### BibTeX

@MISC{Lewis_,

author = {P. A. W. Lewis and G. S. Shedler},

title = {},

year = {}

}

### OpenURL

### Abstract

Abstract: Central problems in the performance evaluation of computer systems are the description of the behavior of the system and characterization of the workload. One approach to these problems comprises the interactive combination of data-analytic procedures with probability modeling. This paper describes methods, both old and new, for the statistical analysis of non-stationary univariate stochastic point processes and sequences of positive random variables. Such processes are frequently encountered in computer systems. As an illustration of the methodology an analysis is given of the stochastic point process of transactions initiated in a running data base system. On the basis of the statistical analysis, a non-homogeneous Poisson process model for the transaction initiation process is postulated for periods of high system activity and found to be an adequate characterization of the data. For periods of lower system activity, the transaction initiation process has a complex structure, with more clustering evident. Overall models of this type have application to the validation of proposed data base subsystem models.

### Citations

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(Show Context)
Citation Context ...s in (0, t] as a function of t, which is a nonparametric estimate of the integrated rate function (dotted curve). It is by no means linear, and the Kolmogorov-Smirnov test statistic (see Eqs. (7) and =-=(8)-=- has value 6.048. This, we surmise, is significantly large even if the Poisson hypothesis were not true. In Table 8 we give the successive test statistics U, for successively more complicated exponent... |

302 | The Statistical Analysis of Time Series - Anderson - 1994 |

289 | Introduction to Stochastic Processes - Činlar - 1975 |

222 |
The Statistical Analysis of Series of Events
- Cox, Lewis
- 1966
(Show Context)
Citation Context ...delineated in a subsequent section of this paper) is a nonhomogeneous Poisson process (NHPP). This could be appropriate here since the transaction initiation process is a superposition (Cox and Lewis =-=[5]-=-, Ch. 8; Cinlar [6]) of inputs from a number of sources (users). 3. Because each user’s activity is likely to consist of a (random) number of transactions after initial sign-on, some clustering in the... |

155 | On the Kolmogorov-Smirnov tests for normality with mean and variance unknown - Lilliefors - 1967 |

113 | On some global measures of the deviations of density function estimates - Bickel, Rosenblatt - 1973 |

30 | On the Kolmogorov-Smirnov test for the exponential distribution with unkown meam - Lilliefors - 1969 |

29 |
A cluster process representation of a self-exciting process
- Hawkes, Oakes
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(Show Context)
Citation Context ...the point process analog of an autoregressive system. This would not be inconsistent with our findings, since (linear) self-exciting procesSes are special types of cluster processes (Hawkes and Oakes =-=[25]-=-). One problem with the above interpretation of the cyclic effect is that we would expect more oscillatory effect during high activity periods than during low activity periods. However, just the oppos... |

28 |
Stochastic models for earthquake occurrences
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(Show Context)
Citation Context ...ls of the structure of the low-activity process, but because the intervals are more dispersed than for a Poisson process, there is consistency with a cluster process hypothesis (Lewis [7], Vere-Jones =-=[20]-=- ). Note, too, that a cluster process will look more and more like a Poisson process as activity increases and this is consistent with the finding that the high-activity data are approximately Poisson... |

10 | Some results on tests for Poisson processes - Lewis - 1965 |

10 |
Remarks on the theory, computation and application of the spectral analysis of series of events, J Sound and Vib
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(Show Context)
Citation Context ... events in any finite set of non-overlapping intervals are assumed to be independent random variables. There are other equivalent definitions, and also minimal definitions; see Gnedenko and Kovalenko =-=[21]-=- and Cinlar [ 131. The following theorem (cf. Cinlar [ 131 ) establishes that a homogeneous Poisson process of rate 1 can be obtained by transformation of the time scale of an NHPP, via the inverse of... |

8 | Empirically Derived Micromodels for Sequences of Page Exceptions - Lewis, Shedler - 1973 |

4 |
Superposition of Point Processes,” in Stochastic Point Processes, edited by
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(Show Context)
Citation Context ...nential polynomial rate function the likelihood of n events in the period (0, to] at times r, < r, < . . . < t, is, from Eq. (4), where n sm = ty, m = 0, 1;. ., r. i=l The observations {ti} enter Eq. =-=(6)-=- only through (n, Sti, St:, . . ., St;), and it can be shown from the exponential form of Eq. (6) that these are a set of sufficient statistics for the set of parameters ao, aI, a%; . ., a,. There is,... |

3 | The Statistical Analysis of Dependencies - Cox - 1972 |

3 | Statistical Analysis of Non-Homogeneous Poisson Processes - Brown - 1972 |

2 | Estimation and Testing of an Exponential Polynomial Rate Function - MacLean - 1974 |

1 |
Stochastic Modeling of Processor Scheduline with Application to Data Base Management Systems
- Lavenberg, Shedler
- 1976
(Show Context)
Citation Context ... number of events in the process in the interval (t, t + SI, the assumptions for a NHPP with rate function h(t) are that, as s + 0, Pr{N(s; r) = 0} = 1 - h(t)s + o(s), Pr{N(s: r) = 1) = h(t)s + o(s), =-=(2)-=- and that the random variable N(s; t) is statistically independent of the number and position of events in (0, t] . As a consequence of Eq. (2), Pr{ N ( s; t) 2 2) = o( s) . The survivor function for ... |

1 |
SASE-IV: An Improved Program for the Statistical Analysis
- Lewis, Katcher, et al.
(Show Context)
Citation Context ...nspecified units) of to = P. A. W. LEWIS AND G. S. SHEDLER IBM. J. RES. DEVELOP.s11936.6066. Much of the statistical analysis was done using the experimental SASE-IV program (Lewis, Katcher, and Weis =-=[4]-=-) for analyzing series of events. SASE-IV has a maximum input of 1999 events; this accounts for the length of the period under study. This high system activity period was selected after an initial ove... |

1 |
Non-homogeneous Branching Poisson Processes
- Lewis
- 1967
(Show Context)
Citation Context ...ly to consist of a (random) number of transactions after initial sign-on, some clustering in the data might be expected. An appropriate model here is the nonhomogeneous Poisson cluster process (Lewis =-=[7]-=- ). In this process an initial primary (main) event generates a finite sequence of secondary (subsidiary) events; the complete process is then the superposition of the primary and secondary events, wh... |

1 |
Recent Results in the Statistical Analysis of Unvariate Point Processes,” in Stochastic Point Proce.sses, edited by P
- Lewis
- 1972
(Show Context)
Citation Context ... on the basis of a homoNonhomogeneous Poisson process model The nonhomogeneous Poisson process model for a series of events N, is discussed in a statistical context by Cox and Lewis [5], Ch. 3, Lewis =-=[9]-=-, Cox [lo], and Brown [ 1 I]. A very detailed mathematical account is given in Gnedenko and Kovalenko [ 121 ; a recent treatment is by Cinlar [ 131. Like the homogeneous Poisson process, the nonhomoge... |

1 | Introduction to Queuing Theory, tr. by - Gnedenko, Kovalenko - 1969 |

1 | Fitting Equations to Datu: Computer Analysis of Multifactor Data for Scientists and Engineers - Daniel, Wood |

1 | Empirical Sampling Study of a Goodness of Fit Statistic for Density Function Estimation,” Naval Postgraduate School Report NPS55Lw7.5031 - Lewis, Liu, et al. - 1975 |

1 |
Mutually Exciting Point Processes,” in Stochastic Point Processes, edited by
- Hawkes
(Show Context)
Citation Context ...is a possibility that what we are seeing is the effect of congestion in the system (e.g., DL/I component), and the data may perhaps be best described by something like a self-exciting process (Hawkes =-=[24]-=-), which is the point process analog of an autoregressive system. This would not be inconsistent with our findings, since (linear) self-exciting procesSes are special types of cluster processes (Hawke... |