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1 Multi-Hop Diversity C. Dong, L.-L. Yang, Senior Member, IEEE , and L. Hanzo, Fellow, IEEE School of ECS, University of Southampton, SO17 1BJ, UK E-mail: cd2g09,lly ,[email protected] n.ac.uk, http://www-mobile.e cs.soton.ac .uk/ Abstract The concept of multi-hop diversity is proposed, where all the nodes of a multi-hop link are assumed to have buffers for temporarily storing their received packets. During each time-slot, the best hop having, for example, the highest signal-to-noise ratio (SNR), is selected from the set of those hops that hav e pack ets awaitin g trans miss ion in the buffer . The packet is then transmit ted over the best hop. This hop-selection procedure yields selection diversity, but it requires the global channel knowledge of the hops’ channel quality. In this paper, we assume having perfect channel knowledge and focus our attention on the principles and performance bounds of the error probability and outage probability. We also analysis the delay time. Our studies show that relying on multiple hops has the potential of providing a signicant diversity gain, which may be exploited for enhancing the reliability of wireless multi-hop communications. Index T erms Wireless multi-hop communication, cooperative communication, diversity , relay , performance analysis. I. I NTRODUCTION In wireless multi-hop communications, source nodes send information to the correspond- ing destination nodes via intermediate relay nodes, which results in a range of advantages over conventional single-hop communicatio ns. Typically , these advantages may include an impro ved ener gy-e fc ienc y and exte nded cov erag e, impro ved link perfo rman ce, enha nced throughput, simplicity and high-exibility of network planning, etc. [1–3]. Owing to their advantages, multi-hop communications have drawn a lot of attention and have been investi- gated from different perspectives, as evidenced by numerous references, such as, [1–11]. July 22, 2011 DRAFT
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Multi-Hop Diversity

C. Dong, L.-L. Yang, Senior Member, IEEE , and L. Hanzo, Fellow, IEEE 

School of ECS, University of Southampton, SO17 1BJ, UK

E-mail: cd2g09,lly,[email protected], http://www-mobile.ecs.soton.ac.uk/ 

Abstract

The concept of multi-hop diversity is proposed, where all the nodes of a multi-hop link are

assumed to have buffers for temporarily storing their received packets. During each time-slot, the

best hop having, for example, the highest signal-to-noise ratio (SNR), is selected from the set of 

those hops that have packets awaiting transmission in the buffer. The packet is then transmitted

over the best hop. This hop-selection procedure yields selection diversity, but it requires the global

channel knowledge of the hops’ channel quality. In this paper, we assume having perfect channel

knowledge and focus our attention on the principles and performance bounds of the error probability

and outage probability. We also analysis the delay time. Our studies show that relying on multiple

hops has the potential of providing a significant diversity gain, which may be exploited for enhancing

the reliability of wireless multi-hop communications.

Index Terms

Wireless multi-hop communication, cooperative communication, diversity, relay, performance

analysis.

I. INTRODUCTION

In wireless multi-hop communications, source nodes send information to the correspond-

ing destination nodes via intermediate relay nodes, which results in a range of advantages

over conventional single-hop communications. Typically, these advantages may include an

improved energy-efficiency and extended coverage, improved link performance, enhanced

throughput, simplicity and high-flexibility of network planning, etc. [1–3]. Owing to their

advantages, multi-hop communications have drawn a lot of attention and have been investi-

gated from different perspectives, as evidenced by numerous references, such as, [1–11].

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In the context of the research of multi-hop links, it has typically been assumed that

information is transmitted from a source node to the destination node one node by one node

successively without any store-and-wait stage at the intermediate relay nodes [4, 5, 8]. For

convenience of description, we refer to this scheme as the conventional multi-hop transmission

scheme in our forthcoming discourse. In this conventional multi-hop scheme, information is

transmitted over a hop during its scheduled time-slot regardless of its link quality quantified,

for example, by its SNR. Hence, the overall reliability of a multi-hop link is dominated by

that of the weakest hop and a route outage occurs, once an outage occurs in any of the

invoked hops. As a result, the route error/outage performance of a multi-hop link usually

degrades, as the number of hops increases. In order to improve the performance of multi-hop

links, recently, novel signaling schemes have been proposed [3, 12, 13], which require the

nodes to have a store-and-wait capability. For example, in [12, 13], adaptive modulation and

coding (AMC) combined with automatic repeat request (ARQ) schemes has been invoked in

cooperative decode-and-forward (DF) communications. Very recently, the authors in [3] have

employed AMC for dual-hop cooperative communications relying on a regenerative relay

node, where the AMC mode of both the hops may be configured independently.

In this contribution, we view the independently fading multiple hops of links as an adap-

tively configurable resource that may be exploited for achieving a diversity gain. To the best

of our knowledge, the multi-hop diversity concept, which exploits the independent fading of 

communication hops for attaining diversity, has never been investigated in the open literature.

The multi-hop diversity is achieved by assuming that every node of a multi-hop link has a

buffer for temporarily storing the packets received. During a given time-slot, the highest-

quality hop is activated from the set of hops having packets in their buffers to send, which

hence results in selection diversity. Intuitively, the implementation of the proposed multi-hop

diversity scheme requires global channel knowledge about all the hops. In this paper, however,

we focus our attention on the basic principles and theoretical performance bounds under

the idealized simplifying assumption that this global channel knowledge may be acquired,

whenever required. Specifically, we analyze the error and outage performance of multi-hop

links, when either buffers of limited or unlimited size are used. Furthermore, simulations are

employed for study and also for verifying the accuracy of our analytical results. Note that,

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the terminology of multi-hop diversity has also been used in [14, 15]. However, the multi-hop

diversity considered in [14, 15] and that defined in this paper have different meaning. In [14,

15], it is assumed that a receiving node can receive signals from several other nodes. In this

case, the multi-hop diversity may be obtained at the receiving node by combining the signals

received from the different nodes transmitted the same information.

Our studies and performance results in this paper demonstrate that independently fading

multiple hops have the potential of providing significant diversity gain for improving the

reliability of communications. The error/outage performance may be improved, as the buffer

size increases. However, the maximum attainable multi-hop diversity may be approached,

provided that each of the nodes has moderate buffer size.

Naturally, once a source node completes its transmission, an increased buffering-induced

delay is imposed, which is higher than that of the conventional multi-hop transmission

scheme [16]. We analysis the probability density function(PDF) of delay time. Although

the average delay time for one packet is larger than before, the delay time dose not increased

when transmission sufficiently large amounts of data. This is because the multi-hop diversity

scheme transmits one packet over one hop per time-slot, identically to the conventional multi-

hop transmission scheme.

The remainder of this paper is organized as follows. The next section presents the system

model of multi-hop links. Section III analyzes the error probabilities, outage probabilities

and delay time. In Section IV, we provide the numerical and simulation results. Finally,

conclusions are summarized in Section V.

I I . SYSTEM MODEL OF MULTI-H OP LINKS

The system model under consideration is a typical multi-hop wireless link [8, 16], which

is shown in Fig. 1. The multi-hop link consists of  (L + 1) nodes, a source node S  (node

0), (L − 1) relay nodes R1, R2, · · · , RL−1 and a destination node D (node L). The distance

from S  to D is DSD . We assume that every relay node distribute equally which means the

distance between two neighbour node is d = DSD

L. The source node S  sends information to

the destination node D via L hops with the aid of the (L − 1) relay nodes. At the relay

nodes, the classic decode-and-forward (DF) protocol is employed for relaying the signals.

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For convenience, we denote the signal transmitted by the source node S  of node 0 by x0

and its estimate at the destination node D of node L by xL = x̂0, while the signal estimated

at the lth relay node by xl, l = 1, . . . , L − 1. When operated at packet level, they are

correspondingly represented by xxx0, xxxl, l = 1, . . . , L − 1, and xxxL = x̂xx0. In this paper, we

assume that the signals are transmitted on the basis of time-slots having a duration of  T 

seconds. The channels of the L hops are assumed to experience block-based flat Rayleigh

fading, where the complex-valued fading envelop of a hop remains constant within a time-

slot but is independently faded for different time-slots. The pathloss is directly proportional

to d−α, where α is from 2 to 6. We assume that total transmission power from S  to D is

1, no matter how many hop. The transmission power for every node is 1L

when hop number

is L. Based on the above assumptions, when the (l − 1)st node transmits a packet xxxl−1, the

observations received by node l can be expressed as

yyyl = hlxxxl−1d−αC + nnnl, l = 1, 2, . . . , L (1)

where hl represents the channel gain of the lth hop from node (L − 1) to node L, while

nnnl is the Gaussian noise added at node l. The channel gain hl is complex Gaussian with

zero mean and E [|hl|2] = 1. The transmission power is E [|xl|

2] = 1L

. The noise samples in

nnnl, l = 1, . . . , L , obey the complex Gaussian distribution with zero mean and a common

variance of  σ2 = 1/(2γ SD) per dimension, where γ SD denotes the received average SNR

when hop number is 1. So E [|n|2] is same for every node, no matter how many hop. And C 

is a constant factor when L is defined. It is obvious that when hop number is L, the received

average SNR will enlarge Lα−1 times.

In this paper, we focus on the principles of multi-hop diversity as well as on its best achiev-

able error and outage performance. For this reason, some idealized simplifying assumptionsare adopted, which are summarized as follows:

• Binary phase-shift keying (BPSK) basedband modulation is assumed for signal trans-

mission. Hence, we have xxxl ∈ {+1, −1}.

• The source node always has packets to send, hence the multi-hop link operates in its

steady state.

• The source node S  and destination D can store an infinite number of packets, respec-

tively. By contrast, each of the (L − 1) relay nodes can only store at most B packets.

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• The fading processes of the L hops of the multi-hop link are independent, while the

fading of a given hop remains constant within a packet duration, but it is independently

faded from one packet to another.

• The distance between S  to D is 1. Factor C  is 1.

• The total transmission power for all node is 1.

• There is a central control unit (CCU), which evaluates and exploits the global knowledge

about the channels of the L hops. Based on the global channel knowledge of the L hops

within a given time-slot, the CCU decides which of the L nodes, i.e., nodes 0, 1, . . . , L−

1, transmits and informs the corresponding receive node without a delay and without

errors.

• A receive node employs ideal channel state information (CSI) for carrying out coherent

detection.

• All the L hops follow the same block-based flat Rayleigh fading.

Under the above assumptions, packets are transmitted over the multi-hop link based on

the following strategy. Among those hops having at least one packet stored in the buffer

awaiting transmission, the CCU first decides which is the most reliable hop according to

the instantaneous SNR values. Then, one packet is transmitted over the most reliable hop

using a time-slot. According to this strategy, packets are transmitted obeying the time-division

principles and hence transmitting a packet from the source node S  to the destination node

D requires L time-slots.

Below we analyze the lower bounds for the BER and outage probability of the multi-hop

link under the above-mentioned assumptions in Sections III-A and III-B, while the accurate

BER and outage probability are analysed in Sections III-C and III-D. Finally, we analyse the

delay time in III-E.

III. PERFORMANCE ANALYSIS

In this section, we first derive the lower-bounds for the BER and outage probability of the

multi-hop link shown in Fig. 1. Then, the accurate BER and outage probability are analyzed.

Finally, delay time will be analysed.

Based on Fig. 1 and on the operational principles of the multi-hop link, as described in

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Section II, we can now infer that, when every relay node has a buffer of size of  B packets,

the following events may occur. Firstly, the buffer of a relay node may be empty at some

instants. In this case, this relay node cannot be the transmit node, since it has no data to

transmit. Secondly, the buffer of a relay node may be full at some instants. Then, This relay

node cannot be the receiving node, since it cannot accept further packets. In these cases the

CCU has to choose a hop for transmission from a reduced number of hops, which results

in an increased BER and outage probability due to the reduced selection diversity gain.

Therefore, our lower-bounds of BER and outage probability are derived by loosening the

above-mentioned constraints and assuming that each relay node has an unlimited buffer size

and that a node always has packets to transmit, whenever it is instructed by the CCU to

transmit. By contrast, for accurate analysis of the BER and outage probability, the above-

mentioned constraints will be considered in Sections III-C and III-D. Finally, delay time is

analysed in III-E.

 A. Lower-Bound Bit Error Rate

In order to derive the lower-bound BER, we first derive the single-hop BER, P L,e, under the

assumptions that every relay node has an infinite buffer and that a node always has packets

prepared to send. Then, the lower-bound of the end-to-end BER, P L,E , of the multi-hop link 

shown in Fig. 1 is derived. The subscript ‘L’ in P L,e and P L,E  stands for the lower-bound.

Considering a time-slot, when the SNR is given by γ , the conditional BER of BPSK

modulation is given by [17]

P L,e(γ ) = Q 

2γ 

(2)

where Q(x) is the Gaussian Q-function, which can alternatively be defined [18] by Q(x) =

π−1 π/20

exp(−x2/(2 sin2 θ))dθ. According to the operational principles of the multi-hop link 

as described in Section II, the SNR γ  in (2) is given by

γ  = max{γ 1, γ 2, . . . , γ  L} (3)

where γ l is the SNR of the lth hop within the time-slot considered, which is given by

γ l = |hl|2γ h, l = 1, 2, . . . , L (4)

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where, γ h is average received SNR. For L hop, γ h = γ SDLα−1. We can readily show that γ l

has the probability density function f (γ l) = γ −1h e−γ l/γ h, l = 1, . . . , L. Furthermore, the PDF

of  γ  defined in (3) can be derived as [18]

f (γ ) = ddγ 

 γ 

0

f (γ l)dγ lL

=L

γ hexp

γ 

γ h

1 − exp

γ 

γ h

L−1(5)

The average single-hop BER P L,e can be obtained by averaging P L,e(γ ) of (2) with respect

to the PDF of (5), yielding

P L,e =  ∞

0

P L,e(γ )f (γ )dγ  (6)

Upon substituting both (2) associated with the Q-function and (5) into the above equation,

as well as carrying out some further simplifications, we arrive at

P L,e =LL−1l=0

(−1)l

l + 1

L − 1

l

1

π

 π/20

sin2 θ

γ h/(l + 1) + sin2 θdθ (7)

Finally, after completing the integration with the aid of (5A.11) in [18], the single-hop lower-

bound BER is given by

P L,e =L2

L−1l=0

(−1)l

l + 1

L − 1l

1 − 

γ hl + 1 + γ h

=1

2

Ll=0

L

l

(−1)l

 γ h

l + γ h(8)

Having obtained the single-hop lower-bound BER P L,e, the lower-bound end-to-end BER

P L,E  can be obtained from [19]. It is

P L,E  =1

2−

1

2(1 − 2P L,e)L (9)

=1

2−

Ln=0

(−1)n2n−1P nL,e (10)

where the single-hop lower-bound BER P L,e is given by (8).

From (9), we can immediately infer that

P L,E  ≈LP L,e (11)

when P L,e is sufficiently small. Since P L,e is the BER of a typical selection combining (SC)

diversity scheme associated with Lth-order diversity, (8) implies that, provided that each of 

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the relay nodes has some buffer, a multi-hop link is capable of attaining L diversity order,

in comparison to the classic multi-hop link, which does not rely on any buffering at the

relay nodes. Note that, for relay nodes operating without buffers, we can readily show that

the BER or the approximate BER of an L-hop link can also be expressed by (9) or (11)

upon replacing P L,e by P e = (1 − 

γ h/(1 + γ h))/2, which is the BER of a BPSK scheme

communicating over Rayleigh fading channels [17].

  B. Lower-Bound Outage Probability

The outage probability is the probability of the event that the maximal SNR of the L hops

is lower than a pre-set threshold. When this event occurs, either no data is transmitted on

the multi-hop link in order to guarantee the minimum required BER, or the BER becomes

higher than a predicted value, if data is still transmitted. Given a threshold γ T , the lower-

bound outage probability is given by

P L,O =

 γ T 0

f (γ )dγ  (12)

When substituting (5) into this equation, we can readily obtain

P L,O =

1 − exp

−γ T γ hL

=

Ll=0

(−1)lL

l

exp

−lγ T γ h

(13)

which is simply the probability that each of the L hops has an SNR lower than γ T .

In contrast to the above multi-hop diversity scheme, an outage occurs in the conventional

L-hop transmission scheme, when one out of the L hops has an SNR below the threshold

γ T . Therefore, the outage probability can be expressed as

P O = 1 − [P (γ l > γ T )]L = 1 −  ∞

γ T 

f (γ l)dγ lL

(14)

Applying the PDF of  f (γ l) into this equation yields

P O = 1 − exp

Lγ T γ h

(15)

Furthermore, it can readily be shown that we have

limγ h→∞

log(P L,O)

log(P O)= L (16)

which means that, if the SNR γ h per hop is high, the outage probability of the L-hop diversity

scheme decreases L times faster than that of the conventional L-hop transmission scheme.

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This property also explains that our proposed transmission scheme is capable of achieving

an Lth-order diversity.

In summary, above we have derived the lower-bound BER and the outage probability of the

multi-hop link, where relay nodes employ buffers for storing packets. It can be shown that an

Lth-order diversity may be attainable by an L-hop link over an L-hop link communicating

based on the conventional multi-hop transmission scheme. Below, we derive the accurate

expressions of BER and outage probability for L-hop links by taking into account the

scenarios, when some of the buffers are either empty or full.

C. Accurate Bit Error Rate

First, the L-hop link is forced to choose the best one from the set of  m hops in order

to send information, when (L − m) out of the L hops do not have information to transmit.

This happens either when some of the transmit nodes’ buffers are empty or when some of 

the receive nodes’ buffers are full. In this case, based on (8), the BER is given by

P e(m) =1

2

ml=0

(−1)l

m

l

 γ h

l + γ h, m = 1, . . . , L (17)

Let us express P m

, m = 1, . . . , L , the probability of the event that only m out of the L hops

can transmit. Then, the average BER of one hop in L-hop link can be formulated as

P e =L

m=1

P mP e(m) (18)

Hence, what we need for evaluation of  P e is first of all the probabilities {P m}, which can

be derived by treating the packet transmissions over the L-hop link as a Markov process.

Let us assume that the buffer size of every relay node is B packets. Let the number of 

packets that the relays R1, R2, . . . , RL−1 hold be b1, b2, . . . , bL−1, where bl = 0, 1, . . . , B.

Then, the states of the L-hop link can be defined in terms of the number of packets stored

in the buffers of the (L − 1) relay nodes as

S i =

b(i)1 , b

(i)2 , · · · , b

(i)L−1

T , i = 0, 1, . . . , N  − 1 (19)

where b(i)l denotes the number of packets held by the lth relay, when the L-hop link is

at state i, N  = (B + 1)L−1 is the total number of states, which are collected into a set

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S  = {S 0, S 1, . . . , S  N −1}. For example, let us consider a three-hop link having the parameters

L = 3 and B = 2. Then, there are in total N  = 32 = 9 states, which form a set

S  =S 0 = [0, 0]T , S 1 = [0, 1]T , S 2 = [0, 2]T , S 3 = [1, 0]T , S 4 = [1, 1]T ,

S 5 = [1, 2]T , S 6 = [2, 0]T , S 7 = [2, 1]T , S 8 = [2, 2]T 

From the N  states, a state transition matrix denoted by T T T  can be populated by the state

transition probabilities {P ij = P (s(t + 1) = S  j |s(t) = S i), i , j = 0, 1, . . . , N  − 1}.

Specifically, the state transition matrix of the above example is given by

T T T  =

0 0 0 1 0 0 0 0 0

1/2 0 0 0 1/2 0 0 0 0

0 1/2 0 0 0 1/2 0 0 0

0 1/2 0 0 0 0 1/2 0 0

0 0 1/3 1/3 0 0 0 1/3 0

0 0 0 0 1/2 0 0 0 1/2

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1/2 1/2 0 0

0 0 0 0 0 0 0 1 0

(20)

From this example, we can see that, although the transition matrix T T T  may be large, it can,

however, be formed easily. A possible algorithm derived for forming the matrix T T T  can be

formulated as follows:

1) The (B + 1)L−1 × (B + 1)L−1 matrix T T T  is first initialized with zero elements.

2) For row i, i = 0, 1, . . . , (B + 1)L−1 − 1, which corresponds to the ith state S i =

b(i)1 , b

(i)2 , · · · , b

(i)L−1

, the following operations are executed:

• If b(i)1 +1 ≤ B, the column corresponding to the output state

b(i)1 + 1, b(i)2 , · · · , b(i)L−1

T is set to one;

• For l = 1, 2, . . . , L − 2, if  b(i)l − 1 ≥ 0 and b(i)l+1 + 1 ≤ B, the column corresponding

to the output state

b(i)1 , . . . , b

(i)l − 1, bil+1 + 1, · · · , b

(i)L−1

T is set to one;

• If b(i)L−1−1 ≥ 0, the column corresponding to the output state

b(i)1 , b(i)2 , · · · , b(i)L−1 − 1

T is set to one.

3) Each of the rows is divided by the number of ones in the row.

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The state transition matrix T T T  has the following properties:

• Matrix T T T  is a sparse matrix. Every row has at most L number of nonzero elements,

while the other at least (B + 1)L−1 − L elements are zero elements;

• The sum of the probabilities in each row is one;

• The number of nonzero elements in a row represents the number of hops that may be

chosen by the CCU for transmission.

Having obtained the state transition matrix T T T , the steady-state probabilities can be com-

puted by the formula [?]

πππ = T T T T πππ (21)

where πππ =

π0, π1, . . . , π(B+1)L−1−1

T , and πi is the steady-state probability that the L-

hop link is at state S i. Equation (21) shows that πππ is the right eigenvector of matrix T T T T 

corresponding to an eigenvalue one. Therefore, πππ can be derived with the aid of classic

methods derived for solving the eigenvector problem [20].

From πππ, we can compute the probability P m, m = 1, 2, . . . , L , by adding those entries in

πππ, which correspond to the specific rows of  T T T  that have m nonzero entries. Finally, given

{P m}, the accurate (or steady-state) BER of the L-hop link can be computed using (18).

  D. Accurate Outage Probability

From the derivation of (13), we can infer that, when m out of the L-hops are available

for transmission, the conditional outage probability is given by

P O(m) =

1 − exp

γ T γ h

m, m = 1, 2, . . . , L (22)

The accurate (or steady-state) outage probability of the L-hop link can be expressed as

P O =L

m=1

P mP O(m) (23)

where {P m} are the probabilities derived in Section III-C.

  E. Delay Probability Density Function

Delay time is the packet transmission time from the source node to destination node. In

conventional communication method, the packet is transmitted from one node to another node

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via intermediate node without stay. Thus, the delay time of a packet is L time slot for L hop

communication. In multi-hop diversity communication, we get lower BER and lower Outage

probability. But we pay delay time, because the packet may stay at intermediate node. There

are two type of delay time. For one packet or for a large amount packets. For a large amount

packet, the average delay time is L. Now, we try to make sure the minimum delay time, the

maximum delay time, and the PDF of delay time for one packet.

The minimum delay for L hop is L time slot.

delaymin = L (24)

In this case, the test packet is transmitted via all relay node without any stay. The maximum

delay time is

delaymax =L−1i=1

B(L − i) +L−1i=1

Bi − L + 2 = L2B − BL − L + 2. (25)

It happens at the worst case. At that case, after source transmits a packet to relay 1, which

costs 1 time slot, all buffer in relay node are full. To transmit all packet in all buffer except

test packet to destination, it costsL−1

i=1 B(L − i) − L + 1 time slot. That’s because that to

transmit one packet from relay i to destination costs (L−1) time slot. And then, the test packet

is transmitted from relay 1 to relay L − 1 which costs L − 2 time slot. At that time, source

transmits packets which make all relay node full. This deployment costsL−1

i=1 Bi − L + 1

time slot. Finally, the test packet is transmitted from relay L − 1 to destination. So totally,

the transmission time is 1 +L−1

i=1 B(L − i) − L + 1 + L − 2 +L−1

i=1 Bi − L + 1 + 1 which

is (25). The process is shown in Fig. 2.

Now, let’s try to find the PDF of delay time. First, we define some matrix. In Fig. 2-(b),

if there are not B packet in every node but bi packet in relay node i, we write it as b(b)

i .

The subscript ’(b)’ stands for before test packet. If system transmitted all packet except test

packet to destination, it costsL−1

i=1 b(b)i (L − i) − L + 1 time slot. We name this time as ’clear

time’. In Fig. 2-(e), if there are not B packet in every node but bi packet in relay node i,

we write it as b(a)i . The subscript ’(a)’ stands for after test packet. From Fig. 2-(d) to every

node has b(a)i packet, it costsL−1

i=1 b(a)i i − L + 1 time slot. We name this time as ’deploy

time’. Before the test packet came to system, as 2-(a), the state of system is statei. After the

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test packet received by destination node, as 2-(f), the state of system is state j. The distance

time from statei to state j is ’clear time’ plus ’L’ and ’deploy time’.

distance(i, j) =L−1

i=1

b(b)i (L − i) − L + 1 +

L−1

i=1

b(a)i i − L + 1 + L (26)

=L−1i=1

b(b)i (L − i) +L−1i=1

b(a)i i − L + 2 (27)

Now we can make a distance time matrix between every state. For example, when hop number

is 3 and buffer size is 2, the distance matrix DisDisDis is

DisDisDis =

state i/j 00 10 20 01 11 21 02 12 22

00 3 4 5 5 6 7 7 8 9

10 5 6 7 7 8 9 9 10 11

20 7 8 9 9 10 11 11 12 13

01 4 5 6 6 7 8 8 9 10

11 6 7 8 8 9 10 10 11 12

21 8 9 10 10 11 12 12 13 14

02 5 6 7 7 8 9 9 10 11

12 7 8 9 9 10 11 11 12 1322 9 10 11 11 12 13 13 14 15

(28)

and we define matrix nnn in which all element is n and the size is same as DisDisDis. As mentioned

before, we can calculate the probability of every state through (21). Now we make a new

matrix π̄ππ, and put all elements into diagonal of  π̄ππ. For example, when hop number is 3 and

buffer size is 2, the probability of every state are P (00) = 239

, P (10) = 539

, P (20) = 539

,

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P (01) = 439

, P (11) = 939

, P (21) = 539

, P (02) = 339

, P (12) = 439

, P (22) = 239

. So π̄ππ is

π̄ππ =1

39

2 0 0 0 0 0 0 0 0

0 5 0 0 0 0 0 0 0

0 0 5 0 0 0 0 0 0

0 0 0 4 0 0 0 0 0

0 0 0 0 9 0 0 0 0

0 0 0 0 0 5 0 0 0

0 0 0 0 0 0 3 0 0

0 0 0 0 0 0 0 4 0

0 0 0 0 0 0 0 0 2

(29)

In(21), the element is the transition probability from one state to another state when choose

different channel. If we only choose the first channel, from source to Relay 1, we can write

another markov transition matrix T T T (1). The subscript ’(1)’ stands for we choose the first

channel. Similarly, if we choose the last channel, we can write another markov transition

matrix T T T (L). The subscript ’(L)’ stands for we choose the last channel. If the delay time is

ddd,

P (d)P (d)P (d) = π̄ππT T T (1)LT T T d−2(DisDisDis (d − 2d − 2d − 2))T T T (L) (30)

where is Hadamard product. T T T (1) lets the test packet into system. Because we choose 1st

channel among L channels, we should multiple L here to make sure entry probability is 1.

During time slot 2 to d − 1, test packet moves from relay 1 to relayL − 1. DisDisDis (d − 2d − 2d − 2)

means we only count the distance from start state to current state is d − 2. Finally, T T T (L)

means the test packet move to destination.

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F. Diversity Order 

In this subsection, let us derive diversity order from (9) and (14).

limγ h→∞

1

2

L

l=0

L

l(−1)l 1 −

l

l + γ h(31)

≈1

2

Ll=0

L

l

(−1)l(1 +

1

2

−l

γ h+

∞k=2

(2k − 3)!!

(2k)!!

(−l)k

γ kh) (32)

=1

2

Ll=0

L

l

(−1)l(

∞k=0

ck(l

γ h)k) (33)

=1

2

Ll=0

L

l

(−1)l(

Lk=0

ck(l

γ h)k) (34)

where ck is coefficient [21](1.112.3). We ignore k > L because they are too small. Now we

proof the kth term, k ≤ L is zero. First, it is obvious:

Ll=0

L

l

(−1)l = 0 (35)

When k = k + 1 and L > k,

Ll=0

L

l

(−1)llk (36)

= −LL−1l=0

L − 1

l

(−1)l(l + 1)k−1 = 0 (37)

So, (31) is

1

2

Ll=0

L

l

(−1)l(cL(

l

γ h)L) (38)

=cL

2γ LhL!(−1)L (39)

So the BER diversity order in L hop case is L. If write (9) as series, it is not difficult to findthe outage diversity order of L hop is L base on (36) and (38).

From (18) and (23) we can find the diversity order for infinity buffer L hops is 1. That is

because when SNR is very high, the dominate part is m = 1 because it has the highest BER

or outage probability.

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IV. PERFORMANCE RESULTS

In this section, we provide numerical and/or simulation results for both the BER and

outage probability of multi-hop links, in order to illustrate the effect of buffer size of relay

nodes on the achievable multi-hop diversity gain. Note that the lower bound end-to-end BER

and lower-bound outage probability of multi-hop links may be conveniently evaluated from

the formulas of (10) and (13), respectively. However, since the state transition matrix T T T  has

(B + 1)L−1 × (B + 1)L−1 elements, the evaluation of the accurate end-to-end BER and the

accurate outage probability becomes extremely hard even for a moderate value of  B, when

the scenarios of  L = 4 and a high number of hops are considered.

The first set of results characterizes the relationship between the end-to-end BER perfor-

mance and the buffer size of relay nodes. The results are plotted versus average SNR per

bit, which is actually the average SNR per bit for all hop for BPSK. In Figs. 3, 4 and 5

multi-hop scenarios having two, three and four hops are characterized, respectively. In all

these figures, the corresponding lower-bound end-to-end BER is provided. Furthermore, for

comparison, the corresponding end-to-end BER of the conventional multi-hop transmission

scheme [8, 16] is also shown in Figs. 3, 4 and 5. Note that, for the two-hop link considered in

Fig. 3, the end-to-end BER of the conventional scheme is the same as that of the multi-hop

diversity scheme marked with ‘B = 1’. From the results shown in these figures, we may

draw the following observations. Firstly, multi-hop diversity is attainable by equipping each

of the relay nodes (also the source node that was assumed to have an infinite buffer.) with a

buffer. As expected, the diversity gain improves, as the buffer size of relay nodes increases,

until reaching the lower-bound end-to-end BER corresponding to infinite buffers. Secondly,

the lower-bound end-to-end BER may be approached by employing buffers of reasonable

size, which depends on the actual SNR. Typically, for the two-, three- and four-hop links

considered, using the buffers of size B = 32 or 128 packets is capable of achieving most of 

the multi-hop diversity available. Thirdly, even when the relay nodes employ buffers of small

size, the achievable multi-hop diversity may still be significant. Furthermore, for the three-

and four-hop links, multi-hop diversity is attainable, even when each of the buffers can only

store a single packet, i.e., for B = 1, as seen in Figs. 4 and 5. In Fig. 4, when B = 1, the

multi-hop diversity scheme proposed in this paper is capable of yielding an approximately

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4 dB performance gain in comparison to the conventional multi-hop transmission scheme.

Observe in Fig. 5, for B = 1, that the multi-hop diversity scheme may obtain in excess of 

4 dB of SNR gain over the conventional multi-hop transmission scheme.

The reason for the multi-hop diversity scheme’s ability to significantly outperform the con-

ventional scheme is explicit. In the multi-hop diversity scheme proposed in this paper, the end-

to-end BER performance improves as the number of hops increases owing to the increased

multi-hop diversity. By contrast, in the conventional multi-hop transmission scheme [8, 16],

the end-to-end BER performance degrades as the number of hops increases, since the packets

are transmitted from one node to another successively.

The second set of results characterizes the outage probability performance of multi-hop

links, as shown in Figs. 6, 7 and 8 for the two-, four- and eight-hop links, respectively,

when relay nodes having various buffer size are considered. Note that, in our numerical

computations and simulations, the threshold γ T  was adjusted to maintain a BER of  0.01 for

a single-hop link. The corresponding lower-bound outage probability evaluated using (13) for

the two-, four- and eight-hop links, respectively, are shown in Figs. 6, 7 and 8, while in Fig. 6,

the accurate outage probability evaluated by (23) is depicted. Furthermore, the corresponding

outage probability of the conventional multi-hop links is also provided for comparison. From

these results, we can draw similar observations, as those drawn from Figs. 3, 4 and 5. A

significant multi-hop diversity gain is attainable, when the relay nodes employ buffers of a

sufficiently high size. Hence, the multi-hop diversity transmission scheme proposed in this

contribution outperforms the conventional multi-hop transmission scheme [8, 16].

In fig 10, we show the delay PDF from simulation and 30. For convenience, we put

different hop number and buffer cases in one fig and all of them have the maximum delay

time around 100. Certainly, the delay time will increase with larger L and B, but we can

trade-off delay time and BER/Outage probability.

In fig ??, we show the diversity order. The diversity order is L for L hop with infinity

buffer. For finite buffer, the diversity order is 1.

V. CONCLUSIONS

In this contribution, we have proposed and investigated a multi-hop diversity scheme.

The BER and outage probability of multi-hop links have been analyzed and a range of 

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formulas have been obtained, when assuming that BPSK signals are transmitted over all the

hops, which experience iid Rayleigh fading. Our analysis and performance results show

that exploiting the resource of multiple hops results in a significant diversity gain. The

multi-hop diversity scheme significantly outperforms the conventional multi-hop transmission

arrangement in terms of the BER/outage performance, when sufficiently large buffers are

considered. Furthermore, the maximum multi-hop diversity can usually be approached, when

each relay node has a buffer of moderate size. After that, the delay PDF is given.

Our future research will consider the multi-hop diversity gain of various modulation

schemes under practical constraints of realistic channels. Furthermore, the delay characteris-

tics and other related statistics of multi-hop diversity schemes will also be addressed.

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Fig. 1. System model for a multi-hop wireless link, where source S  sends message to destination D via (L−1) intermediate

relays.

Fig. 2. The maximum delay time case. (a) All relay nodes are full and Relay 1 stores B−1 packet. (b) Source transmits test

packet to Relay 1. It costs 1 time slot. (c) All packet before test packet transmit to Destination. It costsP

L−1

i=1B(L−i)−L+1

time slot. (d) The test packet is transmitted from Relay 1 to Relay L− 1. It costs L− 2 time slot. (e) All packet after test

packet deploy into relay nodes. After that, all node are full. It costsP

L−1

i=1Bi−L+ 1 time slot. (f) Finally, the test packet

is transmitted from relay L− 1 to Destination node. Total it costs L2B − LB − L + 2 time slot as mentioned in (25).

Fig. 3. BER versus average SNR per hop performance of two-hop links with relays having various buffer size, when

communicating over Rayleigh fading channels.

Fig. 4. BER versus average SNR per hop performance of four-hop links with relays having various buffer size, when

communicating over Rayleigh fading channels.

Fig. 5. BER versus average SNR per hop performance of eight-hop links with relays having various buffer size, when

communicating over Rayleigh fading channels.

Fig. 6. Outage probability versus average SNR per hop performance of two-hop links with relays having various buffer

size, when communicating over Rayleigh fading channels.

Fig. 7. Outage probability versus average SNR per hop performance of three-hop links with relays having various buffer

size, when communicating over Rayleigh fading channels.

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Fig. 8. Outage probability versus average SNR per hop performance of four-hop links with relays having various buffer

size, when communicating over Rayleigh fading channels.

Fig. 9. The delay PDF of L=2,3,4,8. Because the maximum delay time is L2B−BL−L+ 2, we choose a certain buffer

size to make maximum delay time is around 100 for different hop number.

Fig. 10. This case shows the diversity order. The diversity order is L for L hop with infinity buffer. For finite buffer, the

diversity order is 1.

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