# 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 fluid-flow 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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**ABSTRACT:**In this paper, we develop a performance modeling technique for analyzing the time varying network layer queuing behavior of multihop wireless networks with constant bit rate (CBR) traffic. Our approach is a hybrid of a time varying adjacency matrix and a fluid flow queuing network model. Mobile network topology is modeled using time varying adja- cency matrix, while node queues are modeled using fluid flow based differential equations which are solved using numerical methods. Numerical and simulation experiments show that this new approach can provide reasonably accurate results. Moreover, when compared to the computation time required in a standard discrete event simulator, the fluid flow based model is shown to be a more scalable tool. Finally, an illustrative example of our modeling technique application is given to show its capability of capturing the time varying network performance as a function of traffic load, node mobility and wireless link quality.IEEE Transactions on Vehicular Technology 08/2014; · 2.06 Impact Factor -
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**ABSTRACT:**IEEE 802.15.4 Low Rate Wireless Personal Area Network (LR-WPAN) intends to provide low-power ubiquitous communications between devices. However, energy utilization by devices (or nodes) in the network often impacts the application functionality and lifetime due to limited battery capacity. For instance, in the typical multihop LR-WPAN used for data detection and monitoring, the traffic flow often converges to the data sink and such a many-to-one pattern typically results in energy imbalance. Previous research has proposed many approaches to solve this issue. However, a common limitation is about using the idealized energy model, such as “first order radio model”, which is the idealized estimation for RF transmission energy cost of the node. In this paper, we develop a realistic and representative LR-WPAN RF transceiver energy model from the measured data on Chipcon CC2420 and corresponding power control policy. Further, with this energy model, we formulate an optimized energy balanced chain (OEBC) model to maximize the network lifetime. Finally, the OEBC model is applied to develop the network deployment strategy by optimizing node placement, node density and traffic flow distribution.Computing, Networking and Communications (ICNC), 2013 International Conference on; 01/2013 -
##### Conference Paper: Improving WLAN throughput via reactive jamming in the presence of hidden terminals

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**ABSTRACT:**In the area of performance analysis of wireless networks, one critical issue is the hidden terminal problem, which is considered as one of the severest reasons for the degradation of network performance. In this paper, we incorporate reactive jamming scheme with distributed coordination function (DCF) in IEEE 802.11 based wireless local area networks (WLANs) to improve network throughput in the presence of hidden terminals. In the proposed protocol, we schedule access point (AP) to broadcast jamming signal reactively to hinder the simultaneous transmission of hidden terminals. Both analytical and numerical results show that our reactive jamming based protocol can constantly improve WLAN throughput for a wide range of conditions, compared with the traditional RTS/CTS.Wireless Communications and Networking Conference (WCNC), 2013 IEEE; 01/2013

Page 1

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 Nebraska-Lincoln, 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 fluid-flow 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 ad-hoc networks

(VANET) [2], or mobile ad-hoc 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 Multi-University Research Initiative (MURI) grant W911NF-

07-1-0318.

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

978-1-4244-9328-9/10/$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

fluid-flow 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 steady-state 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, two-state 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 non-negative 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: Non-identical 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 two-node deterministic service

system with its equivalent model.

The general M nodes network hybrid model combining

fluid-flow 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 end-to-

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-

to-end 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 Runge-Kutta 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 end-to-end 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 re-routing 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

end-to-enddelayofthetrafficfromnode1destinedfornode

2. During the time t ≥ 500 sec, link 1-2 breaks and x2

has to go through the relay node 3 to reach the destination.

Hence, the end-to-end delay consists of propagation delay

in link 1-3 and link 3-2 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

end-to-end 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

end-to-end 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 two-state Markov process with on-off

(connected-disconnected) 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

end-to-end 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 (1-5). Then, during

the time interval 2124 ≤ t < 2136 sec, a large transient

increase in x5

(4-5) breaks, the traffic x5

reach the destination. This event also results in the increase

of end-to-end delay of the packets x5

utilization of node 1. At t = 2136 sec link (4-5) recovers

and the x5

Starting from t = 2161 sec, link (1-5) breaks and the rout-

ing protocol redirects the packet x5

5. Notice that at t = 2177 sec the link (3-5) is disconnected

and it causes the x5

3 to node 2 resulting in a further increase in the end-to-end

delay. At t = 2185 sec the direct link (1-5) 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 (4-5).

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

end-to-end 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 Runge-Kutta

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 Runge-Kutta algorithm with maximum

error e−α. According to the expression for C(n,p,α) in

[27], with the pre-defined 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 M-node 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 Duo-Core 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 time-varying 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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