Performance Analysis of Buffered CSMA/CD Systems

Wireless Personal Communications 18: 45–65, 2001.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

Performance Analysis of Buffered CSMA/CD Systems

TAO WANNortel Networks, Richardson, TX, U.S.A.

ASRAR U. SHEIKHDepartment of Electrical Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, SaudiArabia

Abstract. This paper presents a new approximation approach, called tagged user approach (TUA), for thequeueing analysis of slotted CSMA/CD systems with a finite user population having either finite or infinite userbuffer capacity. By assuming a symmetric channel, the performance analysis of the overall system is determinedby the performance of an arbitrarily selected user, called the tagged user. With the further assumption that eachuser always operates with its equilibrium probability, the packet service time distribution can be obtained using astate flow graph by studying the channel contention scheme. The obtained packet service time distribution is thenapplied to the queueing analysis of the tagged user using the standard queueing theory. That is, the analysis ofthe channel contention scheme and the analysis of the user queueing behavior are decoupled in this paper. Hence,TUA can be applied to complicated systems such as those where each user buffer may have bulk packet arrivalsand random service discipline as long as the corresponding queueing theory is available. Since a state transitionprobability matrix is not required in TUA, the required amount of computation is quite small. Simulation resultsverify the high accuracy of the analysis.

Keywords: multiple access, CSMA/CD, channel throughput, queueing, packet transmission, equilibrium pointanalysis.

1. Introduction

Carrier sense multiple access with collision detection (CSMA/CD) is a widely used randomaccess protocol in local area networks (LANs) such as Ethernet, the most representative localarea network in the world. Although CSMA/CD is not suitable for application in wirelessnetworks directly, some of the multiple access protocols employed in wireless data networks,such as tone sense multiaccess protocol with partial collision detection (TSMA/PCD) [1] anddigital sense multiple access with collision detection (DSMA/CD), the medium access control(MAC) protocol in Cellular Digital Packet Data (CDPD) System [2], actually borrow theprinciple of CSMA/CD. Needless to say, the same techniques for the analysis of CSMA/CDprotocols can be applied to the analysis of TSMA/PCD and DSMA/CD in a straightforwardway.

There are many publications dedicated to the analysis of CSMA/CD. Some of them makethe assumptions that each user has a buffer capacity of only one packet [3, 15, 16, 18], orthe user population is infinite [14] in order to simplify the mathematical complexity of theanalysis. In the former case, each user is called an unbuffered user and can only be in oneof two states: busy or idle. Thus, the system analysis is simplified. In the latter case, thestrong Poisson assumption leads to a constant offered load for the overall system and noconcern is given to individual users. Although these assumptions are introduced for analytical

46 Tao Wan and Asrar U. Sheikh

convenience, they do not apply to the practical systems that have a finite user population andeach user has a finite buffer capacity.

For the exact performance analysis of a random multiple access system with a finite popu-lation of N users and a finite buffer capacity of L packets (FUFB system), an N-dimensionalMarkov chain with state vector b = (b1, b2, . . . , bN) is required, where bi , i = 1, 2, . . . , N ,is the number of packets in the buffer of the ith station. This makes analysis intractable for anFUFB system with even a small user population. So far, no attempts have been made for theexact analysis of FUFB systems with more than three users.

It is therefore desirable that some approximation techniques be used to restrict the size ofstate space in the analysis of FUFB CSMA/CD systems. To this end, redundant informationin the system is explored to simplify the analysis. One assumption employed by all approxim-ation approaches is that the systems are symmetric. That is, the statistical behavior of all userswith the same parameters in a system is identical. Then, the users with the same parametersare classified into one group. Therefore, one can focus on an arbitrary user in each group,called the tagged user of that group, and study its behavior under the impact of other userssharing the common channel. As soon as the behavior of the tagged user for each group isknown, the behavior of all the other users in the system becomes known.

A system is a heterogeneous system if it consists of more than one group of users. Other-wise, it is homogeneous. So far, only the approximation analysis on homogeneous multipleaccess systems has been addressed in the literature. One approach toward simplifying the ana-lysis of the system is to describe the system using a state vector n = (n1, n2, . . . , nL), whereni , i = 0, 1, . . . , L is the number of stations having i packets in their queues. Obviously,n0 = N −∑L

i=1 ni , and the total number of possible states in the state space is (N+LL ). When

L < N , the latter approach results in a considerable reduction in the size of state space [4].A widely used approximation approach is to describe the overall system using two Markov

chains. One is used to describe the number of busy users in the system since only busy usersmay contend for the channel; the other one describes the number of packets in an arbitraryuser, called the tagged user, since all users have identical statistical behavior because ofchannel symmetry [5]. By analyzing the coupled Markov chains, the system performance iseventually evaluated. The required size of the state space is roughly (N +1)(L+1). Since thebusy or idle status of the tagged user is indicated by its user Markov chain, it is not necessaryto include the tagged user in the channel Markov chain. As a result, the size of state space canbe further reduced to N(L + 1) [4, 8]. A quite different approach for the analysis of bufferedmultiple access protocols is proposed in [9, 10], where the statistical behavior of the system isobtained using the characteristics of the restricted occupancy urn model. It defines the state asthe total number of packets in the system. The number of possible states of the system is hencereduced to NL+ 1. In the above approaches, the Markov chain is still required to analyze thesimplified models and the state transition probability matrices have to be calculated. Even withthe simplified models, the analyses become difficult for a sufficiently large user population orbuffer capacity. Perhaps the only approach for the analysis of buffered CSMA/CD systemswithout calculation of state transition probability matrix is equilibrium point analysis (EPA)[7]. The calculation amount of EPA is small and the accuracy of EPA is quite good. However,because of modeling complexity, it is difficult to extend EPA to more complicated systemssuch as heterogeneous systems or homogeneous systems with bulk packet arrivals.

It should be mentioned that the analysis complexity depends on not only the size of thesystem state space, but also its dimensions. The state spaces in [4, 5, 8] have two dimensions.Although the state spaces in [9, 10] seem to be one-dimensional, they are actually two-

Performance Analysis of Buffered CSMA/CD Systems 47

dimensional. This is the same as using a one-dimensional vector to express multi-dimensionalvectors by appropriate mathematical mapping.

It is interesting to note that although each user in an FUFB system is a queueing system,almost all of the analysis approaches do not use the results already existing in the field ofqueueing theory. We find in [7] a strong statement that the problem of interfering queuescannot be solved using the classic queueing theory. Needless to say, this statement refersto the exact analysis only. For approximation analysis, the classical queueing theory can beused if the packet service time distribution can be obtained by simplifying the system modelproperly.

Several attempts have been made to apply the classical queueing theory to buffered systems[5, 11, 12, 17]. Protocols such as URN and ALOHA are studied in [12]. In [5, 11, 17], CSMAand CSMA/CD systems are analyzed by treating each user as an M/G/1 queue. The packetservice time distribution is obtained and then applied to the user queueing analysis. However,the analytical complexity prohibits them from being widely used in the analysis of bufferedmultiple access protocols. The difficulty in deriving the packet service time distribution is thehurdle in applying the classical queueing theory to the analysis of buffered random multipleaccess protocols.

It is also noticed that almost all available approaches do not decouple the analysis ofchannel contention and the analysis of user queueing behavior. As a result, they cannot beextended to systems with different queueing behavior such as bulk arrival process.

This paper presents a new approximation approach, called the tagged user approach (TUA),for the queueing analysis of slotted CSMA/CD systems. Homogeneous systems are analyzed,although this approach can be extended to heterogeneous systems. By assuming system sym-metry, we focus the analysis on the queueing behavior of an arbitrarily selected user, calledthe tagged user of the system. By further assuming that each user always operates with itsequilibrium probability, the packet service time distribution of the tagged user is obtainedusing a state flow graph [19] by considering the behaviors of other users in the system. Theobtained packet service time distribution is then applied to the queueing analysis of the taggeduser using the classical queueing theory. The performance of the overall system becomesknown after the performance of the tagged user is analyzed.

The simplicity of TUA is due to the fact that the analysis of channel contention for obtain-ing packet service time distribution and the analysis of user queueing behavior are decoupled.Hence, systems with complicated user queueing behavior can be analyzed using TUA as longas the corresponding standard queueing theory is determined.

2. System Model

The system employed here is a homogeneous slotted non-persistent CSMA/CD system witha finite population of N users having a finite buffer capacity of L packets or an infinite buffercapacity. Variable packet size is allowed, but all packets require an integer number of time slotsfor transmission. Denoting hk as the probability that a packet requires k slots of transmissiontime, the probability generating function (PGF) H(z) of packet transmission time is given by

H(z) =∞∑k=1

hkzk. (1)


For a discrete time system with finite capacity, we need to specify a rule concerning theacceptance of a message that arrives in the last slot of a service time when the system isfull. Here, the late arrival model with delayed replacement rule is used [13]. Regrading thetransmission principle, we use the defer first transmission (DFT) rule where new packetsand deferred packets are assigned with a same probability p. However, TUA can be appliedto the analysis of the system using the immediate first transmission (IFT) rule with propermodification.

The system operates as follows. Time is divided into slots of length normalized to 1. Allpacket transmissions start and end at the slot boundary. A busy user senses the channel ineach slot with probability p. If the channel is sensed idle, the station starts transmissionimmediately. If the channel is sensed busy, the station backs off and senses the channel again inthe next slot with probability p. The channel is noise-free. The only reason for unsuccessfultransmission is collision, which occurs when more than one user starts transmission at thesame slot. No packet can survive the collision. The stations experiencing a collision willdetect the collision and abort the transmission withD slots. All packets are of the same length.Packets arrive in the queue of each station following a Bernoulli process such that a messagearrives in each slot with probability λ and no messages arrive in a slot with probability 1 − λ.A user would receive the confirmation and depart the packet immediately after the packet issuccessfully transmitted. However, this model can also handle the system where the acknow-ledgement delay is not ignorable after a minor modification, as described in the numericalexample in Section 6.1. The queue in each user is FCFS. The arriving packets finding thequeue full get dropped and never return.

3. System Analysis Using TUA

In this section, we will first derive the packet service time distribution of the tagged user byfocusing on the channel contention scheme (CSMA/CD) using a state flow graph. Then, wewill analyze the queueing behavior of the tagged user using the classical queueing theorybased on the obtained packet service time distribution. Finally, the analysis is completed bycombining the two parts of analysis.

3.1. DISTRIBUTION OF PACKET SERVICE TIME

The state flow graph of the tagged user is illustrated in Figure 1, where S0, Stx , Sbk, and Seare packet-ready state, transmission-start state, backoff state, and departure state of the taggeduser, respectively. Since the DFT rule is used, the user will switch from S0 to Sbk immediately.At Sbk, the user will issue a transmission with probability pstx or defer for one more slot withprobability 1 − pstx, where pstx is given by

pstx = ppI (2)

since the tagged user will sense the channel with probability p and the channel will be sensedidle with a probability denoted by pI . Denoting psu as the probability that a packet of thetagged user will be transmitted successfully in one transmission attempt, the transfer functionfromXtx to Se is therefore given by psuH(z) because of immediate acknowledgement. Since atransmitted packet from the tagged user would experience a collision with probability 1 −psuand all transmitting users would detect the collision and abort the transmission in D slots, thetransfer function from Stx to Sbk is (1 − psu)z

D.


Figure 1. State flow graph of tagged user.

From Figure 1, the packet service time is just the time duration from a user entering S0

to exiting Se. The probability generating function (PGF) B(z) of the packet service time istherefore the transfer function from S0 to Se. Using Mason’s formula [20], it is given by

B(z) = pstxpsuH(z)

1 − (1 − pstx)z− pstx(1 − psu)zD, (3)

where

psu = (1 − pbp)N−1 (4)

since each other user would be busy with probability pb and each busy user at Sbk would sensethe channel with permission probability p.

Clearly, B(z) should be solely determined by user busy probability pb with all other para-meters given since only busy users would contend for the channel for transmission. However,Equations (2)–(10) imply that B(z) is a function of two unknowns pI and pb. Hence, we needto derive the relationship between pI and pb.

3.1.1. Probability of Channel Sensed IdleEven with the above assumptions, it is hard to find a relationship between pI and pb. Here,we introduce an approximation method for finding pI . That is, analysis errors of TUA onCSMA/CD will be contributed by two sources, one from the general assumptions employedand the other from the approximation analysis on pI .

In order to determine pI , the behavior of the channel has to be analyzed. Employing theclassical scheme for the analysis of CSMA/CD, the time duration in the channel is thoughtto consist of a sequence of cycles. Each cycle consists of an idle duration followed by a busyduration which corresponds with a successful transmission or a collision. Since the tagged userhas packets available for transmission at this moment, it is busy with probability 1. All otherusers in the system are busy with probability pb, since they operate with their equilibrium


probabilities by assumption. Consequently, when the tagged user is busy, the mean length ofidle duration in each cycle is given by

E(I) = 1

1 − (1 − p)(1 − ppb)N−1− 1 . (5)

Denoting qu and qu as the probabilities that the busy duration in a cycle corresponds to asuccessful packet transmission and the packet is contributed by the tagged user and all otherusers, respectively, we have

qu = p(1 − pbp)N−1

1 − (1 − p)(1 − ppb)N−1

= [E(I)+ 1]ppsu(6)

and similarly

qu = [E(I)+ 1](N − 1)ppb(1 − p)(1 − pbp)N−2. (7)

Hence, the busy duration in a cycle would correspond to a collision with probability 1−qu−qu.Denoting pctu as the probability that this collision would involve the tagged user, we then have

pctu = p[1 − (1 − ppb)N−1]

1 − (1 − p)(1 − ppb)N−1

= p[1 − (1 − ppb)N−1][E(I)+ 1] . (8)

In each cycle, the tagged user would sense the channel idle in each idle slot. In addition, itwould sense the channel idle in the first slot of each busy duration which may correspond toa successful transmission or a collision. Hence, in each cycle the average number of possibleslots in which the tagged user would sense the channel idle is E(I)+ 1.

Obviously, in slotted CSMA/CD protocol, a user never senses the channel during its owntransmission time except in the first slot of the transmission in which it has to sense the channelin order to start the transmission. By definition, the probability that the busy duration in a cyclecorresponds to a collision that does not involve the tagged user is 1−qu−qu−pctu. Hence, ineach cycle, the average number of slots in which the tagged user is able to sense the channelis given by E(I)+ 1 + (T − 1)qu + (D − 1)(1 − qu − qu − pctu), where

T = H ′(1) (9)

is the mean packet transmission time.Therefore, the probability pI is finally obtained as

pI = E(I)+ 1

E(I)+ 1 + (T − 1)qu + (D − 1)(1 − qu − qu − pctu). (10)

Equations (5)–(10) establish an approximation relationship between pI and pb. That is, theobtained Equations (1)–(10) imply that B(z) is solely determined by user busy probability pb.Let p0 be the probability that a user buffer is empty, clearly we have

pb + p0 = 1 . (11)

Hence, we can also conclude that packet service time distribution B(z) is solely determinedby user idle probability p0.


3.2. QUEUEING ANALYSIS OF THE TAGGED USER

In Section 3.1, a relationship between packet service time distribution B(z) and user idleprobability p0 is obtained by studying channel contention. In this section, we will apply B(z)on the queueing analysis of the tagged user using the standard queueing theory. The taggeduser is treated as a Geo/G/1 or Geo/G/1/L queueing system, depending on its buffer capacitybeing infinite or finite. It is always true that probability p0 is determined by packet servicetime distribution B(z) in any queueing system. This relationship will be obtained using theavailable queueing theory in [13] straightforwardly. Combining the relationships obtained inSection 3.1 and this section is sufficient to determine the performance of the tagged user andthe overall system. Both finite and infinite buffer systems will be covered.

3.2.1. Finite Buffer CapacityIn this case, the queueing model of the tagged user is Geo/G/1/L. Let ak be the probabilitythat k packets arrive at the tagged user buffer during a packet service time. The PGF of thenumber of packets arriving at the buffer during a packet service time is given by

A(z) = B(1 − λ+ λz) . (12)

Denote πi , i = 0, 1, . . . , L−1, as the probability that a leaving packet sees i packets in queue.Since a leaving packet holds one space before it leaves, it can see at most L− 1 packets in thequeue. Hence, the following equations can be obtained [13]:

π ′k = π0ak +

k+1∑j=1

πjak−j+1 , 0 ≤ k ≤ L− 2 , (13)

L−1∑i=0

πi = 1 . (14)

An efficient algorithm for computing πk, 0 ≤ k ≤ L− 1, can be found in [13] by defining

π ′k = πk/π0, k = 0, 1, . . . , L− 1 . (15)

Then, it is found π ′k, 0 ≤ k ≤ L− 1 can be recursively calculated as follows:

π ′0 = 1

π ′k+1 = 1

a0

π ′

k −k∑j=1

π ′jak−j+1 − ak

, 0 ≤ k ≤ L− 2 .

(16)

After π ′k, 0 ≤ k ≤ L− 1 is known, the probability π0 is found using the total probability law

(14) and (15) as

π0 =(L−1∑i=0

π ′k

)−1

. (17)


Let pk, 0 ≤ k ≤ L, be the probability that there are k packets present in the system at anarbitrary time (slot boundary). Using the results of the Geo/G/1/K system [13] again, we have

pk = πk

π0 + ρ, 0 ≤ k ≤ L− 1

pL = 1 − 1

π0 + ρ,

(18)

where

ρ = λb ,

b = B ′(1) .(19)

It is clear that b is the mean packet service time of each user.Equations (3)–(19) are sufficient for the performance analysis of the CSMA/CD system

with finite buffer capacity.

3.2.2. Infinite Buffer CapacityIn this case, the proper queueing model of the tagged user is Geo/G/1. Denoting πk and pk,0 ≤ k ≤ ∞, as the probability that a leaving packet sees k packets in the queue and theprobability that there are k packets present in the queueing system at any time, respectively, itis found [13] that their PGFs, denoted by %(z) and P(z) respectively, are given by

P(z) = %(z) = (1 − ρ)(1 − z)B(1 − λ+ λz)

B(1 − λ+ λz)− z, (20)

where ρ is still given by Equation (19). It is noticed that P(z) = %(z) holds only for theinfinite buffer system. In this case, we have

p0 = 1 − ρ . (21)

It is found that the analysis for the packet service time distribution in Section 3.1 and theanalysis of the queueing behavior of the tagged user in Section 3.2 are decoupled. Hence, ifuser queueing behavior is changed such as bulk packet arrival and last-come-first-serve servicediscipline, only the queueing theory on the tagged user needs to be changed correspondingly.On the other hand, if a different transmission principle is employed, only the analysis of packetservice time distribution has to be changed.

TUA can be extended to heterogeneous systems which consist of more than one group ofusers. In this case, a tagged user is selected randomly for each group. Considering a systemwithM groups, let Bi(z) be the PGF of the packet service time and pi,0 be the idle probabilityof the tagged user in group i. Then, using the state flow graph for the packet service timedistribution analysis of each tagged user, we have

Bi(z) = fi[p1,0, p2,0, . . . , pM,0], i = 1, 2, . . . , M , (22)

where fi[.] denotes a general function relationship. Using the classical queueing theory forthe queueing analysis of each tagged user, we also have

pi,0 = gi[Bi(z)], i = 1, 2, . . . , M , (23)

where gi[.] denotes a general function relationship. By solving the 2M equations in (22) and(23), the 2M unknowns, Bi(z) and pi,0, i = 1, 2, . . . , M, can be solved. Therefore, theheterogeneous system can be analyzed using TUA.


4. Performance Evaluation

In this section, the system performance is evaluated using the queueing analysis results ob-tained in Section 3. The performance measurements are focused on channel throughput, userqueue length, packet response time, and channel inter-departure time.

4.1. FINITE BUFFER SYSTEM

4.1.1. Packet Blocking ProbabilityFor the finite buffer system, the packet blocking probability, denoted by pB , is an importantperformance measure. Since an arriving packet will be rejected if it sees a full buffer, whichoccurs with probability pL, using Equation (18) we have

pB = pL = 1 − 1

π0 + ρ. (24)

4.1.2. ThroughputThroughput is defined as the average number of successfully transmitted information bits ineach bit time. In this paper, channel throughput and user throughput are defined. The channelthroughput is a measurement of the channel utilization rate. The user throughput is just thethroughput of each user queue. We emphasize again here that the statistical behavior of eachuser in the system is identical and equal to the behavior of the tagged user.

Since all of the packets accepted into the user queue would be transmitted successfullysooner or later, the user throughput is equal to the average information bits accepted by thequeue in one bit time. Hence, the user throughput, denoted by θ , is given by

θ = λT (1 − pB) (25)

since the mean rate in which packets are accepted into a user buffer is λ(1 −pB). The channelis shared by N users. The channel throughput, denoted by θ , is just the throughput contributedby all users. Hence, it is given by

+ = NλT (1 − pB) . (26)

Another expression for user throughput θ is given by

θ = pbT

b(27)

which is explained as the average effective packet transmission time of T slots out of anaverage packet service time of b0 slots timed by the percentage time pb when the user is busy.

4.1.3. Queue LengthDenote the random variable lq as the number of packets in the user buffer. The probabilitydistribution function of the queue length is already given in Equation (18). The PGF P(z) ofqueue length at any time is given by

P(z) =L∑k=0

pkzk , (28)


and the mean queue length is given by

E(lq) =L∑k=0

kpk . (29)

4.1.4. Waiting Time and Response TimeThe waiting time is the time duration from a packet entering the queue to its moving to thehead of queue and becoming ready for transmission. The system or packet response time isthe time duration from a packet entering the queue to its departure from the queue. Hence, thepacket response time is equal to waiting time plus the packet service time.

Denote the random variables tw and tr as the packet waiting time and response time in timeslots, respectively. From [13], the PGFW(z) and the meanE(tw) of the waiting time are givenby

W(z) = π0(1 − z)

1 − λ− z + λB(z)

(1 −

[λB(z)

z− 1 + λ

]L)+ [B(z)]L−1

L−1∑j=0

πj

(λ

z− 1 + λ

)L−j, (30)

and

E(tw) = W ′(1) = E(lq)

λ(1 − pB)− b . (31)

By definition, the PGF R(z) and the mean E(tr) of the packet response time are given by

R(z) = W(z)B(z) (32)

and

E(tr) = R′(1) = E(lq)

λ(1 − pB). (33)

4.1.5. Output Process of the ChannelThe analysis of the distribution of the inter-departure time in the channel is important inthe context of the interconnected local area networks or multihop networks. For example, achannel output process may constitute part of the input processes to other adjacent networks.The channel output process is therefore important in order to analyze the performance ofinterconnected networks.

One very difficult problem is how to obtain the distribution of the channel inter-departuretime. To our knowledge, there are no publications available so far on the analysis of thedistribution of channel inter-departure time of buffered multiple access protocols. Seriouslyspeaking, the state evolutions of all users are required in order to determine the channeloutput process Och(z). Only the equilibrium probabilities of all users are not sufficient.Unfortunately, this information is not available in TUA.

Using the assumption that each user always operates with its equilibrium probability dis-tribution, an approximation analysis of the channel output process is proposed here using thestate flow graph in Figure 2. The packet departure times are selected as the imbedded pointsfor the analysis. In Figure 2, g0 and ge are the nth and (n + 1)th packet departure pointsfrom the channel, respectively, and gidle is the status when the channel is idle. The channelinter-departure time is just the transition time from g0 to ge.


Figure 2. State flow graph of the channel.

After the nth packet departure, the channel becomes idle and enters gidle. If there is only oneuser which senses the channel at this slot, this user would initialize a successful transmission.The channel status will enter ge immediately after this successful transmission. Therefore, wehave

x1(z) = Npbp(1 − pbp)N−1H(z) . (34)

When no user senses the channel at this slot, the channel remains idle for one more slot andstays in gidle. If more than two users sense the channel at this slot, a collision will occur andthe channel keeps busy for D slots before returning to idle. Hence, we have

x2(z) = (1 − pbp)Nz+ [1 − (1 − pbp)

N − pbp(1 − pbp)N−1]zD . (35)

Therefore, the channel output process Och(z) is given by

Och(z) = x1(z)

1 − x2(z)= Npbp(1 − pbp)

N−1H(z)

1 − ((1 − pbp)Nz + [1 − (1 − pbp)N − pbp(1 − pbp)N−1]zD) . (36)

4.2. INFINITE BUFFER SYSTEM

One problem associated with the infinite buffer system is system stability. A finite buffersystem is always stable, although it may have one globally stable equilibrium point or twolocally stable equilibrium points. However, an infinite buffer system may be unstable in thesense that the mean packet arrival rate is larger than or equal to the mean packet service ratesuch that the buffer length increases without limit. In this section, we assume that the systemis stable.

Since all of the arriving packets have to be transmitted in this case, the channel throughputis given by

+ = NλT . (37)

The PGF of the queue length is already given in Equation (20). Hence, the mean queue lengthis given by

E(lq) = P ′(1) = ρ + λ2b(2) − λρ

2(1 − ρ), (38)

where ρ and b are still given by Equation (19), and b(2) is given by

b(2) = B ′′(1)+ B ′(1) . (39)


From [13], the PGF R(z) and the mean E(tr) of the packet response time are given by

R(z) = (1 − ρ)(1 − z)B(z)

1 − λ− z + B(z), (40)

and

E(tr) = R′(1) = E(lq)

λ, (41)

respectively. The PGF and mean of the waiting time can be obtained similarly if needed.

5. The Algorithm

We have sufficient equations for the system queueing analysis. However, it is almost im-possible to solve them analytically. For this purpose, an iterative numerical algorithm isrequired. Based on the discussions above, the key for the queueing analysis of the tagged useris to solve p0 and B(z) simultaneously. After that, the performance analysis of the overallsystem is straightforward.

In fact, an algorithm for solving p0 and B(z) can be obtained by following the analysissteps in Section 3 as follows:

(i) select a very small positive number δ, say δ = 1.0e − 8, and set an arbitrary initial valuebetween zero and one for p0;

(ii) calculate B(z) using Equations (1)–(11);(iii) calculate pk, 0 ≤ k ≤ L using Equations (12)–(19) to obtain a new estimate p0 of the

probability for the finite buffer system, or compute a new estimate of the probability p0

using Equations (19) and (21) for the infinite buffer system;(iv) if |p0 − p0| ≥ δ, let p0 = p0 and go to step (ii). Otherwise, p0 and B(z) are obtained.

5.1. SYSTEM EQUILIBRIUM POINTS

In this section, we discuss briefly the equilibrium points at which the system is supposed tooperate. In general, an infinite buffer system may be stable or unstable. In the former case,the system has only one globally stable equilibrium point. In the latter case, the system has noequilibrium point. When using the numerical algorithm above, the system always convergesto its globally stable equilibrium point for any initial value of p0 if the system is stable.Otherwise, the abnormal phenomenon occurs during the computation of p0 such that theestimate of p0 either diverges or converges to a value outside the region [0, 1]. Hence, thestability of the infinite buffer system can be determined using TUA based on the numericalresults.

A finite buffer system is always stable. However, it may be globally stable or locally stable.In the former case, the system has only one globally stable equilibrium point. In the latter case,the system has two locally stable equilibrium points, where one (corresponding to higherthroughput and shorter delay) is called the desired point and the other the saturation point.The latter case is also called bi-stable in some papers [7]. TUA can be used to determinethese equilibrium points as follows. Use the numerical algorithm above to compute the systemperformance measurements by setting initial values of p0 to 0 and 1. The results for initialvalue p0 = 0 and p0 = 1 would correspond to those of the saturated point and the desiredpoint, respectively, if the system is bi-stable. Otherwise, the results for any initial value of


p0 would be the same and correspond to that of the only equilibrium point if the system isglobally stable.

6. Numerical Results

In this section, three numerical examples are presented. In the first one, TUA is applied tothe CSMA/CD system model considered in [7] where delayed acknowledgement is assumed.We shall explain how to analyze this kind of system using TUA where immediate acknow-ledgement is assumed. In the second example, we apply TUA to the optimal CSMA/CDsystem in [8] where the user permission probability p is no longer a constant, but changesdynamically with the number of busy users in the system. Although TUA analysis is focusedon constant permission probability, it can be applied to the system here with little modification.In Example 3, we apply TUA to the CSMA system with fixed packet length in [6], which isa special kind of CSMA/CD system. The analysis results of this system using EPA are alsoavailable in [6]. It is found that a more reasonable analysis result is obtained using TUAthan using EPA. In addition, the simulation results available in the original publications arecompared with the analysis results of TUA in these examples. The analytical and simulationresults fit quite well. It should be mentioned that the analytical results of the channel outputprocess are not compared with the simulation. However, if the analysis accuracy of channelthroughput and packet delay is good, so is the accuracy of channel output process since theiranalyses are based on the same assumptions.

6.1. EXAMPLE 1: A NORMAL CSMA/CD SYSTEM

The system model in [7] is considered here with N = 100, λ = 0.00035, and a fixedpacket length of 20 slots. The buffer capacity and user permission probability are variable.In this model, it is assumed that the channel will be sensed idle one slot after the end of apacket transmission. That is, the channel propagation delay is one slot. However, immediateacknowledgement is assumed in the TUA analysis above. To apply the analysis to systemswith delayed acknowledgement, we modify the system model as follows.

Because of the one-slot propagation delay, no users will sense the channel idle immediatelyafter a successful transmission. That is, when a packet with a length of τ slots is transmitted,the channel will be sensed idle τ + 1 slots later if the transmission is successful. This isequivalent to a packet with length τ + 1 slots in a system with immediate acknowledgement.Hence, if the PGF of packet length in the system with one slot propagation delay isH ◦(z), theH(z) in the equivalent model with immediate acknowledgement is given by

H(z) = zH ◦(z) . (42)

However, if the channel throughput for the equivalent model is +, the practical channelthroughput +◦ should be given by

+◦ = +(T − 1)

T, (43)

where T is given by Equation (9), since each packet in the equivalent model has an averagelength of T slots, among which only T − 1 slots contain information data.

Here, we have H(z) = z21 since the packet length is fixed at a 20-slot transmission time.Similarly, since a collision will be detected by transmitting users in one slot and one more slotis needed because of propagation delay, we have D = 2 in the equivalent model.


Figure 3. Channel throughput of a CSMA/CD system.

The channel throughput is given in Figure 3 for different buffer capacities and permissionprobabilities, where the lines are the results obtained using TUA and the markers are thesimulation results from [7]. When the permission probability is small enough, the system hasonly one equilibrium point, as indicated by the solid lines. However, when the permissionprobability is sufficiently large, the system has two equilibrium points. One corresponds to ahigher throughput, as indicated by dashed lines. The other corresponds to channel congestion.In this case, channel throughput is low, as indicated by solid lines.

The packet response time is illustrated in Figure 4. The results show that the analysis andsimulation fit very well. The analytical results for the systems using EPA are also available in[7]. Comparing the results obtained using TUA with those using EPA, it is found that TUAand EPA have similar accuracy.

6.2. EXAMPLE 2: AN OPTIMAL CSMA/CD SYSTEM

Here, we consider a system without propagation delay, that is,D = 1. The user population andpacket size are fixed and given by N = 10 and H(z) = z40. The buffer capacity, permissionprobability, and packet generating rate are variable.

We suppose that during a contention period each user would attempt to sense the channelin a slot with probability 1/n◦, where n◦ is the number of busy users at the beginning of theslot. It is proved in [8] that if the above scheme is employed for determining user permissionprobability, the system performance is optimal in the sense of maximum channel throughput.

To apply TUA to an optimal CSMA/CD system, the analysis approach above has to bemodified. Now, we determine the permission probability for the tagged user. According tothe assumptions used in TUA, the tagged user would see each of the other users busy withprobability pb. For our analysis, the tagged user is busy with probability 1. Hence, there are


Figure 4. The system response time of packets of a CSMA/CD system.

Figure 5. Mean queue length versus total packet generating rate of an optimal CSMA/CD system.

n◦ = 1 + (N − 1)pb busy users on average in the system when the tagged user contends forthe channel. By the scheme above, the permission probability of the tagged user is given by

p = 1

1 + (N − 1)pb. (44)

Comparing the above analysis, we not only have one more unknown parameter p, but alsoone more Equation (44). Hence, the system can be solved. The numerical algorithm developedabove can be applied in this case with proper modification.


Figure 6. Mean packet response time (in the unit of packet transmission time) versus total packet generating rateof an optimal CSMA/CD system.

Figure 7. Packet blocking probability versus total packet generating rate of an optimal CSMA/CD system.

The numerical results are given in Figures 5, 6, and 7, where the lines correspond to theanalysis results using TUA and the markers to simulation results from [8]. Although the userpopulation is small, the accuracy of TUA is still very good. In fact, the errors between TUAanalysis and simulation may be less than what is shown in the diagrams since the simulationitself does not correspond to an optimal system, as shown in Figure 10 of [8]. Seriouslyspeaking, an optimal CSMA/CD system should have constant throughput for a sufficientlylarge packet response time. However, the throughput decreases very slowly with the increaseof packet response time in the simulation (see Figure 10 of [8]). Nevertheless, the model of


Figure 8. Channel throughput versus total packet generating rate of an optimal CSMA/CD system.

TUA here is optimal, as illustrated by Figure 8, where the channel throughput increases towarda constant for sufficiently large Nλ.

6.3. EXAMPLE 3: A NORMAL CSMA SYSTEM

The CSMA system corresponding to Figure 10.5 of [7] is considered here. We have N = 100and λ = 0.00035. The packet transmission time is 20 slots, like in Example 1. Since a slotpropagation delay is assumed in the system, we have H(z) = z21. In the CSMA system, auser would know about a collision only after a packet is transmitted. Hence, we have D = 21.

The numerical results are given in Figures 9 and 10, where the lines and markers cor-respond to the results obtained using TUA and the simulation from [7], respectively. It isfound that the system has only one equilibrium point for sufficiently small or large permissionprobability p, and two equilibrium points for the permission probabilities in between. Thisis because too many slots are wasted in each collision. When p is sufficiently large, frequentcollisions result in channel congestion. In fact, this is also true for the CSMA/CD system.However, only a few slots are wasted in each collision in the CSMA/CD system. In the widerange of values of p, the CSMA/CD system would be bi-stable. The system status wouldoscillate between these two equilibrium points. Only for very large p would the system be-come congested with Probability 1. For instance, in Example 1, the CSMA/CD system wouldbecome congested with Probability 1 for some sufficiently large p beyond the range of thediagrams.

The analysis results of this model using EPA is available in [7]. The results show that thesystem with a buffer capacity of 1, 8, and 32 packets is bi-stable for p beyond points C, B,and A, respectively. This is quite different from the results obtained using TUA. Based on theanalysis above, we believe that the results using TUA are more reasonable.


Figure 9. Channel throughput versus transmission permission probability in a CSMA system.

Figure 10. Packet response time versus transmission permission probability in a CSMA system.

7. Conclusions

In this paper, a new approach called TUA is proposed for the performance analysis of slottedCSMA/CD systems with finite user population and infinite or finite buffer capacity, whereeach user is treated as a Geo/G/1 or Geo/G/1/L queueing system, respectively. By assumingthat the channel is symmetric and each user always operates at its equilibrium point, the packetservice time is obtained using a state flow graph, which is then applied to the queueing analysisof the tagged user. Finally, the performance analysis of the overall system is obtained based onthe analysis results of the tagged user. Not only can TUA be applied to the queueing analysis


of the system, but it can also be applied to the stability analysis. One advantage of TUA isthat the analysis of the channel contention scheme and the analysis of the queueing behaviorof the tagged user are decoupled. Hence, TUA can be extended to more complicated systems.The simulation confirms that the accuracy of TUA is good.

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Tao Wan received a B.Sc. degree from Southwest Jiaotong University, Chengdu, China, in1984, a M.Sc. degree from Chongqing University, Chongqing, China, in 1987, and a Ph.D.degree from Carleton University, Ottawa, Canada, in 1999, all in electrical engineering. He iscurrently with Nortel Networks. His research interests include performance analysis and QoSof 3G wireless networks.

Asrar U.H. Sheikh graduated from the University of Engineering and Technology, Lahore,Pakistan with first class honours and received M.Sc. and Ph.D. degree from the University ofBirmingham, England, in 1966 and 1969 respectively. After completing teaching assignmentsin several countries, he returned to Birmingham as a research fellow in 1975. He worked atCarleton University from 1981 to 1997, first as associate professor and later as a professorand associate chairmman for graduate studies. He was the founder director of PCS ResearchLaboratory at Carleton University. Before taking position of Bugshan/Bell Lab Chair at KingFahd University of Petroleum and Minerals in April 2000, he was a professor and associatehead of the Department of Electronic and Information Engineering at Hong Kong Polytech-nic University. He is also the founder director of Wireless Information Systems Research(WISR) Centre. He has published over 166 papers in international journals and conferenceproceedings. Dr. Sheikh is a co-recipient of Paul Adorian Premium from IERE (London) forhis work on impulse noise characterization. He was awarded teaching achievement awards in1984 and 1986, and Research Achievement Award in 1994, all by Carleton University. He hasorganized and chaired many techical sessions at several international conferences. Dr. Sheikh


is actively involved in several international conferences mainly as a member of TechnicalProgram Committees. He chaired the Technical Program of VTC’98. He is on the editorialboard of Wireless Personal Communications. He was a co-guest editor of the Special Issue ofWPC on Interference. Dr. Sheikh has been consultant to many private companies and govern-ment agencies. His current interests are in signal processing in communications, mitigation,spread spectrum, and 3G systems. His other interests include helping developing countries ineducation and research. He had assignments under UNDP’s sustained Development Program.He is a fellow of the IEE and a senior member of the IEEE.

Performance Analysis of Buffered CSMA/CD Systems

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