The investigation of a structured Markovian Traffic Source Model
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The Investigation of a Structured Markovian Tra c Source Model
Stephen Bates
y and Steve McLaughliny
Processing Group Department of Electrical and Electronic Engineering The University of Edinburgh
email sb@
y Signal
May 25, 1995
The simulation of broadband services has proven to be di cult and this is mainly due to the nature of ATM tra c. One problem involves de ning the characteristics of this tra c and adjusting the parameters of a model to match them. In this paper we present a structured Markovian tra c source model and show how its microscopic parameters can be derived from the macroscopic parameters of the tra c. The moments of the cell inter-arrival rate distribution function are calculated in terms of model parameters and we verify that accurate tra c moment estimation is possible.
h 2 6 6 6 4
oa1 a i oi ei oi ti oi oi oi
oa2 ai ai e i ai tiai oi a i oi t i titi aiti ei ti
oaN
i
oi e i tiei
aiei ei ei
3 7 7 7 5
(1) (2)
However the hierarchical structure of the model suggests that sparsity can be introduced. Then, by making states Oi , Ai and Ti recurrent and de ning Ti as an associated state
(3)
3 2 0y 0 0 7 oi oi 6 6 0 aiai 0y 0 7 6 7 6 0 0 titi tiei 7 : 5 4 y 0y 0y 0 0
37/2Biblioteka transition time matrix it is possible to completely de ne the model, 2 3 0 0 oi oi oi a i 6 0 aiai aiti 0 7 7 Pi = 6 0 6 7 4 0 titi tiei 5 0 ei oi ei ai ei ti Ti
Abstract
1 Introduction
Modelling Broadband-ISDN services has been an important and popular area of research because of the ongoing investigation of the Asynchronous Transfer Mode (ATM). There are problems involved with this task and many of them are due to the nature of ATM tra c. One such problem is building models that adequately describe this tra c and deriving values for the microscopic parameters of a model from the macroscopic parameters of the tra c is therefore desirable. Akiramu et al 1] presented the results of such a derivation for an ON-OFF Markov process. The model considered in this paper is a structured Markovian tra c source model 2] which possesses several properties which make it suitable for modelling communication systems which possess a hierarchy of protocol data unit (PDU) modulation. In sections 2 and 3 the model is introduced and the moments of the inter-arrival time distribution function are derived in terms of a state recurrence pdf and the parameters of the model. The derivations for the rst two moments produces a variance estimator. In subsection 3.4 the modelling of long-tailed state recurrence pdfs is considered and the implications for large deviation theory analysis are discussed. In section 4 a series of results are given which illustrate how the estimator can be used to predict model behaviour. Section 5 presents conclusions on this work and goes on to discuss some further areas of research. 37/1
2 Structured Markovian model
Markov processes are commonly employed when modelling ATM tra c streams. Popular models include the ON-OFF (or 2 state) model and the Markov Modulated Poisson Process (e.g., 1], 3]). In these models each state has a constant cell transmission rate associated with it and the times between state transitions are determined using a Poisson process. Variable bit-rate simulation is achieved by multiplexing several of these sources together. However the structured Markovian model (SMM) produces tra c by a di erent method which introduced in this section and section 3. If the system under consideration contains N ATM services then the SMM possesses a state space, S , of size N m. m is de ned as the number of states per tra c type and can be chosen to correspond to the number of levels modulating cell transmission. To clarify this consider the example of a general ATM VBR service. At the cell level it is common to nd bursts of cells being transmitted at, or near, the maximum bit-rate (or more correctly cell transmission rate) of the service. These bursts will occur within an active session in which cells are being transmitted at the average cell transmission rate, Actr . At a higher level these active sessions are contained within the time domain which also contains periods when the source is not connected to the network. It is obvious that a hierarchy exists among these events and this forms the basis for the structure of the model. The rst step in model construction is to de ne the states within it and from the hierarchical example above it is possible to determine m = 4. Oi = Non-active state for tra c type i Ai = Active state for tra c type i Ti = Burst regulation state for tra c type i Ei = Transmission state for tra c type i In state Ei one cell is transmitted, therefore, for tra c type i, the time between cell transmissions, is regulated by states Oi , Ai and Ti . Transitions between tra c types i and j ,(i 6= j ),can only occur through states Oi and Oj , fi; j 2 1; 2; ; N g. This implies that the state transition matrix can then be reduced to a state transition vector, Po , and N state transition matrices of size m m, Pi fi : i 2 1; 2; ; N g as shown below. P0 Pi
Stephen Bates
y and Steve McLaughliny
Processing Group Department of Electrical and Electronic Engineering The University of Edinburgh
email sb@
y Signal
May 25, 1995
The simulation of broadband services has proven to be di cult and this is mainly due to the nature of ATM tra c. One problem involves de ning the characteristics of this tra c and adjusting the parameters of a model to match them. In this paper we present a structured Markovian tra c source model and show how its microscopic parameters can be derived from the macroscopic parameters of the tra c. The moments of the cell inter-arrival rate distribution function are calculated in terms of model parameters and we verify that accurate tra c moment estimation is possible.
h 2 6 6 6 4
oa1 a i oi ei oi ti oi oi oi
oa2 ai ai e i ai tiai oi a i oi t i titi aiti ei ti
oaN
i
oi e i tiei
aiei ei ei
3 7 7 7 5
(1) (2)
However the hierarchical structure of the model suggests that sparsity can be introduced. Then, by making states Oi , Ai and Ti recurrent and de ning Ti as an associated state
(3)
3 2 0y 0 0 7 oi oi 6 6 0 aiai 0y 0 7 6 7 6 0 0 titi tiei 7 : 5 4 y 0y 0y 0 0
37/2Biblioteka transition time matrix it is possible to completely de ne the model, 2 3 0 0 oi oi oi a i 6 0 aiai aiti 0 7 7 Pi = 6 0 6 7 4 0 titi tiei 5 0 ei oi ei ai ei ti Ti
Abstract
1 Introduction
Modelling Broadband-ISDN services has been an important and popular area of research because of the ongoing investigation of the Asynchronous Transfer Mode (ATM). There are problems involved with this task and many of them are due to the nature of ATM tra c. One such problem is building models that adequately describe this tra c and deriving values for the microscopic parameters of a model from the macroscopic parameters of the tra c is therefore desirable. Akiramu et al 1] presented the results of such a derivation for an ON-OFF Markov process. The model considered in this paper is a structured Markovian tra c source model 2] which possesses several properties which make it suitable for modelling communication systems which possess a hierarchy of protocol data unit (PDU) modulation. In sections 2 and 3 the model is introduced and the moments of the inter-arrival time distribution function are derived in terms of a state recurrence pdf and the parameters of the model. The derivations for the rst two moments produces a variance estimator. In subsection 3.4 the modelling of long-tailed state recurrence pdfs is considered and the implications for large deviation theory analysis are discussed. In section 4 a series of results are given which illustrate how the estimator can be used to predict model behaviour. Section 5 presents conclusions on this work and goes on to discuss some further areas of research. 37/1
2 Structured Markovian model
Markov processes are commonly employed when modelling ATM tra c streams. Popular models include the ON-OFF (or 2 state) model and the Markov Modulated Poisson Process (e.g., 1], 3]). In these models each state has a constant cell transmission rate associated with it and the times between state transitions are determined using a Poisson process. Variable bit-rate simulation is achieved by multiplexing several of these sources together. However the structured Markovian model (SMM) produces tra c by a di erent method which introduced in this section and section 3. If the system under consideration contains N ATM services then the SMM possesses a state space, S , of size N m. m is de ned as the number of states per tra c type and can be chosen to correspond to the number of levels modulating cell transmission. To clarify this consider the example of a general ATM VBR service. At the cell level it is common to nd bursts of cells being transmitted at, or near, the maximum bit-rate (or more correctly cell transmission rate) of the service. These bursts will occur within an active session in which cells are being transmitted at the average cell transmission rate, Actr . At a higher level these active sessions are contained within the time domain which also contains periods when the source is not connected to the network. It is obvious that a hierarchy exists among these events and this forms the basis for the structure of the model. The rst step in model construction is to de ne the states within it and from the hierarchical example above it is possible to determine m = 4. Oi = Non-active state for tra c type i Ai = Active state for tra c type i Ti = Burst regulation state for tra c type i Ei = Transmission state for tra c type i In state Ei one cell is transmitted, therefore, for tra c type i, the time between cell transmissions, is regulated by states Oi , Ai and Ti . Transitions between tra c types i and j ,(i 6= j ),can only occur through states Oi and Oj , fi; j 2 1; 2; ; N g. This implies that the state transition matrix can then be reduced to a state transition vector, Po , and N state transition matrices of size m m, Pi fi : i 2 1; 2; ; N g as shown below. P0 Pi