Analytical Modeling of Multi-hop IEEE 802.15.4 Networks

Analytical Modeling of Multi-hop IEEE 802.15.4 Networks

Piergiuseppe Di Marco, Student Member, IEEE, Pangun Park, Member, IEEE, Carlo Fischione, Member, IEEE, and Karl Henrik Johansson, Senior Member, IEEE

Abstract—Many of existing analytical studies of the IEEE 802.15.4 medium access control (MAC) protocol are not adequate because they are often based on assumptions such as homogeneous traffic and ideal carrier sensing, which are far from reality for multi-hop networks, particularly in the presence of mobility. In this paper, a new generalized analysis of the unslotted IEEE 802.15.4 MAC is presented. The analysis considers the effects induced by heterogeneous traffic due to multi-hop routing and different traffic generation patterns among the nodes of the net- work and the hidden terminals due to reduced carrier-sensing capabilities. The complex relation between MAC and routing protocols is modeled, and novel results on this interaction are de- rived. For various network configurations, conditions under which routing decisions based on packet loss probability or delay lead to an unbalanced distribution of the traffic load across multi-hop paths are studied. It is shown that these routing decisions tend to direct traffic toward nodes with high packet generation rates, with potential catastrophic effects for the node’s energy consumption. It is concluded that heterogeneous traffic and limited carrier-sensing range play an essential role on the performance and that routing should account for the presence of dominant nodes to balance the traffic distribution across the network.

Index Terms—Hidden terminals, IEEE 802.15.4, medium access control (MAC), multi-hop.

I. INTRODUCTION

I

Manuscript received July 7, 2011; revised December 18, 2011 and April 16, 2012; accepted April 25, 2012. Date of publication May 30, 2012; date of current version September 11, 2012. This work was supported in part by the Swedish Foundation for Strategic Research; by the Swedish Research Council; by the Swedish Governmental Agency for Innovation Systems; and by the European Union Project FeedNetBack, Project Hycon2, and Project Hydrobionets. The review of this paper was coordinated by Dr. H. Jiang.

P. Di Marco, C. Fischione, and K. H. Johansson are with the School of Electrical Engineering, Royal Institute of Technology, 100 44 Stockholm, Sweden (e-mail: pidm@kth.se; carlofi@kth.se; kallej@kth.se).

P. Park was with the School of Electrical Engineering, Royal Institute of Technology, 100 44 Stockholm, Sweden. He is now with the University of California, Berkeley, CA 94720 USA (e-mail: pgpark@kth.se).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2012.2201221

While the performance of single-hop IEEE 802.15.4 star net- works has been thoroughly investigated, there is not yet a clear understanding of the performance over multi-hop networks. For single-hop scenarios, many papers in the literature have pro- posed models for capturing the behavior of the IEEE 802.15.4 MAC, with saturated or unsaturated traffic, acknowledgements, and retransmissions [3]–[8]. The model presented in [5] is also validated through real test-bed experiments in [9]. These studies are based on extensions of the Markov chain model originally proposed by Bianchi for the IEEE 802.11 MAC protocol [10]. Based on these analytical models, the perfor- mance of the protocol can be improved by opportunely tun- ing the MAC parameters [9] or the duty cycle of the nodes [11]. In [12], the basic assumptions of the aforementioned Markov models are discussed, and the range of application is determined.

However, in all the proposed contributions, traffic is assumed to be homogeneous from node to node, in both saturated and unsaturated scenarios. This assumption is a major limitation in at least three important situations.

1) In single-hop networks, nodes may have different traffic generation rates as a result of the different services they provide, such as control applications with varying sam- pling rates.

2) In multi-hop networks, the traffic load varies according to the routing along the paths. Some nodes may experience heavier cross traffic, thus transmitting more packets than nodes that are traversed by fewer routing paths. It follows that the traffic is not homogeneous, regardless that the nodes generate their own packets at the same rate.

3) In networks with hidden terminals, the traffic sensed by the nodes is different from node to node, even when every node generates the same traffic. This is because some nodes may not perceive the ongoing transmissions of other nodes.

In the situations previously mentioned, which we believe are the most interesting and common, existing analytical studies of the IEEE 802.15.4 MAC are not adequate. In this paper, we attempt to overcome these limitations and propose a compre- hensive analysis.

The remainder of this paper is organized as follows: In Section II, the related work and the original contribution are summarized. In Section III, we introduce the system model. In Section IV, we derive an analytical model of IEEE 802.15.4 MAC for multi-hop networks with heterogeneous traffic and hidden terminals. The accuracy of the model is evaluated by

0018-9545/$31.00 © 2012 IEEE

Monte Carlo simulations in Section V. Section VI concludes this paper and prospects our future work.

II. RELATEDWORKS ANDORIGINALCONTRIBUTION

The single-hop homogeneous models in [3]–[8] have been recently extended to cover some of the aspects of multi-hop heterogeneous networks. However, a comprehensive approach is still missing. Studies for single-hop IEEE 802.11 and IEEE 802.15.4 with heterogeneous traffic can be found in [13]–[15], where traffic classes are considered, but no hidden terminals.

The effects of hidden terminals in homogeneous single-hop networks have been studied in [16] and [17]. In [18], multi-hop communication is modeled for IEEE 802.11 networks under single traffic flow. In [19], the work in [18] has been extended to multiple nonsaturated flows. In [20], a model for saturated traffic flows in IEEE 802.11 networks is presented. However, we note that these models cannot be directly applied to IEEE 802.15.4 networks due to the different access mechanisms of IEEE 802.11 MAC. In [21], a Markov chain model is presented for multi-hop IEEE 802.15.4 networks, but the model is limited to nodes that communicate to the coordinator through an inter- mediate relay node, which is assumed as not generating traffic and not competing for channel access. Therefore, to the best of our knowledge, there is still no analytical study in the literature that investigates the effect of routing over multi-hop networks using IEEE 802.15.4 MAC.

In this paper, we propose a novel analytical study that considers jointly routing and MAC, and we highlight the interdependence between routing decisions and end-to-end performance indicators. We provide an accurate model for small-scale networks and an approximate model that yields effective analysis of the performance for large-scale networks.

We show how the IEEE 802.15.4 MAC may influence remark- ably the routing alternatives. We study different performance indicators of IEEE 802.15.4 MAC over multi-hop networks, i.e., end-to-end reliability, end-to-end delay, and energy con- sumption. Specifically, the effect of the carrier-sensing range of nodes in different routing paths depends on the traffic. Thus, a different distribution of traffic load in the network determines different performances in terms of reliability, delay, and energy consumption of the routing. Based on the analysis proposed in this paper, we study conditions in which routing performance becomes critical for the load distribution and stability of the network.

Although there exists many routing protocol proposals in the literature, there is not yet a definite solution. Lively research activities and standardization efforts are ongoing, including the Routing Over Low power and Lossy networks (ROLL) working group of the Internet Engineering Task Force (IETF) [2], which is defining a routing protocol on top of the physical layer and MAC of IEEE 802.15.4. Therefore, we believe that our study may have an impact on the ongoing standardization.

III. SYSTEMMODEL

Consider a network of N nodes, V₀, . . . , V_N that use the unslotted IEEE 802.15.4 MAC. We focus on this MAC modal-

Fig. 1. Examples of (a) a single-hop topology, (b) a multi-hop topology with single end device, and (c) a multi-hop topology with multiple end devices for IEEE 802.15.4 networks. The dash-dotted area Ω4delimits the carrier-sensing range of node V4, i.e., the largest set of nodes that can be heard by V4while doing the CCA. The shape of Ω4is irregular, because the carrier-sensing range may not always be isotropic. Our analysis incorporates any shape.

ity because it is common and relevant for the ROLL routing standardization [2].¹ In the following, we illustrate the system model by considering three topologies, as shown in Fig. 1.

However, the analytical results that we derive in this paper are general and not limited to a specific topology.

The topology in Fig. 1(a) refers to a single-hop (star) network where nodes forward their packets²with single-hop communi- cation to the root node V0. In star networks, we denote by l the link between Vl and V0, l = 1, . . . , N . The topologies in Fig. 1(b) and (c) are examples of multi-hop networks in which nodes forward traffic according to the uplink routing policy to V₀. In multi-hop networks, we label by l, l = 1, . . . , G, the link between a pair of communicating nodes Viand Vj, where G is the number of such pairs.

For every node V_i, we define a neighborhood set Ω_i, which contains all the nodes in the carrier-sensing range of V_i (de- limited by dash-dotted lines in the examples in Fig. 1). The carrier-sensing range is the set of nodes that can be heard by a node while performing the IEEE 802.15.4 clear channel assessment (CCA), which we describe later on. We denote by

|Ωi| the cardinality of Ωi. Note that the carrier-sensing radius is not necessarily isotropic. We assume that the channel is symmetric, so that, if Vk ∈ Ωi, then Vi ∈ Ωk, which is natural when transmitting and receiving over similar frequencies. For each link (Vi, Vj), we define Ωj\i= Ωj− Ωi as the hidden node set of Viwith respect to Vj, i.e., all nodes that are in the carrier-sensing range of receiver Vj but that do not belong to the carrier-sensing range of transmitter V_i.

As a reference routing protocol, we consider the specifica- tions of IETF ROLL [2]. The root node generates a destination- oriented directed acyclic graph (DODAG). In a DODAG, all edges are oriented such that no cycles exist. Directional routes in the network are indicated by arrows in Fig. 1. We define a parent set Γi⊂ Ωi, which contains all nodes that may be next-hop nodes of Vi, and a children set Δi⊂ Ωi, which contains all nodes that have Vi as next-hop node. The knowl- edge of the topological sets Ωi and Γi is then specified by IETF ROLL.

1The model derivation for the slotted mechanism follows similar steps as those presented in this paper, without a significant increase in complexity.

2Throughout this paper, we refer to packets as the MAC protocol data units.

TABLE I

MAINSYMBOLSUSED IN THEPAPER

We consider two multi-hop topologies in Fig. 1(b) and (c) to illustrate our analysis. In Fig. 1(b), there is one end-device V₇ and two main paths to the destination. Node V₇ may decide to route its packets either through nodes V₄ or V₁, which forward traffic also from nodes V₂and V₅, or through V₆ and V₃. Note that route V₆− V3 is less loaded in terms of traffic forwarding. In addition, we study the more complex routing graph in Fig. 1(c), where multiple end devices (V4, V5, V6, and V7) may decide to route their packets either through nodes V1, V2, or V3 to destination V0. Coherently with the IETF ROLL specifications, we assume homogeneous link quality between a node and each one of its selected parents, because typically the parent set includes only nodes that can be reached with a guaranteed link quality. The actual forwarding decision is then based on routing metrics such as maximum end-to-end reliability or minimum end-to-end delay, which depend on the link performance at the MAC layer. As we show in Section V, the interaction between the MAC layer and routing decisions varies substantially between the routing paths, according to the carrier-sensing ranges.

In the next section, we introduce the general model for multi- hop unslotted IEEE 802.15.4 MAC, and we derive the basic relations with the routing policy. A list of the main symbols used in this paper is reported in Table I.

IV. MULTI-HOPUNSLOTTEDIEEE 802.15.4 MEDIUMACCESSCONTROL

In this section, a generalized model of a heterogeneous unslotted IEEE 802.15.4 network is proposed. The analysis aims at deriving the network performance indicators, i.e., the re- liability as probability of successful packet reception, the delay

for successfully received packets, and the average node energy consumption. We first analyze the single-hop case, and then, we generalize the model equations to the multi-hop case. The main contribution of such a model, with respect to the Markov chain model in [5] or [22], is the presence of heterogeneous traffic with different node packet generation rates, hidden terminals, and multi-hop routing.

A. CSMA/CA Mechanism of Unslotted IEEE 802.15.4

Consider a node trying to transmit. In the unslotted carrier- sense multiple access with collision avoidance (CSMA/CA) of IEEE 802.15.4, first, the MAC layer initializes three variables, i.e., the number of backoffs N B, the backoff exponent BE, and the retransmissions counter RT . The default initialization is N B = 0, BE = macM inBE, and RT = 0.

Then, the MAC layer delays for a random number of complete backoff periods in the range [0, 2^BE− 1] units aU nitBackof f P eriod. When the backoff period is zero, the node performs a CCA. If the CCA is idle, then the node begins the packet transmission. The node experiences a delay of aT urnaroundT ime to turn around from listening to transmit- ting mode. If the CCA fails due to busy channel, the MAC layer increases the value of both N B and BE by one up to a max- imum value macM axCSM ABackof f s and macM axBE, respectively. Hence, the values of N B and BE depend on the number of CCA failures of a packet. Once BE reaches macM axBE, it remains at the value of macM axBE until it is reset. If N B exceeds macM axCSM ABackof f s, we assume that the packet is discarded due to channel access failure. Other- wise, the CSMA/CA algorithm generates a random number of backoff periods and repeats the process. If the channel access is successful, the node starts transmitting packets and waits for ACK. The reception of the corresponding ACK is interpreted as successful packet transmission. If the node fails to receive ACK due to collision or ACK timeout, the variable RT is increased by one up to macM axF rameRetries. If RT is less than macM axF rameRetries, the MAC layer initializes BE = macM inBE and follows the CSMA/CA mechanism to reaccess the channel. Otherwise, the packet is discarded due to the retry limit.

In the rest of this paper, we denote the IEEE 802.15.4 MAC parameters by W₀= 2^{macM inBE}, m₀= macM inBE, m_b= macM axBE, m = macM axCSM ABackof f s, n = macM axF rameRetries, and S_b= aU nitBackof f P eriod.

B. Single-Hop Network Model

In this section, we develop new results for single-hop networks, which we then extend for multi-hop networks in Section IV-F. In particular, we derive the probability τl that node Vl attempts a CCA in a randomly chosen time unit, the probability αl that CCA is busy at link l, and the probability Pcoll,lthat a transmitted packet encounters a collision at link l.

Let sl(t), c_l(t), and r_l(t) be the stochastic processes repre- senting the backoff stage, the state of the backoff counter, and the state of retransmission counter, respectively, which node V_l experiences at time t. By assuming independent probability that

Fig. 2. Markov chain model of the CSMA/CA algorithm of a transmitting node of link l for unslotted IEEE 802.15.4 MAC. Compared with existing works from the literature, this chain models the hidden terminal problem, heterogeneous traffic, and different packet generation rates per node.

nodes start sensing, the stationary probability τlthat Vlattempts carrier sensing in a randomly chosen slot time is constant. Then, the triple (s_l(t), c_l(t), r_l(t)) is the 3-D per-link Markov chain in Fig. 2, where we use (i, k, j) to denote a particular state.

The Markov chain consists of four main parts corresponding to the idle state, backoff states, CCA states, and packet trans- mission states. The idle state corresponds to the idle-queue state when the node is waiting for the next packet generation time. The states from (i, Wm− 1, j) to (i, W0− 1, j) represent the backoff states. The states (i, 0, j) represent the CCA. The states (−1, k, j) and (−2, k, j) correspond to the successful transmission and packet collision, respectively.

The generation of unsaturated traffic at node Vl is modeled by a packet generation probability in idle state ql, i.e., the probability of generating a new packet in each time unit when the node is in idle state. Moreover, to include queueing effects of node buffers, we consider the probabilities of having a packet ready to be transmitted after the node has successfully sent a packet q_succ,l, after a packet has been discarded due to channel access failure q_cf,lor due to retry limit q_cr,l. The expressions of the packet generation probabilities are derived in Appendix A.

We define the packet successful transmission time L_sand the packet collision time Lcas

Ls= L + tack+ Lack+ IF S

Lc= L + tm,ack (1)

where t_ack is the ACK waiting time, IF S is the interframe spacing, and t_m,ack is the timeout of the ACK (see details

in [1]). Packets are assumed to be all of the same length, consistently with the previous literature.

In the proposed Markov chain, Sb is the unit time for all state transitions and corresponds to the transmission time of 20 symbols [1]. When performing CCA, a node is listening in RX mode for a duration of 8 symbols. Then, the nodes take a time of 12 symbols (aT urnaroundT ime) to turn around from RX mode to TX mode before starting the transmission of the packet, which makes a total time of 20 symbols (Sb) for a successful CCA. The length of packet L and that of acknowledgement L_ack are given as multiplies of Sb. Therefore, it is possible to conclude that S_b is accurate enough to capture the main characteristics of the unslotted mechanism for a transmitting node.

By finding the stationary probabilities for each chain, we can derive the probability τ_lthat node V_lattempts CCA. Then, we couple all the per-link Markov chains to obtain a set of equations giving the network operating point, i.e., the busy channel probabilities αl and the collision probabilities Pcoll,l, for l = 1, . . . , N .

We define b^(l)_i,k,j= limt→∞Pr[sl(t) = i, cl(t) = k, rl(t) = j], i∈ (−2, m), k ∈ (0, max(Wi− 1, Ls− 1, Lc− 1)), j ∈ (0, n), as the stationary distribution of the Markov chain of Fig. 2. We remark that these probabilities are associated to each link l. Then, we have the following result:

Proposition 4.1: Suppose that the probability to start sensing for every node is independent of the number of retransmissions suffered. Let α_l be the probability that CCA is busy, and let P_coll,lbe the probability that a transmitted packet encounters a collision, for l = 1, . . . , N . Then, the probability τ_lthat a node V_lattempts CCA in a randomly chosen time unit is

τ_l=

1− α^m+1_l 1− αl

 1− y_lⁿ⁺¹ 1− yl



b^(l)_0,0,0 (2)

where (3), shown at the bottom of the next page, holds, and y_l= P_coll,l(1− α^m+1_l ).

Proof: See Appendix B. 

The probability τ_lgiven by the previous proposition depends on the probability αlthat CCA is busy and the probability Pcoll,l

that a transmitted packet encounters a collision. We study these two probabilities next.

We derive the busy channel probability as follows:

αl= αpkt,l+ αack,l (4)

where αpkt,lis the probability that node Vlsenses the channel and finds it occupied by a packet transmission in the neighbor- hood Ωl, whereas αack,lis the probability of finding the channel busy due to ACK transmission from V0.

The probability that node Vl finds the channel busy due to a packet transmission is the combination of two events: 1) At least one node accesses the channel in one of the previous L time units, and 2) at least one of the nodes that accessed the channel found it clear. We would like to remark here a major difference with the Markov chain model proposed in [5] and [9]. In homogeneous networks with full sensing range, the busy channel probability is network information; it is the same for all

the nodes. In heterogeneous networks, it depends on the access and busy channel probabilities of every node in the neighbor- hood. This introduces substantial analytical challenges.

Denote bySlthe event that node Vlis sensing, and byTlthe event that node Vlis transmitting. Denote also byFlthe event that there is at least one transmission in Ωl. Then

αpkt,l= Pr[Fl|Sl] =

|Ωl|−1 i=1

Cl,i



j=1

Pr

 _i



k=1

Tkj|Sl

 (5)

where

C_l,i=

|Ωl| − 1 i

 .

Index k accounts for the events of simultaneous transmissions in the channel, and index j enumerates the combinations of events in which a number i of channel accesses are performed in the network simultaneously. Therefore, index kjrefers to the node in the kth position in the jth combination of i elements out of Ωlso that

Pr

 _i



k=1

Tkj|Sl



= L

i k=1

τkj

1−

i k=1

αkj

_|Ωl|



h=i+1

(1− τhj).

To illustrate (5), we consider Fig. 1(a) and assume that there are two contending nodes in the neighborhood of V₄, Ω₄= {V0, V₃, V₅}. Note that V0does not generate packets. Then, the event of busy channel for node V4, is given by the sum of three contributions.

1) Only node V₃accessed the channel and found it clear. The probability of this event is Lτ₃(1− τ5)(1− α3).

2) Only node V₅ accessed the channel and found it clear. Similarly to the previous case, the probability is Lτ5(1− τ3)(1− α5).

3) Both nodes accessed the channel and at least one node found it clear. Note that V5 may not belong to Ω3

in this case. This probability is upper bounded by Lτ3τ5(1− α3α5).

Equation (5) follows as a generalization of this example.

The computation of the correlation among the busy channel probabilities is not an easy task. We use the upper bound

(1−_i

k=1αkj) because it represents a worst-case model sce- nario for the busy channel probability (uncorrelated busy chan- nel events). As we show in Section V, this upper bound also provides a good approximation in the case of perfect sensing (maximum correlation). The reason is that this assumption affects only events in which two or more nodes are listening to the channel in the same time unit. However, in the case of homogeneous networks with full sensing range, the term (1−i

k=1α_kj) can be replaced by the accurate expression (1− α).

A busy channel assessment due to ACKs depends on the probability of successful packet reception in Ω₀. Let R_hbe the reliability of link h. It follows that

αack,l= Lack



h∈Ω0,h=l

qhRh (6)

where L_ack is the length of the ACK, and q_h is the packet generation rate of node V_h. By summing up (5) and (6), we can compute α_lin (4).

We now turn our attention to the collision probability P_coll,l, i.e., the probability that the packet transmission from node Vl to the root node V0 encounters one or more simultaneous packet transmissions. Note that these transmissions may fully or partially overlap due to the limited size of the packets.

There are two main reasons of packet collisions for Vl. 1) Collision due to turnaround time: At least one node in Ω_l

senses the channel idle while V_lis in its turnaround after CCA, or at least one node in Ω_lis in its turnaround after CCA while Vlsenses the channel idle.

2) Collision due to hidden nodes: At least one node in Ω_0\l (hidden node) has started a packet transmission in one of the previous L time units or before Vl ends its transmission.

We define byAlthe event of collision due to turnaround time in Ω_land by Blthe event of collision due to hidden nodes in Ω_0\l. Therefore, the collision probability P_coll,lis given by

Pcoll,l = Pr[Al] + Pr[Bl]− Pr[Al] Pr[Bl]. (7) Here, the probability of eventAlis given by the probability that at least one node in Ωlaccesses the channel and finds it free in

b^(l)_0,0,0=

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

 1 2

1−(2αl)^m+1

1−2αl W₀+^1−α

m+1

1−αl l

1−yⁿ⁺¹_l

1−yl +(L_s(1−Pcoll,l)+L_cP_coll,l) (1−α^m+1)^1−y

n+1

1−yl l

+^1−q_q^cf,l

l

α^m+1_l (1−y_lⁿ⁺¹)

1−yl +^1−q_q^cr,l

l yⁿ⁺¹_l +^1−qsucc,l_q

l (1−Pcoll,l)(¹−α^m+1_l )(1−yⁿ⁺¹_l ) 1−yl

₋₁

, if m≤ ¯m = m_b− m0

 12

1−(2αl)^m+1¯

1−2αl W0+^1−α

¯ m+1

1−αll +(2^mb+1)α^m+1_l^{¯ 1−α}

m− ¯m

1−αl l

1−yⁿ⁺¹l

1−yl +(Ls(1− Pcoll,l)+LcPcoll,l) (1− α^m+1)

×^1−y_1−y^n+1l_l +^1−q_q^cf,l

l

α^m+1_l (¹−ylⁿ⁺¹)

1−yl +^1−q_q^cr,l

l y_lⁿ⁺¹+^1−q_q^succ,l

l (1− Pcoll,l)(¹−α^m+1l )(¹−ylⁿ⁺¹)

1−yl

₋₁

, otherwise

(3)

the same time unit (2aT urnaroundT ime≈ Sb), i.e.,

Pr[Al] =

|Ωl|−1 i=0

Cl,i



j=1

i k=1

τ_kj

1−

i k=1

α_{kj |Ωl|}



h=i+1

(1− τhj).

Similarly, the probability of eventBlis given by

Pr[Bl] = 2L

|Ω0\l|−1 i=0

Cl,i



j=1

i k=1

τ_kj

1−

i k=1

α_{kj |Ω0\l|}



h=i+1

(1− τhj).

In the following sections, we use these results to derive the expressions of the reliability, the delay for successfully received packets, and the energy consumption.

C. Reliability

In this section, we derive an expression of the reliability for each link of the network. We use the delivery ratio as a measure of the reliability. In the IEEE 802.15.4 CSMA/CA, packets are discarded due to either of the following reasons:

1) channel access failure or 2) retry limit. Channel access failure happens when a packet fails to obtain clear channel within m + 1 backoffs. Furthermore, a packet is discarded if the transmission fails due to repeated collisions after n + 1 attempts. Following the Markov model shown in Fig. 2, the probability that the packet is discarded due to channel access failure is

P_cf,l= α^m+1_l

n j=0

P_coll,l

1− α^m+1_l j

= α^m+1_l

 1−

P_coll,l

1− α^m+1_l n+1 1− Pcoll,l

1− α^m+1_l  . (8)

The probability of a packet discarded due to retry limit is Pcr,l=

Pcoll,l

1− α^m+1_l n+1

. (9)

Therefore, by using (8) and (9), the reliability is

Rl= 1− Pcf,l− Pcr,l. (10) The expressions of the carrier-sensing probability τ_lin (2), the busy channel probability α_lin (4), and the reliability R_lin (10), for l = 1, . . . , N , form a system of nonlinear equations that can be solved through numerical methods [23]. The solu- tion of these equations provides us with the link reliability for the single-hop networks.

D. Delay

The total delay experienced in a successful packet transmis- sion for node Vlcan be derived as Dl= D^s_l + D_l^q, where D_l^s is the service time for a successfully received packet, and D^q_l is the queueing delay.

The service time D^s_l is defined as the time interval from the instant the packet is ready to be transmitted until an ACK for such a packet is received. If a packet is dropped due to either

the limited number of backoffs m or the finite retry limit n, its delay is not included into the derivation.

Let D_l,j^s be the delay for a node that sends a packet success- fully after j unsuccessful attempts. From the Markov chain of Fig. 2, we derive the expected value of the delay D_l^s

E [D^sl] =

n j=0

Pr[Cj|C] E D^s_l,j

(11)

where

E D^s_l,j

= L_s+ j L_c+

j h=0

E[Th]. (12)

This the backoff stage delay, and Lsand Lcare the time periods in the number of time units for successful packet transmission and collided packet transmission in (1). EventCj denotes the occurrence of a successful packet transmission at attempt j + 1, given j previous unsuccessful transmissions, whereas event C denotes the occurrence of a successful packet transmission within n attempts. We then have

Pr[Cj|C] = P_coll,l^j 

1− α^m+1_l j

_n

k=0

P_coll,l

1− α^m+1_l k

=

1− Pcoll,l

1− α^m+1_l 

P_coll,l^j 

1− α^m+1_l j

1− P_coll,l

1− α^m+1_l n+1

(13) where we recall that Pcoll,l is the collision probability, and 1− α^m+1_l is the probability of successful channel access within the maximum number of m backoff stages. Note that the probability of event Cj is normalized by considering all the possible events of successful attemptsC.

Let T_h,i be the random time to obtain a successful CCA from the selected backoff counter value in backoff level i. By following a similar approach as that for the characterization of D^s_l, we see that the expected total backoff delay is mod- eled byE[Th] =_m

i=0Pr[Di|D] E[Th,i], where Th,i= (1 + i) Tsc+_i

k=0T_h,k^b , Tsc is the sensing time in the unslotted mechanism, and T_h,k^b is the backoff time at attempt k. Since T_h,k^b is uniformly distributed in [0, Wk− 1], we can rewrite the expected backoff delayE[Th] as

E[Th] = Tsc+

m i=0

Pr[Di|D]

i Tsc+

i k=0

Wk− 1 2 Sb

. (14) Event Di denotes the occurrence of a busy channel for i consecutive times and then that of an idle channel at the i + 1th time. The probability of Di is conditioned to the successful sensing event within m attemptsD, given that the node senses an idle channel in CCA. It follows that

Pr[Di|D] = αⁱ_l

m

k=0α^k_l =αⁱ_l(1− αl)

1− α^m+1_l . (15)