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## Analytic solution of stochastic activity networks with exponential and deterministic activities (1993)

Venue: | The University of Arizona |

Citations: | 15 - 0 self |

### Citations

458 |
Iterative solution of large linear systems.
- Young
- 1971
(Show Context)
Citation Context ...as shown in equation 2.5 3.2.4 solveP() Once the P matrix is obtained, the system of linear equations is solved as shown in equation 2.1. For this purpose, the Successive Over Relaxation (SOR) method =-=[37, 39]-=-, which solves the system of linear equations iteratively, is used. In this method, the following equations are solved: ~ A = B X i ~i = 1: The rst equation has many possible solutions. One solution i... |

386 |
Queueing Systems. Volume 1: Theory
- Kleinrock
- 1975
(Show Context)
Citation Context ...al generator matrix (rate matrix) S and an initial state probability distribution. Once S is speci ed, the stationary distribution for the state occupancy probability vector can be found. As shown in =-=[14]-=-, when the Markov process is irreducible and aperiodic, the stationary distribution is the same as the limiting 17sdistribution (also called the steady-state distribution) and is independent of the in... |

337 |
Input versus Output Queueing on a Space Division Packet Switch
- Karol, Hluchyj, et al.
- 1987
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Citation Context ...TRANSMISSION levels. Various switch architectures have been proposed in the literature. To achieve a low loss probability for an arriving cell, di erent bu ering schemes are employed. For example, in =-=[13]-=-, a thorough comparison of input versus output queueing on an N N nonblocking space-division packet switch is presented. In [16], a bu ering scheme is presented for the overload control in which after... |

306 | A class of generalized stochastic Petri nets for the performance analysis of multiprocessor systems
- Marsan, Balbo, et al.
- 1984
(Show Context)
Citation Context ...cess is generated from the constructed model, and generated stochastic process is solved for the desired performance variables using mathematical techniques. Petri nets [31] and stochastic extensions =-=[29, 26, 22, 8]-=-, called stochastic Petri nets (SPNs), were introduced as a graphical framework for model construction and performance variable speci cation. Systems modeled using SPNs can be solved by either analysi... |

154 |
On Petri nets with deterministic and exponentially distributed firing times
- Marsan, Chiola
- 1986
(Show Context)
Citation Context ...he exponential distribution has very high variance, it might be inaccurate to model a deterministic time (which has zero variance) as an exponentially distributed time. M. Ajmone Marsan and G. Chiola =-=[23]-=- developed a mathematical technique in deterministic and stochastic Petri nets (DSPNs), which allowed analytic solution of models with exponentially distributed times along with deterministic times. H... |

135 | Reduced base model construction methods for stochastic activity networks - Sanders, Meyer - 1989 |

113 |
The randomization technique as a modeling tool and solution procedure for transient Markov processes.
- Gross, Miller
- 1984
(Show Context)
Citation Context ...n for the state occupancy probability vector can be found by any of the following methods: 1. Solve the equations: X i S = 0 i = 1: 2. Find the embedded Markov chain subordinated to a Poisson process =-=[12, 11]-=- as follows. Let, s = max j Sii j bP = S s where, b P is the state transition probability matrix for the subordinated discrete time Markov process (henceforth, a discrete time Markov process will be r... |

100 |
Stochastic activity networks: structure, behavior and application
- Meyer, Movaghar, et al.
- 1985
(Show Context)
Citation Context ...and G. Chiola, and improved by Christoph Lindemann, for stochastic activity networks (SANs). SANs are mathematically equivalent to SPNs, but have a hierarchical modeling capability were introduced in =-=[34, 25, 27]-=-. SANs have more modeling power than SPNs since they have constructs called \input gates" and \output gates" which help to model the complex mechanics of a system. The next section brie y reviews SANs... |

98 |
The analysis of non-Markovian stochastic processes, Free Gambr. PkA Soc.,
- Cox
- 1955
(Show Context)
Citation Context ...eterministic activities [5]. Another direction in the research is towards the solution for more than one concurrently-enabled deterministic activities. A method of supplementary variables is shown in =-=[7]-=- to solve this problem. In [10], the numerical solution of this method is presented using the Uniformization technique. Finally, the other direction in the researchistowards the general solution techn... |

83 |
Computing Poisson Probabilities.
- Fox, Glynn
- 1988
(Show Context)
Citation Context ... written as Pi, = ui e QT D = 1X s=0 (s) qT(s) ! D:sTo truncate the in nite series, a small error tolerance >0isintroduced. The left and right truncation points, L and R respectively, are found as in =-=[9]-=- such that, and 1 , L,1 X s=0 RX s=0 qT(s) < 2 qT(s) < 2 : Using these left and right truncation points and considering the fact that the maximum possible value for the elements of (s) is 1, the upper... |

73 |
Performability Modeling with Stochastic Activity Networks,”
- Movaghar, Meyer
- 1984
(Show Context)
Citation Context ...and G. Chiola, and improved by Christoph Lindemann, for stochastic activity networks (SANs). SANs are mathematically equivalent to SPNs, but have a hierarchical modeling capability were introduced in =-=[34, 25, 27]-=-. SANs have more modeling power than SPNs since they have constructs called \input gates" and \output gates" which help to model the complex mechanics of a system. The next section brie y reviews SANs... |

62 |
Extended Stochastic Petri Nets: Applications and Analysis,”
- Dugan, Nicola, et al.
- 1985
(Show Context)
Citation Context ...cess is generated from the constructed model, and generated stochastic process is solved for the desired performance variables using mathematical techniques. Petri nets [31] and stochastic extensions =-=[29, 26, 22, 8]-=-, called stochastic Petri nets (SPNs), were introduced as a graphical framework for model construction and performance variable speci cation. Systems modeled using SPNs can be solved by either analysi... |

57 | Performability modeling with UltraSAN
- Freire, Johnson, et al.
- 1991
(Show Context)
Citation Context ...imple SANs, there may be one-to-one mapping between markings and states and in that case di erent markings can be considered as di erent states. The RBMC technique has been incorporated into UltraSAN =-=[6]-=-. 1.2 Research Objective As mentioned earlier, for analytic solution of SANs, the RBMC technique generates a stochastic process (reduced base model). For solution to proceed, one of the 14snecessary c... |

54 |
On the integration of delay and throughput measures in distributed pro cessing
- Molloy
- 1981
(Show Context)
Citation Context ...cess is generated from the constructed model, and generated stochastic process is solved for the desired performance variables using mathematical techniques. Petri nets [31] and stochastic extensions =-=[29, 26, 22, 8]-=-, called stochastic Petri nets (SPNs), were introduced as a graphical framework for model construction and performance variable speci cation. Systems modeled using SPNs can be solved by either analysi... |

48 |
Analysis of stochastic petri nets by the method of supplementary variables
- German, Lindemann
- 1994
(Show Context)
Citation Context ...nother direction in the research is towards the solution for more than one concurrently-enabled deterministic activities. A method of supplementary variables is shown in [7] to solve this problem. In =-=[10]-=-, the numerical solution of this method is presented using the Uniformization technique. Finally, the other direction in the researchistowards the general solution technique which should be fast enoug... |

46 |
Transient analysis of deterministic and stochastic Petri nets
- Choi, Kulkarni, et al.
- 1993
(Show Context)
Citation Context ...ponents of the state space, the numerical solution is needed for only one of them. Another direction in the research istowards the transient solution for the systems with the deterministic activities =-=[5]-=-. Another direction in the research is towards the solution for more than one concurrently-enabled deterministic activities. A method of supplementary variables is shown in [7] to solve this problem. ... |

45 |
Construction and solution of performability models based on stochastic activity networks
- Sanders
- 1988
(Show Context)
Citation Context ...and G. Chiola, and improved by Christoph Lindemann, for stochastic activity networks (SANs). SANs are mathematically equivalent to SPNs, but have a hierarchical modeling capability were introduced in =-=[34, 25, 27]-=-. SANs have more modeling power than SPNs since they have constructs called \input gates" and \output gates" which help to model the complex mechanics of a system. The next section brie y reviews SANs... |

37 |
An improved numerical algorithm for calculating steady-state solutions of deterministic and stochastic Petri net models. Performance Evaluation
- Lindemann
- 1993
(Show Context)
Citation Context ...egration technique they used as part of the solution procedure was so slow that it restricted the applicability of the technique to very small problems. Using the same basic idea, Christoph Lindemann =-=[18, 19]-=- applied uniformization instead of numerical integration and achieved signi cant improvement in e ciency. This thesis implements the mathematical technique, introduced by M. Ajmone Marsan and G. Chiol... |

16 |
Les Reseaux de Petri Stochastiques et Leur Application a I’evaluation des Systemes Informatiques. These de Docteur Ingegneur, Cnam,
- Natkin
- 1980
(Show Context)
Citation Context |

15 |
Overload control in a finite message storage buffer,
- Li
- 1989
(Show Context)
Citation Context ... arriving cell, di erent bu ering schemes are employed. For example, in [13], a thorough comparison of input versus output queueing on an N N nonblocking space-division packet switch is presented. In =-=[16]-=-, a bu ering scheme is presented for the overload control in which after some threshold (L 2), the bu er is reserved only for the higher priority cells until the number of cells in the bu er goes down... |

8 | Alternative methods for incorporating nonexponential distributions into stochastic Petri nets - Chen, Bruell, et al. |

8 |
Numerical transient solution of finite Markovian queueing systems
- Trivedi
- 1992
(Show Context)
Citation Context ...state occupancy probability vector, and (S) =k , (S) k = the norm of , (S). If Pi, is the ith row of the P matrix and P s i, is the ith row obtained using the steady state detection, then as shown in =-=[28]-=-, k Pi, , P s i, k +2 (S): 2 By selecting (S) =4, an error bound of can be achieved even if the steady state detection is employed. As suggested in [28], at the best, it can be ensured that the norm o... |

8 |
Matrix representations and analytical solution methods for stochastic activity networks
- Tvedt
- 1990
(Show Context)
Citation Context ...as shown in equation 2.5 3.2.4 solveP() Once the P matrix is obtained, the system of linear equations is solved as shown in equation 2.1. For this purpose, the Successive Over Relaxation (SOR) method =-=[37, 39]-=-, which solves the system of linear equations iteratively, is used. In this method, the following equations are solved: ~ A = B X i ~i = 1: The rst equation has many possible solutions. One solution i... |

6 |
chains as an aid in the study of marko processes
- Jensen, \Marko
- 1953
(Show Context)
Citation Context ...n for the state occupancy probability vector can be found by any of the following methods: 1. Solve the equations: X i S = 0 i = 1: 2. Find the embedded Markov chain subordinated to a Poisson process =-=[12, 11]-=- as follows. Let, s = max j Sii j bP = S s where, b P is the state transition probability matrix for the subordinated discrete time Markov process (henceforth, a discrete time Markov process will be r... |

5 | DSPNexpress: A Software Package for the E cient Solution of Deterministic and Stochastic Petri Nets. Performance Evaluation - Lindemann - 1994 |

4 |
Markovian models and numerical analysis of computer system behavior
- Wallace, Rosenberg
- 1966
(Show Context)
Citation Context ...ii j; and (2.12) bP = Q q + I: (2.13) Amultiplicative factor 1.02 in the de nition of q in equation 2.12 ensures that the embedded Markov chain with the transition probability matrix b P is aperiodic =-=[38]-=-. The matrix Q can be written as, Q = q( b P , I):sSo, e QT = e (bP ,I)qT = e bPqT e ,IqT = e bPqT I e ,qT = e bPqT e ,qT : The matrix exponent in the transient analysis can thus be found as in [12, 1... |

3 |
Priority queueing strategies and bu#er allocation protocols for tra#c control at an ATM integrated broadband switching system
- Lin, Silvester
- 1991
(Show Context)
Citation Context ...d for the overload control in which after some threshold (L 2), the bu er is reserved only for the higher priority cells until the number of cells in the bu er goes down to another level L 1 <L 2. In =-=[17]-=-, various priority queueing strategies and bu er allocation protocols for tra c control for the ATM switching system are considered. The use of ATM helps not only to cater to di erent \bandwidth on de... |

2 |
Exploiting Isomorphisms and Special Structures in the Analysis of Markov Regenerative Stochastic Petri Nets
- Lindemann
- 1995
(Show Context)
Citation Context ...x multiplications when R is high but the system can reach the steady-state early. 72s5.2 Further Work Signi cant activities are going on to improve the speed of the solution technique. For example in =-=[20]-=-, the graph analysis is done on the generated state space to detect the isomorphisms and special structures. Once a special structure is detected then the closed form solution can be used and the nume... |

2 |
Sti ness-Tolerant Methods for Transient Analysis of Sti Markov Chains
- Malhotra, Muppala, et al.
- 1994
(Show Context)
Citation Context ...te detection is employed. As suggested in [28], at the best, it can be ensured that the norm of the di erence between the successive iterations is within the bound =4. 5 : 49sHowever, as suggested in =-=[21]-=-, if the rate of convergence is too slow, the change in the elements of the vector between the successive iterations might be smaller than the required error tolerance. In this situation, the steady-s... |

2 |
Kommunication mit atomaten
- Petri
- 1962
(Show Context)
Citation Context ...ytic solution, a stochastic process is generated from the constructed model, and generated stochastic process is solved for the desired performance variables using mathematical techniques. Petri nets =-=[31]-=- and stochastic extensions [29, 26, 22, 8], called stochastic Petri nets (SPNs), were introduced as a graphical framework for model construction and performance variable speci cation. Systems modeled ... |

2 | Design and implementation of a reduced base model construction technique for stochastic activity networks
- Rai
- 1990
(Show Context)
Citation Context ...thod to improve the speed of the solution procedure. 3.1 Changes in Reduced Base Model Construction Technique The existing reduced base model construction (RBMC) technique, as implemented in UltraSAN =-=[32]-=-, generates all the stable markings and the rate of transition from one marking to another when all timed activities in the model are exponentially distributed. As shown in equations 2.10 and 2.11, fo... |

2 |
Simple Decision Rules for Acceptance of Mixed Tra c Streams
- Schoute
- 1988
(Show Context)
Citation Context ...ich can provide capability of exible bandwidth assignment, consists of switching elements which communicate with each other via high speed networks to obtain virtual circuits and transfer data. As in =-=[36]-=-, a switch can be viewed 51sat di erent levels of abstraction. The \CONNECTION" level deals with the call acceptance and rejection, the \ACTION" level deals with the bursty characteristics of the tra ... |

1 |
Optimal Queueing Policies for Fast Packet Switching of Mixed Tra c
- Awater, Schoute
- 1993
(Show Context)
Citation Context ...sic information packet) service time is deterministic. Basic ATM concepts are reviewed in Section 4.1. The particular switch architecture we considered, called the Low Delay OrLow Loss (LDOLL) switch =-=[1]-=-, is described in Section 4.2. The UltraSAN model for this switch is presented in Section 4.3, followed by a discussion of the results obtained using the implementation described in the precious chapt... |

1 |
Performance Improvement ofFast Packet Switching by LDOLL Queueing
- Awater, Schoute
- 1992
(Show Context)
Citation Context ...lated. A system with Q = 10, and N = 4 is also considered for a solution by simulation without burstiness in the tra c. The bursty tra c and variable bit rate tra c are considered using simulation in =-=[2]-=- for Q = 50. 54sIn this study, an analytic solution of the LDOLL switch with Q =16andN =8is considered with bursty tra c of di erent burst lengths and average channel utilization (burst duty cycle). L... |

1 |
Recommendation I.361, ATM layer Speci cation for B-ISDN
- Draft
- 1990
(Show Context)
Citation Context ...performance requirements are discussed. The tradeo between a low loss and a low delay is discussed along with a \short hold" mode for bursty communications. In the CCITT SG XVIII ISDN experts meeting =-=[3]-=-, it was decided that the ATM cell header should have a cell loss priority (CLP) bit. The CLP bit helps to distinguish between two types of cells. Hence two classes of tra c are introduced, one requir... |

1 | Improving The E ciency of The Analysis of DSPN Models - Marsan, Chiola, et al. - 1990 |

1 |
Tadanobu Okada and Kiyohiro Noguchi, \Flow Control Schemes and Delay/Loss Tradeo in ATM Networks
- Ohnishi
- 1988
(Show Context)
Citation Context ... control for the ATM switching system are considered. The use of ATM helps not only to cater to di erent \bandwidth on demand" requirements, but also to cater to di erent performance requirements. In =-=[30]-=-, di erent services which can be o ered by the ATM for di erent performance requirements are discussed. The tradeo between a low loss and a low delay is discussed along with a \short hold" mode for bu... |

1 |
Comparison of Some Tra c Models for ATM Performance Studies
- Ramaswami, Rumsewicz, et al.
- 1991
(Show Context)
Citation Context ...tch: 1. The arrival process, 2. The bu ering policy, and 3. The service policy. The bu ering and service policies were explained previously. Several arrival processes are considered in the literature =-=[33]-=-. For example, the Bernoulli process is an arrival process in which cell arrivals on each input ports are independent and identical. To model burstiness, a two state Markov Modulated Poisson Process (... |