Cell-rate modelling for accelerated simulation of ATM at the burst level

J.M.Pitts

Indexing terms: ATM networks, Queueing theory, Cell-rate modelling

Abstract: The paper presents a new method for the accelerated simulation of ATM networks. The method uses queue and traffic models that manipulate cell-rate information rather than the arrival times of cells. The cell-rate queueing model described in the paper has been validated by comparison with fluid-flow analysis and discrete time queueing analysis with batch geometric arrivals. Results show that the method accurately models the burst-scale component of ATM queueing. Comparisons with cell-by-cell simulation show speed increases of up to five orders of magnitude. This enables the low cell loss probabilities required of ATM networks to be measured within reasonable computing times. The method focuses processing effort on the traffic behaviour which dominates the cell loss, i.e. when the total input rate temporarily exceeds the queue service rate. It produces better speed gains at lower utilisations, and hence at lower cell loss probabilities.

1 Introduction

The problem with a conventional cell-by-cell approach to ATM simulation is that the computing time neces- sary to measure the levels of cell loss required in ATM networks is excessive. For cell loss probabilities of the order of and assuming 100 ps for each simulated cell, about 1 day of computer time on average is required for each lost cell (these values are typical for a desktop workstation based on a SUN SPARC proces- sor). Published results from cell-by-cell simulations rarely extend below (see [l]). To measure lower cell loss probabilities a number of different accelerated sim- ulation techniques have been developed for ATM stud- ies. These can be classified into implementation, measurement and modelling techniques (see [2]).

The implementation category addresses the program- ming methods used to build the simulator. Use of con- current techniques for cell-by-cell simulation does result in modest speed increases. The main limitation on the achievable speedup is that ATM cell-by-cell sim- ulation has a high communications to processing ratio [31. 0 IEE, 1995 IEE Proceedings online no. 19952284 Paper first received 6th February and in revised form 9th August 1995 The author is with the Department of Electronic Engineering, Queen Mary & Westfield College, Mile End Road, London El 4NS, UK

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The measurement category comprises statistical meth- ods to improve the accuracy of performance measures and to estimate tail probability distributions. One of the most promising is the RESTART (repetitive simu- lation trials after reaching thresholds) method [4,5]. This reduces the number of cells that need to be simu- lated by conditioning rare evcnts (c.g. ccll loss) on a less rare event (e.g. a threshold level in the queue) and repeatedly simulating from this threshold. The effi- ciency of RESTART is affected by restoring the system for retrials, but can reduce simulation times by orders of magnitude.

The modelling category comprises hybrid techniques, and alternative models for the queueing mechanism. Both areas depend on the decomposition of a problem into parts. For hybrid techniques, this partitioning is into separate subsystems which are treated by analysis or by simulation. For alternative modelling methods, both the traffic and the queueing behaviour are decom- posed hierarchically into distinct components. In ATM these are the connection, burst and cell components

ATM queueing behaviour is governed by two differ- ent correlation effects in the cell arrival process [8,9]. The periodic nature of cell emissions by sources in an active state results in a negative correlation within a period of the order of the minimum cell interarrival time. However, for periods greater than this, there is a positive correlation in the cell arrival process because of the duration of active states. The positive correlation has a significant effect on queue congestion when the aggregate arrival rate temporarily exceeds the buffer service rate. This is called burst scale congestion, and it is the key factor causing cell loss. The negative correla- tion has the effect that queue congestion, when the aggregate arrival rate is less than the buffer service rate, is due to the relative phasing of cells from differ- ent input streams. This is called cell scale congestion and is the main factor contributing to cell delay varia- tion.

The distinction in queueing behaviour is clearly seen in a graph (see Fig. 1) of cell loss probability against buffer capacity [8]. In [1,9,10] it is shown that the queueing analysis can be separated into these two dis- tinct components, with models tailored either to the cell or burst scales.

This paper describes an accelerated simulation tech- nique, developed specifically for ATM teletraffic stud- ies, which takes advantage of the nature of ATM queueing behaviour. It models only the burst scale component by manipulating cell rates in an alternative model of the queueing mechanism. In this model an event marks a change in the cell rate of a connection.

~ ~ 7 1 .

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The combined cell rate of all connections is compared with the service rate of the queue to determine whether the queue is increasing or decreasing in size or losing cells. Cell-rate simulation handles many cells per event, rather than one cell per event in the case of the cell-by- cell technique. This reduces the processing time required per simulated cell, though at the expense of algorithmic complexity.

buffer capacity

cell scale component cell loss

p robab I I I t y (log scale)

burst scale component f-

Fig. 1 Cell und burst scale components of queueing behaviour

The model is described in detail in Section 2: how traffic is characterised, the equations which describe the queue operation and how measurements are made. The main implementation issues are discussed in Section 3. Section 4 summarises results from two validation stud- ies which compare the cell-rate method with fluid-flow analysis and with a direct probabilistic analysis of batch geometric arrivals to a discrete time queue. This Section also presents results of a comparison with cell- by-cell simulation. These results show that cell-rate simulation accurately models the burst-scale compo- nent of ATM queueing behaviour and that it achieves speed increases of up to five orders of magnitude when compared with cell-by-cell sirnulation. The speed improvement increases in proportion to the average number of cells in a fixed rate burst, and also increases the lower the utilisation and hence also the lower the cell loss. Extensions to the basic simulation technique are described in Section 5, covering the implementation of an approximation to the partial buffer sharing algo- rithm for space priorities, and the measurement of excess-rate cell distributions.

The cell-rate simulation technique was first reported in [ll]. A simplified version of the method was subse- quently developed as part of a UK SERC project [12], and a method similar to this simplified version was developed independently by the CEC RACE project R1084 (MIME) [13]. The main difference between these later methods and the author’s is concerned with how changes to input cell rates are handled in the operation of the queue. This difference makes the implementation of the later methods less complicated but at the expense of approximating the burst-scale queueing behaviour.

A full report of the author’s cell-rate simulation method is to be found in 1141. The method has been validated by comparison with mathematical analysis in [ 15,161.

2 Cell-rate simulation modelling

2. ir Modelling the traffic In cell-rate simulation, an event is defined as a change in the cell rate of a connection. Thus the basic unit of traffic is a ‘burst’ of cells, described as a cell rate last-

380

ing for a particular time period during which the inter- cell time does not vary (Fig. 2). The time instant between bursts of different fixed cell rates is an event, i.e. the end of one burst marks the beginning of the next. This differs from cell level modelling, where an event marks a cell arrival or service.

cell rate

burst c

Fig.2 The basic unit of truflic. the fixed-rate burst of cells

loss

r

Fig.3 Queue model

2.2 Operation o f the queue A queue is described by two parameters: the buffer capacity C,, and the cell service rate O,,,. The state of a queue at any moment in time is determined by the combination of the input rates from all virtual channels (VCs), the current size of the queue and the queue parameter values. The combined cell rate of all connec- tions is compared with the service rate of the queue to determine whether the queue is increasing or decreasing or losing cells.

Therflow of traffic through a queue is described by input, output, queueing and loss cell rates (Fig. 3). These rates are denoted I(i, e), O(i, e), Q(i, e) and L(i, e), respectively, where I { 1, ..., n ) indicates the ith VC and e (0, 1, 2, ...> indicates the eth event at the queue. The number of cells queued is denoted C(i, ie). Over any time period all cells input to the queue must Ibe accounted for; they are either served, queued or lcjst. Thus at any time the cell rates for each VC, and for all VCs, must balance:

I ( i , e ) = O(z, e ) + Q(i, e ) + L(z, e ) Itot(e) = Otot(e) + Qtot(e) + Liot(e)

’ (1)

(2) where Itu,(e) = ZJ(i, e) , and similar definitions apply for

I(z, e), O(i, e) and L(i, e) are always non-negative, but Q(i, e) may take any value (positive indicates increasing queue size; negative indicates decreasing queue size). The treatment of individual VCs introduces complexity into the model, particularly for the queueing and loss rates. These rates, although mutually exclusive for the

OtuAe)? QtqAe), &ot(e) and CtuAe).

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aggregate queueing behaviour, can occur simultane- ously for individual VCs. However, balance eqns. 1 and 2 always apply.

2.3 The empty queue A queue begins to form when the total input rate exceeds the server rate: Z,,,(e) > Qmux. If the combined cell rate of all VCs on the input to an empty queue is less than the service rate, then the queue size C,,,(e) remains at zero and the VC output cell rates equal their input rates. The ability of a queue to be of zero size, yet have VCs of nonzero cell rate, is because of the restriction placed on VC traffic: during a burst of cells, the intercell time does not vary. Under the same condi- tions, with a single traffic source as input, a queue modelled at the cell level would also be of zero size: each arriving cell enters service before the next one arrives. With more than one source multiplexed through the queue, queueing occurs if cells from differ- ent sources arrive simultaneously (i.e. within the period of one cell service time). However, this ceZZ scale con- gestion is not modelled by the cell-rate technique.

2.4 The build-up of cells in a queue If the total input is greater than the server capacity, then the queue begins to increase in size; the server capacity is shared among the VCs in proportion to their input rates, and the queueing rate is the excess of input over output cell rates.

Let e = j be the event when one or more VCs change their input rates (Itot(j) > Om, and C,,,(j) = 0); then

(3)

While the queue is not empty, the total output rate is always equal to the server capacity, and the total queueing rate equals the difference between the total input rate and the server capacity.

The next event to occur at the queue will be either a state change because the queue has filled up, or a change to one or more of the VC input rates. In either case the number of cells in the queue is given by

C ( ~ , J + 1) ( 5 )

Ctot(j + 1) 1 { [ t ( j + 1) - t ( j ) ] Q t o t ( j ) } + Ctot( j ) (6) where t(j) is the time at event j .

MJ’ + 1) - t(j)lQ(z,j)) + C(Z,J)

2.5 When the queue becomes full The time that the queue becomes full is predicted from the queueing rate and the buffer capacity Cmux:

Cmax - Ctot(e) ( t ( e + 1) = t ( e ) + This event is not necessarily associated with any

change to the input rates. When the queue becomes full, the total queueing rate is zero and the excess input rate is now the loss rate:

L t o t ( j ) I t o t ( j ) - Omaz (8)

This equation always applies when the queue is full, regardless of what has happened previously to any of the VCs, because it describes the aggregate behaviour of the queue. However, the calculation of the individ- ual VC loss and queueing rates is more complicated.

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2.6 How changes to input rates affect the queue The effect of a change to the input rate of a VC is not immediately apparent on the output, if there are cells queued. Such an input change for a VC (at e = j + l ) first appears on the output (denoted by event e = k) after those cells, which were in the queue at e = j+ l , have been served. At the time of the input change, only the queueing rate changes, by an amount equal to the change in input rate:

Q( i , j + 1) = Q ( Z , j ) + I ( i , j + 1) - I ( i , j ) (9) Itot(j+l) will not necessarily equal Itot(i), so at the

associated output event e = k, the output rates for all VCs must correspond to the new balance of input rates at e = j+ 1. The notation O(i,m:n) is used to denote the output rate for VC i at event e = y1 as a result of an input rate change (for any VC) at event e = m. Note that e = y1 is not necessarily the next event after e = m (there may be other input rate changes after t(m) and before t(n)). At e = k, the output changes to O(ij+l:k), where

and the queueing rate changes by an amount equal to the change in output rate:

Q(i, k ) = Q(i , k - 1) + O(i , k - 1) - O ( Z , J + 1: k ) (11) Thus input changes propagate through to the queue

output, during which time the queueing rate of each VC temporarily goes positive or negative depending on whether the VC’s proportion of the total input rate has increased or decreased as a result of the change.

Consider now the case when the queue is full at e = j+ l , i.e. C,,,(j+l) = Cmux, and there is a change in the input rate (such that Itot(j+l) > Om,,) which will cause a change to the output rates at e = k. The cell rate flowing into the queue (and being served or queued) is O(ij+l:k) and is given by eqn. 10. Thus the excess of I(ij+l) over O(ij+l:k) is lost:

L(i ,J + 1) = q 2 , j + 1) - O(2, j + 1: k ) (12) The output rate, O(ij+l), is the same as O(ij) and

does not change until e = k (assuming that no other changes are already propagating through the queue). Each VC’s queueing rate is

Q ( i , j + 1) = O ( Z , ~ + 1: k ) - O ( Z , ~ f 1) (13) This effects a new balance of cells queued corre-

sponding to the balance of O(ij+l:k). Note that the combined queueing rates must still total zero for a full queue.

An example of input changes propagating through a full queue is shown in Figs. 4-6. Fig. 4 shows two VCs of equal input, output and loss rates (we assume that the rates are in cells per second). The queue is full and the queueing rates (both individual and aggregate) are zero. VC2 increases its input rate to 90, and the situa- tion soon after this event is depicted in Fig. 5. The loss rates have changed, as have the rates being accepted onto the queue, but the output rates remain as before. The input changes are propagating through the queue, so for VC1 the queueing rate is -10 and for VC2 it is +lo, the aggregate queueing rate being zero. A little while later, the changes appear on the output, the indi- vidual queueing rates return to zero, and the situation is as shown in Fig. 6.

38 1

vc 1

__+ - vc 2

Fig. 4 output and loss rates

Propagation through a full queue: two virtual channels with input,

. .. .. . . .. . .. , . ,

Fig.5 rate

Propagation through a full queue: after VC2 increases its input

Fig.6 Pvopagation through a full queue: effects of queueing rate changes

2.7 When the queue decreases in size The queue decreases in size when the total input cell rate becomes less than the server capacity, i.e. Itot(e) < Omax and hence Qtot(e) < 0. The time that the queue becomes empty may be predicted by substituting 0 for C,,, in eqn. 7. While the queue is decreasing in size, the total output cell rate remains equal to the server capacity until the queue becomes empty, and so the individual VC output rates are greater than their input rates (the extra cells for output being available because of the nonempty, decreasing, queue).

382

2.8 Measurements of performance Cell loss probability (CLP) measurements at a queue are estimated from cell loss ratios which are calculated by totalling the number of cells lost and dividing by the total number of cells input to the queue:

e

The same form of the equation can be used on a per VC basis at a queue.

3 implementing a cell-rate simulator

Unlike conventional discrete event simulators, process- ing is not handled event by event in a cell-rate simula- tor. This is because many VC events happen simultaneously and must be treated together. For example, if simultaneous input rate changes are proc- essed sequentially, then Itot(e) will only be correct when the last of the input rate changes is processed. Much processing effort would be wasted, and spurious future events generated. Thus the event list is specially struc- tured to enable all events happening simultaneously at a particular queue to be processed efficiently. This processing comprises three parts: first, measurements are recorded for the duration since the last group of simultaneous events; secondly, the current state of the queue is obtained, taking into account all changes of input cell rates; thirdly, the appropriate equations are used to calculate queueing, loss and output cell rates, and to predict queue empty and queue full events.

The correct application of the cell-rate modelling equations requires knowledge of the state of the queue, in terms of size (empty, full or in-between) and how the total input rate compares with the queue service rate (greater than, less than or equal). Thus 3 x 3 = 9 possi- ble states are defined. However, knowledge of the cur- rent state of the queue is not enough. The cell-rate queue model, because it models the propagation of rate changes through the queue, must take account of the state of the queue at the previous event. So with nine possible states for current and previous events, there are 9 x 9 = 81 potential transitions, of which 39 are actually possible. These 39 transitions are not imple- mented on their own but are grouped according to the processing required into four main procedures (see

The implementation of these transitions is reasonably complex. However, if the previous state of the queue is neglected, then an alternative cell-rate modelling approach is possible based on a simpler implementa- tion with just nine different queue states. However, the propagation of input rate changes through the queue can no longer be modelled, and so this approach approximates the burst level queueing behaviour. Two projects, working independently, have adopted this approximate cell-rate modelling approach (UK SERC project GR-E76261 [12] and CEC RACE project R1084 MIME [13]).

4 Results

4. ‘I Comparison with analysis Two different analysis methods have been used for comparison with the cell-rate approach described in this paper: a direct probabilistic method using discrete

~ 4 1 ) .

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renewal arrival processes; and fluid-flow theory using homogeneous on-off sources.

The first approach is made possible by having equiv- alent traffic sources: discrete renewal arrival processes are transformed into a special class of general modu- lated deterministic processes which can then be used in cell-rate simulation. For comparison, a single source model is used; this ensures that direct equivalence of the input traffic is maintained.

Fig. 7 shows a graph of the results for a Bernoulli batch arrival process at loadings of 30%, 60% and 90%. The solid lines show the analytical results for the cell loss probability, and the markers show the results from each simulation run as a 95% confidence interval about the mean. Full details are reported in [15].

l O O r

1

0 10 ’ 20 30 40 50 queue capacity

Fig.7 (Reproduced from Electron. Lett., 1992 28, (2), p. 107, Fig. 3)

Cell loss probabilities for Bernoulli hatch input

delay, ms

Fig. 8 (i) C = 4, N = 10 (ii) C = 8, N = 20 (iii) C = 3, N = 5 (Reproduced from Electron. Lett., 1994, 30, (3), p. 186, Fig. 1)

Cell loss probabilities for limited delay model

4.2 Comparison with cell-by-cell simulation Baiocchi et al. [l] study the loss performance of an ATM multiplexer loaded with a homogeneous traffic mix of on-off sources representing a high-speed LAN interconnect. This multiplexing scenario was adopted for the comparison of simulation methods because it clearly displays the cell and burst scale components of queueing behaviour over a range of parameter values for typical ATM sources. The reference source described in [I] was used (peak rate of 10Mbit/s, activ- ity factor (meadpeak) of 0.1 and mean burst length of 16250 bytes).

Cell loss measurements were obtained from both sim- ulation methods for traffic loadings of 40% and 80% of net output capacity (54 and 109 sources, respectively), with buffer capacities up to 80. The results are shown in Fig. 9, displaying clearly the cell and burst scale components. As expected, the cell-rate method did not show the cell scale component.

U U P

lo-! lb io i o i o 50 i o 70 i o buffer capacity

Fig. 9 A 80%. cell-bv-cell

Cell-rate and cell-by-cell simulation results for cell loss

A 80%; cell rate W 40%, cell-by-cell 0 40%. cell rate

The second approach complements the first by addressing multiple sources in a typical multiplexing scenario. In this case, the fluid-flow analysis reported by Tucker in [17] and applied to packet-speech multi- plexing was used for comparison. This is a develop- ment of the method used by Anick et al. [18]; Tucker incorporates finite buffer capacity into his analytical model.

N homogeneous on-off sources load a multiplexer of channel capacity C (where C is in units of the on rate of a source). Tucker specifies the buffer capacity in terms of the maximum delay. Fig. 8 shows results for different combinations of C and N . The solid lines show analytical results from [17], and the markers show the results from each simulation run as a 95% confi- dence interval about the mean. Full details are reported in [16].

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average burst length, bytes Fig. 10 Comparison of cell processing speeds A 30% + 40% 80%

4.3 Speed-up over cell-b y-cell simulation The computation speed was compared by measuring the number of cells processed per second of computing time. Fig. 10 shows results from simulations for a buffer capacity of 15 cells and loadings of 30%, 40% and 80%, with the burst length being varied over five orders of magnitude. Note that the speed improvement was better for lower utilisations.

5 The cell-rate simulation modelling technique has been extended to increase the capabilities of the technique in its application to ATM networks and to enhance the measurements that can be made during a simulation.

Extensions to the basic technique

383

5.1 Modelling space priority The cell loss priority (CLP) bit in the header of an ATM cell allows two levels of space priority to be defined. A study of buffer algorithms for implementing the space priority mechanism [ 191 recommends the par- tial buffer sharing (PBS) algorithm because it combines good performance with implementation simplicity. The PBS algorithm divides the buffer into two parts: when the queue size is less than the threshold, cells of either high or low priority are allowed into the buffer; but at or above the threshold, only high priority cells are admitted.

Section 2 of this paper described the equations used to model an ATM buffer at the burst level. However, this was for a finite buffer without any priority mecha- nism. These equations have been modified to provide an approximation to the PBS algorithm, modelled at the burst level [20]. The approximation is based on the assumption that there is no burst level queueing above the threshold, something that would only occur if the total input rate of high priority cells exceeded the serv- ice rate of the queue. Since priority schemes are only of benefit when the high priority traffic is a small propor- tion of the total, an excess high priority cell rate is very unlikely and so this assumption appears to be a reason- able one.

Thus changes to the modelling equations are required only at the threshold. The approximation treats a full queue in the simulation model as the ATM buffer reaching the PBS threshold. In this state, all high prior- ity traffic is admitted into the buffer, and any service capacity which is not used by the high priority traffic is shared out between the low priority input rates:

If the high priority input rate does exceed the queue service rate, then queueing between the threshold size and the actual full size of the buffer is neglected, and the buffer is treated as though it is full; all low priority traffic is lost, and the service capacity is shared between the high priority input rates:

O(i3.I + 1: k )hp = I(2,j + l ) h p ( Om,' ) ( 1 6 ) Itoh,, ( j + 1)

In both cases, eqn. 12 can be applied with the appro- priate priority rates to calculate the low or high prior- ity loss. This approximation of PBS has been validated against analytical results for cell loss [21].

5.2 Measurement of excess-rate cell distribution Section 2.8 described how cell loss ratios are calculated using measurements from the simulator. Another useful measure is the excess-rate cell distribution. A burst level queue increases in size if the input rate to the queue exceeds the service rate of the queue. Over the time period for which this condition holds (an excess- rate period), the number of excess-rate cells is equal to the amount by which the queue increases in size (if the buffer is full, then the number of excess-rate cells is equal to the number of cells lost during the time period). Thus it is just the queueing (or loss) rate multi- plied by the time period, and can be expressed as number of excess rate cells

384

for all consecutive events e for which Itot(e-l) > Om,,. The frequency of occurrence of each measurement value is recorded in a histogram to provide an estimate of the distribution of excess-rate cells.

Similarly, the distribution of the available cell time slots is based on measurements of number of available time slots

= c ( O m a z - Itot(e - l ) ) ( t ( e ) - t ( e - 1)) (18) e

for all consecutive events e for which Itot(e-l) < Om,, These time slots are those available for reducing the size of the burst level queue.

These measurements have a number of uses. They can be used in the analysis described in [22] to calculate the cell loss probability for a buffer of any size. This analysis can be implemented as a submodel within a bufferless cell rate simulation model in a hybrid tool for even greater acceleration than that available with the basic cell rate simulation technique. Also, the distri- butions can be used to validate connection admission control schemes based on bufferless burst level analyti- cal models.

6 Conclusions

Cell-rate modelling is a new accelerated simulation method that addresses the problem of measuring rare events, such as cell loss in ATM, in a computationally efficient way. It does this by using queue and traffic models that manipulate cell-rate information rather than the arrival times of each individual cell. The cell- rate queueing model described in this paper has been validated as an accurate model of queueing by compar- ison with fluid-flow analysis and discrete-time queueing analysis with batch geometric arrivals. It accurately models the burst-scale component of queueing in ATM buffers, which is the key behaviour determining cell loss.

In comparison with cell-by-cell simulation, cell-rate simulation shows significant speed increases of up to five orders of magnitude. This enables the low cell loss probabilities required of ATM networks to be meas- ured within reasonable computing times. The speed improvement increases in proportion to the average number of cells in a fixed rate burst, and also increases the lower the utilisation and hence also the lower the cell loss. This is because it focuses processing effort on the traffic behaviour which dominates the cell loss. Thus the technique performs better in the situations for which it was designed and for which cell-by-cell simula- tion is inadequate.

Cell-rate simulation is compatible with other promis- ing accelerated simulation techniques, such as parallel processing implementations and statistical techniques such as RESTART. It is more suited to concurrency than cell-by-cell simulation because of the lower com- munications to processing ratio: fewer events and more processing per event. And, in contrast to statistical techniques, it gives the benefits of acceleration across the full range of applications of simulation, whether in production runs for cell loss measurement in specific scenarios or in network experiments addressing the interaction with traffic control mechanisms or traffic management software.

The cell-rate queueing model has been modified to model space priorities based on partial buffer sharing, and the simulation technique is currently being

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extended to model usage parameter control schemes based on the leaky bucket algorithm. The propagation of cell-rate events is being investigated for development of an appropriate parallel processing scheme [23].

7 Acknowledgments

The author gratefully acknowledges support from the CEC RACE programme, projects R1022, ‘Technology for ATD’ and R2059, ‘Integrated communications management’, and the UK EPSRC project GR/J41635.

This paper is a revised and extended version of ‘An accelerated simulation technique for modelling burst scale queueing behaviour in ATM’, originally published in ‘The fundamental role of teletraffic in the evolution of telecommunications networks’, J. Labetoulle and J.W. Roberts (Eds.), pp. 777-786, copyright 1994, and reprinted with kind permission from Elsevier Science B.V. Amsterdam, The Netherlands.

The author would like to acknowledge the contribu- tions of P. Fonseca and N. Akwiwu to the work on the extensions to the cell-rate simulation technique reported in Section 5.

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21 FONSECA, P., PITTS, J.M., and CUTHBERT, L.G.: ‘Exact fluid-flow analysis of single ON/OFF source feeding an ATM buffer with space priority’, Electron. Lett., 1995, 31, (13), pp. 1028-1029

22 PITTS, J.M.: ‘New for old: reusing ATM cell scale analysis at the burst scale’, 12th IEE UK Teletraffic Symposium, 1995

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