T.E. Stern and A.I. Elwalid (1991). Analysis of separable Markov-modulated rate models for information-handling systems. Adv. Appl. Prob. 23 105-139.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....general and can accurately represent a wide range of bursty network traffic, originating from various applications. Plain Markov modulated rate processes (featuring exponentially distributed state sojourns) have been successfully employed for representing voice [4,10] video [6,7,11,12] and data [3,13] traffic. The additional generality offered by the semiMarkovian models, namely the ability of specifying general probability distribution functions for the state durations, permits accurate modeling of traffic subject to shaping (e.g. leaky bucket based regulation) or throttle control. It ....

....chain with a transition probability matrix ] ij p P . The sojourns at a state i are independently and identically distributed random variables, of a general probability distribution function (PDF) i F . An important special case is that of the well known MMRP traffic models (see, e.g. [13]) where all sojourn times are exponentially distributed. 3.2. Top level architecture for traffic generation According to the discussion in Section 2, the traffic generation hardware does not incorporate direct means for simulating sophisticated traffic models. Its only function is to read from ....

T. E. Stern, and A. I. Elwalid, "Analysis of separable Markov-modulated rate models for informationhandling systems". Adv. Appl. Prob., Vol. 23, 105-139, 1991.

.... Mitra and Sondhi [8] Although P 1;0 (x) does not correspond to a probability distribution function, there is a wide spread use of stochastic fluid flow models for the performance analysis of statistical multiplexers in the ATM context (e.g. see Anick, Mitra and Sondhi [8] Stern and Elwalid [35] and Elwalid and Mitra [16] In the next section, a mathematical framework is presented to solve the MAP D 1 K system in case an arbitrary rational approximation B a (s) is imposed. Then performance assessment of Erlang and Pad e approximations in the analysis of the MAP D 1 K queue is ....

.... above can be found in the paper by Golub and Van Loan [17] Actually, in case the incoming MAP is made up of a superposition of many independent MAP s, these eigenvalue eigenvector pairs can be computed via simpler partial problems via the extension of the techniques used by Stern and Elwalid [35] in the context of stochastic fluid models. We also note that the extension of the proposed technique in [35] to the particular case of a superposition of 2 state MMPP s is examined by Akar [4] Once the boundary vector y 0;K is computed, the solution to the differential equations for W c is easy ....

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T. E. Stern and A. I. Elwalid. Analysis of separable Markov-modulated rate models for information-handling systems. Adv. Appl. Prob., 23:105--139, 1991.

....sent by either an individual user or all users. We denote the overall loss probability PL , and the loss probability of useri as P . P max corresponds to the maximum loss probability experienced by an individual user (user N ) in the service class. Note that traditional models, e.g. 1] [19], have focused on expressions for 0 7803 7476 2 02 17.00 (c) 2002 IEEE. the (overall) overflow probability rather than the loss probability. The latter is in our mind a more realistic measure of the performance that users experience. As outlined in Section III, deriving expressions for the loss ....

....system or an N source system, we proceed to derive general expressions for an N source system, from which we subsequently specialize to the two source system by letting N equal to 2. The analysis of systems that involve Markov modulated traffic sources is by now a mature area, e.g. 1] 15] [19], and we rely on this existing body of work to develop our model. The main difference between those works and ours is that ours focus on loss as opposed to overflow probability and, most important, on evaluating both individual and aggregate performance. Our initial model consists of N ....

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T. E. Stern and A. I. Elwalid. Analysis of separable Markov-modulated rate models for information-handling systems. Adv. Appl. Prob., 23:105-- 139, 1991.

....evaluated the system performance using elegant simple expressions for a multiplexer with infinite buffer space and homogeneous arrivals. Tucker[25] considered the finite buffer case. The methodology was generalized to the case of heterogeneous Markov Modulated Rate Processes in a series of papers [24, 6]. The work presented in this paper is related to the methodology presented Hong, Perros, and Yamashita [11] and Makhamreh, McDonald and Georganas [16] In [11] the authors used an 2 off on Figure 1: The Interrupted Bernoulli Process aggregation method for approximating the superposition ....

T. E. Stern and A. I. Elwalid. Analysis of Separable Markov-Modulated Rate Models for Information-Handling Systems. Adv. Appl. Prob., 23:105--139, 1991.

....number of bits sent by either an individual user or all users. We denote the aggregate loss probability ## , and the loss probability of user # as # # # . # # ### corresponds to the maximum loss probability experienced by an individual user (user # ) Note that traditional models, e.g. 1] [22], have focused on expressions for the (aggregate) overflow probability rather than the loss probability. The latter is in our mind a more realistic measure of the performance that users experience, and is also necessary to accurately capture the impact of traffic parameters on the performance ....

....an # source system, we initially proceed TECHNICAL REPORT UNIVERSITY OF PENNSYLVANIA 4 with derivations for an # source system, which we subsequently specialize to the two source case. The analysis of systems that involve Markov modulated traffic sources is by now a mature area, e.g. 1] 17] [22], and we rely on this existing body of work to develop our model. The main difference between those works and ours is our focus on loss as opposed to overflow probability and, most important, on evaluating both individual and aggregate performance. Our initial model consists of # independent ....

[Article contains additional citation context not shown here]

T. E. Stern and A. I. Elwalid. Analysis of separable Markovmodulated rate models for information-handling systems. Adv. Appl. Prob., 23:105--139, 1991.

....evaluated the system performance using elegant simple expressions for a multiplexer with infinite buffer space and homogeneous arrivals. Tucker[14] considered the finite buffer case. The methodology was generalized to the case of heterogeneous Markov Modulated Rate Processes in a series of papers [13, 4]. The work presented in this paper is related to the methodology presented Hong, Perros, and Yamashita [7] and Makhamreh, McDonald and Georganas [9] In [7] the authors used an aggregation method for approximating the superposition process. They first obtained the exact probability transition ....

T. E. Stern and A. I. Elwalid. Analysis of separable Markov-modulated rate models for information-handling systems. Adv. Appl. Prob., 23:105-- 139, 1991.

....and silence periods. Subsequent work extended the model in various directions, such as heterogeneous source characteristics, several source states to account for various activity levels, or activity periods with a general Markovian structure, see for instance Kosten [20] and Stern Elwalid [32]. Under traditional statistical assumptions, it turns out that the tail of the backlog distribution typically exhibits exponential decay. In recent years, empirical findings have triggered a strong interest in fluid models with non Markovian activity periods. Extensive measurements indicate that ....

Stern, T.E., Elwalid, A.I. (1991). Analysis of separable Markov-modulated rate models for information-handling systems. Adv. Appl. Prob. 23, 105-- 139.

....to study such queueing systems via approximating stochastic fluid processes. To a large extent, this view is already taken when people directly introduce Markov modulated fluid models to represent discrete queueing systems, e.g. see Anick, Mitra and Sondhi [1] Mitra [32] Stern and Elwalid [39], Elwalid and Mitra [14] Rogers [37] Kulkarni [24] and references cited there. Nevertheless, it is useful to make the connection between the models clear. Once we see the connection between the fluid models and the associated queueing models, we realize that it may not actually be necessary to ....

....of D D 1 queues in a random environment are often appropriate models for emerging high speed communication networks and manufacturing systems. Thus there is especially strong motivation for the stochastic fluid models in some contexts, e.g. as in Anick et al. 1] Mitra [32] Stern and Elwalid [39] and Elwalid and Mitra [14] Nevertheless, by virtue of the fluid limits, it is possible to represent these fluid models as limits of MMPP M 1 queues and thus apply algorithms for the queueing models to solve the fluid models. This may be counter to intuition until one realizes that Poisson ....

Stern, T. E. and Elwalid, A. I. (1991) Analysis of separable Markovmodulated rate models for information handling systems. Adv. Appl. Prob. 23, 105--139.

....to have a particular sign structure. An analysis without any restriction on the sign structure of the drift vector has been given in the survey paper [12] Background processes (on a finite state space) of a more general type than birth death processes have been considered in many papers, notably [19,20,25,31,14,15,4,17,21]. However, in the birth death context some structural properties prevail which are lost in more general contexts, see [12] As noted above, few results are available in the literature dealing with (variants of) our model when N is countably infinite, as we shall assume in this paper. We know of ....

T.E. Stern and A.I. Elwalid, Analysis of separable Markov-modulated rate models for information-handling systems. Adv. Appl. Probab. 23 (1991) 105-139.

....model may be viewed as a member of a class of models known as fluid queues. In the literature on such models the rate of change of the content of the fluid reservoir is usually determined by the current state of an autonomous stochastic process evolving in the background, see, e.g. 3] 15] and [20]. As far as we know, the model at hand is the first to be analysed in which the dependence works both ways, that is, the evolution of the regulating process is no longer autonomous, but depends on the state of the fluid reservoir (unless 1 = 2 ) An important motivation for studying the present ....

....distribution of (X t , C t ) t # 0 (and related quantities of interest) when the waiting room is unbounded. The analysis amounts to solving a system of di#erential di#erence equations resembling the system of equations appearing in Markov modulated fluid flow models (see, e.g. 3] 8] [20], 21] and [9] An approach similar to that for the infinite credit reservoir does not seem to lead to tractable results when the reservoir has finite capacity. By way of introduction to our approach to this case, we will present, in Section 4, an alternative analysis for the model with infinite ....

T.E. Stern and A.I. Elwalid, Analysis of separable Markov-modulated rate models for information-handling systems. Adv. Appl. Probab. 23 (1991) 105--139.

....component is modulated according to a Markov process evolving in the background. Combining this modelling approach with the uid approximation for information ows leads to Markov modulated uid queues. Fluid models driven by truncated birth death processes has been analyzed by many authors ([2, 6, 11, 16, 17, 19]) In such models, a usual procedure for calculating the distribution of the exact bu er occupancy based on a spectral expansion, as in [8] gives a solution in terms of the eigenvalues and the eigenvectors of the underlying tridiagonal matrix. The diculty associated with this procedure is the ....

....(u) T and R = diag(r 0 ; r 1 ; r N ) and hence, R 1 Q T = 0 B B B B B B B B B B B B B B 0 r 0 1 r 0 0 r 1 1 1 r 1 2 r 1 . N r N 1 N 1 r N N r N 1 C C C C C C C C C C C C C C A (N 1) N 1) 2. 3) Mitra [16] and Stern and Elwalid [19] have shown that R 1 Q T has exactly N negative eigenvalues, N 1 positive eigenvalues and one zero eigenvalue, where N is the cardinality of the set S fj 2 S : r j 0g; and N is that of S fj 2 S : r j 0g: Suppose j ; j = 0; 1; N , are the eigenvalues of the matrix R ....

Stern, T.E. and Elwalid, A.I., \ Analysis of separable Markov- modulated rate models for information-handling systems ". Adv. Appl. Probab., 23(1991), 105-139.

....Markov process. The main reason why the class of MMFMs has attracted so much attention is that they are relevant for modelling certain phenomena in telecommunication networks or otherwise shown to give good approximations for the actual behaviour of network trac ( 1] 6] 7] 10] 11] [14]) In such applications the bursts of data are usually transmitted in many small sized data packets or cells. Here, the use of uid models is particularly useful, since the variations on the cell level are almost neglegible compared to those on the more important burst level. In [2] and [13] the ....

....recursively as follows: B 0 (s) 1 B 1 (s) s r 0 B 0 (s) 0 B n (s) s r n 1 )B n 1 (s) r n 2 r n 1 B n 2 (s) 0; n = 2; N BN 1 (s) s r N )BN (s) r N 1 r N BN 1 (s) 0: 2.8) 3 and dm Fm (0) m 2 S. Mitra [10] and Stern and Elwalid [14] show that R 1 Q T has exactly N negative eigenvalues, N 1 positive eigenvalues and one zero eigenvalue, where N is the cardinality of the set S fj 2 S : r j 0g and N is that of S fj 2 S : r j 0g. That is, j 0; j = 0; 1; N 1; N = 0 and j 0; j = N 1; ....

Stern, T.E. and Elwalid, A.I., Analysis of separable Markov- modulated rate models for information-handling systems. Adv. Appl. Probab., 23(1991), 105-139. 18

....of computation to determine the eigenvalues z i , eigenvectors #(i) and the coe#cients c i . For large systems which are composed of small independent subsystems, a decomposition technique is computationally feasible for calculating all these eigenvalues and eigenvectors (see [16] and also [10] Certain bounds and estimates on the coe#cients are also available (see [16] 2] and [10] However, it is often found that, on the one hand, the amount of computation is considerably, and on the other, most of the terms in the above expansion are insignificant. Calculating the ....

....coe#cients c i . For large systems which are composed of small independent subsystems, a decomposition technique is computationally feasible for calculating all these eigenvalues and eigenvectors (see [16] and also [10] Certain bounds and estimates on the coe#cients are also available (see [16], 2] and [10] However, it is often found that, on the one hand, the amount of computation is considerably, and on the other, most of the terms in the above expansion are insignificant. Calculating the asymptotic behavior of the bu#er overflow probability is more useful for systems with large ....

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T. E. Stern and A. I. Elwalid, "Analysis of separable Markov-modulated rate models for information-handling systems," Adv. Appl. Prob., vol 23 (1991) 105139. 24

....in [21] we consider the case where Q is the rate matrix of birth death process and obtain closed form expressions for the eigenvalues and the left and right eigenvectors. Numerical algorithms are available for determining eigenvalues and eigenvectors when Q is irreducible and time reversible (see [24]) Let Z be the state space of Q and E. The conditions for the reduction of the eigenvalue problem are the following: 1. Z can be decomposed into two classes Z 1 and Z 2 such that no state in Z 1 is accessible from any state in Z 2 and no state in Z 2 is accessible from any state in Z 1 . 2. Z is ....

T. E. Stern and A. I. Elwalid. Analysis of separable Markov-modulated rate models for information-handling systems. Adv. Appl. Prob., 23:105--139, March 1991.

....have shown that the constant ff is usually of the order one. This led many authors to believe that the following simple approximation is true IP[Q x] e Gamma x : 2. 3) Based on this result, admission control policies derived on the concept of effective bandwidth have been developed; see [20, 45, 51, 50, 72, 103, 102]. For this reason equation (2.3) is usually called [25] the effective bandwidth (EB) approximation. However, as discussed in [25] the EB approximation may often be highly inaccurate. This is usually the situation when many sources (N) are multiplexed; in this case it was shown that ff e ....

T. E. Stern and A. I. Elwalid. Analysis of a separable markov-modulated rate model for information-handling systems. Adv. Appl. Prob., 23:105--139, 1991.

.... under multiplexing in the sense that when several independent exponential on off sources are combined, the tail distribution P [W x] still decays exponentially fast [12] Both situations discussed so far are instances of a class of Markov modulated fluid models which has been extensively studied [1, 12, 23, 27] since the seminal work of Kosten [20] A fairly comprehensive theory has been developed for such sources, and algorithms are now available for the numerical evaluation of the tail distribution P [W x] for all values of x [2, 27] On the other hand, the situation is quite different when at least ....

.... modulated fluid models which has been extensively studied [1, 12, 23, 27] since the seminal work of Kosten [20] A fairly comprehensive theory has been developed for such sources, and algorithms are now available for the numerical evaluation of the tail distribution P [W x] for all values of x [2, 27]. On the other hand, the situation is quite different when at least one of the on off sources has heavy tailed on periods. The need for considering such models with heavy tailed components can be traced back to recent measurements of network traffic [21] which exhibit long range dependence and ....

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T. E. Stern and A. I. Elwalid, "Analysis of separable Markov-modulated rate models for information-handling systems", Adv. Appl. Prob., 23:105-139, 1991.

....of these papers consider such models in stationary regime. Anick et al. 1] and Kosten [2] analyzed the fluid model for several on off input sources controlled by a two state homogeneous Markov process. Mitra [3] and [4] generalizes this model by considering multiple on off inputs and outputs. In [5] Stern and Elwalid considered such models for separable Markov modulated rate process which lead to a solution of the equilibrium equations expressed as a sum of terms in Kronecker product form. In [6] Igelnik et al. derive a new approach, based on the use of interpolating polynomials, for the ....

T. E. Stern and A. I. Elwalid. Analysis of separable Markov-modulated rate models for information-handling systems. Adv. Appl. Prob., 23:105--139, 1991.

....the buffer content. This paper, and a new paper of Kosten [36] have contributed much towards establishing the fluid queue fed by on off sources as a key model for representing traffic behaviour in modern communication networks. However, in these studies, and in important successors like [41] and [54], the on and off periods are exponentially distributed or determined by some Markov process, leading to an input process that is short range dependent (see the definition in Subsection 2.1) and to exponential behaviour of the tail of the buffer content distribution. It came as a shock when ....

T.E. Stern and A.I. Elwalid. Analysis of separable Markov-modulated rate models for information-handling systems. Advances in Applied Probability, 23:105--139, 1991.

....to upper and lower bounds on the queue length distributions and to an upper bound on the delay distributions of the GPS system. The rest of the section is organized as follows. In 4. 1, we study a single queue statistical multiplexing system with a modulated service process using the techniques of [EM92, SE91, EM95], which lays the foundation for the lower and upper bound procedures proposed in 4.2 and 4.3. In 4.2, we derive the lower bounds (which are the simpler of the two to derive) and in 4.3 we describe the procedure for obtaining the upper bounds, the proofs of the proposed bounds are relegated to ....

.... the above two expressions, it follows that, as far as the buffer content is concerned, the term (s) Sigma (s) t) can be considered either as a portion of capacity removed from the its maximum value c, or as an additional arrival rate (although the waiting time is different in the two cases) [SE91]. Therefore, the system is equivalent to a statistical multiplexing system with constant service rate c, the input of which is provided by the superposition of the Markov modulated fluid sources (M (a) a) and (M (s) s) Thus, the analysis of the queue length 9 distribution ....

T. Stern, A. Elwalid, Analysis of Separable Markov-Modulated Rate Models for InformationHandling Systems, Adv. Applied Probability, Vol. 23, pp. 105-139, 1991. 36

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T.E. Stern and A.I. Elwalid (1991). Analysis of separable Markov-modulated rate models for information-handling systems. Adv. Appl. Prob. 23 105-139.

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T. E. Stern and A. I. Elwalid, "Analysis of separable Markov-modulated rate models for information-handling systems", Adv. of Appl. Prob. 23, pp. 105-139, 1991.

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T. E. Stern and A. I. Elwalid. Analysis of Separable Markov-Modulated Rate Models for Information-Handling Systems. Adv. Appl. Prob., 23:105--139, 1991.

No context found.

T. E. Stern and A. I. Elwalid. Analysis of separable Markov-modulated rate models for information-handling systems. Adv. Appl. Prob., 23:105--139, 1991.

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T. E. Stern and A. I. Elwalid. Analysis of separable Markov-modulated rate models for information-handling systems. Advances in Applied Probability, 23(1):105--139, March 1991.

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T. Stern and A. Elwalid. Analysis of separable Markov-modulated rate models for information-handling systems. Advances in Applied Probability, 23: 105--139, 1991. 23

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