a two-level stochastic approximation for admission control and bandwidth allocation

222 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 18, NO. 2, FEBRUARY 2000

A Two-Level Stochastic Approximation forAdmission Control and Bandwidth Allocation

Franco Davoli, Senior Member, IEEEand Piergiulio Maryni, Member, IEEE

Abstract—In an access node to a multiservice network [e.g., abase station in an integrated services cellular wireless network orthe optical line terminal (OLT) in a broad-band passive opticalnetwork (PON)], the output link bandwidth is adaptively assignedto different users and dynamically shared between isochronous(guaranteed bandwidth) and asynchronous traffic types. Thebandwidth allocation is effected by an admission controller,whose goal is to minimize the refusal rate of connection requestsas well as the loss probability of cells queued in a finite buffer.Optimal admission control strategies are approximated by meansof backpropagation feedforward neural networks, acting on theembedded Markov chain of the connection dynamics; the neuralnetworks operate in conjunction with a higher level bandwidthallocation controller, which performs a stochastic optimizationalgorithm. The case of unknown, slowly varying input ratesis explicitly considered. Numerical results are presented thatevaluate the approximation and the ability to adapt to parametervariations.

Index Terms—Access control, broad-band communication,communication system control, multimedia communication,multiplexing, neural networks, resource management.

I. INTRODUCTION

A PECULIAR aspect of high-speed integrated servicesnetworks is the necessity of carrying a great variety of

traffic classes, characterized by different statistical behaviorand performance requirements [as expressed by quality ofservice (QoS) indexes]. A great deal of attention is being paidto satisfy all such requirements (and at the same time to ensureefficiency in the use of resources) within the asynchronoustransfer mode (ATM) [1], or within Internet protocol (IP)networks with integrated or differentiated services [2]–[5].Since a mix of different transfer modes is very likely toremain within the existing networks, at least in the mediumterm, multiplexing devices will be used to provide a physicaland/or medium access control (MAC) layer structure, capableof carrying traffic of asynchronous nature and nonstringentQoS requirements [like best-effort IP packets or ATM cellsgenerated by available bit rate (ABR) services], as well asisochronous, connection-oriented, and real-time-constrainedstreams. In particular, even “hybrid” time-division multiplexing(TDM), where slots in a frame can be dynamically assigned todifferent traffic types, can maintain a position in the user accessarea. Actually, TDM still plays a major role in application areassuch as cellular mobile radio networks [6], [7] and is likely to

Manuscript received February 15, 1999; revised August 1, 1999.The authors are with the Department of Communications, Computer and

Systems Science, DIST-University of Genova, 16145 Genova, Italy (e-mail:[email protected]; [email protected]).

Publisher Item Identifier S 0733-8716(00)00510-2.

be used, though in combination with code-division multipleaccess (CDMA), also in the forthcoming third-generationwireless networks [8], [9]. A further example is constituted bypassive optical networks (PON’s), used in the distribution ofmultimedia services (see, e.g., [10] and [11]), where a TDMframe carries ATM cells in the uplink and downlink betweenoptical network units (ONU’s) in the users’ premises and theoptical line termination (OLT) network interface. This situationmay be considered as a possible application of the model andof the control techniques developed in the paper. However,more generally, we do not explicitly need a framed channelin our model: we can consider the presence of a scheduler,which dynamically assigns the link capacity among differentflows, according to some scheduling algorithm that ensuresservicing of the isochronous streams within the required timelimits (see, e.g., [12] and [13]). In general, in order to allocatethe available link capacity among different traffic classes andensure QoS, while at the same time obtaining an efficient use ofthe bandwidth resources, some form of control must be exertedon the multiplexer. In this respect, several attempts have beenmade to allocate the available bandwidth between different ser-vice classes (e.g., isochronous and asynchronous), and to findthe “optimal” allocation policies in order to minimize a costrelated to the basic QoS indicators (e.g., blocking probabilityfor isochronous connections and packet loss rate or delay forasynchronous packets) [14]–[17].

The purpose of the present paper is to investigate the use ofneural networks and stochastic approximation algorithms thatallow the practical implementation of optimal admission con-trol and bandwidth allocation strategies in a hybrid multiplexerof the type mentioned above, serving multiple users, with dif-ferent traffic types. Actually, a first investigation in this sensehas been made in [18], though in the case of a single user, and ahierarchical neural network controller in the multiuser case wasconsidered in [19]. However, the latter was characterized by anonstationary control and a finite horizon optimization, whereasa quite different (and more realistic) setting is considered inthe present paper, by seeking stationary control strategies overan infinite horizon and by explicitly considering time-varyingtraffic parameters. In particular, we are interested here in theinclusion of the neural network controller in a two-level hierar-chical scheme over an infinite time horizon, where bandwidthis allocated among a number of user sites, which in turn inde-pendently perform admission control. As will be apparent in thefollowing, the use of neural approximations in this kind of prob-lems, whose dynamics is characterized by finite state Markovchains, presents a practical interest mainly in the adaptive con-trol case (i.e., when the controller has to adjust its parameters to

0733–8716/00$10.00 © 2000 IEEE

DAVOLI AND MARYNI: TWO-LEVEL STOCHASTIC APPROXIMATION 223

follow slowly varying traffic statistics). Neural networks havebeen widely used in access control and bandwidth allocationproblems in the literature, especially in the context of ATM net-works (e.g., [20]–[23]).

The paper is organized as follows. In Section II, we formallydefine the system model. The approximation of the optimalstrategies for a single user by means of a neural controlleris outlined in Section III, where nonconstant arrival rates inboth isochronous and asynchronous traffic are considered. InSection IV, we illustrate in more detail how the bandwidth can bedivided among multiple users, and in Section V some numericalresults are reported. Section VI draws the final conclusions.

II. THE SYSTEM MODEL

We consider an access multiplexer, handling a slotted channelshared among users. Each user ( ) generatestwo types of traffic: a real-time isochronous and connection-ori-ented one, with average connection request arrival ratere-quests/s and average holding time s, and an asynchronousconnectionless one (cell switched), with average arrival ratecells/frame, with fixed length (one slot). The “frame” time inour case is just the time required by an isochronous sourceto generate cells. The cells originating from the asynchronousconnectionless traffic can be stored in a buffer, which has a fi-nite capacity . As a matter of fact, we can think in terms ofan ATM multiplexer handling continuous bit rate (CBR) trafficon a guaranteed rate basis, as well as ABR traffic; we considerthe former only at the call level, whereas we are interested inthe cell-level behavior of the latter. The ABR cells may be orig-inated, in turn, by the segmentation of variable length packets;in the case, for instance, of TCP/IP data, limiting the cell lossis desirable to reduce retransmission of lost packets. The arrivalrates of both isochronous and asynchronous traffic are not sup-posed to be constant but slowly varying within a given range(slowly with respect to connection dynamics). The dependenceon time of and will be always supposed to hold, evenif, for the sake of notational simplicity, it is not explicitly an-notated. Isochronous calls are subject to a probabilistic admis-sion control scheme. More specifically, given that userhasalready isochronous connections in progress, a new connec-tion request is accepted with probability . The acceptanceprobability values for all possible realizations ofare collectedin the vector .

Fig. 1 depicts a skeleton of the system model. Theisochronous CBR traffic requires continuation of service (cells every s) until the end of the connection and, in order toguarantee this rate, pre-emption is performed with respect tothe asynchronous traffic. Arrivals in the queue of the latter willbe modeled as the superposition of geometric distributions, awidely used approach in modeling cell arrivals at ATM switches(see, e.g., [24]). Connection request interarrival and holdingtimes will be supposed to have exponential distribution; we canthen represent the connection dynamics as a continuous-timebirth–death process. The goal of the assignment strategy shouldbe that of deciding how to share the channel among theusersand how to share the amount of transmission resources as-signed to each user between the two traffic types. It can be seen,

Fig. 1. The skeleton of the system model.

Fig. 2. Graphical description of the two-level partitioning.

therefore, as a hierarchical control task, where the resources ofa large system are first partitioned intosubsystems, one peruser, and then further partitioned among two traffic types. Wecall the first partitioning task multiuser partitioning, and thesecond one multiservice partitioning. Multiservice partitioning,for user , is achieved by means of the acceptance probabilityvector .

Fig. 2 illustrates this double partitioning (i.e., hierarchical)scheme. In the following, the transfer capacity of the systemwill not be described in terms of bits/s, but rather in terms ofthe maximum number of isochronous calls (CBR connec-tions) that can be simultaneously in progress with all the systembandwidth and of the service rate seen by the asynchronoustraffic, given the number of isochronous calls in progress. Inparticular, if at time instant there are isochronous calls inprogress (for a given subsystem), then the service rate seenby the asynchronous traffic (for the same subsystem) is mod-eled as being constant with rate . Fig. 3 gives a pic-torial representation of this way of describing the system ca-pacity. In the simplest case, can be equal toif is the number of bandwidth units (cells per reference time

) used by an isochronous connection and is the capacityassigned to the th user subsystem. In other situations (e.g.,those where the asynchronous flows from different users con-tend for the available bandwidth), can capture the effectof a given channel access control mechanism by representingthe maximum service rate achievable.

Thus, the result of the multiuser partitioning task is tosubdivide the total “capacity” of simultaneous isochronouscalls among the users; the generic user will be, therefore,assigned a capacity equivalent to isochronous calls, with

. Fig. 4 shows a pictorial example of such a


Fig. 3. Graphical description of the system capacity (� refers to the ratebefore partitioning).

resource partitioning between two subsystems (i.e., ).Since in multiuser partitioning we will need to use resultswhich pertain to the multiservice one, we start first by lookingat the latter in more detail.

III. M ULTISERVICE PARTITIONING

Let us now address the multiservice partitioning, the controlmechanism that, for each subsystem (i.e., for each user), allowssharing the networking resources between the two traffic types.In the following, we will assume that the generic user(

) is assigned a capacity of isochronous connec-tions as a result of the multiuser partitioning. The isochronoustraffic is therefore represented by a controlled /queueing system; we consider the embedded Markov chain thatarises from looking only at the instants at which connectionsare set up and torn down. We can do so, since we suppose ourallocation strategy to act explicitly only on the acceptance ofisochronous traffic connection attempts. Cell traffic is simplyassigned to slots that the isochronous connections do not utilize:No explicit control is done on the acceptance of cells. Moreover,we assume that cell dynamics run on a faster time scale with re-spect to the isochronous traffic (as seen at the call level): thisimplies that we can think of the queue process in terms of a sta-tionary one. In other words, we suppose the time between twosuccessive events of the birth–death process characterizing theisochronous traffic to be large enough to make the transient be-havior of the queue process negligible.

A similar way of reasoning has been followed in [25]:the approximation is applied to the joint stationary distribu-tion of the two traffic types. When in the system there are,say, calls in progress, we will model the cell queue as aGeom one (see the Appendix).Therefore, in our context, the state of the whole subsystem ata certain time instant is represented only by the numberofcalls in progress ( ). Letrepresent a discrete-time variable, counting the occurrenceof an event in the isochronous traffic process for the genericuser (i.e., a connection has been set up or has ended), and

let be the number of calls in progress at time(actually,the notation should be and ; however, for the sakeof notational simplicity, we have dropped the subscript).Whenever an incoming connection request for a circuit appears,it is either accepted with probability or it is blocked(refused) with probability . Since the channel isonly able to carry isochronous connections, we will have

. Thus represents a randomized controlstrategy that maps into a probability of call acceptance, andwhose choice will be one of the main goals of our treatment.Since we suppose that asynchronous traffic cells occupy slotsunused by the isochronous traffic, this strategy plays the roleof a “probabilistic movable boundary” and realizes a partitionof the available capacity. Notice that, since we consider timevarying input rates, the functions (i.e., the vectors withcomponents turn out to be alsotime varying (not explicitly annotated for the sake of notationalsimplicity). However, we suppose input rate variations tobe slow enough, with respect to isochronous call dynamics,to let the isochronous process reach stationarity. Under thisassumption, if we look now at the call dynamics (for a givenvalue of the input rates), we will see that the processis acontrolled birth–death one, with upward rate anddownward rate , as depicted in Fig. 5.

We compute the strategy by minimizing a local cost func-tion with the following structure:

(1)

whereand weighting coefficients;

and two generic functions, whose choice willbe specified;blocking probability for isochronouscalls;cell loss probability for asynchronoustraffic.

All this is related to the th user. In the following we willconsider

(2)

where is the packet loss probability given that() isochronous calls are in progress (see the Appendix); in

regard to , we will consider two different forms, namely

(3a)

(3b)

In the above expressions, represents the probability ofhaving isochronous calls in progress for user(i.e., the prob-ability of being in state ). The form of (3a) “enforces” a prob-abilistic control strategy, owing to the way the control functionenters the cost, and its rationale is to minimize the blockingprobabilities by always trying to accept with probability 1 at


Fig. 4. Graphical description of the partitioning among subsystems (M = 2).

each state value. On the other hand, (3b) contains the averageblocking probability

(4)

as a whole, and it comes into effect only when this quantityexceeds a given threshold . In practice, this form representsa penalty function, trying to enforce a constraint on the averageblocking probability; notice that, whenever , (3b) wouldlead to a deterministic control strategy. In the following we willdenote the two cost functions as and .

By looking at Fig. 5, with some trivial calculations, it isstraightforward to come up with the following recursive way ofcomputing :

Fig. 5. Isochronous call dynamics for userm.

(5)


If we had fixed input rates, we would have now all the informa-tion for computing the optimum control law by using, forexample, a gradient descent algorithm. Since we have variableinput rates (and therefore variable ), it is reasonable to em-ploy a more complex framework by introducing a neural networkcontroller. Our aim is to keep the local admission controller assimple as possible by approximating our probabilistic admissioncontrol strategies over a range of input rates, to avoid solving amathematical programming problem for each new value of therates that is observed. The latter may be tracked over time, bymeans of some estimation mechanism (see, e.g., [26]).

A. The Neural Controller

Let us constrain the control strategies to take on a fixed struc-ture of the form

(6)

whereinput/output mapping of a multilayer feedforwardneural network;

vector of the synaptic weights to be determined.

We have indicated explicitly the dependence of the acceptancestrategy on the input rates. The function takes into accountconstraints on the output of the neural network, that is

(7)

which means that we cannot accept more calls than the numberof slots we have in each frame. If we now substitute (6) in (5),(2), (3a), and (3b) in (1), the cost function takes on the form

. Let us now describe in some detail the neural net-works that implement our strategies and, hence, the finalcontrol strategies. In general, one such network is composed of

layers and, in the generic layer, neural units are active.The input/output mapping of theth neural unit of theth layeris given by (we drop the index for the sake of simplicity)

(8)

(9)

whereoutput variable of the neural unit;

sigmoidal activation function;

, weight and bias coefficients.

All these coefficients are the components of the vectorap-pearing in the control law (6); the variable coincides with ,

coincides with , coincides with , and the vari-able yields the output generating the control strategies

. The structure of the neural network is presented in Fig. 6,where a single hidden layer is considered. In this respect, thefollowing remarks, also related with the numerical results to bediscussed in Section V, can be made.

(a)

(b)

Fig. 6. The neural network structure. (a) General structure. (b) Detail of aneuron.

• It is “almost always” possible to say that the same level ofapproximation obtained with more than one hidden layercan be achieved with a single hidden layer, as is the caseof the net proposed in the paper. This is a consequence ofthe Weierstrass property possessed by nets with a singlehidden layer and linear output function (see, among others,[27]). The term “almost always” stems from the fact that,in certain particular optimization problems, it is possibleto show that nets with two hidden layers can perform tasksthat cannot be achieved by nets with a single hidden layer[28].

• As concerns the number of neural units present in thesingle hidden layer, we have deemed of little significance(given the net’s simplicity) the use of “pruning” or other,anyhow heuristic, techniques. We have therefore chosenonly to perform various experiments on the number ofneurons. In practice, we have kept increasing the numberof neurons, as far as the cost value became practically in-sensitive to the number used. This is clearly a “trial anderror” rule, but we considered its application to be reason-able in this case.


We focus our attention on methods of the gradient type, as,when applied to neural networks, they are well suited to dis-tributed computation. The gradient algorithm can be written asfollows:

(10)

where is the descent step size, which is generally madedependent from the iteration number. More specifically, wehave chosen , where and aretwo constant values. Moreover, the last term in (10) representsthe so-called “momentum term,” which is usually introduced intraining neural networks, in order to accelerate the convergence( is a suitable positive scalar). It is worth noting that thealgorithm in (10) actually implements a stochastic approxima-tion with respect to the input rates (see [29] and [30]), whichconsists in iterating a classical gradient algorithm by pickingrandomly, at each descent iteration, the values of the input rates;the dependence of the gradient on the realization of , and

at the th iteration has been therefore indicated.The neural network is then trained by following the classical

backpropagation algorithm [31].

IV. M ULTIUSER PARTITIONING

The goal of the multiuser partitioning task is to decide howusers have to share a common resource. More in particular,

we need to identify how to subdivide a total bandwidth equiva-lent to isochronous calls. In doing this, we will not take intoaccount the exact values of the input rates, but only their rangeof variation (or their probabilistic distribution). Our aim is tofind a simple “average” partitioning over an infinite horizon (inpractice, a partitioning that can last for a relatively long time,with respect to fluctuations in the input rate). We now define aglobal cost as the sum of all the local ones

(11)

where the dependence on and has not been explicitlyindicated. Cost (11) should be averaged with respect to the inputrates, and then minimized with respect to and ,

. In order to avoid averaging over the input rates, weprefer to derive the optimal values of for each given valueof by means of a stochastic approximation technique (seeagain [29]).

Let us further define the minimum cost, with respect to thecontrol strategy

(12)

In this framework, we can use a classic dynamic programmingalgorithm in a resource partitioning context [32]. Let us definethe cost matrix , each element of which represents the min-imum average local cost of user , if capacity isassigned. In order to compute the minimum average local costs

, we can use a fast technique such as the stochasticgradient descent one, as in the previous section, namely

(13)

whereiteration dependent descent step-size, thatalso takes on the form ;

, input rate values at iteration(see again [29]).

The matrix has dimensions , wheretotal capacity (in terms of number of CBR connec-tions);total number of users.

Let us further define a cost-to-go function in the following way:

(14)

The cost-to-go has to be interpreted as follows. If for usersthere are resource units (i.e, bandwidth units) left

and user uses, say, resource units, then for usersthere will be resource units left. Thus, the op-

timum combination of and is the one shown in (14), whichhas to be computed for each, . Going backwardfrom user to user 1, it is possible, by means of (14) to com-pute the cost-to-go matrix. Once this matrix has been com-puted, the following forward step allows to find the optimumallocation:

(15)

The result of the dynamic programming algorithm is the optimalset , such that , which mini-mizes (11).

V. NUMERICAL RESULTS

In this section we will give some numerical results relativeto the allocation strategy described in the previous sections. Re-sults from both cost functions will be presented. The parameterswhich characterize the system are the global capacityand, forthe generic user , the buffer size for asyn-chronous traffic , the range of variation of the input rates

, , , , their distribution within suchrange, the two cost weighting factors and , the structureof the neural networks (number of levelsand number of neu-rons per level ), the learning rate factors and


, and, in regard to , the parameter . For both caseswe considered

Network:

Neural Net:

Cost:

for all ; in regard to , we considered itas a linear function of the number of isochronous CBR calls.Each CBR call corresponds to four asynchronous servers (i.e.,

). Input rates are sup-posed to vary with uniform distribution within given ranges.More in particular, with respect to input rates, we present re-sults from three main cases (for both costs and ). In thefirst case (case I), the range of input rates is the same for all users(i.e., , and , are the same for all

); in the second case (case II), all users have thesame range for isochronous arrivals only; in the third case (caseIII) half of the users use a range while the other half uses an-other range, different with respect to both types of traffic. Thesethree cases, along with the actual values adopted, are depicted inFig. 7, where the superscriptshave been dropped in the inputrates on the axes.

With such figures, the multiuser partitioning algorithm gives,for both costs ( and ), the following bandwidth unit as-signments.

Case I 5 5

Case II 6 4

Case III 7 3

It is interesting to notice that, with both cost forms, mini-mizing the sum of the average costs seems to always lead togive more resources to those users that have lower input rates.In other words, the cost reduction obtained by giving more re-sources to the first set of users (Users ) is larger than thecost increase obtained by giving less resources to the second setof users (Users ). This is mainly because of the role playedby the cell loss probability [the one given in (2)]; this behaviorcan, hence, be partially driven through the weighting parame-ters and .

In regard to the multiservice partitioning, let us consider firstthe results obtained by using the cost function and then theones relative to .

A. Cost function

Figs. 8–10 show the behavior of the neural network approx-imation for a generic user within each of the groups that havethe same capacity assignment, for the three cases of input rateranges (we have dropped the index). The output of the neural

Fig. 7. The input rates configurations (the numbers within the graph representthe users over which the range of values of� and� is applicable).

Fig. 8. Gradient descent (� ) versus neural network approximate (~�). Case I.

networks (one per user) have been compared with the ones ob-tained by the application of an exact gradient method, wherethe descent is based on the given values of the input rates. Thevalues of the components of the acceptance probabilityareplotted versus the stateof isochronous calls in progress and


Fig. 9. Gradient descent (� ) versus neural network approximate (~�). Case II.

the isochronous call input rate . For the sake of plot read-ability, has been kept fixed to its average value (i.e.,

), even if the neural network has beentrained within the range given above. It is worth highlightingthat the neural network has been trained only once for the wholeset of CBR input rates (i.e., ) and therefore the values ofhave been simply obtained as outputs of the neural network(giving the values of , and as input). In regard to thegradient method, in order to compute, the descent algorithmhas been performed for each value of (and ). The plots,and many others not reported here for lack of space, show thatthe neural network approximation is very good. On the otherhand, the neural network has needed about 10 000 (offline) stepsto be trained, while the gradient method needs about 500 (on-line) descent steps to give good results. Under this point of view,choosing between the two mechanisms is mainly a matter ofavailable (online) processing power. The quantity of memoryneeded by the neural network is, in this case, very little sincevery few neurons are needed. Experiments with larger scale sys-tems have shown that the increase in memory size needed is rea-sonable (i.e., smaller than linear). In order to show that the smalldifferences between the “optimal” functions obtained with theexact gradient descent and those obtained with the neural net-works give rise to small differences also in the cost functionsand in their single elements, we show in Fig. 11 the differencein cell loss probabilities in the two situations; the differences be-tween the complete cost functions is shown in Fig. 12 (we havelimited the comparison to case II; all other cases give a similarbehavior). Notice that in this last figure the average cost is eval-

Fig. 10. Gradient descent (� ) versus neural network approximate (~�). CaseIII.

Fig. 11. Average cell loss probabilities (optimal versus approximate). Case II.


Fig. 12. Cost computed with optimal versus approximate values of�. Case II.

Fig. 13. Average cell loss probabilities. Controlled (optimal) versusuncontrolled (�(s) = 1; 8 s) situation. Case II.

Fig. 14. Gradient descent� ) versus neural network approximate (~�). Case I.

Fig. 15. Gradient descent (� ) versus neural network approximate (~�). CaseII.

uated over the entire range of values of both input rates. More-over, Fig. 13 highlights the difference in cell loss probabilitiesbetween the controlled and the uncontrolled ( )case (always with respect to case II).

B. Cost Function

Within this cost we have . This means that, inregard to the isochronous part, this cost “comes into being” onlywhen the average blocking probability exceeds 1% of the totalincoming calls. Figs. 14–16 show the behavior of the neural net-work approximations for the same cases considered above. Theplots have to be interpreted exactly as the ones obtained withcost . The “tendency” toward a deterministic behavior of theacceptance strategies is apparent. In regard to the evaluation ofthe performance in terms of the cost functions, in Figs. 17–19


Fig. 16. Gradient descent (� ) versus neural network approximate (~�). CaseIII.

Fig. 17. Average cell loss probabilities (optimal versus approximate). Case II.

the set of comparisons is carried out as above, showing a goodlevel of accuracy.

Fig. 18. Cost computed with optimal versus approximate values of�. Case II.

Fig. 19. Average cell loss probabilities. Controlled (optimal) versusuncontrolled (�(s) = 1; 8 s) situation. Case II.


VI. CONCLUSION

A two-level bandwidth allocation and admission controlstrategy has been defined in the paper, in the context of anaccess multiplexer to a multiuser, multiservice broad-bandtelecommunication network. This task is accomplished bypartitioning the bandwidth among a set of users and, for eachuser, by approximating optimal admission control strategiesfor two traffic types by means of multilayer feedforward neuralnetworks. Numerical results have shown the capacity to followvariations in traffic parameters by means of a fast and simpleadaptation mechanism.

APPENDIX

THE GEOM QUEUE

In a synchronous Geom queue (see [24, Ap-pendix]) servers are available to mutually independent cus-tomers, each requesting service with probabilityand positions (including the places for the customers inservice) are available to hold the customers in the system. Thesystem is synchronous in the sense that the time axis is slotted,each slot lasting one (deterministic) service time of a customer.All arrivals and departures take place at slot boundaries. Theprobability distribution of customer requestsin a generic slotis binomial

(16)

with mean value .An assumption which simplifies computations is that within

the queue, first customers in service (at most) are removedand then new customers are stored. Ifdenotes the (discrete)slot time, the system evolution is described by

(17)in which and represent the customers in the queue and thenew customers requesting service, respectively, at the beginningof slot .

Let us denote with the probability of having customersin the system at a generic time(i.e., ); thenthe balance equations (from which thecan be obtained) arederived by considering the following three cases.

Case I)

(18)

Case IIa) and

(19)

Case IIb) and

(20)

Once the are known, it is possible to compute the averagecarried load of the servers in the following way:

(21)

The loss probability is then computed by consideringthat the average load offered to the queue is

(22)

The model fits in our system for the generic userby setting (i.e., ),

(hence ) and (see Figs. 3 and4) representing the maximum possible value (before resourcepartitioning) for the cell output rate.


The computation of is performed for all the values thatcan assume. If , then .

REFERENCES

[1] M. Schwartz,Broadband Integrated Networks. Upper Saddle River,NJ: Prentice Hall, 1996.

[2] P. P. White and J. Crowcroft, “The Integrated Services in the Internet:State of the art,”Proc. IEEE, vol. 85, pp. 1934–1946, Dec. 1997.

[3] R. Braden, D. Clark, and S. Shenker. (1994, June) Integrated Servicesin the Internet architecture: An overview. Internet Engineering TaskForce. Request for Comments, RFC 1633. [Online] Available FTP:ds.internic.net/rfc/rfc1633.txt.

[4] S. Blake, D. Black, M. Carlson, E. Davies, Z. Wang, and W. Weiss.(1998, Dec.) An architecture for differentiated services. Internet Engi-neering Task Force. Request for Comments, RFC 2475. [Online] Avail-able FTP: ds.internic.net/rfc/rfc2475.txt.

[5] T. Li and Y. Rekhter. (1998, Oct.) A provider architecture for differ-entiated services and traffic engineering (PASTE). Internet EngineeringTask Force. Request for Comments, RFC 2430. [Online] Available FTP:ds.internic.net/rfc/rfc2430.txt.

[6] J. E. Padgett, C. G. Günther, and T. Hattori, “Overview of wireless per-sonal communications,”IEEE Commun. Mag., vol. 33, pp. 28–41, 1995.

[7] D. D. Falconer, F. Adachi, and B. Gudmundson, “Time division multipleaccess methods for wireless personal communications,”IEEE Commun.Mag., vol. 33, pp. 50–57, 1995.

[8] L Ortigoza-Guerrero and A. H. Aghvami, “A distributed dynamic re-source allocation for a hybrid TDMA/CDMA system,”IEEE Trans. Veh.Technol., vol. 47, pp. 1162–1178, Nov. 1998.

[9] A. Samukic, “UMTS universal mobile telecommunications system:Development of standards for the third generation,”IEEE Trans. Veh.Technol., vol. 47, pp. 1099–1104, Nov. 1998.

[10] I. Van de Voorde and G. Van der Plas, “Full service optical access net-works: ATM transport on passive optical networks,”IEEE Commun.Mag. , vol. 35, pp. 70–75, Apr. 1997.

[11] N. J. Frigo, “Local access optical networks,”IEEE Network, vol. 10, pp.32–36, Nov./Dec. 1996.

[12] A. Parek and R. Gallager, “A generalized processor sharing approachto flow control—the single node case,”IEEE/ACM Trans. Networking,vol. 1, pp. 366–375, 1993.

[13] , “A generalized processor sharing approach to flow control—Themultiple node case,”IEEE/ACM Trans. Networking, vol. 2, pp. 137–150,1993.

[14] B. S. Maglaris and M. Schwartz, “Optimal fixed frame multiplexing inintegrated line- and packet-switched communication networks,”IEEETrans. Inform. Theory, vol. IT-28, pp. 273–283, 1982.

[15] K. W. Ross and D. H. K. Tsang, “Optimal circuit access policies inan ISDN environment: A markov decision approach,”IEEE Trans.Commun., vol. 37, pp. 934–939, 1989.

[16] G. Meempat and M. Sundareshan, “Optimal channel allocation policiesfor access control of circuit-switched traffic in ISDN environments,”IEEE/ACM Trans. Networking, vol. 5, pp. 291–304, Apr. 1997.

[17] R. Bolla and F. Davoli, “Control of multirate synchronous streams inhybrid TDM access networks,”IEEE/ACM Trans. Networking, vol. 5,pp. 291–304, Apr. 1997.

[18] R. Bolla, F. Davoli, P. Maryni, and T. Parisini, “An adaptive neural net-work admission controller for dynamic bandwidth allocation,”IEEETrans. Syst., Man, Cybern. B (Special Issue on Artificial Neural Net-works), vol. 28, pp. 592–601, Aug. 1998.

[19] A. Karbowski, M. Małowidzki, R. Bolla, F. Davoli, and T. Parisini, “Ahierarchical neural optimization structure for access control and resourcesharing in integrated communication networks,” inProc. IFAC Symp.Large Scale Systems Theory and Applications, London, U.K., July 1995,pp. 889–894.

[20] IEEE Commun. Mag. (Special Issue on Neurocomputing In High-SpeedNetworks), vol. 33, Oct 1995.

[21] A. Hiramatsu, “Integration of ATM call admission control and link ca-pacity control by distributed neural networks,”IEEE J. Select. AreasCommun., vol. 9, pp. 1131–1138, Sept. 1991.

[22] E. Nordstrom, J. Carlstrom, O. Gallino, and L. Asplund, “Neural net-works for adaptive traffic control in ATM networks,”IEEE Commun.Mag., vol. 33, pp. 43–49, Oct. 1995.

[23] C. K. Tham and W. S. Soh, “Multi-service connection admission controlusing modular neural networks,” inProc. INFOCOM 98, San Francisco,CA, Mar./Apr. 1998.

[24] A. Pattavina, Switching Theory—Architecture and Performance inBroadband ATM Networks. Chichester, U.K.: Wiley, 1998.

[25] S. Ghani and M. Schwartz, “A decomposition approximation for theanalysis of voice/data integration,”IEEE Trans. Commun., vol. 42, pp.2441–2452, 1994.

[26] R. Warfield, S. Chan, A. Konheim, and A. Guillaume, “Real-time trafficestimation in ATM networks,” inProc. ITC’94, 1994, pp. 907–916.

[27] M. Leshno, V. Ya, and A. Pinkus, “Multilayer feedforward networkswith a nonpolynomial activation function can approximate any func-tion,” Neural Networks, vol. 6, pp. 861–867, 1993.

[28] E. D. Sontag, “Feedback stabilization using two-hidden-layer nets,”IEEE Trans. Neural Networks, vol. 3, pp. 981–990, 1992.

[29] H. J. Kushner and G. G. Yin,Stochastic Approximation Algorithms andApplications. New York: Springer-Verlag, 1997.

[30] T. Parisini and R. Zoppoli, “Neural approximations for infi-nite—Horizon optimal control of nonlinear stochastic systems,”IEEE Trans. Neural Networks, vol. 9, pp. 1388–1408, 1998.

[31] D. E. Rumelhart and J. L. McClelland,Parallel Distributed Pro-cessing. Cambridge, MA: MIT Press, 1986.

[32] K. Ross,Multiservice Loss Models in Broadband TelecommunicationNetworks. London, U.K.: Springer-Verlag, 1995.

Franco Davoli (M’90–SM’99) received the Laureadegree in electronic engineering from the Universityof Genoa, Genoa, Italy, in 1975.

In 1985 he became Associate Professor andin 1990 Full Professor of TelecommunicationNetworks at the University of Genoa, where he iswith the Department of Communications, Computerand Systems Science (DIST). From 1989 to 1991and from 1994 to 1996, he was also with theUniversity of Parma, Italy, where he taught a class intelecommunication networks. In 1994 and 1999, he

guest edited two special issues on wireless communications for theEuropeanTransactions on Telecommunications (ETT). He has coauthored over 140scientific publications in international journals and international conferenceproceedings. His current research interests are in bandwidth allocation,admission control and routing in high-speed multiservice networks, mobilecellular networks, and multimedia communications and services.

Dr. Davoli is a member of the Editorial Board of theInternational Journal ofCommunication SystemsandStudies in Informatics and Control.

Piergiulio Maryni (M’96) was born in Genoa, Italy,in 1965. He received the Laurea degree in electronicengineering and the Ph.D. degree in telecommunica-tions from the University of Genoa, Genoa, Italy, in1990, and 1994, respectively.

In 1993, he was with the Center for Telecommu-nications Research (CTR) at Columbia University,New York, as a Visiting Scholar, and in 1994 hewas with the University of Parma, Italy, where hetaught a class in digital signal processing. Since1997 he has been a Postdoctoral Research Fellow at

the Department of Communications, Computer and Systems Science (DIST),University of Genoa. His main research interests are in bandwidth allocationand admission control, both in wired and wireless networks, multimediacommunications and services, and ATM load estimation.

a two-level stochastic approximation for admission control and bandwidth allocation

Documents