O
backpressure	B-Protocol
routing	I-Protocol
can	O
be	O
used	O
for	O
networks	O
with	O
probabilistic	O
channel	O
errors	O
,	O
including	O
networks	O
that	O
exploit	O
the	O
wireless	O
broadcast	O
advantage	O
via	O
multi-receiver	O
diversity	O
.	O
</s>
<s>
Note	O
that	O
backpressure	O
does	O
not	O
require	O
First-in-First-Out	O
(	O
FIFO	B-Operating_System
)	O
service	O
at	O
the	O
queues	O
.	O
</s>
<s>
that	O
Last-in-First-Out	O
(	O
LIFO	B-Application
)	O
service	O
can	O
dramatically	O
improve	O
delay	O
for	O
the	O
vast	O
majority	O
of	O
packets	O
,	O
</s>
<s>
Consider	O
a	O
multi-hop	B-General_Concept
network	I-General_Concept
with	O
N	O
nodes	O
,	O
as	O
described	O
in	O
the	O
above	O
section	O
.	O
</s>
<s>
The	O
conditional	O
Lyapunov	B-Algorithm
drift	I-Algorithm
is	O
defined	O
:	O
</s>
<s>
Thus	O
,	O
the	O
drift	O
of	O
a	O
quadratic	O
Lyapunov	O
function	O
is	O
less	O
than	O
or	O
equal	O
to	O
a	O
constant	O
B	O
for	O
all	O
slots	O
t	O
.	O
This	O
fact	O
,	O
together	O
with	O
the	O
assumption	O
that	O
queue	O
arrivals	O
have	O
bounded	O
second	O
moments	O
,	O
imply	O
the	O
following	O
for	O
all	O
network	O
queues:M	O
.	O
J	O
.	O
Neely	O
,	O
"	O
Queue	O
Stability	O
and	O
Probability	O
1	O
Convergence	O
via	O
Lyapunov	B-Algorithm
Optimization	I-Algorithm
,	O
"	O
Journal	O
of	O
Applied	O
Mathematics	O
,	O
vol	O
.	O
</s>
<s>
Backpressure	O
has	O
been	O
shown	O
to	O
work	O
in	O
conjunction	O
with	O
flow	O
control	O
via	O
a	O
drift-plus-penalty	B-Algorithm
technique	O
.	O
</s>
