Dimensioning of play-out buffers for real-time services in a B-ISDN

Computer Communications 21 (1998) 980–995

Dimensioning of play-out buffers for real-time services in a B-ISDN N. Ble´fari-Melazzi a,*, V. Eramo b, M. Listanti c a

University of Roma ‘Tor Vergata’, Dipartimento di Ingegneria Elettronica, Via della Ricerca Scientifica, 00133 Rome, Italy b Fondazione Ugo Bordoni, Rome, Italy c University of Roma ‘La Sapienza’, Dip. INFOCOM, Rome, Italy Received 22 October 1997; received in revised form 25 March 1998; accepted 26 March 1998

Abstract A large fraction of traffic that the future B-ISDN will probably transport is made up of real-time services with stringent delay and delay jitter requirements. Since ATM networks do not provide time transparent links (i.e. constant delay links), a delay equalisation has to be provided in the adaptation layer or in user equipments. In this paper, we first present a source model analysis and then propose an analytical model that allows the evaluation of the end-to-end delay and of the relevant jitter; finally, we focus on the dimensioning of play-out buffers. The proposed model is validated with simulation results and we found a good agreement between analytical and simulation results. To make the study analytically tractable we use rather simple traffic source models and we make suitable simplifying assumptions. These assumptions are fairly general but the source models are still far from representing accurately certain kinds of real traffic. For this reason, we also carry out an additional simulation study (supported by some analytical arguments), by using real experimental MPEG and LAN traffic traces, to assess the system performance in real scenarios. q 1998 Elsevier Science B.V. Keywords: B-ISDN/ATM; Tandem queueing systems; Tagged and background traffic; End-to-end analysis; Delay jitter; Play-out buffer dimensioning; Analytical modelling; MPEG and LAN traffic

1. Introduction In the heterogeneous ATM framework it is necessary to take into account many performance measures such as mean loss probability, number of consecutive losses, loss fairness, mean and maximum delay, quantiles of the delay and delay jitter. In fact, some user applications are loss sensitive, some are delay sensitive and some are both loss and delay sensitive [1,10]. In particular, a large fraction of traffic of emerging high speed networks will be made up of real-time, isochronous services such as video, voice and multimedia applications. These services have stringent delay and delay jitter requirements. For instance, the MPEG codified video, which is expected to be the major compression algorithm to be used in video applications, can tolerate very small delay variations; if the network does not provide a time transparent link between data source and decoder, then it is the latter that has to equalise the delay; this means that some memory must be allocated in the decoder for holding data before it is * Corresponding author. Tel.: +39 6 7259 7450; fax: +39 6 487 3300; e-mail: [email protected]

0140-3664/98/$19.00 q 1998 Elsevier Science B.V. All rights reserved PII S 01 40 - 36 6 4( 9 8) 0 01 7 1- 6

decoded. Set-top manufacturers will need to add memory chips and the end-to-end delay will increase [9]. Such added delay is completely acceptable for non-real-time traffic but the real-time traffic has stringent constraints from this point of view. The B-ISDN will provide delay equalisation together with other functionalities in the adaptation layer. For instance, the AAL1 has been designed to support constant bit rate (CBR) services and provides a constant delay (and a FEC) across network connections. Similar considerations can be made also for loss impairments; many applications tolerate only very small loss probabilities and the network operator has to guarantee such performance since retransmission procedures are either not convenient or not usable at all (e.g. real-time services). There is currently a debate in the standard bodies and industry forums on which AAL is to be used for each specific service (e.g. MPEG) and on which adaptation functionalities must be performed by network equipments and which by user equipments [9]. In any case there is a need to evaluate end-to-end performance. Most of the performance evaluation studies carried out up to now have been concerned with the analysis of the access stage of the network. Only a few papers deal with the

N. Ble´fari-Melazzi et al./Computer Communications 21 (1998) 980–995

characterisation of the traffic streams within the ATM trunk network. The characterisation of the traffic within the ATM trunk network can be applied either to the evaluation of the loss experienced by a traffic stream passing through the network or to the analysis of the end-to-end delay and delay jitter. In Ref. [3] we faced the loss analysis of a cascade of switching or multiplexing buffered elements (BE). In this paper, we first evaluate the end-to-end delay and delay jitter and then we focus on the dimensioning of play-out buffers. The latter is a very interesting point since it has implications on the design of both AAL and specific user equipments functionalities. To the best of our knowledge, only few results are available in the literature on the characterisation of the traffic in the inner network stages (among them, Refs. [7,8,11–14]). The study of the burst expansion phenomenon [12] and of the inter-departure time distribution of a tagged stream multiplexed with a background traffic [13] has shown that, under suitable hypothesis, burstiness and mean waiting times decrease at each crossed node. The inter-departure time distribution converges to a limiting distribution. The limit distribution depends only on the tagged stream bandwidth and on the background traffic. The initial statistical characteristics of the tagged stream affect only the convergence speed [8,11]. In Refs. [8,11] the authors assume a renewal tagged source and a renewal background traffic (in the end-to-end analysis, a correlated background traffic is considered only in the analysis of the access stage in Ref. [11]). However, the model applications are computationally affordable when the load offered by the tagged sources is high, e.g. no result is provided for values of the traffic offered by tagged streams lower than the 10% of the channel capacity. Moreover, in Ref. [11] only results concerning periodic tagged sources (or CBR) are provided. Therefore the applicability of these models covers only a part of realistic source scenarios. In this paper, we present an analytical model for the endto-end delay and jitter calculus, solved by means of a very efficient algorithmic procedure. The delay that we consider is only the one suffered in the network queueing systems plus the time needed to transmit a cell but not the propagation delay. The latter can be trivially added to the former. The model is based on the same hypothesis made in Refs. [8,11] and can therefore be regarded as a simple extension of the aforementioned models, but, by decreasing the computational complexity, we extend the model applicability to cover more realistic traffic sources, i.e. sources with average bit rate considerably lower than the link bit rate and VBR ones. Under the assumption of ‘low traffic sources’, we also provide a useful approximation of the total delay distribution experienced by the tagged streams in a cascade of ATM nodes. This approximate expression depends only on the characteristics of the background traffic and on the number of crossed stages. The proposed model is validated

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with simulation techniques and we found a very good agreement between analytical and simulation results. Real-time, isochronous services need the delay jitter to be eliminated; this is achieved by means of the play-out procedure. According to this procedure, successive data units are stored in a buffer and then they are played out to the receiver at a rate that matches the emission rate (this is done, for instance, by using the timestamp mechanism [9]). In this work, we also introduce a model for the dimensioning of the play-out buffer for real-time services, located at the receiving site. Under the assumption of independence of the delays experienced by the cells of the tagged stream, a simple expression of the buffer size needed to obtain a given integrity of the reconstructed signal is found. Moreover, an overestimate of the error implied by this independence assumption is given. Also in this case a very good agreement between analytical and simulation results has been observed. Finally, in order to have a feeling on the distance between the renewal and independence assumptions and some significant real traffic scenarios, we carry out also a simulation study by using real experimental MPEG and LAN traces. This study is supported by analytical arguments. In this paper, we assume that all the queues shared by the traffic are FIFO queues. Other scheduling policies (for example, Refs. [2,6]) may have a very significant impact on the delay jitter, since one of their aims is to limit its effects. A last assumption is that only open-loop or preventive congestion control schemes are used. Reactive control schemes significantly modify the traffic characteristics inside the network and further studies are required to assess the network performance in such cases. The numerical values of the source parameters and of the buffer sizes used in this work have only a significance of a case study. The main conclusions hold independently of the particular values. As for the organisation of the paper, Section 2 is devoted to the description of the modelling framework. In Section 3 we present the model for the end-to-end delay analysis, while in Section 4 we address the dimensioning of the play-out buffers. In Section 5 we validate the model by using simulation results and present a performance evaluation study. Section 6 deals with real experimental sources. The conclusions are drawn in Section 7 and some analytical results are given in the Appendix. 2. Modelling framework This section aims at describing the model used for: (i) the evaluation of the delay experienced by the ATM cells of a tagged stream through a cascade of ATM nodes; (ii) the analysis of the play-out procedure aiming at offering a circuit emulation service to isochronous communications. As far as the ATM network operation is concerned, it is assumed that: •

the cell transmission time (slot) is deterministic (the ATM cells have a fixed size);

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• • •

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cells are neither lost nor misinserted by the network; cells belonging to the same stream follows a fixed route and their sequence integrity is preserved; the temporal axis is divided into slots (the slot being the time needed to transmit a cell), and cell arrivals and departures occur at the end of the slots; an incoming cell that finds the buffer empty is immediately served.

2.1. Delay equaliser We refer to a cell stream generated by a source and carried through the ATM network by a single VC. The cells may encounter different traffic conditions in shared buffers and therefore may experience different delays. If some cells experience a shorter delay than the preceding ones we have a clumping effect. Instead, if some cells experience a larger delay than the preceding ones, we have a dispersion effect [7]. A circuit emulation service, supporting isochronous communications, needs the delay jitter to be eliminated; this is achieved by means of the play-out procedure. According to this procedure, successive data units are stored in a buffer and then they are played out to the receiver at a rate that matches the emission rate. This procedure tries to provide the receiver with a noninterrupted flow of cells avoiding the gaps due to delay jitter. In more detail, the first cell of a connection is delayed in the play-out buffer by a time equal to D. The time D is also equal to the maximum time shifting between the received and the reconstructed (equalised) signal. As long as the play-out buffer is not empty, successive cells are delivered according to the application rate. If the application is CBR, the application rate is fixed and the time between two successive cells is simply the inverse of the cell rate of the connection; if the application is VBR, this rate varies. In the latter case, the source must send time stamps that indicate to the receiving site what time its system clock should be read at the instant the time stamp is received [9]. In practice, in order to provide a time transparent link, the cells must be delayed in a differential way (according to the actual delay that they have suffered) with respect to a given maximum equalisation delay (equal to D) so as to

achieve a constant delay. For both CBR and VBR sources, the delay equaliser must be synchronised with the emission source. The play-out procedure is implemented by a delay equaliser located at the receiver. We let: t i ¼ the emission time (at the source) of the ith cell of the considered stream; d i T ¼ the total delay suffered in the network by the ith cell of the considered stream; t out,i ¼ arrival time of the ith cell to the delay equaliser; trout,i ¼ epoch at which the cell is delivered by the delay equaliser to the user; D ¼ maximum equalisation (or reconstruction) delay; K ¼ play-out buffer size. The block diagram of the delay equaliser is shown in Fig. 1. The jittered cell stream arrives at the receiving site. Cells that have suffered in the network a delay greater than D are discarded by the delay controller, the remaining ones enter the play-out buffer and, after a time equal to their respective equalisation delay, are delivered to the user at the original application rate. Fig. 2 shows the play-out procedure. The emission times at the source and the arrival times at the delay equaliser of the cells are reported on the ordinate and abscissa axes, respectively. In this framework, the 458 axis starting from origin represents the emission axis, the other 458 axis, D shifted from the previous one, represents the reconstruction axis. The solid lines indicate the delay suffered by each cell in the network, whereas the dotted ones indicate the play-out (or equalisation) delays, i.e. the time spent by each cell within the play-out buffer. Obviously, the sum of these two contributions is equal to D. Those cells for which d i T . D are dropped by the delay controller since their arrival time is successive to their reconstruction instant: their delay is too great to be equalised, given the chosen value of D. The dimensioning of the delay equaliser consists of setting the values of the parameter D and of the play-out buffer size K with the following constraints: •

the fraction of cells lost because of excessive delay or because of play-out buffer overflows must remain below a given threshold;

Fig. 1. Block diagram of the delay equaliser.

N. Ble´fari-Melazzi et al./Computer Communications 21 (1998) 980–995

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Fig. 2. The play-out procedure.

•

possible user-requirements in terms of the overall maximum delay suffered by the information flow must be taken into account.

Such dimensioning requires obviously the knowledge of the amount of the delay suffered by the cells through the network and of the delay jitter introduced by the network within the cell stream. Note that if D is chosen equal to the total maximum delay that a cell can suffer from its emission by the source to its delivery to the destination system, then it never happens that d i T . D and no cells are dropped by the delay controller. Such a worst case dimensioning of D eliminates one of the above two causes of loss phenomena but can imply a too-high delay seen by the user; in practice, then, the choice of D is made by trading off delay and loss: if we decrease D, the delay perceived by the user decreases, but the fraction of cells lost because of excessive delay increases. 2.2. Tandem queues and traffic source models The dimensioning of the play-out buffer requires the statistical characterisation of the cell stream arriving at the delay equaliser. Such characterisation requires the end-toend delay and delay jitter analysis. In order to evaluate the delay performance of a whole virtual circuit we model the latter with N tandem queues (or BEs) in series. We assume that all the BEs in the network are characterised by the same values of the service time, which is deterministic, and by an infinite buffer size. Then each BE is modelled as a discrete time single server queue. A tagged cell stream enters the first BE and proceeds through all the tandem BEs. Each BE is also loaded with

a background traffic that, for the sake of simplicity, is assumed to be the same for all the BEs. However the model that we are going to introduce allows to take into account different kinds of background traffic for each tandem queues. The background traffic immediately leaves the system, i.e. it does not proceed through the tandem queues but it is renewed at each stage. Of course this is a simplifying assumption needed to make the model analytically tractable. This setting is depicted in Fig. 3 together with the assumed model of the switches. However we have considered also a simulation setting in which the background traffic is not renewed at each stage and a given fraction of this traffic proceeds through the tandem queues. The service discipline is FIFO; as far as the tagged and background cells that arrive in the same slot are concerned, we will consider different priority rules, as done in Ref. [11]: Rule 1: the tagged stream cell has priority over the background traffic cell that arrives in the same slot; Rule 2: the background traffic cell has priority over the tagged stream cell that arrives in the same slot; Rule 3: random: the server chooses at random between the tagged stream cell and the background traffic cell that arrive in the same slot. Let us now introduce the source model of the tagged and background traffics. The time unit is the transmission time of a cell on the output link (slot). The tagged source arrival process is assumed to be a renewal one. We let: a ¼ inter-arrival times (in slots), i.e. the time elapsing between the arrival of the first bit of a cell and the first bit of the following one;

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Fig. 3. Path of a tagged stream through a cascade of BEs and model of a switch.

a(k) ¼ Pr{a ¼ k slots}.The generating function of a(k) is given by: A(z) ¼

` X

a(k)zk

(1)

k¼1

and its mean value is: 1 dA(z) l ¼ A¯ ¼ lT dz z ¼ 1

(2)

where l T is the mean inter-arrival frequency. To make the analysis computationally tractable, the cell inter-arrival times must be bounded; the tail of the distribution a(k) is then truncated to a value a m and the tail probabilities are swept to the truncation point. The value a m is chosen in such a way that: Pr{a $ k} # e, for k . a m, with e equal to a very small value (e.g. 1 3 10 ¹9). We assume, as in Ref. [8], that the probability mass function (PMF) of the inter-arrival times is a weighted sum of D PMFs, {a i}, (with weights p i), i ¼ 1, 2, …, D: a(k) ¼ Pr{a ¼ k} ¼

D X i¼1

pi ai (k) ¼

D X

pi Pr{ai ¼ k}

(3)

i¼1

where D X

pi ¼ 1

0 # pi # 1, i ¼ 1, 2, …, D

(4)

i¼1

The above arrival process can be represented by means of a discrete time Markov chain (shown in Fig. 4). When the state variable of this Markov chain is in the state t, with t [ [0,a m ¹ 1], the next arrival is due after t time slots. The transition probabilities, i.e. the probability that the state

variable, s, assumes the value t at the time instant n þ 1, given that at the preceding time instant, n, it was equal to j, are given by:    s ¼t Pr n þ 1 sn ¼ j 8 0 if j Þ 0 and j Þ t þ 1 > > > < ¼ 1 if j ¼ t þ 1 ð5Þ > > > : a(t þ 1) ¼ Pr{a ¼ t þ 1} if j ¼ 0 This arrival model has been used [8] to generate an On–Off emission process such that in the On period the emission is a periodic process. In the On state the time that has to elapse before the next arrival is taken into account and the process can pass from the On state to the Off state only in the slots where there are cell arrivals. In this model there is a cell arrival in correspondence with the transition from the Off state to the On state and the packetization delay is not taken into account (i.e. the cell is emitted immediately after the latter transition without waiting for the time needed to packet the information and to form the cell). Similar or even simpler source models have also been used in Refs. [18–20]. In order to consider a more realistic source and to take into account the packetization delay, we introduce a slight improvement with respect to the model considered in Refs. [8,11] (see the Markov chain depicted in Fig. 5). Our source model is then nothing other than a simple extension of the model used in Refs. [8,11]. We also found that the numerical results obtainable with these two models are almost equal to each other [4] so the reason to introduce such

Fig. 4. Discrete time Markov chain of the general arrival process.

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(note that since the service time is assumed to be the time unit, Eq. (8) is also equal to the mean load offered to a queueing system with this normalised service time). The mean offered load of this stream (that will be the tagged source) is then r T ¼ l T. Finally, if in Eq. (3) we let D ¼ 1, p 1 ¼ 1 and a 1(k) ¼ d(k ¹ T), where d(…) is the delta Dirac function, we get a simple periodic source with a period equal to T. The background cell arrivals are modelled as an independent and identically distributed (i.i.d) sequence representing the batch arrivals of the background traffic in each slot, with b ¼ number of arrivals. We let b(i), Pr{b ¼ i}, and the generating function of b{i} is:

Fig. 5. Discrete time Markov chain of the On–Off arrival process.

extension is motivated only by the willingness to use a more correct model of the real way of operation. The comparison between these models is not shown here both for space reasons and because we think that it would not be so interesting. Also in our source model, the Markov chain should have an infinite number of state. To make the analysis tractable we truncate the number of states, with the same approach described above (Eq. (2)), to a value labelled a m,1. The source model is obtained by letting, in Eq. (3), D ¼ 2, p 1 ¼ 1 ¹ p off/on (and obviously p 2 ¼ p off/on); a 1(k) represents the PMF of the inter-arrival times in the On state and it is assumed to be a deterministic one (as in Ref. [8], i.e. the arrival process in the On state is a CBR one), while the PMF of the Off time a 2(k) is given by: a2 (k) ¼ Pr{a2 ¼ k} ¼ Pr{t þ a1 þ 1 ¼ k} 8 < pon=off (1 ¹ pon=off )k ¹ 2 if k $ 2 ¼ : 0 otherwise

(6)

where t is length of the Off period, with ( pon=off (1 ¹ pon=off )k ¹ 2 if k $ 2 Pr{t ¼ k ¹ 1} ¼ 0 otherwise

B(z) ¼

` X

b(i)zi

and its mean is: lB ¼

dB(z) l dz z ¼ 1

r ¼ lT þ lB

(11)

(since the service time is the time unit) and we assume r , 1. Finally, we let b n be the number of background traffic cells that arrives in the same slot in which the nth tagged cell arrives and that are served before the said cell; it can be shown that this quantity does not depend on the index n and that the generating functions of the variable b depends on which of the priority rules defined above is assumed: Rule 1 : bn (z) ¼ b(z) ¼ 1 Rule 2 : bn (z) ¼ b(z) ¼ B(z)

p off/on ¼ F(U1 ); U ¼ mean number of arrivals in the On period; p on/off ¼ F(M1 ); M ¼ mean length of the Off period (in slots); L ¼ UA 1; L ¼ mean length of the On period (in slots); A 1 ¼ mean inter-arrival time in the On period.

Rule 3 : bn (z) ¼ b(z) ¼

` X m¼0

(12) 1 1 ¹ zm þ 1 Pr{b ¼ m} mþ1 1¹z

The proof of the above relations can be found in Ref. [4].

3. End-to-end delay analysis

The resulting PMF of the inter-arrival times is: Pr{a ¼ k} ¼ Poff=on Pr{t þ a1 þ 1 ¼ k} ð7Þ

The overall mean inter-arrival frequency of this source is: 1 lT ¼ p off=on þ A1 pon=off

(10)

where l B is the mean number of arrivals in one slot; the mean offered load of this stream (that will be the background traffic) is r B ¼ l B. We assume a statistically independence between tagged and background traffic; the overall mean offered load, r, is given by:

and where

þ (1 ¹ pon=off )Pr{a1 ¼ k}

(9)

i¼0

(8)

In order to evaluate the end-to-end delay it is necessary to characterise the departure process of the tagged stream from a BE. In fact, modelling the inter-arrival process of the tagged stream at queue i þ 1 with the inter-departure time distribution at stage i makes it possible to iterate the analysis indefinitely. We will approximate the departure process from the first queue with a renewal process. In this way, the process loading the second queue is a superposition of a renewal process and of the background process; since we assumed that the

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N. Ble´fari-Melazzi et al./Computer Communications 21 (1998) 980–995

background process is always the same at each stage (i.e. it is renewed), the analysis of the second queue is entirely similar to the analysis of the first one and so it is the analysis of all the tandem queues. We also assume that the delay suffered by the tagged stream cells in each of the tandem queues is statistically independent from the delay suffered in the other queues. This assumption is valid if the mean offered traffic of the tagged stream is much less than that relevant to the background traffic. This is a realistic case since it is not likely that a single source uses a great fraction of the link capacity. The above assumption is valid also if the overall mean offered load is small. It is to be noted that in Refs. [8,11] the authors make the same assumption but their model applications are affordable computationally when the load offered by the tagged sources is high, e.g. no result is provided for traffic offered by tagged streams lower than the 10% of the channel capacity. Our approach allows one to consider more realistic ATM sources, i.e. sources with average bit rate considerably lower than the link bit rate for which the independence hypothesis is much more verified. The characterisation of the departure process, the queueing analysis of a system loaded with the above arrival streams and, finally, the evaluation of the end-to-end delay are rather cumbersome. In Ref. [4] we carry out this study and develop a very efficient algorithmic procedure to solve it. The main results of this study are presented in Appendix A. 4. Dimensioning of the play-out buffers The first problem that we have to address is the evaluation of the time shifting between the emission and the reconstructed signals, D, i.e. the delay suffered in the tandem queues plus the delay suffered in the delay equaliser. Such a value must be such that the percentage, 100e, of tagged cells that arrives after D is below a given threshold. We introduce the binary variable a i that assumes a value equal to 0 if the ith cell arrives before D, and assumes a value equal to 1 otherwise. For the evaluation of D we point out that cells arriving after D are such that t out,i . trout,i (see Section 2.1 and Fig. 2). Since t out,i ¼ ti þ d i T and trout,i ¼ ti þ D then t out,i . trout,i becomes d i T . D. We will then take for D the smallest value such that: T . D} # e pftm ; Pr{ai ¼ 1} ¼ Pr{dss

d sTs

(13)

is the total time spent in the steady state by a where tagged stream cell in all the tandem queues (this is evaluated in Appendix A). The delay equaliser performance are mainly characterised by its loss probability. As explained in Section 2.1 and Fig. 1, the loss is due to two phenomena: 1. excessive delay; 2. buffer overflow.

To describe the cell loss we introduce the binary variable g i that is equal to 0 if the ith cell is lost and is equal to 1 otherwise. We also let P eq,K be the overall loss probability of a delay equaliser with a buffer size equal to K, and P b,K the loss probability due only to the play-out buffer overflow (we assume that when this buffer is full, and a tagged cell arrives in the same slot in which another cell is extracted from the buffer, the former is not lost). By applying the total probability theorem we get:    gi ¼ 0 eq, K ¼ Pr{gi ¼ 0} ¼ Pr Pr{ai ¼ 1} P ai ¼ 1    gi ¼ 0 Pr{ai ¼ 0} ¼ Pftm þ Pb, K (1 ¹ Pftm ) þ Pr ai ¼ 0 ð14Þ Eq. (14) derives from the fact that a cell arriving after D is lost (Pr{(g i ¼ 0)/(a i ¼ 1)} ¼ 1), otherwise a loss occurs if, when it arrives, the play-out buffer is full (Pr{(g i ¼ 0)/ (a i ¼ 0)} ¼ P b,K). Due to the complexity of the evaluation of the loss probability, we will evaluate the survivor function of the playout buffer occupancy seen by the ith tagged cell arriving in the delay equaliser, by assuming an infinite play-out buffer size. It is well known that this measure is an overestimate of the loss probability and it is very close to the latter for small values of such quantity. We have then to evaluate the quantity:    dbi $ K (15) Psup, K ¼ Pr ai ¼ 0 where db i is the stochastic variable that represents the buffer occupancy seen by the ith cell. If we neglect the cells arriving after D (a reasonable assumption since the number of these cells must be very small) and if we use the ‘low traffic sources’ assumption (that implies that the delays suffered by the tagged stream cells in all the tandem queues can be considered statistically independent from each other), after some passages we have the following approximation of Eq. (15) (see Appendix B): sup, K > Papp

r T Pr{tout, i , tout, Pr{t(K) þ dss , D} i ¹ K} ¼ r ftm } Pr{tout, i # tout, 1 ¹ P i

(16)

where t (K) ¼ t i ¹ t i¹k . Finally it can be shown [4] that Eq. (16) is an overestimate of Eq. (15) and that the absolute error, e appr. (i.e. the difference between Eqs. (16) and (15)) implied by our approximation is bounded by: K eappr: ¼ KPftm Psup, app

(17)

5. Model validation and performance evaluation In this section, we present some results on the dimensioning of the delay equaliser, i.e. the dimensioning of the

N. Ble´fari-Melazzi et al./Computer Communications 21 (1998) 980–995

parameters D and K. To validate the model a simulation package has been developed. Such a tool emulates in detail the operation of the system at hand without making use of the simplifying assumptions discussed above. The relative confidence intervals of the simulation results will not be shown for the sake of neatness, but they are always less then 5%. The aim of these simulations is then to show that the assumed hypotheses are not too restrictive and that the proposed analytical model can be used effectively for the performance analysis of the system at hand, with the assumed traffic source model. The model validation has been performed for several values of the system parameters. In the following we will present only a sub-set of the obtained results. However, we stress that the conclusions that we will present, with reference to the following set of system parameters, are also valid for all the other cases tested. 5.1. On-off sources We assume that the tagged source is modelled by the Markov chain showed in Fig. 5. In the On period, the emission process is a periodic one with a period equal to five time slots (i.e. it emits one cell every five time slots); the mean number of arrivals in the On period is U ¼ 1, the mean length of the Off period is M ¼ 25 slots, and the mean length of the On period is L ¼ 5 slots. The tagged stream mean offered load is then equal to r T ¼ 0.03333. We also assume a geometrically distributed background traffic with a mean offered load equal to r B ¼ 0.8. Priority rule 1 is adopted. Table 1 shows the values of the reconstruction delay, D, such that the probability of cell loss (obviously of the tagged stream) due to excessive delay is less than 1 3 10 ¹7, as a function of the number of crossed tandem queues. In other words, we consider a play-out buffer located after a different number of crossed queues. As can be expected, D increases when the path (measured in number of hops) of the tagged stream is longer, i.e. when the tagged stream crosses a greater number of queues. Figs. 6 and 7 show the survivor function of the play-out buffer queue length seen by the tagged source; the play-out buffer is located after just one network queue (with D ¼ 84) and after seven tandem queues (with D ¼ 155), respectively; the loss probability suffered by the tagged stream (obtained with simulations) is also shown. The model results (relevant

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Fig. 6. Survivor function of the occupancy of a play-out buffer (and loss probability) located after one network queue.

to the survivor function of the queue length) are compared with the simulation ones. The main comments about these figures are the following: (i) notwithstanding the approximations that we have introduced, our model is in very good agreement with simulation results; (ii) the survivor function of the play-out queue length with an infinite buffer size is a close overestimate of the loss probability suffered by the tagged stream; (iii) as the tagged stream proceeds through the tandem queues, the buffer occupancy of the play-out buffer increases, but not in a significant measure (compare Figs. 6 and 7); a play-out buffer size of just a few cells is sufficient to keep the loss probability under very small values. Fig. 8 shows the loss probability suffered by the tagged stream in the play-out buffer as a function of the latter buffer size for different values of the number of crossed queues (with the values of D shown in Table 1).

Table 1 Reconstruction delay (in slots) as a function of the number of crossed queues Number of 1 crossed queues (path length in hops)

2

3

4

5

6

7

D

99

112

124

135

145

155

84

Fig. 7. Survivor function of the occupancy of a play-out buffer (and loss probability) located after seven tandem queues.

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N. Ble´fari-Melazzi et al./Computer Communications 21 (1998) 980–995

Fig. 8. Loss probability suffered by the tagged stream in the play-out buffer as a function of the latter buffer size.

Fig. 10. Survivor function of the delay (expressed in slots) suffered by the tagged source stream after four queues.

5.2. Periodic sources In this section, we consider CBR or periodic tagged sources with a period equal to T. By considering a cascade of N queues it can be shown [4] that the minimum play-out buffer size required to avoid cell losses due to play-out buffer overflows is given by: KT ¼ d(D ¹ N)=Te

(18)

It is useful to introduce the following proposition. Proposition 1: given two CBR tagged source with a period equal to T 1 and T 2 respectively and with T 1 . T 2, by keeping fixed all other system and source parameters, the following relations hold: KT1 # KT2

(19)

T T Pr{dss, T1 $ h} # Pr{dss, T2 $ h} ;h

(20)

where dTss,T represents the total delay suffered by a tagged stream cell. In other words, the play-out performance worsens (its buffer occupancy increases) when the source period decreases. Proof: see Ref. [4]. To illustrate the above proposition let us consider three CBR tagged sources with a period equal to T 1 ¼ 28, T 2 ¼ 35 and T 3 ¼ 42 slots. The background traffic is geometrically distributed and its mean offered load is equal to r B ¼ 0.8. Priority rule 1 is adopted. We consider a cascade of four tandem queues. Figs. 9 and 10 show the survivor function of the delay (expressed in slots) suffered by the tagged source stream after one queue and after four queues (the latter comprehends the delay suffered in all the four queues). These figures show also the survivor function of the delay seen by a CBR source when T → ` (i.e. when the tagged stream offered load tends to zero), and confirm Proposition 1 (in the latter case, the survivor function is the one seen by the first of all the cells of the background traffic that arrive at the end of each slot). Table 2 shows the value of D such that the probability that a cell arrives after D is equal to 1 3 10 ¹7, for different values of T and as a function of the number of crossed queues. The limit value D(`), obtained Table 2 Value of D such that the probability that a cell arrives after D is equal to 1 3 10 ¹7 for different values of T and as a function of the number of crossed queues

Fig. 9. Survivor function of the delay (expressed in slots) suffered by the tagged source stream after one queue.

Number of crossed queues (path length in hops)

1

2

3

4

D(T 1) D(T 2) D(T 3) D(`)

84 81 79 72

99 95 93 85

111 107 105 96

123 118 116 106

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N. Ble´fari-Melazzi et al./Computer Communications 21 (1998) 980–995 Table 3 Minimum value of the play-out buffer size to avoid loss due to overflow as a function of the number of crossed queues Number of crossed queues (path length in hops)

1

K(T 1) K(T 2) K(T 3)

3 3 2

2 4 3 3

3 4 3 3

4 5 4 3

by assuming that r T is negligibly small (T → `), is also displayed. Table 3 refers to the minimum value of the play-out buffer size to avoid loss due to overflow as a function of the number of crossed queues with the values of D shown in Table 2. These tables confirm that, as expected, the performance worsens when the period of the tagged source decreases. This result will be used in the following section.

6. Real, experimental sources The source models considered up to now are fairly general, nevertheless they are still far from representing accurately certain kinds of real traffic. In this section, we carry out a simulation study by using real experimental MPEG, LAN and Internet traces and we support the simulation campaign with some analytical arguments. The MPEG traffic, representing a service with stringent delay and delay jitter requirements, will be the tagged source; three different kinds of background traffic will be alternatively considered: a geometrically distributed traffic (see Section 2.2), a LAN trace and an Internet trace. We point out that the results presented in this section are only related loosely to those of the previous one. The problem is that it does not seem possible, for the time being, to evaluate analytically the performance of the system at hand with more realistic traffic sources. Nevertheless, we believe that it can be interesting to give a hint on the performance of ‘real’ systems. For this reason we did our best to provide an analytical setting when it is possible to do so (in the previous sections) and we resort to simulation techniques when known and viable analytical approaches ‘break down’ (in this section). However, in the following, some of the analytical results above obtained are used whenever this is possible. We considered as tagged sources different real MPEG sequences [15]. These sequences are characterised by the number of cells generated for each frame. We assume that each ATM cell carries 48 bytes of coded video. The ATM cells produced by the video codec in a frame time are assumed to be uniformly spaced within the frame itself (see Ref. [5] for the reasons for this choice and for further details on MPEG source modelling). The frame duration is 1/25 s. The link capacity is assumed to be 150 Mbit s ¹1. A frame lasts 14151 slots.

Table 4 Minimum values of D such that the probability that a tagged cell arrives after D slots is less than 1 3 10 ¹3 as a function of the number of crossed queues Number of crossed queues (path length in hops)

1

2

3

4

D

30

40

49

57

The mean offered load, r T, of the considered tagged sources varies for the different sources from 4 3 10 ¹3 to 9 3 10 ¹3. The following results refer to one of the above sources, but we have verified that similar qualitative results are obtained with the remaining ones. In the first experiment, the background traffic is geometrically distributed with a mean offered load equal to r B ¼ 0.8. Priority rule 1 is adopted. The MPEG source analysed in the following has r T ¼ 4.99 3 10 ¹3. The maximum number of cells per frame is 488. The peak bit rate is 5.17 Mbit s ¹1 and the mean bit rate is 748.83 kbit s ¹1. We consider a cascade of four tandem queues. Even if the analytical model developed in the previous sections cannot be directly used to evaluate the performance of a MPEG tagged source, we can proceed as follows. Since the mean offered load of an MPEG sequence is much less than the background traffic one, the total delay suffered by the tagged source cells can be evaluated with our model by assuming that r T is negligibly small. Table 4 shows the minimum values of D such that the probability that a tagged cell arrives after D slots is less than 1 3 10 ¹3, obtained with such approximation. Note that, in this table, the probability that a tagged cell arrives after D slots has been chosen in the order of 1 3 10 ¹3 since, with the MPEG traffic sources, we cannot use the full analytical model but we have to employ simulation techniques. On the contrary, in the previous Table 1, we used full analytical techniques and thus we could constrain the above probability to be less than 1 3 10 ¹7. Table 5 displays the probability that a tagged cell arrives after D slots, comparing simulation and model results and by using the values of D given in Table 4. The agreement is quite good (the worst case in this table presents a relative error in the order of 20%). In alternative to the above methodology, since an MPEG source emits its cells periodically in each frame and varies this period frame by frame, we can use the results obtained in Section 5.2 to carry out a conservative dimensioning of the play-out buffer. We have shown that, for periodic sources, the performance worsen as the bit rate of the source increases (Proposition 1). We can then dimension the playout buffer on the basis of the peak bit rate of the MPEG source, i.e. by using Proposition 1. To verify this approach, we first dimension D (on the basis of the peak bit rate of the MPEG source) in such a way that the probability that a tagged cell arrives after D slots is less than 1 3 10 ¹7 (as a function of the number of

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N. Ble´fari-Melazzi et al./Computer Communications 21 (1998) 980–995

Table 5 Probability that a tagged cell arrives after D slots: comparison between simulation and model results Number of crossed queues (path length in hops)

1

2

Model Simulation

9.95 3 10 1.18 3 10 ¹3 ¹4

crossed queues, see Table 6) and then we evaluate the minimum value of the play-out buffer size needed to avoid cell losses due to buffer overflow, as a function of the number of crossed queues with the values of D shown in Table 6 (see Table 7). Table 7 compares simulation results (relevant to a sequence composed of about 7 000 000 cells), peak bit rate dimensioning results and analytical results (obtained by assuming that r T is negligibly small). Table 7 shows that the peak bit rate dimensioning provides the same results obtained by assuming that r T is negligibly small and both of them are a close overestimate of the real results. The above results show also that, even if we consider MPEG sources, a small play-out buffer size is needed to avoid cell loss due to buffer overflow; in addition, this result does not vary significantly as the number of crossed queues increases. In the previous results we have assumed an uncorrelated background traffic. This hypothesis has been adopted to make our study analytically tractable. However, it is well known that correlation effects play an important role in assessing the queueing performance. For this reason we evaluate the performance of the play-out buffer when the background traffic is a real, experimental LAN (Ethernet) traffic generated at Bellcore in 1989 (the trace is available at ftp://thumper.bellcore.com); 99.5% of the encapsulated packets carried by the Ethernet PDUs were IP, supporting not only a major portion of the internal traffic but also all traffic to and from the Internet and all of Bellcore. Such traffic is characterised by long-term correlations that have motivated a large research effort to find suitable traffic models (e.g. self similar models) (for example, Ref. [16]). Modelling correlated traffic is a very demanding task; only approximate analytical models for the performance evaluation of a single queue loaded by such traffic are available [17] while, to the best of our knowledge, a cascade of queues has not yet been analysed at all. We then use simulation techniques. Since the traffic carried in the above LAN is too small to load significantly a 150 Mbit s ¹1 link and thus to play the role of background traffic, we divided the sequence in S pieces, Table 6 Minimum values of D such that the probability that a tagged cell arrives after D slots is less than 1 3 10 ¹7 as a function of the number of crossed queues Number of crossed queues (path length in hops)

1

2

3

4

D

72

85

96

106

3

9.82 3 10 1.20 3 10 ¹3 ¹4

4

9.00 3 10 1.14 3 10 ¹3

8.75 3 10 ¹4 1.14 3 10 ¹3

¹4

obtaining S different sequences and then we superposed (or multiplexed) them. This can be seen as a case in which S traffic streams coming from S different LANs are multiplexed together in the same ATM link. The number S has been chosen in such a way that the mean offered load of the background traffic in the considered link is equal to r B ¼ 0.76. The tagged source is assumed to be the above MPEG one. We have dimensioned D in such a way that the probability that a tagged cell arrives after D slots (after one network queue) is less than 1 3 10 ¹7. The resulting value is D ¼ 764 slots. Fig. 11 depicts the loss probability suffered by the tagged stream in the play-out buffer (located after one network queue and with D ¼ 764 slots) as a function of the playout buffer size. Both the value D ¼ 764 slots and Fig. 11 (compare them with Tables 6 and 7) show that the performance of the delay equaliser when the network traffic is correlated are worse than in the case of a renewal background traffic. Finally, Fig. 12 considers the same setting as that relevant to Fig. 11 and shows the loss probability suffered by the tagged stream in the play-out buffer located after one network queue and after two tandem network queues; the differences from the previous case are the following. •

•

• •

The background traffic is now an Internet trace, 1 h long, of wide-area traffic between Digital Equipment Corporation and the rest of the world, generated in March, 1995 (the trace is available at http://www.acm.org/sigcomm/ITA/). The link capacity is assumed to be 34 Mbit s ¹1 because the mean traffic generated in such trace is even lower than the one relevant to the LAN trace and, even by adopting the above approach of dividing the available trace in different pieces, we could not load significantly a 150 Mbit s ¹1 link. r B is equal to 0.76. The MPEG source playing the role of tagged traffic is equal to the previous one, but since the link capacity is now 34 Mbit s ¹1, the resulting r T is 2.2 3 10 ¹2.

Table 7 Minimum value of the play-out buffer size to avoid loss due to overflow as a function of the number of crossed queues Number of crossed queues (path length in hops)

1

2

3

4

K (simulated results) 3 K (peak bit rate dimensioning) 3 K (model results, T → `) 4

3 4 4

4 4 5

4 5 5

N. Ble´fari-Melazzi et al./Computer Communications 21 (1998) 980–995

991

7. Conclusions

Fig. 11. Loss probability suffered by the tagged stream in the play-out buffer as a function of the latter buffer size.

•

D is dimensioned in such a way that the probability that a tagged cell arrives after D slots is less than 1 3 10 ¹5. The resulting value is D ¼ 5630 slots when the play-out buffer is located after one network queue and D ¼ 8046 slots when the play-out buffer is located after two tandem network queues.

The performances of the delay equaliser are even worse than in the previous case. The main reason for this behaviour is that when the link capacity decreases (34 Mbit s ¹1 instead of 150 Mbit s ¹1) the delay jitter increases (significantly). Another reason is that, as stated in Proposition 1, the play-out performance worsens when the mean offered load of the tagged traffic increases; in this case r T is 4.41 times greater than in the case depicted in Fig. 11.

The analytical study of tandem queues is a difficult problem. For this reason we have adopted suitable assumptions to make this study tractable (even if it remains cumbersome). In this work we have extended previous literature results, relaxed some hypotheses made there and provided analytical techniques to evaluate the play-out buffer performance. All the ‘analytical part’ of this work has been carried out on the basis of simplifying hypotheses. We are aware that both the assumed source models and the above assumptions, although fairly general, are still far from representing accurately certain real environments. For this reason we carried out a simulation study by using real experimental MPEG, LAN and Internet traces, supporting the simulation campaign with some analytical arguments. The latter results are only loosely related to the ‘analytical’ ones. The problem is that it does not seem possible, for the time being, to evaluate analytically the performance of the system at hand with more realistic traffic sources. Nevertheless, we believe that it can be interesting to give a hint on the performance of the system with the latter traffic sources. For this reason we did our best to provide an analytical setting when it is possible to do so and we resorted to simulation techniques when known and viable analytical approach ‘break down’. We are also aware that further work is needed to assess the network performance under more real traffic assumptions; however, we believe that this work could be a useful starting point for such complex investigation. Finally, we point out that we did not present here some passages and some proofs obtained in Ref. [4]. This choice is motivated by the need to limit the length of the paper. However, the report [4], even if it has not been reviewed, is available and can be requested from any of the authors (eventually via e-mail).

Appendix A In this section, we first analyse a queueing system loaded with the tagged source and with the background traffic (at the first stage of the tandem queues) and we evaluate the PMF of the inter-departure times of the tagged source; then we approximate this departure process with a renewal process, so that the above analysis can be applied also to the second stage and, by iterating this procedure, to all the tandem queues. Finally, we evaluate the total time spent by a tagged stream cell in one queueing system and the total time spent by a tagged stream cell in all the tandem queues. We let: Fig. 12. Loss probability suffered by the tagged stream in the play-out buffer as a function of the latter buffer size.

d0, n (k) ¼ Pr{d0, n ¼ k} d0, ss (k) ¼ Pr{d0, ss ¼ k}

(A.1a)

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N. Ble´fari-Melazzi et al./Computer Communications 21 (1998) 980–995

dh, n (k) ¼ Pr{dh, n ¼ k} dh, ss (k) ¼ Pr{dh, ss ¼ k}

(A.1b)

expressed in matrix form are:

di (k) ¼ Pr{di ¼ k} dss (k) ¼ Pr{dss ¼ k}

(A.1c)

P¼H p P

Eq. (A.1a) are the transient, d 0,n(k), and the steady state, d 0,ss(k), PMFs of the stochastic variable representing the queue length seen by the first of all the cells that arrive at the end of each slot in the queueing system at hand, by assuming that the system is in a time slot where there is a tagged stream cell arrival. Eq. (A.1b) are the transient, d h,n(k), and steady state, d h,ss(k), PMFs of the stochastic variable representing the queue length seen by the first of all the cells that arrive at the end of each slot, by assuming the system is in the hth time slots after a tagged stream cell arrival. Eq. (A.1c) are the transient, d i(k), and steady state, d s s(k), PMFs of the stochastic variable representing the queue length seen by the first of all the cells that arrive at the end of each slot. We have to evaluate:

8 a(n)b(j ¹ k) > > > > > > b(0) þ b(1) > > < ¼ b(j ¹ k þ 1) > > > > b(0) > > > > : 0

p ¼ 0 and j $ k k ¼ 0 and j ¼ 0 and p ¼ n þ 1 p ¼ n þ 1 and j $ k and j $ 1

2

Ai ¼ a(i) p A

p ¼ n þ 1 and k ¼ j þ 1 otherwise

The above transition probabilities can be obtained by observing that: •

• •

the number of cells in the system immediately after the (i þ 1)th slot is given by the sum of the cells in the system immediately after the ith slot minus one if there is a cell being served in the (i þ 1)th slot; the state transition of the tagged process is determined by Eq. (5); we assumed that tagged and background processes are statistically independent.

By letting Pðn;jÞ ¼ Pr{s ¼ n,d s s ¼ j} be the steady state probabilities of the Markov chain, the equilibrium equations

b(0) b(1)

6 6 0 6 6 A9 ¼ 6 6 6 6 4

b(g)

b(0) 0

0 2 6 6 6 6 C9 ¼ 6 6 6 6 4

b(0) þ b(1) b(2) b(0)

b(1)

0

b(0)

3

7 7 7 7 7 7 7 b(1) 7 5 b(0)

b(g þ 1)

0 0

b(0)

3

7 7 7 7 7 7 7 b(2) 7 5 b(1)

where A9 and C9 are the transpose of the matrices A and C. The parameter g (i.e. the truncation point of the above matrices) is chosen by iteratively increasing its value until a convergence, with a pre-specified accuracy, is reached. The truncation of the above vectors is acceptable since, if r , 1 (as assumed), then: lim P(i, g) ¼ 0

(A.2)

(A.3)

where eg9 is a row vector with components equal to one, whose dimension is g9 ¼ a m 3 (g þ 1) and 3 2 3 3 2 (0) 2 P(i, 0) A1 C 0 0 P 7 6 7 7 6 6 7 6 A2 0 C 6 P(1) 7 6 P(i, 1) 7 7 6 7 7 6 6 7 6 7 7 6 6 7 7 P(i) ¼ 6 7 H ¼6 P¼6 7 6 7 7 6 6 7 6 7 7 6 6 6 7 7 6 6 C7 5 4 5 5 4 4 P(i, g) Aam 0 0 P(am ¹ 1)

1. the steady state PMFs of Eq. (A.1a); 2. the PMF of the inter-departure times of the tagged source in steady state. As far as point 1 is concerned, we have to consider a bidimensional Markov chain with a state variable given by (S i,D i) (the first component of the latter variable represents the state of the tagged process in the ith slot; the second component is the delay seen by the first of all the cells that arrive at the end of the ith slot). The transition probabilities are:    si þ 1 ¼ n, di þ 1 ¼ j Pr si ¼ p, di ¼ k

eg9 p P ¼ 1

g→`

for i ¼ 0, 1, …, am ¹ 1

The equation system (A.3) is of order a m 3 (g þ 1); the relevant computational complexity increases when the tagged stream mean offered load decreases (and a m increases) and when the overall mean offered load increases (high values of g). For such cases, we have developed an iterative algorithm that has no numerical instability and can be used for a wide range of the values of a m and g [4]. Since we are interested only in the evaluation of the vector P(0) we use, instead of the second equation in Eq. (A.3), the normalising condition: l¼

g X

P(0, j)

(A.4)

j¼0

To evaluate P(0) we start from Eq. (A.3), which can be rewritten as the combination of the following two

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N. Ble´fari-Melazzi et al./Computer Communications 21 (1998) 980–995

algebra:

relations: (i)

P ¼ Ai þ 1 p P

(0)

þC p P

(i þ 1)

0 # i # am ¹ 2

P(am ¹ 1) ¼ Aam p P(0)

(A.5)

` X

Dbm þ 1 (z) ¼ z ¹ 1

Pr{dmb þ b ¼ i}zi

i¼0

(A.6)

þ (1 ¹ z ¹ 1 )Pr{dmb þ b ¼ 0}

From Eqs. (A.6) and (A.5), for i ¼ a m ¹ 2, we get

ðA:11Þ

b

P(am ¹ 2) ¼ Dam ¹ 1 p P(0)

Since D m and B are independent statistically stochastic variables, we can write Eq. (A.11) as:

where:

Dbm þ 1 (z) ¼ z ¹ 1 Dbm (z)B(z) þ (1 ¹ z ¹ 1 )Dbm (0)B(0)

Dam ¹ 1 ¼ Aam ¹ 1 C p Aam

P(am ¹ 3) ¼ Dam ¹ 2 p P(0)

Thanks to Eq. (A.12) we can derive the generating function D sbs(z) of D sbs, representing the queue length in equilibrium seen by the first of all cells that arrive at the end of each slot. In fact, since D sbs(z) is the limit of Dbm (z) and Dm þb1(z) as n → `, Eq. (A.12) becomes:

where:

Dbss (z) ¼

From Eq. (A.7), for i ¼ a m ¹ 3 and P(am ¹ 2) ¼ Dam ¹ 1 p p(0) , we get

Dam ¹ 2 ¼ Aam ¹ 2 þ C p Dam ¹ 1

(i ¼ am ¹ 2, …, 0)

Dbss (z) ¼ (1 ¹ lb )

where (i ¼ am ¹ 2, …, 1)

Di ¼ Ai þ C p Di þ 1

(A.7)

From Eq. (A.7), for i ¼ 0, we can write: P(0) ¼ D1 p P(0)

D1 ¼

am X

C i ¹ 1 p Ai

(A.8)

i¼1

Finally, P(0) can be evaluated from Eqs. (A.8) and (A.4). For the evaluation of the steady state PMFs of Eq. (A.1a), we can then write:    Pr{s ¼ 0, dss ¼ k} P(0, k) d ¼k ¼ ¼ d0, ss (k) ¼ Pr ss Pr{s ¼ 0} l s¼0 (A.9) If we let D sbs be the stochastic variable, representing the queue length seen by the first of all the cells that arrive at the end of each slot when there are no tagged stream cells (and d sbs be a realisation of this stochastic variable), then, when the tagged stream mean offered load is significantly smaller than the overall mean offered load, the second equations of Eq. (A.1a)Eq. (A.1b)Eq. (A.1c) tend to d sbs(k) ¼ Pr{d sbs ¼ k}. For the transient study it is sufficient to take into account that dmb þ 1 ¼ max(dmb þ b ¹ 1:0) dbmþ1

b

(A.13)

Eq. (A.13) contains the unknown D sbs(0) which can be determined by means of the additional equation D sbs(1) ¼ 1. Therefore, we get for the generating function of D sbs:

In a similar way, we obtain P(i) ¼ Di þ 1 p P(0)

(1 ¹ z ¹ 1 )Dbss (0)B(0) 1 ¹ z ¹ 1 B(z)

(A.12)

(A.10)

and d m are the realisations of the stochastic where variables D m b, Dbmþ1 representing respectively the queue length seen by the first of all the cells that arrive at the end of each slot in the mth and m þ 1th slots. By taking the z-transform of both members of the Eq. (A.10) and by denoting with D m b(z), D m þb1(z) the generating function of the PMF of D m b, D m þb1, we obtain after some

1 ¹ z¹1 1 ¹ z ¹ 1 B(z)

(A.14)

Let us now turn our attention to the evaluation of the PMF of the inter-departure times of the tagged source in steady state (point 2 introduced above). We use an approach similar to Ref. [11] and we take into account the different priority rules of Eq. (12) (Appendix 2.2). Let c n ¼ a þ d 0,nþ1 ¹ d 0,n þ b nþ1 ¹ b n denote the interdeparture time between the nth and (n þ 1)th cells and C n(z) the generating function of the PMF of C n. We can write: Cn (z) ¼ ED0, n þ 1 , D0, n , bn , A [aa þ d0, n þ 1 ¹ d0, n þ bn þ 1 ¹ bn ] ¼ b(z)ED0, n þ 1 , D0, n , bn , A [aa þ d0, n þ 1 ¹ d0, n ¹ bn ]

ðA:15Þ

By conditioning on a¼k

bn ¼ i3

d0, n ¼ i2

Eq. (A.15) yields: Cn (z) ¼ b(z)

am X ` X

z ¹ i3

k ¼ 1 i3 ¼ 1

` X i3 ¼ 1

z ¹ i2 E D



0, n þ 1

d0, n , bn , A

3 [zd0, n þ 1 ]Pr{d0, n ¼ i2 }Pr{bn ¼ i3 }Pr{a ¼ k} ðA:16Þ It can also be shown [4] that ED



0, n þ 1

[zd0, n þ 1 ] ¼ (z ¹ 1 B(z))k ¹ 1 zi2 Q(z, i3 )

d0, n , bn , A

þ (z ¹ 1)b(z)

kX ¹1

(z ¹ 1 B(z))m ¹ 1 (1 ¹ z ¹ 1 )b(0)Pki2¹, i3m, 0

m¼1

ðA:17Þ

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N. Ble´fari-Melazzi et al./Computer Communications 21 (1998) 980–995

where

where



  dm, n ¼ o i2 , I ¹ 3, 0 ¼ Pr Pm d0, n ¼ i2 , bn ¼ i3 Q(z, i3 ) ¼



` X

zi Pr

i¼0

 Pr

  bn ¼ i Pr{b ¼ i} b ¼ i3 Pr{bn ¼ i3 }

Css (z) ¼ zb(z)b(1) (z)

  ( 1 bn ¼ i3 ¼ Rule 2 : Pr b ¼ k ¹ i2 0 



(B(z))k ¹ 1 Pr{a ¼ k}

Rule 3 : Pr

k¼1

þ (z ¹ 1)b(z)

am X

where kX ¹1

(z ¹ 1 B(z)) ¹ m

mX ¹ 1 mX ¹1

z ¹ i3 z ¹ i2 Pr{d0, ss ¼ i2 }

i2 ¼ 0 i3 ¼ 0

m¼1

3 Pr{bn ¼ i3 }b(0)Pim2 , i3 , 0 b(1) (z) ¼

` X

ðA:20Þ

z ¹ i3 Pr{bn ¼ i3 }Q(z, i3 )

(A.21)

i3 ¼ 0

An efficient algorithm for the evaluation of the quantities i2 , i3 , 0 is the following (the relevant proofs are presented in Pm Ref. [4]). For 2 # m # a m we have i2 , i3 , k Pm

  dm, n ¼ k ¼ Pr d0, n ¼ i2 , bn ¼ i3

T ¼ dss

  1 bn ¼ i3 ¼ b ¼ k ¹ i2 k ¹ i2 þ 1

N X

wiss

with wiss ¼ d0,i ss þ b þ 1

k¼0 1 # k # am ¹ 1 ¹ m

DTss (z) ¼

N Y

Wssi

Wssi (z) ¼ zDi0, ss (z)b(z)

i¼1

Eq. (A.22) has to be initialised by the following relation, valid for 0 # k # a m ¹ 2:

P1i2 , i3 , k ¼ Pr



q.e.d X

  d1, n ¼ k d0, n ¼ i2 , bn ¼ i3

8    > < Pr bn ¼ i3 b ¼ k ¹ i2 Pr{b ¼ k ¹ i2 } ¼ > : 0

(A.23)

By introducing the relevant generating functions, since the variables Di0,ss and b are statistically independent from each other and since we assumed that the delays suffered by the tagged stream cells in all the tandem queues are statistically independent from each other, we can finally write:

(A.22)

h¼0

otherwise

i¼1



8 i2 , i3 , 0 i2 , i3 , 1 i2 , i3 , 0 > b(0)(Pm > ¹ 1 þ Pm ¹ 1 Þ þ b(1)Pm ¹ 1 < kX ¼1 ¼ i2 , i3 , h > > b(k þ 1 ¹ h)Pm : ¹1

if i3 ¼ k ¹ i2

Now, the arrival process at the second queueing system of the tandem queues is the superposition of the background traffic (that is a renewal process) and of the departure process of the first queue; by approximating the latter with a renewal process with a PMF of the inter-arrival times given by EQ. (A.19), the above analysis can be applied to the second system and, by iterating the procedure, to all the tandem queues. Finally, by indicating with w sis ¼ di0,ss þ b þ 1 (b is the stochastic variable whose generating function is given in Eq. (12)) the total time spent by a tagged stream cell in the ith queueing system in steady state and with d sTs the total time spent by a tagged stream cell in all the tandem queues, we have:

(B(z))k ¹ 1 Sk (z)Pr{a ¼ k} ðA:19Þ

k¼1

Sk (z) ¼



depends on which of the priority rules is assumed:    ( 1 if i3 ¼ 0 bn ¼ i3 ¼ Rule 1 : Pr b ¼ k ¹ i2 0 otherwise

(A.18)

and E[…] denotes the expected value of the quantities inside the brackets. In equilibrium, from Eqs. (A.17) and (A.18) and after some algebra, we obtain for the generating function C s s(z) of the PDF of the inter-departure times: am X

bn ¼ i 3 b ¼ k ¹ i2

for i2 ¼ 0, 1, …, k i3 ¼ 0, 1, …, k ¹ i2 otherwise

(A.24)

N. Ble´fari-Melazzi et al./Computer Communications 21 (1998) 980–995

995

Appendix B

References

In this section we evaluate Eq. (15), i.e.    dbi $ K sup, K ¼ Pr P ai ¼ 0

[1] ATM Forum. ATM User-Network Interface Specification, Version 4.0, December 1995. [2] A. Baiocchi, N. Ble´fari-Melazzi, F. Cuomo, M. Listanti, Achieving statistical gain in ATM networks with the same complexity as peak allocation, IEEE INFOCOM’94, Toronto, 14–16 June 1994, pp. 374– 382. [3] N. Ble´fari-Melazzi, On the modification experienced by a bursty traffic stream passing through a B-ISDN and on the congestion control problem, Computer Communications 20 (4) (1900) 246–260. [4] N. Ble´fari-Melazzi, V. Eramo, M. Listanti, End-to-end delay and jitter calculus applied to the dimensioning of play-out buffers in ATM networks, Internal Report No. 6/003/96, INFOCOM Dpartment, University of Roma La Sapienza. [5] N. Ble´fari-Melazzi, Modelling and analysis of MPEG video sources for performance evaluation of broadband integrated networks, International Zu¨rich Seminar 96, 19–23 February 1996, Zu¨rich (in B. Plattner (Ed.), Broadband Communications, Lecture Notes on Computer Science, Springer, 1996). [6] P.E. Boyer, F.M. Guillemin, M.J. Servel, J.P. Coudreuse, Spacing cells protects and enhances utilization of ATM network links, IEEE Network 6 (5) (1992) 38–49. [7] COST 242 Mid-Term Seminar, Cell Delay Variation in ATM Networks, L’Aquila, Italy, 27–28 September 1994. [8] M. D’Ambrosio, R. Melen, On the modification of the cell streams within an ATM network, IEEE Globecom 93, Houston, TX, November 29–December 2, 1993, pp. 1334–1340. [9] S. Dixit, P. Skelly, MPEG-2 over ATM for video dial tone networks: issues and strategies, IEEE Network, 9(5) (1995). [10] ITU-T Study Group 13. Recommendation I.371: Traffic Control and Congestion Control in B-ISDN, Geneva 1994. [11] W. Matragi, K. Sohraby, C. Bisdikian, Jitter calculus in ATM Networks: Single and Multiple Node Case, IEEE INFOCOM ’94, Session 2C, pp. 232–251. [12] I. Norros, Deformation of a traffic stream passing through several ATM multiplexers, Technical Research Center of Finland, Telecommunication Laboratory, doc 224TD(91), April 91. [13] Y. Ohba, M. Murata, H. Miyahara, Analysis of interdeparture processes for bursty traffic in ATM networks, EEE Journal on Selected Areas in Communications, 9(3) (1991). [14] J. Roberts, F. Guillemin, Jitter in ATM networks and its impact on peak rate enforcement, Performance Evaluation 16 (1992) 35–48. [15] O. Rose, Statistical properties of MPEG video traffic and their impact on traffic modeling in ATM systems, Proceedings of the 20th Annual Conference on Local Computer Networks, Minneapolis, 15–18 October 1995. [16] W. Leland, M. Taqqu, W. Willinger, D. Wilson, On the self similar nature of ethernet traffic, IEEE/ACM Transactions on Networking 2 (1) (1994) 1–15. [17] R. Addie, M. Zukerman, T. Neame, Performance of a single server queue with self-similar input, ICC’95, pp. 18–22, June 1995. [18] T.A. El Batt, Sherif El-Henaoui, S. Shaheen, Jitter Recovery Strategies for Multimedia Traffic in ATM Networks, IEEE Globecom 97, Phoenix, November 4–8, 1997, pp. 1202–1206. [19] H.Naser, A. Leon Garcia, A simulation study of delay and delay variation in ATM networks, Part I: CBR Traffic, IEEE Infocom 96 San Francisco, March 26–28, 1996, pp. 393–400. [20] I. Cidon, R. Gue´rin, A. Khamisy, K.N. Sivarajan, Cell versus message level performances in ATM networks, Telecommunication Systems 5 (1996) 223–239.

(B.1)

where db i is the stochastic variable that represents the buffer occupancy seen by the ith cell. If we neglect the cells arriving after D, then instead of Eq. (B.1) we can evaluate:    dbi $ K sup, K Papp ¼ Pr ai ¼ 0 ai ¹ 1 ¼ 0, …, ai ¹ k ¼ 0 (B.2) Because of the introduced conditionings, the events {db i $ K} and {t out,i , trout,i¹K} implicates one another and have the same probability measure. In steady state, Eq. (B.2) becomes: K 1. for K ¼ 0: Psup, app ¼ 1 2. for K $ 1:   r sup, K tout, i ,tout, i ¹ K Papp ¼ Pr

 ¼ Pr



ai ¼ 0, ai ¹ 1 ¼ 0, ai ¹ K ¼ 0



T t( K) þ dss ,D



ai ¼ 0, ai ¹ 1 ¼ 0, …, ai ¹ K

ðB:3Þ

where t (K) ¼ t i ¹ t i¹k. Since the tagged source is a renewal process the PMF t (K) is obtained by convolving k stochastic variables i.i.d. Thanks to the ‘low traffic sources’ assumption (which implies that the delays suffered by the tagged stream cells in all the tandem queues can be considered statistically independent from each other) Eq. (B.3) becomes:  (K)   T ,D t þ dss K > Pr Psup, app ai ¼ 0 

  T ,d t(K) þ dss ¼ Pr r tout, i # tout, i ¼

r r Pr{tout, i , tout, i ¹ K , tout, i # tout, i } r Pr{tout, i # tout, i # tout, i }

(B.4)

Since r r {tout, i , tout, i ¹ K } # {tout, i # tout, i }

it follows that r r r {tout, i , tout, i ¹ K } ∩ {tout, i # tout, i } ; {tout, i , tout, i ¹ K }

and we finally get: K Psup, app >

q.e.d X

r T Pr{tout, i , tout, Pr{t(K) þ dss , d} i¹K} ¼ r ftm Pr{tout, i # tout, i } 1¹P

(B.5)