Performance analysis of delay-loss priority mechanism using Markov modulated arrival stream

M.H. Lee Y.S.Mun B.G. Kim

Indexing terms: ATM network, MMPP, Priority control, Quality of service, Stochastic integral, Traflic control

Abstract: The authors propose a new mechanism to handle delay ancl loss requirements of ATM traffic simultaneously. It is designed to satisfy the different levels of cdl-loss performance for two classes of heteroganeous nonreal-time ATM traffics as well as the delay and loss requirements of real-time traffics simultaneously. The cell arrivals of input streams are modelled by MMPP. Both a real-time buffer and a nonreal-time buffer are served by a common server. The buffer occupancies of each traffic are controlled by the partial buffer sharing scheme. The steady-state probabilities of queue-length distributions are obtained using the stochastic integral approach.

1 Introduction

Asynchronous transfer mode (ATM) is recognised as the standard switching; and multiplexing technique to support various types of existing and new multimedia services in a unified ne .work with greater flexibility and efficiency. It transfers the user information in small fixed-size packets through a very high speed link. Such characteristics make the congestion control and resource allocation in ATM network complicated. To cope with these, many novel traffic management strate- gies have been suggested such as call admission control (CAC), usage parameter control (UPC), and reactive control mechanisms [l I.

The priority control mechanism, which is part of UPC, is an effective method to support multiple classes of services having dif'ferent quality of service (QOS) requirements. In this mechanism various traffics are classified and prioriti sed according to their cell loss and/or delay requirements [l]. Basically there are two ways to utilise priorities. The first, known as a loss pri- ority or space priority mechanism, includes a scheme to selectively discards cells based on their loss-rate requirement, such as the push-out scheme, the partial buffer sharing (PBS) scheme, and the threshold push- D IEE, 1997 IEE Proceedings online no. 19971508 Paper first received 26th February 1996 and in revised form 7th July 1997 M.H. Lee is with the Department of Multimedia Science, Chungnam Sanup University, Namjang ri San 29, Hongsung-eup, Hongsung-gun, Chungnam-do 350-800, Kore,i Y.S. Mun and B.G. Kim are with the School of Computing, Soongsil University, 1-1 Sangdo-5 dong Dongjak-ku, Seoul 156-743, Korea

out scheme [24] . Then, once cells are accepted, schemes, known as delay priority or service priority mechanisms, are needed to schedule the transmission priority of each cell. Various strategies such as head of line (HOL), minimum laxity threshold (MLT), and queue length threshold (QLT) have been studied [5]. However, these schemes for selective discarding or pri- ority scheduling consider the only one aspect of QOS requirements, either the cell-loss rate or the delay requirement. To meet the loss rate and delay require- ments simultaneously, nested threshold cell discarding with multiple buffers (NTCD/MB) was proposed in [6, 71. In NTCDiMB different classes of real-time traffics are considered but it does not support multiple classes of nonreal-time traffics. The space priority is not con- trolled for nonreal-time traffics. Consequently it does not meet different levels of loss-rate requirement for heterogeneous nonreal-time traffics. In this paper we propose a delay-loss priority control mechanism that is designed to satisfy different loss-rate requirements of heterogeneous nonreal-time traffics in addition to both the delay and loss-rate of real-time traffics.

Meanwhile, as ATM traffic is the superposition of many traffics from different sources, it is important to model them properly. In [2-51 the Poisson or Bernoulli process was considered for a large number of independ- ent traffic sources. But these are not suitable for voice or video which have the correlation in their cell interar- rival time [8]. Pure Poisson process was also found inadequate to model bursty data traffic such as LAN type [9]. For this reason the bulk Bernoulli process (BBP), Markov modulated Bernoulli process (MMBP), Markov modulated Poisson process (MMPP) or Markov modulated fluid flow (MMFF) model was used in [6-8, 10, 111. We consider MMPP following the approach in [lo].

2 Earlier works

2.1 Priority control First we compare three selective discarding schemes with each other [2-41. In the push-out scheme cells are admitted into the buffer regardless of their priorities until it becomes full. Once full, newly arriving low-pri- ority cells are prohibited in the buffer and discarded, but newly arriving high-priority cells replace low-prior- ity cells within the buffer. The PBS scheme accepts cells into the buffer regardless of their priorities up to a pre- assigned threshold. Beyond this threshold, only high- priority cells are allowed to enter. The threshold push- out scheme works in the same way as the push-out

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scheme until the buffer is full. Once full, class 1 cells push out class 2 cells if the number of class 2 cells exceeds a threshold value and vice versa. The push-out scheme has slightly better performance than PBS but it needs a more complicated algorithm for determining which cell is to be pushed out. The advantage of the threshold push-out scheme is its ability to adjust the loss rates between traffic classes, unlike the previous push-out scheme. It also support a higher maximum load than PBS for a given buffer size.

Three priority scheduling schemes can be compared [5]. HOL always transmits delay-sensitive cells before loss-sensitive cells. But it degrades the performance of loss-sensitive cells too much when the load of delay- sensitive cells is great. In MLT, the laxity of a cell is defined as the number of time slots remaining before its delivery deadline has expired. When the minimum lax- ity of all the cells in the queue is less than or equal to a threshold, priority is given to real-time cells. Otherwise it is given to nonreal-time cells. Under QLT, loss-sensi- tive cells are transmitted first when the number of loss- sensitive cells in the buffer exceeds a threshold, or when there is no delay-sensitive cell. Otherwise priority is given to delay-sensitive cells. It has the same perform- ance as MLT but is simpler.

To control cell loss and delay at the same time, NTCD/MB uses a real-time buffer and a nonreal-time buffer [6, 71. The cell discarding is controlled by PBS scheme for the real-time traffic only. The priority scheduling between real-time and nonreal-time traffic is achieved by guaranteeing a different fraction of band- width to each buffer. NTCD/MB can satisfy both the delay and the loss rate of real-time traffics, but it does not meet the different levels of loss-rate requirements for the multiple classes of nonreal-time traffics.

2.2 Traffic model The traffic models can be classified into the following six categories [12].

2.2.1 Renewal traffic models.: In these the cell interarrival time for each individual traffic is assumed to have independent, identical distribution. They have been used for a long time because of their mathemati- cal simplicity. But, with few exceptions, the superposi- tion of independent renewal process does not produce a renewal process. Also it does not capture the autocor- relation so it cannot explain traffic burstiness, which is the major characteristics of the broadband network and LAN. Poisson, Bernoulli or a phase-type renewal process is used in this model. The Poisson process can be used for an ideal data source or a large number of independent traffic streams. The Bernoulli process is a discrete version of Poisson process. Also the bulk Ber- noulli process can be defined by specifying the distribu- tion of the bulk size, independent of the bulk interarrival time that follows Bernoulli distribution. It was found to be comparable to MMPP in accuracy and provides bounds. Also it is based on closed-form expression [9, 11-13].

2.2.2 Markov and Markov-renewal traffic models: These introduce dependency into the random sequence of interarrival time and they can capture traf- fic burstiness. In a Markov traffic model interarrival times are exponentially distributed and their rate parameters depend on the current state. On the other hand, Markov-renewal traffic models are more general.

312

Interarrival times can be arbitrarily distributed and, moreover, these distributions depend on both current state and previous states [12].

2.2.3 Markov modulated traffic models: In these an auxiliary Markov process M is defined and the probability law of arrivals is controlled or modulated by the state of this process. The most commonly used Markov modulated model is the Markov modulated Poisson process (MMPP) where arrivals occur accord- ing to a Poisson process at rate & in state k of M. The sojourn time of each state follows a Bernoulli or Pos- sion process. A two-state MMPP is sometimes called an interrupted Poisson process (IPP) and has been widely used to model a voice traffic source. Also the basic MMPP can be extended to aggregations of inde- pendent MMPP sources. On the other hand, in the transition modulated process, the rate is determined by a state transition rather than a state itself [lo, 12, 131.

2.2.4 Fluid traffic models: In these traffic is viewed as a stream of fluid which is characterised by a flow rate like bits per second. A traffic count is replaced by traffic volume. The waiting time becomes the time it takes to serve the current buffer and the loss rate is calculated in terms of overflow volume. So these models enable a saving in the CPU time needed for simulation. Typical fluid models assume the source is bursty. Many generalisations described for MMPP can be applied to these models as well, including multiple sources or multiple classes of sources [12].

2.2.5 Autoregressive traffic models: These define the next random variable in the sequence as an explicit function of previous ones within a time win- dow. Such models are particularly useful for modelling VBR-coded video. The sequence of bit rates comprising a video scene may be modelled by an autoregressive scheme while scene changes can be modelled by some modulating mechanisms such as a Markov chain. The linear autoregressive models utilise the recursive rela- tion to generate the next random element in the sequence from previous one and, because of this sim- plicity, it is useful for modelling autocorrelated traffics using empirical data. The transform-expand sample (TES) models try to capture both marginals and auto- correlations of traffics from empirical data [12, 131.

2.2.6 Self-similar traffic models: Traffics such as packet traffic over ethernet have been reported to show burstiness across an extremely wide range of timescales differently from conventional telephone traffics. That is, for this traffic, similar looking traffic bursts are evi- dent at every timescale ranging from a few milliseconds to minutes and hours. These traffic models try to cap- ture this ‘self-similarity’ using the so-called Hurst parameter [9, 121.

3 Delay-loss priority control mechanism

Our delay-loss priority control mechanism consists of a real-time buffer and a nonreal-time buffer. Each arriv- ing cell enters one of these buffers according to their delay requirements. They are served by a common server which spends the predetermined portions of bandwidth to each buffer when both buffers have the cell. When one of these is empty, the entire bandwidth is allocated to a nonempty buffer. Thereby it may pro- vide guaranteed performance to both real-time and

IEE Proc -Commun, Vol 144, No. 5, October 1997

nonreal-time traffics. This function is similar to NTCDIMB. In addition, two levels of loss performance for nonreal-time traffii: as well as real-time traffic are desired. For this purpose buffer occupancies of each arriving cell are controlled using PBS in both of these two buffers as in Fig. I.

f BNR

nonreal-time traffic

INRH and NRL) ___*

T real-time traffic 1 (RH and RL) - E BR

Delay-loss prcorzty control mechanism Fig. 1

We consider delay-sensitive real-time traffic and loss- sensitive nonreal-time traffic, both of which have, respectively, two classes of arrival streams with differ- ent loss-rate requirements. Therefore there exist four classes of stream as shown in Fig. 1; real-high, real- low, nonreal-high and nonreal-low. Each stream is modelled by a 2-state MMPP which can be used to rep- resent a single source or a superposition of multiple sources. When each arrival process is modelled by MMPP the superposed process can be modelled by extended MMPP of thc larger state space [lo, 12-16].

4 Analysis

We denote two real-time streams as A, B and two non- real-time streams as C', D. Each stream stays at state

5 , 6) and YD(t) = j (j = 7, 8) with cell arrival rate dg, Ah, Ai and Aj in each state., respectively. Each stream tran- sits between its two states with rates (ylz, y21), (y34, y44, (yS6, y65) and (y7x, ys7), respectively. We assume real- time buffer of size KR - 1 and nonreal-time buffer of size KN - 1. And 0 .indicates the threshold of each buffer. Let a pair of Zlt(t) (0 s ZR(t) s KR) and Zdt) (0 s Zdt) I KN) denote the system state at time t. Then the limiting distribution of Markov process { YA(t), Ye(& Ydt), Yo(& ZR(t), Zd t )} can be defined by the steady-state probabilitjc .(g, h, i, j , qR, qN).

Now we obtain the long-term probabilities PA(qR) and Pe(qR) that stream A and stream B, respectively, see the real-time subsystem in state qR. Also the long- term probabilities PdqN) and PD(qN) which stream C and stream D, respectively, see the nonreal-time subsys- tem in state qN are obtained. The expressions for these probabilities can be derived in the Appendix (Section 8.1). Similarly the long-term probabilities P(qR) and P(qN) that real-time traffic and nonreal-time traffic see state q R and state qN, respectively, are derived in the Appendix (Section 8.1).

To obtain these long-term probabilities define $g, h,

YA(t) = g (g = 1, 2), YB(t) = h (h = 3 , 4), Y,(t) = i (i =

i , j , qR, q N ) and calculate $g, qR) , 4h, q R ) , n(i, q N ) , and a, qN) as follows:

4 9 , 4 R ) = XI; &cj4g, h, i , i 4R, 4 N )

T ( h 7 q R ) = c 5 , c 3 c . 7 T ( 9 7 i , j 7 ~ R I qN) r(2, qN) = ~ g ~ h ~ j ~ ( g , h, i , j , qR, qN)

n(j," = ~ , C h W S , h, i , i qR, 4 N ) (1)

Using the instantaneous transition rate matrix for our delay-loss priority control system, as given in the Appendix (Section 8.2), the steady-state equation for the delay-loss priority system becomes

where ?G = (41 , 3, 5 , 7, 0, 0), $1, 3, 5, 8, 0, 0), ..., 4 2 , 4 ,6, 8, KR, KN)). By solving the set of equations with the condition ze = 1, the probability $g, h, i , j , qR, q N ) is easily calculated. Based on this steady-state probabil- ity, the cell-loss rate can be computed as follows. Here we assume that the streams A and C have higher prior- ities than B and D, respectively.

nQ=O (2)

4 N = o N

R O S S ( D ) = A7778+X8787

( 3 ) The average delay for real-time traffic can be calculated using Little's law as follows:

Here AR is the average arrival rate of real-time traffic, E(nR) is the average queue length for the real-time buffer, AR is obtained from eqn. 15 and E(nR) from the steady-state probability.

5 Numerical examples

In this Section, through numerical examples, the effec- tiveness of the dealy-loss priority mechanism is con- firmed. For this purpose consider three traffic profiles TI, T2 and T3 which have the following characteristics, respectively:

T1 : A1 = A 2 = A3 = A4 = 0.2 A5 A6 A7 A S I 0 . 1 5

Y l 2 = 721 = 734 = Y43 = 0.4 756 = 7 6 5 = 778 = 787 = 0.2

T2 : A1 A2 A3 1 A4 1 0 . 2 A5 A6 0.35, Ai' = = 0.15

Y l 2 = Y21 = 7 3 4 = 7 4 3 = 0.4 756 = 765 778 = 7 8 7 = 0.2

T3 : A1 0.05, A2 = 0.35, A3 = Aq = 0.2 A5 = A6 = A7 = As = 0.15

7 1 2 = Y21 = 7 3 4 = Y43 = 0.4

756 7 6 5 = 778 = 787 = 0.2 Fig. 2 shows the effect of nonreal-time buffer size and the threshold on the nonreal-time cell-loss rates. Note that the different levels of cell-loss performance can be satisfied for two classes of nonreal-time traffic by using

loads the loss rate of can be supported for low-pri- ority traffic. However, excellent loss-rate performace may be obtained for high-priority traffic. The loss-rate

PBS scheme. At the buffer size 300 and low traffic

313 IEE Proc.-Commun., Vol. 144, N s . 5, October 1997

performance of nonreal-time traffic is very sensitive to its load because only a small portion of the total band- width is given to nonreal-time traffic.

10 ‘ h

I I I

50 100 150 200 250 300 buffer size

Fig.2 -+- T1, no control T1, high

-A- T1,low - -X- - T2, no control - -t- - T2, high - - O - - T2, low

Loss rates of nonreal-time traffic against buffer size

lo-’

threshold Loss rates of real-time low-priority traflic against threshold Fig.3

-4- TI, a = 0.4 -B- T1, a = 0.5 -A- T1, a = 0.7 -X- T3, a = 0.4 -t- T3, a = 0.5 -0- T3, a = 0.7

In Fig. 3 one can see the effect of the capacity frac- tion a and the real-time threshold on the loss rate of the real-time low priority. For the smaller value of a, the effect of the threshold change is greatly reduced. Also it is noted that the burstiness of high-priority traf- fic reduces those effects greatly.

0.OL

0.02

n 20 30 40 50

t hresho Ld Fig. 4 -e- TI, a = 0.4

Delay of real-time high-priority traffic against threshold

-E- T1, a = 0.5 -A- T1, a = 0.7 -x- T3, a = 0.4 -*- T3, a = 0.5 -0- T3, a = 0.7

Fig. 4 indicates that for larger value of a the change of real-time buffer threshold has almost no effect on delay. At smaller a, the real-time buffer threshold should be determined according to the delay require- ment.

Now the implementation of our model is considered. It takes 120 - 180 iterations to obtain the converged solution for the equations. And the number of opera- tions required are estimated at about 480,000 opera- tions per each iteration for real-time buffer size 50 and nonreal-time buffer size 100. The computational cost is formidable.

6 Concluding remarks

We have proposed a delay-loss priority control mecha- nism. The proposed delay-loss priority control mecha- nism can satisfy the different level of the cell-loss performance for nonreal-time traffics and the delay requirement for real-time traffic simultaneously. One of the major advantages of the model is achieving out- standing cell-loss rate for the nonreal-time traffics of high priority by imposing priorities on the nonreal-time traffics. We assumed the Markov modulated Poisson process for each heterogeneous traffic and extended it to the superposed traffic. The stochastic integral approach was used to derive our cell-loss rate expres- sion.

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References

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8 Appendix

8.1 Derivations of long-term probabilities First the long-term probability PA(qR) is obtained using the result in [lo]

t P A ( ~ R ) = lim -- lim -

t+cc NA(t) t+cc t 1 if 2RI:t) = qR 0 otherwise uqR = {

Eqn. 6 indicates the average arrival rate of stream A ( 5 )

(9)

- A1721 + A2712 - 7 1 2 + 7 2 1

(13) A3743 + A4734 -t

Y34 + 7 4 3 Also

IEE Proc-Commun.. Vol. 144, No. 5, October 1997