Self-adaptive random-access protocols for WDM passive star networks

G.I. Papadimitriou D.G. Maritsas

indexing terms: Automata based random access protocol, Learning automaton, Transmission probability, WDM passive star network

Abstract: A learning automata based random access protocol for WDM passive star networks is introduced. The proposed protocol makes use of learning automata to achieve a high throughput and a low delay under any load conditions. An array of learning automata that determines the transmission probability of each wavelength is placed at each station. After each slot the trans- mission probability of each wavelength is modified according to the network feedback information. The asymptotic behaviour of the system which consists of the automata and the network is analysed and it is proved that under any load con- ditions, the transmission probability asymp- totically tends to take its optimum value. Extensive simulation results are presented which indicate that the use of the proposed learning automata based scheme leads to a significant improvement of the network‘s performance.

1 Introduction

The emerging new generation of local area networks is characterised by its increasing bandwidth demands which have led to the utilisation of optical fibre as a transmis- sion medium. A first attempt to implement fibre optic networks was simply to replace the well known copper wire by an optical fibre. Due to the limited speed of the stations’ electronic circuits, single-channel optical net- works such as FDDI, Fasnet, Expressnet, etc. did not prove capable of supporting gigabit data rates.

The introduction of the wavelength division multi- plexing (WDM) technique [l , 23 solved this problem by dividing the available optical bandwidth into multiple channels of lower bandwidth that can be easily supported by the stations’ circuits. Both multiplexing and demulti- plexing of the multiple channels are performed in the optical domain using totally passive optical devices without the need for optical to electronic translation and vice versa. In this way, the WDM technique allows the implementation of purely optical networks that are

0 IEE, 1995 Paper 1866E (C3, C4), first received 24th February and in revised form 23rd December 1994 G.I. Papadimitriou is with the Department of Computer Engineering, University of Patras, GR 26500, Patras, Greece and the Computer Technology Institute, PO Box 1122, GR 261 10, Patras, Greece D.G. Maritsas is with the Department of Computer Engineering, Uni- versity of Patras, GR 26500, Patras and the Computer Technology Institute, PO Box 1122, GR 26110, Patras, Greece

306

environment 1 I I

Fig. 1 Learning automaton that interacts with stochastic environment

capable of providing gigabit data rates using present-day optical and electronic technology.

Passive star networks (Fig. 1) the most commonly used WDM networks, use a passive star coupler to broadcast all inputs to all outputs. A passive star network using tunable lasers and fixed optical filters that operates under the slotted ALOHA protocol [3, 41 is considered in this paper. The key issue in such a network is the determination of the transmission probability p with which each steady station transmits at the beginning of each time slot. Since the optimum value of p depends on the network‘s load, a fixed choice of p leads to a significant decrease of the networks performance under variable load conditions.

In this paper, a new random access protocol which makes use of learning automata [ S , 61 to dynamically determine the transmission probability p is introduced. The asymptotic behaviour of the system consisting of the automata and the network is analysed, and it is proved that under any load conditions the transmission prob- ability asymptotically tends to take its optimum value. Furthermore, extensive simulation results are presented that indicate that the use of the proposed learning automata based protocol leads to a significant improve- ment of the network’s performance.

2

The network under consideration is a passive star network. Several variants of this topology can be found in the literature [1-4, 8-10]; the architecture considered in this paper is the one proposed in References 3 and 4.

Each transmitter is provided with a tunable laser that can be tuned at each one of the W available wavelengths, Fig. 2. Optical fibres are used to connect the output of each laser to one of the input ports of a N x N (where N is the number of stations) star coupler at the network

WDM passive star networks using the slotted ALOHA protocol

This work was supported by the Computer Tech- nology Institute, Patras, Greece.

IEE Proc.-Comput. Digit. Tech., Vol. 142, No. 4, July 1995

hub. Thus, at each output port of the coupler all the wavelengths are available. Each output port of the star coupler is connected to the corresponding receiver by

station U, station U-

station u2 station u2 d=l hyf passive F H Z I optical f , detector laser

couder

station u w ‘ ’ I station UN

4-P Fig. 2 optical filters

W D M passive star network using tunable lasers and fixed

means of an optical fibre. At each receiver the optical signal is fed to a fixed optical filter which passes only one wavelength.

A protocol commonly used in a network of this archi- tecture is the slotted ALOHA. According to this proto- col, when a station has a packet to transmit, it tunes its laser at the receiver’s wavelength and transmits the packet with transmission probability p . If two or more stations concurrently transmit on the same wavelength, all the transmitted packets are destroyed. Such a situ- ation is called a collision. Each colliding station senses the collision [7] and retransmits the collided packet at the next time slot with probability p.

Assume that at a time t, M stations are waiting to transmit on a specific wavelength, say A,. The probability that a successful transmission will take place at this wavelength is

P,&) = Mp(1 - p y - 1

It is known that the right-hand side of this relation is maximised when p = 1/M. Thus, for users waiting to transmit on wavelength Ai, the optimum value of trans- mission probability depends on the total number of these users, which in turn depends on the offered load. When the network operates under variable load conditions and the transmission probability p is fixed, the throughput of the network is decreased, since the protocol is not capable of being adapted to the load changes.

Unfortunately, the network usually operates under variable load conditions owing to the following two reasons.

(i) Traffic in gigabit LANs is highly bursty. Data traffic, which constitutes most of load, is intrinsically bursty. As the network speed increases, the peak rate increases faster than the average, thus making tafic become even more bursty.

(ii) Even if the load offered to a WDM passive star network using the ALOHA protocol is stable, the load of each specific wavelength is time variable. Assume that instantaneously, a large number of packets are trying to be transmitted on a specific wavelength, say A,. This wavelength suffers from a large number of collisions, which decreases its throughput. Therefore the transmis- sion rate probably becomes lower than the arrival rate at Ai . Consequently, the number of packets waiting for transmission at Ai further increases, causing a vicious circle. This situation, termed ‘wavelength overloading’, leads to a dramatic decrease of the network’s throughput.

IEE Proc.-Comput. Digit. Tech., Vol. 142, No. 4, July 1995

In this paper, a new self-adaptive protocol which dynamically determines the transmission probability p , is introduced. Due to its adaptivity the proposed protocol is capable of operating efficiently under any load condi- tions.

3 LABRA protocol

The optimum transmission probability differs from wave- length to wavelength and depends on the number of packets waiting to be transmitted on each wavelength. According to the LABRA protocol, each station uses an array of learning automata to determine the transmission probability of each wavelength. The LABRA protocol and its hardware implementation (Fig. 3) are described in what follows.

to star coupler

@-; -------:

col detection

Fig. 3 - data ~~~- control signals

Internal architecture of station using LABRA protocol

3.1 Determination of transmission probability The set of stations is defined as U = {U,, u2, ..., U,} where N is the number of stations. The set of wavelengths is defined as A = { A l r A,, . . . , A,} where W is the number of wavelengths. An array of W learning automata LA, , L A , , . . . , L A , is placed at each station uk . Each automa- ton LA, corresponds to a specific wavelength Ai and at any time instant contains the transmission probability P i t ) of the A, wavelength.

At each time slot t, each station uk which is ready to transmit on A,, transmits the packet with probability P,(t). The transmission is postponed with probability 1 - Pit). If the transmission is postponed or the transmit-

ted packet is destroyed owing to a collision, the same operation is repeated at the next time slot.

3.2 Feedback mechanism Each station is provided with a WDM demultiplexer which separates the different wavelengths. Each one of the separated wavelengths is detected for collision. The collision detection operation can be implemented either by computing the checksum of the packet’s header or by measuring the optical power of the signal by means of a

307

photofet. At each time slot, the collision detection information of each wavelength A,, is fed to the wavelength-specific learning automaton LA, which updates the transmission probability P i t ) by means of a learning algorithm. Since the feedback is common for all the stations, it follows that the corresponding automata of all the stations always contain the same transmission probabilities.

The role of the WDM demultiplexer is twofold. Besides providing the automata with feedback informa- tion it also replaces the fixed optical filter. If station ut passes only l,, it is only required to connect the corres- ponding output of the WDM demultiplexer to the receiv- ing buffer (Fig. 3).

In this study, the end-toend propagation delay is assumed to be negligible. Therefore the network feedback information is immediately available for use. Some remarks concerning the application of the LABRA proto- col to networks with large end-to-end propagation delay are given in Section 6.

3.3 Probability updating scheme The key issue of the proposed protocol is the determi- nation of the transmission probabilities P i t ) for i = 1, . . . , W . A high transmission probability P i t ) would lead to a large number of collisions at A i . On the other hand, a relatively small Pit) would lead to large number of idle slots at A,. In both cases, the throughput of the A, wave- length decreases. The throughput of the A, wavelength is maximised when the transmission probability takes an optimum value which depends on the number of users waiting to transmit on this wavelength.

If M users are ready to transmit on l , at time slot t, the probability that a successful transmission will take place at A, is

The probability that A, will be idle during the time slot t is

(2) Furthermore, the probability that a collision will take place at A, is

Pi'&) = (1 - P A t ) ) M

P&) = 1 - P&) - P,(t)

= 1 - (1 - - MPXtXl - (3)

As discussed earlier, the success probability Pa(?) is max- imised when P i t ) = 1/M. Consequently, one must choose P i t ) = 1/M to maximise the throughput of the Ai wave- length. However, the value of M is unknown. Therefore there is a serious problem on how to determine the optimum value of Pi t ) .

According to the LABRA protocol, a learning automaton is used to determine the transmission prob- ability of each wavelength. The Occurrence of an idle slot is probably due to a small value of the transmission probability Pit). Therefore Pit) must be increased. A suc- cessful transmission implies that the number of packets waiting for transmission at A, is probably decreased. Therefore the transmission probability P i t ) must be increased. For these reasons, if A, was idle or a successful transmission took place during the last time slot, the LA, automaton increases the transmission probability Pit).

On the other hand, the Occurrence of a collision during the last time slot is probably due to a high value of the transmission probability. Therefore the LA,

308

automaton decreases the transmission probability P i t ) when a collision occurs at A,.

Assume that SLOT,@) E {IDLE, SUCCESS, COLLISION} denotes the state of A, during the t time slot. The general probability updating scheme which must be used by the automata of the LABRA protocol is the following

P,(t + 1) = Pit) + A1

if SLOTJt) = IDLE or SUCCESS (4) Pi(? + 1) = P i t ) - A2

if SLOTi t ) = COLLISION

where 0 < A1 < 1 - Pit) and 0 < A, < Pit). Our aim is to appropriately choose A1 and Az so that the transmis- sion probability Pit) asymptotically tends to be equal to 1/M. The design of such an updating scheme is possible, and is based on the following two remarks.

Remark A: When P,(t) = 1/M, the reward probability dit) = P,(t) + PA?) is approximately equal to 2e-l . From eqns. 1 and 2 it is derived that if Pit) = 1/M then limM-m dit) = 2e-' = 0.736. Furthermore, for small values of M (M 2 2), the reward probability d i t ) takes values very close to 2e- '. (Eqns. 1 and 2 give M = 2 =- dit) = 0.750, M = 4 - d i t ) = 0.738, M = 5 =-d i t ) = 0.737m etc.).

M = 3 * dit) = 0.741,

Remark E : For any wavelength Ai (with M 2 2), the reward probability d i t ) is a monotonically decreasing function of the transmission probability Pit).

Proof: It suffices to show that the first derivative of the function

D,(P,) = (1 - PJM + MPAl - I'dM-'

is negative for Pi E (0, 1) and M 2 2. We have

DI(P,) = [(l - PJM + MPXl - PJM-']'

= -(M - 1)MPil - P y - 2 < 0 Remark A guarantees that the unknown optimum value of Pit) (Pit) = 1/M) corresponds to a known value of d i t ) = 2e-' = 0.736 = U, irrespective of the value of M. Remark B implies that the wanted value of d,(t) = U can be achieved by increasing or decreasing the value of Pit). By analysing the general updating scheme (eqn. 4) we have

= dit)Al - (1 - d i W *

= (Ai + AzXdXt) - + Ad) (5)

To asymptotically converge to the point dit) = U, the probability updating scheme must satisfy the following three properties (where adit) = dit + 1) - d i t ) )

(a) if d i t ) > U then ErGPktll > 0 and consequently - .. .- - . ~ ~ i c i ~ t ) ] <'O

(b) if dit) < U then ECSPit)] < 0 and consequently

(c) If dit) = U then ECGPit)] = 0 and consequently ECGdifll > 0

ECGdXt)] = 0

Eqn. 5 guarantees that all these properties are satisfied when A2/(A1 + A2) = U or equivalently A1 = ((1 - o)/u)A, = hA2 h = (1 - u)/u = (1 - 2e-lX2e-l = 0.359. with

I E E Proc.-Cornput. Digit. Tech., Vol. 142, No. 4, July 1995

If A, = A, the general updating scheme (expr. 4) becomes

Pi(t + 1 ) = P i t ) + hA if SLOT&) = IDLE or SUCCESS

Pi(t + 1 ) = P i t ) - A if SLoTdt) = COLLISION

where 0 < A < (1 - Pit))/h and 0 < A < Pi t ) . It remains to choose the appropriate value of A. We selected A = LPi(t)(l - Pi t ) ) with 0 c: L < 1. This choice of A leads to the following updating scheme

Pi(t + 1) = P,(t) + hLPit)(l - P i t ) )

if SLOT&) = IDLE or SUCCESS

Pi(t + 1) = P i t ) - LPitX1 - Pi t ) ) if SLOT&) = COLLISION

where L E (0, 1). This scheme has a serious disadvantage. When the

transmission probability P i t ) takes values in the neigh- bourhood of 0 or 1, the probability updating step becomes very small. This leads to a serious loss of the automaton’s adaptivity and consequently to a serious decrease of the network’s performance. To eliminate this disadvantage, the probability updating scheme is modi- fied in the following way

Pi(t + 1) = P i t ) + hLPAt)(l - P i t ) ) + d2(1 - Pit))’ if SLOTXt) = IDLE or SUCCESS

Pi(t + 1) = P i t ) - LPAt)(l - P i t ) ) - bP(P,{t))’ if SLOT&) = COLLISION

where L E (0, 1) and a, b E (0, l/L).

4 Performance analysis

The asymptotic behaviour of the probability updating scheme is analysed and it is proved that for small values of the L parameter, the transmission probability P i t ) asymptotically tends to take its optimum value. The pro- position is formally expressed as follows.

Theorem 1 : Assume that the following probability updat- ing scheme is used

Pi(t + 1) = P i t ) + hLPAt)(l - Pit)) + aP(1 - Pit))’

if SLOTit) = IDLE or SUCCESS

Pi(t + 1) = P i t ) - LPi(tX1 - P i t ) ) - bL2(Pdt))’ if SLOT&) = COLLISION

where L ~(0, 1) and a, b E (0, l/L). If M packets are waiting to be transmitted on Ai, then

limpit) = 1/M L-tO 1 - r n

The number M of packets waiting to be transmitted on a specific wavelength, say Ai, is slowly varying with time. However, theorem 1 guarantees that the LABRA proto- col always tends to satisfy the condition Pdt) = 1/M.

Proof: The proof is given in Section 8.1.

IEE Proc.-Comput. Digit. Tech., Vol. 142, No. 4, July 199s

5 Simulation results

In the following, the proposed LABRA protocol is com- pared to the well known slotted ALOHA protocol [3,4]. This protocol was chosen for two reasons: both proto- cols are applied to WDM passive star networks using tunable lasers and fixed receivers; and the LABRA proto- col can be considered as an extension of the slotted ALOHA protocol. Therefore a performance comparison between the two protocols will clearly demonstrate the performance improvement through the use of the new learning automata based scheme.

The two protocols under comparison were simulated to be applied to four different networks ( N , , N , , N , and N,). (The simulator does not simulate the mathematical model that is described in Section 4; it does simulate the operation of a real network using the LABRA or the S.ALOHA protocol.) The number of stations N and the number of wavelengths W of each simulated network, were taken to be as follows:

network N , : N = 8, W = 4 network N , : N = 12, W = 4 network N , : N = 16, W = 8 network N,: N = 24, W = 8

For all networks, the total bandwidth was taken to be equal to lOGbit/s. The packet size was equal to IO00 bits, while the queue size was three packets. The fixed receiver of each station ui passes only the ,Ij wavelength, with j = [i/Wl. Each station is assumed to have only packet-switched traffic. All the stations have the same arrival rate of packets, with packet arrivals following the Poisson model.

The internal parameters of the automata used by the LABRA protocol were taken to be: L = 0.3, a = 0.1 and b = 3.0. These values were selected because they give satisfactory results in all the simulated networks. For each simulated network, the slotted ALOHA protocol was simulated for three different values of the fixed trans- mission probability P i . In any case, the values of Pi were appropriately selected so that it is guaranteed that there is no any other value of Pi which leads to a considerably higher throughput of the slotted ALOHA protocol. For each one of the four simulated networks, the fixed values of Pi were chosen as follows. N , : Pi = 0.20, 0.25 and 0.30; N,: Pi = 0.15, 0.20 and 0.25; N , : Pi = 0.15, 0.20 and 0.25; N,: Pi = 0.10,0.15 and 0.20.

We have used the following two broadly used per- formance metrics to compare the LABRA protocol with the slotted ALOHA one:

(i) delay against throughput characteristic (ii) throughput against offered load characteristic

The delay against throughput characteristics of the LABRA and slotted ALOHA protocols when applied to networks N I , N , , N , and N , are shown at Figs. 4, 6, 8 and 10, respectively, while the throughput against offered load characteristics of the two protocols when applied to the networks are shown at Figs. 5, 7, 9 and 11, respec- tively. For each point of these curves, the simulation time was 4OOO00 slots. Therefore the confidence level of the simulation results is very high and consequently the curves are smooth.

From the graphs it becomes clear that under any load conditions the LABRA protocol achieves a higher throughput and a lower delay than the slotted ALOHA one. One might expect that for some values of load, for example when the average number of ready stations per

309

wavelength is equal to l/Pi the slotted ALOHA protocol would achieve a higher throughput than the LABRA one. However, even when the total load offered to the network is fixed, the load offered at each specific wavelength is time variant owing to the ‘wavelength overloading’ described in Section 2. Therefore the throughput of the LABRA protocol remains higher than the one of the

25

20

5 15

5 s 10 n

5

0 005 010 015 020 025 030 035

throughput per wavelength per slot

Fig. 4 Delay against throughput characteristics of LABRA and .%ALOHA protocols when they operate in network N , N = 8; W = 4 I LABON 2 S.ALOHA R, = 0.25 3 S.ALOHA P, = 0.30 4 %ALOHA P, = 0.20

4 035

030

f e 025 U - Q 020 3

a 015

0 010 020 030 040 050 060 070 080 090 10

offered load per wavelength per slot

Fig. 5 S.ALOHA protocols when they operate in network N , CUNeS as in Fig. 4 captlon

Throughput agarnst offered load characteristics of LABRA and

0’ 0 005 010 015 020 025 0 3 0 0.35

thrwghput per wavelength per slot

Fig. 6 Delay against throughput characteristics of LABRA and %ALOHA protocols when they operate in network N, N = 12; W = 4 I LABRA 2 S.ALOHA R, = 0.20 3 S.ALOHA Ri = 0.15 4 S.ALOHA R, = 0.25

310

slotted ALOHA under any load conditions. This applies to both high and low load conditions, wether the load is variable or fixed.

6 Conclusion

We have presented a learning automata based random- access protocol that achieves a high performance under any load conditions.

An analysis of the asymptotic behaviour of the LABRA protocol was presented. It was proved that the transmission probability of each wavelength, asymp-

f cn 025 E

offered lood per wavelength per slot

Fig. 7 S.ALOHA protocols when they operate in network N, CUNCS as in Fig. 6 caption

Throughput against offered load characteristics of LABRA and

0 005 010 015 020 025 030 035

throughput per wavelength per slot

Fig. 8 Delay against throughput characteristics of LABRA and S.ALOHA protocols when they operate in network N, N = 16; W = 8 Curves identified as in Fig. 6 caption

0 10

e ‘ ‘0 010 020 030 040 050 O M ) 070 080 090 10

offered load per wavelength per slot

Fig. 9 S.ALOHA protocols when they operate in network N, N = 16; W = 8 Curves identified as in Fig. 6 caption

Throughput against offered load characteristics ofLABRA and

IEE Proc.-Cornput. Digit. Tech., Vol. 142, No. 4, July 1995

totically tends to take its optimum value. Furthermore, extensive simulation results were presented which indic- ate that a significant performance improvement is achieved when the proposed learning automata based protocol is used.

0' 0 005 010 0.15 020 0.25 030 035

throughput per Wavelength per slot

Fig. 10 S.ALOHA protocols when they operate in network N, N = 24; W = 8 1 LABRA 2 SALOHA Pi = 0.15

Delay against throughput characteristics of LABRA and

3 S.ALOHA pi = 0.10 4 S.ALOHA P, = 0.20

5 0 3 5 VI 1

030 /- 7

6 ' 015

I ! l / 010 \ 0 010 020 030 040 050 060 070 080 090 IO

f oy., " " ' -' ' ' ' " " " " ' ' ' " ' ' " ' ' " ' " " " " " ' , , 1

offered load per wavelength per slot

Fig. 11 Throughput offered load characteristics of LABRA and %ALOHA protocols when they operate in network N, Curves identified as in Fig. IO caption

In this study, the end-to-end propagation delay was assumed to be negligible. However, the LABRA protocol can be applied to networks with large end-to-end propa- gation delay by making use of pipelining [8, l o ] . We are currently working to this direction.

7 References

1 BRACKETT, C.A.: 'Dense wavelength division multiplexing network: principles and applications', IEEE J . Sel. Areas Commun., 1990,8, (6), pp. 948-964

2 HENRY, P.S.: 'Very high capacity lightwave networks'. Proceedings of IEEE ICC, 1988, pp. 1206-1209

3 GANZ, A., and KOREN, Z.: 'WDM passive star - protocols and performance analysis'. Proceedings of IEEE INFOCOM91, April 1991, Bal Harbour, FL., USA

4 GANZ, A.: 'End-toend protocols for WDM star networks'. Pre- sented at IFIP WG6.1-WG6.4 Workshop on protocols for high-speed networks, Zurich, Switzerland, May 1989

5 NARENDRA, K.S., and THATHACHAR, M.A.L.: 'Learning automata: An introduction' (Prentice Hall, Englewood Cliffs, NJ, 1989)

6 NARENDRA, K.S., and THATHACHAR, M.A.L.: 'On the hehav- ior of a learning automaton in a changing environment with appli- cation to telephone traffic routing', IEEE Trans., 1980, SMC-IO, (5). pp. 262-269

IEE Proc.-Cornput. Digit. Tech., Vol. 142, No. 4, July 1995

7 SIVALINGAM, K.M., BOGINENI, K., and DOWD, P.W.: 'Pre- allocation media access control orotocols for multide access WDM photonic networks'. Presented' at SIGCOMM92 August 1992 Baltimore, MD, USA

8 CHEN, M.-S., W N O , N.R., and RAMASWAMI, R.: 'A mcdia a m s s protocol for packet switched wavelength division multiaccess metropolitan network', IEEE J . Sel. Area Commun., 1990, 8, (6). pp. 1048-1057

9 DONO, N.R., GREEN, P.E., LIU, K., RAMASWAMI, R., and TONG, F.F.-K.: 'A wavelength division multiple access network for computer communication', IEEE J . Sel. Areas Commun., 1990, 8, (a), pp. 983-994

10 PAPADIMITRIOU, G.I., and MARITSAS, D.G.: 'WDM passive star networks: Receiver collision avoidance algorithms using multi- feedback learning automata'. Prsented at IEEE 17th conference on Local computer networks, September 1992, Minneapolis, Minnesota, USA

11 NORMAN. M.F.: 'Markovian learninn PIO~SSCS'. SIAM Reo., - . 1974.16, pp. 143-162

Publishers, Moscow, 1977) 12 NIKOLSKY, S.M.: 'A course of mathematical analysis' (MIR

8 Appendix

8.7 Proof of theorem 7 : As shown in Section 3, the function Dip,) which expresses the reward probability as a function of the transmission probability Pi is defined as follows

Di(Pi) = ( 1 - P i p + M P d l - There exists a P: with 0 < Pf < 1 such that

(i) DLP:) = U (6)

(ii) (Pi - Pr)(DXP,) - U) < 0 for all Pi # P: (7) and

Remark A of Section 3 ensures that P:

DLPXt)), we have

1/M. Now define GP&) = P & + 1 ) - PAC). With d&) =

E[GP&) I P M I = d&)(hLPdt)(l - PAC)) + aL'(1 - Pdt))')

+ ( 1 - ddt))( - LPXtXl - Pdt)) - bP(Pdt))') = ql + h)PLt)(l - P4t)Xddt) - U )

+ L'(dXtX1 - Pdt))' - b(l - dXt))(Pdt))')

To prove theorem 1 use the following theorem due to Norman [ll] also presented in Reference 6. Let { ~ ( t ) } , > , ~ be a stationary Markov process dependent on a constant parameter 0 E CO, 11. Each x(t) E I where I is a subset of the real line. Let Gx(t) = x(t + 1) - x(t). The following are assumed to hold

(1) I is compact

(4) EC I 8 4 4 l3 1x0) = YI = 0(e3) where

(2) ~ ~ a ~ ( t ) I x(t) = y] = e&) + o(e2) (3) E[ I w t ) 12 I x(t ) = y] = + o(e2)

O(e5 sup - c co for k = 2, 3 y s l @

and

( 5 ) &) has a Lipschitz derivative in I (6) b(y) is Lipschitz in I

The following theorem concerns the behaviour of { ~ ( t ) } for small values of the parameter 8.

311

Theorem B : If assumptions ( 1 ) to (6) hold, +) has a unique root y* in I and do/dyI ,= , < 0, then

(a) var [x( t ) I x(0) = x ] = O(0) uniformly for all x E I and t 2 0

(b) for any X E I the differential equation dy(s)/ds = o(fis)) has a unique solution y(s) = fiz, x) with y(0) = x and E [ x ( t ) I x(0) = x ] = Ate) + O(0) uniformly for all x E I and t 3 0.

(c) (x(t) - y(t0))/,/(0) has a normal distribution with zero mean and finite variance as 0 -+ 0 and t0 -t 03.

To apply this theorem to the proof of Theorem 1 identify x(t) with Pit). 0 with L and I with (0, 1). If d, = Dipi), we have

ECGPAt) I Pit) = Pil

= I q ( 1 + h)Pi(l - Pi)(di - U)

= Lo(Pi)

+ L[adi(l - Pi)2 - b(1 - di)(PJz])

E [ I @it) 1’ I pit) = Pil

= L’((PX1 - Pi))’(l + dXhz - 1))) + ~ 3 ( 2 ~ h d ~ ~ i i - ~ 3 3 + 2 ~ 1 - d , x ~ 3 3 ( 1

+ L?(a2dX1 - Pi)4 + b2(1 - di)(Pi)4)

= CqPJ + O(L?)

I G P X ~ ) i 3 I P i t ) = p i ]

= P(di[hPAl - Pi) + ,541 - + ( 1 - di) x [P,(l - Pi ) + Lb(Pi)2]3)

= o(~3)

The functions o(P i ) and b(Pi) are defined as follows

o(Pi) = ( 1 + h)Pi1 - Padi - U )

+ ,!.(addl - Pi)’ - b(1 - di)(Pi)2)

WPi) = (PA1 - PJ)’(l + ddhz - 1))

It is immediately seen that assumptions ( 1 ) to (4) are satisfied. It can also be proved that b(Pi) and o’(Pi) are Lipschitz in (0, 1 ) by showing that their first derivatives @‘(Pi) and w”(Pi), respectively, are bounded [12] for Pi E

It remains to show that o(PJ has a unique root e near the point P: (defined by eqns. 6 and 7) and that

Since 4 0 ) = La > 0, 41) = -Lb < 0 and o(PJ is a continuous function it follows that o(P i ) has at least one root e in the unit interval.

(0, 1 ) .

do(Pi)/dPiIP,=c < 0.

Since

o(Pi) = ( 1 + h)PXl - Pi)(di - U )

+ ,!.(addl - Pi)’ - b(1 - di)(Pi)’)

and L can be arbitrarily small, there are only three prob- able roots of o(PJ in the (0, 1) interval.

(i) one root in the neighbourhood of point Pi = 0 (ii) a second root in the neighbourhood of point Pi = 1 (iii) a third root in the neighbourhood of point Pi =

Pf

The relation o(Q = 0 can be written as follows (where 4 = D x m

(1 + h)e(1 - p;Xd: - U)

= - L(d( 1 - Py - b( 1 - rnP1)’) (8)

Assume that o(PJ has a root Pi in the neighbourhood of 0. In this case eqn. 8 is an absurdity since the left-hand side is positive while the right-hand side is negative. Con- sequently, w(PJ does not have a root in the neighbour- hood of 0. In the same way, if o(P i ) has a root < in the neighbourhood of 1 , then eqn. 8 is an absurdity. The left- hand side is negative while the right-hand side is positive. Therefore o(PJ does not have a root in the neighbour- hood of 1 . Consequently, o(Pi) has a unique root in the unit interval. This root is in the neighbourhood of Pt . Note that can be as close to Pf as desired by selecting a small enough parameter L. Now, defining 4 = Dip;) and d;. = Dr(c) , we have

P , = K do(Pi)/dPi 1

= ( 1 + h)[(d; - 11x1 - 2 P 3 + c ( 1 - e x , ]

+ ~ [ d , ( i - e)’ - 2 4 1 - e)d; + b<,(m2 - 26(1- 4)m < ( 1 + h ) [ ( 4 - 0x1 - 2 c ) + c(1 - K)diV] (9)

From the proof of Remark B it is known that 4. < 0. Consequently, c(1 - eM, < 0. Selecting a small enough parameter L, the quantity (4 - U ) can be as close to 0 as desired, so (4 - u)(1 - 2 c ) + c(1 - ex, < 0. In this case, expr. 9 becomes

do(Pi)/dPi I < 0 Q.E.D. Pi=K

It has been shown that d P i ) has a unique root in the neighbourhood of Pt with do(Pi)/dPilPi=fi < 0. Setting PAT) = c , the differential equation dPis) /dr = o(PAr)) is satisfied (0 = 0). Thus, Pi.) = e is a solution of the above differential equation. From theorem B, it is derived that this solution is unique, thus all the solutions starting in (0, 1 ) of the differential equation converge to the point P i s ) = e N Pf N 1/M. According to theorem B, we have limI+m E [ P i t ) ] = P: + O(L) and var [ P i t ) ] = O(L) for all t. Consequently,

lim Pit) = 1/M L-rO I’m

Q.E.D.

312 I E E Proc.-Comput. Digit. Tech., Vol. 142, No. 4, July 1995