real-time communications using tdma-based multi-access protocol

Real-time communications using TDMA-based multi-access protocol

F. Simonota, Y.Q. Songb

aESSTIN-Universite´ Henri PoincareNancy 1, Parc R. Bentz, 54500 Vandoeuvre, FrancebCRIN-Universite´ Henri PoincareNancy 1, 2 av. de la foreˆt de Haye, 54516 Vandoeuvre, France

Received 26 July 1994; accepted 9 February 1996

Abstract

In this paper, we study first the performance issues, especially message sojourn time and message overflow, of the TDMA protocol andthen show how to use these results to design protocol parameters in order to meet the soft real-time requirements. We consider a single queuewith multiple output (b messages) and finite buffer capacity (denoted byK), K andb being two protocol parameters that a designer canchoose. Assuming Poisson fixed-length message arrivals, the exact expressions for the distribution function of the message sojourn time andthe overflow probability are derived. It has been shown that these results can be applied to various particular cases (e.g. the infinite buffercapacity case) and therefore appear as a versatile solution for TDMA schemes. Moreover, the monotonicity for the overflow probability isproved, which gives us a theoretical base for the dimensioning task. Using the stated results, the design and implementation of real-timeprotocols based on TDMA can be achieved. An example is examined emphasizing the choice of the parametersK andb according to time andoverflow constraints.q 1997 Elsevier Science B.V.

Keywords:Real-time communication; Multi-access protocol; TDMA; Performance evaluation; Deadline miss probability; Dimensioning

1. Introduction

The introduction of networking technology into distribu-ted real-time applications, such as process-control and inter-active multimedia, brought attention to the performancestudy of many time-constrained multi-access protocolssuch as Timed-token-based and TDMA-based[12,18,31,34] or even CSMA-based by introducing somedeterminist mechanisms [15]. As the messages exchangedin a real-time application are often time-constrained andcommonly characterized by adeadline, the key problemfor a real-time communication system is how to transmitsuch a message before its deadline. According to timingrequirements, the distributed real-time applications can beclassified into two classes:hard real-timeandsoft real-time.The former focuses on meeting strict time constraints in apredictable way, while the latter requires to meet the dead-lines of messages with a certain probability. For both theclasses, taking into account the performance parameters ofthe subjected communication system during functional spe-cification (design phase) is of prime importance for obtain-ing a satisfying implementation (good behaviour, highquality of service, etc.) [33].

Many telecommunication networks and local area net-works use multiple-access protocols based on TDMA, forinstance, the subframe switching reported in Foschini et al.

[6], loop communication systems [10] and circuit-switchedTDMA used by local, metropolitan and satellite networksproviding multiple-access support to real-time services [23],and the MAC protocol dealing with time-constrained trafficin FIP fieldbus standards [1].

Since most of the existing systems have not been initiallydesigned for real-time communication (for instance, onlyFIFO scheduling is available in these systems, no advancedscheduling algorithms like Earlest Deadline First or RateMonotonic are introduced), we study the QoS (Quality ofService) in terms of thedeadline miss probability. Takinginto account the fact that all buffers have a finite capacity inany actual system, themessage overflow probabilityis con-sidered as another QoS parameter. It is worth noting that themessage sojourn time in a finite-capacity send queue isalways upper bounded, which was not the case in manytraditional analyses in which the finite buffers are approxi-mated by infinite ones.

In this paper, we study first the performance issue ofTDMA and then, using these results, we show how todimension the protocol parameters to meet soft real-timerequirements with the best effort. Readers interested inhard real-time aspects should refer to [29,17,16].

TDMA is a predetermined-assignment multiple-accessprotocol to a single transmission channel. This communica-tion channel is divided into equal-duration time slots. The

Computer Communications 20 (1997) 435±448

0140-3664/97/$17.00q 1997 Elsevier Science B.V. All rights reservedPII S0140-3664(97)00045-5

slots are allocated to all stations in the network in a cyclicmanner.

The basic TDMA orfixed-assignedTDMA has three dif-ferent allocation policies:

• single slot per station per cycle,• multiple contiguous slots per station per cycle,• multiple non-contiguous slots per station per cycle.

The queue-size and message delay performance of TDMAschemes have already been investigated by many research-ers under various assumptions. To make our discussioneasier, we divide the message delay into three parts: RMD(Residual Message Delay), DMD (Discret Message Delay)and TD (Transmission Delay) (see Fig. 1 and Fig. 2).Related work is briefly described below.

(1) Under the assumption ofunlimited buffer capacity,the queue-size analysis has been more or less completed.Chu and Konheim [4], Kobayashi and Konheim [11], Lam[13] and Rubin [19] have derived the generating function ofthe queue length for a TDMA scheme under which a singleslot per frame is allocated to each station. The messagedelay distribution has been derived for a single slot perframe TDMA model by Lam [13] and Rubin [19]. For themultiple slot per frame TDMA scheme, an approximateanalysis was carried out by Rubin [20]. The average delayexpression is given by Hayes [7], Foschini et al. [6] and Koand Davis [10] with Poisson arrival and contiguous slots.Using a different approach, Bruneel [3] tried to extend theprevious work and to obtain the generating function of themessage delay for a general uncorrelated message arrivalprocess, i.e. the numbers of arrivals at a station during thesubsequent frames are described as a sequence of i.i.d.(independent identically distributed) random variables.Under the same assumption, Lee and Liang [14] have ana-lysed the DMDþ TD. Rubin and Zhang [23] have studiedthe circuit-switched TDMA in which we find some interest-ing results under the assumption of a geometric or aBernoulli arrival stream (i.e. a discrete version of the com-pound Poisson process). Bounds for the DMDþ TD havebeen derived in [28]. More complete results can be found in[24]. In all the above-mentioned works, theplease waitstrategy [8] is considered. Another strategy calledcomeright in is addressed by Rom and Sidi with non-contiguousslot assignment [21]. Some practical applications of theavailable results can be found in Rubin [23] for thecircuit-switched TDMA used in integrated service commu-nication, in Song [27] and in Simonot et al. [25] for the time-critical protocol proposed by FIP standards [1].

(2) For limited buffer capacity, we can find some resultsconcerning queue-size in the work of Simonot et al. [25], of

Tijms [32] and of Bhat [2]. As for the delay analysis, recentwork by Simonot et al. [26] gives an exact expression of themessage sojourn time distribution by assuming Poissonfixed-length arrivals and finite buffer capacity with multiplecontiguous outputs.

We are interested in the sojourn time probability distribu-tion of the average message(the meaning of this averagemessage will be explained in Section 2) in the buffer and theoverflow probability due to limited buffer size. Since thesetwo parameters are of major interest to users, we believe thatan efficient implementation of a soft real-time communica-tion system needs accurate knowledge about both messagedelay and overflow probability. This problem has been themain motivation for the present paper.

The rest of this paper is organized as follows. Section 2consists of a description of the queuing model and messagesojourn time which constitutes the most useful performanceindicator for real-time network design. Other operatingcharacteristics, such as overflow probability, are alsogiven. Section 3 is dedicated to the reporting of some the-oretical results about the probability distribution of thesojourn time and of some useful properties about overflow,first under a general input flow assumption and then underPoisson arrival. Traditional parameters such as theLaplace–Stieltjes transform and the mean value of thesojourn time are also reported as one of the direct conse-quences of the main result. Numerical insights are also per-formed with simulation to emphasize analytical results.Application to the infinite buffer capacity case is workedout and the classical result of Ko and Davis [10] is derived.Section 4 provides an example showing how to design softreal-time protocols based on TDMA schemes. Finally, Sec-tion 5 summarizes the work done and sketches the potentialapplication fields of our results. Some future research direc-tions are also indicated.

2. Modelling

Consider a TDMA scheme with multiple output and finitebuffer capacity. In such a system, time is divided into timeslots of constant length. A fixed number, sayb, of slots isallocated to each station in a periodical way. For a givenstation, a frame of lengthT is defined as the time periodbetween the ends of two adjacent groups ofb transmissionslots. It is assumed that the stations have at their disposal a

Fig. 1. TDMA model.

Fig. 2. Delay evolution in TDMA.

436 F. Simonot, Y.Q. Song/Computer Communications 20 (1997) 435–448

finite buffer of capacityK for the temporary storage ofmessages awaiting transmission. All the messages arrivingand finding the buffer full are lost. The messages have thesame constant length of one slot each and are transmitted ona FIFO basis.

2.1. Model description

Generally, the analysis of the message delay in theTDMA, in which the input flows corresponding to the sta-tions are mutually independent, can be reduced to the studyof one station which can be modelled as a simple queuingmodel (see Fig. 1 and Fig. 2).

Fig. 1 shows a queuing model with a finite bufferK andwith an intermittent server. The reader should not be sur-prised by Fig. 1 in which the model is built with two finitebuffers, one withK places and another withb places. In fact,in a TDMA scheme, once the switcher closes, we are surethat at mostb messages can be served from the consideredstation. So, for the convenience of the sojourn time investi-gation, we can add a virtual buffer of capacityb followingthe switcher and consider that this batch of messages will betemporarily stored in this virtual buffer for at most theduration of one frame.

Fig. 2 shows the evolution of the whole message delay insuch a model for the particular case corresponding tobcontiguous slots. The behaviour of such a system can bedescribed as follows:

• The arriving messages of fixed length (1 slot) are firststored in the buffer, the capacity of which is limited toKmessages. Each message which upon arrival finds a fullbuffer is lost and does not influence the system anylonger.

• At fixed epochsn¼ 0,T, 2T, …, messages are taken outfrom the buffer and are transmitted. Any interval)nT, (nþ 1)T) can be a frame.

• At mostb messages can be transmitted during a frameT.Obviously, 1# b # K.

• The transmission of a message can only start at thebeginning of a frame so that a message arriving duringthe nth frame must wait until the beginning of the nextframe (nþ 1)T. This delay is referred to as residualmessage delay (RMD).

2.2. Operating characteristics

We now describe the parameters which will be investi-gated later in this paper. LetX(t) be the number of messagesin the buffer at timet; Xn ¼ X(nT) is the number of messagesin the buffer at the end of thenth frame, just before trans-mission; Yn ¼ X(nTþ ) is the number of messages at thebeginning of the(nþ 1)th frame, just after transmission;n(t) is the number of arrivals during [0, t];An ¼ n(nT) ¹ n((n¹ 1)T) is the number of arrivals duringthe nth frame((n¹ 1)T,nT). With these notations in hand,

we are able to write the following relations:

Xnþ 1 ¼ min(Yn þ Anþ 1,K) (1)

Yn ¼ max(0,Xn ¹ b) (2)

From Eq. (1) and Eq. (2) we derive the classical recursiveequations:

Ynþ 1 ¼ min(Yn þ Anþ 1,K) ¹ min(Yn þ Anþ 1, b) (3)

Xnþ 1 ¼ min(max(0,Xn ¹ b) þ Anþ 1, K) (4)

Moreover, fornT , t # (nþ 1)T:

X(t) ¼ min(K,Yn þ n(t) ¹ n(nT)) (5)

For the type of communication system considered, twocharacteristics play a fundamental role for the evaluationof performance, namely:

• the message sojourn time,• the message overflow.

The following section is devoted to a detailed study of thesetwo characteristics, in order to achieve an optimal choice ofK andb. The message sojourn time is figured out since it isof prime importance for real-time communication systems.Considering the complexity of the investigation, we willonly report some results of practical interest withoutdetailed proofs (see [26] for more details). The messageoverflow is another performance indicator of interest forusers.

Before proceeding further, let us define the averagemessage.

2.2.1. Average messageWe refer to the model of Fig. 2; among the entering

messages we pick up a message at random; this selectedmessage defines what has been called theaverage message.Moreover, we call a batch the set of messages arriving dur-ing a frame, and introduce the eventCm as the averagemessage comes from a batch of size m.

Remark. Instead of considering the average message, we canalso consider theworst message, i.e. the message whichsuffers the longest sojourn in each batch.

2.2.2. Message sojourn timeThe message sojourn time only depends on the number of

messages that the average message finds upon arrival. Aswas stated at the beginning of this section, the number ofmessages at the beginning of thenth frame (at timenTþ)obeys the recursive Eq. (3) which is well known in storagesystems.

Now let R be the rank of the average message among itsbatch of arrivals andT ¹ t be the residual delay, i.e.t is theelapsed time between the beginning of the current frame andthe arrival epochu of the average message (see Fig. 2).Formally, t can be written ast ¼ u¹ [u=T]T (where [a]

437F. Simonot, Y.Q. Song/Computer Communications 20 (1997) 435–448

means the integer part ofa). The sojourn time of the averagemessageW in the buffer is then given by the sum:

W¼Rþ Y¹ 1

b

� �

T þ T ¹ t ¼ÿRþ Y

bT ¹ t½ (6)

whereR¹ 1þ Y is the number of messages in the bufferseen by the average message and ]a[ means inf{n; n $ a}.

2.2.3. Message overflowMessage overflow can occur if we consider a finite buf-

fer—a realistic viewpoint. The user’s requirements for QoSshould include a criteria about the message overflow prob-ability. Some results can be found in [32]. In the followingsection we give some proofs concerning basic results. Mes-sage rejection can be performed in many ways: FIFO, LIFO,or other imaginable policies. We recall that we restrict our-selves to FIFO.

3. Message overflow and sojourn time evaluation

3.1. Basic results for the overflow probability

In this subsection we prove that the overflow probabilityis non-increasing inK andb.

3.1.1. Overflow probabilityFor a communication system with parameters(K,b), let

us define the following quantities:

• the cumulative overflowCK,b(t), i.e. the number of mes-sages rejected during]0, t];

• the overflow of the nth frame:RK,b(n) ¼ CK,b(nT) ¹ CK,b((n¹ 1)T) which can be writ-ten RK,b(n) ¼ max(0, Yn¹ 1 þ An ¹ K), bearing in mindthatYn¹ 1 represents the level in the buffer at the begin-ning of thenth frame;

• the overflow probability pK,b defined aspK,b ¼ limt→`CK,b(t)=n(t) (when this limit exists).

To proceed further we need some additional assumptions:from now on, we assume that the r.v.’sAn are i.i.d., noticingthat this condition is fulfilled particularly when the arrivalstream of messagesn ¼ (n(t)) possesses stationary indepen-dent increments. With theAn i.i.d., (Xn) and (Yn) becomediscrete Markov chains (finite ifK , þ `), and under mildassumptions on the d.f. ofAn, these chains admit a long rundistribution. The overflow probabilitypK,b obeys the classi-cal properties:

P1: pK, b ¼ lim t→`CK,b(t)=n(t) exists with probability 1and definespK,b.P2: Denoting(pj)1#j#K the long run distribution of theMarkov chain (Xn), pK,b turns intopK,b ¼ 1¹ limn→`E[min(b, Xn)]=E[A] or explicitly:

pK, b ¼ 1¹1

E[A]

∑b¹ 1

j ¼ 0jpj þ b

∑Kj ¼ b

pj

!

P3:pK,b may also be expressed with the overflow of thenth frame through the formula:

pK,b ¼1

E(A)limn→`

E[RK,b(n)]

3.1.2. Comparison of sample pathsLet S1 and S2 be two communication systems with

respective parameters(K1,b1) and (K2, b2) where1 # bi # Ki # þ `, and with respective arrival streamsn1 ¼ (n1(t)) andn2 ¼ (n2(t)) possessing the same probabilitydistribution, and denote byX1 ¼ (X1(t)) and X2 ¼ (X2(t))the number of messages in the buffers at timet for S1

and S2.In the sequel we adopt the common approach of establish-

ing stochastic ordering relations by making sample pathcomparisons. Consider the stochastic processesX1(t) andX2(t) generated by the systemsS1 and S2 for a commonarrival streamn ¼ (n(t)) with the same probability distribu-tion asn1 or n2 and with constant initial levelsX1(0þ) andX2(0þ), hence the two stochastic processesX1 and X2 aredefined on the same probability space and possess the sameprobability distribution asX1 and X2. For any elementaryevent q, we can compare the sample paths (X1(t;q)) and(X2(t;q)) of the stochastic processesX1 and X2 describingthe number of messages in the buffer at timet and for thesame arrival streamn ¼ (n(t)). Referring to Eq. (1), Eq. (2),Eq. (3), Eq. (4) and Eq. (5) controlling the evolution of theseprocesses, we can assert:

Lemma 3.1. IfX1(0þ ) $ X2(0þ ), b1 # b2 and K1 $ K2 thenX1(t q) $ X2(t q) for any t andq.

The proof is by induction:

• for t # T we can see thatX1(t q) $ X2(t q) sinceX1(0þ ) $ X2(0þ );

• if X1(t q) $ X2(t q) for any t # nT thenX1(nTq) $ X2(nTq), since b1 # b2 we haveX1(nTþ q) $ X2(nTþ q), therefore X1(t q) $ X2(t q)for any t # (nþ 1)T due toK1 $ K2. Q.E.D.This resultis not really surprising!

Returning to the original processesX1 ¼ (X1(t)) andX2 ¼ (X2(t)), this lemma can be translated in terms of sto-chastic ordering:X1(0þ ) $ X2(0þ ), b1 # b2 and K1 $ K2

imply X2(t) #st X1(t) for anyt, where # st means ‘‘stochas-

tically smaller than’’, namelyP(X2(t) . x) # P(X1(t) . x)for any x (see [30,22] for further information).

3.1.3. Monotonicity for the overflow probabilityThe initial conditionXi(0þ) of Xi does not influence the

overflow probability, hence, without restriction we mayassume thatX1(0þ ) $ X2(0þ ). Define Yin ¼ Xi(nTþ ),Xin ¼ Xi(nT), An ¼ n(nT) ¹ n((n¹ 1)T) andRi

Ki ,bi(n) ¼ max(0, Xi

n¹ 1 þ An ¹ Ki), for i ¼ 1,2.First, consider the case whenb1 # b2 and


K1 ¼ K2 ¼ K , `. The lemma impliesR2K,b2

(n) # R1K,b1

(n)for any n; Ri

Ki ,bi(n) andRi

Ki ,bi(n) have the same probability

distribution; as a consequence E[R2K, b(n)] ¼

E[R2K, b(n)] # E[R1

K,b(n)] ¼ E[R1K,b(n)]; reference to P3

yields the desired result,pK, b1$ pK,b2

.If b1 ¼ b2 ¼ b and K1 $ K2, Lemma 3.1 shows that

X2n # X1

n; then E[min(b, X2n)] ¼ E[min(b, X2

n)]#E[min(b, X1

n)] ¼ E[min(b,X1n)]; a look at P2 convinces that

pK1,b # pK2,b.We can now state the main result of this section:

Theorem 3.1. If b1 # b2 and K1 # K2 thenpK1,b1$ pK2,b2

.

3.2. Sojourn time

As mentioned in Section 1, the main available results onlyconcern the average values and generating function expres-sions. We provide here a more operational result, the p.d.f.of the sojourn time. Of course, the average values and gen-erating function expression are also derived to show theversatility of our results.

3.2.1. Probability distribution of the sojourn timeReferring to Eq. (6), we note that it is impossible to deal

separately witht and ](Rþ Y)=b[ for the calculation of thedistribution function ofW, since they are both dependent onR and hence not independent, which prescribes the use ofconditional probability.

Let F(w) be the distribution function ofW; conditioningon R, Y andCm yields:

F(w) ¼ P[W # w] ¼∑K ¹ b

j ¼ 0

∑m$1

∑min(m,K ¹ j)

r ¼ 1

[P(W # wlR¼ r,Cm,Y¼ j)P(R¼ r,CmlY¼ j)P(Y¼ j)] ð7Þ

In this formula three quantities appear:(1) P(Y¼ j) ¼ qj where(q0,q1, …,qK ¹ b) is the stationary

distribution of the Markov chain(Yn)n$0, which can becomputed separately using ordinary procedures [32].

(2) P(R¼ r, CmlY¼ j), which is unknown and depends onthe way the average message is selected. In wordsP(R¼ r, CmlY¼ j) means: the probability of the averagemessage having rankr among its arrival batch ofm mes-sages, knowing that there arej messages at the beginning ofthe frame.

We reason as follows: letM be a great number of batchescorresponding toY¼ j; since the average message isselected at random (among the entering messages) the num-ber of messages belonging to a batch of sizemand with rankr is approximatelyM 3 am, and the number of enteringmessages is approximatelyM

∑m$1min(m,K ¹ j)am since

at most K ¹ j messages can enter the buffer. LettingMtend to infinity, by the law of large numbers we get:

P(R¼ r, CmlY¼ j) ¼am

mj

where for 0# j # K ¹ b:

mj ¼∑m$1

min(m,K ¹ j)am ¼ K ¹ j ¹∑K ¹ j

k¼ 0(K ¹ j ¹ k)ak

(3)P(W # wlR¼ r, Cm,Y¼ j) ¼ P(t $ ¹ wþarj TlR¼ r,Cm, Y¼ j) ¼ 1¹ P(t # ¹ wþarj TlR¼ r, Cm,Y¼ j), witharj ¼ ](r þ j)=b[.

It remains to deal with the conditional distribution of theresidual message delay:

P(t # tlR¼ r,Cm, Y¼ j)

In general, the calculation of this conditional distribution isvery difficult. But if (n(t)) is a Poisson process with intensityl, we have:

P[An ¼ m] ¼ am ¼ e¹ lT (lT)m

m!

It is worth pointing out that from now on all our results willrely on this assumption. We can then claim the followingproposition (see [26,5] for the proof).

Proposition 3.2. For1 # r # min(m,K ¹ j):

P(t # tlR¼ r,Cm, Y¼ j) ¼

0 if t # 0

1b(r,m¹ r þ 1)

∫t=T0

ur ¹ 1(1¹ u)m¹ rdu if 0 , t , T

1 if t $ T

ð8Þ

8

>>>>>><

>>>>>>:

whereb(p,q) denotes theb function defined by

b(p,q) ¼

∫1

0up¹ 1(1¹ u)q¹ 1du

A simple derivation of the above formula yields the con-ditional p.d.f. denoted byf r

m:

f mr (t) ¼

tr ¹ 1(T ¹ t)m¹ r

Tmb(r ,m¹ r þ 1)x)0,T((t) (9)

wherex)0,T((t) is the indicator of the set)0,T(.Let us examine the p.d.f. of the sojourn time denoted by

f(w). Substitution in Eq. (7) followed by a derivation yields:

f (w) ¼∑K ¹ b

j ¼ 0qjm

¹ 1j

∑m$1

am

∑min(m,K ¹ j)

r ¼ 1f mr (arj T ¹ w)

Inverting the order of summation, after a trite calculation weget a much simpler form of the p.d.f. ofWand state the mainresult:

Theorem 3.2.

f (w) ¼ l∑K ¹ b

j ¼ 0qjm

¹ 1j

∑K ¹ j

r ¼ 1

e¹l(arj T ¹ w) (l(arj T ¹ w))r ¹ 1

(r ¹ 1)!x)arj T ¹ T,arj T((w) ð10Þ


The distribution function can be restored through:

F(w) ¼ P(W , w) ¼

∫w0

f (u)du (11)

This result is the key to deriving the usual operating char-acteristics of the system. For instance, in a real-time appli-cation, one would like to know the percentage of messagesthat are transmitted successfully (prior to their deadlines),which can be easily obtained through the probability distri-bution of the sojourn time. This rather intricate formula is anexplicit expression of the p.d.f. which enables us, as a rule,to figure out any quantity depending onW, especially theLaplace-Stieltjes transform ofW as well as the expectation.This is performed below.

3.2.2. Consequences

3.2.2.1. Laplace–Stieltjes transform of W. LetLw(s) ¼ E(esw) be the Laplace–Stieltjes transform ofW; itcan be computed by conditioning and hence following thesame procedure as in the proof of Proposition 3.2 or by theintegration of eswf (w).

Lw(s) ¼ l∑K ¹ b

j ¼ 0qjm

¹ 1j

∑K ¹ j

r ¼ 1esarj T

∫T0

e¹ (l þ s)u (lu)r ¹ 1

(r ¹ 1)!du (12)

We note that the integral in Eq. (12) can be reduced toan incompleteG function and hence put into a finite sum,since:

Ga(r þ 1) ¼

∫a0

e¹ t tr

r!dt ¼ 1¹

∑r

k¼ 0e¹ a ak

k!

3.2.2.2. Average sojourn time.The derivation of the aboveLaplace–Stieltjes transform ofW results in:

E(W) ¼ T∑K ¹ b

j ¼ 0qjm

¹ 1j

∑K ¹ j

r ¼ 1arjGlT(r) ¹

rlT

GlT(r þ 1)� �

(13)

3.2.3. Infinite capacityThe general formulae stated above for any buffer capacity

also provide the results corresponding to the infinite buffercapacity case.

This model appears in many communication systemssuch as subframe switching [6], loop communication sys-tems [10] and more recently circuit-switched TDMA usedby local, metropolitan and satellite networks providing mul-tiple-access support to real-time services [23]. The calcula-tion of E(W) in this case has been performed by Ko andDavis [10] with Poisson input, and by Bruneel [3] undermore general input assumptions. Only the DMD has beeninvestigated. The calculation of the RMD has not been

performed, but is arbitrarily replaced byT/2 which corre-sponds to the case of Poisson input. We note that the DMDonly is not enough to assess the performance of such amodel, due to the fact that when the offered load is small,the DMD is often zero.

AssumingE(A) ¼ lT , b to ensure a steady state, thenmj ¼

∑m$1min(m,K ¹ j)am ¼

∑m$1mam ¼lT remember-

ing that the arrival stream of messages is a Poisson processwith parameterl.

The expectation ofW is derived and compared with theresults given by [10] and [3].

Eq. (13) withK ¼ þ ` yields:

E(W) ¼1l

∑j$0

qj

∑r$1

arjGlT(r) ¹r

lTGlT(r þ 1)

� �

Standard but lengthy calculations result in:

E(W) ¼T2b

þTE(Y)

bþ

lT2

2b

þlT ¹ b

lb

∑b¹ 1

k¼ 1

bk

bk ¹ 1

∏b¹ 1

n¼ 1

bk ¹ zn

1¹ zn

with b¼ exp(2ip=b).This formula fits exactly with the result given by Ko and

Davis [10] and Bruneel [3]. Moreover, this result can begreatly simplified. In fact, it can be proved thatE(W) ¼ T=2þ E(Y)=l. And it is well known that:

E(Y) ¼(lT)2 ¹ b(b¹ 1)

2(b¹ lT)þ

∑b¹ 1

n¼ 1

11¹ zn

wherez1, z2, …, zb¹1 denote the roots ofzb ¹ exp(lT(z¹ 1))lying inside the unit circle. Therefore:

E(W) ¼T2

þ(lT)2 ¹ b(b¹ 1)

2l(b¹ lT)þ

1l

∑b¹ 1

n¼ 1

11¹ zn

(14)

which is better suited for practical calculations.

3.2.4. Numerical resultsBased on the preceding formulae, some numerical results

concerning the probability distribution and the averagevalue of the sojourn time are derived. The first step consistsin the calculation of the steady state distribution of thenumber of messages in the buffer at the beginning of eachframe, i.e.(qi)0#i#K ¹ b. These probabilities are obtained bysolving a set ofK ¹ b linear equations using efficient itera-tive procedures such as the over-relaxation method [32].With these qi in hand, the achievement off(w), F(W)andE(W) becomes a simple routine.

3.2.4.3. Probability distribution function of W. To get abetter insight into the preceding results, the p.d.f. off(w)is shown in Fig. 3, Fig. 4, Fig. 5 and Fig. 6 with differentvalues of parameters (K, b, l).


First, we note that formulaf(w) is a linear combination ofdiscontinous functions, the discontinuties being located inthe set 0,T, 2T, …. The shape off(w) is easily understoodwhenl is very small or very large.

• Whenl is very small, the probability that more than one

message arrives during a frame is negligible, and henceat the beginning of a frame the system is empty, there-fore the sojourn time of this message reduce toT ¹ t, thePoisson assumption ensurest is uniformly distributed on)0,T( and the same is true forT ¹ t. This fact can beclearly seen in Fig. 3 and Fig. 5.

Fig. 3. P.d.f. of sojourn time forK ¼ 63 andb¼ 2.



• If l is very large, the system is always full, hence at thebeginning of a frame there areK ¹ b messages. During aframe a great number of messages arrive but only thefirst b among these messages are allowed to enter thesystem within a short time interval at the beginning ofthe frame. LettingK ¼ bqþ r with 0 # r , b, we see

that the sojourn time of theb¹ r messages among theb messages allowed to enter the system reduces toW < ((K ¹ bþ b¹ r)=(b))T ¼ qT¼ [(K)=(b)]T. For thelastr messages among theb allowed to enter, the sojourntime is given by W < [(K)=(b)]T þ T. The ratio ofmessages leaving at time[(K)=(b)]T is ((b¹ r)=(b)),




and the ratio of messages leaving at time[(K)=(b)]T þ Tis ((r)=(b)) ¼ ((K)=(b)) ¹ [(K)=(b)] (see Fig. 4 and Fig. 6).

Since the d.f.F(w) is of particular interest for designing areal-time communication system, some graphical represen-tations are provided in Fig. 7, Fig. 8, Fig. 9 and Fig. 10.

3.2.4.4. Average sojourn time.The calculated values aswell as the simulated values of the average message sojourntime E(W) versus the normalized offered loadr¼ lT=b areshown in Fig. 11 for buffer capacityK ¼ 63, and in Fig. 12for buffer capacityK ¼ 11, both for four different values ofoutputb per frameT.

Fig. 7. Distribution function of sojourn time(K ¼ 63,b¼ 2).



A computer simulation of the model is also performed inorder to validate the theoretical results. Using QNAP2, aQueuing Network Analysis Package, this objective is easilyachieved.

From these two figures it is easily seen that the simulationresults strongly agree with the theoretical results.

4. Dimensioning

This section is devoted to showing, through an example,how the preceding results can be used to dimension TDMA-based multi-access protocols for meeting the real-time con-straints with best effort.


Fig. 10. Distribution function of sojourn time(K ¼ 11, b¼ 5).


Recall that the two main constraints arising concern:

• the message overflow probability, which must be smallenough;

• the message queuing time, which must be less than themessage life time.

So, for a given load generated by a userr ¼ E(An) ¼ lT, Kandb must be chosen according to the following constraints:

• C1: overflow probabilityPK,b # e1;• C2: deadline miss probabilityPK,b(W . w0) # e2;

where the values ofe1, e2 andw0 are given.

Fig. 11. Average message sojourn timeE(W)/T versus normalized offered loadr ¼lT=b for K ¼ 63, b¼ 1, 2,5, 10.

Fig. 12. Average message sojourn timeE(W)/T versus normalized offered loadr ¼lT=b for K ¼ 11, b¼ 1, 2,5, 10.


For a Poisson arrival process, the overflow probabilityturns into:

PK, b ¼ 1¹1

lT

∑b¹ 1

j ¼ 0jpj þ b

∑Kj ¼ b

pj

!

(15)

To provide a general guide for dimensioning, we summarizesome useful properties of the overflow probability as well asa relationship betweenK andb regarding the deadline missprobability.

Property 4.1. From Theorem 3.1, we have:PK,b is decreas-ing when b and/or K is increasing.

Property 4.2. IflT , b thenlimK→`PK,b ¼ 0.

Property 4.3. If K¼ b, from Eq. (15), we have:

PK, b ¼E(A¹ K)þ

E(A)¼ 1¹

KlT

þ1

lT

∑K ¹ 1

j ¼ 0(K ¹ j)aj (16)

Property 4.4. The condition: PK,b(WK, b . w0) # e2 is ful-filled when

K ,w0

T

h i

b (17)

with w0 $ 2T.The proof of Property 4.4 is straightforward. As a mes-

sage waits in a queue of capacityK and at mostb messagescan be transmitted each timeT, the last message in the queuewill wait WK,b which is upper bounded by:

WK, b #Kb

� �

þ 1

� �

T

ReplacingWK,b by w0 and taking the upper bound, we obtainEq. (17).

Now let us consider the following configuration corre-sponding to lT ¼ 3 and the life time of the tagged

messagew0 þ TD¼ 3T þ TD, with the following require-ments:

·PK, b # 10¹ 5

·P(W . w0) # 10¹ 2

We look for suitable values ofb andK.Fig. 13 shows how can we find the values ofK andb. The

procedure is rather simple. One follows the steps below:(1) Determining a closed search region. Since

[r] þ 1 # b # T, we havebmin ¼ [r] þ 1 andbmax¼ T (theduration of one slot is 1). From Property 3.1, we obtain thevalue ofKmaxby calculatingPK, bmin

from K ¼ bmin until Kmax

for which PKmax,bmin# e1 begins to be met. We note that

K , b is not an interesting case since the transmission med-ium will be in the starving state. The intersection ofb $ [r] þ 1, b # T and K # Kmax form a closed searchregion of the suitable values of (K,b).

(2) Determining the low boundary for C1. We calculate,using Algorithm 4.1, the smallest values ofK and b withwhich the overflow constraint is still satisfied. The lowboundary of the overflow constraint is thus obtained.Although the upper region of the low boundary within thesearch region gives all interesting values of (K,b) satisfyingthe overflow constraint C1, it is not certain that the timeconstraint C2 is also satisfied.

(3) Determining the region within which both C1 and C2are met. We draw the lineK ¼ [w0=T]b. If the line goesacross the search region determined by the above step, itdivides the above region into two domains: domain I anddomain II (see Fig. 13). We know from Property 4.4 that thedeadline constraint C2 is satisfied for any pair (K,b) ifb # K , [w0=T]b. That means that all pairs (K,b) takingvalues within domain I fulfil both C1 and C2. Since theupper bound given by Eq. (7) is not tight, suitable valuesof (K,b) can also be found within domain II. But this willneed the calculation ofPK, b(WK,b . w0) for each chosenpair (K,b) using Eq. (10) in order that C2 is respected.

Fig. 13. Illustration of the choice ofK andb.


Algorithm 4.1. Calculation of the low boundary for C1

K ¼ Kmax

b¼ bmin

WHILE (b # T AND K$ b) DOb¼ bþ 1WHILE (PK, b # e1) DOK ¼ K ¹ 1PK,b ¼ 1¹ (1=lT)(

∑b¹ 10 jpj þ b

∑Kb pj)

ENDWHILEK ¼ K þ 1PRINT (K,b)ENDWHILE

From Fig. 13, it is clearly seen that although any pair(K,b) in domain I fulfills both overflow and deadline con-straints, one may want to choose some ‘‘optimal’’ values.For instance,K ¼ 15 andb¼ 5 is a better choice thanK ¼ 15and b¼ 6, since the former requires less transmissionresources than the latter.

From the practical point of view, the optimization criter-ion can be very different. One may want to minimize eitherK or b according to the type of network. Generally, if aWAN is used, as the transmission medium is relativelyexpensive compared with the station resource (memory,for instance), a minimizedb is desirable. However, if aLAN is used, the resource of the station may be relativelyexpensive compared with the transmission medium, sominimizing K may be preferred. This is particularly truefor the fieldbus networks in which the cost of the commu-nication chips must be low enough to equip the field deviceslike sensors and actuators.

In our example, Fig. 13 shows that choosingb¼ 4and referring to the d.f. ofW, for K ¼ 21 we getP(W . 3) ¼ 4:243 10¹ 3. Therefore, the pair(K ¼ 21,b¼ 4) is suitable which requires a minimum transmissionmedium capacity.

5. Conclusion

Our interest in areas like process control and interactivemultimedia has led us to consider multiple-access protocolsoffering real-time support.

In this account a versatile solution for the sojourn time ofa message, valid for TDMA as well as TDM, has beensuggested. Hence, various sparse results on message delaycan be gathered in the same setting. Moreover, the proof ofmonotonicity for the overflow probability and the derivationof exact formulae for the sojourn time provide ground for anaccurate design of real-time protocols used in multimediaand process-control areas.

By using a short exemple, the interest of theoreticalresults for practical purposes, such as the determination ofthe buffer size and station bandwidth allocation, has beenemphasized, contributing to reducing the traditional gapbetween theoreticians and practitioners.

A glance at Proposition 3.2 shows that the derived solu-tion for the sojourn time is far from universal, since theresults rely heavily upon the nice properties of the Poissonprocess. Therefore it remains to deal with more generalarrival streams, for instance a non-homogeneous Poissonprocess and a cyclo-stationary process, which are at presentunder study.

Another possible extension consists of the study of theworst messagesojourn time instead of considering the aver-age message. This will be interesting when the temperalconstraints over message delivery are more strict.

Our results could also be useful for the buffer design ofsome switching systems or concentrators (Hub) based on theTDM scheme [9], since all our results are also available forTDM [26]. The same approach applies to the analysis of FIP[1] which supports both periodic and sporadic traffic andwhere the protocol for sporadic traffic is modelled by aTDMA with multiple non-contiguous output slots.

References

[1] Afnor, NF C46-602: Application Layer of FIP, NF C46-603: DataLink Layer of FIP, Association Franc¸aise de Normalisation, ParisLa Defense, France, 1992.

[2] U.N. Bhat, Elements of Applied Stochastic Processes, 2nd edition,Wiley, 1984.

[3] H. Bruneel, Message delay in TDMA channels with contiguousoutput, IEEE Trans. on Commun. COM-34 (1986) 681–684.

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communications, International Telemetry Conference, Los Angeles,1978.

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[8] J.F. Hayes, Modeling and Analysis of Computer CommunicationsNetworks, Plenum, 1984.

[9] J.Y. Hui, E. Arthurs, A broadband packet-switch for integrated trans-port, IEEE J. on Selected Areas in Commun. SAC-5 (8) (1987) 1264–1273.

[10] K.T. Ko, B.R. Davis, Delay analysis for a TDMA channel with con-tiguous output and Poisson message arrival, IEEE Trans. on Commun.COM-32 (1984) 707–709.

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[12] J.F. Kurose, M. Schwartz, Y. Yemini, Multiple-access protocols andtime-constrained communication, Computing Surveys ACM 16 (1)43–70, 1984.

[13] S.S. Lam, Delay analysis of a time division multiple access (TDMA),IEEE Trans. on Commun. COM-25 (1977) 1489–1494.

[14] H.W. Lee, L. Liang, A generalized analysis of message delay inSTDMA, Computer Networks and ISDN systems 19 (1990) 11–24.

[15] G. Le Lann, N. Rivierre, Real-time communications over broadcastnetworks: The CSMA-DCR and the DOD/CSMA-CD protocols,RTS’94, Paris, 1994, pp. 67–84.

[16] N. Malcolm, W. Zhao, Hard real-time communication in multiple-access networks, J. of Real-time Systems 8 (1995) 35–77.


[17] F. Panzieri, R. Davoli, Real time systems: A tutorial, Lecture notes inComputer Science, No. 729, Springer-Verlag, 1993, pp. 435–462.

[18] K. Ramamritham, J.A. Stankovic, Time-constrained communicationprotocols for hard real-time systems, 6th IEEE Workshop on Real-time Operating Systems and Software, Pittsburg, PA, 1989.

[19] I. Rubin, Message delay in FDMA and TDMA communicationchannels, IEEE Trans. on Commun. COM-27 (1979) 769–777.

[20] I. Rubin, Access-control disciplines for multi-access communicationchannels: Reservation and TDMA schemes, IEEE Trans. Inform.Theory IT-25 (1979) 516–536.

[21] R. Rom, M. Sidi, Message delay distribution in generalized timedivision multiple access (TDMA), Prob. in the Engineering and Infor-mation Sciences 4 (1990) 187–202.

[22] S.M. Ross, Stochastic Processes, Wiley, 1983.[23] I. Rubin, Z. Zhang, Message delay and queuing-size analysis for

circuit-switched TDMA systems, IEEE Trans. on Commun. COM-39 (6) (1991) 905–914.

[24] I. Rubin, Z. Zhang, Message delay analysis for TDMA schemes usingcontiguous-slot assignments, IEEE Trans. on Commun. COM-40 (4)(1992) 730–737.

[25] F. Simonot, Y.Q. Song, J.P. Thomesse, Queuing analysis of messageexchange in fieldbus FIP, Mathematical and Intelligent Models inSystem Simulation, The International Association for Mathematicsand Computers in Simulation (IMACS), J.C. Baltzez A.G. ScientificPublishing Co., 1991, pp. 95–101.

[26] F. Simonot, Y.Q. Song, J.P. Thomesse, On message sojourn time inTDM schemes with any buffer capacity, IEEE Trans. on Commun. 43(2/3/4) part 2 (1995) 1013–1021.

[27] Y.Q. Song, Etude de performances de FIP, aide au dimensionnementd’applications, PhD Dissertation (in French), Institut National Poly-technique de Lorraine, 1991.

[28] B. Steyaert, H. Bruneel, A general analysis of the packet delay inTDMA channels with contiguous-slot assignments, Proceedings ofICC’91, Denver, CO, 1990, pp. 1539–1543.

[29] J.A. Stankovic, Misconceptions about real-time computing, IEEEComputer 21 (10) (1988) 10–19.

[30] D. Stoyan, Comparison methods for queues and other stochastic mod-els, Wiley, 1983.

[31] H. Takagi, Queuing analysis of polling models, Computing SurveysACM 20 (1988) 5–28.

[32] H.C. Tijms, Stochastic Modelling and Analysis: A ComputationalApproach, Wiley, 1986.

[33] K. Tindell, A. Burns, A. Wellings, Analysis of hard real-timecommunications, J. of Real-time Systems 9 (1995) 147–171.

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Francois Simonot received the B.Sc. degreein mathematics, the M.Sc. degree in statisticsand the Ph.D. degree in applied mathematicsfrom the University of Nancy 1, in 1970,1971 and 1981 respectively. Since 1974, hehas been teaching pure and applied mathe-matics, especially probability, at Universityof Nancy 1 and Institut national polytechni-que de Lorraine in Nancy. He has previouslyworked on probability metrics, robustness of

stochastic systems and approximation of Markov chains. His currentresearch includes queuing systems, multiple access protocols and per-formance analysis of computer communication networks.

Ye-Qiong Song received the B.S. degree fromthe Beijing University of Posts and Telecom-munications in 1984, the Mastere degreefrom the Ecole Nationale Supe´rieure desTelecommunications de Paris in 1987, bothin telecommunications engineering, the DEAdegree from the University of Paris 6 in1988, and the Ph.D. degree from the InstitutNational Polytechnique de Lorraine in 1991,in computer science. He worked on the mod-

elling and performance evaluation of the FIP fieldbus using queuingtheory and simulation. His current research interests include modellingand performance evaluation of local area networks and multiple accessprotocols when used in real-time environments. He is now an associateprofessor at the University of Nancy 1.


real-time communications using tdma-based multi-access protocol

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