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* Corresponding author. E-mail addresses: gp@csd.auth.gr (G.I. Papadimitriou), apombo@csd.auth.gr (A.S. Pomportsis). Neurocomputing 35 (2000) 177}188 Letters On the use of stochastic estimator learning automata in time division multiple access systems: A methodology Georgios I. Papadimitriou*, Andreas S. Pomportsis Department of Informatics, Aristotle University, Box 888, 54006 Thessaloniki, Greece Received 11 February 2000; revised 1 June 2000; accepted 2 June 2000 Abstract Due to its "xed assignment nature, the well-known TDMA protocol su!ers from poor performance when the o!ered tra$c is bursty. In this paper, a new time division multiple access protocol which is capable of operating e$ciently under bursty tra$c conditions is introduced. According to the proposed protocol, the station which grants permission to transmit at each time slot is selected by means of stochastic estimator learning automata. The system which consists of the automata and the network is analyzed and it is proved that the probability of selecting an idle station asymptotically tends to be minimized. Therefore, the number of idle slots is drastically reduced and consequently, the network throughput is improved. Further- more, due to the use of a stochastic estimator, the automata are capable of being rapidly adapted to the sharp changes of the dynamic bursty tra$c environment. Extensive simulation results are presented which indicate that the proposed protocol achieves a signi"cantly higher performance than other well-known time division multiple access protocols when operating under bursty tra$c conditions. ( 2000 Published by Elsevier Science B.V. All rights reserved. Keywords: Time division multiple access; Bursty tra$c; Learning automata; Stochastic estimator 1. Introduction The key issue in broadcast networks is how to determine who gets to use the channel. A broad range of demand assignment, random access and "xed assignment 0925-2312/00/$ - see front matter ( 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 2 3 1 2 ( 0 0 ) 0 0 3 2 0 - 9

On the use of stochastic estimator learning automata in time division multiple access systems: A methodology

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*Corresponding author.E-mail addresses: [email protected] (G.I. Papadimitriou), [email protected] (A.S. Pomportsis).

Neurocomputing 35 (2000) 177}188

Letters

On the use of stochastic estimator learning automata in timedivision multiple access systems: A methodology

Georgios I. Papadimitriou*, Andreas S. PomportsisDepartment of Informatics, Aristotle University, Box 888, 54006 Thessaloniki, Greece

Received 11 February 2000; revised 1 June 2000; accepted 2 June 2000

Abstract

Due to its "xed assignment nature, the well-known TDMA protocol su!ers from poorperformance when the o!ered tra$c is bursty. In this paper, a new time division multiple accessprotocol which is capable of operating e$ciently under bursty tra$c conditions is introduced.According to the proposed protocol, the station which grants permission to transmit at eachtime slot is selected by means of stochastic estimator learning automata. The system whichconsists of the automata and the network is analyzed and it is proved that the probability ofselecting an idle station asymptotically tends to be minimized. Therefore, the number of idleslots is drastically reduced and consequently, the network throughput is improved. Further-more, due to the use of a stochastic estimator, the automata are capable of being rapidlyadapted to the sharp changes of the dynamic bursty tra$c environment. Extensive simulationresults are presented which indicate that the proposed protocol achieves a signi"cantlyhigher performance than other well-known time division multiple access protocols whenoperating under bursty tra$c conditions. ( 2000 Published by Elsevier Science B.V. All rightsreserved.

Keywords: Time division multiple access; Bursty tra$c; Learning automata; Stochasticestimator

1. Introduction

The key issue in broadcast networks is how to determine who gets to use thechannel. A broad range of demand assignment, random access and "xed assignment

0925-2312/00/$ - see front matter ( 2000 Published by Elsevier Science B.V. All rights reserved.PII: S 0 9 2 5 - 2 3 1 2 ( 0 0 ) 0 0 3 2 0 - 9

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protocols have been proposed as solutions to this problem. Demand assignmentprotocols * such as token ring, token bus and DQDB [19,16] * are based ona signaling procedure which allows certain network entities to be informed about thetransmission and networking needs and demands of the network stations. Randomaccess protocols* such as ALOHA, CSMA and CSMA/CD [19,16]* are charac-terized by the fact that stations contend for access to the communications channel,in accordance with an algorithm that can lead to colliding transmissions. All thecollided packets are scheduled for retransmission. Fixed assignment protocols *such as TDMA [1,5,14}18], RTDMA [3,4] and FDMA [19] * assign a "xedportion of the available bandwidth to each station. In this way, collisions are avoided.Due to the absence of collisions, protocols of this family achieve a high performancewhen the tra$c of each station is stable and a priori known. However, when thetra$c is bursty, "xed assignment protocols are not capable of being adapted to thesharp changes of the stations' tra$c. Therefore, their performance is dramaticallydegraded.

In this paper, a new time division multiple access protocol which is capable ofoperating e$ciently under bursty tra$c conditions is introduced. According to theproposed protocol, the station which grants permission to transmit at each time slot isselected by means of learning automata [7}12].

The system which consists of the automata and the network is analyzed and it isproved that the probability of selecting an idle station asymptotically tends to beminimized. Therefore, the number of idle slots is drastically reduced and consequently,the network throughput is improved. Furthermore, due to the use of a stochasticestimator [9], the automata are capable of being rapidly adapted to the sharp changesof the dynamic bursty tra$c environment.

The proposed stochastic-estimator-based time division multiple access (SE-TDMA)protocol is applicable to a broad range of broadcast network architectures, includingbus, star and wireless LANs. This paper focuses on the theoretical aspects of SE-TDMA rather than on its application to speci"c network architectures.

The paper is organized as follows: In Section 2, the proposed SE-TDMA protocolis introduced, while the stochastic estimator learning automaton is presented inSection 3. In Section 4, extensive simulation results are presented which indicate thesuperiority of the SE-TDMA protocol over other well-known TDMA protocols.Finally, concluding remarks are given in Section 5.

2. The SE-TDMA protocol

N stations (u1,2, u

N) are assumed to be connected on a broadcast medium and are

trying to get access to this medium. According to the SE-TDMA protocol, all thestations are provided with a stochastic estimator learning automaton (SELA) [4], thatdecides which station grants permission to transmit at each time slot. All the stationsuse the same learning algorithm and * due to the broadcast nature of the network* the feedback information is common for all of them. Furthermore, all the stationsuse the same random number generator and the same seed. Therefore, at each time

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slot, all the stations arrive at the same decision on which station grants permission totransmit [4]. Thus, the protocol is collision-free.

Since the o!ered tra$c is bursty, when the selected station has a packet to transmit,it is probable that this station will have packets to transmit in the near future.Therefore, when the selected station transmits a packet, then the automaton is fedwith a reward. On the other hand, when the selected station has no packet to transmit,it is probable that this station will remain idle in the near future. In this case, theautomaton is fed with a penalty. According to this scheme, stations which had packetstransmissions in the near past are selected more frequently than other stations.Furthermore, due to the use of a stochastic estimator, stations that have not beenselected recently, have the opportunity to be selected. In this way, the fairness of theprotocol is signi"cantly improved. Furthermore, the protocol is capable of beingadapted to the sharp changes of the dynamic bursty tra$c environment.

The set of automaton's actions is de"ned as A"Ma1,2, a

NN, where N is the number

of stations. At each time slot t, the learning automaton randomly selects an actiona(t) according to a probability distribution P(t)"Mp

1(t),2, p

N(t)N. If a(t)"a

i, then

station uigrants permission to transmit during time slot t. The result of this decision

(packet transmission or idle slot) is used as feedback information in the following way:(i) b(t)"1 (reward) if the selected station u

itransmitted a packet during time slot t.

(ii) b(t)"0 (penalty) if slot t was idle, since the selected station ui

had no packetsto transmit.

The automaton takes into account the feedback information and updates theprobability distribution P(t).

3. The stochastic estimator learning automaton

SELA [9] is a learning automaton which keeps estimates of the environmentalcharacteristics in order to achieve an accurate convergence. The estimates of thereward probabilities of actions are computed stochastically. So, they are not strictlydependent on the environmental responses. The dependence between the stochasticestimates and the deterministic estimator's contents is more relaxed when the latterare old and probably invalid. In this way, actions that have not been selected recently,have the opportunity to be estimated as `optimala, to increase their choice probabil-ity, and, consequently, to be selected. Thus, the automaton is capable of being adaptedto the environmental changes, since the estimator is always recently updated.

The SELA learning automaton is de"ned as quintuple SA,B, P,E,¹T whereA"Ma

1,2, a

rN is the set of the r o!ered actions (24r(R). The action selected at

time instant t is denoted by a(t). B"M0,1N is the input set of the possible environ-mental responses. `1a denotes a reward and `0a denotes a penalty response. Theenvironmental response at time instant t is denoted by b(t). P is a probabilitydistribution over the set of actions. We have P(t)"Mp

1(t),2, p

r(t)N, where p

i(t) is the

probability of selecting action aiat time instant t. E is the estimator, which at any

time instant t, contains the estimated environmental characteristics. We de"neE(t)"(D@(t),M(t),G(t)) where D@(t)"Md@

1(t),2, d@

r(t)N is the deterministic estimator

G.I. Papadimitriou, A.S. Pomportsis / Neurocomputing 35 (2000) 177}188 179

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vector, which, at any time instant t, contains the current deterministic estimates of thereward probabilities of actions. The deterministic estimate d@

i(t) of the reward prob-

ability of each action ai(i"1,2, r) is de"ned as follows:

d@i(t)"

+Wk/1

wki(t)

=, (1)

where= is an integer internal parameter of the automaton called `learning windowaand wk

i(t) for k"1,2,= are the environmental responses received during the= last

times that action aiwas selected. M(t)"Mm

1(t),2, m

r(t)N is the oldness vector, which

at any time instant t contains the time passed from the last time each actionwas selected. Time is counted in number of iterations. We de"ne: m

i(t)"t!maxqMq :

q4t and a(q)"aiN. G(t)"Mg

1(t),2, g

r(t)N is the stochastic estimator vector, which,

at any time instant t, contains the current stochastic estimates of the reward probabil-ities of the actions. The current stochastic estimate g

i(t) of each action a

iis de"ned as

follows:

gi(t)"d@

i(t)#N(0, p2

i(t)) where p

i(t)"minMam

i(t), p

.!9N. (2)

N(0, p2i(t)) denotes a random number selected with a normal probability distribution,

with zero mean and a variance equal to p2i(t). a is an internal automaton's parameter

that determines how rapidly the stochastic estimates become independent from thedeterministic ones. p

.!9is the maximum permitted value of p

i(t). It bounds the

stochastic estimates in order not to increase in"nitely. ¹ is the learning algorithm. Itsalgorithmic description is presented below:

Step 1: Select an action a(t)"ak

according to the probability distribution P (t).Step 2: Receive the feedback b(t)3M0,1N from the environment.Step 3: Compute the deterministic estimate d@

k(t) of the reward probability of action

ak, as it is given by relation (1).Step 4: Update the oldness vector by setting m

k(t)"0 and m

i(t) :"m

i(t!1)#1 for

all iOk.Step 5: For each action a

i( for i"1,2, r) compute the new stochastic estimate g

i(t),

as it is given by relation (2).Step 6: Select the `optimala action a

mthat has the highest stochastic estimate of

reward probability. Thus, gm(t)"max

iMg

i(t)N.

Step 7: Update the probability vector in the following way: (i) For every actionai

with iOm and pi(t)51/n, set p

i(t#1) :"p

i(t)!1/n. (n is the `resolution para-

metera of the automaton which determines the step size of the probability updating.)(ii) For the `optimala action a

mset p

m(t#1) :"1!+

iEmpi(t#1).

Step 8: Go to Step 1.It can be noted that the above learning algorithm does not terminate. Since the

network operation is continuous, the learning algorithm has to run continuously inorder to be adapted to the environmental changes. A termination of the learningalgorithm would lead to "xed probability distribution P(t). In this case, the protocolwould be unable of being adapted to the environmental changes.

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Theorem. If di

is the reward probability of action ai, for i"1,2,N and

R(t)"+Ni/1

dipi(t), then E[R(t#1)DP(t)]'R(t) for all t and all p

i(t)3(0, 1)

(i"1,2,N).

Proof. The proof is given in [9]. Although, an S-model environment is considered in[9], the proof is also valid for a P-model environment, since the stochastic estimatesgi(t) are continuous random variables symmetrically distributed about their means

di(for i"1,2, N). (Note: In an S-model environment, the environmental responses

can take any value in the [0,1] interval, while in a P-model one, the environmentalresponses take values from the set M0,1N.) h

R(t) represents the probability that slot t is not idle. Since at each time slot, R(t)is increased, it follows that the number of idle slots asymptotically tends to beminimized. Therefore, the throughput of a network operating under the SE-TDMAprotocol tends to be maximized.

4. Simulation results

In the following, the proposed SE-TDMA protocol is compared to TDMA[1,5,14}18] and RTDMA [3,4]; two representative time division multiple accessprotocols. The protocols which are under comparison were simulated to be applied tothree di!erent networks (N

1,N

2and N

3) under bursty tra$c conditions. We used an

event-driven simulator, running on a Pentium III machine. The bursty tra$c wasmodelled in a way similar to the ones presented in [2,6]. Each node can be in one ofthe two states S

0and S

1. When a node is in state S

0then it has no packet arrivals.

When a node is in state S1

then at each time slot it has a packet arrival withprobability Z. Given a station is in state S

0at time slot t, the probability that this

station will transit to state S1

at the next time slot is P01

. The transition probabilityfrom state S

1to state S

0is P

10. It can be shown that, when the load o!ered to the

network is R packets/slot and the mean burst length is B slots then the transitionprobabilities are P

10"1/B and P

01"R/(B(NZ!R)). Each station is provided with

a waiting queue. Its capacity is equal to Q packets. When, a packet arrival takes placein a full queue, then the arriving packet is lost.

The number of users N, the queue capacity Q, the mean burst length B and thepacket arrival probability Z of each active station, were taken to be as follows:

(a) Network N1

: N"10, Q"10, B"10, Z"1.0.(b) Network N

2: N"20, Q"15, B"10, Z"1.0.

(c) Network N3

: N"5, Q"3, B"1000, Z"0.8.

We have used the following two broadly used performance metrics in order tocompare the three protocols:

(1) The delay versus throughput characteristic.(2) The throughput versus o!ered load characteristic.

G.I. Papadimitriou, A.S. Pomportsis / Neurocomputing 35 (2000) 177}188 181

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Fig. 1. The delay versus throughput characteristics of SE-TDMA, TDMA and RTDMA when applied tonetwork N

1.

Fig. 2. The throughput versus load characteristics of SE-TDMA, TDMA and RTDMA when applied tonetwork N

1.

The delay versus throughput characteristics of the compared protocols when theyare applied to networks N

1, N

2and N

3are shown in Figs. 1, 3 and 5, respectively. The

throughput versus o!ered load characteristics of the compared protocols when theyare applied to networks N

1, N

2and N

3are shown in Figs. 2, 4 and 6, respectively. The

presented results are averages for all network stations. In the simulation of protocolSE-TDMA, we have used the optimal values of the internal parameters of SELA(a,=, n, p

.!9) for each one of the networks. These values have been selected by testing

a broad range of values and selecting those values that result in a high throughput-delay performance. For Figs. 1}6 the tra$c is assumed to be symmetric. Thus, theo!ered load is the same for all the stations. The performance of the three protocolsunder unbalanced tra$c will be studied at the end of this section.

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Fig. 3. The delay versus throughput characteristics of SE-TDMA, TDMA and RTDMA when applied tonetwork N

2.

Fig. 4. The throughput versus load characteristics of SE-TDMA, TDMA and RTDMA when applied tonetwork N

2.

Fig. 5. The delay versus throughput characteristics of SE-TDMA, TDMA and RTDMA when applied tonetwork N

3.

G.I. Papadimitriou, A.S. Pomportsis / Neurocomputing 35 (2000) 177}188 183

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Fig. 6. The throughput versus load characteristics of SE-TDMA, TDMA and RTDMA when applied tonetwork N

3.

Fig. 7. The throughput of protocols SE-TDMA, TDMA and RTDMA as a function of the queuecapacity Q.

From the comparative graphs, it becomes clear that SE-TDMA achieves a signi"-cantly higher delay-throughput and throughput-load performance than protocolsTDMA and RTDMA, when operating under bursty tra$c conditions.

In order to study how the queue capacity Q and burst size B a!ect the performanceof the protocols which are under comparison, graphs illustrating network throughputas a function of these parameters are presented in Figs. 7 and 8, respectively. In bothgraphs, a 10-station network is considered.

In Fig. 7, the mean burst size B is constant (B"10 packets), while the queuecapacity is gradually increased from 2 to 20 packets. For any value of the queuecapacity, the SE-TDMA achieves a signi"cantly higher throughput than protocolsTDMA and RTDMA. The throughput advantage of SE-TDMA over the two otherprotocols slowly decreases as the queue capacity increase, because a queue of very

184 G.I. Papadimitriou, A.S. Pomportsis / Neurocomputing 35 (2000) 177}188

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Fig. 8. The throughput of protocols SE-TDMA, TDMA and RTDMA as a function of the mean burstsize B.

high capacity can avoid some of the packet losses which are due to the burstiness oftra$c. However, large queues do not solve the performance problem of protocolsTDMA and RTDMA, because they lead to large queueing delays.

A constant queue capacity is considered in Fig. 8, while the mean burst size isgradually increased from 5 to 50 packets. It is clear that the throughput improvementwhich is achieved by the use of SE-TDMA is higher when the o!ered tra$c is morebursty (i.e. when the mean burst length is high). The throughput of the SE-TDMAprotocol is practically una!ected by the burstiness of the o!ered tra$c, because thisprotocol is based on the network feedback information in order to dynamicallyallocate the available bandwidth to those stations that actually have packets totransmit. On the other hand, protocols TDMA and RTDMA statically allocate thebandwidth to the stations without taking into account the network feedback informa-tion. Therefore, a signi"cant amount of bandwidth is allocated to idle stations,resulting in a decrease of the network throughput. As the throughput becomes morebursty, the number of idle stations is increased, resulting in further performancedegradation of TDMA and RTDMA. Under these tra$c conditions, the throughputof the proposed SE-TDMA protocol is practically una!ected because the bandwidthis dynamically allocate to those stations which actually have packets to transmit.

For the same reasons, the SE-TDMA protocol outperforms TDMA and RTDMAwhen the o!ered tra$c is unbalanced. The performance (in terms of mean throughputand mean packet delay) of the protocols which are under comparison when operatingunder unbalanced tra$c is presented in Table 1. Three tra$c scenarios are con-sidered. In all cases, a 10-station network is considered with a queue capacity equal to10 packets and a mean burst size equal to 10 packets. The overall tra$c o!ered to thenetwork is equal to 0.5 packets/slot. In the "rst scenario the tra$c is balanced. All thestations have a tra$c of 0.05 packets/slot. According to the second scenario the tra$cis unbalanced. Five stations have a tra$c of 0.08 packets/slot, while the rest of thestations have a tra$c of 0.02 packets/slot. Finally, in the third scenario, it is assumed

G.I. Papadimitriou, A.S. Pomportsis / Neurocomputing 35 (2000) 177}188 185

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Table 1Throughput and mean packet delay of protocols SE-TDMA, TDMA and RTDMA under unbalancedtra$c

SE-TDMA TDMA RTDMA

Scenario 1 (Balanced tra$c) Throughput (packets/slot) 0.442 0.328 0.323Delay (slots) 1.27 48.7 52.8

Scenario 2 (Unbalanced tra$c) Throughput (packets/slot) 0.441 0.314 0.309Delay (slots) 12.4 50.7 54.6

Scenario 3 (Highly unbalanced Throughput (packets/slot) 0.468 0.163 0.162tra$c) Delay (slots) 8.5 72.2 73.7

Fig. 9. The standard deviation of the packet delay of protocols SE-TDMA, TDMA and RTDMA forvarious values of the network load.

that one of the stations have a tra$c 0.41 packets/slot, while the rest of them havea tra$c of 0.01 packets/slot. Thus, the tra$c is highly unbalanced. In all scenarios, thetra$c values are periodically reassigned to the stations every 1000 slots by rotatingthem by one place. The mean throughput of the network and the mean packet delaywhich is achieved by each one of the three protocols which are under comparisonwhen operating under scenarios 1, 2 and 3 are presented in Table 1. Due to the reasonsdiscussed in the previous paragraph, protocol SE-TDMA is practically una!ected bypresence of unbalanced tra$c. On the other hand, the presence of unbalanced tra$cleads to a performance degradation of protocols TDMA and RTDMA.

Finally, the fairness of the three protocols which are under comparison is examined.We have used the standard deviation of the packet delay as performance metric formeasuring the fairness of the three protocols [13]. A 10-station network with a queuecapacity Q equal to 10 packets and mean burst size of 10 packets is considered inFig. 9. The standard deviation of the packet delay is measured for various values of theo!ered load. It becomes clear that SE-TDMA achieves a signi"cantly lower standarddeviation of packet delay than protocols. TDMA and RTDMA. SE-TDMA o!ers the

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available bandwidth to those stations that actually need it. However, due to the use ofthe stochastic estimator, stations that have not been selected recently have theopportunity to be estimated as `optimala, to increase their choice probability, andconsequently, to grant permission to transmit. In this way, the fairness of the protocolis signi"cantly improved.

5. Conclusion

This paper has presented a new time division multiple access protocol for broadcastnetworks. According to the proposed SE-TDMA protocol, the station which grantspermission to transmit at each time slot is selected by means of learning automata,which are capable of being adapted to the changes of the stations' tra$c. Therefore,the new protocol is capable of achieving a low delay and a high throughput in thedynamic bursty tra$c environment.

The main characteristics of the SE-TDMA protocol are summarized below:

(a) It achieves a high performance, even when the o!ered tra$c is bursty.(b) It is self-adaptive. Thus, when the tra$c conditions change, the choice prob-

abilities of the stations are rapidly adapted to the new tra$c conditions.(c) No centralized control of the stations is required, since the protocol is fully

distributed.(d) It is fault-tolerant, since its operation is not a!ected by a possible node failure.(e) No signi"cant increase of the implementation cost is introduced. The only

additional hardware* in relation to TDMA or RTDMA* is a processor whichimplements the learning algorithm.

The use of learning automata o!ers a new highly promising approach to the designof self-adaptive multiaccess protocols for broadcast networks. We are currentlyworking in this direction.

Acknowledgements

The authors wish to thank reviewers for their useful and insightful comments.

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