An Estimation Technique for Cell Loss Ratio in ... - Semantic Scholar

An Estimation Technique for Cell Loss Ratio in ATM Networks with Bursty Multiclass Sources Ping Wang Byung G. Kimy Computer Science Department University of Massachusetts at Lowell [email protected] [email protected]

Abstract

the multiclass arrival processes into a single Markov process but approaches rather dierently to achieve better computational complexity. In computing CLR, the trac behavior is described in terms of a regenerative network cycles consisting of idle and busy periods. The mathematical relationship between the aggregate arrival process and the buer availability is simpli ed by a newly introduced stochastic random variable, the rst passage arrival, which is de ned as the number of total arrivals in a rst passage time. Numerical and simulation results show reasonable agreements for various trac loads with small to medium buer sizes. The rest of this paper is organized as follows. In section 2, we describe the approximation technique for multiclass sources and discuss the relationship between the superposition process of arrival streams and the buer uctuation. In section 3, we introduce a new random variable, the rst passage arrival and the computations of cell losses and arrivals. Numerical results are presented in section 4 to demonstrate the accuracy by comparing the results to existing approaches and simulation results. Finally, concluding remarks are presented in section 5.

Estimation of cell loss ratios in a multiclass ATM network turns out to be a computationally intensive problem. In this paper, a computationally ecient algorithm is developed through a series of approximations. Multiple classes of bursty input sources are rst approximated into an equivalent single class model. Based on the single-class model, the relationship between the arrival process and the buer sizes is established by way of a renewal process alternating between overload and underload periods. Using variables associated with the renewal process, a typical two-dimensional model involving the arrival process and the buer sizes is simpli ed into a onedimensional model. A further approximation is made by estimating the available buer space at beginnings of overload periods. Numerical and simulation results show a good match at moderate buer sizes.

1 Introduction

A class of homogeneous ATM bursty sources has been modeled by a Fluid-Flow model [1, 10], a Markov-Modulated Possion Process (MMPP) [8], and Markov-Modulated Deterministic Process (MMDP) [12]. These source models are used to estimate the Cell Loss Ratio (CLR) in an ATM multiplexer with 2a computational complexity ranging from O((NK )3 ) to O(N ) (where N denotes the number of sources and K is buer capacity). By assuming an in nite-capacity buer, the Fluid-Flow model can reduce the computational complexity to O(N 2) and O(N ). When multiple classes of sources are involved, two approaches are oftenly used. First, the original trac is modeled by a multi-dimensional Markov process. The equilibrium state probabilities are then computed to obtain CLR [6]. This approach has a low distortion, but the computational complexity may be prohibitably high. Approximation techniques are presented in [2]. In the second approach, multiple classes of arrival processes are aggregated in order to reduce the dimensionality of the original superposition process [5, 3]. However, the aggregation process itself incurs both information loss and computational complexity. For example, the complexity of the aggregation process in [5, 7, 4] is ranged from O(2 ) to (N 4 ). The approximation technique in this paper also aggregates

2 The Models

2.1 Source Model

Consider an ATM multiplexer with N homogeneous bursty sources, and a nite buer with a capacity of K cells. De ne a slot as the time required to transmit a cell. A bursty source alternates between burst and silence periods. In a burst period, cells are regularly generated at every D slots. No cells are generated in a silence period. Let us call the contiguous D-slot interval a frame, which represents the inter-arrival time in slots. Without loss of generality, we assume that D is an integer. The frame length is seen equivalent to the capacity of the multiplexer since it represents the maximum number of simultaneous bursts that can be supported without buering. This source model is referred to as an Interrupted Deterministic Process (IDP). Now, consider a superposition of two classes with N IDP sources (i = 1; 2, and N = N1 + N2 ). Without loss of generality, we assume that D1  D2 and that D1 is selected as the frame size of the superposition process. The transition matrix of the superposition process is then given by a Kronecker product of individual class matrices. The size of this transi Now with Ascend Communications, Inc., Westford, MA 01886 y This research was partially supported by the Electronics and tion matrix exponentially grows as the number of classes. In order to keep the dimensionality of the superposition process Telecommunications Research Institute, Taejon, S. Korea i

N

1

period X (t) > 0. By properties of Markov processes, each idle or busy period is independent and identically distributed. In an idle period, any occupied buer space will be continuously released (regenerated). When X (t)  D, the multiplexer is underloaded. Therefore, buer contents are being released until either the buer is emptied or the end of the underload period is reached. A special case is when X (t) = D during which the buer contents are unchanged. When X (t) > D, the multiplexer is overloaded. The buer contents continue increasing to a full buer or until the overload period ends. As soon as the buer comes full, new arrivals are discarded and the multiplexer enters a loss period. Note that not all overload periods produce cell losses. Let (t) and (t) de ned as the total number of cells that are lost due to buer over ow and the total number of cells that are received until time t. Suppose there are I (t) network cycles up to time t. Let  and  denote the number of lost cells and the received cells in the i-th cycle, respectively. Both  and  are i:i:d: random variables. From the strong law of large number, with the probability of 1, the CLR is equal to P  ( t ) =1  = E [] : CLR = lim (4) P !1 (t) = lim !1 =1  E [] Consequently, computations of the average losses and arrivals in a network cycle are required. In this case of heterogeneous trac, the overall cell loss ratio is of more interest than class-dependent CLR [9]. The cell loss ratios of individual classes can be estimated by the following heuristic observation: the percentage of losses will be proportional to the distribution of excess cell arrivals among individual classes. Let q denote the fraction of excess cell arrivals belonging to class i. Then the individual class cell loss ratios are approximated by ]q = E []E [Z ] CLR = EE[[ (5) ] E [ ]E [Z ] where E [ ] and E [Z ] are average numbers of cell arrivals and excess cell arrivals in individual classes, E [Z ] is average overall excess cell arrivals, over network cycles. Therefore, additional E [Z ]; E [Z ] and E [ ] have to be computed [11].

low, Hong et al. performed the aggregation across diagonal lines of the matrix after generating the complete transition probability matrix [5]. Here, we avoid generating the complete transition probability matrix and construct the matrix by aggregating birth rates and death rates across the diagonal lines of the superposition process. This will reduce the aggregation cost to less than O(N 2). Assume that each class is characterized by a discrete-time Markov birth-death process X . Let X denote the number of class i bursts in an arbitrary frame (0  X  N , i = 1; 2). Let p (i; j ) be de ned as P fX1 = i; X2 = j j X1 + X2 = kg (k = 0; 1;


; N ). The superposed two-dimensional process is approximated by a one-dimensional Markov process by aggregating states with X1 + X2 = k to a single state k. The transition rates among aggregate states are given by birth and death rates of  and de ned as i

i

i

i

k

k

X

 = k

X



=

k

j

j

i

(1)

i

k

[i1 + j2 ]p (i; j ); (k = 0; 1;


; N ) k

+=

i

[(N1 ? i)1 + (N2 ? j )2 ]p (i; j ); k

+=

i

k

2 1 ? 0 0 0


66 1 1 ? 1 ? 1 1


Q = 64 .. .. .. .. . . . .

0 0 .. . 0


1?

0

3 7 7 7 5

N

:

(2)

k

i

i

k

j

i

i

i

i

k

k

I

I

i

Let denote the cell arrival rate at the aggregate state k. The aggregate arrival process is seen to generate cells over a frame with the frame-based state transition probabilities given in (2). It can then be shown that X D1 )p (i; j ); (k = 0; 1;


; N ): (3)

= (i + j D 2 += i

i

i

t

i

0

i

I

k

where  and  are birth and death rates of the class i. Namely, the superposed transition matrix of the dimension (N1 + 1)(N2 + 1)  (N1 + 1)(N2 + 1) is simpli ed to an (N + 1)  (N + 1) tri-diagonal matrix, i

i

i

i

i

k

i

i

We call this a Frame-based Markov-Modulated Deterministic Process (FMMDP). This FMMDP is fully speci ed with the following parameters: the frame size D (which can be directly related to the aggregate source peak rate), the ag- 3.1 First Passage Time and First Passage gregate birth and death rate vectors f g, f g, the aggreArrival gate arrival rate vector f g, and the number of multiplexed The rst passage time, T ; represents the time that the sources N . Note that the aggregation process above can be extended process starts at state i until it rst enters state j . Let A denote the rst passage arrival de ned as the total number of to cases with more than two classes of sources. cells generated in the rst passage time T . Let the arrival function G ( ) be de ned as the number of arrivals in state i 2.2 Multiplexer Model in successive  frames. Then, A is the sum of G (T ) when Consider a (N + 1)-state Frame-based Markov-Modulated the state k traverses each of states in T and the state k arrival process, X (t), which has a D-slot frame. The arrival lasts for T frames. The numbers of cells that arrived during process according to the FMMDP generates a cluster of cells an underload and an overload periods are equal to the rst in a frame. The size of the cluster is determined by the state passage arrivals A +1 and A +1 . Note that transition of X (t) which denotes the number of active bursty sources in probabilities of multiplexer states are given by an ideal situation. The behavior of the multiplexer can be P fX (n + 1) = i + 1jX (n) = ig =  =( + ); described by a series of network cycles consisting of idle and busy periods. In an idle period, X (t) = 0, whereas in a busy P fX (n + 1) = i ? 1jX (n) = ig = =( + ): (6)

3 The Analysis

k

k

k

i;j

i;j

i;j

i

i;j

k

k

i;j

k

D;D

2

D

;D

i

i

i

i

i

i

Also, the number of consecutive frames in state i is denoted The expected losses, E [(Z ? B )+ ], can be derived over an by L and can easily been seen to be geometrically distributed overload period as with p = 1 ? ( + ), so that 1 X X E [(Z ? B )+ ] = P fB = kg nP fZ ? k = ng P fL = kg = p ?1q (7) i

i

i

i

K

k i

i

i

where q = 1 ? p . It is necessary to point out that the arrival function G ( ) can also account for the quantities other than the number of arrivals, such as the excess cell arrivals when the multiplexer is in an overload state, or the residual bandwidth when the multiplexer is in an underload state. Therefore while the rst passage time is seen as one-dimensional accumulation when the multiplexer transits from one state to the rst entry of another state, this new stochastic random variable, the rst passage arrival, extends to two-dimensional accumulation. The Fig. 1 illustrates some rst passage arrivals with arrival function de ned as the number of arrivals. i

k

i

where

4=

A 45 A 32

7

A 04

6 5

A 43

4

A 34

3

A 45 A 32

A 34

A 23

2

A 23

A 12

1

t

Figure 1: Some rst passage arrivals

We start with E [] rst. We only consider burst-level losses which occur when more than D sources are in bursts (i.e. overload periods). Suppose, in the i-th busy period, there are M i:i:d overload periods. Let Z (j = 1; 2;


; M ) denote the total excess input (the number of cells which arrived beyond the multiplexer capacity) in the j -th overload period. Then X (8)  = (Z ? B )+ ij

i

Mi

i

j

=1

ij

j

where B denotes the available buer space immediately prior to the j -th overload period. Its mean is approximated by Ward equation as E [] = E [M ]E [(Z ? B )+ ]: (9) Let K denote the number of times that the multiplexer at state i leaves the state i and enters state i +1 before it nally leaves the state i for state i ? 1. Its distribution is given by P fK = kg =  (1 ?  ); (k  0) (10) where  =  =( + ). An overload only occurs when the multiplexer enters state (D +1). Thus, E [M ] is expressed as j

i

k i

i

i

i

i

i

i

E [M ] = E [K1 ]


E [K ] = D

Y D

: =1

i

i

i

=0

P fB = kg

1 X =1

(k ? i)+ P fZ = ig:

(12) (13)

i

We rst examine the eectiveness of the new estimation technique by comparing results with those from other analyses for homogeneous trac. As a source model, parameters corresponding to LAN trac are chosen so that the peak data rate P = 12Mb=s, burstiness (the ratio of peak to average rate) b = 10, and the mean burst length L = 307cells. In Fig. 2, CLRs from the Fluid-Flow [1] and the two-state MMDP analyses [12] are compared with the projected results from the technique presented in this paper. Also plotted are simulation results. The multiplexer has a trac load of 0:5 and 0:8, respectively. It is observed that for the range of buer sizes, the projection results show a better match with simulation. For a heterogeneous trac environment, we consider the LAN trac sources mixed with a class of more bursty sources. For such source, trac parameters are that P = 12Mb=s, b = 22:99, and L = 2604cells. The aggregate CLR for both classes of trac is plotted in Fig. 3 as the trac mixes are varied. In this case, the trac load is 0:5. Again, the projected results demonstrate a good match with simulation results. To further increase the heterogeneity of the sources involved, we consider typical image sources as the 2nd class with the peak rate of P = 2Mb=s. When the multiplexer load is kept at 0.5 with varying degrees of trac mixes from 0.1 to 0.9, the analytical results show a bimodal behavior (Fig. 4). Matches are shown to be better at either end of the curve than in the middle. This is from the fact that as sources are approximated into a single process, a certain information distortion causes CLR estimation errors. At both ends of the curve, the heterogeneity of the overall input trac is smallest, whereas larger dierences exist elsewhere. Such approximation distortion is re ected in this CLR estimation.

3.2 Cell Losses and Arrivals i

K

4 Numerical Results

A 01

0

n

Here, 4 is seen as the average buer space left over at the end of an overload period. From equation (12), we need to compute three quantities to obtain the expected number of cell losses in a network cycle: E [Z ], E [B ], and 4. Both E [Z ] and E [B ] are computed and approximated via means of the rst passage arrival. The 4 can then be further approximated and estimated in a closedform. Due to the space limitation, we refer the readers to [11] for details of the derivations. The expected number of cell arrivals in a network cycle, E [], is directly derived from the rst passage arrival during time the multiplexer transits from a busy period to an idle period [11, Eq. 5.4]. Subsequently, the cell loss ratio is obtained by the equation (4).

10

8

X k

X(t)

9

=1

=0

= E [Z ] ? E [B ] + 4

i

(11)

3

Comparison among analyses and simulation results 10e-0 C=159Mb/s P=12Mb/s b=10 L=307cells N=60 10e-1

95% Conf & 10% Interv Projection results Fluid-Flow model Fluid-Flow Asym Two-state MMDP Comparison between analysis and simulation 10e-1 95% conf & 10% interv projection results class 1 results class 2 results

Cell Loss Rates

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10e-4 C=159Mb/s P=12Mb/s Class 1: b1=10 L1=307cells N1=15 Class 2: b2=22.99 L2=2604cells N2=103 Class 1: 25% load class 2: 75% load

Comparison among analyses and simulation results C=159Mb/s P=12Mb/s b=10 L=307cells N=96

95% Conf & 10% Interv Projection results Fluid-Flow model Fluid-Flow Asym Two-state MMDP

10e-5 0

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Comparison between analysis and simulation 10e-1 95% conf & 10% interv projection results class 1 results class 2 results

Cell Loss Rates

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10e-4 C=159Mb/s P=12Mb/s Class 1: b1=10 L1=307cells N1=30 Class 2: b2=22.99 L2=2604cells N2=69 Class 1: 50% load class 2: 50% load

Figure 2: CLR vs. buer capacity, comparison of new approach, existing approaches and simulation results

10e-5 0

50

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150 200 Buffer Capacity

250

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350

Comparison between analysis and simulation

In section 2.2, it was pointed out that the individual class CLRs can be estimated by predicting the distribution of excess cell arrivals among individual classes. Our experiments show that when the dierence among individual class peak rates becomes signi cant, more cells in a cluster of arrivals (in a frame) will be correlated to each other. These cells in a cluster will likely belong to the same sources. This correlation relationship needs to be addressed when estimating the distribution of excess cell arrivals among individual classes. We analyzed this problem with two classes to show that the new approximation technique can also be used to estimate the individual class CLRs when a right method is used to predict the the excess cell arrivals among individual classes (Fig. 5). The two classes involved are dierent on their peak rates only. The experiment had equal load for both classes but varied the peak rate rato from 1 to 6. The results exhibit a reasonably good match. Figure 3: CLR vs. buer capacity, comparison of new apIn terms of computational eciency, all quantities involved proach and simulation results (E [M ], E [Z ], 4, and E []) either only require a linear cost, or can be estimated in a closed-form. The aggregation cost to approximate the multiclasses sources into an equivalent (N + 1)-state Frame-based Markov-Modulated Deterministic Process will be in the order of O(N 2) for  classes. This 10e-1

95% conf & 10% interv projection results class 1 results class 2 results

Cell Loss Rates

10e-2

10e-3

10e-4

C=159Mb/s P=12Mb/s Class 1: b1=10 L1=307cells N1=45 Class 2: b2=22.99 L2=2604cells N2=34 Class 1: 75% load class 2: 25% load

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Comparison between analysis and simulation results

Comparison between analysis and simulation results

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95% conf & 10% interv projection results

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overall: 95% conf & 10% interv overall projection results class 1: 95% conf & 10% interv class 1 projection results class 2: 95% conf & 10% interv class 2 projection results

10e-3

10e-4 C=26.5Mb/s K=50cells Class 1: P1=12Mb/s b1=10 L1=307cells N1=1-9 Class 2: P2=2Mb/s b2=22.99 L2=2604cells N2=124-14

C=26.5Mb/s K=50cells b=10 L=307cells Class 1: P1=12Mb/s N1=5 Class 2: P2=12 6 4 3 2.4 2Mb/s N2=5-30 Class 1: 50% load class 2: 50% load

10e-5

10e-5 0.1/0.9 0.2/0.8 0.3/0.7 0.4/0.6 0.5/0.5 0.6/0.4 0.7/0.3 0.8/0.2 0.9/0.1 Ratio of traffic loads between classes

0

1

2 3 4 5 Ratio of peak rates between classes

6

7

Figure 4: CLR vs. trac load ratio, comparison of new ap- Figure 5: Individual CLRs vs. peak rate ratio, individual proach and simulation results CLRs compare to simulation results Death Fluid Queue. IEEE Transactions on Communications, quadratic complexity is incurred during the aggregation of 43(5), May 1995. individual birth and death rates (Eq. 1).2 Therefore the overall computational complexity is O(N ) for the estimation [4] Khaled M. Elasyed and Harray G. Perros. Analysis of an algorithm. ATM Statistical Multiplexer with Heterogeneous Markovian

5 Conclusion

[5]

In this paper, we studied an ATM multiplexer with a nite buer space subject to dierent classes of bursty sources. A new approach is presented to evaluate the loss performance of ATM multiplexer. The introduced aggregation method on individual class birth and death rates largely reduces the aggregation complexity. The rst passage arrival distributions along with underload and overload periods apparently allow us to establish more direct relationship between the aggregated (N + 1)-state Markov-Modulated arrival process and multiplexer buer states. As a result of this relationship, an ecient algorithm is obtained to estimate the cell loss ratios. Several numerical results demonstrate a good agreement with simulation results at certain buer range. However, more studies are required, with more classes involved and dierent trac parameters and scenarios, to further investigate the numerical relationship among the heterogeneity of input trac, loss performance and estimation complexity.

[6] [7] [8] [9]

References

[10]

[1] D. Anick, D. Mitra, and M. M. Sondhi. Stochastic Theory of a Data-Handling System with Multiple Sources. The Bell [11] System Technical Journal, 61(8), October 1982. [2] A. Baiocchi, N. Blefari-Melazzi, A. Roveri, and F. Salvatore. Stochastic Fluid Analysis of an ATM Multiplexer Loaded with Heterogeneous On-O Sources: an Eective Computational [12] Approach. In IEEE INFOCOM, 1992. [3] Soren Blaabjerg, Hakan Andersson, and Hakan Andersson. Approximating the Heterogeneous Fluid Queue with a Birth-

5

On/O Sources and Applications to Call Admission Control. Manuscript, 1996. S. Hong, H.G. Perros, and H. Yamashita. A discrete-time queueing model of the shared buer ATM switch with bursty arrivals. Telecommunication Systems, 2:1{20, 1993. L. Kosten. Stochastic Theory of Data Handling systems with Groups of Multiple Sources. In Performance of ComputerCommunications, IFIP, North-Holland, 1984. I. I. Makhamreh, D. McDonald, and N. D. Georganas. Approximate Analysis of a Packet Switch with Finite Output Buering and Imbalanced Correlated Trac. In International Conference on Communications (ICC), 1994. Ramesh Nagarajan, James F. Kurose, and Don Towsley. Approximation Techniques for Computing Packet Loss in FiniteBuered Voice Multiplexers. IEEE Journal on Selected Areas in Communications, 9(3), April 1991. I. Norros and J.T. Virtamo. Who Loses Cells in the Case of Burst Scale Congestion? In Teletrac and Datatrac in a Period of Change (ITC-13), Copenhagen, Denmark, June 1991. Roger C. F. Tucker. Accurate Method for Analysis of a Packet-Speech Multiplexer with Limited Delay. IEEE Transaction on Communications, 36(4), April 1988. Ping Wang. Computational Algorithms for Cell Loss Ratio in ATM Networks with Bursty Sources. PhD thesis, Department of Computer Science, University of Massachusetts, Lowell, August 1996. Tao Yang and Danny H. K. Tsang. A Novel Approach to Estimating the Cell Loss Probability in an ATM Multiplexer Loaded with Homogeneous On-O Sources. IEEE Transactions on Communications, 43(1), January 1995.