performance analysis of an atm statistical multiplexer with batch arrivals

Performance analysis of an ATM statistical multiplexer with batch arrivals

M. Abbas Z.A. Ahmad

Indexing t e r m : ATM statistical multiplexer, Queueing theory

Abstract: The authors propose a method for evaluating the impact of bursty traffic sources on the performance of an ATM multiplexer which can handle packetised voice, data and video traffic. The multiplexer is modelled as a single- server queue with batch phase type Markov renewal input. The authors describe an analytical approach to the solutions of such queueing systems with finite capacity using the matrix analytic method. The methodology was chosen because its ability to model exactly complex non- Markovian models having correlated, nonrenewal input processes. It is nearly impossible to obtain exact expressions using the classical method of generating functions. A study of different mixed traffic assumptions indicates that bursty traffic does have a very significant effect on the loss performance.

1 Introduction

The asynchronous transfer mode (ATM) has been selected as the technique to be used for switching and multiplexing in future B-ISDN networks. In an ATM network, all voice, data and video signals are digitised, coded, packetised and multiplexed together, before being transmitted to a high-speed link. The packet (or better known as cell) streams from these sources are usually very different from the Poisson process. The superposed process of such non-Poisson streams is correlated, nonrenewal and complex. Analysis of a complex non- Markovian model by using classical methods of generating functions [l] is nearly impossible and sometimes will not yield exact expressions. Under such situ- ations, we can replace the input process by a more analytically tractable model which accurately approx- imates the input process.

Neuts [Z] introduced a versatile Markovian point process, also known as the N-process [3], which is analytically simple and possesses properties that make it suitable for the approximation of complicated nonrenewal processes. Systematic and detailed studies on the matrix-analytic method and its related references can be found in Neuts’ book [4]. Several researchers [3, 51 have effectively used the matrix-analytic approach to solve queueing problems including complex non-Markovian

0 IEE, 1994 Paper 11301 (En, first received 27th August 1992 and in revised form 4th January 1994 The authors are with the Faculty of Electrical Engineering, Universiti Teknologi Malaysia, Jalan Semarak, 54100 Kuala Lumpur, Malaysia

190

processes. Sometimes, we encounter some inverse matrix manipulations of matrices have large dimensions and their computation normally requires a high-performance computer with large memory. However, there are several articles published [l, 61 that provide solutions on how to compute the matrices more effectively and thus reduce the amount of memory and disk space required.

2 Phase-type (PH) distributions

First, to avoid confusion, the notations which will be used throughout this paper are defined. Generally, we will avoid declaring dimensions of matrices and vectors by adopting a convention. Notably, we write Y x Z only if it is defined, that is, the number of columns of Y is equal to the number of rows of Z. Likewise, Y + Z is defined only if Y and Z are of the same dimensions. The column of row vector whose entry is one is denoted by e, the dimension of which is usually self-evident from the context and hence suppresses the notation unless other- wise mentioned. Both row and column vectors are denoted by italic and bold face lower case letters. The identity matrix of order k, denoted by Z,, is a diagonal matrix with all the diagonal entries equal to one. The symbol ‘0’ may be a scalar or a matrix according to the context.

In this Section, we will only give a brief introduction to phase (PH) distributions. A detailed discussion can be found in Neuts [4]. Consider an (m + 1)-state continuous-parameter Markov chain for which the states { 1, . . . , m} are transient and the state {m + 1) is absorbing; i.e. a process in state m + 1 must stay there forever. Suppose that the initial probability vector is given by (a, a,+ t), with m + = 1. The infinitesimal generator of this Markov chain is given by

where Te + P = 0, is a non-negative vector, and the matrix T is nonsingular with T, < 0, for 1 d i G m, and T j b 0, for i # j. The pair (a, T ) is called a representation of F( .). We may assume without loss of generality that U,,, = 0, so that F( .) does not have a jump at 0. The

The first author would like to thank Prof. H. Akimaru and T. Okuda of Toyohashi University of Technology (TUT), Dr. Z. Niu of Fujitsu Lab. Ltd., Dr. N. Yunus of Celcom and Muzlifah M. Ali of Universiti Teknologi Malaysia (UTM) for their valuable comments and continuous encour- agements on this work. Part of this work was done at TUT with the support of UTM.

I E E Proc.-Commun., Vol. 141, No. 3, June I994

stationary probability vector z of Q* is obtained by solving the equations zQ* = 0, ze = 1.

If we model the arrival process as a PH-reneval process (which can also be modelled as departure process of phase-type), the inter-arrival time probability distribution function is of the form

(2)

(3)

F(x) = 1 - K exp ( T x ) ~ for x 3 0

F*(s) = 4 s I - T)-'P with Laplace Stieltjes transform (LST)

The PH-distributions are considered versatile probability distributions in which a number of well-known probability distributions can be included as special cases:

(i) For the exponential distribution with parameter a, the infinitesimal generator is given by [ i' i] and a, = 1, a2 = 0 so that F( .) has the simple representation (1, -I).

(ii) The generalised Erland distribution obtained by the convolution of k exponential distributions with parameters I,, . . . , I h

has as one of its representations the pair (K, T ) given by

K = [1 0 " ' 01

' . . -Ik-, A,-, 0 - I , O I

0 -I1 I , '. ' ... ... 0 ... ...

with

r 01

3 Phase-Markov renewal process (PH-MRP)

The phase Markov renewal process (PH-MRP) is a direct extension [7] of the phase type renewal process introduced by Neuts [8]. It is also similar to the general Markov amval process (also known as MAP) [SI. However, instead of referring to it as MAP, the notations used in this paper to analyse the PH-MRP are based on those of Niu [l].

Since plural absorbing states are assumed, the PH-MRP can represent non-renewal process such as the Markov modulated Poisson process (MMPP) whereas the PH-RP can only represent renewal processes due to the assumption of only one absorbing state. Consider a continuous Markov process with state space 11, ..., m, m + 1, . .., m + n} for which the states {1, ..., m} are transient and the states {m + 1, . . . , m + n} are absorbing. We assume that starting at any transient state, absorption into a state in {m + 1, . . . , m + n} is almost certain. Then the infinitesimal generator of such a Markov process has the form [; ': 1, where T is an m x m matrix with T j < 0 and T j 3 0 for i # j , such that T-' exists. is a non-negative m x n matrix and satisfies Te + P e , = 0, where e (or eo) is an m x 1 (or n x 1) column vector with all elements equal to 1.

We will consider the Markov renewal process which is obtained by instantaneously restarting the Markov process after each absorption. The state transition prob-

distributions

IEE Prof.-Commun., Vol. 141, No. 3, June 1994

ability matrix between an epoch visiting the absorbing state and the epoch instantaneously restarted after this epoch is defined as K (see Fig. 1). From this definition, it is clear that K is an n x m rectangular matrix.

absorblng states

transient states

Fig. 1 Mmkov renewal process Markov renewal

More recently, modelling of packetised voice and data traffic has required consideration of more complicated arrival processes than the Poisson process. It is now well known [lo] that the interarrival times in the packet streams are strongly correlated, The Markov modulated Poisson process (MMPP) was used in Reference 10 to approximate the superposition of packetised voice processes. The MMPP was chosen because it is simple, versatile, analytically tractable and can approximately match certain statistical properties of the original traffic.

For instance, we can represent a 2-phase MMPP in the form of PH-MRP as follows:

K = [i y ] = I2 (unit matrix of order 2)

= q 2 - q1 - 1 2 q1 - q2 1

(4)

Another interesting property is that the superposition of several PH-MRPs forms a new PH-MRP. For example, if we consider superposed voice and data as two 2-phase MMPPs whose representations are given as (K,, T u , c) and ( K ~ , T d , q) respectively, a new PH-MRP with the representation ( K ~ , T s , c) will be given by

T, = To @ 12 +- 12 8 Td

c = E 8 1 2 , 1 2 @ Cl (5 ) where @ denotes the Kronecker's product [ll].

4 Quasi birth-and-death (QBD) process

A quasi birth-and-death (QBD) process is a Markov process on the state space R = {(i, j ) : i 2 0, 1 Q j Q m}, with infinitesimal generator Q, given by a tridiagonal matrix in which all the blocks may also be in matrix form, i.e.

'40 Bo Ml A , 4

M2 A2 B, 1 ' " Mi-, ". 4 - 1 , . A i . . ' B 1 . . ]

. . (6)

191

The matrices Mi, Ai and Bi denote the transition rates from level i to level i - 1, i and i + 1 respectively. The performance measures of a queueing model can be obtained by solving the system of equations p Q = 0 and the normalising condition pe = 1, where the vector p denotes the stationary probabilities of the underlying system. Therefore, in this case, Aoe + Boe = M l e + A l e + Ble = (M, + A , + B2)e = 0.

5 First-in-first-out discipline with batch

Queueing models with infinite buffer using the matrix- analytic approach have been investigated widely by several researchers [l, 31. The model where the queue has only finite number of waiting places, seems realistic especially in systems where buffering memory is limited and where loss probability is an important performance measure of the system. Esaki [12] has made a study which shows that the cell loss rate is the dominant parameter compared to average queueing delay. To make a comparison between different models, it is

necessary to model an uncontrolled queueing sy tem such as one with no-priority control in which all c l L s of traffic are treated equally. Such an uncontrolled system is known as a queue model with first-in-first-out (FIFO) discipline. This model is frequently analysed by many researchers and discussed in many textbooks [13, 141. Our paper, however, gives a different example whereby, we assumed that one of the traffic classes (for example, class-1) arrives in batches of fixed size. There are two batch strategies that can be considered when the number of cells in an arrival exceeds the remaining buffer capacity of the system:

(i) PBAS (partial batch acceptance strategy): the batch fills the remaining room in the system, and the remaining cells in the batch are rejected.

(ii) WBAS (whole batch acceptance strategy): all cells in the batch are rejected.

We denote by i the number of customers (both class-1 and class-2) in the queueing system. The state-transition diagram of its birth-and-death (BD) process with batch arrival is shown in Fig. 2. In the following example, we

examples

"0 Vl 'U-1 'K-2

Fig. 2 PBAS d e l with batch size x = 2

simply choose PBAS with batch size 2 for no specific

Q =

192

reasons. The infinitesimal generator, Q, is given by,

A0 Do vo Ml A1 Dl Vl ... M2 A2 D2 v2 . . . . . . .

M K - 2 A K - 2 D K - 2 vK-2

M K - l 'K-1 'K-1

MK

This is a modified version of the standard QBD process. The generator Q can be interpreted as follows: matrices Vi and Di denote the arrival rate matrices of class-1 and class-2 cells respectively. If the cell arrival is from class-1, there will be a level jump (i + i + 2) when the queue length i Q K - 2 due to the batch arrivals of size 2 and only one step up when i = K - 1 due to the discard of one of the batch arrival: Matrix indicates the superposition of class-1 (which now consists one part of the batch) and class-2 traffic. If the cell arrival is from class-2, the level increases only by one step (i + i + 1). On the other hand, the matrix Mi denotes the departure rate matrix of the cell in service at the level i, i.e., the transition matrix from level i to (i - 1). In order for the process to have no transition from level i, neither cell arrival from class-1 and class-2 nor departure from the queue is allowed. In other words, the only possible tran- sitions are the changes of the phases during level i. This can be represented by the matrix Ai which satisfies

A o e + D o e + Voe = 0

M i e + A i e + Die + Vie = 0 (1 Q i Q K - 2)

M,- , e + A,- ,e + BK- le = 0

M K e + A K e = O (8) Since the cell length in every kind of call in an ATM system is fixed, the service time of the statistical multiplexer is a unit distribution. In all the analytical models, treated in this paper, it is necessary that the service time distribution has the phase-type structure. We approximate the unit distribution by the Erlang distribution (J, S, So), defined by

(9)

SO = (0, ..., 0, p)' If the Erlang distribution has an appropriate number of phases, k, then the Erland distribution is a good approximation to a unit distribution.

5.1 Queueing model-/: My' + MJPHJlIK (FIFO) In this model, i is defined as the number of cells (both class-1 and class-2) in the queueing system and h as the phase of PH service time at an arbitrary time. Then, (i, h) forms a continuous-time Markov chain (CTMC) on the state space

(10) Then by partitioning the state space into following set of levels:

R = ((0) U (i, h); 1 Q i Q K. 1 i h i k}

level 0 = ((0)) level i = {(i, l), ..., (i, k)) (1 Q i Q K ) (11)

we can characterise the underlying queueing model as a modified QBD process whose idinitesimal generator is given above.

In this model, we defined the matrices M, , A, and Bi as follows. A("), A(') and A, are the Poisson arrival rates of

I E E ProcXommun., Vol. 141, No. 3, June 1994

class-1, class3 and the superposition of class-1 and class3 cells respectively.

A0 = -1,

A i = - A S I k + S ( 1 g i g K - 1 ) A , = S

Vo = 1‘”/3

Vi = l(’)Ik Do = A(2)b

(1 Q i < K )

Di = A(’)Ik (I Q i < K) B,-, = AsIk (1 d i < K ) Ml =So

M i = Sob (1 Q i < K) (12) Then we partitioned the stationary probability vector p according to the levels as follows:

P = (PO>Pl....?PK-l,PX) (13)

Pi = CPXl), pi(% . . . , PXW (1 Q i < K ) (14) pih) denotes the stationary probability that there are i cells in the system and in service phase h at an arbitrary time.

where po is a scalar but p i is a vector given by

52 Queueing model-ll: PH-MRPY’ -+ MJPHflJK (FIFO)

In model-11, we defined i as the number of cells (both class-1 and class-2) in the queueing system, jl as the phase of PH-MRP, represented by (a,, T,, q) and h as the phase of PH service time at an arbitrary time. The state space of this CTMC can be defined as

a = {(O, jl) U (i,jl, h); 1 < i Q K , 1 C j , Q r , , 1 Q h d k }

(15) We can then partition the state space into the following levels :

level 0 = {(O, I), . . ., (0, r,)}

level i = { ( i , 1, l), ..., (i, 1, k), ..., (i, r l , l), ..., (i, r , , k)} (1 Q i g K) (16)

A . = T, - 1(231,1

A i = ( T ~ - 1 ‘ 2 ’ I , , ) ~ I k + I , , ~ S (1 Q i g K - 1)

AX = (TI G K 1 ) €3 I k I , ,@ S

Vo = q K i €3 b Vi = GK, €3 (1 Q i C K ) Do = 1‘2)1,1 €3 b Di = A(’)& €3 It (1 Q i Q K )

BK- , = (GK, + 1(’)I,,) €3 Ik M , = I,, @So

M I = I,, €3 Sob (1 Q i C K ) (17) Again, we partitioned the stationary probability vector of this model into the appropriate levels

P = ( P O 7 P19 ...? P K - l r PK)

Po = CPo(l), . . ., PO(‘1)l

IEE P?oc.-Commun., Vol. 141, No. 3, June I994

(1 Q i Q K) (18)

5.3 Queueing model-Ill: PH-MRP?’

In model-111, i is defined as the number of cells (both class-1 and class-2) in the queueing system, jm as the phase of PH-MRP, (n = 1, 2) represented by (K,,, Tm, c) and h as the phase of PH service time at an arbitrary time. Then we constructed the CTMC of model-I11 with the state space

-+ PH-MRPJPHf 1/K (FIFO)

a = {(0,jl, j 2 ) U (i,il, j 2 , h); 1 Q i Q K ,

1 Q j l ~ r l , 1 Q j z ~ r 2 , 1 Q h Q k } (19)

We partitioned the state space into the following set of levels :

level 0 = {(0, 1, I), . . . , (0, 1, rz) , . . . , (0, rlr 11, . . ., (0, rl, r2)l

(i, 1, r2, 11, . . . , ( i , 1, r2 , k), . . . , (i. rl, 1, I), . .., (i. rl, 1, k), ..., (i, rl, r2, I), . . . , (i. rl, r2 , k)}

level i = {(i, 1, 1, I), ..., (i, 1, 1, k), ...,

(1 Q i Q K ) (20)

5.4 Performance measures for FIFO discipline A stationary probability distribution, p, is given by the following recursion formulas:

POFO = 0

P1 = P o Fl

pi = p i - 2 U, + p I - , F I (2 C i Q K ) (23)

193

embedded coding technique on class-2 sources to separate the more significant bits and less significant bits into two separate cells: most significant part cell and less significant part cell which contained more significant bits and less significant bits of the source samples [3] . However, this paper is only concerned with a nonpriority scheme, whereas an analysis of multiple priority scheme has been described by Abbas and Ahmad [16].

In Figs. 3 and 4, we assume that cells from a class-1 source arrive deterministically at intervals of

5.4.1 Cell loss rates for class-1 traffic: The cell loss rates of the class-1 cells are then given by

L'" = 4h"e (25) where

is the stationary probability vector that there are K cells in the system just prior to a class-1 arrival. C:=opi Vi is none other than the average arrival rate of class-1.

5.42 Cell loss rates for class-2 traffic: The cell loss rates of class-2 cells are then given by

(27) L'2' = *pe

4(y =

where

(28) C Pi Di

i = 0

is the stationary probability vector that there are K cells in the system just prior to a class-2 cell arrival. It is clear that pi Di is the average arrival rate of class-2 traffic entering the system.

6 Example results

In this Section, some numerical examples are presented to illustrate the effectiveness of evaluating loss performance using different arrival assumptions. We approx- imated the queueing model that uses PH-MRP as the arrival process by a 2-state MMPP represented by ( I l , I,, q l , q,). We will then use Akimaru and Okuda's (AO) method of calculating these MMPP parameters since the CPU time takes less than a second whilst pro- viding good accuracy of approximation [l5]. The com- monly used ATM parameters are: the cell length is assumed to be 53 bytes (5 bytes in header, 4 bytes in ATM adaptation layer (AAL) and 44 bytes for payload), and the transmission line speed is 156 Mb/s. The buffer capacity plus the server space totals 15 cells (K). We consider that the number of phases of the Erlang distribution (k) is 30.

Normally, a bursty source depicts shorter active periods and longer silence periods typical of voice and video traffic characteristics. In some cases, data traffic which arrives in batches is also very bursty. In the following example, class-1 and class-2 sources are assumed to be bursty sources. Furthermore, class-2 source traffic arrives in a batch size of 2 since it is possible to apply an

194

I"

0 5 10 15 20 25 30 35 40 45 50 number of doss-2 sources

Fig. 3 Number of class-1 sources = 5 ~ model-I . . . . . . . - _ _ _ model-I11

Comparison ofcell loss rates for class-2 with different models

model-I1

0- 0 5 10 15 20 25 30 35 40 45 50

number of class-2 sources

Fig. 4 Numkr of class-1 sou~cu = 5 ~ model-I . . . . . . . _ _ _ _ model-111

Comparison of cell loss ratio with different models

model-I1

T = 0.02 ms, during ON periods, and that no cells arrive during OFF periods. These ON and OFF periods alter- nate according to exponential distributions with rates +A1 = 2 ms and 4;h = 30 ms respectively. The parameters of class-2 are taken to be 4;' = 4 ms, 4;; = 60 ms and T = 0.02 ms which mean that two cells are generated simultaneously during this period. As shown in Fig. 3, it is dangerous to assume a bursty source from Poisson arrivals because as illustrated by models I and 11, the loss rates are significantly less than for model411 which takes into consideration the effects of its burstiness. If we look at the ratio of both classes as depicted in Fig. 4, the ratio is constant with the same loss rates for model-I.

I E E Proc.-Commun., Vol. I l l , No. 3, June 1994

However, in model-11, the losses are different due to their different characteristics. An addition of class-2 sources causes an increase in its losses. The curve for model-I11 decreases due to the increasing load of class-2 which effectively causes higher losses at constant load of class-1 sources.

7 Conclusion

The authors have described an analytical tool for evaluation of complex nonrenewal classes of mixed traffic inputs. It was observed that the correlated effects of the traffic had a significant effect on its performance. A statistical multiplexer which handles bursty input traffic had higher loss rates as compared to a simple Poisson arrival model. Thus, the paper confirms that there is a need for developing more accurate queueing models that are able to capture the various types of traffic arrivals. This method has also been successfully applied on priority schemes to handle multiple classes [16, 171 in an ATM network.

8 References

1 NIU, Z.: ‘Studies on mixed loss and delay systems in telecommuni- cations networks’. PhD thesis, Toyohashi University of Technology, 1992

2 NEUTS, M.F.: ‘Renewal processes of phase type’, Naval Research Logist. Q., 1976, U, pp. 445-454

3 RAMASWAMI, V.: ‘The N/G/I queue and its detailed analysis’, Adv. Appl. Probab., 1980,12, pp. 222-261

4 NEUTS. M.F.: ‘Matrix-geometric solution in stochastic models: an algorithmic approach’ (The John Hopkins University Press, Balti- more, MD, 1981)

5 MACHIHARA, F.: ‘On ovedow processes from the PH, + PH,/M/S/K queue with two independent PH-renewal inputs’,

Paform. E d . , 1988.8, pp. 243-253 6 YAMADA, H., and MACHIHARA, F.: ‘Performance analysis of a

statistical multiplexer with control on input and/or service process’, Perform. Eual., 1991,14, pp. 21-41

7 MACHIHARA, F.: ‘Completion time of service unit interrupted by PH-Markov renewal customers and its applications’. 12th Int. Tele- trait Congress, Torino, 1988, Paper 5.4B.5

8 NEUTS, M.F.: ‘A versatile Markovian point process’, J. Appl. Probab., 1979.16, pp. 764-779

9 LUCANTONI, D.M., MEIR-HELLSTERN, K.S., and NEUTS, M.F.: ‘A single-server queue with server vacations and a class of non-renewal amval processes’, Ado. Appl. Probab., 1990, 22, pp. 676-705

10 HEFFES, H., and LUCANTONI, D.M.: ‘A Markov modulated characterization of packetized voice and data traffic related statistical multiulexer oerformance’, IEEE J . Sel. Areas Commun., 1986, 4, (6), pp. 856-867

1 I BELLMAN, R.: ‘Introduction to matrix analysis’ (Mffiraw-Hill Book, Co., 1960)

12 ESAKI, H.: %all admission control method in ATM networks’. Proc. ICC ‘92, Paper 354.4, pp. 1628-1663

13 ALLEN, A.O.: ‘Statistics, and queueing theory with computer science applications’ (Academic Press, Inc., 1990). 2nd edn.

14 KLEINROCK, L.: ‘Queueing systems. Vol. 1: Theory’ (John Wiley & Sons, 1975)

15 OKUDA, T., and AKIMARU, H.: ‘A simplified performance evaluation for packetized voice systems’, IEICE Trans., 1990, E73, (a), pp. 936-941

16 ABBAS, M., and AHMAD, Z.A.: ‘Performance evaluation of select- ive cell discarding control in ATM networks’. Proc. 1992 Singapore Int. Conf. on Communication Systems and Int. Sym. on Information Theory and its Applications (ICCS/ISITA ’92), Singapore, Nov. 1992, pp. 142-146

17 ABBAS, M., and AHMAD, Z.A.: ‘Analysis of a partial buffer sharing control with bursty traffic in ATM networks’. Proc. 1992 IEEE Region 10 Int. Conf. (TENCON ’92), Melbourne, Nov. 1992, pp. 16-20

IEE Proc.-Comun., Vol. 141, No. 3, June 1994 195

performance analysis of an atm statistical multiplexer with batch arrivals

Documents