els for Markovian Arrival Processes (MAPs), as these processes are able to match correlations and burstiness, characteristics that are inherent to IP traffic. Using.
Approximate Model for Merging Markovian Arrival Processes Irena Atov, Richard J. Harris Abstract— Traffic-based decomposition models encompass procedures required for modelling of the basic network operations of superposition, departure and splitting, arising due to the common sharing of the resources and routing decisions taking place in packetswitched networks. It is desirable to study such models for Markovian Arrival Processes (MAPs), as these processes are able to match correlations and burstiness, characteristics that are inherent to IP traffic. Using the method of exact superposing of MAPs has limitations, as the computational complexity dramatically increases in practical cases. In order to keep the computational efforts required to a minimum, in analyzing queueing networks using the method of decomposition, one has to use a MAP of small order (e.g., MAP-2) to represent the intermediate node, as well as, the offered traffic inputs in the network. In this paper, we propose an approximate model for evaluation of the exact superposed process of a number of independent MAP processes as a MAP of order two, which provides good accuracy across wide range of burstiness parameters for the individual traffic processes and across various traffic load scenarios. Keywords— Traffic Modelling, Process.
Markovian Arrival
I. Introduction It is very important that the chosen traffic model for the planning process be able to capture burstiness and correlations, as real IP traffic exhibits these characteristics. Traditionally, for ease of tractability a renewal traffic model has been used in network planning processes and thus the correlation structure of the external and internal flows was neglected. However, it is well known that, except for the M/M/1 and M/G/1/1 systems, the departure process of any single-server queue represents a nonrenewal point process [4]. Moreover, it has been demonstrated that correlations of the arrival processes have a significant impact on the performance measures, especially for bursty input traffic [1]. The MAPs are able to match correlated and/or bursty arrival processes, which may also have selfsimilar properties and long-range dependence [3]. Moreover, the MAPs are analytically tractable models, which is demonstrated in various efficient computational procedures of the matrix-analytic approach for queueing systems. Despite these capabilities, MAPs have not been used in planning processes due to the associated state-space problem, which occurs when the basic network operations of merging, departure and splitting, owing to common sharing of the The authors are with the Centre for Advanced Technology in Telecommunications, Department of Electrical and Computer Systems Engineering, RMIT University, Melbourne, Australia. E-mail: {irena,richard}@catt.rmit.edu.au
resources and routing decisions taking place in the network, are being modelled. Recently, in [4] a traffic decomposition method for analysis of open queueing networks, where traffic arrivals occur according to a MAP process and service times have a phase-type (PH) distribution, has been developed. However, only finite MAP approximations for the output process of a MAP/PH/1(/K) queue have been studied, while exact methodologies for superposing and splitting of MAPs have been applied. Using the method of exact superposing has limitations, as the computational complexity dramatically increases in practical cases. In order to keep the computational effort required to a minimum, in analyzing queueing networks using the method of decomposition, one has to use a MAP of small order (e.g., MAP2) to represent the intermediate node, as well as, the offered traffic inputs in the network. For this purpose, development of a parameter estimation method to approximate the original superposed process as a MAP-2 is required. While performance analysis for queueing systems incorporating Markov-based traffic models (i.e., MAPs, MMPPs, and PH-type processes) is a welldeveloped research area, much less progress has been made on parameter estimation for these models. Moreover, the literature has mainly focused on methods for estimating the parameters of PH-type renewal processes [12], [13], [15] and MMPPs [18], [17], [19], [20], [21], [25], thus leaving the problem of parameter estimation for MAPs largely unexplored. In this paper we propose a model that provides a good approximation of the exact superposed MAP process by a MAP of order two. Conceptually, the model is similar to that of Rossiter [20] and Gusella [21] for parameter estimation of MMPP-2 in that it is a moment based method and treats processes with only two states. The greatest challenge with these methods is to select the descriptors of the original process, such that the approximate model derived from these descriptors will have the ability to match the impact that the original process has on a queue. The proposed model, combined with the existing models for finite MAP approximations for the departure process from a queue, provide the necessary ingredients for an efficient decomposition method based on MAPs. This enables us to overcome the state-space problem while capturing the correlation statistics that have impact on the queueing performance, and thus, incorporate these traffic processes into the planning procedures. In Section II we first introduce the MAPs formally
Approximate Model for Merging Markovian Arrival Processes and discuss the most important properties of such processes. Section III briefly overviews the three basic network operations i.e., superposition, departure and splitting of MAPs and exposes the state-space problem associated with the exact method for superposition of MAPs. The proposed approximate model for evaluating the exact superposed process as a MAP of order two is discussed in Section IV. In Section V we investigate the accuracy of the proposed model over a range of test cases, using matrix analytic techniques and, finally, Section VI concludes the paper.
T and T +j denote any two intervals k lags apart in the sequence of interarrival times. The interarrival time distribution function of a MAP is given by: F (t) = 1 −
. . . −qmm a11 a12 . . . a1m a21 a22 . . . a2m (1) D1 = . .. .. .. .. . . . am1 am2 . . . amm Pm Pm with Q = D0 + D1 and qii = j=1,j6=i qij + j=1 aij . D1 is a non-negative matrix, with elements that give the transition rates of the observed (or marked) transitions: passing through an observed transition triggers an arrival event. D0 has negative diagonal elements and non-negative off-diagonal elements representing the rates of the hidden transitions. It is required that Q 6= D0 , which implies that D0 is invertible and that the arrival process does not terminate. For a MAP the steady state probability vector π is defined by: πQ = 0 , πe = 1 (2) qm1
qm2
where e is a column vector of ones of appropriate dimension. In the following, we give the formulae of some relevant characteristics of a MAP. The mean arrival rate and squared coefficient of variation of a MAP are defined as [10]: λ
=
c2
=
1 = πD1 e , E[T ] E[T 2 ] − 1 = 2λπ(−D0 )−1 e − 1 (E[T ])2
(3) (4)
where T denotes the interarrival time of the MAP process. MAP process has a non-zero lag correlation structure as its interarrival times are correlated. The non-zero lag coefficients of correlation ρ(k) (k > 0) for a MAP can be derived from [10], [5]: ρ(k)
=
E[T T +k ] − E[T ]2 = E[T 2 ] − E[T ]2 k λπ (−D0 )−1 D1 (−D0 )−1 − 1 (5) 2λπ(−D0 )−1 e − 1
(6)
which leads to the following expression for the ith moment of the interarrival time [4]: E[T i ] = i!
II. The Markovian arrival process A MAP is a process which counts transitions of a finite continuous-time Markov chain (CTMC) [2]. Such a CTMC has an irreducible infinitesimal generator Q which is split into two matrices D0 and D1 as follows: −q11 q12 . . . q1m q21 −q22 . . . q2m D0 = . , . .. .. .. .. . .
1 πD1 eD0 t e πD1 e
1 πD1 (−D0 )−i e . πD1 e
(7)
The moments of the counting function N (t) of a MAP, i.e., the mean E[N (t)] and the variance Var[N (t)] of the number of arrivals in (0, t] are given by [4], [7]: E[N (t)] Var[N (t)]
= πD1 e · t ,
(8)
= (1 + 2πD1 e)E[N (t)] − 2πD1 (eπ + Q)−1 D1 et − Qt
2πD1 (I − e
−2
)(eπ + Q)
(9) πD1 e .
Many other arrival processes represent special cases of MAPs. Such examples are Poisson and Markov modulated Poisson (MMPP) processes, and - most importantly for our purposes - the superpositions of independent MAPs [4], [23]. III. Modelling Internal Flows From the offered traffic processes to the network and the given routing information, characterisation of the total traffic on every link in the network i.e., internal flows can be obtained by applying the methods for superposition, departure and splitting of MAPs, as described below. Departure flow from a queue The characterisation of the departure process of MAP/PH/1 queues is required in order to use these as input processes for queues further downstream in the queueing network. For a MAP/PH/1 queue with a finite buffer, the output MAP process has a dimension given by the product of the maximum buffer size, and the dimensions of the matrix descriptors of the input MAP and the PH-server. The size of this exact representation may be too large for practical use and, thus, approximate techniques which reduce the size of these representations become necessary. In the infinite buffer case no such exact finite description exists. In fact, the output process is an infinite MAP, which, due to its infinite size, becomes impractical for further use in network analysis. In order to solve this problem, in [8] truncation techniques for the infinite output MAP of a MAP/PH/1 queue have been studied and a family of MAP approximations to the departure process are proposed. In [14] similar approach has been proposed for a MAP/MAP/1 queueing system. Another approach for modelling of the departure process, presented in [4], [5], achieves a more compact MAP representation by choosing the parameters of the output
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St.Petersburg, Russia
MAP so that it reflects the busy-period behaviour of the queue. The latter approach, due to the more compact MAP representations, is preferred in the analysis of large communication networks. Splitting a MAP probabilistically The probabilistic splitting of a MAP (A0 , A1 ) with probability p, results in two MAPs (B0 , B1 ) and (C0 , C1 ) defined as [14], [4]: (B0 , B1 ) = (A0 + (1 − p)A1 , pA1 ), (C0 , C1 ) = (A0 + pA1 , (1 − p)A1 ).
(10)
This method will be used to characterise the flow being routed from some other node to the queue being examined. Superposition of independent MAPs This method is used for superposition of individual traffic streams that are being assigned different routes through the network but are traversing the same link and thus entering the same queue. The superposition of independent MAP arrival processes is also a MAP. Consider n MAP arrival processes, each characterised (i) (i) by (D0 , D1 ), respectively. Their superposition pro(s) (s) cess is a new MAP (D0 , D1 ), for which an exact representation is given by [23]: (s)
D0
(s) D1
(1)
= D0 =
(1) D1
(2)
⊕ D0 ⊕
(2) D1
(n)
⊕ . . . ⊕ D0 ⊕ ... ⊕
,
(11)
(n) D1
where ⊕ represents the Kronecker sum. It follows that the exact superposition of n MAPs, each of order mi , will result in a MAP of order k = Πni=1 mi . This exact representation suffers from the “curse of dimensionality", especially when considered as a tool in the analysis of a network. For practical applications in network design problems, one needs to reduce the complexity of having to solve queues with a large number of arrival processes. Therefore, the exact superposed process may be approximated by a simpler process that captures important characteristics of the original process as closely as possible. The simplest model that has the potential to approximate a MAP with a large number of states accurately is the MAP of order two (MAP-2). MAP-2 is defined by six parameters and, thus, the problem is reduced to choosing the parameters of the MAP-2 using six metrics of the superposed process. In the following, we propose a model, which enables the superposed process of a number of independent MAPs to be approximated by a MAP-2 process. IV. Approximate model for merging MAPs It is an open research issue as to which combination of characteristics should be used to obtain a good and efficient match with the original superposed process. At this point it is useful to consider some well-known moment based methods for MMPP-2 that have been proposed in the literature. Our intention here is not to discuss the proposed methodologies for estimation of the models’ defined characteristic parameter set from
empirical data but to evaluate the type of characteristic parameters that have been used for matching of the original process. The method of Rossiter [20] relies on the counting process N (t), the number of events in the interval (0, t]. Specifically, he makes use of the following set of four characteristics {λ, c2 , I, C} and showed how the parameters of an MMPP-2 can be uniquely determined from this set, provided that the condition I > c2 > 1 is satisfied. The chosen characteristics in his model include: mean arrival rate λ, squared coefficient of variation of the interarrival times c2 , the limiting index of dispersion for counts I = limt→∞ Var[N(t)]/E[N(t)], and the covariance of the counts in intervals (0, t] and (t, 2t] for t → ∞ i.e., C = limt→∞ C(t) with C(t) = Cov[N(t), N(2t)−N(t)]. Gusella [21] employs a similar method where, in addition to the first three characteristics of the Rossiter’s method λ, c2 , I, he makes use of the time constant µ = r1 + r2 of MMPP-2 as a fourth characteristic parameter. The parameters r1 and r2 define the rates of transition from state 1 to state 2 of the MMPP-2 and vice versa. He develops a procedure that derives the MMPP-2 parameter set from the values of the four metrics. Heffes and Lucantoni [25] develop a moment method in which the MMPP-2 parameter set is derived by matching the following four characteristics of the original (superposed) process: the mean arrival rate λ, the index of dispersion for counts at time t0 I(t0 ) = Var[N(t0 )]/E[N(t0 )], the limiting index of dispersion for counts I, and the third central moment of the counting function at time t0 E[(N(t0 ) − E[N(t0 )])3 ]. All the above models share the view that, in order to accurately approximate the correlation characteristics of the original (superposed) process, it is important that the approximate process provide a good match to the variance-time curve. This observation directly follows from known classical results [22], [25] as it is known that the variance-time curve completely defines the correlation structure for a process i.e., if C(t) = Cov[N(t), N(2t) − N(t)] then C(t) =
Var[N(2t)] − Var[N(t)] . 2
In addition to the above observation another useful result for our modelling purposes was presented by Whitt in [16]. Namely, Whitt showed empirically that the effect of the third moment of the interarrival time distribution on the average number in the GI/G/1 queue increases significantly with the increase of the squared coefficient of variation of the interarrival times of the arrival process. In our modelling approach, we take account of the above-mentioned observations. Specifically, through experimentation we have investigated the issue of which set of characteristic parameters would achieve the closest match to the superposed process for a wide
Approximate Model for Merging Markovian Arrival Processes range of burstiness parameters for the component processes and traffic intensities at the queue in which they are merged. In our proposed approximate model for merging MAPs, we choose the following six characteristics to match the superposed process (D0 , D1 ) with Q = D0 + D1 , viz : 1. The mean arrival rate, λ, given by (3), 2. The squared coefficient of variation of the interarrival time, c2 , given by (4), 3. The third moment of the interarrival time distribution E[T3 ], which follows from (7) for i = 3, 4. The index of dispersion for counts I(t) at time t = t0 , I(t0 )
=
Var[N (t0 )] = (1 + 2πD1 e) − E[N (t0 )] 2πD1 (eπ + Q)−1 D1 et0 − (12) πD1 e 2πD1 (I − eQt0 )(eπ + Q)−2 πD1 e , πD1 et0
5. The limiting index of dispersion for counts, I, given by [4], [7]: I
=
lim I(t) = 1 + 2πD1 e −
t→∞
(13)
2 πD1 (eπ + Q)−1 D1 e , πD1 e 6. The covariance of the counts in intervals (0, t] and (t, 2t] for t = t0 , given by [7]: C(t0 ) = π(M1 (t0 ))2 e − (πM1 (t0 )e)(πeQt0 M1 (t0 )e) (14) where M1 (t0 ) represents the first moment matrix of the counts during an interval (0, t0 ] and is given by: M1 (t0 )
= λt0 eπ + (eπ − Q)−1 D1 eπ + e(πD1 (eπ − Q)−1 ) − 2λeπ .
In this approach, a certain degree of arbitrariness is introduced by having the freedom to choose the time point t0 . Ideally, we have to select t0 such that a good fit to the variance-time curve over the entire range of t is achieved. Heffes and Lucantoni discussed the issue of a good choice for t0 for their mapping model for MMPP-2 [25]. Later, in [6] Heindl proposed an approach for selection of t0 for Heffes and Lucantoni’s model when it is specifically applied to the problem of matching a superposed process of a number of MMPP2 processes. Following his approach, we have experimentally studied the issue of a good choice for t0 in our model and our experiments showed that the following value for t0 provides a good fit to the variance-time curve of the superposed process of n MAP processes: n m m 1 1 X X X (15) t0 = (k) (k) 2n q + a i=1 k=1
j=1,j6=i
ij
ij
where m is the smallest order of the component processes to be merged1 . 1 This
result has been tested for component processes of order
The parameter matrices of the exact superposed MAP (D0 , D1 ) are easily obtained from their counterparts of the n MAP processes to be merged (D0 (i) , D1 (i) ) i ∈ {1, . . . , n} by applying (11). Its steady state probability vector π is then obtained from (2). Given the parameters D0 , D1 , π of the exact superposed process, one can then compute the six characteristics that need to be matched by the approximate MAP-2 process, by applying the formaulae given above. Then, using the equations for the six characteristics of the two-state MAP and the given values for the six characteristics of the superposed process, we have a nonlinear system of six equations and six unknowns, which can be solved to obtain the required unknowns i.e., the six rates of the approximate MAP-2 process (A0 , A1 ). In vector notation, we want to find one or more six-dimensional solution vectors x such that f (x) = c where f = [f1 , f2 , . . . , f6 ] is the six-dimensional vectorvalued function whose components fi , i = {1, . . . , 6}, are the individual equations to be satisfied simultaneously. The six-dimensional solution vector x consists of components which represent the six defining parameters of the approximate MAP-2 process, that is x= [q12 , q21 , a11 , a12 , a21 , a22 ]. The offset vector c= [c1 , c2 , . . . , c6 ] consists of components which have the values of the six characteristics computed for the exact superposed process. In general, there is no guarantee that there is a MAP-2 that matches the six characteristics of the superposed process exactly. Even if there is an exact match, we have a nonlinear system of equations to deal with. Hence, most often the matching has to be done approximately, as it has been the case with all matching methods for MMPP-2 discussed earlier. The advantage of modelling the superposed process as a MAP-2 instead of MMPP-2 is that we can match six characteristics, which are able to capture the properties of the original process that heavily influence the performance of a queueing system, instead of four and thus achieve a closer match to the original process. Albeit this is at the expense of increased modelling complexity. Specifically, the closed-form expressions of the chosen six characteristics for MAP-2 are quite complex and do not lend themselves easily to efficient inversion procedures for derivation of the MAP-2 parameter set. Therefore, for the solution of the nonlinear system of equations we use the approach of solving the following minimisation problem: P6 Minimise
z=
2
[fi (x) − ci ] P6 2 i=1 ci
i=1
(16)
where z has to be less than or equal to when a solution for f (x) = c is found (where is a sufficiently small value). For the solution of the above minimisaless than or equal to four. However, this requires further investigation for the case where MAPs of larger order than m = 4 are superposed.
NEW2AN 2004 tion problem, we have applied the direction set Powell’s method. In this way, the solution would converge to what is only a local minimum. Techniques for finding the global minimum, such as Simulated Annealing, can be employed however at the expense of significantly increased computational effort which, for our purposes, becomes impractical. We have found that the local minimum solution approach is quite satisfactory for this purpose. The local minimum solution approach has been traditionally applied in parameter estimation techniques for MMPPs based on maximum likelihood methods where the objective function is maximized instead. Thus, it is to be expected that more than one solution for MAP-2 exists. In our analysis, we have found that solutions for which the value of z is equal to or close to zero i.e., 0 ≤ z ≤ (where = 10−10 ) proves to be a useful approximation. Some comments follow. From the test examples in Section V we shall see that the increased parametrization of MAP-2, as opposed to MMPP-2, leads to an increased modelling accuracy. Although our proposed model conveys queueing performance better than the existing MMPP-2 models, it requires a much higher computational effort than that of the MMPP-2 matching models. With this in mind, our contribution can be regarded in terms of identifying the set of traffic characteristics that effectively captures the properties of the superposed traffic which are important in regard to queueing performance. Further work is needed to refine this method so that an efficient inversion algorithm can be developed for estimation of the MAP-2 parameter set. V. Numerical Results The specific model we analyse, to test the accuracy of the merging approximate model, is the ΣMAP / PH / 1 system with a PH(2)- type service time distribution and total number of n MAP-2 processes (n = 2, 4, 8) as component arrival processes that are merged as they enter the system. The choice for n = 2, 4, 8 stems from the fact that for networks of queues it is primarily important that the characterisation method be accurate when a small number of MAPs are superposed (e.g., two to ten). The performance of this system was analysed by feeding the exact superposed MAP (D0 , D1 ) and the approximate MAP-2 (A0 , A1 ) into the system, independently, for various traffic inputs. Specifically, ten different correlated arrival processes were used as component processes to create 135 different test cases. The cases we use have been set up to test the proposed method over a fairly extreme set of circumstances i.e., over extremes of combinations of arrival rate and variability. The combinations of arrival rate which have been considered for the case of n = 2 start from equal rates for the two processes to combinations where one process has two to twenty times the mean arrival rate of the other. The combinations of squared coefficient of variation of the interarrival times that we consider are all combinations of different pairs from
St.Petersburg, Russia {1.5, 2.5, 5, 8.5, 10}. Our research suggests that the actual service time distribution used in this type of investigation is relatively unimportant and, therefore, Erlang-2 service times have been chosen for convenience in computing the exact queueing results. The tests for n = 2 considered all combinations of pairs of arrival processes, resulting in 45 different superposed processes and test scenarios. In addition, appropriate scaling method was applied to vary the traffic intensities at the queue for eight different values starting from 0.25 to 0.95 for a step size of 0.1. The test scenario for n = 4 (n = 8) considred twice (four times) the traffic load of each combination considered in the test for n = 2. Overall results for the 1080 different traffic load scenarios were broken into three groups, each representing the case scenario which involed n = 2, 4, 8 traffic input processes, respectively. The accuracy of the approximate model was measured against the exact one by comparing the resulting performance measures of the system obtainable using matrix geometric techniques. The underlying infinite Markov chain of MAP/PH/1 queue can be considered as a special case of a quasi-birth-and-death process (QBD) [9] where each level of the QBD state space corresponds to a specific number of customers in the queueing system. For this system, the steady state probability vector Ψ can be computed by using matrix-geometric solution methods, like the LRapproach [11]. Having obtained Ψ, many performance measures of the queue can be computed easily. In our analysis, we make use of the mean and the variance of the total waiting time in the queueing system (including service). For comparion, we study the previously discussed MMPP-2 matching models for the same test cases. In his PhD thesis, Rossiter [20] extensively analysed the accuracy of his matching model to that proposed by Heffes and Lucantoni [25]. He concluded that both models perform very similarly. Therefore, in our analysis we consider the model of Rositer and the model of Gusella only. The latter, was slightly adapted in order to suit the need of matching superposed process of a number of MAPs. Namely, Gusella was primarily concerned with the estimation of the MMPP-2 parameter set from empirical data and did not discuss the possibility of using this model to the case of superposing of arrival processes. In the case when number of MMPP processes are merged his model can readily be applied by using previously obtained result by Heffes [24]. Namely, starting from the definition of a time constant Z 1 ∞ C(t) dt µ= ν 0 where C(t) is the covariance function and ν = limt→∞ Var[N(t)], Heffes derives the following useful result for MMPP processes [24]: ν I =1+2 µ . λ The above result, enables derivation of the time constant for an MMPP process from the moments of the
Approximate Model for Merging Markovian Arrival Processes counting function i.e., the mean arrival rate λ, the variance of number of arrivals in a long time interval ν (which can easily be derived from (9)), and the limiting index of dispersion for counts I. Although the above result does not directly apply to a MAP process, we make use of this approach as the nature of the whole excercise is an approximation. Note that, the other three characteristics used by Gusella and, also the whole set of characteristics used by Rossiter, can easily be computed from the respective formulae given for MAPs in this paper. TABLE I Averaged absolute RE (%) for the 1st and 2nd central moments of the queue waiting time for various sets of 2 input traffic streams.
ρ 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 ARE
MAP-2 App. E[W] Var[W] 1.10 4.59 2.71 9.18 2.82 10.6 1.88 8.63 0.81 5.84 0.21 3.70 0.17 1.96 0.12 0.60 1.23 5.63
Rossiter E[W] Var[W] 6.82 24.7 12.1 34.9 13.9 38.8 12.1 36.9 9.52 31.1 7.95 23.0 4.31 14.1 0.96 4.75 8.46 26.1
Gusella E[W] Var[W] 5.98 15.9 8.97 22.4 10.8 20.9 7.91 17.6 6.69 14.9 4.82 10.4 3.86 7.80 0.73 2.07 6.22 14.0
TABLE II Averaged absolute RE (%) for the 1st and 2nd central moments of the queue waiting time for various sets of 4 input traffic streams.
ρ 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 ARE
MAP-2 App. E[W] Var[W] 1.16 9.24 3.12 16.3 3.38 19.5 2.86 18.3 2.04 12.7 1.63 8.26 1.32 4.72 0.69 1.52 2.03 11.3
Rossiter E[W] Var[W] 4.82 21.2 10.3 38.4 15.9 49.9 16.7 50.4 18.6 51.3 14.5 42.2 8.25 27.9 2.59 10.1 11.5 36.4
Gusella E[W] Var[W] 3.37 12.1 6.82 23.7 11.6 31.9 12.5 26.9 10.7 19.2 8.89 11.2 4.96 10.7 1.68 8.53 7.57 18.0
As an illustration, we show in Table I, Table II, and Table III the advantage of modelling a MAP process of higher order as an approximate MAP-2 process, versus an approximate MMPP-2 as derived from the models by Rossiter and Gusella, respectively. Table I shows results for mapping the superposed process of two MAP-2 traffic streams (i.e., MAP-4) into equivalent approximate process by applying the three models that we consider. All the combinations of pairs of input traffic streams (as discussed earlier)
TABLE III Averaged absolute RE (%) for the 1st and 2nd central moments of the queue waiting time for various sets of 8 input traffic streams.
ρ 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 ARE
MAP-2 App. E[W] Var[W] 0.88 5.42 2.58 18.3 5.06 19.0 5.98 21.1 4.15 17.2 2.81 12.3 1.67 8.81 0.98 6.01 3.01 13.5
Rossiter E[W] Var[W] 3.57 12.4 6.44 28.4 12.5 45.2 18.9 56.2 22.7 59.4 22.1 55.1 14.3 39.8 4.01 14.9 13.1 38.9
Gusella E[W] Var[W] 2.39 8.65 4.96 20.1 10.5 35.3 14.6 44.1 18.7 44.8 13.8 39.4 4.83 13.0 2.93 3.76 9.09 26.1
were considered for each value of the queue utilization (0.25 ≤ ρ ≤ 0.95). The results shown in the table represent the average over each set of the absolute value of the relative percentage error, given by: | approximation − exact| · 100 exact for the mean and the variance of the queue waiting time. Similarly, Table II and Table III show results for mapping the superposed process of four MAP-2 traffic streams (i.e., MAP-16) and eight MAP-2 traffic streams (i.e., MAP-256), respectively, into the equivalent MAP-2 and MMPP-2 by using the approximate models. The averaged absolute relative percentage error (ARE) with respect to the exact model, reflects good accuracy of the approximate MAP-2 model across various traffic load scenarios and across wide range of burstiness characteristics for the individual traffic processes. The relative error of 3.013 % for the mean delay in the case when a MAP of order 256 is approximated by a MAP-2 (see Table III) shows that the approximation model is capable of capturing and conveying the queueing performance of the original process very well. A similar observation follows for the accuracy of the MAP-2 model with respect to the variance of the delay. This shows that we can successfully solve the state-space problem associated with the exact superposing while capturing the correlation statistics that have an impact on the queueing performance. In addition, the results show that the proposed moment based MAP-2 model outperforms the moment based MMPP-2 models by Rossiter and Gusella, respectively. However, considering the efficiency of the inversion algorithms offered by these models, the results suggest that the method of Gusella can be used as an alternative for the MAP-2 model in cases where higher efficiency is required, as the results show that it performs better than the method of Rossiter.
NEW2AN 2004
St.Petersburg, Russia VI. Conclusions
The motivation for using MAP process as a traffic descriptor in the analysis and planning functions of IP-based networks is the fact that complex arrival patterns, as those appearing in the Internet, can best be described using MAPs. In order these processes to be integrated into practical network applications approximate models for the superposing operation of these processes are necessary. In this paper, we have presented an approximate mapping model for evaluation of the exact superposed process of a number of independent MAP processes as a MAP of order two. It is a moment based model which captures six characteristics of the exact aggregate process and translates the captures statistics into an equivalent MAP-2 process by solving a minimisation problem. The numerical results show good accuracy across wide range of burstiness parameters for the individual traffic processes and various traffic load scenarios. The proposed model enables us to overcome the state-space problem associated with the superposing of MAPs while capturing the correlation statistics that have impact on the queueing performance. This model can be used as one of the main building blocks in a MAP-based decomposition model, which is required for the performance analysis and planning functions of IP-based networks. Our ongoing work is on the refinment of the proposed model in this paper with a goal to develop an efficient inversion algorithm for the estimation of the MAP-2 parameter set. In addition, our focus is on the integration of a MAP-based decomposition method into a planning tool for multiservice IP networks, as discussed in [26]. Acknowledgments The authors wish to acknowledge Prof. Peter Taylor of The University of Melbourne for the valuable discussions, which largely inspired this work. We also thank David Green and Bartek Wydrowski for their assistance in relation to some of the computational aspects of this work. Finally, the comments by Bill Lloyd-Smith are greatfully acknowledged. References [1] M. Livny, B. Melamed and A.K. Tsiolis, “The Impact of Autocorrelation on Queueing Systems", Management Science, vol. 39, pp. 322-339, 1993. [2] M.F. Neuts, “A Versatile Markovian Point Process", Journal of Applied Probability, vol. 16, pp. 764-779, 1979. [3] A. Horváth, G.I. Rózsa and M. Telek, “A MAP Fitting Method to Approximate Real Traffic Behavior", Proceedigns of 8th IFIP Workshop on Performance Modelling and Evaluation of ATM and IP Networks, Ilkley, UK, 2000. [4] A. Heindl, Traffic-Based Decomposition of General Queueing Networks with Correlated Input Processes, PhD Thesis, Technische Universität Berlin, 2001. [5] A. Heindl, and M. Telek, “Output Models of MAP/PH/1(/K) Queues for an Efficient Network Decomposition" Performance Evaluation, Vol. 49, No. 1-4, pp.321-339, 2002. [6] A. Heindl, “Decomposition of general queueing networks with MMPP inputs and finite buffers based on SMPs and MMPPs", Proc. 4th Int. Workshop on Queueing Networks
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