IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 7, JULY 2013 1467

Random Access Scheme to Improve Broadcast ReliabilityYun Han Bae

Abstract—A simple CSMA-based random access scheme toimprove broadcast reliability is proposed in this letter. Weconsider successful delivery probability as a broadcast reliabilitymeasure. Hinted from the result [3] on the optimal accessprobability in the slotted-Aloha, the access probability in a-persistent CSMA is designed to improve the successful deliveryprobability. An analytical model is also presented to evaluatethe performance of the proposed scheme. The results showthat the proposed scheme significantly improves the successfuldelivery probability of broadcast messages, in comparison withthe existing ones.

Index Terms—Broadcasting, a-persistent CSMA, performanceanalysis, successful delivery probability, slotted-Aloha.

I. INTRODUCTION

THE number of applications of random access-basedbroadcasting has continuously been increasing. Various

examples can be found in ad hoc networks and vehicularnetworks using 802.11-based broadcasting, and reliable dis-semination of multimedia information over mobile networks,etc.

In general, broadcast data are transmitted without anycontrol frames. Therefore, e.g., according to IEEE 802.11protocol, any acknowledgement is not transmitted by any ofthe receivers of the message and thus there is no MAC-level recovery or retransmission of the message. As a result,the reliability of broadcast service is severely reduced com-pared to unicast service which can use various error recoverymechanisms such as control frames (RTS/CTS, ACK) andretransmission. In this letter, we are interested in the followingfundamental issue: how can we improve broadcast reliabilityunder simple random access-based broadcasting ?

We consider a-persistent CSMA(carrier sense multipleaccess)-based broadcasting where each node tries to accessa channel with a common access probability a in order tosend its broadcast messages. It is very meaningful to considera-persistent CSMA protocol in that its variants constitute aheart of contemporary wireless medium access technologyand besides the performance of many CSMA protocols usedin 802.11 DCF and 802.15.4 can be evaluated by studyingthe performance of their corresponding a-persistent CSMAprotocol. The following assumptions are made: a broadcastmessage is neither acknowledged nor retransmitted; broadcastmessage has a strict delivery deadline within which it shouldbe delivered. It is natural to consider successful delivery

Manuscript received March 18, 2013. The associate editor coordinating thereview of this letter and approving it for publication was I.-R. Chen.

This research was supported by a 2012 Research Grant from SangmyungUniversity.

Y. H. Bae is with the Department of Mathematics Education, SangmyungUniversity, Seoul, Korea (e-mail: yhbae@smu.ac.kr).

Digital Object Identifier 10.1109/LCOMM.2013.051313.130613

probability, which accounts for actual packet delivery ratiofor broadcast messages with a given delivery deadline, as abroadcast reliability measure.

System throughput has been widely used as a performancemeasure for evaluating the efficiency of random access proto-cols. As for broadcast service, system throughput is not alwaysa sound measure in some scenarios, e.g., delay-constrainedbroadcasting in vehicular network as considered in [4], [5].This motivates us to consider the successful delivery prob-ability. The focus of this letter is to determine the accessprobability a maximizing the successful delivery probability.In essence, the basic idea used for determining the accessprobability is based on the result [3] in which the optimalaccess probability in the slotted-Aloha-based broadcasting isinvestigated.

It is essential to accurately understand the performance ofunderlying MAC protocols when they are used for broad-casting service. In response to this need, several analyticalmodels [1]-[6] were recently proposed. In [1], [2], the authorsproposed analytical models to evaluate the performance ofbroadcast service in IEEE 802.11 DCF, including both sat-uration throughput and packet delivery ratio. But, the packetdelivery ratio defined there is not a measure considering thepacket delivery deadline. Meanwhile, in the area of IEEE802.11p-based vehicular network, several researchers [4], [5],[6] proposed analytical models to evaluate the successfuldelivery probability of safety-related messages with a strictdelivery deadline, which shares a common interest with thiswork. However, they did not focus on the issue on how tooptimally determine the access probability for the purpose ofimproving the performance, but instead analyzed the perfor-mance of 802.11P/WAVE protocol itself, while taking channelswitching effect into account. Therefore, this letter deals withhow to determine the access probability of nodes in order toimprove the successful delivery probability under a-persistentCSMA-based broadcasting and further an analytical model forobtaining the successful delivery probability is presented.

II. SYSTEM MODEL

Consider a wireless network with a single shared channelconsisting of N nodes. The time is divided into equal mini-slots (or backoff slots) with an unit length. Each node triesto access the channel in order to send broadcast messagesto its recipients based on a-persistent CSMA protocol. Un-der a-persistent CSMA protocol, each node probabilisticallydetermines whether to access the channel or not based ona common access probability a. The length of a message isassumed to be L(mini-slots).

It is assumed that a broadcast message has its own deliverydeadline T (mini-slots), the duration from the moment of its

1089-7798/13$31.00 c© 2013 IEEE

1468 IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 7, JULY 2013

arrival at the head of the queue to its transmission completionpoint, within which it should be delivered. The followingadditional assumptions are made in this letter: (1) the radiochannel is ideal, so the only source of loss is collision; (2)all SUs are within their communication range (no hiddennode); (3) a broadcast message is neither acknowledged norretransmitted. Thus, a message will be regarded as being deliv-ered successfully if it is transmitted within the given deliverydeadline T with no collision. Otherwise, if the message iscollided with others or is not transmitted during the deliverydeadline, it will be dropped from the queue as it is useless.

III. SUCCESSFUL DELIVERY PROBABILITY

In this section, we obtain successful delivery probability Ps,which is defined as the probability that a broadcast messagewill be successfully transmitted within the delivery deadline Tsince its arrival at the head of queue. Assume that each nodeis saturated, that is, it always has another message to sendafter completing the transmission of a message. The numberN of nodes in the network is known priori.

We choose an arbitrary node as a tagged node. Considera message at the head of the tagged node’s queue. Toderive the successful delivery probability, we introduce anabsorbing Markov chain {xk : k = 0, 1, 2, · · · , T } wherexk ∈ {1, 2, · · · , T−L, 0ab, 1ab}. Here, xk = i, 1 ≤ i ≤ T−Lrepresents that the message is not transmitted during k-steptransitions and the elapsed time since its arrival at the headof the queue is i(mini-slots); the states 0ab, 1ab are absorbingstates, where the state 0ab represents that the message has beentransmitted within the given delivery deadline and the state 1abrepresents that the message has not been transmitted within thegiven delivery deadline and thus is dropped from the queue.Since ‘xk > T −L’ implies that the message will not be suc-cessfully transmitted within the given delivery deadline T , wedo not need the additional states {T−L+1, T−L+2, · · · , T }.We also note that the Markov chain terminates in at most T−Lsteps.

The states 1, 2, · · · , T − L, 0ab, 1ab are listed in lexico-graphic order. Let P be the one-step transition probabilitymatrix of the Markov chain {xk : k = 0, 1, 2, · · · , T − L}.Then, the transition probability matrix P has the followingcanonical form:

P =

[T t0 I

], (1)

where T is a square matrix of order T −L, t is a (T −L)×2

matrix such that T1+t

[11

]= 1, here 1 is a column vector

of ones with dimension T − L, 0 is a zero vector whoseelements are all zeros, and I is a 2×2 identity matrix. Supposethat the Markov chain is currently being in xk = i, 1 ≤ i ≤T −L. The elements of the component matrices of P are thengiven as follows:

• For 1 ≤ i ≤ T − 2L

– If all the nodes (including the tagged node) do notattempt to transmit their packet, then xk+1 = i+ 1.Therefore,

P (xk+1 = i+ 1|xk = i) = Ti,i+1 = (1− a)N .

– If the tagged node does not attempt its transmissionand at least one of the other N − 1 nodes attempt totransmit, then xk+1 = i+ L. Therefore,

P (xk+1 = i+ L|xk = i) = Ti,i+L

= (1− a)× (1− (1− a)N−1).

– If the tagged node attempts to transmit its message,then xk+1 = 0ab. Therefore,

P (xk+1 = 0ab|xk = i) = ti,0ab= a.

• For T − 2L+ 1 ≤ i ≤ T − L− 1

– If all the nodes (including the tagged node) do notattempt to transmit their packet, then xk+1 = i+ 1.Therefore,

P (xk+1 = i+ 1|xk = i) = Ti,i+1 = (1− a)N .

– If the tagged node does not attempt its transmissionand at least one of the other N − 1 nodes attemptto transmit, then the residual time to the deliverydeadline T is less than L and therefore the taggednode’s message cannot be delivered within the de-livery deadline. Thus, xk+1 = 1ab and

P (xk+1 = 1ab|xk = i) = ti,1ab

= (1 − a)× (1− (1 − a)N−1).

– If the tagged node attempts to transmit its message,then xk+1 = 0ab. Therefore,

P (xk+1 = 0ab|xk = i) = ti,0ab= a.

• For i = T − L

– If the tagged node attempts to transmit, then xk+1 =0ab. Otherwise, since the residual time is not enoughto accommodate the tagged node’s transmissionwithin the given delivery deadline, xk+1 = 1ab.Therefore,

P (xk+1 = 0ab|xk = T − L) = tT−L,0ab= a

P (xk+1 = 1ab|xk = T − L) = tT−L,1ab= 1− a.

Let K be the number of steps required until the Markovchain reaches the absorbing states 0ab or 1ab, starting from aninitial state x0 = 1. Note that since the absorbing states arereachable within at most T−L transitions, K ∈ {1, 2, · · · , T−L}. Then, K follows a phase (PH) type distribution withrepresentation (η,T), where η is an initial distribution and,noting that x0 = 1, η = [1, 0, · · · , 0]. Expressing the sub-matrix t as [t0 t1] where t0, t1 are column vectors, theprobability mass function of K is then given by

P (K = k) = ηTk−1(t0 + t1), 1 ≤ k ≤ T − L. (2)

Now we are ready to derive the successful delivery probabil-ity Ps. Denote by E the event that the message is successfullytransmitted. Let F be the event that the message is transmittedwithin the delivery deadline T . Then, Ps is obtained as

Ps = P (E ∩ F ) = P (E|F )P (F )

= (1− a)N−1 · ηT−L∑k=1

Tk−1t0. (3)

BAE et al.: RANDOM ACCESS SCHEME TO IMPROVE BROADCAST RELIABILITY 1469

IV. CHANNEL ACCESS PROBABILITY

A. Optimal Access probability

We now have obtained the successful delivery probabilitygiven that each node uses access probability a. The optimalaccess probability a∗opt maximizing Ps (given by Eq.(3)) isthen obtained as

a∗opt = argmaxa∈(0,1)Ps. (4)

However, finding the solution of the above problem involvesa huge computation complexity due to the calculation of Ps.Instead, we provide a suboptimal access probability and then asimple heuristic method for determining the access probabilityis finally proposed.

B. Suboptimal Access probability

1) Optimal access probability for the slotted-Aloha [3]:The optimal access probability for the slotted-Aloha whichmaximizes the successful delivery probability was presentedin [3]. In this work, the basic idea for determining suboptimalaccess probability for CSMA is to apply the result in [3].For completeness, we briefly review the result [3]. For theslotted-Aloha where time is equally divided by the packettransmission time L (mini-slots), each node attempts to trans-mit its message at every L mini-slots with a common accessprobability a1 and the message delivery deadline is T (mini-slots), the successful delivery probability is then given by

P alos = (1− a)N−1

D∑k=1

a(1− a)k−1, (5)

where D = �TL �. It is easy to see that Eq.(3) reduces to Eq.(5)

in the case of slotted-Aloha. The optimal access probabilitya∗ maximizing Eq.(5) is given by (Theorem 1 in [3])

a∗ = 1− (N − 1

N − 1 +D)

1D . (6)

We further note that as D → ∞, a∗ ↘ 0 and Ps ↗ 1.2) Application to a-persistent CSMA: Let P alo

s (a) (givenby Eq.(5)) and P cs

s (a) (given by Eq.(3)) be the successful de-livery probabilities for slotted-Aloha and a-persistent CSMA,respectively, when each node uses the access probability a. Inthe slotted-Aloha above, it is worth noting that the quantityD can be regarded as the maximum possible number oftransmission attempts which each node may have within thegiven delivery deadline T . Note that D is constant for theslotted-Aloha. By abuse of notation, we use a∗(D) insteadof a∗ to emphasize its dependence on D. The followingobservations are made from the above facts:

• In the case of CSMA protocol, the maximum numberD of transmission opportunities which each node hasvaries from �T

L � (if all the consecutive slots are full oftransmissions during the time T ) and T (if there is notransmission during the time T ).

• If the access probability a∗(D) (given by Eq.(6)) isused in CSMA protocol, then the successful delivery

1The slotted-Aloha considered in this work is slightly different from pureslotted-Aloha in that pure slotted-Aloha uses different access probabilitydepending on packet whether it is new or backlogged.

probability P css (a∗(D)) is obtained from Eq.(3). Then, it

is easy to see that for the access probability a∗(D) andD = �T

L �, P alos (a∗(D)) ≤ P cs

s (a∗(D)). The reasoningis as follows: with the use of the optimal a∗(D), theprobability that a message is successfully transmittedwithin the possible transmission attempts D is same forboth protocols; but, in case of CSMA protocol, there isadditional possibility for the message to be successfullytransmitted as long as its delivery deadline T does notexpire after the possible D attempts .

• For �TL � ≤ D ≤ T , it is likely that P cs

s (a∗(D)) increaseswith the increase of D as long as the probability that themessage is not transmitted within the delivery deadline Tis negligible. However, it seems that P cs

s (a∗) decreaseswith the increase of D after a certain value becausespreading out transmission attempts over time may leadto the excess in the delivery deadline. Thus, it is expectedthat there is an optimal D∗ maximizing P cs

s (a∗(D∗)).From the above observations, the problem to find the

suboptimal access probability a∗sub maximizing P css (a) (given

by Eq.(3)) can be formulated as follows:

D∗ = argmaxD∈[�TL �,T ]P

css (1 − (

N − 1

N − 1 +D)

1D ), (7)

where P css (a) is given by Eq.(3). Thus, the suboptimal access

probability a∗sub(D∗) for CSMA is obtained as

a∗sub(D∗) = 1− (

N − 1

N − 1 +D∗ )1

D∗ . (8)

The solution for the above problem can be solved by anexhaustive search for integer values of D. This approach isalso not desirable because it still involves high computationcomplexity. Instead, a simple heuristic method is proposed inthe next subsection.

C. A Heuristic Method

A heuristic method to determine the access probability a isas follows:

1) First, calculate the mean length G of a generic slot wherea generic slot is either a unit backoff slot with probability(1 − a)N or a busy slot with probability 1 − (1 − a)N .Thus, the mean length G of a generic slot is given by

G = (1 − a)N + (1 − (1− a)N ) · L. (9)

It is easy to see that, for the access probability given byEq.(6), as D goes from 0 to ∞, G decreases from L to1 .

2) Let D = TG

. From Eq.(6) and Eq. (9), we have

D =T

( N−1N−1+D )

ND + (1− ( N−1

N−1+D )ND )L

. (10)

Let f(D) = T

( N−1N−1+D )

ND +(1−( N−1

N−1+D )ND )L

, which is the

right-hand side of Eq.(10). Then, it is easy to see thefollowings: (1) f(TL ) > T

L ; (2) f(T ) < T ; (3) notingthat G decreases with the increase of D, f(D) increaseswith D. These results imply that Eq.(10) has at leastone solution (fixed point) in the interval (�T

L �, T ). The

1470 IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 7, JULY 2013

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 55000.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

D

succ

essf

ul d

eliv

ery

prob

abili

ty P

s

CSMA with access probability a*(D) (analysis)

CSMA with access probability a*(D) (simulation)

CSMA with aheu* (D

heu* ) and D

heu* =2723

(a) Successful delivery probability vs. D, (N = 20)

5 10 15 20 25 300.4

0.5

0.6

0.7

0.8

0.9

1

the number N of nodes

succ

essf

ul d

eliv

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abili

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s

CSMA with aheu* (D

heu* ) − analysis

CSMA with athrou

− analysis

optimal slotted Aloha with a*([T/L]) − analysis

CSMA with aheu* (D

heu* ) − simulation

CSMA with athrou

− simulation

optimal slotted Aloha with a*([T/L]) − simulation

(b) Comparison of Ps among the three protocols

5 10 15 20 25 300.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

the number N of nodes

syst

em th

roug

hput

throughput−maximizing CSMACSMA with access probability given by Eq.(11)

(c) Comparison of system throughput

Fig. 1. Performance of the proposed scheme.

solution D∗heu of Eq.(10) can be found numerically as fol-

lows: for integer values of D ∈ {�TL �, �T

L �+ 1, · · · , T },D∗

heu = argminD(D − f(D)). Finally, the access prob-ability is determined by

a∗heu = 1− (N − 1

N − 1 +D∗heu

)1

D∗heu . (11)

V. NUMERICAL RESULTS

To show the performance of our proposed scheme, wepresent numerical results which are also validated by ex-tensive simulations developed in Matlab. The followingparameters are used: channel data rate=6 Mbps; back-off slot length=9μs; PHY header=20 μs; MAC header=28Bytes; DIFS=34 μs; message payload=128 Bytes; messagedelivery deadline T=50ms; the length of a busy slot=(PHY+MAC) headers+payload+DIFS. Then, T = 5555 andL = 30 in the unit of backoff slot.

Fig.1(a) shows the successful delivery probability P css for

the varying D, when N = 20 and the access probabilitya∗(D) (given by Eq.(6)) is used. With the increase of D,P css increases up to a certain D value and then decreases as

explained in Section IV-B. We therefore see that there existsD∗ satisfying Eq.(7). It is worth noting that P cs

s amountsto 0.96 in the neighborhood of suboptimal access probabilitya∗sub(D

∗). The vertical dotted line in Fig.1(a) represents thesuccessful delivery probability for the case that each node usesthe access probability a∗heu (given by Eq.(11)). It is worthyto note that the heuristic method achieves the performancealmost close to the suboptimal one, which justifies the use ofthe heuristic method, instead of the suboptimal one.

Fig.1(b) compares the performance of the following threeprotocols; (1) CSMA with a∗heu (given by Eq.11), (2) theoptimal slotted Aloha with a∗(�T

L �) where a∗(�TL �) and the

successful delivery probability are given by Eq.(6) and Eq.(5),respectively, (3) throughput-maximizing CSMA with athrouwhere athrou is the access probability maximizing the systemthroughput. Here, the system throughput S of a-persistentCSMA can be obtained by S = LNa(1−a)N−1

(1−a)N+L(1−(1−a)N ) andthe access probability athrou is the one maximizing thisquantity. We notice that the proposed CSMA with a∗heu is ableto significantly improve the successful delivery probability

in comparison with both the optimal slotted-Aloha and thethroughput-maximizing CSMA. It is interesting to see that the

optimal slotted-Aloha outperforms the throughput-maximizingCSMA when the number of contending nodes is small (N ≤10), but with the increase of N the situation is reversed.Fig.1(c) compares system throughput between the proposedCSMA with a∗heu and the throughput-maximizing CSMA withathrou. The proposed scheme has lower throughput than thethroughput-maximizing CSMA, which is the result of usingthe small access probability so as to maximize the successfuldelivery probability. But, the throughput gap becomes lesswith the increase of N . Finally, we see in Fig.1(a) andFig.1(b) that analytical results match well with simulationresults, where each node generates 10000 messages duringeach simulation run.

VI. CONCLUSIONS

This letter presented a simple random access scheme whichsignificantly improves broadcast reliability. As a broadcastreliability measure, the successful delivery probability whichaccounts for the actual delivery ratio of broadcast messageswith a given delivery deadline, was considered and analyzedmathematically.

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