A time dependent performance model for multihop wireless networks with CBR traffic
ABSTRACT In this paper, we develop a performance modeling technique for analyzing the time varying network layer queueing behavior of multihop wireless networks with constant bit rate traffic. Our approach is a hybrid of fluid flow queueing modeling and a time varying connectivity matrix. Network queues are modeled using fluidflow based differential equation models which are solved using numerical methods, while node mobility is modeled using deterministic or stochastic modeling of adjacency matrix elements. Numerical and simulation experiments show that the new approach can provide reasonably accurate results with significant improvements in the computation time compared to standard simulation tools.
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Article: A survey on wireless mesh networks
[show abstract] [hide abstract]
ABSTRACT: Wireless mesh networks (WMNs) have emerged as a key technology for nextgeneration wireless networking. Because of their advantages over other wireless networks, WMNs are undergoing rapid progress and inspiring numerous applications. However, many technical issues still exist in this field. In order to provide a better understanding of the research challenges of WMNs, this article presents a detailed investigation of current stateoftheart protocols and algorithms for WMNs. Open research issues in all protocol layers are also discussed, with an objective to spark new research interests in this field.IEEE Communications Magazine 10/2005; · 3.66 Impact Factor  SourceAvailable from: psu.edu[show abstract] [hide abstract]
ABSTRACT: A mobile ad hoc network (MANET), sometimes called a mobile mesh network, is a selfconfiguring network of mobile devices connected by wireless links. The Ad hoc networks are a new wireless networking paradigm for mobile hosts. Unlike traditional mobile wireless networks, ad hoc networks do not rely on any fixed infrastructure. Instead, hosts rely on each other to keep the network connected. It represent complex distributed systems that comprise wireless mobile nodes that can freely and dynamically selforganize into arbitrary and temporary, ‘‘adhoc’’ network topologies, allowing people and devices to seamlessly internetwork in areas with no preexisting communication infrastructure.Ad hoc networking concept is not a new one, having been around in various forms for over 20 years. Traditionally, tactical networks have been the only communication networking application that followed the adhoc paradigm. Recently, the introduction of new technologies such as the Bluetooth, IEEE 802.11 and Hyperlan are helping enable eventual commercial MANET deployments outside the military domain. These recent evolutions have been generating a renewed and growing interest in the research and development of MANET. This paper attempts to provide a comprehensive overview of this dynamic field. It first explains the important role that mobile ad hoc networks play in the evolution of future wireless technologies. Then, it reviews the latest research activities in these areas of MANET_s characteristics, capabilities and applications.International Journal of Computer Applications. 01/2010;  SourceAvailable from: Kishor S Trivedi
Conference Proceeding: The effect of detection and restoration times for error recovery in communication networks
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ABSTRACT: Detection and restoration times are often ignored when modeling network reliability. We develop Markov regenerative reward models (MRRM) to capture the effects of detection and restoration phases of protection switching network recovery. The states of the MRRM represent conditions of network resources while the statetransitions represent occurrences of failure, repair, detection and restoration. Reward rates, assigned to states of the MRRM are computed based on a performance model (M/D/1/L+1 queue) that accounts for contention. We compare our model with ones that ignore these parameters and show significant differences, in particular for transient measuresMilitary Communications Conference, 1995. MILCOM '95, Conference Record, IEEE; 12/1995 · 0.43 Impact Factor
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A Time Dependent Performance Model for Multihop Wireless Networks with
CBR Traffic
Kunjie Xu, Siriluck Tipmongkonsilp
David Tipper, Prashant Krishnamurthy
Graduate Telecommunications and Networking Program
University of Pittsburgh, Pittsburgh, PA
kux1@pitt.edu, sirilucktip@hotmail.com
{tipper, prashant}@tele.pitt.edu
Yi Qian
Department of Computer
& Electronics Engineering
University of NebraskaLincoln, NE
yqian2@unl.edu
Abstract
In this paper, we develop a performance modeling tech
nique for analyzing the time varying network layer queue
ingbehaviorofmultihopwirelessnetworkswithconstantbit
rate traffic. Our approach is a hybrid of fluid flow queue
ing modeling and a time varying connectivity matrix. Net
work queues are modeled using fluidflow based differential
equation models which are solved using numerical meth
ods, while node mobility is modeled using deterministic or
stochastic modeling of adjacency matrix elements. Numeri
cal and simulation experiments show that the new approach
can provide reasonably accurate results with significant im
provements in the computation time compared to standard
simulation tools.
1. Introduction
Interest in multihop wireless networks, such as wire
less mesh networks (WMN) [1], vehicular adhoc networks
(VANET) [2], or mobile adhoc networks (MANET) [3],
has increased significantly in recent years. Multihop wire
less networks are expected to become an important part of
the communications landscape and may work in a fully au
tonomous scenario or as an extension to an infrastructure
network. In multihop wireless networks, nodes must coop
erate with one another to dynamically establish routes using
wireless links. Routes may involve multiple hops with each
node acting as a router. Since network nodes may move
arbitrarily, the network topology is expected to change fre
quently and unpredictably. Hence, multihop wireless net
works require highly adaptive protocols and efficient fail
This research was funded in part by the US Army Research Office
under the MultiUniversity Research Initiative (MURI) grant W911NF
0710318.
ure recovery strategies to deal with the frequent topology
changes. Multihop wireless networks also inherit the tradi
tional problems of wireless communications and network
ing (e.g., asymmetric channels and signal propagation, links
that are of poor quality in comparison to wired links, hidden
terminal and exposed terminal problems, etc.), which when
combined with the other unique aspects, such as mobility
and lack of infrastructure make their design and develop
ment challenging [1] [2] [3].
Fundamental to the design of efficient multihop wire
less networks is the ability to estimate and predict the per
formance of such networks. Traditionally, multihop wire
less network performance has been evaluated using discrete
event based computer simulations. Most of the literature
adopts a steady state simulation methodology, where for a
given network scenario the simulation is conducted over a
fixed period of time, with multiple runs from different ran
dom number seeds and collected data being averaged over
the runs. Additionally, the observations gathered during the
transient period in each run are usually eliminated to avoid
initialization bias.
A weakness of most of the literature on the performance
of multihop wireless networks is that steady state analysis
techniques are used even though transient or nonstationary
periods will occur in the network, especially after a link
or node failure. The importance of this transient behav
ior after link/node failures has been illustrated in several
network technologies including circuit switched networks
[4], packet switched data networks [5], packet based sig
naling networks [6], cellular networks [7], and MANETs
[8]. This work, taken together, demonstrates that the dom
inant factor on network performance after a dynamic event
such as a link failure is the transient or nonstationary con
gestion period. Due to the mobility of nodes and their
limited battery life, link and node failures are common in
multihop wireless networks. Thus, one would expect that
9781424493289/10/$26.00 ©2010 IEEE
Page 2
transient/nonstationary conditions to occur often and likely
dominate the performance behavior [8]. Hence, routing
schemes, QoS mechanisms, and congestion control tech
niques designed and evaluated via steady state analysis may
not make optimum use of network resources after a failure
or during nonstationary periods.
Simulation studies of time varying behavior for such net
works are possible [9], though computationally difficult. To
study nonstationary behavior, the measurements of quan
tities observed over small intervals or at specific points in
time are important. Therefore, the time average is not a
proper approach and ensemble averages are more appropri
ate. The idea is to construct ensemble average curves of
quantities of interest across a set of statistically identical but
distinct independent simulation runs, along with the calcu
lated confidence interval. With many such points collected
at different time instants, the behavior of the system can be
shown as a function of time. However, to assure the accu
rate portrayal of the actual system, a large number of runs
are required resulting in large amounts of CPU time and
scalability issues. In summary, while significant progress
has been made towards developing multihop wireless net
work protocols [1, 2, 3] and developing simulation tools
[10, 11, 12, 13] to estimate their performance, relatively lit
tle work has appeared on performance models which cap
ture their time varying behavior.
In this paper, we develop a new performance modeling
techniquethatfocusesonthedynamicsofmultihopwireless
networks, which includes both time varying and steady state
behavior. Network queues are modeled using fluid flow
based differential equations, which can be solved by us
ing numerical integration methods. Node connectivity, rep
resenting topology changes, are integrated into the model
using either deterministic or stochastic based connectivity
matrix modeling techniques. Our approach is shown to be
more scalable than nonstationary discrete event simulations
while allowing the same insight into the interaction of net
work nodes/protocols.
The rest of the paper is organized as follows. Section
II provides the details of our modeling approach. Section
III presents numerical results from both our modeling ap
proach and for comparison standard simulation. Section IV
studies the computational time complexity of our proposed
modeling technique and gives numerical results illustrating
the advantages of our method in comparison to simulation.
Our conclusions and future works are given in Section V.
2 MODELING DYNAMIC BEHAVIOR
The network topology in multihop wireless networks can
change dynamically depending on the link connectivity be
tween each node pair. When nodes in the networks are al
lowed to move arbitrarily, it will lead to frequent changes
in the topology of the queueing network model. For that
reason, we propose a hybrid approach to approximate the
network performance using two components: (1) a time
varying adjacency matrix model to represent the network
topology and (2) a fluid flow model based set of differential
equations which model the time dependent queueing behav
ior of each node. In developing a performance model of the
network, we start with modeling a single queue and then
generalize to a arbitrary queue in a network. We adopt a
fluidflow based approximation technique to describe the
time varying behavior of the queue at each network node,
with the help of an approximation concept. Specifically, the
Pointwise Stationary Fluid Flow Approximation (PSFFA)
method [8] [18] [19], which models the average number in
the system at a queue by one or more nonlinear differential
equations which are derived from a pointwise mapping of
the steadystate queueing relationship and can be solved nu
merically using standard numerical integration techniques.
2.1Network Topology Modeling
Consider a multihop wireless network consisting of M
nodes, the network topology in terms of connectivity at any
time t is modeled with a M x M adjacency matrix denoted
as A(t).
⎡
⎣
⎧
⎩
radio range which is a function of a variety of factors, such
as the distance between the nodes, antenna radiation pat
tern, power level, geographic terrain, propagation environ
ment, interference, receiver sensitivity, speed, etc. In gen
eral, the range of every node is different and the connectiv
ity among nodes can be asymmetric (i.e., aij(t) = 1 while
aji(t) = 0). Given an initial placement of the network
nodes, the dynamic network topology due to node mobility
is reflected in the adjacency matrix by changes in the aij(t)
values with time. Information about node movement and
connectivity can be determined from experimentally gath
ered trace data, a discrete event simulation of a mobility
model (e.g., random waypoint) or stochastic/probabilistic
models of mobility effects on link connectivity [16] [17].
In the trace based approach, the data is mined for the link
connectivity information versus time.
approach, a mobility model is used to create the network
topology dynamics. Specifically, every node pair is checked
A(t)=
⎢⎢⎢
a11(t)
a21(t)
...
aM1(t)
a12(t)
a22(t)
...
aM2(t)
...
...
a1M(t)
a2M(t)
...
aMM(t) ...
⎤
⎦
⎥⎥⎥
whereaij(t) =
⎨
1
if node i and j are directly connected
at time t
otherwise
0
The connectivity between two nodes is determined by their
In the simulation
Page 3
for the possible connectivity change based on their current
speeds and directions. Note, that changes in speed, direc
tion and power level are also considered events. The event
times are placed in chronological order and as time evolves
the pair wise connectivity calculation is repeated for every
event time and the matrix is changed accordingly. In this
way, the adjacency matrix can reflect the topology change
dynamically. Note that no traffic need be generated just
the node mobility and range of the nodes, this can be ac
complished using existing tools such as the BonnMotion
simulator [14]. A computationally simpler approach is to
model mobility by directly manipulating the elements of
the adjacency matrix according to a planned experiment or
a stochastic/probabilistic model (as shown in Section III).
Such a probabilistic model can be developed either from
the mobility model assumptions and analysis [15] or from
fitting a statistical model to data gathered from a test bed or
simulation (for example, twostate MMPP [17]).
2.2 Fluid Flow Model Background
Consider a single server first in first out queueing system
with a nonstationary arrival process, where λ(t) represents
the ensemble average arrival rate at time t. The model is de
veloped by focusing on the dynamics of the packet queues
at the transmission link. With x(t) defined as the state vari
able representing the ensemble average number in the sys
tem at time t, ˙ x = dx/dt is the rate of change of the state
variable with respect to time. According to the flow conser
vation principle, the rate of change of the average number
in the system equals to the difference between the flow in
and the flow out of the system at time t, denoted by fin(t)
and fout(t) resulting in:
˙ x(t) = −fout(t) + fin(t)
(1)
For infinite waiting space queues, the flow in equals the ar
rival rate fin(t) = λ(t). The flow out can be related to
the ensemble average utilization of the server as fout(t) =
μCρ(t), where 1/μ refers to average packet length, C de
fines the server capacity and ρ(t) is the server utilization.
Thus μC represents the average service rate in number of
packets per unit time. This equation can model a wide range
of queueing systems as shown in [5] and [19]. The fluid
flow equation can then be written in terms of the average
arrival rate and the departure rate as:
˙ x(t) = −μCρ(t) + λ(t)
(2)
The utilization ρ(t) in (2) depends on stochastic model
ing assumptions of the queue under study such as traffic
arrival process and service time distribution. In general,
the exact expression of ρ(t) is difficult to determine. In
[19] we proposed the function be determined by an approx
imation approach  matching the equilibrium point in the
differential equation at particular time instant to the cor
responding steady state queueing theory result. To adopt
thismappingapproach, weassumethattheserverutilization
function can be approximated by the nonnegative function
G(x(t)), which represent the ensemble average utilization
of the server at time t as a function of the state variable x(t).
Then (2) can be written as:
˙ x(t) = −μC(G(x(t))) + λ(t)
In general, the function G(x(t)) is nonlinear and we can
solve for x(t) in (3) using numerical methods as follows.
Given an initial condition of the state variable at time zero
as x(0) and an approximation of the arrival rate as a con
stant λ over a small time step [0,Δt] (i.e., λ = λ(Δt/2)),
equation (3) can be solved numerically for state variable at
the end of the time interval x(Δt), which then becomes an
initial condition for the next time step [Δt,2Δt]. The ar
rival rate for the new time step is adjusted and the procedure
is repeated for each time interval along the time horizon.
Numerical studies in [18] have shown that results from the
PSFFA model are reasonably accurate. Here we derive the
PSFFA model for the queueing system of interest beginning
with the general multiple traffic stream case.
(3)
2.3Multiple Traffic Stream Fluid Flow
Model
Consider the case of a single queue with S differ
ent traffic streams, each with the average arrival rate of
λ1(t),λ2(t),...,λS(t) respectively as illustrated in Fig. 1.
λ1
λ2
λS
μC
Figure 1. Queuing model with S traffic
streams
The aggregate traffic can be considered as one arrival pro
cess λT(t) =
semble average number of stream l in the system at time
t, the total average number in the system is defined as
xT(t) =?S
˙ xT(t) = −μC(G(xT(t))) + λT(t)
We note that the flow conservation principle also applies
to each traffic stream. Therefore, a state model can also
be developed for each stream with G(xl(t),xT(t)), as the
average utilization function of the link by stream l traffic, as
?S
l=1λl(t). Let xl(t) represents the en
l=1xl(t), and the state model in (3) becomes:
(4)
Page 4
a function of the total average number in the system xTand
the average number of stream l packets in the system xl.
˙ xl(t) = −μC(Gl(xl(t),xT(t))) + λl(t) ∀l = 1,2,...,S
(5)
Thus the transmission link can be described by a set of S
coupled state equations, each representing the traffic behav
ior of its own stream. Next we tailor this model to the case
of periodic traffic streams.
2.4 Modeling CBR Traffic
We focus on modeling a queue with constant bit rate in
put traffic streams and consider two cases: (1) where all
CBR streams have the same data rate and (2) when the CBR
streams have different data rates. The two cases are studied
in turn below.
Case I: Identical Sources
Consider a queuing model as in Fig. 1 where all the traf
fic sources have the same period (although not necessarily
insynchronization)andthepacketshavethesamesize. This
model is often referred to as the S ∗D/D/1 queue. The pe
riod of the S input streams is measured in server time units
D, and the server utilization equals to ρ = S/D which must
have ρ < 1 for stability.
Following [23] and [25], we model lack of synchroniza
tion among the streams by assuming that the first arrival of
each flow is independently and uniformly distributed over
the first arrival period interval. Let Ltdenote the number of
customers present at time t and Qt(r) = Pr{Lt> r} is the
survival function or complementarydistribution ofthe num
ber in the system at arbitrary instant t. Further let A(u,t)
denote the number of arrivals in an interval (u,t). As noted
in [23] the survival function can be written as:
Qt(r) =
∞
?
s=1
ps(r)π0(r,s)
(6)
where ps(r) = Pr{A(t − s,t) = r + s} and π0(r,s) =
Pr{system empty at t−s r+s arrivals in (t−s,t)}. Not
ing that the binomial distribution provides the probability of
arrivals during an interval. Then Qt(r) [25] can be written
as:
Qt(r) =
S−r
?
s=1
??
?D − S + r
S
r + s
??s
D
?r+s?
??
1 −s
D
?S−r−s
D − s
for 0 ≤ r < S
(7)
where the first three terms in the sum represent the number
of arrivals and the rightmost term represents the probabil
ity the system is initially empty given r + s arrivals. The
total average number in the system xT can be found using
Qt(r) [23], since the mean of nonnegative variable xTcan
be computed as xT=?S−1
S−1
?
?D − S + r
The above formula can be used to numerically determine
xTfor a given traffic mix (i.e., S,D,ρ ). We apply a curve
fitting approach using the (ρ,xT) data from (8) to find the
utilization function G(.). The resulting G(xT(t)) is in the
form of a polynomial (i.e., G(xT(t)) = axn
... + k) and can be substituted back into the general fluid
flow model (4).
Determining the state model for each traffic stream
l at a queue, following the approach of steady state
equilibrium matching with ˙ xT(t) = 0, from (4) and
(5), we find λT(t)= μCG(xT(t)) and λl(t)
μCGl(xl(t),xT(t)) ∀l = 1,2,...,S, respectively. Solv
ing these equations along with G(xT(t)) determined from
data fitting, the average utilization for stream l packets,
Gl(xl(t),xT(t)) can be determined as
xl
xT
xl
xT
This can be substituted into (5) to provide the multiple traf
fic stream fluid flow model.
Case II: Nonidentical Sources
Now we consider the case where a group of sources with
different rates are multiplexed on a transmission link where
the total bit rate is less than the transmission capacity to
ensure stability. Suppose there are m types of sources with
Niof type i generating traffic with the arrival period of Di
time units, for i = 1,2,...,m. This type of queue may
be referred to as a N1D1+ N2D2+ ... + NmDm/D/1
queue. Similar to Case I, we rely on numerical analysis of
the queue length distribution to find the utilization function
G(.). Following the same approach as before, we focus on
determining Qt(r) using (6).
Determining ps(r) in (6), we note for an interval of
length s, a type i stream will generate [s/Di] arrivals
and possibly one additional arrival with probability αi =
s/Di− [s/Di]. Considering all Niof type i sources, the
total number of arrivals equals to Ni[s/Di]+kiwhere kiis
a random variable representing additional arrivals for each
traffic type with the following distribution:
r=0Qt(r) [24]. Therefore, for the
S ∗ D/D/1 queue, xTis given by:
S−r
?
xT=
r=0s=1
??
S
r + s
??s
D
?r+s?
??
1 −s
D
?S−r−s
D − s
for 0 ≤ r < S
(8)
T+ bxn−1
T
+
=
G(xl(t),xT(t)) =
∗ G(xT(t))
?axn
=
T+ bxn−1
T
+ ... + k?
(9)
bsi(k) = Pr{ki= k}
=
0
? ?Ni
k
?αk
i(1 − αi)Ni−kfor 0 ≤ k ≤ Ni
otherwise
(10)
Page 5
Let qs(k) be defined as the distribution of?kifor all m
?
types, it is given by:
qs(k) =
?
iki=k
m
?
i=1
bsi(ki)
for 0 ≤ k ≤
?
Ni (11)
Then the probability ps(r) of r + s arrivals during time in
terval (t − s,t) can be determined from qs(k) as:
⎧
⎩
The conditional probability of queue being idle at the be
ginning of the interval π0(r,s) in (6) is difficult to derive.
However, bounds for probabilities Qt(r) can be obtained
from the fact that (1 − ρ) ≤ π0≤ 1, where ρ corresponds
to the server utilization?Ni/Di. Substituting in (6), we
∞
?
A tighter upper bound has been derived in [26] as:
ps(k) =
⎨
qs(r + s −?
0
otherwise
iNi[s/Di])
for?
iNi[s/Di] ≤ r + s ≤?
iNi([s/Di] + 1)
(12)
have the bounds:
(1 − ρ)
s=1
ps(r) ≤ Qt(r) ≤
∞
?
s=1
ps(r)
(13)
Qt(r) ≤
∞
?
?
s=1
?
?
?
ki=r+s−ds
?
m
?
Ni− ki
Di(1 − αsi)
i=1
bsi(ki)1 −
m
?
i=1
?+??
(14)
with (x)+= max{0,x} and ds =?Ni[s/Di]. Noting
(13) or (14) to find the corresponding bounds for xT. After
that, curve fitting the data pair (ρ,xT) results in the poly
nomial utilization function G(xT(t)), which can be substi
tuted back into the general fluid flow model (5). The re
sult is two fluid flow models, one providing a lower bound
and the other an upper bound. As an alternative, one can
simplify the computation by assuming homogeneous traf
fic. For the lower bound, traffic from all sources are fixed
with period of Dmax= max{Di} and for the upper bound
the traffic period equals to Dmin= min{Di}. We can ap
ply Dmaxand Dmininto (8) from S ∗D/D/1 to obtain the
upper and lower bounds of the data pair (ρ,xT) and then
find out the utilization function for each bound by curve fit
ting. Once the bounding fluid models for the total traffic
flow xT are found, one can similarly find upper and lower
bounding fluid models for each traffic stream using (9).
that xT =?S−1
r=0Qt(r), one can use function Qt(r) from
2.5 Networks with Periodic Traffic
Consider a multihop wireless network consisting of M
nodes with all traffic being constant bit rate. An arbitrary
node i is shown in Fig. 2. At each node, there are M − 1
possible traffic types, sorted into M − 1 classes based on
their destinations. We assume that packets are generated at
the node i destined for node j according to a determinis
tic process, (which can be nonstationary), with mean rate
γj
the queueing system at node i destined for node j at time
t. We denote the average packet length with 1/μ and let Ci
denote the transmission capacity of node i. When consider
ing the network as a whole, (5) must be modified to clearly
identify the source node i and destination node j for each
state variable xj
through intermediate nodes when a direct link is not acces
sible. We define aij(t) in the adjacency matrix to represent
node connectivity, as described in section 2.1. Similarly, in
order to model multihop routing, we define the routing vari
able rj
the routing algorithm, with rj
destined to node j is routed through network node k at time
t and rj
i(t). Let xj
i(t) denote the average number of packets in
i(t), as well as to model traffic being routed
ik(t) as a zero/one indicator variable determined by
ik(t) = 1 if traffic from node i
ik(t) = 0 otherwise.
flow from node 1
flow from node M
μC
Routing
Controller
flow from node 2
flow to node 1
flow to node M
flow to node 2
Figure 2. An arbitrary node i queueing model.
Note that, the flow out of node i to node k of a particular
traffic class j will depend upon the existence of a direct link
between i and k and the routing variables for traffic class j.
Hence one must modify the flow out term in (3) to incorpo
rate aik(t) and rj
class j traffic to node k is given by
ik(t). Specifically, the flow out of node i of
class j traffic flow out of node i to node k
= μCi(Gj
i(xj
i(t),xT(t)))(aik(t)rj
ik(t))
(15)
The flow of class j traffic into the node i queue will consist
of traffic generated at node i with rate γj
of class j traffic to node i from other network nodes. For
example, the flow of class j traffic into node i from node l
is given by
i(t) and the flow
class j traffic flow into node i from node l
= μCl(Gj
l(xj
l(t),xT(t)))(ali(t)rj
li(t))
(16)
In interconnecting queues, the literature [20][22] indi
cates that the output from a queueing system with deter
ministic service time should be treated as a delayed input to
the next stage. This idea is applicable to our model, where
the input to the next stage is a superposition of the delayed
Page 6
input streams from the nearby nodes plus any external arriv
ing traffic. We illustrate the concept by considering a sim
plified two stage tandem queue model as in Fig. 3(a)(b), let
xi(t), λi(t) and Gi(t) be the average number in the system,
total average arrival rate and average utilization at node i,
respectively. Then, λl(t) = γ1(t) is the arrival rate to the
first queue, and μCGl(t) is the departure rate from the first
queue. The departure rate then becomes the input to the sec
ond queue with a deterministic propagation delay of service
periodDstimeunits, thatisλ2(t) = μCG1(t−Ds)+γ2(t).
We can then write a set of fluid flow equations at node 1 and
node 2, for Fig. 3(a)(b) as:
˙ x1(t) = −μCG1(t) + γ1(t)
˙ x2(t) = −μCG2(t) + γ2(t) + μCG1(t − Ds)
γ2
(17)
node 2
γ1
(a) Original System
node 1
node 2
(b) Equivalent model
γ2
γ1
Delay = D
Figure 3. A twonode deterministic service
system with its equivalent model.
The general M nodes network hybrid model combining
fluidflow model with routing and connectivity is obtained
by summing the flow in and out over all possible nodes:
˙ xj
i(t) = −μCi(Gj
i(xj
i(t),xT(t))
M
?
k=1,k?=i
aik(t)rj
ik(t) +
γj
i(t) +
M
?
l=1,l?=i,j
?
μCl(Gj
l(xj
l(t − Ds),xT(t − Ds)))
?
(aik(t)rj
li(t))
∀i,j = 1,2,...,M
(18)
For a queue with the superposition of periodic arrival
streams, the server utilization function G(.) can be written
in the form of polynomial expression given in (9). There
fore, the final network fluid flow model is
˙ xj
i(t) = −μCixj
?
?axn
i(t)
xT(t)
?axn
T(t) + bxn−1
T
(t) + ... + k?
μClxj
xT(t − Ds)
M
k=1,k?=i
aik(t)rj
ik(t) + γj
i(t) +
M
?
l=1,l?=i,j
i(t − Ds)
T(t − Ds) + bxn−1
T
(t − Ds) + ... + k?(aik(t)rj
∀i,j = 1,2,...,M
li(t))
(19)
Given a routing algorithm, connectivity model and traffic
information, this model can be solved numerically for the
mean queue lengths as a function of time using any standard
numerical integration technique.
2.6 Additional Performance Metrics
In addition to the mean queue length versus time, other
performance metrics can be evaluated from the model. Here
for the sake of brevity we discuss estimation of the endto
end delay only. Typically, a packet is forwarded from the
source via a path which may include several intermediate
nodes until it reaches the destination. As a result, the end
toend delay is the sum of the delays experienced at each
node along the way. The delay at a node consists of the
processing time, the queuing delay, the transmission time
on the link and the propagation time over the link to the
next node. In general, the queueing and transmission delay
are the main factors.
From Little’s theorem, the average number in the system
is equivalent to the product of the average arrival rate and
the average time in the system. If x denotes the average
number in the system, λ is the average arrival rate and W is
the average waiting time, then x = λW. With the assump
tion of constant arrival rate over a small step, the change in
average waiting time can be related to the rate of change in
average number in the system˙W = ˙ x/λ. Now consider a
path P of j−1 hops from source node 1 to destination node
j (class j traffic), given by (1,2), (2,3)...(j − 1,j), where
(i,i+1) represents a link on the path, for ∀i = 1,2...j −1.
The average node delay at node i for class j traffic on link
(i,i + 1) is denoted by Wj
P can be written as WP(t) =?j−1
change of this path delay is given by:
i(t) and the total latency of path
i=1Wj
i(t). However, as
we are interested in the time dependent behavior, the rate of
˙WP(t) =
j−1
?
i=1
˙Wj
i=
j−1
?
i=1
˙ xj
λj
i(t)
i(t)
(20)
according to (18), λj
i(t) can be calculated by
λj
i(t) = γj
i(t) +
M
?
xT(t − Ds)))(aik(t)rj
l=1,l?=i,j
?
μCl(Gj
l(xj
l(t − Ds),
li(t))
?
(21)
3NUMERICAL RESULTS
We solved the fluid flow based model, presented in the
previous section, using the fifth order RungeKutta numer
ical integration routine with variable time step Δt in Mat
lab. For comparison purposes, an equivalent discrete event
Page 7
simulation is built using OPNET 14.5 [11]. In our OPNET
model, each queue of the node is configured as a first in first
out (FIFO) queue with infinite buffer size and each traffic
stream is buffered at a different subqueue. In addition, we
use minimum hop routing for both the fluid flow model and
the discrete event simulation. The discrete event simulation
results are the average of 4096 runs using the nonstationary
simulation methodology of [9] discussed in the introduc
tion.
A simple simulation scenario of three nodes with pre
determined connectivity change between nodes at each time
interval during the length of simulation time as illustrated
in Fig. 4(a)  (f), is studied first. This topology is used to
evaluate the accuracy of our proposed model. In this setup,
when the direct link is no longer available, traffic must be
rerouted through relay nodes and uses some available por
tion of the shared link capacity. We set the capacity for all
nodes Ci= 104bps with packet length 1/μ = 1250 bytes,
so that the average service rate 1/Dsis normalized to one
packet per second.
12
3
12
3
(a) t<100(b) 100<=t<200(c) 200<=t<300
12
3
12
3
12
3
(d) 300<=t<400 (e) 400<=t<500 (f) 500<=t
12
3
Figure 4. Three node network connectivity
scenario.
For case I, the rate of externally arriving packets of each
node pair are set to γ2
packets per second but are not synchronized (i.e., the first
arrival from each stream is determined from a uniform [0,5]
random variable). As described in Section II.C case I, we
first compute the data pair (xT,ρ) from (8). Curve fitting
the data pair (ρ,xT), the server utilization function is deter
mined as G(xT) = 0.0832x3
This is then used in (19) to model the network. Fig. 5
shows the results of the effect of topology change on the
average number in system and endtoend delay for traffic
at node 1 destined for node 2. For the time interval t < 100
sec, all nodes are directly connected. Nodes go through an
initial transient period and reaches steady state. For time
100 ≤ t < 200 sec, the link between node 1 and 3 breaks
and traffic going through this link has to go through the re
lay node 2. But traffic x2
terval 300 ≤ t < 400 sec, the link between node 2 and
node 3 breaks, leading to traffic rerouting and an increase
in the number of x2
server utilization of node 1, the average queuing time of
1= γ3
1= γ1
2= γ3
2= γ1
3= γ2
3= 0.2
T− 0.4353x2
T+ 1.0843xT.
1is not affected. For the time in
1packets in the system. Due to higher
each packet at node 1 increases and we can see the rise in
endtoenddelayofthetrafficfromnode1destinedfornode
2. During the time t ≥ 500 sec, link 12 breaks and x2
has to go through the relay node 3 to reach the destination.
Hence, the endtoend delay consists of propagation delay
in link 13 and link 32 as well as the queueing delay at node
1 and node 3. The behavior of the other network nodes and
traffic streams are similar and not shown here for the sake
of brevity.
1
0100 200300 400500 600
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
t (second)
(a)
Average Number of Packets
Simulation
Hybrid Modeling
X1
2
0 100200 300400500 600
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
t (second)
(b)
End−to−end Delay (second)
Simulation
Hybrid Modeling
X1
2
Figure 5. Average number of packets and
endtoend delay of x2
1with S ∗D/D/1 queue.
Next we tested the performance of the fluid flow model
with case II, the rates of externally arrivals are set as: γ2
0.16 γ3
packets per second. The baseline case when all links are
working results in each node being a N1D1+ N2D2/D/1
queue with D1 = 1/0.16 = 6.25 and D2 = 1/0.2 = 5.
Following the procedure described in section II.D case II,
we calculate the data pair (xT,ρ) for the lower bound from
the left hand term in (13) and the upper bound from (14).
The resulting lower and upper bounds utilization functions
are Glower(xT) = −63.5754x3
and Gupper(xT) = 0.2061x3
utilization functions are then used in (19) to form two mod
els of the network. Fig. 6 shows the behavior of the traffic at
node 2 destined for node 3 (i.e., x3
are connected through direct link, the bounds are tight, but
when the direct link breaks, a wider gap occurs between
bounds. The first reason is that the server utilization ρ of the
1=
1= 0.2 γ1
2= 0.16 γ3
2= 0.2 γ1
3= 0.16 γ2
3= 0.2
T+ 49.7702x2
T−0.6099x2
T− 7.8764xT
T+1.1257xT. The
2). Here, while all nodes
Page 8
relay node becomes higher so that the lower bound gener
ated by (13) drops off. Secondly, (13) does not make use of
any information of the kicombination to tighten the bound.
Another way to possibly reduce the gap between bounds is
by assuming homogeneous traffic, as discussed in section
2.4 case II and as implemented in the next scenario.
0 100 200300 400 500 600
0
0.1
0.2
0.3
0.4
0.5
0.6
t (second)
(a)
Average Number of Packets
Simulation
Hybrid Modeling
X2
3
0 100200300400 500600
0.5
1
1.5
2
2.5
t (second)
(b)
End−to−end Delay (second)
Simulation
Hybrid Modeling
X2
3
Figure 6. Average number of packets and
endtoend delay of traffic destined for node
3 at node 2
Next we consider a five node network with the random
waypoint mobility (RWM) model. Recently in [16] and
[17], it is shown that the RWM model link connectivity
can be modeled as a twostate Markov process with onoff
(connecteddisconnected) transition, with both link on and
off durations following exponential distributions. This link
connectivity model can reproduce the average link stability
statistics of RWM model without requiring a detailed node
mobility simulation. In this experiment, each link is expo
nentially distributed on duration with a mean of 50 seconds
and off duration with a mean of 10 seconds. The link ca
pacity and packet length remain the same as ones in the
three node network scenario. The externally arrival rates
are assigned as: γ3
0.22 γ5
ing behavior of the network, we randomly select the time
interval [2100 2200] from a 6000 seconds of the total ex
periment duration. The link connectivity as determined by
the RWM for the time interval is shown in Fig. 7. In the
following discussion, we focus on the traffic from node 1
1= 0.25 γ5
1= 0.16 γ5
2= 0.18 γ5
3=
4= 0.25. In order to illustrate the typical time vary
destined for node 5 and all the routes of class 5 traffic going
through node 1 are marked by dotted lines in the figure.
1
4
32
5
1
4
32
5
1
4
32
5
1
4
32
5
1
4
32
5
1
4
32
5
(a) 2100<=t<2115
(b) 2115<=t<2124(c) 2124<=t<2131
(d) 2131<=t<2136(e) 2136<=t<2161 (f) 2161<=t<2166
1
4
32
5
1
4
32
5
1
4
32
5
(g) 2166<=t<2177
(h) 2177<=t<2185(i) 2185<=t<=2200
Figure 7. Typical RWM connectivity scenario
for five node network.
Fig. 8 shows the average number in system x5
endtoend delay of the packets from node 1 destined for
node 5 during the time interval [2100 2200]. Initially, ev
ery packet goes through the direct link (15). Then, during
the time interval 2124 ≤ t < 2136 sec, a large transient
increase in x5
(45) breaks, the traffic x5
reach the destination. This event also results in the increase
of endtoend delay of the packets x5
utilization of node 1. At t = 2136 sec link (45) recovers
and the x5
Starting from t = 2161 sec, link (15) breaks and the rout
ing protocol redirects the packet x5
5. Notice that at t = 2177 sec the link (35) is disconnected
and it causes the x5
3 to node 2 resulting in a further increase in the endtoend
delay. At t = 2185 sec the direct link (15) is working again
and the traffic is rerouted over the direct link resulting in a
decrease in the delay.
The fluid flow numerical results presented in Fig. 8, are
calculated by assuming “homogeneous traffic” instead of
(13) (14). For the lower bound, all sources are assumed
to have homogeneous traffic with the period of Dmax =
max{Di} = 1/0.16 = 6.25, while for the upper bound,
the homogeneous traffic becomes Dmin = min{Di} =
1/0.25 = 4. Following the procedure in Section II.D
case I, we obtain the utilization function for both bounds
as Glower(xT) = 0.0889x3
Gupper(xT) = 0.0743x3
are used in (19) to form the two bounding models of the
network. From Fig. 8, we notice that the lower and upper
1and the
1occurs due to traffic rerouting, when the link
4needs to go through node 1 to
1, because of the higher
4traffic is rerouted back to the direct link (45).
1through node 3 to node
1traffic to take one more hop from node
T− 0.4496x2
T− 0.4159x2
T+ 1.0706xT,
T+ 1.0993xT, which
Page 9
bounds are tighter than ones in three node network scenario.
Hence, the “homogeneous traffic” approach might be better
choice than (13) and (14), when the?m
0.7
iNiDi/D/1 queue
has close arrival rates.
2100 2110212021302140
t (second)
(a)
2150 2160 21702180 21902200
0.1
0.2
0.3
0.4
0.5
0.6
Average Number of Packets
Simulation
Hybrid Modeling
X1
5
21002110 21202130 2140
t (second)
(b)
215021602170218021902200
1
1.5
2
2.5
3
3.5
4
End−to−end Delay (second)
Simulation
Hybrid Modeling
X1
5
Figure 8. Average number of packets and
endtoend delay of traffic at node 1 destined
for node 5.
4COMPUTATION TIME COMPLEXITY
4.1 Complexity Analysis
The proposed hybrid modeling approach is composed of
a fluid flow queueing model together with a time varying
connectivity matrix. At each time interval, network met
rics are evaluated by integrating a set of fluid flow model
based differential equations with a specific connectivity ma
trix. Similarly to other areas of complexity theory, the ex
act number of arithmetic operations required for solving the
differential equations with one step time by RungeKutta
algorithm is hard to determine but an upper bound on the
complexity order can be obtained.
Let dt refer to the time step size of solving the differ
ential equations and T be the length of the time interval of
interest. Then T/dt represents the number of total times to
solve the set of differential equations. Let K denote the av
erage time to execute one arithmetic operation on a CPU.
Following [27], C(n,p,α) denotes the upper bound on the
number of arithmetic operations required within one step
time, so that n differential equations can be solved by a
p−th order explicit RungeKutta algorithm with maximum
error e−α. According to the expression for C(n,p,α) in
[27], with the predefined value of p and α, C increases lin
early with n. As a result, only considering the varying of n
in K ·(T/dt)·C(n,p,α), the computation time complexity
of our hybrid modeling TCHis upper bounded by O(n).
For an Mnode wireless network, regardless of the number
of externally arriving traffic flows, the number of differen
tial equations n is M(M − 1). Therefore, the upper bound
of TCHis O(M(M − 1)).
4.2 Comparison by Sample Networks
To further evaluate the complexity we numerically de
termined the computational time for a set of sample net
work implementations. We constructed both OPNET dis
crete event simulation and the proposed hybrid fluid flow
model for network settings of three nodes, four nodes, five
nodes and thirteen nodes networks, as shown in Fig. 9. The
numerical results are shown in Table I when both hybrid
modeling and simulation were run on a laptop with Intel
T7400 2.16GHz DuoCore Processor with 2GB memory.
For the discrete event simulation the time given is the to
tal time to execute 4096 runs. This result shows that for the
hybrid modeling, the numerical computation time is propor
tional to the number of differential equations. However, the
computation time required to complete the OPNET discrete
event simulation increases with the total number of events
which is a complex function of the amount of traffic, topol
ogy changes and accuracy desired.
1
4
32
5
1
43
212
3
3
1
4
2
5
12
6
8
7
9
10
11
13
(a) 3 Nodes(b) 4 Nodes (c) 5 Nodes(d) 13 Nodes
Figure 9. Topologies of sample networks
Table 1. Computation Time Comparison (in
second)
# # Flows of
Externally
Arriving
4
6
8
11
# Diff.
Equation
Simulation
(sec)
Hybrid
Model
(sec)
0.21
0.48
0.83
6.62
Nodes
3
4
5
13
6
12
20
156
123.1
1173.7
11350.5
389723.4
Page 10
5Conclusion
In this paper, we propose a performance modeling tech
nique to represent time varying behavior of multihop wire
less networks with CBR traffic, using time varying connec
tivity matrix modeling and numerical method based queue
ing analysis. Network queues are modeled using fluid flow
based differential equations and solved using numerical in
tegration routines, while topology change is integrated into
the connectivity matrix using deterministic or probabilistic
modeling techniques. The proposed hybrid modeling ap
proach can generally be applied to a wide range of queue
ing systems. Numerical results using the proposed model
have been given in comparison with results from traditional
discrete event simulations. The computation time required
by both approaches is also shown side by side. We be
lieve that this hybrid modeling approach is a proper tool
for evaluating the timevarying behavior of multihop wire
less networks. With the computation time saved from the
fluid flow based hybrid modeling method, it is a tremendous
gain in flexibility for modeling complex networks. Future
work includes validating the model with testbed measure
ment results and developing additional features (e.g., multi
rate links) to increase the fidelity of the model.
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