## Characterizing the BMAP/MAP/1 departure process via the ETAQA truncation (2005)

Citations: | 9 - 6 self |

### BibTeX

@MISC{Zhang05characterizingthe,

author = {Qi Zhang and Armin Heindl and Evgenia Smirni},

title = {Characterizing the BMAP/MAP/1 departure process via the ETAQA truncation},

year = {2005}

}

### OpenURL

### Abstract

We propose a family of finite approximations for the departure process of a BMAP/MAP/1 queue. The departure process approximations are derived via an exact aggregate solution technique (called ETAQA) applied to M/G/1-type Markov processes. The proposed approximations are indexed by a parameter n (n> 1), which determines the size of the output model as n + 1 block levels of the M/G/1-type process. This output approximation preserves exactly the marginal distribution of the true departure process and the lag correlations of the interdeparture times up to lag n − 2. Experimental results support the applicability of the proposed approximation in traffic-based decomposition of queueing networks.

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(Show Context)
Citation Context ...per. Let for be the stationary probability vectors (of dimension ) for states in . For the computation of the stationary probability vector defined by and , matrix-analytic methods have been proposed =-=[14]-=-. Commonly, the subvectors are determined using Ramaswami’s recursive formula [15], which is based on matrix , the key element to matrix-analytic methods and solution of Matrix has an important probab... |

125 |
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Citation Context ... the process first entering through state , given that it starts from state of [14, page 81]. Iterative algorithms are used to calculate , with the cyclic reduction algorithm being the most efficient =-=[9]-=-. To formulate Ramaswami’s formula, we define the matrices where we additionally set . Note that (7) then takes the form . Ramaswami’s formula defines the following recursive relation: . for . . .. (5... |

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Citation Context ...stomers (or packets) may arrive in batches, significantly impacting queueing behavior. Correlated flows with batches are prevalently represented by so-called Batch Markovian Arrival Processes (BMAPs, =-=[11]-=-). Special cases of BMAPs include Poisson processes, phase-type (PH) renewal processes, Markov-modulated Poisson processes (MMPPs), and Markovian arrival processes (MAPs). A MAP is a BMAP with a batch... |

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Citation Context ...even when the finite representation is not a proper MAP. We then deal with a (correlated) matrix-exponential (ME) sequence, by which correlated flows are described in linear-algebraic queueing theory =-=[10]-=- 2 . 2 We did not explicitly state this in [6], although this also holds for the output approximations to the MAP/MAP/1 queue based on ETAQA. 2sRelated work Characteristics of departure processes of B... |

26 |
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Citation Context ...n of the interbatch arrival process with interevent time are given by MAP BMAP (2) MAP MAP BMAP (3) The lag- coefficients of correlation of the (stationary) interbatch arrival process are computed as =-=[13]-=- corr Var MAP BMAP BMAP MAP BMAP where and denote two interbatch times lags apart. In our experiments of Section 4, we will also consider a BMAP correlation structure taking into account the zero inte... |

25 | Decomposition of general queueing networks with MMPP inputs and customer losses. Performance Evaluation 51(2-4
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Citation Context ...e is analyzed in isolation, internal traffic descriptors are constructed for departure processes to be fed into downstream queues as arrival processes. We remark that traffic-based decomposition (see =-=[4]-=- for an introduction) often is the only alternative to simulation of queueing networks, as classic analytic techniques cease to apply in the light of service/arrival correlations and/or batch arrivals... |

14 | Modeling IP traffic using the batch markovian arrival process. Performance Evaluation - Klemm, Lindemann, et al. - 2005 |

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de Liefvoort. Approximation models of feed-forward g/g/1/n queueing networks with correlated arrivals. Perfomance Evaluation
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Citation Context ...ve . In fact, representation (13)/(14) defines a matrix-exponential (ME) process. Such correlated sequences of matrix exponentials are generalizations of MAPs used in linear-algebraic queueing theory =-=[10, 12]-=-. The scope of this paper does not allow us to formally introduce ME processes. In general, matrices related to ME processes lack the (local) physical interpretability of the rate matrices of MAPs. ME... |

12 | Exact aggregate solutions for M/G/1-type Markov processes
- Riska, Smirni
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(Show Context)
Citation Context ...uthors presented a truncation model for the MAP/MAP/1 departure process. This truncation model is based on an aggregation technique for the solution of Quasi-Birth-Death (QBD) processes, called ETAQA =-=[16, 18]-=-. Recall that the (generally non-renewal) departure process of a MAP/MAP/1 queue can be described exactly as a MAP with an infinite number of states, where the QBD process of the MAP/MAP/1 queue is fi... |

10 |
Lag correlations of approximating departure processes of MAP/PH/1 queues
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Citation Context ...arious truncation models, which also capture the interdeparture distribution and the first lag coefficients of correlation of the departure process, have been proposed for single-server queues (e.g., =-=[3, 19, 8]-=-). But to the best of the authors’ knowledge, batch arrival processes have not yet been considered in such traffic-based decomposition techniques. Among these alternative approaches in the literature,... |

10 | MAMsolver: A matrix analytic methods tool
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Citation Context ...es in the system, we also show the ACF of the departure process from the second server, i.e., at point “C” in Figure 1. All analytic results are obtained via MAMSolver, a matrix-analytic methods tool =-=[17]-=-. To assess the quality of the approximations, simulation results are also presented. The simulation space is 100M requests. Each simulation was run 10 times with 10 different random number generator ... |

9 |
Characterizing traffic streams in networks of MAP/MAP/1 queues
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(Show Context)
Citation Context ...arious truncation models, which also capture the interdeparture distribution and the first lag coefficients of correlation of the departure process, have been proposed for single-server queues (e.g., =-=[3, 19, 8]-=-). But to the best of the authors’ knowledge, batch arrival processes have not yet been considered in such traffic-based decomposition techniques. Among these alternative approaches in the literature,... |

8 | ETAQA truncation models for the MAP/MAP/1 departure process
- Heindl, Zhang, et al.
- 2004
(Show Context)
Citation Context ...partially supported by the National Science Foundation under grants ITR-0428330 and CCR-0098278 and ACI-0090221. 1sThe ETAQA approach to characterize departure processes This paper extends results in =-=[6]-=-, where the authors presented a truncation model for the MAP/MAP/1 departure process. This truncation model is based on an aggregation technique for the solution of Quasi-Birth-Death (QBD) processes, ... |

8 | M/G/1-Type Markov Processes: A Tutorial
- Riska, Smirni
(Show Context)
Citation Context ...uthors presented a truncation model for the MAP/MAP/1 departure process. This truncation model is based on an aggregation technique for the solution of Quasi-Birth-Death (QBD) processes, called ETAQA =-=[16, 18]-=-. Recall that the (generally non-renewal) departure process of a MAP/MAP/1 queue can be described exactly as a MAP with an infinite number of states, where the QBD process of the MAP/MAP/1 queue is fi... |

7 | de Liefvoort. The correlation region of second-order MAPs with application to queueing network decomposition - Heindl, Mitchell, et al. - 2003 |

6 | Departure Processes from MAP/PH/1 Queues
- Green
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(Show Context)
Citation Context .... .. if if . (11)s3.1 Exact departure process Starting from the infinitesimal generator (5), we give the exact departure process of a BMAP/MAP/1 queue as a MAP of infinite order. By “filtration” (see =-=[2]-=-), i.e., by collecting in matrix “backward” transitions of that correspond to departures, we arrive at the following MAP representation: . . . . . . .. The infinite order is of course impractical for ... |

4 |
A stable recursion for the steady-state vector in Markov chains of M/G/1 type
- Ramaswami
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(Show Context)
Citation Context ... For the computation of the stationary probability vector defined by and , matrix-analytic methods have been proposed [14]. Commonly, the subvectors are determined using Ramaswami’s recursive formula =-=[15]-=-, which is based on matrix , the key element to matrix-analytic methods and solution of Matrix has an important probabilistic interpretation: an entry in expresses the conditional probability of the p... |

3 |
de Liefvoort. Characterization of the departure process from an ME/ME/1 queue
- Kumaran, Mitchell, et al.
- 2004
(Show Context)
Citation Context ...arious truncation models, which also capture the interdeparture distribution and the first lag coefficients of correlation of the departure process, have been proposed for single-server queues (e.g., =-=[3, 19, 8]-=-). But to the best of the authors’ knowledge, batch arrival processes have not yet been considered in such traffic-based decomposition techniques. Among these alternative approaches in the literature,... |

2 |
Departure processes of BMAP/G/1 queues
- Ferng, Chang
(Show Context)
Citation Context ...tate this in [6], although this also holds for the output approximations to the MAP/MAP/1 queue based on ETAQA. 2sRelated work Characteristics of departure processes of BMAP/G/1 queues are studied in =-=[1]-=-. Algorithms (and explicit formulae) to compute various measures, including the moments and covariances of the interdeparture times, are developed for different types of queues. While general service ... |