Abstract:
A computer is programmed to fit exponential models to upper percentile subsets of observed measurements for performance metrics collected as attributes of a computer system. The subsets are defined from sets chosen to reduce model bias due to expected variations in system performance, e.g. those resulting from temporal usage patterns induced by end users and/or workload scheduling. Measurement levels corresponding to high cumulative probability, indicative of likely performance anomalies, are extrapolated from the fitted models generated from measurements of lower cumulative probability. These levels are used to establish and to automatically set warning and alert thresholds which signal to (human) administrators when performance anomalies are observed.

Description:
BACKGROUND 
   Detection and management of performance issues in complex computing systems has traditionally been accomplished by applying thresholds that are fixed, against system-specific metric values that are collected over time.  FIG. 1  illustrates a fixed threshold  100  that has been set to value  75  for a metric (e.g. disk reads per second) whose measured value normally varies in a sinusoidal manner depending on the hour of the day, as shown by line  101 . Systems using a fixed threshold make simple arithmetic comparison of current metric value against the fixed threshold and alert administrators when the threshold is exceeded (over or under, depending on metric semantics). In the example shown in  FIG. 1 , such a system generates a false alert at 12 PM when a measurement  102  of the metric is at value  80  even though this value is less than normal (which is shown by line  101 ) for that hour of the day. The system also fails to generate an alert at 12 AM when the measurement  103  of the metric is at value  70  even though this value is greater than normal. 
   In addition to missed alerts and false alerts, systems using fixed thresholds for detection of performance anomalies suffer from a number of other shortcomings. In particular such systems are labor-intensive, error-prone, and subjective. Fixed threshold systems are labor-intensive because extensive configuration (and re-configuration) by administrators is often required to be done manually, to initialize and set up the detection mechanisms. Fixed threshold systems are error-prone in that they fail to adjust to expected fluctuations in performance and frequently either fail to signal real problems or signal falsely. Moreover, fixed thresholds are subjective in that every system must be individually configured, often in the absence of accurate historical information, so administrators must make educated (or arbitrary) guesses. 
   U.S. Pat. No. 6,675,128 granted to Hellerstein on Jan. 6, 2004, entitled “Methods And Apparatus For Performance Management Using Self-Adjusting Model-Based Policies” is incorporated by reference herein in its entirety as background. This patent describes using models of measurement variables to provide self-adjusting policies that reduce the administrative overhead of specifying thresholds and provide a means for pro-active management by automatically constructing warning thresholds based on the probability of an alarm occurring within a time horizon. Hellerstein&#39;s method includes components for model construction, threshold construction, policy evaluation, and action taking. Hellerstein&#39;s thresholds are computed dynamically, based on historical data, metric models, and separately specified policies for false alarms and warnings. Hellerstein describes an example in which a metric model is used to determine the metric&#39;s 95th percentile, for the time interval in which the control policy is being evaluated, which is used as the alarm threshold. Hellerstein does not appear to be interested in using a model to determine very high significance thresholds. 
   U.S. Pat. No. 6,675,128 does not appear to explicitly describe how a metric model is to be constructed. Hellerstein states that a model constructor  230  is used to estimate the values of unknown constants in models based on historical values of measurement data  215 . Hellerstein further states that the operation of component  230  is well understood, as disclosed in the literature on time series forecasting, e.g., G. E. P. Box and G. M. Jenkins, “Time Series Analysis,” Prentice Hall, 1977. 
   SUMMARY 
   A computer is programmed to fit exponential tail models to upper percentile subsets of observed measurements for performance metrics collected as attributes of a system under observation, such as an email application or a database application. The performance metric can be any metric of such a system that is indicative of the system&#39;s performance. 
   The subsets are defined, from a predetermined percentile range (e.g. 95%–99%), in sets of measurements that are obtained by partitioning a time series to reduce model bias due to expected variations in the observed system&#39;s performance, e.g. variations resulting from temporal usage patterns induced by human end users and/or workload scheduling. The time series itself is extracted from measurements being generated by the system under observation, by use of a static or moving time window identified by the administrator as a baseline. 
   The fitted models obtained from the subsets are extrapolated beyond the upper limit of the predetermined percentile range (e.g. to a percentile greater than 99%) to establish and automatically set warning and alert thresholds to levels of high statistical significance (e.g. 3 nines or 4 nines significance) that inform human administrators when performance anomalies are observed in the performance metrics. 
   Exclusion of measurements above the upper limit of the predetermined percentile range during subset definition helps eliminate statistical outliers, and therefore makes the fitted model more trustworthy than in the prior art. Moreover, exclusion of measurements below the lower limit of the predetermined percentile range eliminates the need to model the remainder of the probability density function, whose shape may vary depending on the metric. One embodiment characterizes the bulk of the density function using simple computed percentiles (e.g. 25%, 50%, 75%, 90%, 95%.) 
   Extrapolation of a fitted model beyond the upper limit of the predetermined percentile range eliminates the need to collect and process a large number of measurements that would be otherwise required in the absence of extrapolation to identify values that have the administrator-selected high significance (for use as thresholds). 

   
     BRIEF DESCRIPTION OF THE FIGURES 
       FIG. 1  illustrates, in a graph, use of a fixed threshold by a prior art computer system that generates false alerts, as well as fails to generate alerts when appropriate. 
       FIG. 2A  illustrates a graphical user interface through which a computer programmed in accordance with the invention receives from a human administrator, a periodicity in time that is to be used in partitioning measurements into sets. 
       FIG. 2B  illustrates, in a graph, actual measurements (real data) of a system performance metric (e.g. SQL query executions per second) which exhibits a first pattern across each weekday (e.g. hour of the day), and a second pattern across each week (i.e. weekends decrease significantly as compared to weekdays). 
       FIGS. 3A–3C  illustrate, in flow charts, a method in accordance with the invention that uses an exponential tail to automatically set thresholds for system performance metrics. 
       FIG. 4A  represents fitting, in accordance with the invention, of an exponential decay function  402  to the tail portion  401  of an idealized distribution of measurements. 
       FIG. 4B  illustrates, in a graph, multiple measurements of a system performance metric, from which measurements  402  that found to be in a predetermined percentile range are selected for use in curve fitting. 
       FIG. 4C  illustrates, in a graph, a fitted portion  411  and an extrapolated portion  412  of an exponential tail  410  that models the subset of measurements  402 . 
       FIG. 4D  illustrates, in a graph, with −logQ (wherein Q is [100−percentile]/100) on the x axis, and the measurements (of a system performance metric) on the y axis, the concept of fitting a straight line  422  to the measurements  421 , to determine two parameters (slope and intercept) of an exponential tail model. 
       FIG. 4E  illustrates, in a graph, how a fitted portion of the exponential tail, when extrapolated, identifies a value  434  that corresponds to an administrator-selected significance  433 , for use as a threshold. 
       FIG. 4F  illustrates a graphical user interface through which a computer programmed in accordance with the invention receives from a user a significance level for use in setting a threshold. 
       FIGS. 5A and 5B  illustrate, in block diagrams, hardware and software portions of a computer that performs the method illustrated in  FIG. 3A–3C . 
       FIG. 6  illustrates, in a block diagram, a programmed computer of one illustrative embodiment which implements a parallel implementation that estimates threshold from measurements using an exponential tail. 
   

   DETAILED DESCRIPTION 
   A computer  250  ( FIG. 2A ) is programmed, in some embodiments of the invention, to use measurements of a performance metric of a system, to set thresholds for alerting an administrator (e.g. human) at one or more levels (such as alerts and warnings). The measurements are typically numeric values collected and/or computed at regular time intervals and can thus be thought of as a time series. The measurements are generated by the system under observation asynchronously relative to the methods of setting thresholds and alarming that are performed in computer  250 , i.e. to use the measurements. Depending on the embodiment, the system under observation may be either internal to or external to computer  250 . 
   Such a time series M is expressed as a set of 3-tuples composed of an identifier, a timestamp and a metric value as follows:
 
M1={&lt;id1, t1, v1&gt;, &lt;id1, t2, v2&gt;, . . . , &lt;id1, tn, vn&gt;}
 
(wherein first tuple includes: id1 which uniquely identifies the specific system (such as a specific database) and the specific metric, t1 which represents a specific timestamp and v1 which represents a specific value of the metric). Note that the term “data source identifier” as used herein refers to a single identifier that uniquely identifies each of a specific system (e.g. a database server or an email server) and a specific metric (e.g. SQL executions per second or email messages sent per minute). The above definition of time series M can be extended to include multivariate time series; that is a vector of metric values sharing a common timestamp having been measured simultaneously.
 
   A tuple of a measurement as described above can contain other types of members, e.g. instead of (or in addition to) the timestamp, a member called “Workload Type” is used in an alternative embodiment of the tuple. This alternative embodiment uses measurements that identify the numeric value “v” in addition to the indicator of the kind of work the system was doing at that time, e.g. “OLTP”, “Batch” and “Idle”. In yet another alternative embodiment, the measurements themselves do not identify the type of work (e.g. the tuple could be same as in the previous paragraph), but instead computer  250  is programmed to identify which workload type is associated with each measurement, based on in which time period the measurement&#39;s time falls during which the system under observation was in a given workload state. 
   In some embodiments, all measurements are stored in computer  250  in a single table in a relational database, and each of “id”, “t” and “v” is a column in this table. Note that a single identifier “id” is not used in other embodiments (which use a combination of multiple identifiers, such as a column that identifies an ORACLE database and another column that identifies a specific RAC instance of the ORACLE database). Note that the source identifier “id1” has the same value in all the measurements listed above for metric M1, because it is the identifier of metric M1. For a different metric, the source identifier is different. 
   There are a number of sources for such time series in computer  250  that is programmed with software for the ORACLE Database, version 10 g available from Oracle Corporation, such as (1) V$SYSMETRIC virtual table (2) DBA_HIST_SYSMETRIC_HISTORY view both of which are available in the Server, and (3) MGMT_METRICS_RAW table available in Enterprise Manager (“EM”) Repository. In alternative embodiments, the metric time series is actually a time series of statistical aggregates from a raw data time series. Examples of aggregate time series that are used in a few embodiments are (1) DBA_HIST_SYSMETRIC_SUMMARY in the Server, which is a snapshot-level aggregation of V$SYSMETRIC and (2) MGMT_METRICS — 1 HOUR in Enterprise Manager Repository, which is an hourly aggregation of MGMT_METRICS_RAW. 
   In one embodiment, metrics for which thresholds are computed and set are as follows. 
   
     
       
             
             
           
         
             
                 
             
             
               Metric ID 
             
             
                 
             
           
           
             
                 
               Performance Metrics 
             
             
               2106 
               SQL Service Response Time 
             
             
               2109 
               Response Time Per Txn 
             
             
               2123 
               Database Time Per Sec 
             
             
                 
               Workload Volume Metrics 
             
             
               2003 
               User Transaction Per Sec 
             
             
               2004 
               Physical Reads Per Sec 
             
             
               2006 
               Physical Writes Per Sec 
             
             
               2016 
               Redo Generated Per Sec 
             
             
               2026 
               User Calls Per Sec 
             
             
               2058 
               Network Traffic Volume Per Sec 
             
             
               2103 
               Current Logons Count 
             
             
               2121 
               Executions Per Sec 
             
             
                 
               Workload Type Metrics 
             
             
               2031 
               Logical Reads Per Txn 
             
             
               2045 
               Total Parse Count Per Txn 
             
             
               2066 
               Enqueue Requests Per Txn 
             
             
               2072 
               DB Block Changes Per Txn 
             
             
                 
             
           
        
       
     
   
   In another embodiment, thresholds are set for the following metrics in the manner described herein. 
   
     
       
             
             
           
         
             
                 
             
             
               Metric ID 
               Metric Name 
             
             
                 
             
           
           
             
               2000 
               Buffer Cache Hit Ratio 
             
             
               2003 
               User Transaction Per Sec 
             
             
               2004 
               Physical Reads Per Sec 
             
             
               2006 
               Physical Writes Per Sec 
             
             
               2017 
               Redo Generated Per Txn 
             
             
               2018 
               Logons Per Sec 
             
             
               2019 
               Logons Per Txn 
             
             
               2022 
               User Commits Per Sec 
             
             
               2025 
               User Rollbacks Percentage 
             
             
               2026 
               User Calls Per Sec 
             
             
               2027 
               User Calls Per Txn 
             
             
               2031 
               Logical Reads Per Txn 
             
             
               2034 
               Redo Writes Per Sec 
             
             
               2044 
               Total Parse Count Per Sec 
             
             
               2045 
               Total Parse Count Per Txn 
             
             
               2050 
               Cursor Cache Hit Ratio 
             
             
               2054 
               Execute Without Parse Ratio 
             
             
               2057 
               Host CPU Utilization (%) 
             
             
               2058 
               Network Traffic Volume Per Sec 
             
             
               2066 
               Enqueue Requests Per Txn 
             
             
               2072 
               DB Block Changes Per Txn 
             
             
               2075 
               CPU Usage Per Sec 
             
             
               2076 
               CPU Usage Per Txn 
             
             
               2103 
               Current Logons Count 
             
             
               2106 
               SQL Service Response Time 
             
             
               2107 
               Database Wait Time Ratio 
             
             
               2108 
               Database CPU Time Ratio 
             
             
               2109 
               Response Time Per Txn 
             
             
               2120 
               Executions Per Txn 
             
             
               2121 
               Executions Per Sec 
             
             
               2123 
               Database Time Per Sec 
             
             
                 
             
           
        
       
     
   
   Some data sources (such as a database or other system) produce a measurement&#39;s time series over intervals and thus with two timestamps (a “begin time” and “end time”). In this case the two timestamps are converted by computer  250  into a single timestamp using a midpoint between them. The use of midpoint timestamps in some embodiments is based on the assumption that interval-based time series from a common data source will have equal-sized intervals. When this is not the case, then weighted computations using interval size as the weighting factor are used (in other embodiments) to generate the single timestamp. Still other embodiments use the end time as the representative time stamp for the interval. 
   Computer  250  is programmed in several embodiments to extract certain of the above-described measurements (see act  301  in  FIG. 3A ) based on a “window” of time. The window is specified by an administrator, e.g. “trailing 21 days” through a field  252  (see  FIG. 2A ) in a graphical user interface in some embodiments of computer  250 . The administrator-specified window is used by computer  250  in forming sets of measurements (by partitioning as discussed below) from the administrator-specified time period which is called a “baseline” period. A baseline period B is a period of time over which the behavior of one or more metrics is used for threshold generation as described herein for some embodiments. Specifically, computer  250  is programmed to compare thresholds that are computed from measurements over the baseline period, to measurements observed at a future time. The baseline period is a non-empty period of time expressed as disjoint collection of any number of time intervals:
 
 B={[t   1   ,t   2 ), [ t   3   ,t   4 ), [ t   2n−1   ,t   2n )}
 
   where t j ≦t j+1  and
 
[ t   j   ,t   j+1 )∩[ t   k   ,t   k+1 )=φ if  j≠k  (i.e. non-overlapping)
 
In the above definition for B, the multiple time intervals are expressed as half-closed to ensure that any given timestamp can belong to at most one of the member intervals of a baseline period. Note that any collection of overlapping time intervals can be reconstructed into an equivalent baseline period by merging overlapping intervals. In one embodiment the baseline period consists of a single time interval (such as “trailing 21 days”).
 
   Note that computer  250 , when configured by administrator to use a moving window baseline adapts thresholds to slowly evolving systems (e.g. if 10 email users are being added every month), by computing thresholds using measurements from a window of a fixed length that moves over time, wherein measurements from only the last N days (relative to today) are available for partitioning (N is illustrated in field  252  in  FIG. 2A  as being 21 days). Such a moving window baseline period is automatically and regularly computed by computer  250 , based on the window length specified in the user input in field  252 . Use of moving window baseline periods has the advantage that system evolution over time is automatically incorporated into baseline characterization. 
   A static baseline period is an ad hoc collection of non-overlapping time intervals provided by the user (i.e. administrator). As an example, a static baseline may be selected from a drop-down list by an administrator by clicking one of the predetermined baselines in field  253  in  FIG. 2A , the period between Thanksgiving and Christmas of a particular year (e.g. 2003), which is of particular relevance to e-commerce businesses. A static baseline may be predetermined using knowledge outside the scope of the computer system but with clear meaning for the user (i.e. the human administrator), by clicking on link  254  (labeled “Manage Static Metric Baselines”) which opens a new screen for input therein. An example of such knowledge is a company&#39;s Month End Inventory period, which is a 20-hour period occurring sometime in the last weekend of the month depending on the level of business during the month. Static baseline periods are useful when there is a known time period that is desirable for use as a basis for comparison. 
   Measurements from a metric&#39;s time series M are extracted by computer  250 , from measurements being generated by the system under observation, if their timestamps fall within one of the time intervals in the administrator-selected baseline B. The just-described intersection between the time series M and baseline period B, yields a baselined time series characterized as follows (assuming a 2-tuple representation of each measurement):
 
 M∘B={&lt;t   m   ,v   m &gt;} where &lt; t   m   ,v   m &gt;∈M and  t   j   ≦t   m   &lt;t   j−) ∈B  
 
Hence, a computer of several embodiments is programmed to form a baselined time series for a metric M and baseline B by identifying all 2-tuple s in M whose timestamps lie within one of the time intervals in B.
 
   In one embodiment, a baselined time series is extracted by intersecting time intervals of baseline B with a persisted store of historical measurements of metric M in a SQL relational table. Other embodiments extract data from in-memory sources and/or non-relational formats (e.g. XML). The historical measurement data is characterized in one illustrative embodiment as in a type declaration for [raw measurement data] in Appendix A. This illustrative embodiment instantiates such an historical measurement data store as a simple relational table with one column for each attribute of [raw measurement data]. This embodiment makes use of an abstract characterization of an interval of time, for example as in the type declaration for [time_interval] in Appendix A. Hence, this illustrative embodiment implements relational tables based on the type definitions of Appendix A over which SQL queries of the type outlined in Appendix A are executed to obtain the result sets containing measurement data as intersected with the time intervals, i.e. the baselined time series for this embodiment. Hence, measurement data is extracted in this embodiment only if the measurement&#39;s timestamp lies within one of the time intervals of baseline B. 
   Computer  250  is further programmed to map any measurement of a baselined time series into one of a fixed “set” of values, to implement partitioning (e.g. based on time or events) as follows:
 
f:t m →P where {&lt;t m ,v m &gt;}∈M∘B
 
A number of partitions P are therefore generated from the baselined time series (in acts  302 A– 302 A that are performed in parallel in some embodiments for the respective metrics A-Z), to allow a human administrator (i.e. user) to slice and dice a dataset of the baselined time series, e.g. in a manner similar to the GROUP BY construct in SQL. One example of partitioning functions is hour-of-the-day partition which may be specified in a field  251  in  FIG. 2A . This function “f” maps any timestamp into an integer between 0 and 23. Similarly, the function day-of-the-week maps any timestamp to the strings “Monday” . . . “Sunday”. Note that the product of two partitioning functions is itself a partitioning function. Hence, the function Same hour and day is simply the product of Same hour of the day and Same day of the week. Such partition functions are used by computer  250  to extract measurements of a “set” (over which threshold values are computed) from both static baselines and moving window baselines according to a temporal partitioning scheme specified by the user.
 
   Hence, computer  250  automatically partitions all available measurements of a system performance metric (such as disk reads per second) into a number of sets, based on a predetermined scheme for partitioning the measurements, e.g. based on the time of observation (also called “time partitioning”). Depending on the embodiment, a partitioning scheme may be hard-coded into computer  250 , or supplied by a human (as described in reference to  FIG. 2A ). Hence, such embodiments estimate and apply different thresholds for different sets. New measurements are compared (by a computer in which the thresholds are set which may or may not be same as the computer that sets the thresholds) to thresholds that are estimated from respective sets into which the new measurements would belong, based on timestamps of the new measurements. 
   For example, if a metric normally varies sinusoidally over 24 hours as illustrated in  FIG. 1 , and if the time partitioning is operator-selected to be made by the hour of the day, then the computer automatically selects measurements made during a given hour of each day, across several days, to form one “set.” For example, measurements on each day of the week, Monday-Friday which are made during the lunch hour in the respective days are automatically selected by computer  250 , to form one set. Moreover, measurements made during the midnight hour in the same five days are automatically selected by computer  250  to form another set. In this manner, when the time partitioning is by the hour of the day, a total of twenty-four sets are formed by the computer of such embodiments (because there are 24 hours in a day), and each set contains measurements that are made over a number of days. Note that in order to partition measurements into their respective “sets,” it is necessary to know a timestamp at which each measurement was made (for time-based partitioning). 
   Instead of partitioning a baselined time series into sets using a time-based partitioning scheme, other embodiments may use event-based partitioning schemes (such as when a batch job starts and when the batch job ends). Also, the above-described hour of the day partitioning scheme does not take into account variability in measurements at a larger scale, e.g. measurements during weekdays being higher than measurements during weekends, as illustrated in  FIG. 2B . A time partitioning by the hour of the day over weekdays and weekends accounts for such variability, but requires partitioning the available measurements into a total of 48 sets as follows: 24 sets containing measurements in the weekdays and another 24 sets containing measurements in the weekends. 
   Another time partitioning scheme, for systems whose performance is strongly correlated with employees&#39; work hours, apportions all measurements into just two sets, one set containing measurements during the day (e.g. 8 AM to 8 PM) and another set containing measurements during the night (e.g. 8 PM to 8 AM). The larger scale variability is accounted for if the time partitioning is done by, day and night over weekdays and weekends, which requires a total of 4 sets. Yet another time partitioning, for metrics that are strongly correlated to the different days in a week, is by the day of the week, wherein a total of 7 sets are formed. If partitioning by day and night, per day of the week a total of 14 sets are formed. 
   Note that time partitions that are used in some embodiments are defined by the periodicity of usage of the systems by humans and/or by scheduled jobs. For example, the weekday and weekend partition based on human usage may be implemented in such embodiments as having 5 workweek days and 2 weekend days for normal weeks in the year, and only 4 workweek days and 3 weekend days in weeks that have a long weekend, such as the Memorial Day weekend. Similarly, the day and night partition of some embodiments implements changes made to clocks on account of day light savings. As another example, jobs are also scheduled at periodic intervals, such as every Monday regardless of whether the Monday is a work day or a holiday in a long weekend. 
   Some embodiments may impose a time partition scheme on a time series of measurements using a computer program function as follows. Specifically, a function “TimeGroup” receives as input variables “date-time” and “time-partitionining-scheme” and returns as output a classification of “time-group” of the input “date-time” according to the input “time-partitioning scheme”. Some embodiments call such a function in the context of SQL queries that sample raw time series data to group time series observations for purposes of statistical calculations, e.g. using the SQL GROUP BY clause. Such functions are used as partitioning functions (for example time based or event based) in some embodiments. 
   One illustrative embodiment partitions the baselined time series by a combination of a human operator&#39;s selection for a day grouping and a week grouping, using a scheme that concatenates string tokens representing a day code for the input date-time variable, with string tokens representing a week code for the input date-time variable. Hence, one embodiment supports the following nine schemes for time-based partitioning of the baselined time series: 
                                                             DAY                WEEK   H   N   X                       D   HD   ND   XD           W   HW   NW   XW           X   HX   NX   XX                        
In the above table, daily scheme code values are shown in columns, selected from the set {‘H’, ‘N’, and ‘X’ }, where: ‘H’ means group by hour of day, assign a different code for each hour; ‘N’ means group day hours together and night hours together; and ‘X’ means group times together. Moreover, weekly scheme code values are shown in rows, selected from the set {‘D’, ‘W’, and ‘X’ }; where: ‘D’ means group times by day of week; ‘W’ means group weekdays together and weekends together; and ‘X’ means group all times from all days of week together. One such embodiment allows the operator to select only those schemes from the above table, for which there may be sufficient data in the baselined time series.
 
   Computer  250  when executing the function “TimeGroup” takes as input a timestamp (including the date), and the operator-selected daily grouping and hourly grouping (which may be input as two separate tokens or as a single token depending on the embodiment), and returns an identifier of a “set” to which this input timestamp belongs. In this sentence, the word “token” represents a string or a number (or any other data type) that encodes the partitioning scheme. Hence, inputs to this Function “TimeGroup” are:
 
Date-time input variable=timestamp, e.g. from measurement timestamps Time partitioning input variable=daily scheme code+weekly scheme code
 
Hence, the output of this Function “TimeGroup” is:
 
Output variable=daily group code string+‘:’+weekly group code string
 
   Daily group code string values used in one embodiment are: (1) ‘00’–‘23’ representing the hour of day of the date input variable if the daily scheme code is ‘H’; (2) ‘DY’ representing the daytime group if the date input variable timestamp is between 7 am and 6:59 pm and the daily scheme code is ‘N’; (3) ‘NT’ representing the nighttime group if the date input variable timestamp is between 7 pm and 6:59 am and the daily scheme code is ‘N’; and (4) ‘XX’ when the daily scheme code is ‘X’ 
   In one embodiment the hourly group code string is obtained using a specific format mask with the Oracle TO_CHAR function: 
   
     
       
             
             
           
         
             
                 
                 
             
           
           
             
                 
               Hour code = string representation of: 
             
             
                 
               TO_CHAR(date-time input variable, ‘HH24’) 
             
             
                 
                 
             
           
        
       
     
   
   Weekly group code string values used in one embodiment are: (1) ‘00’–‘06’ representing an encoding of the day of the week for the date input variable timestamp when the weekly scheme code is ‘D’; (2) ‘WE’ representing a weekend group when the date input variable timestamp is between 12:00 am Saturday and 11:59 pm Sunday and the weekly scheme code is ‘W’; (3) ‘WD’ representing a weekday group when the date input variable timestamp is between 12:00 am Monday and 11:59 pm Friday and the weekly scheme code is ‘W’; and (4) ‘XX’ for all values of the date input variable timestamp when the weekly scheme code is ‘X’. 
   In one embodiment the weekly group code string is obtained using a specific format mask with the Oracle TO_CHAR date function: 
                                           Day of week code = string representation of:           TO_CHAR(date-time input variable, ‘D’)                        
Note that such a day of the week code is normalized in one embodiment, to account for differences in day of week numbering in different countries of the world (e.g. the 1 st  day of the week in the US is Sunday but in Great Britain is Monday).
 
   In an example under this coding scheme the input variable value ‘HX’ indicates that the times should be partitioned so as to group together time stamps with the same hour of day and not to make any group separation based on the day of the week. There are 24 possible encodings for this scheme, for example as may be represented by the following set of 5-character strings: 
                                           {           ‘00:XX’, ‘01:XX’, ‘02:XX’, ‘03:XX’, ‘04:XX’, ‘05:XX’, ‘06:XX’,           ‘07:XX’, ’08:XX’, ‘09:XX’, ‘10:XX’, ‘11:XX’, ‘12:XX’, ‘13:XX’,           ‘14:XX’, ‘15:XX’, ’16:XX’, ‘17:XX’, ‘18:XX’, ‘19:XX’, ‘20:XX’,           ‘21:XX’, ‘22:XX’, ‘23:XX’           }                        
Similarly for the coding scheme input variable ‘XD’ there are seven possible 5-character time group codes returned:
 
   
     
       
             
             
           
         
             
                 
                 
             
           
           
             
                 
               { 
             
             
                 
               ‘XX:00’, ‘XX:01’, ‘XX:02’, ‘XX:03’, ‘XX:04’, ‘XX:05’, ‘XX:06’ 
             
             
                 
               } 
             
             
                 
                 
             
           
        
       
     
   
   After such partitioning, the measurements in each set are used to compute basic percentiles and optionally other statistics such as mean and standard deviation for each metric A-Z (see acts  303 A– 303 Z in  FIG. 3A ). Specifically, the measurements are initially sorted in the ascending order of their value (although the descending order may be used in other embodiments), as per act  311  in  FIG. 3B . Any sorting method known in the art may be used. In some embodiments, a sorting method built into the database ORACLE 10 g available from Oracle Corporation is used. The sorted measurements are illustrated by an idealized plot  401  in  FIG. 4A  the probability density (along the y axis) of the measurements, as a function of the value of the measurements (along the x axis). 
   Then, one or more statistics and percentiles (such as minimum, maximum, average, standard deviation, 25 th  percentile, median, 75 th  percentile, 95 th  percentile and 99 th  percentile) are computed and stored persistently in a database, in some embodiments (see act  312  in  FIG. 3B ). Such a computation includes identification of the upper and lower limits of a predetermined percentile range (e.g. 95% and 99%), simply by inspection of the sorted measurements. Specifically, if “n” is the total number of measurements in a set, then 0.95 n (rounded to higher whole number) identifies the rank of the measurement for which the value is at the 95 th  percentile. 
   Next, a subset of measurements  412  ( FIG. 4B ) that happen to fall in the predetermined percentile range are thereafter selected (in act  313  in  FIG. 3B ), for use in curve fitting only a tail  404  (of curve  401 ; see  FIG. 4A ). A “tail” of a distribution conventionally refers to the extreme regions of the distribution—both left and right. However, most embodiments of the invention use only the right tail (i.e. at the high end of the metric) of an exponential distribution which is defined by the following function (which is valid only for x≧0 and is at value 0 for x&lt;0):
 
 f ( x )=λ e   −λx .
 
A cumulative function F for the function f in the above equation is as follows (which is valid only for x≧0 and is at value 0 for x&lt;0):
 
 F ( x )=1 −e   −λx .
 
Note that alternative embodiments may use any long-tailed probability density function f(x) that is bounded by two exponentially decaying functions as follows:
 
 ae   −ax   ≦f ( x )≦ be   −bx  
 
wherein there exists some x 0  for which this inequality holds for all x&gt;x 0 .
 
   Regardless of which function is used to model a tail, several embodiments eliminate the need (during curve fitting) to model the remainder of the probability density function (i.e. outside of tail  404 ). Specifically, a remainder that is below the predetermined percentile range is different for each of the following distributions: Exponential Distribution, Weibull Distribution, Lognormal Distribution, and Gamma Distribution. An engineering approximation is made as illustrated by the example shown in  FIG. 4B . Specifically, a bottom end of a percentile range, which is used to identify a subset of measurements, is predetermined to be sufficiently high, e.g. 95%. Hence, measurements  411  ( FIG. 4B ) whose values are within the bottom 95% in the set are excluded from the subset to be used for curve fitting. In an example with one thousand measurements in a set, the bottom nine hundred fifty measurements are excluded. 
   Moreover, in the example of  FIG. 4B , a top end of the percentile range is selected to be 99%, which means that measurements  413  whose values are within the top 1% in the set (also called “outliers”) are excluded from the subset. In the example of 1000 measurements, the top 10 measurements (i.e. up to 10 outliers) are excluded. Therefore, the subset contains the remaining measurements  412 , which are forty in number. Note that 1020 measurements per partition are accumulated in 17 days, if a measurement is made once a minute (60 measurements/hour), and if the baselined time series is partitioned by the hour of the day. 
   Hence, a top end (e.g. 99%) of a percentile range to be used in curve fitting (a top end of tail portion  404 M; e.g. at  2500  in  FIG. 4C ) is predetermined to minimize the potential for inclusion of statistical outliers (i.e. abnormal values) in measurements to which a curve is to be fitted, increasing the accuracy of the estimate. A bottom end (e.g. 95%) of the percentile range is predetermined to exclude a large number of (more than a majority of) measurements whose use would require curve fitting of a complete probability distribution function (which may be arbitrarily complex depending on the metric, and not necessarily exponential). 
   In using only a predetermined percentile range of measurements, a large number of measurements remain unused, and moreover the number of measurements used becomes small. For example, if a measurement is generated once every 5 minutes, then there are 12 measurements in each hour, and 85 days are required to accumulate  1020  measurements. For such a metric, a lower bottom end (e.g. 85%) is used in some embodiments for the predetermined percentile range, to increase the number of measurements in the predetermined percentile range which in turn reduces the size of the set. In an alternative embodiment, a coarser time partitioning is used, e.g. group by day-night, there are only two sets and 1000 measurements are accumulated in a week (for 5 minute interval measurements). Pseudocode for selecting a subset of measurements, for use in curve fitting, is illustrated in Appendix B. 
   After identifying measurements in the subset, computer  250  automatically fits a curve of a predetermined shape to these measurements (see act  314  in  FIG. 3B ), and thereafter stores persistently in a database, the parameters obtained from curve fitting (see act  315  in  FIG. 3B ). The inventors note that many computing system performance metrics exhibit an exponential distribution which has an exponentially decaying tail, and even if exhibiting a non-exponential statistical distribution nonetheless have a region that can be modeled by (or approximated by) an exponentially decaying tail. 
   Hence, as discussed below, an exponentially decaying tail is fitted in many embodiments, to which one or more of the following apply: a) performance issues in otherwise stable systems are unusual, and occur with unexpected frequency; b) performance issues in otherwise stable systems are associated with unusual observations in system performance metrics; c) measurements of system performance metrics, either in raw form or through a transform, have ranges with one-sided tails, e.g. ranging from zero to a large unknown maximum value; d) stable systems often exhibit significant yet expected variations in performance over predictable time periods (e.g. between online and batch processing cycles); e) stable systems evolve over time and this evolution is reflected as changes to expected distributions of measurements of system performance metrics. 
   In several embodiments, the curve being fitted models a portion  404 M ( FIG. 4C ) of exponentially decaying tail  404 . As discussed later, another portion  404 E ( FIG. 4C ) of this same tail  404  which is obtained by extrapolation is used to identify thresholds in accordance with the invention. Fitting of measurements to such a model (per act  314  in  FIG. 3B ) involves determining two parameters namely a decay factor of tail  404  and a location of the tail  404 . 
   As tail  404  decays exponentially, such fitting may be conceptually understood as follows: generate Q=1-percentile for each of measurements  412  in the selected subset, convert Q into the logarithmic domain, and fit the −log Q of measurements  412  to a straight line  432  ( FIG. 4D ). Since a straight line  432  is represented by two unknowns, namely slope 1/λ and intercept β, one may solve for these two unknowns by use of any two of the converted measurements  412 . 
   The two parameters that identify an exponentially decaying tail  404  ( FIG. 4C ) can be computed in computer  250  in any manner well known in the art, e.g. by use of a least squares method as described next. In some embodiments, an ordinary least squares regression fit is performed in a graph of measurement x k  along the ordinate versus −log (Q), along the abscissa as shown in  FIGS. 4D and 4E , wherein Q=(1−k/n), to provide the two parameters of the fitted tail, namely (A) a point at the smallest measurement in the subset x m  and (B) the slope of a line passing through this point. As noted above, the highest values of measurement X (e.g. in the top 1%) in each set were excluded from the subset since these may be due to unusual events (i.e. statistical outliers). Moreover, a majority of the values of measurement X that comprise the remainder of the distribution of measurements are also excluded, e.g. below 95%. Thus, fitting of −log(1−k/n) along the abscissa, versus x k  along the ordinate, is performed within the range m≦k≦l, wherein m≦l≦n. 
   x k —k is the rank of measurement x when sorted in ascending order 
   X—measurement 
   n—highest rank (total number of measurements in a set) 
   m—value of k where the tail&#39;s fitting starts, 0.95*n (in this example) 
   I—value of k where the tail&#39;s fitting ends, 0.99*n 
   Note that the above-described curve fitting method using least squares has certain problems. First, the log-linear relationship is based on an asymptotic approximation and is only valid for large n. A more serious problem is that basic linear regression theory assumes that the abscissa values being fitted are independent. This is not the case for many metrics, because the x k  are correlated. To remedy this problem, the following formulae are programmed into computer  250  of some embodiments, to yield estimates of (B) the slope 1/λ (also referred to as μ) of the fitted line, and (A) the point as determined by log(1−m/n) on the abscissa and β on the ordinate (through which point the line passes): 
   
     
       
         
           
             
               
                 
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   Several embodiments compute the above-listed model parameters in equations (1) and (2) from the above-described subset of measurements in the predetermined percentile range, as illustrated in the pseudo-code in Appendix B below, that is incorporated by reference herein in its entirety. As shown in Appendix B, such embodiments implement a function “exponential_tail_statistics” to compute statistics using as input a variable of type “observation set” and returning as output variable of type “statistics set.” Specifically, one embodiment uses a nested table of Oracle objects as the input type, as illustrated by the type definitions in Appendix B. 
   Such a function is essentially a transform of a set of raw data measurements (in one embodiment having an embedded time partitioning group code) into a set of statistics computed over the time groups identified by code values. One embodiment accomplishes this transformation in a single SQL SELECT statement organized with the following pseudo-code structure, where the quantities being computed are expressions derived from the equations (1) and (2) above, and the Goodness of Fit Formula from inequality (4) above. 
   For purposes of illustrating the efficacy of the invention, the measurements  411  ( FIG. 4B ) which were excluded from curve fitting are converted into the logarithmic domain and shown in  FIG. 4E  as the respective converted measurements  441 . As can be seen from  FIG. 4E , straight line  432  obtained by curve fitting as described in reference to  FIG. 4D  provides a good fit for many measurements even outside of the subset of selected measurements used in curve fitting. Hence, line  432  is used to identify one or more thresholds as discussed next. 
   A fitted curve (exemplified by line  432  in  FIG. 4E ) is used in many embodiments as the basis for establishing statistically significant percentile values by extrapolation (see act  316  in  FIG. 3B ), such as the value of the metric at 3 nines and at 4 nines significance. For example, in some embodiments computer  250  elicits from the user, through a field  499  ( FIG. 4F ) of a graphical user interface, a significance level to be used in setting a threshold. Thereafter, computer  250  transforms this significance level into the metric&#39;s value, using the fitted curve. Conceptually, by going to the logarithmic domain, one can take a measurement&#39;s log to yield a value  443  and apply value  443  at point  6 . 91  on the x axis in  FIG. 4D  to identify a corresponding value  444  at point  3500  on the y axis which is the percentile value. Computer  250  of many embodiments uses the above-described equations (1) and (2) to compute the two unknown parameters of the fitted curve namely slope 1/λ and intercept β (also called “model parameters”). 
   Value  444  that is obtained from using line  432  is thereafter stored by computer  250  (as per act  317  in  FIG. 3B ) as a threshold for use in notifying a human operator. Next, computer  250  checks if all sets obtained by time partitioning in act  301  have been processed in the above-described manner. If not, then control transfers to act  302 . If all sets have been processed in this manner, then the computer  250  waits (as per act  305 ) for a predetermined duration between intervals (over which the measurements in a set are accumulated), followed by returning to act  301 . The predetermined duration is selected to ensure that at least one new measurement has been added to each set. In the above-described example of time partitioning by the hour of the day, the computer waits for a day (24 hours), because there are 24 sets. 
   In the example illustrated in  FIG. 4E , a significance level of 0.999 (1 in 1000, or 3 nines) has the logarithmic value approximately 6.91 (i.e. value  443 ), which yields  3500  (i.e. value  444 ) as the threshold. Use of a fitted exponential tail as described above (i.e. extrapolation) enables establishing thresholds for statistical significance levels that cannot be determined by use of percentiles computed over available measurements. Several embodiments compute an estimate x 0.9999  of a system performance metric for four nines i.e. 0.9999 (1 in 10,000) significance level, based on a subset containing far less than 10,000 measurements. Hence, use of a fitted exponential tail eliminates the need for large numbers of measurements which otherwise require a large amount of time to accumulate. 
   Significance level is expressed in several embodiments in terms of “the number of 9&#39;s”, i.e. a number of 9s following the decimal point in writing the value of a probability p. For any general value p between 0 and 1, the value of p expressed in units of 9s is given by −log 10 (1−p). For α=1−(m/n) which is the fraction at which tail fitting starts, (e.g. α=0.05 when m is the rank at 95 th  percentile), the standard deviation σ of the error in 3 9&#39;s estimate, i.e. the standard error is 1.70/√{square root over (l−m)} while that of the 49&#39;s estimate is 2.70/√{square root over (l−m)}. Note that the values 1.70 and 2.70 are obtained as follows. The standard error σ in units of 9s of estimation of x p  (the pth level of significance), such as an estimated x 0.999  is approximated by the following: 
             1       l   -   m         ⁢       log   ⁡     (     α     1   -   p       )         log   ⁢           ⁢   10             
To consider the number of measurements in a set to be sufficient for 3 9s and 4 9s estimations of the type described herein, some embodiments ensure that there is at least one sigma separation between two adjacent bands of variability around estimates, e.g. a first band, in the variability for a 3 9s estimate, and a second band in the variability for a 4 9s estimate, i.e. 2a &lt;1 (where the value 1 and the value of σ are both in units of 9s). To satisfy this condition, for reasonable 3 9s estimates these embodiments use a sample size of at least 290 measurements and for 4 9s estimates at least 730 measurements. One embodiment uses 700 (i.e. seven hundred) measurements in each set (as shown in Appendix A below), for both three 9s and four 9s estimates used as thresholds.
 
   As noted above, in several embodiments, such values with high significance (3 9s and 4 9s) are stored persistently in a database (as per act  317  in  FIG. 3B ). Next, computer  250  computes and sets thresholds (as per act  304  in  FIG. 3A ) in a process that is independent of the above-described process for computing and storing the percentiles and statistics (as per acts  303 A– 303 Z). Specifically, in act  304 , computer  250  of some embodiments performs acts  311 – 315  as discussed next. In act  311 , the computer waits for a period specified by granularity for setting thresholds (e.g. 1 hour or on a specified event). 
   Next, the computer  250  uses the information previously persisted in the database (i.e. in acts  312 ,  315  and  317 ), and user-specified parameters (e.g. critical at 4 nines based on exponential tail coefficients, or warning at 115% of 2 nines significance level based on ranking) to compute the thresholds. Thereafter, computer  250  invokes an assessment function (see act  313  in  FIG. 3C ) to evaluate the quality of the model parameters. 
   When performing the assessment function, computer  250  decides on whether specific estimated exponential thresholds are sufficient to be used as the basis for alerting. The assessment function in some embodiments is based on a measure of goodness of fit of the fitted exponential tail to the actual measurements, as well as the number of measurements in the subset (called “cardinality”) used in curve fitting. When either goodness of fit or cardinality are insufficient to ensure reasonable confidence in the estimate (as evaluated by application of one or more predetermined rules), the computer of such embodiments is programmed to either unset or not set alert thresholds using these statistics (in accordance with previously specified user preference). Appendix Z below provides pseudocode for an assessment function that is used in some embodiments of the invention. 
   Some embodiments of computer  250  compute a value for the mean μ=1/λ twice for two different portions of the exponential tail, from two different subsets of measurements in two different predetermined percentage ranges. For example, one value for the mean is computed based on measurements in the percentile range 95–97%, whereas another value for the same mean is computed based on measurements in the 97–99% range. To the extent that these two values for the mean agree with one another (e.g. within a predetermined tolerance), the exponential tail identified from the measurements is deemed to be a good fit, and used to set thresholds. If μ 1  and μ 2  are the two estimates (called “half-tail” estimates) of p based on two halves of a subset of measurements, and then their average ½ (μ 1 +μ 2 ) is an overall estimate for μ obtained from the entire subset (called “full tail” estimate). If an exponential distribution applies, the central limit theorem indicates that both μ 1  and μ 2  are independent and normally distributed with mean p and variance (2/(l−m)) μ 2  because ½ (l−m) points are used in each half-tail estimate. Note that μ 2  is mathematically derivable from μ and μ 1 , and is derived therefrom in some embodiments. 
   Hence, one such embodiment uses a chi-squared statistic as a measure of goodness of fit. The smaller the value of this statistic, the better the fit. In this embodiment, the chi-squared statistic value when set to, for example, 3 or 4, results in an acceptance rate of (i.e. a rejection confidence) of 91.7% and 95.4% respectively, for the tail fitting that has been done (i.e. the estimated exponential tail parameters). When using the rejection confidence of 4, such embodiments may use the inequality formula (4) listed above as a criterion, wherein the single bar denotes absolute value. Thus if the sample size of measurements used in curve fitting is (l−m)=40 (i.e. if there are 1000 measurements in a set and the tail is being fitted in the range 95%–99%) these embodiments reject the exponential tail parameters that have been estimated if a half-tail estimate of the mean differs from the full-tail estimate of the mean by more than 32%. 
   On completion of the assessment function, computer  250  has decided whether or not the model parameters are acceptable. If acceptable, the computer  250  sends thresholds (see act  314  in  FIG. 3C ), to a program that generates alarms and/or warnings (such as the Enterprise Manager Agent or the Database server). Note that a threshold is updated, in some embodiments, only if the threshold has changed (i.e. un-changed thresholds are not updated). If not acceptable, computer  250  performs a user-specified action for this situation, e.g. either preserve the previous thresholds or unset previous thresholds (see act  315  in  FIG. 3C ). 
   On completion of percentile computation and setting of thresholds (as per acts  303 A– 303 Z and  304 ), computer  250  waits for a preset duration (as per act  305  in  FIG. 3A ) or for a preset event to occur (e.g. the time becomes the top of the hour), and thereafter checks whether or not a moving window is in use. If a moving window is in use, computer  250  returns to acts  302 A– 302 Z so as to re-partition the measurements, followed by curve fitting to obtain percentiles (as per acts  303 A– 303 Z and  304 – 306 ). If a moving window is not in use, computer  250  returns to act  304  to re-compute the thresholds. 
   Note that application of the thresholds to measurements is not shown in  FIGS. 3A–3C  because this activity is performed asynchronously, relative to the methods of  FIG. 3A–3C . In the example illustrated in  FIGS. 4A and 4D , if measurements  404  and  405  are obtained after value  434  is set as the threshold, then they would be identified as statistically significant events, because they exceed value  434 . 
   Hence, several embodiments of the invention identify metric values that are “unusual” as potential indicators of problems to be alerted about. Here, unusual means statistically significant and not just large in some arbitrary sense. In such embodiments, the computer is programmed to automatically fit previous metric values to the exponential tail of an exponential distribution that is then used to determine the statistical significance of future observations for alerting purposes. Alert thresholds are implemented in the programmed computer based on statistical significance levels at different orders of magnitude, and typical values are: 
   WARNING=0.999 (“three nines”) 
   CRITICAL=0.9999 (“four nines”). 
   Hence, a fitted exponential tail is used to find thresholds (in the same units as the measurements) for the just-described two statistical significance levels in some embodiments, and the thresholds are used to generate and display alerts to system administrators in the usual manner (e.g. as email messages or messages in an application that shows each message in a single line in an array of such lines). In certain embodiments, values at such statistical significance levels are used as boundaries of predetermined ranges of percentiles for use in generating a graphical display over time, as described in the concurrently filed co-pending patent application entitled “GRAPHICAL DISPLAY AND CORRELATION OF SEVERITY SCORES OF SYSTEM METRICS”, by John M. Beresniewicz, Amir Najmi and Jonathan F. Soule. 
   The method illustrated in  FIGS. 3A–3C  is used to program a computer system  500  of the type illustrated in  FIG. 5A  which is discussed next. Specifically, computer system  500  includes a bus  502  ( FIG. 5A ) or other communication mechanism for communicating information, and a processor  505  coupled with bus  502  for processing information. Computer system  500  also includes a main memory  506 , such as a random access memory (RAM) or other dynamic storage device, coupled to bus  502  for storing information and instructions to be executed by processor  505 . 
   Main memory  506  also may be used for storing temporary variables or other intermediate information during execution of instructions to be executed by processor  505 . Computer system  500  further includes a read only memory (ROM)  508  or other static storage device coupled to bus  502  for storing static information and instructions for processor  505 . A storage device  510 , such as a magnetic disk or optical disk, is provided and coupled to bus  502  for storing information and instructions. 
   Computer system  500  may be coupled via bus  502  to a display  512 , such as a cathode ray tube (CRT), for displaying information to a computer user. An input device  514 , including alphanumeric and other keys, is coupled to bus  502  for communicating information and command selections to processor  505 . Another type of user input device is cursor control  516 , such as a mouse, a trackball, or cursor direction keys for communicating direction information and command selections to processor  505  and for controlling cursor movement on display  512 . This input device typically has two degrees of freedom in two axes, a first axis (e.g., x) and a second axis (e.g., y), that allows the device to specify positions in a plane. 
   As described elsewhere herein, transportation planning is performed by computer system  500  in response to processor  505  executing one or more sequences of one or more instructions contained in main memory  506 . Such instructions may be read into main memory  506  from another computer-readable medium, such as storage device  510 . Execution of the sequences of instructions contained in main memory  506  causes processor  505  to perform the process steps described herein and illustrated in  FIGS. 3A–3C . In alternative embodiments, hard-wired circuitry may be used in place of or in combination with software instructions to implement the invention. Thus, embodiments of the invention are not limited to any specific combination of hardware circuitry and software. 
   The term “computer-readable medium” as used herein refers to any medium that participates in providing instructions to processor  505  for execution. Such a medium may take many forms, including but not limited to, non-volatile media, volatile media, and transmission media. Non-volatile media includes, for example, optical or magnetic disks, such as storage device  510 . Volatile media includes dynamic memory, such as main memory  506 . Transmission media includes coaxial cables, copper wire and fiber optics, including the wires that comprise bus  502 . Transmission media can also take the form of acoustic or light waves, such as those generated during radio-wave and infra-red data communications. 
   Common forms of computer-readable media include, for example, a floppy disk, a flexible disk, hard disk, magnetic tape, or any other magnetic medium, a CD-ROM, any other optical medium, punch cards, paper tape, any other physical medium with patterns of holes, a RAM, a PROM, and EPROM, a FLASH-EPROM, any other memory chip or cartridge, a carrier wave (such as an electromagnetic wave) as described hereinafter, or any medium from which a computer can read. 
   Various forms of computer readable media may be involved in carrying one or more sequences of one or more instructions to processor  505  for execution. For example, the instructions may initially be carried on a magnetic disk of a remote computer. The remote computer can load the instructions into its dynamic memory and send the instructions over a telephone line using a modem. A modem local to computer system  500  can receive the data on the telephone line and use an infra-red transmitter to convert the data to an infra-red signal. An infra-red detector can receive the data carried in the infra-red signal and appropriate circuitry can place the data on bus  502 . Bus  502  carries the data to main memory  506 , from which processor  505  retrieves and executes the instructions. The instructions received by main memory  506  may optionally be stored on storage device  510  either before or after execution by processor  505 . 
   Computer system  500  also includes a communication interface  515  coupled to bus  502 . Communication interface  515  provides a two-way data communication coupling to a network link  520  that is connected to a local network  522 . Local network  522  may interconnect multiple computers (as described above). For example, communication interface  515  may be an integrated services digital network (ISDN) card or a modem to provide a data communication connection to a corresponding type of telephone line. As another example, communication interface  515  may be a local area network (LAN) card to provide a data communication connection to a compatible LAN. Wireless links may also be implemented. In any such implementation, communication interface  515  sends and receives electrical, electromagnetic or optical signals that carry digital data streams representing various types of information. 
   Network link  520  (not shown in  FIG. 5A ) typically provides data communication through one or more networks to other data devices. For example, network link  520  (not shown in  FIG. 5A ) may provide a connection through local network  522  to a host computer  525  or to data equipment operated by an Internet Service Provider (ISP)  526 . ISP  526  in turn provides data communication services through the world wide packet data communication network  528  (not shown in  FIG. 5A ) now commonly referred to as the “Internet”. Local network  522  and network  528  (not shown in  FIG. 5A ) both use electrical, electromagnetic or optical signals that carry digital data streams. The signals through the various networks and the signals on network link  520  (not shown in  FIG. 5A ) and through communication interface  515  (not shown in  FIG. 5A ), which carry the digital data to and from computer system  500 , are exemplary forms of carrier waves transporting the information. 
   Computer system  500  can send messages and receive data, including program code, through the network(s), network link  520  and communication interface  515 . In the Internet example, a server  550  might transmit a mission (which is part of a transportation plan) through Internet  528  (not shown in  FIG. 5A ), ISP  526 , local network  522  and communication interface  515 . 
   The instructions for performing the methods of  FIGS. 3A–3C  may be executed by processor  505  as they are received, and/or stored in storage device  510 , or other non-volatile storage for later execution. In this manner, computer system  500  may obtain the just-described instructions and any related data in the form of a carrier wave. 
   Note that  FIG. 5A  is a very low-level representation of many hardware components of a computer system. Several embodiments have one or more additional software components in main memory  506  as shown in  FIG. 5B : Operating System  591  (e.g. Microsoft WINDOWS 2000), Database Server  595  (e.g. Oracle Server v9i2 for the source computer; e.g. Oracle Server v8i for the target computer), Java Development Kit  593  (e.g. JDK v118), graphing program  592  (e.g. Scalable Vector Graphics), and JDBC drivers  594  (e.g. JDBC driver available from Oracle Corporation). 
   One embodiment in accordance with the invention has the following advantages over prior solutions: a) detects performance issues and raises alerts based on statistically significant events, rather than arbitrary or subjective thresholds (and hence this embodiment is superior at detecting truly abnormal situations that may deserve attention); b) is based on sound statistical principles, rather than simplistic arithmetic comparisons; c) is adaptive to both regular expected variations as well as gradual system evolution over time, rather than requiring manual intervention to effect adjustments; d) is simpler to configure as input parameters are metric-independent and thus do not depend on detailed knowledge by users of underlying metrics; e) is robust in that exponential tail modeling can produce reasonable estimates of non-exponential long-tailed distributions and can be computed over relatively sparse sample sizes. 
   The just-described embodiment also provides a) superior alerting: dynamic statistical baselines are expected to significantly improve the accuracy of performance alerting while also reducing exposure to the false positives commonly incurred under fixed threshold schemes; b) improved manageability: fixed thresholds induce management overhead that is proportional to both the number of targets and the number of performance metrics monitored (statistically determined thresholds using dynamic baselines can be configured with a few decisions applied over many targets and metrics); c) technology neutral: the statistical techniques introduced by the project are technology neutral with respect to the monitored target (the functionality is designed as a service that can be leveraged within Oracle Enterprise Manager across targets); d) customer acceptance: customers easily understand the basic concepts and recognize the value provided by self-adjusting statistical thresholds using dynamic baselines; e) market leadership: some smaller vendors have begun to adopt similar techniques. 
   Some embodiments of the invention contain an implementation of estimator computation, as illustrated in Appendix C below. Such embodiments implement a function “extract_compute_statistics” that accepts streams of raw data measurements as input (e.g. as a cursor defined over a table of persisted measurements in order of data source identifier, i.e. one metric&#39;s time series after another metric&#39;s time series) and returns the exponential tail parameters and other statistics computed over groups defined by data source and a group code (e.g. as produced by time partitioning functions as discussed above). In one embodiment such a function takes the form of an Oracle table function with a cursor variable input type and returns a nested table of statistics object type as output. Appendix C illustrates, in pseudo-code, processing logic implemented in certain embodiments. 
   One such embodiment, illustrated in  FIG. 6 , implements a parallelized version of the processing logic. This embodiment declares the table function such that it can execute in parallel in separate Oracle processes using Oracle parallel query features. In this embodiment each computing process generates all statistics for each of some data sources (i.e. some metrics) represented in the input measurement stream. From among all measurements generated by the system under observation (and held in a table  601  in computer  250  shown in  FIG. 6 ), the measurements for a given data source identifier (e.g. a given database and a given metric) are directed to the same compute process, via a parallel-executing cursor  602  (invoked in computer  250 , as shown in  FIG. 6 ). For example, if there are two processes, a first process generates statistics for all sets of the metric disk reads per second from database “A” and a second process generates statistics for all sets of the metric SQL queries per second from database “B”. 
   Specifically, the baseline statistics (including exponential tail estimates) are computed (in steps  603  and  604  shown in  FIG. 6  and performed by computer  250 ) using technologies that allow computation to be spread in parallel across multiple CPUs and even across nodes, e.g. in an Oracle RAC configuration. The statistical computation itself likewise takes advantage of features that enable statistics (such as minimum, maximum, average, standard deviation, and basic percentiles) to be computed in a single SQL SELECT statement. The parallel implementation uses the two PL/SQL functions (illustrated by steps  603  and  604  in  FIG. 6  and described in Appendices B and C) that work together to compute the statistics as described below. 
   Function “EXTRACT_COMPUTE_STATS” (see step  603  in  FIG. 6 ) is declared with the PARALLEL_ENABLE clause. PARALLEL_ENABLE allows parallelism in the input cursor to dictate parallelism in the function execution. That is, if the input cursor named extract_cv is executing in parallel, then the function is invoked in parallel by the same Oracle processes executing over the input cursor. Since it is necessary for any given process to have all measurements from any given data source identifier&#39;s time series, the input cursor&#39;s data is split across processes by a hash function on the data source time series identifier. Every input row for a given time series will be identically hashed and thus assigned to the same parallel slave process for statistics computation. The function is also declared as PIPELINED. This allows it to fetch data in batches, compute results for those batches, and output results prior to fetching more batches. Proper batching of the data by time series is ensured by a CLUSTER declaration that specifies data be clustered by time series (the function is supplied all rows for a given time series before rows from another time series are supplied.) A PL/SQL signature of this function appears below. Note in this signature that datasource_guid referred to in the CLUSTER declaration represents an identifier for the individual metric time series. 
   
     
       
             
             
           
         
             
                 
                 
             
           
           
             
                 
               function extract_compute_stats 
             
             
                 
                  (extract_cv in extract_cvtype 
             
             
                 
                  ,compute_date_in in date := SYSDATE) 
             
             
                 
               return bsln_statistics_set 
             
             
                 
               PIPELINED 
             
             
                 
               CLUSTER extract_cv by (datasource_guid) 
             
             
                 
               PARALLEL_ENABLE 
             
             
                 
                 (PARTITION extract_cv BY HASH(datasource_guid)); 
             
             
                 
                 
             
           
        
       
     
   
   The parallelism of function “EXTRACT_COMPUTE_STATS” is driven by the cursor variable extract_cv and the cursor variable&#39;s parallelism can be driven by setting degree of parallelism at the table level for the main table referenced by the query. The number of parallel slaves executing the function can be tuned from outside the function execution context. 
   A second function called “EXPTAIL_STATS” (see step  604  in  FIG. 6 ) performs the statistical calculations on a batch of time series data and returns statistics rows as output. This function uses analytic functions available in the computer system, and nested inline views to accomplish the full computation within a single SELECT statement. The following analytic functions are used: CUME_DIST, ROW_NUMBER, PERCENTILE_DISC, MAX, MIN, AVG, and STDDEV. The algorithm of the embodiment of this function is documented in Appendix B. The “fit quality” is a number that normalizes the goodness of fit over a range 0–100 for ease of comparison. 
   Numerous modifications and adaptations of the embodiments described herein will become apparent to the skilled artisan in view of this disclosure. 
   Although receipt from a human of an appropriate time partitioning scheme is described above in some embodiments, in alternative embodiments, the computer is programmed to automatically check for several types of periodicities in the measurements (e.g. by applying a correlation function thereof), and use the automatically identified periodicities to partition the available measurements into the appropriate number of sets. Although in some embodiments, the fitted exponentially decaying tail is used to detect and alert anomalous system behavior, other embodiments use the fitted tail to perform other functions, such as system sizing and capacity planning, and establishing service level agreements. 
   Numerous modifications and adaptations of the embodiments described herein are encompassed by the scope of the invention. 
   Although the above description refers to exponential tails that are used in many embodiments, other embodiments of the type described herein may use functions in which the tail is non-exponential but nonetheless a heavy tail (as in a Weibull Distribution, or Lognormal Distribution). 
   
     
       
             
             
           
         
             
                 
               APPENDIX A 
             
             
                 
                 
             
           
           
             
                 
               type [raw measurement data] is object 
             
             
                 
                 (data source identifier 
             
             
                 
                 ,measurement timestamp 
             
             
                 
                 ,measurement value 
             
             
                 
                 ) 
             
             
                 
               type [raw measurement data table] is table of [raw 
             
             
                 
               measurement data] 
             
             
                 
               type [time_interval] is object 
             
             
                 
                  (begin time 
             
             
                 
                  ,end time 
             
             
                 
                  ) 
             
             
                 
               type [set_of_intervals] is table of [time_interval] 
             
             
                 
               SELECT 
             
             
                 
                  [measurement data rows] 
             
             
                 
                FROM 
             
             
                 
                  [raw measurement data table] D 
             
             
                 
                  ,[set of intervals] I 
             
             
                 
               WHERE 
             
             
                 
                 D.measurement timestamp 
             
             
                 
                   BETWEEN I.begin_time AND I.end_time 
             
             
                 
                 
             
           
        
       
     
   
   
     
       
             
           
             
             
           
             
           
             
             
           
             
             
           
             
           
             
             
           
             
           
             
             
           
             
           
             
             
           
             
           
             
             
           
             
           
             
             
           
             
           
             
             
           
             
           
         
             
               APPENDIX B 
             
             
                 
             
           
           
             
               TYPE observation IS OBJECT 
             
           
        
         
             
                   (timeseries id 
               raw(16) 
             
             
                   ,timeperiod id 
               raw(16) 
             
             
                   ,timegroup code 
               raw(21) 
             
             
                   ,obs_time 
               date 
             
             
                   ,obs_value 
               number 
             
             
                   ) 
             
           
        
         
             
               TYPE observation set IS TABLE OF observation 
             
             
               TYPE statistics obj IS OBJECT 
             
           
        
         
             
                (timeperiod id 
               raw(16) 
             
             
                ,timeseries id 
               raw(16) 
             
             
                ,compute date 
               date 
             
             
                ,timegroup code 
               raw(21) 
             
             
                ,sample_count 
               number 
             
             
                ,average 
               number 
             
             
                ,minimum 
               number 
             
             
                ,maximum 
               number 
             
             
                ,standard_deviation 
               number 
             
             
                ,pctile_25 
               number 
             
             
                ,pctile_50 
               number 
             
             
                ,pctile_75 
               number 
             
             
                ,pctile_90 
               number 
             
             
                ,pctile_95 
               number 
             
             
                ,pctile_99 
               number 
             
           
        
         
             
                ,estimator_sample_count 
               number 
             
             
                ,estimator_parm1 
               number 
             
             
                ,estimator_parm2 
               number 
             
             
                ,estimator_fit_quality 
               number 
             
             
                ,estimator_pctile_999 
               number 
             
             
                ,estimator_pctile_9999 
               number 
             
             
                ) 
             
           
        
         
             
               TYPE statistics set IS TABLE OF statistics obj 
             
             
                 SELECT 
             
             
                   [Z.statistics grouping attributes] 
             
             
                  ,[Z.retained full set statistics] 
             
             
                  ,[Compute 
             
             
                   μ, β, μ1, goodness-of-fit-test] 
             
             
                 FROM 
             
             
                   (SELECT [Compute 
             
             
                     Sum_{k=m to j} x_k] 
             
             
                    ,[Y.full set statistics] 
             
             
                    ,[Y.statistics grouping attributes] 
             
             
                   FROM 
             
             
                    (SELECT [Compute 
             
           
        
         
             
                 
               m, l, j, x_j, x_m, x_l, 
             
             
                 
               Sum_{k=m to 1} x_k] 
             
           
        
         
             
                     ,[X.full set statistics] 
             
             
                     ,[X.statistics grouping attributes] 
             
             
                    FROM 
             
             
                     (SELECT 
             
           
        
         
             
                 
                [statistics grouping attributes] 
             
             
               [1] 
               ,[compute: full set statistics] 
             
             
                 
               ,[x_k = measurement value] 
             
             
                 
               ,[k = rank of the measurement] 
             
           
        
         
             
                       FROM 
             
           
        
         
             
                 
                TABLE [input observation set] 
             
           
        
         
             
                      GROUP BY 
             
           
        
         
             
                 
                [statistics grouping attributes] 
             
           
        
         
             
                      ) X 
             
             
                     WHERE 
             
           
        
         
             
                 
               X.Cumulative_distribution &gt;= 
             
           
        
         
             
               tail_low_percentile 
             
             
                      AND X.Cumulative_distribution &lt;= 
             
             
               tail_high_percentile 
             
             
                    ) Y 
             
             
                  ) Z 
             
             
               [1] One embodiment computes the following statistics over the entire 
             
             
               input observation set (i.e. not limited to the “tail” but over all 
             
             
               measurements in a set) grouped by statistics id information using Oracle 
             
             
               analytic functions. 
             
             
               [full set statistics] = 
             
           
        
         
             
                 [ 
                 
             
             
                  sample_count 
               number 
             
             
                 ,average 
               number 
             
             
                 ,minimum 
               number 
             
             
                 ,maximum 
               number 
             
             
                 ,standard deviation 
               number 
             
             
                 ,pctile_25 
               number 
             
             
                 ,pctile_50 
               number 
             
             
                 ,pctile_75 
               number 
             
             
                 ,pctile_90 
               number 
             
             
                 ,pctile_95 
               number 
             
             
                 ,pctile_99 
               number 
             
             
                 ] 
             
           
        
         
             
               [statistics grouping attributes] = 
             
             
                 [ 
             
             
                  timeseries id 
             
             
                 ,timeperiod id 
             
             
                 ,timegroup code 
             
             
                 ] 
             
             
                 
             
           
        
       
     
   
   
     
       
             
           
             
             
           
             
           
             
             
           
             
           
             
             
           
             
           
             
             
           
             
           
             
             
           
             
           
             
             
           
             
           
         
             
               APPENDIX C 
             
             
                 
             
           
           
             
               Signature: 
             
             
               Function extract_compute_statistics 
             
             
                (input cursor of measurement observations in data 
             
             
               source order) 
             
             
               Return 
             
             
                (output statistics data computed by data source and 
             
             
               grouping code) 
             
             
               Input: 
             
           
        
         
             
                 
               [open cursor of measurement observations] 
             
           
        
         
             
               Local variables: 
             
           
        
         
             
                 
               [l_observation_set] nested table of observations 
             
           
        
         
             
               within function 
             
           
        
         
             
                 
               [l_statistics_set] output variable for the 
             
           
        
         
             
               function 
             
             
               Logic: 
             
           
        
         
             
                 
               LOOP 
             
             
                 
                FETCH FROM 
             
             
                 
                 [open cursor of measurement observations] 
             
             
                 
                INTO [l_observation_set] 
             
             
                 
                EXIT loop when cursor empty 
             
             
               [1] 
                IF [compute batch is full] 
             
             
                 
                THEN 
             
             
                 
                 [l_statistics_set] = 
             
           
        
         
             
               exponential_tail_statistics(l_observation_set]) 
             
           
        
         
             
               [2a] 
                 output [l_statistics_set] 
             
             
                 
                END IF 
             
             
                 
               END LOOP 
             
             
                 
               IF [final batch non-empty] 
             
             
                 
               THEN 
             
             
                 
                 [l_statistics_set] = 
             
           
        
         
             
               exponential_tail_statistics(l_observation_set]) 
             
           
        
         
             
               [2b] 
                 output [l_statistics_set] 
             
             
                 
               END IF 
             
             
                 
               CLOSE [open cursor of measurement observations] 
             
             
                 
               RETURN from function 
             
           
        
         
             
               Notes on above characterized embodiment by numbered pseudo-code 
             
             
               location. 
             
             
               [1] In this embodiment compute batches (e.g. a batch for each metric 
             
             
               and/or for each set of a metric) are formed by loading all measurements 
             
             
               for a given data source together into [l_observation_set] before 
             
             
               calling the exponential_tail_statistics function. 
             
             
               [2a], [2b] In this embodiment the function extract_compute_statistics 
             
             
               uses Oracle pipelined function feature to incrementally return statistics 
             
             
               output rows incrementally as batches are processed. Such pipelining 
             
             
               allows incremental batching (e.g. for each metric and/or for each set 
             
             
               of a metric).