End-to-End Delay Modeling for Mobile Ad Hoc Networks: A Quasi-Birth-and-Death Approach

Abstract

Understanding the fundamental end-to-end delay performance in mobile ad hoc
networks (MANETs) is of great importance for supporting Quality of Service
(QoS) guaranteed applications in such networks. While upper bounds and
approximations for end-to-end delay in MANETs have been developed in
literature, which usually introduce significant errors in delay analysis, the
modeling of exact end-to-end delay in MANETs remains a technical challenge.
This is partially due to the highly dynamical behaviors of MANETs, but also due
to the lack of an efficient theoretical framework to capture such dynamics.
This paper demonstrates the potential application of the powerful
Quasi-Birth-and-Death (QBD) theory in tackling the challenging issue of exact
end-to-end delay modeling in MANETs. We first apply the QBD theory to develop
an efficient theoretical framework for capturing the complex dynamics in
MANETs. We then show that with the help of this framework, closed form models
can be derived for the analysis of exact end-to-end delay and also per node
throughput capacity in MANETs. Simulation and numerical results are further
provided to illustrate the efficiency of these QBD theory-based models as well
as our theoretical findings.

Full text

arXiv:1312.7201v1 [cs.NI] 27 Dec 2013
End-to-End Delay Modeling for Mobile Ad Hoc
Networks: A Quasi-Birth-and-Death Approach
JUNTAO GAO1⋆
, YULONG SHEN2†
, XIAOHONG JIANG1‡
, OSAMU
TAKAHASHI1, NORIO SHIRATORI3
1School of Systems Information Science, Future University Hakodate, 041-8655
Japan
2School of Computer Science and Technology, Xidian University, 710071 China
3GITS, Waseda University, Tokyo and RIEC, Tohoku University, Sendai-shi,
980-8579 Japan.
Understanding the fundamental end-to-end delay performance
in mobile ad hoc networks (MANETs) is of great importance
for supporting Quality of Service (QoS) guaranteed applications
in such networks. While upper bounds and approximations for
end-to-end delay in MANETs have been developed in literature,
which usually introduce significant errors in delay analysis, the
modeling of exact end-to-end delay in MANETs remains a tech-
nical challenge. This is partially due to the highly dynamical be-
haviors of MANETs, but also due to the lack of an efficient theo-
retical framework to capture such dynamics. This paper demon-
strates the potential application of the powerful Quasi-Birth-and-
Death (QBD) theory in tackling the challenging issue of exact
end-to-end delay modeling in MANETs. We first apply the QBD
theory to develop an efficient theoretical framework for capturing
the complex dynamics in MANETs. We then show that with the
help of this framework, closed form models can be derived for
the analysis of exact end-to-end delay and also per node through-
put capacity in MANETs. Simulation and numerical results are
further provided to illustrate the efficiency of these QBD theory-
⋆email: gaojuntao223@gmail.com
†email: ylshen@mail.xidian.edu.cn
‡email: jiang@fun.ac.jp
1

based models as well as our theoretical findings.
Key words: Ad hoc networks, routing protocol, delay performance anal-
ysis.
1 INTRODUCTION
Mobile ad hoc networks (MANETs) represent a class of important wireless ad
hoc networks with mobile nodes. Since the flexible and distributed MANETs
are robust and rapidly deployable/reconfigurable, they are highly appealing
for a lot of critical applications [6, 14], like deepspace communication, disas-
ter relief, battlefield communication, outdoor mining, device-to-device com-
munication for traffic offloading in cellular networks, etc. To facilitate the
applications of MANETs in providing Quality of Service (QoS) guaranteed
services, understanding the fundamental delay performance of such networks
is of great importance [18].
Notice that the end-to-end delay, the time it takes a packet to reach its
destination after it is generated at its source, serves as the most fundamental
delay performance for a network. However, the end-to-end delay modeling
in MANETs remains a technical challenge. This is partially due to the highly
dynamical behaviors of MANETs, like node mobility, interference, wireless
channel/traffic contention, packet distributing, packet queueing process in a
node and the complicated packet delivering process among mobile nodes,
but also due to the lack of a theoretical framework to efficiently depict the
complicated network state transitions under these network dynamics. By now,
the available works on end-to-end delay analysis in MANETs mainly focus
on deriving upper bounds or approximations for such delay.
Based on the M/G/1queueing model, some closed-form upper bounds
on the expected end-to-end delay were derived for MANETs with two-hop
relay routing [23]. For MANETs with multi-hop back-pressure routing, the
Lyapunov drift model was adopted to derive an upper bound on the expected
end-to-end delay [4]. For MANETs with multi-hop linear routing, a network
calculus approach was proposed to derive upper bounds on end-to-end delay
distribution [12, 8]. In addition to the delay upper bound results, approxi-
mations to end-to-end delay in MANETs have also been explored recently
[17, 19]. By adopting the polling model, an approximation to expected end-
to-end delay was provided in [17] for a simple two-hop relay MANET con-
sisting of only one source node, one relay node and one destination node. For
2

more general MANETs with multiple source-destination pairs and multi-hop
relay routing, approximations to corresponding end-to-end delay were devel-
oped in [19] based on the elementary probability theory.
It is notable that the results in [23, 4, 12, 8, 17, 19] indicate that although
above upper bound and approximation results are helpful for us to understand
the general delay behaviors in MANETs, they usually introduce significant
errors in end-to-end delay analysis. This is mainly due to the lack of an
efficient theoretical framework to capture the complex network dynamics and
thus the corresponding network state transitions in MANETs. This paper
demonstrates the potential application of the powerful Quasi-Birth-and-Death
(QBD) theory in capturing the network state transitions and thus in tackling
the challenging exact end-to-end delay modeling issue in MANETs.
The main contributions of this paper are as follows:
•We first demonstrate that the QBD theory actually enables a novel and
powerful theoretical framework to be developed to efficiently capture
the main network dynamics and thus the complex network state transi-
tions in two-hop relay MANETs.
•With the help of the theoretical framework, we then show that we are
able to analytically model the exact expected end-to-end delay and also
exact per node throughput capacity in the concerned MANETs.
•Extensive simulation and numerical results are further provided to val-
idate the efficiency of our QBD theory-based models for end-to-end
delay and per node throughput capacity, and to illustrate how end-to-
end delay and throughput capacity in MANETs are affected by some
main network parameters.
The rest of this paper is organized as follows. Section 2 introduces the
system models involved in this study. A QBD-based theoretical framework is
developed in Section 3 to capture network state transitions, based on which
the exact expected end-to-end delay and per node throughput capacity are
then derived in Section 4. Section 5 first provides simulation results to vali-
date the efficiency of our theoretical framework and related delay and capac-
ity results, and then explores the effects of network parameters on delay and
capacity. Finally, we conclude this paper in Section 6.
3

S
m
m
FIGURE 1
A snapshot of a cell partitioned MANET with m= 16.
2 SYSTEM MODELS
In this section, we introduce first the basic network models regarding node
mobility, wireless channel, radio and traffic pattern in the considered MANET,
and then Medium Access Control (MAC) protocol for transmission schedul-
ing to resolve wireless channel contention and interference issues. Finally,
the two-hop relay routing scheme that deals with packet delivery and traffic
contention is discussed.
2.1 Network Models
Node mobility and channel models: As shown in Fig. 1 that we consider
a unit torus MANET partitioned evenly into m×mcells [25, 9, 22]. In the
concerned MANET, there are nnodes moving around according to the i.i.d.
mobility model [15, 23]. We consider the time slotted system, where each
node randomly chooses one cell to move into at the beginning of every time
slot and then stays in it for the whole time slot. We assume that a common
bandwidth limited wireless channel is shared by all nodes for data transmis-
sions. In each time slot, the data transmitted between any two nodes through
the wireless channel is normalized to one packet.
Radio model: Each node employs the same radio power to transmit data
through the common wireless channel. To enable the transmission region of
4

a node (say Sin Fig. 1) to cover its own cell and also its 8neighbor cells
(called coverage cells of the node hereafter), the corresponding radio range r
of the node should be set as r=√8/m. Based on the widely used protocol
model [16, 20, 9, 22], data transmission from a transmitting node (transmitter)
ito a receiving node (receiver) jcan be conducted only if the Euclidean
distance dij between them is less than r(i.e., dij ≤r), while the data can
be successfully received by receiver jonly if dkj ≥(1 + ∆) ·rholds for
any other concurrent transmitter k, here k6=i, j, and ∆≥0is a specified
guard-factor for interference prevention.
Traffic model: Similar to previous work [9], we consider the permutation
traffic pattern in which each node acts as the source of a traffic flow and at
the same time the destination of another traffic flow. Thus, there are in total n
distinct traffic flows in the MANET. Each source node exogenously generates
packets for its destination according to an Bernoulli process with average rate
λ(packets/slot) [23].
2.2 MAC Protocol
For a fair channel access, we consider here a commonly used MAC protocol
for transmission scheduling, based on the idea of equivalent-class (EC) [20,
9, 22]. An EC is defined as a set of cells as illustrated in Fig. 2, where any
two cells in an EC are separated by a horizontal and vertical distance of some
integer multiple of αcells (1≤α≤m). Thus, we have in total α2ECs in
the MANET. Under the EC based MAC protocol (MAC-EC), these ECs are
scheduled to be active alternatively as time evolves. The cells in an active
EC are called active cells and nodes (if any) in an active cell contend fairly to
access the common wireless channel.
To enable as many number of concurrent transmissions to be scheduled as
possible while avoiding interference among these transmissions, the param-
eter αshould be set appropriately. As illustrated in Fig 2 that for the trans-
mission between a transmitter Sand its possible receiver Rto be successful,
the distance between Rand another possible closest concurrent transmitter
W, i.e., (α−2)/m, should satisfy the following condition according to the
protocol model [16]:
(α−2)/m ≥(1 + ∆) ·r, (1)
Notice also that r=√8/m and α≤m, the parameter αshould be deter-
mined as
α= min{⌈(1 + ∆)√8 + 2⌉, m},(2)
5

2 3 44
5 6 7 8
9 10 11 12
m
1 1 1 1
13 14 15 16
1 1 1 1
1 1 1 1
1 1 1 1
S
R
W
)1( !
r
"
#
#
m
FIGURE 2
Illustration of equivalent-classes in a cell partitioned MANET. There are 16
equivalent-classes in this MANET with α= 4. All shaded cells belong to equivalent-
class 1.
where ⌈x⌉takes the least integer value greater than or equal to x.
2.3 Two-Hop Relay Routing
Once a node, say Sin Fig. 2, succeeds in wireless channel contention and
becomes a transmitter, it executes the popular two-hop relay (2HR) routing
protocol defined in Algorithm 1 for packet delivery [11, 5]. With the 2HR
routing, each exogenously generated packet at Sis first distributed out to
relays through wireless broadcast [11, 24], and it is then delivered to its des-
tination Dvia these relays.
Algorithm 1 2HR Routing Protocol
1: Transmitter Sselects to conduct packet-broadcast with probability q,0<
q < 1, and to conduct packet-delivery with probability 1−q;
2: if Sselects packet-broadcast then
3: Sexecutes Procedure 1.1;
4: else
5: Sexecutes Procedure 1.2;
6: end if
6

Procedure 1.1 packet-broadcast
1: if Shas packets in its source-queue then
2: Sdistributes out the head-of-line (HoL) packet of source-queue
through wireless broadcast to all nodes in its coverage cells;
3: Any node, say R, in the coverage cells of Sreserves a copy of that
packet;
4: if Ris not the destination Dthen
5: Rinserts the HoL packet into the end of its relay-queue associated
with D;
6: else
7: if Ris currently requesting the HoL packet then
8: Rkeeps the HoL packet and increases AC K (D)by 1;
9: else
10: Rdiscards that packet;
11: end if
12: end if
13: Smoves that HoL packet out of source-queue and inserts it into the
end of its broadcast-queue;
14: Smoves ahead the remaining packets in its source-queue;
15: else
16: Sremains idle;
17: end if
To facilitate the operation of the 2HR routing protocol, each node, say S,
is equipped with three types of First In First Out(FIFO) queues: one source-
queue, one broadcast-queue and n−2parallel relay-queues (no relay-queue
is needed for node Sitself and its destination node D).
Source-queue: Source-queue stores packets exogenously generated at S
and destined for D. These exogenous packets will be distributed out to relay
nodes later in FIFS way.
Broadcast-queue: Broadcast-queue stores packets from source-queuethat
have already been distributed out by Sbut have not been acknowledged yet
by Dthe reception of them.
Relay-queue: Each node other than Sand Dis assigned with a relay-
queue in Sto store redundant copies of packets distributed out by the source
of that node.
To ensure the in-order packet reception at D, similar to previous work [23]
that Slabels every exogenously generated packet with a unique identification
7

Procedure 1.2 packet-delivery
1: Srandomly selects a node Uas its receiver from nodes in its coverage
cells. Denote the source of Uas V;
2: Sinitiates a handshake with Uto acquire the packet number ACK(U) +
1and thus to know which packet Uis currently requesting;
3: Schecks its corresponding relay-queue/broadcast-queue whether it bears
a packet with I D(V) = ACK(U) + 1;
4: if Sbears such packet then
5: Sdelivers that packet to U;
6: Sclears all packets with I D(V)≤ACK (U)from its corresponding
relay-queue/broadcast-queue;
7: Smoves ahead the remaining packets in its corresponding relay-
queue/broadcast-queue;
8: Uincreases ACK(U)by 1;
9: end if
number ID(S), which increases by 1every time a packet is generated; des-
tination Dalso maintains an acknowledgment number ACK(D)indicating
that Dis currently requesting the packet with ID(S) = AC K (D) + 1 (i.e,
the packets with ID(S)≤ACK(D)have already been received by D).
3 QBD-BASED THEORETICAL FRAMEWORK
In this section, we first present some preliminaries, and then develop a novel
theoretical framework based on the QBD theory to capture the complex net-
work state transitions in the concerned MANETs.
3.1 Preliminaries
We focus on one specific traffic flow from source Sto destination Din our
analysis. Notice that once a packet is generated at S, it first experiences
a queueing process in the source-queue of Sbefore being distributed out
(served), and it then experiences a network delivery process after being dis-
tributed out into the network by Sand before being successfully received
by D. Since Drequests packets in order according to AC K (D), all pack-
ets distributed out by Swill be also delivered (served) in order. Thus, we
can treat the network delivery process as a queueing process of one virtual
network-queue. Notice also that the departure process of source-queue is just
the arrival process of network-queue.
8

To fully depict the two queueing processes in both source-queue and network-
queue, we define following probabilities for a time slot.
•pb:probability that Sbecomes transmitter and also selects to do packet-
broadcast.
•pc(j) : probability that jcopies of a packet exist in the network (includ-
ing the one in S) after the packet is distributed out by Sin the current
time slot, 1≤j≤n−1.
•pr(j) : probability that Dreceives the packet it is currently requesting
given that jcopies of the packet exist in the network, 1≤j≤n−1.
•p0(j) : probability that jcopies of a packet exist in the network after S
becomes transmitter and selects to do packet-broadcast for this packet,
given that Dis out of the coverage cells of Sand network-queue is
empty, 1≤j≤n−1.
•p0(0) : p0(0) = 1 −Pn−1
j=1 p0(j).
•p+
b(j) : probability that Sbecomes transmitter, selects to do packet-
broadcast and also successfully conducts packet-broadcast for one packet;
at the same time, Dreceives the packet it is requesting given that j
copies of that packet exist in the network, 1≤j≤n−1.
•p−
b(j) : probability that Sbecomes transmitter, selects to do packet-
broadcast and also successfully conducts packet-broadcast for one packet;
at the same time, Ddoes not receive the packet it is requesting given
that jcopies of that packet exist in the network, 1≤j≤n−1.
•p+
f(j) : probabilitythatSdoes not successfully conduct packet-broadcast
for any packet; at the same time, Dreceives the packet it is requesting
given that jcopies of that packet exist in the network, 1≤j≤n−1.
•p−
f(j) : probabilitythat Sdoes not successfully conduct packet-broadcast
for any packet; at the same time, Ddoes not receive the packet it
is requesting given that jcopies of that packet exist in the network,
1≤j≤n−1.
The following lemma reveals a nice property about the source-queue and
network-queue, which will help us to evaluate the above probabilities in Lemma 2.
9

Lemma 1: For the considered MANET with MAC-EC protocol for trans-
mission scheduling and 2HR-B protocol for packet delivery, the arrival pro-
cess of network-queue is a Bernoulli process with probability λand it is in-
dependent of the state of source-queue.
Proof. We know from Section 2.1 that the arrival process of source-queue in
Sis a Bernoulli process with probability λ. The service process of source-
queue is actually also a Bernoulli process, because in every time slot Sgets
a chance with constant probability pbto do packet-broadcast to distribute out
a packet in source-queue (or equivalently, the source-queue is served with
probability pbin every time slot). Thus, the source-queue in Sfollows a
Bernoulli/Bernoulli queue, and in equilibrium the packet departure process of
source-queue is also a Bernoulli process with probability λ, which is indepen-
dent of the state of source-queue (i.e., the number of packets in source-queue)
[10]. Because the arrival process of network-queue is just the departure pro-
cess of source-queue, this finishes the proof of this Lemma. 
Lemma 2:
10

pb=qm2
α2n1−m2−1
m2n(3)
pc(j) = nn−2
j−1(m2−9)n−1−j
m2n−(m2−1)nn(m2−9)f(j) +f(j+ 1)o(4)
pr(j) = j(1−q)m2
α2n(n−1)1−m2−1
m2n
−n
m2m2−9
m2n−1(5)
p0(j) = λ·q·n−2
j−1(m2−9)n−j
α2m2n−2pb
f(j)(6)
p0(0) = 1 −λ·q·(m2−9)
α2(n−1)pb1−m2−1
m2n−1(7)
p+
b(j) = (j−1) λ(q−q2)(m4−m2α2)
α4n(n−1)(n−2)pb
·1−2m2−1
m2n
+m2−2
m2n
−n
m2m2−9
m2n−1
+n
m2m2−10
m2n−1(8)
p−
b(j) = λ−p+
b(j)(9)
p+
f(j) = pr(j)−p+
b(j)(10)
p−
f(j) = 1 −p+
b(j)−p−
b(j)−p+
f(j)(11)
where
f(x) = 9x−8x
x(12)
Proof. The proof of Lemma 2 is given in Appendix A. 
Remark: The complex network dynamics of node mobility, interference,
wireless channel and traffic contention are incorporated into the calculation
of the above probabilities as shown in Appendix A. The network dynamics of
packet distributing, packet queueing and delivering processes will be captured
in the following QBD modeling process.
3.2 QBD Modeling
We use L(t)≥0to denote the number of local packets distributed out from S
but not received yet by Duntil time slot t, and use J(t)to denote the number
of copies of the packet Dis currently requesting at time slot tin the network,
11

0≤J(t)≤n−1. As time tevolves, the queueing process of network-queue
follows a two-dimensional QBD process [3, 21]
{(L(t), J(t)), t = 0,1,2,··· },(13)
on state space
{(0,0)} ∪ {(l, j)};l≥1,1≤j≤n−1(14)
where (0,0) corresponds to the empty network-queue state. L(t)increases
by 1if Sdistributes out a packet from its source-queue while Ddoes not
receive the packet it is requesting at slot t,L(t)decreases by 1if Sdoes not
distribute out a packet from its source-queue while Dreceives the packet it is
requesting at slot t, and L(t)keeps unchanged, otherwise.
All states in (14) can be divided into the following subsets
N(0) = {(0,0)}(15)
N(l) = {(l, j)},1≤j≤n−1, l ≥1(16)
where subset N(0) is called level 0and subset N(l)is called level l. It is
notable that when network-queue is in some state of level l(l≥1) at a time
slot, the next state of one-step state transitions could only be somestate in the
same level lor in its adjacent levels l−1and l+ 1.
Based on the queueing process of network-queue and the definitions of
probabilities in Lemma 2, the underlying QBD process of the network-queue
has state transition diagram shown in Fig. 3. In Fig. 3, p+
f(∗)pc(j)denotes
the probability of the transition from some state (3,∗)in level 3to the state
(2, j)in level 2, where the asterisk ‘∗’ means some eligible copy number in
{1,2,···, n −1}.
To facilitate our discussion, we classify the state transitions in Fig. 3 as
intra-level transition (denoted by dotted arrows in Fig. 3) and inter-level
transition (denoted by solid arrows in Fig. 3).
Intra-level Transition: There are two cases regarding the intra-level tran-
sitions, namely, the state transition inside level 0and the state transitions
inside level l(l≥1). For level 0, it has only one state (0,0), which could
only transit to itself. For levels l≥1, they all follow the same intra-level
transitions, i.e., a state (l, j)in level lcould transit to any state (including
itself) in the same level, 1≤j≤n−1.
Inter-level Transition: There are also two cases regarding the inter-level
transitions, namely, transitions between level 0and level 1, and transitions
between level land level l+ 1 (l≥1). The inter-level transitions between
12

)1(
N
)2(
N
)(l
N
1,1
2,1
j
,1
1,1
n
!
!
!
!
1,2
2,2
j
,2
1,2
n
)1()( cb pjp!
)1()(
!npjp cb
)()()( jpjpjp fcb
!
!
)2()( cb pjp!
)( jpb
)()( jpjp cf
!
)( jpf
!
)0(
0
p
)(
0jp
)1()(
!
npjp cf
)2()( cf pjp
!
)1()( cf pjp
!
)( jpb
)()( jpp cf "
!
)1()( cb pjp!
)1()(
!npjp cb
)()()( jpjpjp fcb
!
!
)2()( cb pjp!
0,0
)0(
N
FIGURE 3
State transition diagram for the QBD process of network-queue.
level 0and level 1are simply bi-transitions between state (0,0) and any state
(1, j),1≤j≤n−1. For adjacent levels land l+ 1 (l≥1), they all follow
the same inter-level transitions, i.e., a state (l, j)in level lcould only transit
to the corresponding state (l+ 1, j)in level l+ 1, while a state (l+ 1, j)in
level l+ 1 could transit to any state in level l,1≤j≤n−1.
4 DELAY AND THROUGHPUT CAPACITY
With the help of the QBD-based theoretical framework, we derive the ex-
pected end-to-end delay and also per node throughput capacity for the con-
cerned MANETs.
Definition 1: End-to-end delay Teof a packet is the time elapsed between
the time slot the packet is generated at its source and the time slot it is deliv-
ered to its destination.
Definition 2: Per node throughput capacity µis defined as the maximum
packet arrival rate λevery node in the concerned MANET can stably support.
Before presenting our main result on the expected end-to-end delay, we
13

first derive the per node throughput capacity, with which the input rate the
MANET can stably support and the corresponding end-to-end delay can then
be determined.
Theorem 1: For the considered MANET, its per node throughput capacity
µis given by
µ= min (pb,1
Pn−1
j=1
pc(j)
pr(j))(17)
Proof. In equilibrium, the service rate µsof source-queue is
µs=pb(18)
and the service rate µdof network-queue, i.e., the rate Dreceives its request-
ing packets, is
µd=1
Pn−1
j=1
pc(j)
pr(j)
.(19)
To ensure network stability, packet generation rate λat Sshould satisfy
λ < min{µs, µd}(20)
Thus, the per node throughput capacity µis determined as
µ= min{µs, µd}(21)

Based on above per node throughput capacity result and the QBD-based
theoretical framework, we now establish the following theorem on the ex-
pected end-to-end delay of the concerned MANET.
Theorem 2: For the concerned MANET, where each source node exoge-
nously generates packets according to a Bernoulli process with probability λ
(λ < µ), the expected end-to-end delay E(Te)of a packet is determined as
E(Te) = L1+L2
λ,(22)
where
L1=λ−λ2
pb−λ(23)
L2=y1(I−R)−21
φ(24)
14

R=A0(I−A1−A01v0)−1(25)
[y0,y1] = [y0,y1]B1B0
B2A1+RA2(26)
φ=y0+y1(I−R)−11(27)
v0=pc(1) pc(2) ··· pc(j)··· pc(n−1) (28)
A0= diag p−
b(1), p−
b(2),···, p−
b(j),···, p−
b(n−1)(29)
A1=diagp−
f(1), p−
f(2),···, p−
f(j),···, p−
f(n−1)
+
p+
b(1) p+
b(2) ··· p+
b(j)··· p+
b(n−1) Tv0(30)
A2=B2v0(31)
B0=p0(1) p0(2) ··· p0(j)··· p0(n−1) (32)
B1= [p0(0)] (33)
B2=hp+
f(1) p+
f(2) ··· p+
f(j)··· p+
f(n−1) iT(34)
here Idenotes an identity matrix of size (n−1)×(n−1),1denotes a column
vector of size (n−1) ×1with all elements being 1,y0is a scalar value, and
y1is a row vector of size 1×(n−1).
Proof. From Lemma 1 we know that we can analyze queueing processes of
source-queue and network-queue separately.
First, for the source-queue at S, since it follows a Bernoulli/Bernoulli
queue, we know from [23, 3] that the expected number of packets L1in the
queue is determined as
L1=λ−λ2
pb−λ(35)
Then, for the network-queue, its queueing process follows a QBD process
shown in Fig. 3. The corresponding state transition matrix Qof the transition
diagram in Fig. 3 is given by
Q=








B1B00 0 ···
B2A1A00···
0 A2A1A0···
0 0 A2A1···
.
.
..
.
..
.
..
.
....








(36)
where B1defined in (33) represents the state transition from (0,0) to (0,0);
B0defined in (32) represents the state transitions from (0,0) to (1, j ),1≤
15

j≤n−1;B2defined in (34) represents the state transitions from (1, j)to
(0,0),1≤j≤n−1;A1defined in (30) represents the state transitions from
(l, j)to (l, i),l≥1,1≤j, i ≤n−1;A0defined in (29) represents the state
transitions from (l, j )to the corresponding (l+ 1, j),l≥1,1≤j≤n−1;
A2defined in (31) represents the state transitions from (l, j )to (l−1, i),
l≥2,1≤j, i ≤n−1.
Based on the QBD process theory [3, 21], the queueing process of the
network-queue can be analyzed through two related matrices Rand Gdeter-
mined as:
R=A0(I−A1−A0G)−1(37)
G=A2+A1G+A0G2(38)
where R= (rij )(n−1)×(n−1), the entry rij (1≤i, j ≤n−1) of matrix R
is the expected number that the QBD of network-queue visits state (l+ 1,j)
before it returns to states in N(0) ∪ · ·· ∪ N(l), given that the QBD starts in
state (l,i), and G= (gij )(n−1)×(n−1), the entry gij (1≤i, j ≤n−1) of
matrix Gis the probability that the QBD starts from state (l,i) and visits state
(l−1,j) in a finite time.
Due to the special structure of A2, which is the product of a column vector
B2by a row vector v0, matrix Gcan be calculated as
G=1v0(39)
Based on the results in [3], the expected number of packets L2of network-
queue is given by
L2=y1(I−R)−21
φ,(40)
where y1and φare determined by (26) and (27), respectively.
Finally, by applying Little’s Theorem [7], (22) follows. This finishes the
proof of Theorem 2. 
5 NUMERICAL RESULTS
To validate the QBD-based theoretical results on expected end-to-end delay
and per node throughput capacity, a customized C++ simulator has been de-
veloped to simulate packet generating, distributing and delivering processes
16

FIGURE 4
Expected packet end-to-end delay vs. number of nodes nin MANET.
in the considered MANET⋆. In the simulator, not only the i.i.d. node mobil-
ity model but also the typical random walk [13] and random waypoint [26]
mobility models have been implemented.
•Random Walk Model: At the beginning of each time slot, each node
first independently selects a cell with equal probability 1/9among its
current cell and its 8neighboring cells; it then moves into that cell and
stays in it until the end of that time slot.
•Random Waypoint Model: At the beginning of each time slot, each
node first independently generates a two-element vector [x, y ], where
both elements xand yare uniformly drawn from [1/m, 3/m]; it then
moves along the horizontal and vertical direction of distance xand y,
respectively.
5.1 End-to-End Delay Validation
For networks of different size n, Fig. 4 shows both theoretical and simulation
results on packet end-to-end delay under the settings of m= 16, system load
⋆The program of our simulator is now available online at [1]. Similar to [2], the guard-factor
is set as ∆ = 1.
17

FIGURE 5
Expected packet end-to-end delay vs. system load ρin MANET.
ρ= 0.6 (ρ=λ/µ)and packet-broadcast probability q={0.1,0.3,0.5}.
Unless otherwise mentioned, simulation results are reported with small 95%
confidence intervals. The results in Fig. 4 show clearly that in a wide range
of network scenarios considered here, theoretical results match very nicely
with simulated ones, indicating that our QBD-based theoretical modeling is
really efficient in capturing the expected packet end-to-end delay behavior of
concerned MANETs. From Fig. 4 we can also see that as network size n
increases, packet end-to-end delay increases as well. This is because that in
the concerned MANET with fixed unit area and fixed setting of m= 16, as n
increases the contention for wireless channel access becomes more intensive,
resulting in a lower packet deliveryopportunity and thus a longer packet end-
to-end delay.
For the setting of n= 150, m = 16 and q= 0.4, Fig. 5 shows both the
theoretical and simulation results on packet end-to-end delay when system
load ρchanges from ρ= 0.2to ρ= 0.9. In addition to the i.i.d. mobility
model considered in this paper, the corresponding simulation results for the
random walk and random waypoint mobility models have also been included
in Fig. 5 for comparison. Again, we can see from Fig. 5 that our theoretical
18

FIGURE 6
Per node throughput VS. packet generation rate λin MANET.
delay model is very efficient. It is interesting to see from Fig. 5 that although
our theoretical framework is developed under the i.i.d. mobility model, it can
also nicely capture the general packet end-to-end delay behavior under more
realistic random walk and random waypoint mobility models.
5.2 Throughput Capacity Validation
Another observation of Fig. 5 is that the packet end-to-end delay increases
sharply as system load ρapproaches 1.0(i.e., as packet generation rate λ
approaches per node throughput capacity µ), which serves as an intuitive
verification of our theoretical per node throughput capacity result. To fur-
ther validate our theoretical model on throughput capacity, Fig. 6 provides
the simulation results on the achievable per node throughput, i.e., the av-
erage rate of packet delivery to destination, when packet generation rate λ
increases gradually, where the results of three network scenarios with differ-
ent throughput capacity {n= 150, m = 16, q = 0.4, µ1= 2.37 ×10−4},
{n= 100, m = 16, q = 0.2, µ2= 3.46 ×10−4}and {n= 100, m = 8, q =
0.3, µ3= 7.52 ×10−4}are presented. We can see from Fig. 6 that for each
19

FIGURE 7
Expected packet end-to-end delay TeVS. 2HR-B parameter q.
network scenario there, the corresponding per node throughput first increases
monotonously as λincreases before λreaches the corresponding throughput
capacity (µ1,µ2and µ3), and then per node throughput remains a constant
and does not increase anymore when packet generation rate λgoes beyond
the corresponding theoretical throughput capacity. Thus, our theoretical ca-
pacity model is also efficient in depicting the per node throughput capacity
behavior of the considered MANET.
5.3 Performance Analysis
With the help of the QBD-based theoretical models, we explore how parame-
ter qof 2HR-B routing will affect the packet end-to-end delay Teand per node
throughput capacity µunder given m,nand ρ. The corresponding numerical
results are summarized in Figs. 7 and 8.
We first examine the impact of qon Te. For network settings of m= 16,
ρ= 0.5and n={80,300,500}, Fig. 7 shows that for a given network its
delay Tealways first decreases and then increases as qincreases. This phe-
nomenon can be explained as follows. The end-to-end delay experienced by
20

FIGURE 8
Per node throughput capacity µVS. 2HR-B parameter q.
a packet consists of the time it spends in the source-queue and the time it
spends in the network-queue. An increase in qhas two-fold effects on Te: on
one hand, it decreases the time a packet spends in the source-queue, because
source Shas more chance to do packet-broadcast for packets in its source-
queue, which makes the queue to be served more quickly; on the other hand,
it increases the time a packet spends in the network-queue, because each re-
lay has less chance to do packet-delivery to deliver a packet to destination
D, which makes the network-queue to be served more slowly. Thus, Tede-
creases as qincreases when the first effect dominates the second one, while
Teincreases as qincreases when the second effect dominates the first one.
We next explore how parameter qaffects µ. For network settings of m=
16 and n={80,300,500}, Fig. 8 shows that for a given network its capac-
ity µalways first increases and then decreases as qincreases. Notice that µis
determined by the minimum of service rates of the source-queue and network-
queue. When qis small, µis determined by the service rate of source-queue,
which increases as qincreases. When qis large, µis determined by the service
21

rate of network-queue, which decreases as qincreases. Another observation
from Fig. 8 is that the capacity µof n= 80 is the largest among different n
there. This is because that for a MANET with fixed m= 16, a larger num-
ber of nodes there will cause a more intensive wireless channel contention,
which decrease opportunities of packet transmission and thus the per node
throughput capacity.
6 CONCLUSION
The main finding of this paper is that the Quasi-Birth-and-Death (QBD) pro-
cess can be a promising theory to tackle the challenging issue of analytical
end-to-end delay modeling in MANETs. We demonstrated through a two-hop
relay MANET that QBD theory can help us: 1) to develop a novel theoretical
framework to capture the complicated network state transitions in the highly
dynamic MANET, 2) to analytically model the expected end-to-end delay and
also the per node throughput capacity of the network, and 3) to enable many
important network dynamics like node mobility, wireless channel contention,
interference and traffic contention to be jointly considered in the delay mod-
eling process. It is expected that this work will shed light on end-to-end delay
modeling in general MANETs also.
A PROOF OF LEMMA 2
Calculation of pb:The event corresponding to pbhappens iff the following
sub-events happen simultaneously:
1) Smoves into an active cell; jout of the remaining n−1nodes move
into the same cell with S,0≤j≤n−1; other nodes move into cells other
than that active cell;
2) Sbecomes transmitter after fair wireless channel contention;
3) Sselects to do packet-broadcast after traffic contention (i.e. conducting
packet-broadcast or packet-delivery).
Notice that in a time slot, every node moves according to the i.i.d. mobility
model. Thus, we have
pb=1
α2
n−1
X
j=0 n−1
j 1
m2jm2−1
m2n−1−j1
j+1 q(41)
=qm2
α2n1−m2−1
m2n(42)
22

Calculation of pc(j):According to the definition of pc(j), it is a condi-
tional probability determined as
pc(j) = pb(j)
pb(43)
where pb(j)is the probability that jcopies of a packet exist in the network
after Sbecomes transmitter and selects to do packet-broadcast for that packet,
1≤j≤n−1. The event corresponding to pb(j)happens iff the following
sub-events happen:
1) Smoves into in an active cell; j−1out of the remaining n−2nodes
other than Sand Dmove into the coverage cells of S, among which knodes
are in the same cell with Sand the remaining j−1−knodes are in other
coverage cells of S,0≤k≤j−1; other nodes move into cells other than
the coverage cells of S;
2) Sbecomes transmitter after fair channel contention;
3) Sselects to do packet-broadcast.
Notice that Dcould move either into the same cell with Sor any cell other
than that active cell, we have
pb(j) = 1
α21
m2n−2
j−1j−1
X
k=0j−1
k 1
m2k
8
m2j−1−k
·m2−9
m2n−1−j1
k+ 2q
+m2−1
m2n−2
j−1j−1
X
k=0 j−1
k 1
m2k
·8
m2j−1−km2−9
m2n−1−j1
k+ 1q(44)
=q
α2n−2
j−1m2−9
m2n−1−j
·m2−9
m2jf(j) + 1
m2jf(j+ 1)(45)
where
f(x) = 9x−8x
x(46)
After substituting (45) and (42) into (43) and conducting some basic algebraic
23

calculations, we have
pc(j) = nn−2
j−1(m2−9)n−1−j
m2n−(m2−1)nn(m2−9)f(j) + f(j+1)o(47)
Calculation of pr(j):Notice that in a time slot, Dcould only receive the
packet it is currently requesting from one of the jnodes carrying copies of
that packet. Thus, after similar arguments to the calculation of pb, we have
pr(j) = j(1−q)m2
α2n(n−1)1−m2−1
m2n
−n
m2m2−9
m2n−1
(48)
To calculate the remaining probabilities, we need to construct the arrival
process of network-queue. From Lemma 1, we know that the arrival process
of network-queue is a Bernoulli process with probability λ. The arrival pro-
cess of network-queue can be constructed as follows: once source node S
becomes transmitter and selects to do packet-broadcast (with probability pb),
Ssuccessfully conducts packet-broadcast for one packet with probability λ′,
λ′=λ
pb(49)
Thus, after Sbecomes transmitter and selects to do packet-broadcast, Swill
successfully distribute out one packet with probability λ′.
Calculation of p0(j):The event corresponding to p0(j)happens iff the
following sub-events happen:
1) Smoves into an active cell; Dmoves into any cell other than that active
cell; j−1out of the remaining n−2nodes move into the coverage cells of S,
among which knodes are in the same cell with Sand the remaining j−1−k
nodes are in other coverage cells of S,0≤k≤j−1; other nodes move into
cells other than the coverage cells of S;
2) Sbecomes transmitter after fair contention;
3) Sselects to do packet-broadcast and Sdistributes out a packet.
Then, we have
p0(j) = 1
α2
m2−9
m2n−2
j−1j−1
X
k=0j−1
k 1
m2k
8
m2j−1−k
·m2−9
m2n−1−j1
k+ 1 qλ′(50)
=λ·q·n−2
j−1(m2−9)n−j
α2m2n−2pb
f(j)(51)
24

Calculation of p0(0):From the definition of p0(0), we know that
p0(0) = 1 −
n−1
X
j=1
p0(j)(52)
After substituting (51) into (52), we have
p0(0) = 1 −λ·q·(m2−9)
α2(n−1)pb1−m2−1
m2n−1(53)
Calculation of p+
b(j):The event corresponding to p+
b(j)is composed of
j−1exclusive sub-events, each of which is that: in a time slot Sbecomes
transmitter, selects to do packet-broadcast for one packet; at the same time D
receives the packet it is requesting from a specific relay node (say R) carrying
a copy of that packet. If we denote by p+
bthe probability that one such sub-
event occurs in a time slot, then
p+
b(j) = (j−1)p+
b(54)
The event corresponding to p+
bhappens iff the following sub-events happen:
1) Smoves into an active cell; Rmoves into another active cell; Dmoves
into either the same active cell with Ror other coverage cells of R;kout of
the remaining n−3nodes move into the coverage cells of R, among which
i≥0nodes are in the same cell with R;t≥0of the remaining n−3−k
nodes are in the same active cell with S; other nodes move into cells other
than the active cell of Sand the coverage cells of R;
2) Sand Rboth become transmitters after fair contention in their respec-
tive active cells;
3) Sselects to do packet-broadcast and Sdistributes out a packet; Rselects
to do packet-delivery and Dis selected as its receiver.
25

Then, we have
p+
b=1
α2
m2−α2
m2α2
n−3
X
k=0n−3
k(k
X
i=0 k
i1
m2i
8
m2k−i
·
n−3−k
X
t=0 n−3−k
t 1
m2tm2−10
m2n−3−k−t
·1
m2
1
i+ 2
1
k+ 1 +8
m2
1
i+ 1
1
k+ 1(1−q)1
t+1 qλ′
)(55)
=λ(q−q2)(m4−m2α2)
α4n(n−1)(n−2)pb
·1−2m2−1
m2n
+m2−2
m2n
−n
m2m2−9
m2n−1
+n
m2m2−10
m2n−1(56)
From (56) and (54), (8) follows.
Calculation of p−
b(j):From the definitions of p−
b(j)and p+
b(j), we know
that p−
b(j) + p+
b(j)is the probability that in a time slot Sbecomes trans-
mitter, selects to do packet-broadcast and also successfully conducts packet-
broadcast for one packet. This probability is just λaccording to the arrival
process of network-queue. Thus, p−
b(j)can be calculated as
p−
b(j) = λ−p+
b(j)(57)
Calculation of p+
f(j):By the definition of pr(j), it is easy to see that
p+
f(j)can be calculated as
p+
f(j) = pr(j)−p+
b(j)(58)
Calculation of p−
f(j):From the definitions of p+
b(j),p−
b(j),p+
f(j)and
p−
f(j), we know that
p+
b(j) + p−
b(j) + p+
f(j) + p−
f(j) = 1 (59)
Thus, (11) follows.
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